SIGNAL & SYSTEM
Objective Paper –“Topic wise Updated up to GATE-2019 & IES-2014”
(VERSION : 12|07|19)
GATE / IES
For “Electrical” , “Elect. & Comm.” And “Instrumentation” Engg.
Also useful for:
Public Sector Units & State Engineering Service Examination
This booklet contains Topic Wise…..
GATE (EC/IN/EE) 31 year of problems (Year 1987 to 2019).
IES (EC/EE) 24 year of problems (Year 1991 to 2014).
In-house developed concept building problems.
Total Around 800 number of problems.
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place of trust since 2009…
SIGNAL & SYSTEM
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SYLLABUS: SIGNAL & SYSTEM
GATE-2020
Electronics & Comm.(EC)
Continuous-time signals: Fourier series and Fourier transform representations, sampling theorem and
applications; Discrete-time signals: discrete-time Fourier transform (DTFT), DFT, FFT, Z-transform,
interpolation of discrete-time signals; LTI systems: definition and properties, causality, stability, impulse
response, convolution, poles and zeros, parallel and cascade structure, frequency response, group delay,
phase delay, digital filter design techniques.
Electrical (EE)
Representation of continuous and discrete-time signals, Shifting and scaling operations, Linear Time Invariant
and Causal systems, Fourier series representation of continuous periodic signals, Sampling theorem,
Applications of Fourier Transform, Laplace Transform and z-Transform.
IES-2020
Electronics & Comm.(EC)
Classification of signals and systems: System modelling in terms of differential and difference equations;
State variable representation; Fourier series; Fourier representation; Fourier series; Fourier transforms and
their application to system analysis; Laplace transforms and their application to system analysis; Convolution
and superposition integrals and their applications; Z-transforms and their applications to the analysis and
characterisation of discrete time systems; Random signals and probability, Correlation functions; Spectral
density; Response of linear system to random inputs.
Expert Comments
Comparing to the GATE-EC syllabus GATE-EE syllabus does not contains discrete time analysis
(except the Z transform).
IES-EE syllabus does not contains the “Signal & System”.
SIGNAL & SYSTEM
Table of Contents
1. CONTINUOUS TIME SIGNAL & SYSTEM
1
1.1
SYSTEM’S CLASSIFICATION…………………………………………………………………………………………… 1
1.2
CONTINUOUS SIGNAL…………………………………………………………………………………………………. 7
1.3
PERIODS OF SIGNAL…………………………………………………………………………………………………. 10
1.4
CONVOLUTION THEOREM………………………………………………………………………………………….. 12
1.5
DELTA FUNCTIONS…………………………………………………………………………………………………… 17
1.6
ENERGY, POWER & RMS…………………………………………………………………………………………. 18
1.7
MISCELLANEOUS…………………………………………………………………………………………………….. 21
2. DISCRETE TIME SIGNAL & SYSTEM
26
2.1
SYSTEM’S CLASSIFICATION…………………………………………………………………………………………..26
2.2
MISCELLANEOUS…………………………………………………………………………………………………….. 29
3. FOURIER SERIES
35
3.1
THEORETICAL PROBLEM………………………………………………………………..………………………….. 35
3.2
NUMERICAL PROBLEM……………………………………………………………………………………….…….. 38
4. FOURIER TRANSFORM
45
4.1
THEORETICAL PROBLEM………………………………………………………………………………………….... 45
4.2
NUMERICAL PROBLEM………………………………………………………………………………………….….. 48
5. LAPLACE TRANSFORM
59
5.1
THEORETICAL PROBLEM………………………………………………………………………………………….....59
5.2
NUMERICAL PROBLEM………………………………………………………………………………………….….. 60
6. SAMPLING THEOREM
75
7. Z- TRANSFORM
80
8. DFS/DTFT/DFT/FFT
94
9. RANDOM VARIABLE
98
10. MISCELLANEOUS
103
10.1 LTI SYSTEMS CONTINUOUS AND DISCRETE (TIME DOMAIN) …………………………………………… 111
ANSWER KEYS
113
01
Continuous Time
Signal & System
(A) The impulse response will be
integrable, but may not be absolutely
integrable.
(B) The unit impulse response will have
finite support.
(C) The unit step response will be
absolutely integrable
(D) The unit step response will be bounded.
System’s Classification
(1)
AA [GATE – EC3 – 2014]
Let h(t) denotes the impulse response of a
1
.
S 1
Consider the following three statements.
S1: The system is stable.
causal system with transfer function
S2:
h t 1
is independent of t for t > 0.
h t
(5)
S3: A non causal system with the same
transfer function is stable.
For the above system,
(A) Only S1 and S2 are true
(B) Only S2 and S3 are true
(C) Only S1 and S3 are true
(2)
(3)
(D) S1 , S2 and S3 are true
AC [GATE – EC – 1991]
An excitation is applied to a system at t = T
and its response is zero for t T .
Such a system is a
(A) non-causal system
(B) stable system
(C) causal system
(D) unstable system
(6)
AD [GATE - EE/EC/IN - 2012]
The input x(t) and output y(t) of a system are
related as y(t )
t
AD [IES - EC - 2000]
A continuous - time system is governed by
the equation :
3 y 3 (t ) 2 y 2 (t ) y (t ) x 2 (t ) x (t )
x( τ )cos(3τ )dτ.
{y(t) and x(t) respectively are output and
input }. The system is
The system is :
(A) Time-invariant and stable
(B) Stable and not time-invariant
(C) Time-invariant and not stable
(D) Not time-invariant and not stable
(A) linear and dynamic
(B) linear and non - dynamic
(C) non - linear and dynamic
(D) non - linear and non - dynamic
AD [GATE – EE – 2015]
(4)
AD [IES - EC - 1997]
Which of following represents a stable
system?
1. Impulse response of the system
decreases exponentially
2. Area within the impulse response is
finite.
3. Eigen values of the system are positive
and real.
4. Roots of the characteristic equation of
the system are real and negative
Select the correct answer using the codes
given below:
Codes :
(A) 1 and 4
(B) 1 and 3
(C) 2, 3 and 4
(D) 1, 2 and 4
For linear time invariant systems, that are
Bounded Input Bounded Output stable,
which one of the following statements is
TRUE?
(7)
www.targate.org
AC [GATE – EC – 2008]
The input and output of a continuous time
system are respectively denoted by x (t) and
y (t). Which of the following descriptions
corresponds to a causal system?
Page 1
SIGNAL & SYSTEM
(A) y (t ) x(t 2) x(t 4)
(C) y(t) = 2 x(t - 1) - x(t - 2) - x(t - 4)
(B) y (t ) (t 4) x(t 1)
(D) y(t ) = 2 x(t) + 3.6
AB [IES-EC-2014]
(13) A system is characterized by the inputoutput relation
(C) y (t ) (t 4) x(t 1)
(D) y (t ) (t 5) x(t 5)
(8)
AB [IES - EC - 2004]
If the response of a system to an input does
not depend on the future values of the input,
then which one of following is true for the
system?
(A) It is aperiodic
(B) It is causal
(C) It is anticipatory
(D) It is discrete
(9)
AB [IES - EC - 2000]
Which one of the following systems is a
causal system? [ y(t) is output and u(t) is a
input step function]
(A) y(t) = sin (u (t + 3))
y (t ) x(2t ) x (3t )
For all t, where y(t) is the output and x(t) is
the input. It is
(A) linear and causal
(B) linear and non-causal
(C) non-linear and causal
(D) non-linear and non-causal
AB [GATE – EC2 – 2015]
(14) Input x(t) and output y(t) of an LTI system
are related by the differential equation y"(t)
– y'(t) – 6y(t) = x(t). If the system is neither
causal nor stable, the impulse response h(t)
of the system is
(A)
1 3t
1
e u(t) e2t u(t)
5
5
(B)
1 3t
1
e u(t) e2t u(t)
5
5
(C)
1 3t
1
e u(t) e2t u(t)
5
5
(B) y(t) = 5u (t) + 3u (t - 1)
(C) y(t) = 5u (t) + 3u (t + 1)
(D) y(t) = sin ( u(t -3)) + sin (u (t + 3))
AD [IES - EC - 2006]
(10) Which one of the following is the correct
statement ?
The system characterized by the equation
y(t) = a x(t) + b is
(A) linear for any value of b
(B) linear if b > 0
(C) linear if b < 0
(D) non-linear
AD [GATE - IN - 2009]
(11) For input x(t), an ideal impulse sampling
system produces the output
1 3t
1 2t
(D) e u(t) e u(t)
5
5
AB [GATE – EC – 2005]
(15) Which of the following can be impulse
response of a causal system?
(A)
y (t )
x ( kT ) (t kT ) where (t) is
k
the Dirac delta function.
The system is
(A) Nonlinear and time invariant
(B) Nonlinear and time varying
(C) Linear and time invariant
(D) Linear and time varying
AD [IES - EC - 1998]
(12) Which one of the following system is nonlinear? [y(t) = output; x(t) = input]
(B)
(C)
(A) y(t) = 2x(t - 1) - 3 x(t -2) + x(t - 3)
(B) y(t) = 5 x(t)
Page 2
TARGATE EDUCATION GATE-(EE/EC)
Topic.1 - Continuous Time Signal & System.
AA [GATE – EC – 2008]
(18) Let x(t) be the input and y(t) be the output of
a continuous time system. Match the system
P1 , P2 and P3
properties
with system
(D)
relations R1 , R 2 , R 3 , R 4 .
AD [IES - EC - 1999]
(16) Which one of the following input - output
relationships is that of a linear system?
(A)
Properties
Relations
P1:
Linear
but
NOT
timeinvariant
R 1 : y t t 2 x(t)
P2:
Time-invariant
but NOT linear
R2 : y t t x t
P3:
Linear
and
time-invariant
R3 : y t x t
R 4 : y t x t 5
(A) P1 , R 1 , P2 , R 3 , P3 , R 4
(B) P1 , R 2 , P2 , R 3 , P3 , R 4
(B)
(C) P1 , R 3 , P2 , R 1 , P3 , R 2
(D) P1 , R 1 , P2 , R 2 , P3 , R 3
AA [IES - EC - 2010]
(19) Assertion (A) : A linear system gives a
bounded output if the input is bounded.
(C)
Reason (R) : The roots of the characteristic
equation have all negative real parts and the
response due to initial conditions decays to
zero as time t tends to infinity.
Codes :
(A) Both A and R are true and R is the
correct explanation of A
(D)
(B) Both A and R are true but R is not a
correct explanation of A
(C) A is true but R is false
AD [GATE – EC – 2001]
(17) The impulse response functions of four
linear systems S1 , S2 , S3 and S4 are given
respectively by
h1 ( t ) 1
h3 (t )
;
h2 ( t ) u (t ) ;
u (t )
; h4 (t ) e3t u(t ),
t 1
Where u (t) is the unit step function. Which
of these systems is time invariant, causal,
and stable ?
(A) S1
(B) S2
(C) S3
(D) S4
(D) A is false but R is true
AD [IES - EC - 2010]
(20) Assertion (A) : The system described by
y 2 (t) 2y(t) x 2 (t) x(t) c is a linear
and static system.
Reason (R) : The dynamic system is
characterized by differential equation.
Codes :
(A) Both A and R are true and R is the
correct explanation of A
(B) Both A and R are true but R is not a
correct explanation of A
(C) A is true but R is false
(D) A is false but R is true
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Page 3
SIGNAL & SYSTEM
AD [GATE - EE - 2006]
(21) A continuous-time system is described by
y(t) = e | x (t )| , where y(t) is the output and
x(t) is the input. y(t) is bounded
(A) Only when x(t) is bounded
(B) Only when x(t) is non-negative
(C) Only for t 0 if x(t) is bounded for
t0
(D) Even when x(t) is not bounded
AB [GATE – EC – 2009]
(22) Consider a system whose input x and output
y are related by the equation
y(t )
x(t τ )h(2τ )dτ
Where h(t) is shown in the graph.
Which of the following four properties are
possessed by the system?
(C) At least one system is causal and all
systems are unstable
(D) The majority are unstable and the
majority are causal
AC [GATE - IN - 2011]
(25) Consider a system with input x(t) and output
y(t) related as follows
d t
e x(t )
dt
Which one of the following statements is
TRUE?
(A) The system is nonlinear
(B) The system is time – invariant
(C) The system is stable
(D) The system has memory
y (t )
AC [GATE - IN - 2010]
(26) The input x(t) and the corresponding output
y(t) of a system are related by
BIBO: Bounded input gives a bounded
output.
y (t ) x ( τ ) dτ . The system is
Causal: The system is causal.
(A) Time invariant and causal
LP : The system is low pass.
(B) Time invariant and non causal
LTI : The system is linear and timeinvariant.
(C) Time variant and non causal
5t
(D) Time variant and causal
AAB [IES - EC - 1997]
(27) Let h(t) be the response of a linear system to
a unit impulse δ(t).
(A) Causal, LP
(B) BIBO, LTI
(C) BIBO, Causal, LTI
(D) LP, LTI
AB [GATE - EE - 2010]
(23) The system represented by the input – output
relationship: y(t) =
5t
x ( τ )dτ , t 0 is
(A) Linear and causal
(B) Linear but not causal
(C) Causal but not linear
(D) Neither linear nor causal
AB [GATE - EE - 2009]
(24) A cascade of 3 Linear Time Invariant
systems is causal and unstable. From this,
we conclude that
(A)
Each system in the cascade
individually causal and unstable
is
(B) At least one system is unstable and at
least one system is causal
Page 4
Consider the following statements in this
regard :
1. If the system is causal, h(t) = 0 for t < 0
2. If the system is time-variable, then the
response of the system to an input of δ(t
- T) is h(t - T) for all values of the
constant T.
3. If the system is non-dynamic, then h(t)
is of the from A δ(t), where the
Constant A depends on the system.
Of these statements
(A) 1 and 2 are correct
(B) 1 and 3 are correct
(C) 2 and 3 are correct
(D) 1, 2 and 3 are correct
AD [IES - EC - 2006]
(28) Which one of the following is the correct
statement ?
The continuous time system described by
y(t) = x(t2) is
(A) causal, linear and time-varying
(B) causal, non-linear and time-varying
TARGATE EDUCATION GATE-(EE/EC)
Topic.1 - Continuous Time Signal & System.
(C) non-causal,
invariant
non-linear
and
time-
List I
(Equation)
(D) non-causal and time-variant
(A) 2t
AD [IES - EC - 2007]
(29) Which of the following is correct ? A system
can be completely described by a transfer
function if it is
(A) non-linear and continuous
(B) linear and time-varying
(C) non-linear and time-invariant
dy
4y 2x
dt
(C) 4
d2 y
dy
dx
2 y 3
2
dt
dt
dt
2
AD [IES - EC - 2008]
(30) If v-i characteristic of a circuit is given by
v(t) = t i(t) + 2, the circuit is of which type ?
(A) Linear and time invariant
(B) Linear and time variant
(C) Non-linear and time invariant
(D) Non-linear and time variant
List II
(System Category)
1. Linear, time-invariant and dynamic
AA [IES - EC - 2009]
(31) The output y(t) of a continuous-time system
S for the input x(t) is given by :
2.
Non-linear, time-invariant and dynamic
3.
Linear, time-variable and dynamic
4.
Non-linear, time-variable and dynamic
5. Non-linear, time-invariant and nondynamic
Codes:
t
(B) y
dx
dy
(D) 2ty 4
dt
dt
(D) linear and time invariant
y(t)
x( )d
A
B
C
D
(A)
3
2
1
4
(B)
4
1
5
3
(C)
3
1
5
4
(D)
4
2
1
3
Which one of the following is correct ?
(A) S is linear and time-invariant
(B) S is linear and time-varying
(C) S is non-linear and time-invariant
(D) S is non-linear and time-varying
AC [IES - EC - 2012]
(32) With the following equations, the timeinvariant systems are
1.
AD [GATE - EE - 2008]
(34) The impulse response of a causal linear time
– invariant system is given as h(t). Now
consider the following two statements:
d 2 y (t )
d
2t y (t ) 5 y (t ) x(t )
2
dt
dt
Statement(I): Principle of superposition
holds
2. y (t ) e 2 x (t )
d
3. y (t ) x (t )
dt
4. y (t )
dy
4y 2tx
dt
Statement (II): h(t) = 0 for t < 0
2
Which one of the following statement is
correct?
(A) Statement (I) is correct and Statement
(II) is wrong
d 2 t
[e x(t )]
dt
(A) 1 and 2
(B) 1 and 4
(C) 2 and 3
(D) 3 and 4
AA [IES - EC - 2005]
(33) The governing differential equations
connecting the output y(t) and the input x(t)
of four continuous time systems are given in
the List I and List II respectively. Match List
I (Equation) with List II (System Category)
and select the correct answer using the code
given below the Lists :
(B) Statement (II) is correct and Statement
(I) is wrong
(C) Both Statement (I) and Statement (II)
are wrong
(D) Both Statement (I) and Statement (II)
are correct
AD [GATE - EE - 2008]
(35) A system with input x(t) and output y(t) is
defined by the input – output relation:
www.targate.org
Page 5
SIGNAL & SYSTEM
2t
Y(t)=
x t d
The system will be
(A) Causal, time – invariant and unstable
(B) Causal, time – invariant and stable
(C) Non – Causal, time – invariant and
unstable
(D) Non – causal, time – variant and
unstable
AA [IES - EC - 2005]
(39) Consider the following statements about
linear time-invariant (LTI) continuous time
system :
1. The output signal in an LTI system with
known input and known impulse
response can always be determined.
2. A causal LTI system is always stable.
3. A stable LTI system has an impulse
response, h(t) which has a finite value
when integrated over whole of the time
AD [IES - EC - 1998]
(36) The impulse response of a causal, linear,
time-invariant, continuous-time system is
h(t). The output y(t) of the same system to an
input x(t), where x(t) = 0 for t < -2, is
t
(A) h ( ) x (t ) d
0
t
(B)
h ( ) x ( t ) d
2
t 2
(C)
h ( ) x (t )d
2
t 2
(D)
h ( ) x (t )d
0
AB [IES - EC - 1999]
(37) Which one of the following pairs is NOT
correctly matched? (input x(t) and output
y(t)).
(A) Unstable system:
dy(t )
0.1y(t ) x(t )
dt
dy(t )
2t 2 y(t ) x(t )
dt
(C) Non causal system : y(t) = x(t + 2)
(D) Non dynamic system : y(t) = 3 x2 (t)
(B) Nonlinear system:
AA [IES - EC - 2000]
(38) If the step response of a causal, linear time
invariant system is a(t), then the response of
the system to the general input x(t) would be
t
(A)
0
da ( )
x (t )d
d ( )
t
(B) a (0) x ( t )
0
axis, i.e.,
AA [IES - EC - 1994]
(40) Assertion (A) : A memory less system is
causal
Reason (R) : A system is causal if the
output at any time depends only on values
of input at that time and in the past.
Codes:
(A) Both A and R are true and R is the
correct explanation of A
(B) Both A and R are true but R is not a
correct explanation of A
(C) A is true but R is false
(D) A is false but R is true
A(1-D),(2-B) [GATE – EC – 1997]
(41) Match each of the items 1, 2 on the left with
the most appropriate item A, B, C or D on
the right.
In the case of a linear time invariant system
(1)
Poles pole in
the right half
plane implies
(A)
Exponential
decay of output
(2)
Impulse
response zero
for t 0
implies
(B)
System is
causal
(C)
No stored
energy in the
system
System is
unstable
da ( )
x ( t ) d
d ( )
(D)
x ( )a ( t ) d
0
t
(D) x (0) a ( t )
0
Page 6
is finite.
Which of the statements given above are
correct ?
(A) 1 and 3
(B) 1 and 2
(C) 2 and 3
(D) 1, 2 and 3
t
(C) x (0) a ( t )
h( )d
da ( )
x ( t ) d
d ( )
AB [GATE – EC – 2000]
(42) A system with an input x(t) and output y(t) is
described by the relation: y(t) = tx(t). This
system is
TARGATE EDUCATION GATE-(EE/EC)
Topic.1 - Continuous Time Signal & System.
(A)
(B)
(C)
(D)
linear and time-invariant
linear and time-varying
non-linear and time –invariant
non-linear and time-varying
Continuous Signal
(1)
AA [GATE-EE1-2014]
The function shown in the figure can be
represented as
S6AC [GATE – EE – 2016]
(43) Consider a continuous-time system with
input x (t ) and output y (t ) given by
This system is
(A) linear and time-invariant
(B) non-linear and time-invariant
(C) linear and time-varying
(D) non-linear and time-varying
(A) u (t ) u (t T )
AB [GATE–S2–EC–2017]
(44) The input x(t) and the output y(t) of a
continuous-time system are related as
y (t )
t
t T
(t 2T )
u (t 2T )
T
t
t
(B) u (t ) u(t T ) u(t 2T )
T
u
(t T )
(C) u (t ) u (t T )
T
(t 2T )
u (t )
u (t )
T
(t T )
(t 2T )
u(t T ) 2
(D) u(t )
T
T
u (t 2T )
x (u )du
The system is
(A) linear and time-variant
(B) linear and time-invariant
(C) non-linear and time-variant
(D) non-linear and time-invariant
AC [GATE – EC – 2018]
(45) Let the input be u and the output be y of a
system, and the other parameters are real
constants. Identify which among the
following systems is not a linear system:
(A)
(2)
d3y
d2y
dy
a
a2
a3 y b3u b2
1
dt 3
dt 2
dt
du
d 2u
b1 2
dt
dt
conditions)
(t T )
u (t T )
T
AB [IES - EC - 1997]
Match List-I with List-II and select the
correct answer using the codes given below
the Lists:
List - I
(A)
(with
initial
rest
t
(t )
u() d
(B) y(t ) e
(B)
0
(C) y au b, b 0
(D) y au
AD [GATE-EE-2019]
(46) The symbols, a and T, represent positive
quantities and u(t) is the unit step function.
Which one of the following impulse
responses is NOT the output of a causal
linear time-invariant system?
(A) eat u(t )
(C)
(B) ea(t T )u(t )
(D)
a ( t T )
(C) e
u(t )
at
(D) 1 e u(t )
**********
www.targate.org
Page 7
SIGNAL & SYSTEM
List - II
1. v(t) = u(t + 1)
2. v(t) = u(t) - 2u(t - 1) + 2u(t - 2) - 2u(t - 3)
3. v(t) = u(t - 1) - u(t - 3)
4. Lt v ( t ) ( t 1)
(C)
a0
Codes :
A
(A) 1
(B) 3
(C) 4
(D) 4
(3)
B
2
4
3
3
C
3
1
2
1
D
4
2
1
2
(D)
AC [GATE – EC – 2006]
Which of the following is true?
(A) A finite signal is always bounded
(B) A bounded signal always possess finite
energy
(5)
AD [IES - EC - 2005]
In the graph shown in fig., which one of the
following expresses v(t) ?
(C) A bounded signal is always zero outside
the interval t 0 , t 0 for some t 0
(D) A bounded signal is always finite
(4)
AA [IES - EC - 1999]
If a plot of signal x(t) is as shown in the Fig.
1,
(A) (2t + 6) [u(t-3)+2u(t – 4)]
(B) (–2t – 6) [u(t – 3)+u(t – 4)]
(C) (–2t+6)[u(t – 3) + u(t – 4)]
(D) (2t – 6) [u(t – 3) – u(t – 4)]
(6)
AD [IES - EC - 1991]
The expression for the waveform in terms of
step function is given by
Then the plot of the signal x(1 - t) will be
(A)
(A) v = u(t - 1) - u(t - 2) + u(t - 3)
(B) v = u(t - 1) + u(t - 2) + u(t - 3)
(C) v = u(t - 1) + u(t - 2) - u(t - 3)
(D) v = u(t - 1) + u(t - 2) + u(t - 3) - 3u(t - 4)
(7)
(B)
Page 8
AA [IES - EC - 1991]
If from the function f(t) one forms the
function, (t ) = f(t) + f(-t), them (t ) is
(A) even
(B) odd
(C) neither even nor odd
(D) both even and odd
TARGATE EDUCATION GATE-(EE/EC)
Topic.1 - Continuous Time Signal & System.
(8)
AC [IES - EC - 2010]
The mathematical model of the below shown
signal is
This represents the unit
(A) sinc function
(B) area triangular function
(C) signum function
(D) parabolic function
*********
(A)
(B)
(C)
(D)
(9)
x(t) = u(2 + t)
x(t) = u(t – 2)
x(t) = u(2 – t)
x(t) = u(t – 1)
AB [GATE – EC – 2000]
Let u (t) be the unit step function. Which of
the waveforms in Fig. (A)- (D) corresponds
to the convolution of [u(t ) u (t 1)] with
[u (t ) u(t 2)]?
(A)
(B)
(C)
(D)
AC [IES - EC - 2004]
(10) Consider the following waveform :
Which one of the following gives the correct
description of the waveform shown in the
below diagram?
(A) u(t) + u(t – 1)
(B) u(t) + u(t – 1)u (t – 1)
(C) u(t) + u(t – 1) + (t – 2) u(t – 2)
(D) u(t) + (t – 2) u (t – 2)
AB [IES - EC - 2012]
(11) A signal f(t) is described as
f (t) [1 | t |] when | t | 1
0
when | t | 1
www.targate.org
Page 9
SIGNAL & SYSTEM
Periods of Signal
(1)
(2)
(7)
AA [GATE-EC/EE/IN-2013]
For
a
periodic
signal
v (t ) 30sin100t 10cos300t 6sin (500t ,
π / 4) the fundamental frequency in rad/s is
AA [GATE - IN - 2006]
The Fourier series for a periodic signal is
given as
x ( t ) cos(1.2 πt ) cos(2 πt ) cos(2.8πt )
The fundamental frequency of the signal is
(A) 100
(B) 300
(A) 0.2 Hz
(B) 0.6 Hz
(C) 500
(D) 1500
(C) 1.0 Hz
(D) 1.4 Hz
AC [GATE - IN - 2011]
The continuous – time signal x (t ) sin 0 t
is a periodic signal. However, for its discrete
– time counterpart x[n] = sin 0 n to be
periodic, the necessary condition is
(A) 0 0 2π
(B)
(8)
AB & D [GATE – EC – 1992]
Which of the following signals is/are
periodic?
(A) s(t ) cos 2t cos3t cos5t
(B) s(t ) exp( j8t )
(C) s(t ) exp(7t )sin10t
2π
to be an integer
0
(D) s(t ) cos 2t cos 4t
2π
(C)
to be a ratio of integers
0
(9)
AA [GATE - IN - 2005]
The fundamental period of the sequence x[n]
= 3 sin(1.3πn 0.5 π ) 5 sin(1.2 πn ) is :
(D) none
(3)
The
period
of
AD [GATE - EE - 2010]
the signal x(t) = 8
(C)
π
sin 0.8πt is
4
(A) 0.4 π s
(C) 1.25 s
(4)
Consider
(B) 0.8 π s
(D) 2.5 s
AB [IES - EC - 2007]
two signals x1 (t) e j20 t and
x 2 (t) e( 2 j) t . Which one of the following
statements is correct ?
(A) Both x1(t) and x2(t) are periodic
(B) x1(t) is periodic but x2(t) is not periodic
(C) x2(t) is periodic but x1(t) is not periodic
(D) Neither x1(t) nor x2(t)is periodic
(5)
(A) 20
AA [IES - EC - 2001]
If x1(t) = 2 sin πt + cos 4πt and x2(t) = sin
5πt + 3 sin 13πt, then
(A) x1 and x2 both are periodic
(B) x1 and x2 both are not periodic
(C) x1 is periodic, but x2 is not periodic
AC [GATE - IN - 2009]
The fundamental period of x(t) =
2 sin(2 πt ) 3 sin(3πt ), with t expressed in
seconds, is
(A) 1
(B) 0.67
(C) 2
(D) 3
Page 10
2π
1.3π
(D) 10
AB [IES - EC - 1999]
(10) The period of the function cos (t 1) is
4
(A) 1/8 s
(B) 8 s
(C) 4 s
(D) 1/4 s
AA [IES - EC - 2008]
(11) Which one of the following functions is a
periodic one ?
(A) sin(10t) sin(20 t)
(B) sin(10t) sin(20 t)
(C) sin(10 t) sin(20t)
(D) sin(t) sin(25t)
AD [IES - EC - 2012]
(12) The period of the signal
x(t) 10sin(12 t) 4cos(18 t) is
(D) x1 is not periodic, but x2 is periodic
(6)
2π
1.2π
(B)
(A)
4
(B)
1
6
(C)
1
9
(D)
1
3
A [IES - EC - 1991]
(13) A periodic function of half - wave symmetry
is necessarily
TARGATE EDUCATION GATE-(EE/EC)
Topic.1 - Continuous Time Signal & System.
(A) an even function
(B) an odd function
(C) neither odd nor even
(D) both odd and even
AD [GATE – IN – 2008]
(14) The fundamental period of the discrete-time
signal x n e
(A)
5
j n
6
(A) 0
is
6
5
(C) 6
1 2
(B)
2 10
(B)
12
5
1 4
(C)
2 10
(D) 12
A6[GATE-IN-2015]
(15) The fundamental period of the signal
2t
x t 2cos
cos t , in seconds, is
3
__________s.
AA[GATE-IN-2003]
(16) Given X = (a, b, c, d) as the input, a linear
time invariant system produces an output
y x, x, x,........., repeated N times .
The impulse response of the system is
2
1 4
(D)
2 5
2
2
S4A8 [GATE – IN – 2016]
(20) The fundamental period N0 of the discretetime sinusoid x [ n ] sin 3 01 n is ____ .
4
A1 [GATE – IN – 2017]
(21) A periodic signal x(t) shown in the figure.
The fundamental frequency of the signal x(t) in
Hz is _______.
N 1
(A)
n 4
i 0
(B) u[n] – u[n – N]
AD [GATE – IN – 2018]
(22) Two periodic signals x(t ) and y (t ) have the
same fundamental period of 3 seconds.
Consider the signal z (t ) x (t ) y (2t 1) .
The fundamental period of z (t ) in seconds is
(C) u[n] – u[n – N – 1]
N 1
(D)
n i
i 0
A0[GATE-IN-2016]
(17) If X(s), the Laplace transform of signal x(t)
s 2
is given by X s
, then the
2
s 1 s 3
value of x(t) as t is ___________
S1AB [GATE – EC – 2016]
(18) A continuous-time function x(t) is periodic
with period T. The function is sampled
uniformly with a sampling period T s . In
which one of the following cases is the
sampled signal periodic?
(A)
(B) T 1.2 Ts
(C) Always
(D) Never
(A) 1
(B) 1.5
(C) 2
(D) 3
A6 [GATE-IN-2019]
5
j n
3
j n
4
e
(23) A discrete-time signal x(n) e
is down-sampled to the signal xd(n) such that
xd(n) = x(4n). The fundamental period of the
down-sampled signal xd(n) is ______.
A11.99 to 12.01 [GATE-EC-2019]
(24) Consider the signal
2t
f (t ) 1 2cos(t ) 3sin
4cos t ,
4
3
2
where t is in seconds. Its fundamental time
period, in seconds, is _______.
S4AA [GATE – IN – 2016]
(19) For the periodic signal x(t) shown below with
period T = 8 s, the power in the 10th
harmonic is
www.targate.org
**********
Page 11
SIGNAL & SYSTEM
Convolution theorem
(1)
(2)
(5)
AD [GATE – EC1 – 2015]
The
result
of
the
convolution
x(t)* (t t 0 ) is :
(A) x(t t 0 )
(B) x(t t 0 )
(C) x(t t 0 )
(D) x( t t 0 )
f1(t) 0 for 1 < t < 3
= 0 elsewhere
f2(t) 0 for 5 < t < 7
= 0 elsewhere,
AC [GATE – EE/EC/IN – 2013]
Two systems with impulse responses
h 1 t and h 2 t are connected in cascade.
Then the overall impulse response of the
cascaded system is given by
Then the convolution of f1(t) and f2(t) is zero
everywhere except for
(A) Product of h 1 t and h 2 t
(6)
(B) Sum of h 1 t and h 2 t
(C) Convolution of h 1 t and h 2 t
(D) Subtraction of h 2 t from h 1 t
(3)
AD [GATE – EE – 2015]
The impulse response g(f) of a system, G, is
as shown in figure (A). What is the
maximum value attained by the impulse
response of two cascaded blocks of G as
shown in figure (B).
2
(A)
3
4
(C)
5
(4)
(B)
t (t 1)
u(t 1)
2
(t 1)2
u(t 1)
(C)
2
(D)
Page 12
t 2 1
u(t 1)
2
(B) 3 < t < 5
(C) 5 < t < 21
(D) 6 < t < 10
AA [GATE - EE - 1993]
S(t) is step response and h(t) is impulse
response of a system. Its response y(t) for
any input u(t) is given by
(A)
d t
s(t τ )u( τ )dτ
dt 0
(B)
s(t τ )u(τ )dτ
0
t
τ
s(t τ1 )u (τ1 )dτ
(C)
(D)
d t
h(t τ )u( τ )dτ
dt 0
0 0
AB [GATE - EE - 2002]
Let s(t) be the step response f a linear system
with zero initial conditions. Then the
response of this system to an input u(t) is
(b)
t
3
(B)
4
(D) 1
(A)
s(t τ )u(τ )dτ
(B)
d t
s(t τ )u( τ )dτ
0
dt
0
t
AC [GATE-EC/EE/IN-2013]
The impulse response of a system is
h(t ) t u (t ). For an input u (t 1), the output
is
t2
(A) u (t )
2
(A) 1 < t < 7
t
(7)
(a)
AD [IES - EC - 1998]
If f1(t) and f2(t) are duration-limited signals
such that
(C)
t u ( τ )dτ dτ
s
(
t
τ
)
0
0 1 1
1
(D)
(8)
2
s(t τ ) u(τ )dτ
0
AA [IES - EC - 2008]
Which one of the following is the impulse
response of the system whose step response
is given as c(t) = 0.5 (1 – e–2t) u(t) ?
(A) e 2 t u(t)
(B) 0.5(t) e 2 t u(t)
(C) 0.5(t) 0.5e 2 t u(t)
(D) 0.5e2t u(t)
TARGATE EDUCATION GATE-(EE/EC)
Topic.1 - Continuous Time Signal & System.
(9)
(A) Change the initial condition to y (0)
and the forcing function to 2x(t )
A3.9 to 4.1 [GATE – EC – 2014]
The sequence x[n] = 0.5 n u n is the unit step
sequence, is convolved with itself to obtain
(B) Change the initial condition to 2 y (0)
and the forcing function to x(t )
y[n]. Then
y n is _______.
n
(C) Change the initial condition to j 2 y (0)
and the forcing function to j 2 x ( t )
AB [GATE – EC – 1998]
(10) The unit impulse response of a linear time
invariant system is the unit step function u(z)
For t > 0, the response of the system to an
excitation e at u ( t ), a 0 will be
(D) change the initial condition to 2 y (0)
and the forcing function to 2 x (t )
AA [GATE – EC – 1995]
(15) Let h (t) be the impulse response of a linear
time invariant system. Then the response of
the system for any input u (t) is
(A) a e at
(B) (1 / a )(1 e at )
(C) a (1 e at )
(B)
d t
h()u(t d
dt 0
at
(D) 1 e
A0.4 [GATE – EC3 – 2015]
(11) Consider a continuous-time signal defined as
t
(A)
0
h ( ) u ( t d
sin( t / 2) *
(t 10n)
x (t)
( t / 2) n
(C)
t
t h()u (t d dt
0 0
where ‘*’ denotes the convolution operation
and t is in seconds. The Nyquist sampling
rate (in samples/sec) for x(t) is _____.
(D)
AC [GATE-EE1-2014]
(12) x(t) is nonzero only for T x < t < T 'X , and
similarly , y (t) is nonzero only for T Y < t <
T 'y. Let z (t) be convolution on x (t) and y
(t). Which one of the following statements is
TRUE?
t
0
h 2 ( )u ( t d
AB [GATE – EC – 1990]
(16) The impulse response and the excitation
function of a linear time invariant causal
system are shown in Fig. a and b
respectively. The output of the system at t =
2 sec. is equal to
(A) Z (t) can be non-zero over an unbounded
interval.
(B) Z (t) is nonzero for t < TX + Ty.
(C) Z (t) is zero outside of TX + TY < t < T'X
+ T'Y.
Fig. (a)
(D) Z (t) is nonzero for t >T'X + T'Y.
AA [IES-EC-2013]
(13) The convolution x (n) * δ(n - n0) is equal to
(A) x(n - n0)
(B) x(n + n0)
(C) x(n0)
(D) x(n)
AD [GATE-EC/EE/IN-2013]
(14) A system described by a linear, constant
coefficient, ordinary, first order differential
equation has an exact solution given by y (t )
for t 0, when the forcing function is x (t )
and the initial condition is y (0). If one
wishes to modify the system so that the
solution becomes 2 y (t ) for t 0, we need
to
Fig. (b)
(A) 0
(B) 1/2
(C) 3/2
(D) 1
AA [GATE – EC – 2004]
(17) A rectangular pulse train s (t) as shown in
Fig. 1 is convolved with the signal
cos 2 (4π 103 t ). The convolved signal will
be a
www.targate.org
Page 13
SIGNAL & SYSTEM
(C)
(A) DC
(B) 12 kHz sinusoid
(C) 8 Hz sinusoid
(D) 14 kHz sinusoid
(D)
AA [GATE - EE - 2011]
(18) Given two continuous time signals x(t) = e t
and y ( t ) e 2 t which exist for t 0, the
convolution z(t) = x (t ) * y (t ) is :
(A) e t e 2 t
(C) e t
(B) e3t
(D) e t e 2 t
AB [IES - EC - 2001]
(20) The impulse response of a system consists of
two delta functions as shown in the given
Fig.
AD [GATE-IN-2007]
(19) The signal x(t) and h(t) shown in the figures
are convolved to yield y(t).
The input to the system is a unit amplitude
square pulse of on unit time duration. Which
one of the following diagrams depicts the
correct output?
(A)
Which one of the following
represents the output y(t)?
figures
(B)
(A)
(B)
Page 14
TARGATE EDUCATION GATE-(EE/EC)
Topic.1 - Continuous Time Signal & System.
(C)
t
(A) 2 e u ( t )
2 t
(C) e u ( t )
1 2 t
e u (t )
(B) 2
(D) e t u (t )
AC [GATE - EE - 1998]
(25) The output of a linear time invariant control
system is c(t) for a certain input r(t). If r(t) is
modified by passing it through a block whose
transfer function is e s and then applied to
the system, the modified output of the system
would be
(D)
(A)
r (t )
1 et
(C) r ( t 1) u ( t 1)
AB [GATE – EC – 2001]
(21) The transfer function of a system is given by
H ( s)
(B)
r (t )
1 e t
(D) r ( t ) u ( t 1)
AD [GATE - EE - 2009]
(26) A Linear Time Invariant system with an
impulse response h(t) produces output y(t)
when input x(t) is applied. When the input
x ( t τ ) is a applied to a system with
response h ( t τ ), the output will be
1
. The impulse response of
s ( s 2)
2
(A) y(t)
(B) y(2(t - τ ))
(C) y (t τ )
(D) y ( t 2 τ )
AA [GATE - IN - 2006]
(27) Given x (t ) * x (t ) t exp( 2t )u (t )
the system is
(A) ( t 2 * e 2 t )U ( t )
(B) (t * e 2 t )U ( t )
the function x(t) is
(C) ( t e 2 t )U ( t )
(A) exp( 2 t )u ( t )
(D) (t e 2 t ) U (t )
(B) exp( t ) u ( t )
(* denotes convolution, and U (t) is unit step
function)
(C) t exp( t )u (t )
AC [GATE – EC – 2006]
(22) Let g(t) = p (t) * p(t), where * denotes
convolution and p(t) = u(t) – u(t – 1) with
u(t) being the unit step function
The impulse response of filter matched to the
signal s(t) = g(t) - (t 2)* g (t ) is given as
(A) s (1 t )
(B) s(1 t )
(C) s (t )
(D) s(t)
(D) 0.5 t exp( t ) u ( t )
AD [GATE - IN - 2008]
(28) The step response of a linear time invariant
system is y(t) = 5e 10 t u (t ), where u(t) is the
unit step function. If the output of the system
corresponding to an impulse input (t ) is
h(t), then h(t) is
(A) 50e 10 t u (t )
d
f (t ) [GATE - EE - 1994]
dt
(23) If f(t) is the step-response of a linear timeinvariant system, then its impulse response is
given by .........................
(B) 50 e 10 t ( t )
A h (t )
AC [GATE - EE - 1995]
(24) The impulse response of an initially relaxed
linear system is e 2 t u (t ). To produce a
response of t e 2 t u (t ), the input must be
equal to
(C) 5u (t ) 50e 10 t u (t )
(D) 5 (t ) 50 e 10 t u (t )
AB [IES - EC - 1999]
(29) Fig.1 and Fig.2 show respectively the input
x(t) to a linear time - invariant system and
the impulse response h(t) of the system.
www.targate.org
Page 15
SIGNAL & SYSTEM
S4AA [GATE – EC – 2016]
sin(t ) sin(t )
*
(34) If the signal x(t )
with*
t
t
denoting the convolution operation, then x(t)
is equal to
(A)
sin(t )
t
(B)
(C)
2sin(t )
t
sin(t )
(D)
t
Fig. 1
sin(2t )
2t
2
A31.00 [GATE–S1–EC–2017]
(35) Two discrete-time signals x[n] and h[n] are
both non-zero only for n = 0, 1, 2 and are
zero otherwise. It is given that
x 0 1, x 1 2, x 2 1, h 0 1
Fig. 2
The output of the system is zero everywhere
except for the time - interval
(A) 0 < t < 4
(B) 0 < t < 6
(C) 1 < t < 5
(D) 1 < t < 6
AC [IES - EC - 2008]
(30) The convolution of f(t) with itself is given it
t
be f ( )d . Then what is f(t) ?
0
(A) The unit ramp function
(B) Equal to 1
(C) The unit step function
(D) The unit impulse function
1
1
A e 2t et u (t ) [GATE - EE - 1995]
3
3
(31) The
convolution
of
the
functions
2t
t
and
f1 ( t ) e u ( t )
f 2 (t ) e u (t ) is equal
to......................
AC [GATE - EE - 2007]
(32) If u(t), r(t) denote the unit step and unit ramp
functions respectively and u(t) * r(t) their
convolution, then the function u(t + 1) * r(t –
2) is given by
(A) (1/2) (t – 1)(t – 2)
(B) (1/2) (t – 1)(t – 2)
(C) (1/2) (t – 1)2u(t – 1)
(D) none of the above
AC [IES - EC - 2003]
(33) Two rectangular waveforms of duration T1
and T2 seconds are convolved. What is the
shape of the resulting waveform ?
(A) Triangular
(B) Rectangular
(C) Trapezoidal
(D) Semi-circular
Page 16
Let y[n] be the linear convolution of x[n] and
h[n]. Given that y[1] = 3 and y[2] = 4, the
value of the expression 10y 3 y 4 is
_____ .
AA [GATE–S1–EE–2017]
(36) Let z t q x t * y t , where “*” denotes
convolution. Let c be a positive real-valued
constant. Choose the correct expression for
z(ct).
(A) c.x ct * y ct
(B) x ct * y ct
(C) c.x t * y ct
(D) c.x ct * y t
AD [GATE–S2–EE–2017]
(37) A cascade system having the impulse
responses h1 ( n ) {1, 1} and h2 ( n ) {1, 1} is
shown in the figure below, where symbol
denotes the time origin,
The input sequence x(n) for which the
cascade system produces an output sequence
y ( n ) {1, 2,1, 1, 2, 1} is
(A) x ( n ) {1, 2,1,1}
(B) x ( n ) {1,1, 2, 2}
(C) x ( n ) {1,1,1,1}
(D) x ( n ) {1, 2, 2,1}
TARGATE EDUCATION GATE-(EE/EC)
**********
Topic.1 - Continuous Time Signal & System.
(A) 5 t
Delta Functions
(1)
AA [GATE – EC – 2001]
Let (t) denote the delta function. The value
of the integral
3t
t cos 2 dt
is :
(C)
(8)
(A) 1
(B) 1
(C) 0
(D) / 2
(B) 5 u(t) – C
5
t
C
(D)
5u (t )
C
AB [IES - EC - 2011]
Which one of the following relations is not
correct ?
(A) f (t)(t) f (0)(t)
(2)
(B)
AC [IES-EC-2013]
The value of sin t t dt is
4
(C)
2
(B)
(C)
1
2
(D) 3
( ) d() 1
1
3
(A)
f (t)()d 1
(D) f (t) (t ) f ( ) (t )
AB [GATE - IN - 2010]
(9)
The
integral
AB [GATE - IN - 2011]
(3)
The integral
1
2π
evaluates to
2
t 2 e t / 2 (1 2t )dt is
equal to
(A)
1
e 1/8
8 2π
(C)
1 1/ 2
e
2π
(B)
1
e 1/8
4 2π
(t π / 6)6sin(t )dt
(A) 6
(B) 3
(C) 1.5
(D) 0
AC [GATE – EC – 2002]
(10) Convolution of x (t 5) with impulse
function (t 7) i equal to
(D) 1
AB [IES - EC - 2001]
(A) x(t 12)
(B) x(t 12)
(C) x(t 2)
(D) x(t 2)
(4)
If y (t ) y ( ) x(t )d (t ) x( t ) then
0
y(t) is
(A) u(t)
(B) δ(t)
(C) r(t)
(D) 1
AD [GATE – EC – 2006]
(11) The Dirac delta function (t) is defined as
t 0
1
(A) t
0 otherwise
t0
(B) t
0 otherwise
AA [IES-EC-2013]
(5)
The values of the integral
t0
1
and
(C) t
0 otherwise
2
I=
5t
2
1 ( t ) dt is
1
(A) 0
(B) 1
(C) 42/3
(D) 125/3
t0
and
(D) t
0 otherwise
A e2 [GATE - EE - 1994]
(6)
The value of the integral
6
5
e 2 t (t 1) dt is
equal to -------------------(7)
AD [GATE - EE - 2002]
A current impulse, 5 (t ), is forced through
a capacitor C. The voltage, vc (t ), across the
capacitor is given by
t dt 1
t dt 1
AA [GATE - IN - 2009]
(12) The response of a first order measurement
system to a unit step input is 1 e0.5 t , where
t is in seconds. A ramp of 0.1 units per
second is given as the input to this system.
The error in the measured value after
transients have died down is
www.targate.org
(A) 0.02 units
(B) 0.1 units
(C) 0.2 units
(D) 1 unit
Page 17
SIGNAL & SYSTEM
AC [IES - EC - 2001]
(13) The impulse response of a system is h(t) = δ(t
- 0.5).
If two such systems are cascaded, the
impulse response of the overall system will
be :
(A) 0.5 δ(t - 0.25)
(B) δ(t - 0.25)
(C) δ(t - 1)
(D) 0.5 δ(t - 1)
Energy, Power & RMS
(1)
A2 [GATE – EC1 – 2015]
The waveform of a periodic signal x(t) is
shown in the figure.
S6AA [GATE – EE – 2016]
(14) The value of
e t (2t 2) dt , where (t ) is
the Driac delta function, is
1
(A)
2e
(C)
t 1
A signal g(t) is defined by g(t) = x
.
2
The average power of g(t) is _____.
2
(B)
e
1
e2
(D)
1
2e 2
(2)
AA [GATE – IN – 2018]
1, | t | 2
(15) Consider signal x(t )
. Let (t )
0 | t | 2
denote the unit impulse (Dirac-delta)
function. The value of the integral
5
0
2 x(t 3)(t 4) dt is
s t 8cos 20t 4sin 15t is
2
(3)
(A) 2
(B) 1
(C) 0
(D) 3
AA [GATE – EC – 2005]
The power in the signal
(A) 40
(B) 41
(C) 42
(D) 82
AB [IES - EC - 2001]
The signal x(t) = A cos ( t ) is
(A) an energy signal
(B) a power signal
(C) an energy as well as a power signal
**********
(D) neither an energy nor a power signal
(4)
(5)
AC [GATE - IN - 2009]
The root mean squared value of x(t) =
3 2 sin(t ) cos(2t ) is
(A)
3
(B)
8
(C)
10
(D)
11
AD [IES - EC - 2008]
What is the average power of periodic nonsinusoidal voltages and currents ?
(A) The average power of the fundamental
component alone
(B) The sum of the average powers of the
harmonics excluding the fundamental
(C) The sum of the average powers of the
sinusoidal components including the
fundamental
(D) The sum of the root mean square power
of the sinusoidal components including
the fundamental
Page 18
TARGATE EDUCATION GATE-(EE/EC)
Topic.1 - Continuous Time Signal & System.
(6)
(7)
AD [GATE - EE - 2002]
What is the rms value of the voltage
waveform shown in Fig.?
(A) 200/ π V
(B) 100/ π V
(C) 200 V
(D) 100 V
AA [GATE – EC – 1995]
The RMS value of rectangular wave of
period T, having a value of +V for a
duration. T1 ( T ) and –V for the duration.
T T1 T2 , equals
(A) V
(C)
(8)
V
2
(B) 12 kJ
(C) 13.2 kJ
(D) 14.4 kJ
AB [GATE – EC – 2001]
(11) If a signal f (t) has energy E, the energy of
the signal f (2t) is equal to
(A) E
(B) E/2
(C) 2E
(D) 4E
AB [GATE – EC – 2010]
(12) Consider an angle modulated signal
x(t ) 6 cos[2 10 6 t 2sin(8000t ) +4
T T
(B) 1 2 V
T
T
(D) 1 V
T2
AC [IES-EC-2014]
The current waveform i(t) in a pure resistor
of 20 is shown in the figure
(A) 220 J
cos (8000t )] V. The average power of x(t)
is :
(A) 10 W
(B) 18 W
(C) 20 W
(D) 28 W
AD [IES - EC - 2007]
(13) Which one of the following is the
mathematical representation for the average
power of signal x(t)?
T
The power dissipated in the resistor is
(9)
(A) 135 W
(B) 270 W
(C) 540 W
(D) 14.58 W
AC [IES - EC - 2007]
Which one of the following is correct ?
(A)
1
x(t)dt
T 0
(B)
1T 2
x (t)dt
T 0
(C)
1 T/2
x(t)dt
T T/ 2
1 T/2 2
x (t)dt
T T
T/2
(D) Lt
Energy of a power signal is
(A) Finite
(B) zero
(14)
(C) Infinite
(D) between 1 and 2
A0.408 [GATE – EC1 – 2014]
A periodic variable X is shown in the figure
as a function of time the root mean square
(rms) value of X is --------.
AC [GATE – EC – 2009]
(10) A fully charged mobile phone with a 12 V
battery is good for a 10 minute talk-time.
Assume that, during the talk-time, the battery
delivers a constant current of 2 A and its
voltage drops linearly from 12 V to 10 V as
shown in the Fig. 1. How much energy does
the battery deliver during this talk-time?
www.targate.org
Page 19
SIGNAL & SYSTEM
AC [GATE - IN - 2008]
(15) If a current of
π
6 2 sin(100πt ) 6 2 cos 300πt 4 6 2
A is passed through a true RMS ammeter, the
meter reading will be :
(A) 6 2 A
(B)
126 A
(C) 12 A
(D)
216 A
(A) 14.1 A
(B) 17.3 A
(C) 22.4 A
(D) 30.0 A
AA [GATE - EE - 2005]
(20) For the triangular waveform shown in the
figure, the RMS value of the voltage is equal
to
AB [GATE - EE - 1995]
(16) The rms value of the periodic waveform e(t),
shown in figure is :
(A)
(C)
1
V
6
1
V
3
(B)
1
V
3
(D)
2
V
3
AD [IES - EC - 1995]
(21) In the given Fig., the effective value of the
waveform is
(A)
3
A
2
(B)
2
A
3
(C)
1
3
(D)
2 A
(A) 0.5
AA [GATE - EE - 2005]
(17) The RMS value of the voltage v(t) = 3 + 4
cos(3t ) is
(A) 17 V
(B) 5 V
(C) 7 V
(D) (3 2 2 ) V
AA [GATE - EE - 2004]
(18) The rms value of the periodic waveform
given in Fig. is :
(C)
(B) 2.5
2.5
(D)
50
AA [GATE – IN – 2003]
sin c n
(22) Given x n
, the energy of the
n
signal given by
x n
2
n
c
(B) c
(C) infinite
(D) 2 c
(A)
S3A0.24-0.26 [GATE – EC – 2016]
sin(4t )
(23) The energy of the signal x (t )
is
4t
_______
(A) 2 6 A
(B) 6 2 A
(C)
(D) 1.5 A
4/3 A
AB [GATE - EE - 2004]
(19) The rms value of the resultant current in a
wire which carries a dc current of 10 A and a
sinusoidal alternative current of peak value
20 A is
Page 20
A7.95-8.05 [GATE–S2–EC–2017]
(24) Consider an LTI system with magnitude
response
|f|
, | f | 20
1
| H ( f ) | 20
0,
| f | 20
and phase response
If the input to the system is
TARGATE EDUCATION GATE-(EE/EC)
Topic.1 - Continuous Time Signal & System.
x(t ) 8cos 20t 16sin 40t
4
8
Miscellaneous
(1)
24cos 80
16
AB [GATE – EC3 – 2015]
The impulse response of an LTI system can
be obtained by
(A) differentiating the unit ramp response
then the average power of the output signal
y(t) is _____ .
(B) differentiating the unit step response
A7.0 [GATE–S2–EC–2017]
(25) Consider the parallel combination of two LTI
systems shown in the figure,
(C) integrating the unit ramp response
(D) integrating the unit step response
(2)
AC [IES-EC-2013]
If a continuous time signals x (t) can take on
any value in the continuous interval (-∞,∞) it
is called
(A) Deterministic Signal
The impulse responses of the systems are
(B) Random Signal
h1 (t ) 2 (t 2) 3 (t 1)
(C) Analog Signal
h2 (t ) (t 2)
(D) Digital signal
If the input x(t) is a unit step signal, then the
energy of y(t) is ______ .
A6 [GATE–S2–EE–2017]
(26) The mean square value of the given periodic
waveform f(t) is _______ .
(3)
AD [IES-EC-2013]
The ramp function can be obtained
from the unit impulse at t = 0 by
(A) Differentiating unit impulse function
once
(B) Differentiating unit impulse function
twice
(C) Integrating unit impulse function once
(D) Integrating unit impulse function twice
(4)
AD [GATE – EE – 2018]
(27) The signal energy of the continuous-time
signal
x(t) =[(t -1) u(t -1)]-[(t -2)u(t -2)]-[(t -3)u(t 3)]+[(t -4)u(t -4)] is
A0.155 [GATE – EC3 – 2015]
Consider the function g(t) e t sin(2t)u(t)
where u(t) is the unit step function. The area
under g(t) is _____.
A0.19to0.21 [GATE – EC2 – 2014]
(5)
The value of the integral
sin c 2 5t dt is
(A) 11/3
(B) 7/3
------------
(C) 1/3
(D) 5/3
AA [GATE – EC – 1990]
The response of an initially relaxed linear
constant parameter network to a unit impulse
applied at t = 0 is 4 e 2 t u ( t ). The response of
this network to a unit step function will be:
(6)
**********
(A) 2[1 e 2 t ]u (t )
(B) 4[e t e2 t ]u (t )
(C) sin 2t
(D) (1 4 e 4 t ) u ( t )
www.targate.org
Page 21
SIGNAL & SYSTEM
(7)
AB [GATE – EC – 1991]
The voltage across an impedance in a
network is V(s) = Z(s) I(s), where V(s), Z(s)
and I(s) are the Laplace Transforms of the
corresponding time functions v(t), z(t) and
i(t). The voltage v (t) is :
(A) v(t ) z (t ).i(t )
AA [GATE – EC – 2000]
(11) A linear time invariant system has an
impulse response e 2 t , t 0. If the initial
conditions are zero and the input is e3t , the
output for t > 0 is
3t
2t
(B) e
3t
2t
(D) None
(A) e e
5t
t
(B) v (t ) i ( τ ) z (t τ ) dτ
(C) e e
0
AC [GATE - EE - 2011]
(12) The response h(t) of a linear time invariant
system to an impulse ( t ), under initially
t
(C) v (t ) i ( τ ) z ( t τ ) dτ
0
(D) v(t ) z (t ) i(t )
(8)
AA [GATE – EC – 2004]
A system described by the differential
equation:
d2y
dy
3 2 y x (t )
2
dt
dt
(B) ( e t e 2 t )u (t )
is initially at rest. For input x (t) = 2u (t), the
output
y (t) is
(A) (1 2e t e 2 t )u (t )
(D) e t ( t ) e 2 t u (t )
(C) (1.5 e t 0.5 e 2 t )u (t )
(B) (1 2 e t 2e 2 t )u (t )
(C) (0.5 e t 1.5 e 2 t ) u ( t )
(D) (0.5 2 e t 2 e 2 t )u (t )
(9)
relaxed condition is h(t) e t e 2 t . The
response of this system for a unit step input
u(t) is :
(A) u (t ) e t e 2 t
AD [GATE – EC – 2008]
The impulse response h (t) of a linear timeinvariant continuous time system is describe
by h (t) = exp ( t )u(t ) + exp( t )u ( t ) ,
where u (t) denotes the unit step function,
and and are real constants. This
system is stable if
AB [IES - EC - 2011]
(13) Given the differential equation model of a
physical system, determine the time constant
dx
of the system : 40 2x f (t)
dt
(A) 10
(B) 20
(C) 1/10
(D) 4
AD [IES - EC - 2004]
(14) The impulse response of a linear timeinvariant system is a rectangular pulse of
duration T. It is excited by an input of a pulse
of duration T. What is the filter output
waveform ?
(A) is positive and is positive
(A) Rectangular pulse of duration T
(B) is negative and is negative
(B) Rectangular pulse of duration 2T
(C) is positive and is negative
(C) Triangular pulse of duration T
(D) is negative and is positive
(D) Triangular pulse of duration 2T
AB [GATE – EC – 2010]
(10) A continuous time LTI system is described
by
d 2 y (t )
dy (t )
dx (t )
4
3 y (t ) 2
4 x (t )
2
dt
dt
dt
Assuming zero initial conditions, the
response y(t) of the above system for the
input x(t) = e 2 t u ( t ) is given by
(A) (e t e 3t )u (t )
t
3 t
)u (t )
t
3t
)u (t )
(B) ( e e
(C) ( e e
t
3t
(D) (e e )u (t )
Page 22
AC [GATE – EC – 2008]
(15) A linear, time – invariant, causal continuous
time system has a rational transfer function
with simple poles at s = 2 and s = 4, and
one simple zero at s = 1. A unit step u(t) is
applied at the input of the system. At steady
state, the output has constant value of 1. The
impulse response of this system is
(A) [exp(2t ) exp(4t )]u (t )
(B) [4exp(2t ) 12exp(4t ) exp(t )]
u (t )
(C) [4exp(2t ) 12exp(4t )]u(t )
(D) [0.5exp(2t ) 1.5exp(4t )]u (t )
TARGATE EDUCATION GATE-(EE/EC)
Topic.1 - Continuous Time Signal & System.
AD [GATE – EC – 2011]
(16) An input x(t) = exp(2t )u(t ) (t 6) is
applied to an LTI system with impulse
response h(t) = u(t). The output is :
(A) [1 exp(2t )]u(t ) u(t 6)
AB [GATE - EE - 1996]
(20) The unit impulse response of a system is
given as c(t) = 4 e t 6 e 2 t . The step
response of the same system for t 0 is
equal to
(A) 3e 2 t 4 e t 1
(B) [1 exp(2t )]u(t ) u(t 6)
(B) 3e 2 t 4 e t 1
(C) 0.5[1 exp(2t )]u(t ) u(t 6)
(C) 3e 2 t 4 e t 1
(D) 0.5[1 exp(2t )]u(t ) u(t 6)
(D) 3e 2 t 4 e t 1
AC [GATE - EE - 2003]
(17) A control system is defined by the following
mathematical relationship
AB [GATE - EE - 2003]
(21) A control system with certain excitation is
governed by the following mathematical
equation
d 2x
dx
6
5 x 12(1 e 2 t )
2
dt
dt
d 2 x 1 dx 1
x 10 5e 4 t 2e 5t
dt 2 2 dt 18
The response of the system as t is :
(A) x = 6
(B) x = 2
The natural time constants of the response of
the system are
(C) x = 2.4
(D) x = -2
(A) 2s and 5s
AD [GATE - EE - 2007]
(18) Let a signal a1 sin( 1t 1 ) be applied to a
stable linear time-invariant system. Let the
corresponding steady state output be
represented as a2 F ( 2 t 2 ). Then which
of the following statements is true?
(B) 3s and 6s
(C) 4s and 5s
(D) 1/3s and 1/6s
AD [GATE - EE - 2004]
(22) The unit impulse response of a second orderdamped system starting from rest is given by
(A) F is not necessarily a “sine” or “cosine”
function but must be periodic with
1 2 .
c (t ) 12.5e 6t sin 8t , t 0
The steady-state value of the unit step
response of the system is equal to
(B) F must be a “sine” or “cosine” function
with a1 a2
(C) F must be a “sine” function with
1 2 and 1 2
(D) F must be a “sine” or “cosine” function
with 1 2
(A) 0
(B) 0.25
(C) 0.5
(D) 1.0
AA [GATE - EE - 2008]
(23) A signal x(t) = sin c ( αt ) where α is a real
AC [GATE - EE - 2008]
(19) A signal e sin( t ) is the input to a real
Linear Time Invariant system. Given K and
are constants, the output of the system
αt
will be of the form Ke βt sin(υt ) where
(A) β need not be equal to α but υ equal
to
(B) υ need not be equal to but β equal
to α
(C) β equal to α and υ equal to
(D) β need not be equal to α and υ need
not be equal to
www.targate.org
sin(πx )
constant sin c ( x )
is the input to
πx
a Linear Time invariant system whose
impulse response h(t) = sin c ( βt ) where β
is a real constant. If min(α, β) denotes the
minimum of α and β , and similarly max
(α , β ) denotes the maximum of α and β ,
and K is a constant, which one of the
following statements is true about the output
of the system?
(A) It will be of the form K sin c (γ t ) where
γ min( α , β )
(B) It will be of the form K sin c (γ t ) where
γ max( α , β )
Page 23
SIGNAL & SYSTEM
D. f(t) (g(t) - g(0)) = 0;
(C) It will be of the form K sinc ( αt )
For any arbitrary g(t)
(D) It cannot be a sinc type of signal
AD [GATE - EE - 2008]
(24) A function y(t) satisfies the following
differential equation :
dy (t )
y (t ) (t )
dt
Where (t ) is the delta function. Assuming
zero initial condition, and denoting the unit
step function by u(t), y(t) can be of the form
(A) e
t
(C) e
2 t
(B) e
(D) e u (t )
AD [GATE - IN - 2009]
(25) A linear time – invariant causal system has a
frequency response given in polar form as
1
tan 1 . For input x(t) = sin (t),
2
1
the output is
(A)
1
2
(B)
cos(t )
(B) e
1 t
(C) e u(t )
3.
Impulse
4.
Causal
5.
Sinusoid
Codes:
A
B
C
D
(A)
4
1
5
3
(B)
1
4
5
3
(C)
4
2
5
1
(D)
2
5
4
1
AC [IES - EC - 2001]
(28) If a function f(t) u(t) is shifted to right side
by t0 , then the function can be expressed as
(D) f (t t0 )u(t t0 )
AB [IES - EC - 1997]
(26) The unit step response of a system is given
by (1 e t )u(t ) .The impulse response is
given by
u (t )
Growing exponential
(C) f (t t0 )u(t t0 )
1
π
sin t
4
2
(A) e
2.
(B) f (t )u(t t0 )
1
(C)
sin(t )
2
t
Decaying exponential
(A) f (t t0 )u(t )
1
π
cos t
2
4
(D)
1.
t
t
u (t )
List - II
t
AC [IES - EC - 2007]
(29) The response of a linear, time-invariant
system to a unit step is s(t) = (1 – e-t/RC) u(t),
where u(t) is the unit step. What is the
impulse response of this system ?
(A) e t /RC
(B) e t / RC u(t)
(C) 1/ RC{e t / RC u(t)}
(D) (t)
u (t )
(D) e t u(t )
AA [IES - EC - 1999]
(27) Match the List - I (Functions of f(t)) with
List - II (Characteristic of f(t)) and select the
correct answer using the codes given below
the lists:
AD [IES - EC - 2009]
(30) A signal x1(t) and x2(t) constitute the real and
imaginary parts respectively of a complex
valued signal x(t). Also x1(t) is symmetric (or
even) and x2(t) is anti symmetric (or odd).
What form of waveform does x(t) possess?
(A) Real symmetric
(B) Complex symmetric
List - I
(C) Asymmetric
A. f(t) (1 - u(t)) = 0
(D) Conjugate symmetric
B.
f(t) + Kdf(t)/dt = 0;
k is positive constant
C. f(t) + K
d 2 f (t )
0;
dt 2
AB [IES - EC - 2010]
(31) If the response of LTI continuous time
1 1
system to unit step input is e 2t ,
2 2
then impulse response of the system is
K is positive constant
Page 24
TARGATE EDUCATION GATE-(EE/EC)
Topic.1 - Continuous Time Signal & System.
1 1
(A) e 2t
2
2
2t
(C) 1 e
(B) (e2t )
(D) Constant
AA [IES - EC - 2012]
(32) A linear time-invariant system has an
impulse response of e 2 t , t 0 . If the initial
conditions are zero and the input is e3t , the
output for t > 0 is
(A) e3t e 2t
(B) e5t
(C) e3t e2 t
(D) e t
AB [IES - EC - 1995]
(33) Double integration of a unit step function
would lead to
(A) an impulse
(B) a parabola
(C) a ramp
(D) a doublet
AB [IES - EC - 2007]
(34) The relation between input x(t) and output
y(t) of a continuous time system is given by
dy(t)
3y(t) x(t)
dt
What is the forced response of the system
when x(t) = k (a constant) ?
(A) k
(B) k/3
(C) 3k
(D) 0
-------0000-------
www.targate.org
Page 25
02
Discrete Time
Signal & System
(C) is unstable
(D) stability cannot be assessed from the
given information
System’s Classification
(1)
AA [IES-EC-2014]
A discrete-time system has input x[] and
output y[] satisfying
y[ m ]
m
j
AA [IES - EC - 2008]
n
(5)
x[ j ]
AA [GATE – EC – 2004]
The impulse response h[n] of a linear timeinvariant system is given by
(6)
where x (n) is the input and y (n) is the
output. The above system has the properties
(A) P, S but not Q, R
(B) P, Q, S but not R
(C) P, Q, R, S
(D) Q, R, S but not P
(A) stable but not causal
(B) stable and causal
(C) causal but unstable
(D) unstable and not
AB [GATE – EC – 2011]
A system is defined by its impulse response h
(n) = 2 n u ( n 2). The system is
(7)
(A) stable and causal
(B) causal but not stable
(C) stable but not causal
AA [IES - EC - 2012]
The following equation describes a linear
time-varying discrete time system
(A) y (k 2) ky (k 1) y ( k ) u (k )
(B) y ( k 2) ky 2 ( k 1) y (k ) u ( k )
(C) y (k 2) 3 y ( k 1) 2 y (k ) u (k )
(D) y k 2 y 2 k 1 ky k u k
(D) unstable and non causal
AA [GATE – EC – 1992]
A linear discrete – time system has the
3
characteristic equation, z 0.81 z = 0. The
system
AA [GATE – EC – 2003]
Let P be linearity, Q be time-invariance, R be
causality and S be stability. A discrete time
system has the input-output relationship,
x( n), n 1
y(n) = 0,
n0
x( n 1) n 1
Where u[n] is the unit step sequence. The
above system is
(4)
is an
example of
(A) Invertible system
(B) Memory less system
(C) non-invertible system
(D) Averaging system
h[n] u[n 3] u[n 2] 2u[n 7]
(3)
x[k]
k
The system is :
(A) linear and unstable
(B) linear and stable
(C) non-linear and stable
(D) non-linear and unstable
(2)
A system defined by y[n]
(8)
(A) is stable
(B) is marginally stable
www.targate.org
AA [GATE - EE - 2007]
Consider the discrete-time system shown in
the figure where the impulse response of
G(z)
is
g (0) 0, g (1) g (2) 1,
g (3) g (4) ... =0
Page 26
Topic.2 – Discrete Time Signal & System
AD [IES - EC - 2004]
(12) Match List-I (Equation Connecting Input
x(n) and Output y(n) with List-II (System
Category) and select the correct answer using
the codes given below :
(9)
This system is stable for range of values of K
List-I
(A) [-1, 1/2]
(B) [-1, 1]
A. y(n+2)+y(n+1)+y(n)=2x(n+1)+x(n)
(C) [-1/2, 1]
(D) [-1/2, 2]
B.
n2y2(n)+y(n) = x2(n)
AD [IES - EC - 2005]
Assertion (A) : The discrete time system
described by y[n] = 2 x[n] + 4 x[n – 1] is
unstable, (here y[n] = 2x[n] + 4 x[n – 1] is
unstable, (here y[n] is the output and x[n] the
input)
C. y(n+1) + ny(n) = 4nx(n)
D. y(n+1) y(n) = 4x(n)
List-II
1.
Linear, time-variable, dynamic
2.
Linear, time-invariant, dynamic
Reason (R) : It has an impulse response with
a finite number of non-zero samples.
3.
Non-linear, time-variable, dynamic
4.
Non-linear, time-invariant, dynamic
Codes :
5.
Non-linear, time-variable, memory less
(A) Both A and R are true and R is the
correct explanation of A
Codes :
A
B
C
D
(B) Both A and R are true but R is not a
correct explanation of A
(A) 3
5
2
1
(B) 3
2
5
4
(C) 2
3
5
1
(D) 2
5
1
4
(C) A is true but R is false
(D) A is false but R is true
AB [GATE – EC – 2002]
(10) If the impulse response of a discrete-time
system is h[n] = 5 n u[ n 1], then the
system function H(z) is equal to
(A)
z
and the system is stable
z 5
AB [IES - EC - 2002]
(13) Match List - I (input-output relation) with
List-II (property of the system) and select the
correct answer using the codes given below
the lists :
List - I
A. y(n) x(n)
z
(B)
and the system is stable
z 5
B. y(n) x(n 2 )
C. y(n) x 2 ( n)
z
(C)
and the system is unstable
z 5
(D)
D. y(n x 2 (n)
z
and the system is unstable
z 5
List - II
1. Non linear, non-causal
AC [IES - EC - 2007]
(11) The outputs of two system S1 and S2 for the
2. Linear, non-causal
same input x[n] e jn are 1 and (1)n ,
respectively. Which one of the following
statements is correct ?
4. Non linear, causal
3. Linear, causal
Codes :
(A) Both S1 and S2 are linear time invariant
(LTI) systems
A
B
C
D
(A) 1
4
3
2
(B) S1 is LTI but S2 is not LTI
(B) 3
2
1
4
(C) S1 is not LTI but S2 is LTI
(C) 1
2
3
4
(D) Neither S1 nor S2 is LTI
(D) 3
4
1
2
www.targate.org
Page 27
SIGNAL & SYSTEM
AD [GATE - EE - 2006]
(14) y[n] denotes the output and x[n] denotes the
input of a discrete-time system given by the
difference equation
y[ n ] 0.8 y[ n 1] =
x[ n ] + 1.25 x[ n 1]. Its right-sided impulse
response is
(A) Causal
(B) Unbounded
(C) periodic
(D) Non-negative
AB [IES - EC - 2012]
(15) The discrete time system described by
y(n) x 2 (n) is :
(A) causal and linear
(B) causal and non-linear
(C) non-causal and linear
(D) non-causal and non-linear
AC [GATE – EC – 2010]
(16) The transfer function of a discrete time LTI
system is given by
(C) Stable, non-causal and has memory
(D) Unstable, non-causal and memory
AA [IES - EC - 1999]
(18) Assertion (A) : An LTI discrete system
represented by the difference equation
Y(n + 2) - 5y(n + 1) + 6y(n) = x(n) is
unstable.
Reason (R) : A system is unstable if the
roots of the characteristic equation lie outside
the unit circle.
Codes:
(A) Both A and R are true and R is the
correct explanation of A
(B) Both A and R are true but R is NOT the
correct explanation of A
(C) A is true but R is false
(D) A is false but R is true
AD [IES - EC - 2006]
(19) The impulse response of a system h(n) = an
u(n). What is the condition for the system to
be BIBO stable ?
(A) a is real and positive
(B) a is real and negative
3 1
z
4
H (z) =
3
1
1 z 1 z 2
4
8
2
(C) | a | > 1
(D) | a | < 1
Consider the following statements:
S1 : The system is stable and causal for
ROC: |z| > 1/2
S2 : The system is stable but not causal for
ROC: |z| < ¼
AC [GATE – EC – 2004]
(20) A causal LTI system is described by the
difference equation
2 y[n] y[n 2] 2 x[n] x[n 1]
The system is stable only if
(A) | | 2,| | 2
S3 : The system is neither stable nor causal
for ROC: ¼ < |z| < ½
(B) | | 2,| | 2
Which one of the following statements is
valid?
(D) | | 2, any value of
(A) Both S1 and S2 are true
(B) Both S2 and S3 are true
(C) Both S1 and S3 are true
(D) S1 , S2 and S3 are all true
AC [IES - EC - 2004]
(17) A discrete time system has impulse
Response, h(n) a n u(n 2) , | a | 1 . Which
one of the following statements is correct?
The system is :
(A) Stable, causal and memory less
(B) Unstable, causal and has memory
Page 28
(C) | | 2, any value of
AC [GATE – EC – 2006]
(21) A system with input x[n] and output y[n] is
5
given as y[n] = sin n ( n).
6
The system is :
(A) linear, stable and invertible
(B) non-linear, stable and non-invertible
(C) linear, stable and non-invertible
(D) linear, unstable and invertible
AD [GATE - IN - 2008]
(22) Which one of the following discrete – time
systems is time invariant?
TARGATE EDUCATION GATE-(EE/EC)
Topic.2 – Discrete Time Signal & System
(A) y[ n ] nx[ n ]
(B) y[ n ] x[3 n ]
(C) y[ n ] x[ n ]
(D) y[ n ] x[ n 3]
AD [IES - EC - 2003]
(23) A discrete LTI system is non-casual if its
impulse response is :
(A) a n u(n 2)
(B) a n 2 u(n)
(C) a n 2 u(n)
(D) a n u(n 2)
Miscellaneous
AB [GATE-EE1-2014]
(1)
1
be the Z-transform of a
1 z 3
causal signal x[ n] . Then, the values of x[2]
and x[3] are :
Let X ( z )
(A) 0 and 0
(C) 1 and 0
(B) 0 and 1
(D) 1 and 1
(A) y(n) = n x(2n)
(B) y(n) = x(n2)
AA [GATE – EC2 – 2014]
The input – output relationship of a causal
stable
LTI
system
is
given
as
y n y n 1 x [n ] . If the impulse
response h [n] of this system satisfies the
(C) y(n) = n2x(n)
(D) y(n) = x2(n)
condition
AD [IES - EC - 2007]
(24) Which one of the following systems
described by the following input-output
relations is non-linear ?
(2)
h n 2,
the relationship
n0
AA [GATE–S1–EC–2017]
(25) Consider a single input single output
discrete-time system with x[n] as input and
y[n] as output, where the two are related as
between α and β is
(A) 1 / 2
(B) 1 / 2
n x n , for 0 n 10
y n
otherwise
x n x n 1 ,
(C) 2
(D) 2
(C) It is not causal but stable
AC [GATE – EC – 2010]
Two discrete time systems with impulse
responses
and
h1 [ n ] [ n 1]
h2 [ n ] [ n 2] are connected in cascade.
The overall impulse response of the cascaded
system is
(D) It is neither causal nor stable
(A) [ n 1] [ n 2]
Which one of the following statements is true
about the system:
(3)
(A) It is causal and stable
(B) It is causal but not stable
(B) [n 4]
AD [GATE–S1–EE–2017]
(26) Consider the system with following inputoutput
relation
n
(C) [ n 3]
y n 1 1 x n ,
(D) [n 1] [n 2]
where x[n] is the input and y[n] is the output.
The system is
(4)
(A) invertible and time invariant
(B) invertible and time varying
(C) non-invertible and time invariant
AC [GATE - EE - 2010]
Given the finite length input x[n] and the
corresponding finite length output y[n] of an
LTI system as shown below, the impulse
response h[n] of the system is
(D) non-invertible and time varying
AA [GATE-IN-2019]
(27) The input x[n] and output y[n] of a discretetime
system
are
related
as
y[ n] y[ n 1] x[ n] . The condition on
for which the system is Bounded-Input
Bounded-Output (BIBO) stable is :
(A) | | 1
(B) | | 1
(C) | | 1
(D) | | 3 / 2
(A) h[ n ] {1, 0, 0,1}
(B) h[ n ] {1, 0,1}
(C) h[ n ] {1,1,1,1}
(D) h[ n ] {1,1,1}
(5)
**********
www.targate.org
AB [IES - EC - 1997]
The system described by the difference
equation
Page 29
SIGNAL & SYSTEM
y(n) - 2 y(n - 1) + y(n - 2) = x(n) - x(n - 1)
has y(n) = 0 for n < 0.
(7)
(A) 2
(B) 1
(C) zero
(D) -1
AD [IES - EC - 2005]
The unit sample response of a discrete
11
system is 1
00 0....... . For an input
24
sequence 1 0 1 0 0 0 ….., what is the output
sequence ?
1 1 1 1
(A) 1
0 0 .......
2 4 2 4
1
(B) 1 0
0 0 ........
4
1 5
(C) 2
0 0 0......
2 4
1 5 1 1
(D) 1
0 0 .......
2 4 2 4
Consider
the
AC [GATE - IN - 2007]
discrete – time signal
n
1, n 0.
1
x(n) u (n). where u(n) =
3
0, n 0
Define the signal y(n) as y(n) = x( n ),
n . Then
y ( n) equals.
n
2
3
3
(C)
2
(A)
(8)
(B)
2
3
(D) 3
AA [IES - EC - 2004]
What is the phase angle of the composite
sinusoidal signal resulting from the addition
of
v1 (n) sin[5n] and
v 2 (n) 2 cos[5 n] .
0
(9)
Consider
A9.9to10.1 [GATE – EC2 – 2014]
a
discretetime
signal
for 0 0 10
. If y[n] is the
otherwise
n
x n
0
If x(n) = δ(n), then y(2) will be
(6)
(10)
(A) 54.7
(B) 5
(C)
(D) / 3
AD [GATE – EC1 – 2014]
A discrete time signal x[n] =sin( 2 n ),n
being an integer is
convolution of x [n] with itself, the value of y
[4] is -------.
AC [IES - EC - 2004]
(11) To which one of the following difference
equations,
the
impulse
response
h(n) (n 2) (n 2) corresponds ?
(A) y(n + 2) = x(n) – x(n – 2)
(B) y(n – 2) = x(n) – x(n – 4)
(C) y(n) = x(n+2) – x(n – 2)
(D) y(n) = –x(n+2) + x(n – 2)
AA [IES - EC - 2012]
(12) Match List – I with List – II and select the
correct answer using the code given below
the Lists :
List – I
A. Even signal
B. Causal signal
C. Periodic signal
D. Energy signal
List - II
n
1
1. x(n) u(n)
4
2. x(n) x(n)
3. x(t)u(t)
4. x(n) x(n N)
Code :
A
(A) 2
B
3
C
4
D
1
(B) 1
3
4
2
(C) 2
4
3
1
(D) 1
4
3
2
AD [GATE - IN - 2003]
(13) Given h[n] = [1, 2, 2], f[n] is obtained by
convolving h[n] with itself and g[n] by
correlating h[n] with itself. Which one of the
following statements is true ?
(A) f[n] is causal and its maximum value is
9
(A) Periodic with period .
(B) f[n] is non – causal and its maximum
value is 8
(B) Periodic with period 2 .
(C) g[n] is causal and its maximum value is
9
(C) Periodic with period 2 .
(D) Not periodic.
Page 30
(D) g[n] is non – causal and its maximum
value is 9
TARGATE EDUCATION GATE-(EE/EC)
Topic.2 – Discrete Time Signal & System
AA [IES - EC - 2009]
(14) What is the period of the sinusoidal signal
x(n) 5cos[0.2 n] ?
AC [GATE - EE - 2006]
(18) A discrete real all pass system has a pole at z
= 2 300 : it, therefore,
(A) 10
(B) 5
(A) Also has a pole at 1 / 2 30 0
(C) 1
(D) 0
(B) Has a constant phase response over the
z-plane: are | H ( z ) | = const
AB [GATE - EE - 2008]
(15) Given a sequence x[n], to generate the
sequence y[n] = x[3 – 4n], which one of the
following procedures would be correct ?
(A) First delay x[n] by 3 samples to generate
z1 [ n ], then pick every 4th sample of
z1 [ n ] to generate z2 [ n] , and then finally
time reverse z2 [ n] to obtain y[n]
(B) First advance x[n] by 3 samples to
generate z1 [ n ], then pick every 4th
(C) Is stable only if it is anticausal
(D) Has a constant phase response over the
unit circle: arg | H ( e i ) | = const
AA [GATE - EE - 2006]
(19) x[n] = 0; n < -1, n > 0, x[-1] = -1, x[0] = 2 is
the input and y[n] = 0; n < -1, n > 2, y[-1] = 1 = y[1], y[0] = 3, y[2] = -2 is the output of a
discrete-time LTI system. The system
impulse response h[n] will be
sample of z1 [ n ] to generate z2 [ n] and
(A) h[n] = 0; n<0, n>2, h[0] = 1, h[1] = h[2]
= -1
then finally time reverse z2 [ n] to obtain
y[n]
(B) h[n] = 0; n<-1, n>1, h[-1] = 1, h[0] =
h[1] = 2
(C) First pick every fourth sample of x[n] to
generate v1[ n ] , time reverse v1[ n ] to
(C) h[n] = 0; n<0, n>3, h[0]=-1, h[1] = 2,
h[2]=1
obtain v2 [ n] , and finally advance v2 [ n]
by 3 samples to obtain y[n]
(D) h[n] = 0; n<-2, n>1, h[-2] = h[1] = -2,
h[-1] = -h[0] = 3
(D) First pick every fourth sample of x[n] to
generate v1[ n ] , time reverse v1[ n ] to
obtain v2 [ n] , and finally delay v2 [ n] by
3 samples to obtain y[n].
AA [IES - EC - 1991]
(20) The impulse train shown in the Fig.,
represents the second derivation of a function
f(t). The value of f(t) is
AD [GATE – EC – 2004]
(16) The impulse response h[n] of a linear time
invariant system is given as
2 2, n 1, 1
h[n] 4 2, n 2, 2
0,
otherwise
(A) - t u(t - 1) - t u(t - 2) + t u(t - 3) + t u(t 4) - t u(t - 5) + 2t u (t - 6) - t u(t - 7)
If the input to the above system is the
jn/4
sequence e
, then the output is
(A) 4 2e jn / 4
(C) 4e
jn /4
(B) - t u(t - 1) - t u(t - 2) - t u(t - 3) - t u(t - 4)
+ t u(t - 5)
(B) 4 2 e j n / 4
(D) 4e
j n /4
AD [GATE – EC – 2008]
(17) A discrete time linear shift – invariant system
has an impulse response h[n] with h [0] = 1,
h [1] = -1, h [2] = 2, and zero otherwise. The
system is given an input sequence x[n] with x
[0] = x [2] = 1, and zero otherwise. The
number of nonzero samples in the output
sequence y[n], and the value of y [2] are,
respectively
(A) 5, 2
(B) 6, 2
(C) 6, 1
(D) 5, 3
(C) t u(t - 3) + t u(t - 4) + 2 t u(t - 6)
(D) t u(t + 1) + t u(t + 2) + t u(t + 3) + t u(t +
4) + t u(t + 5) + 2 t u(t + 6) + t u(t + 7)
AC [IES - EC - 1998]
(21) If a(n) is the response of a linear, time invariant, discrete-time system to a unit step
input, then the response of the same system
to a unit impulse input is
(A)
d
[a ( n )]
dn
(B) n a(n)
www.targate.org
Page 31
SIGNAL & SYSTEM
(C) a(n) - a(n - 1)
(C) [3, 9, 8, 14, 7, 5, 2]
(D) a(n + 1) - 2 a(n) + a(n - 1)
(D) None
AB [IES - EC - 2002]
(22) Which one of the systems described by the
following input-output relations is timeinvariant ?
(A) y(n) nx(n)
AA [IES - EC - 2004]
n
(26)
y[n ]
x[k]
k
(B) y(n) x(n) x(n 1)
Which one of the following systems is
inverse of the system given above ?
(C) y(n) x(n)
(A) x[n] = y[n] – y[n – 1]
(D) y(n) x(n)cos 2f0n
(B) x[n] = y[n]
AA [IES - EC - 2004]
(23) Consider the following systems :
(C) x[n] = y[n + 4]
(D) x[n] = ny[n]
AA [IES - EC - 2005]
(27) A signal v(n) is defined by
1. y[k] x[k] a1x[k 1] b1y[k 1]
b2 y[k 2]
n 1
1 ;
v(n) 1 ;
n 1
0 ; n 0and| n | 1
2. y[k] x[k] a1x[k 1] a 2 x[k 2]
3. y[k] x[k 1] a1x[k] a 2 x[k 1]
4. y[k] a1x[k] a 2 x[k 1] b1y[k 2]
What is the value of the composite signal
defined as v[n] + v[–n]?
Which of the systems given above represent
recursive discrete systems?
(A) 0 for all integer values of n
(B) 2 for all integer values of n
(A) 1 and 4
(B) 1 and 2
(C) 1, 2 and 3
(D) 2, 3 and 4
(24) Given
that
AA [IES - EC - 2004]
x1 (t) ek1 t u(t)
and
x 2 (t) e k 2 t u(t) . Which one of the following
gives their convolution?
ek1t e k2 t
(B)
[k 2 k1 ]
(C)
(D) –1 for all integer values of n
AD [IES - EC - 2005]
(28) The lengths of two discrete time sequence
x1(n) and x2 (n) are 5 and 7, respectively.
What is the maximum length of a sequence
x1 (n) * x2 (n) ?
ek1t e k2 t
(A)
[k1 k 2 ]
k1t
(C) 1 for all integer values of n
(A) 5
(B) 6
(C) 7
(D) 11
AA [IES - EC - 2005]
(29) Let x[n] a n u[n]
k2 t
e e
[k1 k 2 ]
h[n] b n u[n]
What is the expression for y[n], for a
discrete-time system?
ek1t e k2 t
(D)
[k 2 k1 ]
AD [IES - EC - 2004]
(25) Which one of the following gives the crosscorrelation [R xy (k)] of two finite length
sequences
x(n) {1,3,1,3}
y(n) {1, 2,1, 3} ?
(A)
a
k
u[k]b n k u[n k]
n
u[k]b n k u[n k]
k
u[n k]b n u[k]
k
(B)
a
k
and
(C)
a
k
(A) [3, 10, 8, 14, 7,5,2]
(D)
(B) [2, 10, 7, 14, 6, 6, 3]
Page 32
a
nk
k
TARGATE EDUCATION GATE-(EE/EC)
u[k]b n k u[n k]
Topic.2 – Discrete Time Signal & System
AA [IES - EC - 2006]
(30) The discrete-time signal x[n] is given as
n 1, 2
1
x(n) 1
n 1, 2
0 n 0and| n | 2
(C) [ n 1] [ n 2]
(D) [ n ] [ n 1] [ n 2]
AA [GATE - IN - 2008]
(34) Consider a discrete – time system for which
the input x[ n ] and the output y[n] are related
Which one of the following is the timeshifted signal y[n] = x[n + 3]
n 1, 2
1,
(A) y[ n] 1,
n 4, 5
0,
n 3n 5 and n 1
0,
(B) y[ n] 1,
1,
1,
(C) y[ n] 0,
1,
1,
(D) y[n] 1,
0,
1
y[ n 1]. If y[ n ] 0 for
3
n < 0 and x[ n ] [ n ], then y[ n ] can be
expressed in terms of the unit step u[ n ] as
as y[ n ] x[ n ]
n
n 1, 2
n 4, 5
n 3n 5 and n 1
n
1
(A) u[n]
3
1
(B) u[n]
3
(C) (3) n u[ n ]
(D) ( 3) n u[ n ]
n 1, 2
n 4, 5
n 3n 5 and n 1
AB [GATE - IN - 2011]
difference
equation
(35) Consider
n 1, 2
the
1
y[ n ] y[ n 1] x[ n ] and suppose that
3
n 4,5
n 3n 5 and n 1
1
x[n] u[n]. Assuming the condition
2
n
of initial rest, the solution for y[n], n 0 is
AA [IES - EC - 2007]
(31) The discrete time signal x(n) is defined by
n
1
1
(A) 3 2
3
2
n 1
1,
x(n) 1,
n 1
0, n 0and| n | 1
n
1
1
(B) 2 3
3
2
Which one of the following is the composite
signal y(n) = x(n) + x(–n) for all integer
values of n ?
(A) 0
(B) 2
(C)
(D)
n
AD [IES - EC - 2012]
(32) The natural response of an LTI system
described by the difference equation
y(n) 1.5y(n 1) 0.5y(n 2) x(n) is
(A) y(n) 0.5u(n) 2(0.5) n u(n)
(B) y(n) 0.5u(n) (0.5) n u(n)
n
n
n
n
2 1 1 1
(C)
3 3 3 2
11 2 1
(D)
3 3 3 2
A*[GATE-EC-1993]
(36) Sketch the waveform (with properly marked
axes) at the output of a matched filter
matched for a signal s(t), of duration T, given
by
A for
st
0 for
(C) y(n) 2u(n) 0.5(0.5) n u(n)
(D) y(n) 2u(n) (0.5) n u(n)
AA [GATE - IN - 2008]
(33) Consider a discrete – time LTI system with
input x[ n ] [ n ] [ n 1] and impulse
response h[n] = [ n ] [ n 1]. The output
of the system will be given by
n
2
0t T
3
2
Tt T
3
(37) The
discrete-time
1
1 2z
is
1 0.5z 1
AD[GATE-IN-2013]
transfer
function
(A) [ n ] [ n 2]
(A) non-minimum phase and unstable
(B) [ n ] [ n 1]
(B) minimum phase and unstable
www.targate.org
Page 33
SIGNAL & SYSTEM
(C) minimum phase and stable
(D) non-minimum phase and stable
A2 [GATE – IN – 2016]
(38) The signal x[n] shown in the figure below is
convolved with itself to get y[n]. The value
of y[−1] is ________ .
AD [GATE – IN – 2018]
(39) Let y[ n] x[ n]* h[n] , where * denotes
convolution and x[ n] and h[ n ] are two
discrete time sequences. Given that the ztransform of y[ n] is Y ( z ) 2 3 z 1 z 2 ,
the z-transform of p[ n] x[n]* h[ n 2] is
(A) 2 3z z 2
(B) 3z z 2
(C) 2 z 2 3z 1
(D) 2 z 2 3z 3 z 4
-------0000-------
Page 34
TARGATE EDUCATION GATE-(EE/EC)
03
Fourier series
Theoretical Problem
(1)
(2)
AA [IES - EC - 1995]
If f(t) = -f(-t) and f(t) satisfies the Dirichlet's
conditions, then f(t) can be expanded in a
Fourier series containing
(A) only sine terms
(B) only cosine terms
(C) cosine terms and a constant terms
(D) sine terms and a constant terms
(5)
AD [GATE – EC –1994]
The Fourier Series of an odd periodic
function, contains only
4.
The amplitude spectrum is continuous
(A) 1, 2 and 4
(B) 2, 3 and 4
(C) 1, 3 and 4
(D) 1, 2 and 3
AB [GATE-EE1-2014]
For a periodic square wave, which one of the
following statements is TRUE?
(A) The Fourier series coefficients do not
exist.
(B) The Fourier series coefficients exist but
the reconstruction converges at most
point.
(B) even harmonics
(C) cosine terms
(C) The Fourier series coefficients exist and
the reconstruction converges at no
points.
(D) sine terms
AA [IES - EC – 2009]
Assertion (A) : There are no convergence
issues with the discrete time Fourier series in
general.
Reason (R) : A discrete-time signal is always
obtained by sampling a continuous-time
signal.
(D) The Fourier series coefficients exist and
the reconstruction converges at every
point.
(6)
AC [GATE – EC – 1996/ 2011]
The trigonometric Fourier series of an even
function of time does not have
Codes :
(A) the dc term
(A) Both A and R are true and R is the
correct explanation of A
(B) cosine terms
(B) Both A and R are true but R is not a
correct explanation of A
(D) odd harmonic terms
(C) A is true but R is false
(C) sine terms
(7)
(D) A is false but R is true
(4)
The evaluation of Fourier coefficients
gets simplified it waveform symmetries
are used
Which of the above statements are correct ?
(A) Odd harmonic
(3)
3.
(A) cosine terms
AD [IES - EC – 2002]
Consider the following statements related to
Fourier series of a periodic waveform :
1.
2.
It expresses the given periodic
waveform as a combination of d.c.
component, sine and cosine waveforms
of different harmonic frequencies.
AA [GATE – EC – 1998]
The trigonometric Fourier series of a periodic
time function can have only
(B) sine terms
(C) cosine and sine terms
(D) d.c. and cosine terms
(8)
The amplitude spectrum is discrete.
www.targate.org
AD [IES - EC - 1991]
About the Fourier series expansion of a
periodic function it can be said that
Page 35
SIGNAL & SYSTEM
(9)
(A) Even functions have only a constant and
cosine terms in their FS expansion
(B) Odd functions have only sine terms in
their FS expansion
(C) Functions with half-wave symmetry
contain only odd harmonics
(D) All the above three
List - II
AA [IES - EC – 2004]
Assertion (A) : A periodic function
satisfying Dirichlet's conditions can be
expanded into a Fourier series.
5. cosine terms of even harmonics can exist.
A
B
C
D
Reason (R) : A Fourier series is a
summation of weighted sine and cosine
waves of the fundamental frequency and its
harmonics
(A) 4
5
3
1
(B) 3
4
1
2
(C) 5
4
2
3
(D) 4
3
2
1
Codes:
(A) Both A and R are true and R is the
correct explanation of A
(B) Both A and R are true but R is not a
correct explanation of A
1. Even harmonics can exist.
2. Odd harmonics can exist.
3. The dc and cosine terms can exist.
4. sine terms can exist.
Codes:
AA [IES - EC - 2000]
(12) Match List-I with list-II and select the correct
answer using the codes given below the
Lists:
(C) A is true but R is false
List-I
(D) A is false but R is true
A. f (t ) f ( t )
AD [IES - EC – 2003]
(10) Assertion (A) : In the exponential fourier
representation of a real-valued periodic
function f(t) of frequency f 0 , the coefficients
of the term e
of each other.
j2 nf 0 t
and e
j2 nf0 t
are negatives
B.
Ce
jn0 t
n
n
C.
f (t )e jt dt
Reason (R) : The discrete magnitude
spectrum of f(t) is even and the phase
spectrum is odd.
Codes:
t
D.
f ( ) f
1
2
(t )d
0
List - II
1. Exponential from of Fourier series
(A) Both A and R are true and R is the
correct explanation of A
2. Fourier transform
(B) Both A and R are true but R is not a
correct explanation of A
3. Convolution integral
(C) A is true but R is false
5. Odd function wave symmetry
(D) A is false but R is true
Codes:
AD [IES - EC - 2000]
(11) Match List - I (Properties) with List - II
(Characteristics of the trigonometric from) in
regard to Fourier series of periodic f(t) and
select the correct answer using the codes
given below the Lists:
List - I
(A) f(t) + f(-t) = 0
(B) f(t) - f(-t) = 0
(C) f(t) + f(t - T/2) = 0
4. z - transform
A
B
C
D
(A) 5
1
2
3
(B) 2
1
5
3
(C) 5
4
2
1
(D) 4
5
1
2
AD [IES - EC – 2003]
(13) Match List I (Nature of Periodic Function)
with List II (Properties of Spectrum
Function) and select the correct answer using
the codes given below the Lists :
(D) f(t) - f(t - T/2) = 0
Page 36
TARGATE EDUCATION GATE-(EE/EC)
Topic.3 – Fourier Series
List - I
(Nature of Periodic Function)
A. Impulse train
x t a 0 a n cosn0 t b n sin n0 t
n 1
If x t x t x t / 0 , we can
conclude that
(A) a n are zero for all n and bn are zero for
n even
B.
Full-wave rectified sine function
C.
2t
4t
sin
cos
6
6
(B) a n are zero for all n and bn are zero for
n odd
(C) a n are zero for n even and bn are zero
for n odd
(D) a n are zero for n odd and bn are zero
for n even
D.
List-II
(Properties of Spectrum Function)
1. Only even harmonics are present
2. Impulse train with strength 1/T
3. 3
1
1
1
1
; 3 ; 1 ; 1
4j
4j
4j
4j
AB [GATE–S1–EC–2017]
(16) Let x(t) be a continuous time periodic signal
with fundamental period T = 1 seconds. Let
a k be the complex Fourier series
coefficients of x(t), where k is integer valued.
Consider the following statements about
x(3t):
4. Only odd harmonics are present
5. Both even and odd harmonics are present
Codes:
A
B
C
D
(A) 5
2
3
4
(B) 2
1
4
3
(C) 5
2
4
3
(D) 2
1
3
4
AA [IES - EC – 2003]
(14) Assertion (A) : A periodic function
satisfying Dirichlet conditions can be
expanded into Fourier series.
Reason (R) : A periodic function can be
reconstructed from
I.
The complex Fourier series coefficients
of x(3t) are a k where k is integer
valued.
II. The complex Fourier series coefficients
of x(3t) are 3a k where k is integer
valued.
III. The fundamental angular frequency of
x(3t) is 6rad / s
For the three statements above, which one of
the following is correct?
(A) only II and III are correct
(B) only I and III are true
(C) only III is true
(D) only I is true
a0
a n cos n0 t b n sin n0 t
2 n 1
n 1
**********
for very large n, excluding infinity.
Codes:
(A) Both A and R are true and R is the
correct explanation of A
(B) Both A and R are true but R is not a
correct explanation of A
(C) A is true but R is false
(D) A is false but R is true
AA [GATE–S1–EC–2017]
(15) A periodic signal x(t) has a trigonometric
Fourier series expansion
www.targate.org
Page 37
SIGNAL & SYSTEM
(A) 3 sin(25t )
Numerical Problem
(1)
(B) 4cos(20t 3) 2sin(710t )
AD [GATE – EE – 2000]
If an a.c voltage wave is corrupted with an
arbitrary number of harmonics, then the
overall voltage waveform differs from its
fundamental component in terms of
(A) only the peak values
(C) exp( | t | sin(25t )
(D) 1
(7)
(B) only the rms values
1, | t | T1
x(t )
T0
0, T1 | t | 2
(C) only the average values
(D) all the three measures(peak, rms and
average values)
(2)
(3)
(4)
The d.c. component of x(t) is
AA [GATE – EC – 2013]
A band-limited signal with a maximum
frequency of 5 kHz is to be sampled.
According to the sampling theorem, the
sampling frequency in kHz which is not valid
is
(A) 5 kHz
(B) 12 kHz
(C) 15 kHz
(D) 20 kHz
AC [GATE – EC – 2003]
A periodic signal x(t) of period T0 is given by
(A) T1 / T0
(B) T1 / (2T0 )
(C) 2T1 / T0
(D) T0 / T1
2
[GATE – EC – 1993]
8
A
(8)
A0.5 [GATE – EE – 2014]
Leg g: [0, ) [0, ) be a function
defined by g(x) = x – [x], where [x]
represents the integer part of x.(That is the
largest integer which is less than or equal to
x). The value of the constant term in the
Fourier series expansion of g(x) is ______
Fourier series of the periodic function (period
2 ) defined by
0,
f ( x)
x,
is
1
cos(n)sin(nx)
n
By putting x in the above, one can
deduce that the sum of the series
Cosine terms if it is even
Q. Sine terms if it is even
1
R.
Cosine terms if it is odd
S.
Sine terms if it is odd
(9)
Which of the above statements are correct?
(5)
(6)
(A) P and S
(B) P and R
(C) Q and S
(D) Q and R
A0.5 [GATE – EC – 2015]
n
Let x n 1 be a periodic signal with
8
period 16. Its DFS coefficients are defined by
1 15
a k x n exp j kn for all k. the
16 n 0
8
value of the coefficient a 31 is ________.
AC [GATE – EC – 2005]
Choose the function f (t ); t , for
which a Fourier series cannot be defined.
Page 38
0 x
1
2 [cos( n cos( nx )
4 1 n
AA [GATE – EC – 2009]
The Fourier series of a real periodic function
has only
P.
x0
1 1 1
..... is
32 52 72
AD [GATE - EE - 1996]
A periodic rectangular signal, x(t) has the
waveform shown in Figure. Frequency of the
fifth harmonic of its spectrum is
(A) 40 Hz
(B) 200 Hz
(C) 250 Hz
(D) 1250 Hz
AB [GATE - EE - 2008]
(10) Let x(t) be a periodic signal with time period
T. Let y(t) = x (t t0 ) x (t t0 ) for some
TARGATE EDUCATION GATE-(EE/EC)
Topic.3 – Fourier Series
The
has
(A)
(B)
(C)
t 0 . The Fourier Series coefficients of y(t) are
denoted by bk . If bk = 0 for all odd k, then
only sine terms with all harmonics.
only cosine terms with all harmonics
only sine terms with even numbered
harmonics
(D) only cosine terms with odd numbered
harmonics
t0 can be equal to
(A) T/8
(C) T/2
(B) T/4
(D) 2T
AA [GATE - EE - 2010]
(11) The second harmonic component of the
periodic waveform given in the figure has an
amplitude of
(A) 0
(B) 1
(C) 2/ π
(D)
A0.50to0.52 [GATE – EC2 – 2014]
(16) Consider the periodic square wave in the
figure shown. The ratio of the power in the
7th harmonic to the power in the 5th harmonic
for this wave form is closest in value to -----.
5
AC [IES - EC - 1994]
(12) The impulse response of a first order system
is K e-2t . If the input signal is sin 2t, then the
steady state response will be given by
K
(A)
sin 2t
4
2 2
1
sin2t
4
K
(C)
sin 2t
4
2 2
AB [GATE – EC – 2002]
(17) Which of the following cannot be the Fourier
series expansion of a periodic signal?
(A) x(t ) 2cos t 3cos3t
(B)
(D)
Fourier series expansion of sgn(cos(t))
(B) x(t ) 2cos t 7cos t
(C) x(t ) cos t 0.5
sin 2t Ke 2 t
4
2 2
1
(D) x(t ) 2cos1.5t sin 3.5t
AB [IES - EC - 2000]
(13) The Fourier series representation of a
periodic current is
AB [GATE – EC – 2009]
2
(18) A function is given by f (t) = sin t cos 2t.
Which of the following is true?
(A) f has frequency components at 0 and 1/2
Hz
[2 6 2 cos(t ) 48 sin(2t )] A
The effective value of the current is
(A) (2 6 24) A
(B) 8 A
(C) 6 A
(D) 2 A
(B) f has frequency components at 0 and 1/
Hz
AA [IES - EC – 2012]
(14) An ideal low-pass filter has a cutoff
frequency of 100 Hz. If the input to the filter
in volts is v(t) = 30 2 sin 1256t, the
magnitude of the output of the filter will be
(A) 0 V
(B) 20 V
(C) 100 V
(D) 200
(C) f has frequency components at 1/2 and
1/ Hz
(D) f has frequency components at 0, 1/2
and 1/ Hz
AD [GATE – EC – 2003]
(19) The fourier series expansion of a real
periodic signal with fundamental frequency
f 0 is given by g p (t )
AD [GATE – EE1 – 2015]
(15) The signum function is given by
x
; x0
sgn(x) | x |
0; x 0
www.targate.org
ce
J 2 nf 0 t
n
. It is
n
given that c3 3 j 5. Then c3 is
(A) 5 j3
(B) 3 j 5
(C) 5 j 3
(D) 3 j5
Page 39
SIGNAL & SYSTEM
AC [GATE – EC – 2005]
(20) The trigonometric Fourier series for the
waveform f (t) shown below contains
series coefficients of W1 and W 2 , for n 1 ,
n odd, are respectively proportional to
(A) | n |3 and | n |2
(A) only cosine terms and zero value for the
dc component
(B) only cosine terms and a positive value
for the de component
(C) only cosine terms and a negative value
for the dc component
(D) only sine terms and a negative value for
the dc component
(B) | n |2 and | n |3
(C) | n |1 and | n |2
(D) | n |4 and | n |2
AC [GATE - EE - 2002]
(24) Fourier Series for the waveform, f(t) shown
in Fig. is
AC [GATE – EC – 1987]
(21) A half-wave rectified sinusoidal waveform
has a peak voltage of 10 V Its average value
and the peak values of the fundamental
component are respectively given by :
(A)
20 10
V,
V
2
(B)
10 10
V
V
2
(C)
10
V ,5V
(D)
20
V ,5V
AD [GATE – EC – 1999]
(22) The Fourier series representation of an
impulse train denoted by
s (t )
(t nT )
0
(A) 82 sin πt 1 sin(3πt ) 1 sin(5πt ) ...
π
9
25
(B) 82 sin πt 1 cos(3πt ) 1 sin(5πt ) ...
π
9
25
(C) 82 cos πt 1 cos(3πt ) 1 cos(5πt ) ...
π
9
25
(D) 82 cos πt 1 sin(3πt ) 1 sin(5πt ) ...
π
9
25
n
AC [GATE - EE - 2006]
(25) x(t) is a real valued function of a real
variable with period T. Its trigonometric
Fourier Series expansion contains no terms
of frequency 2 π (2 k ) / T ; k = 1, 2 ...
Also, no sine terms are present. Then x(t)
satisfies the equation
(A) x(t) = - x(t – T)
is given by
1
j 2nt
(A) exp
T0
T0 n
1
jnt
exp
T0
T0 n
(B)
1
jnt
(C) exp
T0
T0 n
(B) x(T) = x(T – 1) = - x(-t)
(C) x(t) = x(T – t) = - x(t – T/2)
1
j 2nt
exp
T0
T0 n
(D) x(t) = x(t – T) = x(t – T/2)
(D)
AC [GATE – EC – 2000]
(23) One period (0, T) each of two periodic
waveforms W1 and W 2 are shown in Fig. 1
and Fig.2. The magnitudes of the nth Fourier
Page 40
(26) The
Fourier
AD [GATE - EE - 2011]
series expansion f(t) =
a0 an cos nt bn sin nt
of
the
n 1
periodic signal shown below will contain the
following nonzero terms:
TARGATE EDUCATION GATE-(EE/EC)
Topic.3 – Fourier Series
AA [IES - EC - 1991]
(30) The amplitude and phase spectra for the few
harmonics of a periodic signal, f(t) of time
period 1 sec are shown in the Fig., below.
The function f(t) is
(A) a 0 and bn n = 1, 3, 5, ...
(B) a0 and a n , n = 1, 2, 3, .....
(C) a 0 , a n and bn , n = 1, 2, 3 .....
(D) a0 and a n , n = 1, 3, 5, ..........
AD [GATE - EE - 2005]
(27) The Fourier series for the function f(x) =
sin 2 x is :
(A) sin x sin 2 x
(B) 1 cos 2 x
(C) sin 2 x cos 2 x
(D) 0.5 0.5 cos 2x
AA [GATE - EE - 2007]
(28) A signal x(t) is given by
(A) cos 2 t 0.75cos 6 t
4
2
1, T / 4 t 3T / 4
x(t ) 1,3T / 4 t 7T / 4
x(t T )
0.1cos 10 t ...
6
t
t
(B) cos
0.75cos
2 4
6 2
Which among the following gives the
fundamental Fourier term of x(t)?
(A)
4
πt π
cos
π
T 4
t
0.1cos
10 6
(B)
π
πt π
cos
4
2T 4
1
(C) cos 2 t 0.75cos 6 t .
4
2 3
(C)
4 πt π
πT 4
1
0.1cos 10 t .
6 6
(D)
π
πt π
sin
4
2T 4
(D) cos 2 t 0.75cos 6 t
4
2
AC [GATE - EE - 2009]
(29) The Fourier Series coefficients, of a periodic
signal x(t), expressed as x(t) =
k
ak e j 2 πkt /T are given by a 2 = 2- j1;
a 1 0.5 j 0.2; a0 j 2;
a1 0.5 j 0.2; a 2 2 j1; and ak 0;
for | k | 2. Which of the following is true?
0.1cos 10 t
6
AA [IES - EC - 1993]
(31) Which of the following periodic waveforms
will have only odd harmonics of sinusoidal
waveforms?
(A) x(t) has finite energy because only
finitely many coefficients are non-zero
(B) x(T) has zero average value because it is
periodic
(C) The imaginary part of x(t) is constant
(D) The real part of x(t) is even
www.targate.org
1.
Page 41
SIGNAL & SYSTEM
(A) all cosine terms
(B) all sine terms
2.
(C) odd cosine terms
(D) odd sine terms
AA [GATE - IN - 2010]
(34) f(x), shown in the adjoining figure is
represented by
f ( x ) a0 {a n cos( nx ) bn sin( nx )}.
3.
n 1
The value of a 0 is
4.
Select the correct answer using the codes
given below:
Codes:
(A) 1 and 2
(B) 1 and 3
(C) 1 and 4
(D) 2 and 4
AA [IES - EC - 1997]
(32) The amplitude of the first odd harmonic of
the square wave shown in the Fig., is equal to
(A) 0
(B) π / 2
(C) π
(D) 2π
AB [GATE - IN - 2011]
(35) Consider a periodic signal x(t) as shown
below
It has a Fourier series representation x(t) =
4V
(A)
2V
(B)
3
V
(C)
(D) 0
AD [IES - EC - 1997]
(33) A periodic triangular wave is shown in the
Fig. Its Fourier components will consist only
of
ae
j (2 π / T ) kt
k
k
Which one of the following statement is
TRUE?
(A) ak 0, for k odd integer add T = 2
(B) ak 0, for k even integer and T = 2
(C) ak 0, for k even integer and T = 4
(D) ak 0, for k odd integer and T = 4
AB [IES - EC - 1998]
(36) A periodic voltage having the Fourier series
v(t) = 1 + 4 sin ωt + 2 cos ωt volts is applied
across a one-ohm resistors. The power
dissipated in the one-ohm resistor is
Page 42
(A) 1 W
(B) 11 W
(C) 21 W
(D) 24.5 W
TARGATE EDUCATION GATE-(EE/EC)
Topic.3 – Fourier Series
The source has nonzero impedance. Which
one of the following is a possible form of the
output measured across a resistor in the
network?
AA [IES - EC – 2003]
(37) For half-wave (odd) symmetry, with T0 =
periodic of x(t), which one of the following is
correct ?
(A) x(t T0 / 2) x(t)
(B) x (t T0 / 2) x (t)
(C) x (t T0 ) x(t)
(D) x (t T0 ) x(t)
3
(A)
cos( k 0 t k ) , where bk ak , k
4
(B)
b
k
cos( k 0 t k ) , where bk 0, k
k 1
3
(C)
a
k
cos( k 0 t k )
k
cos( k 0 t k )
k 1
2
(D)
a
k 1
S4AB [GATE – EC – 2016]
2
t cos( t ) is the input to
3
(42) A signal 2 cos
an LTI system with the transfer function
H (s) es es
2A(1 cos n)
(B)
n
2A(1 cos n)
(C)
n
2A(1 cos n)
(D)
[(n 1)]
If C k denotes the kth coefficient in the
exponential Fourier series of the output
signal, then C3 is equal to
(A) 0
(B) 1
(C) 2
(D) 3
S8AB [GATE – EE – 2016]
AC [IES - EC – 2010]
(39) Consider the following statements :
Fourier series of any periodic function x(t)
can be obtained if
(43) Let f(x) be a real, periodic function satisfying
f(−x) = −f(x). The general form of its Fourier
series representation would be
(A) f ( x) a0
T0
| x(t) |dt
1.
k
k 1
AA [IES - EC – 2004]
(38) A square wave is defined by
A 0 t T0 / 2
x(t)
A T0 / 2 t T0
It is periodically extended outside this
interval. What is the general coefficient 'an' in
the Fourier expansion of this wave ?
(A) 0
b
(B) f ( x)
0
2.
(D) f ( x)
Which of the above statements is / are correct
a cos(kx)
k 1 k
b sin(kx)
k 1 k
(C) f ( x) a0
Finite number of discontinuous exist
within finite time interval t.
a
a cos(kx)
k 1 2 k
k 0 2k 1
sin(2k 1)x
(A) 1 only
S6AC [GATE – EE – 2016]
(B) 2 only
(44) Suppose x1 (t ) and x2 (t ) have the Fourier
transforms as shown below.
(C) Both 1 and 2
(D) neither 1 and 2
AA [IES - EC – 2012]
(40) A waveform is given by
v(t) = 10 sin 2π 100t. What will be the
magnitude of the second harmonic in its
Fourier series representation ?
(A) 0 V
(B) 20 V
(C) 100 V
(D) 200 V
S1AA [GATE – EC – 2016]
(41) A network consisting of a finite number of
linear resistor (R), inductor (L), and capacitor
(C) elements, connected all in series or all in
parallel, is excited with a source of the form
3
a
k
cos( k 0 t ) , where ak 0 , 0 0 .
k 1
www.targate.org
Which one of the following statements is
TRUE?
Page 43
SIGNAL & SYSTEM
(A) x1 (t ) and x2 (t ) are complex and
x1 ( t ) x 2 ( t ) is also complex with nonzero
imaginary part
120 Hz. Which of the following is an
accurate description of the output?
(B) x1 (t ) and x2 (t ) are real and x1 ( t ) x 2 ( t )
is also real
(C) x1 (t ) and x2 (t ) are complex but
x1 ( t ) x 2 ( t ) is real
(D) x1 (t ) and x2 (t ) are imaginary but
x1 ( t ) x 2 ( t ) is real
AC [GATE–S1–EE–2017]
k
k
(45) Let the signal x t 1 t
2000
k
be passed through an LTI system with
frequency response H , as given in the
figure below
(A) Output is zero.
(B) Output consists of both 50 Hz and 100
Hz frequency components.
(C) Output is a pure sinusoid of frequency
50 Hz.
(D) Output is a square wave of fundamental
frequency 50 Hz.
AA [GATE – EE – 2018]
(48) A continuous-time input signal x(t) is an
eigenfunction of an LTI system, if the output
is
(A) k x (t ) , where k is an eigenvalue
(B) k e jt x(t ) , where k is an eigenvalue and
The Fourier series representation of the
output is given as
(A) 4000 4000cos 2000t 4000cos 4000t
4000cos 2000t 4000cos 4000t
(B) 2000 2000cos 2000t 2000cos 4000t
2000cos 2000t 2000cos 4000t
(C) 4000cos 2000t
(D) 2000cos 2000t
AC [GATE – EC – 2018]
(46) Let x(t ) be a periodic function with period
T 10 . The Fourier series coefficients for
this series are denoted by a k , that is
x(t )
ak e
jk
The same function x(t ) can also be
considered as a periodic function with period
T ' 40 . Let b k be the Fourier series
coefficients when period is taken as T ' . If
k | ak | 16 , then k | bk | is equal to
(B) 64
(D) 4
AC [GATE – IN – 2018]
(47) An ideal square wave with period of 20 ms
shown in the figure, is passed through an
ideal low pass filter with cut-off frequency
Page 44
A9.5-10.5 [GATE – EE – 2018]
(49) The Fourier transform of a continuous-time
signal
x(t)
is
given
by
1
where
X ()
, ,
2
10 j
j 1 and denotes frequency. Then the
value of | ln x(t ) | at t =1 is ________ (up to 1
decimal place). (ln denotes the logarithm to
base e )
AC [GATE-EE-2019]
(50) A periodic function f(t), with a period of 2 ,
is represented as its Fourier series,
2
t
T
k
(A) 256
(C) 16
e j t is a complex exponential signal
(C) x(t ) e jt , where e j t is a complex
exponential signal
(D) k H () , where k is an eigenvalue and
H () is a frequency response of the
system
f (t ) a0 n 1 an cos nt n 1bn sin nt .
A sin t , 0 1
If f (t )
t 2
0,
the Fourier series coefficients a1 and b1 of
f(t) are
A
(A) a1 0; b1 A / (B) a1 ; b1 0
2
A
A
(C) a1 0; b1
(D) a1 ; b1 0
2
-------0000-------
TARGATE EDUCATION GATE-(EE/EC)
04
Fourier Transform
Theoretical Problem
(4)
(1)
AC [IES - EC - 2006]
Match List-I with List-II and select the
correct answer using the code given below
the lists:
List - I
List I(Application of Signals)
A. Fourier series
A. Reconstruction
B.
B. Fourier transform
Over sampling
C. Interpolation
C. Discrete time Fourier series
List II (Definition)
D. Discrete Fourier transform
1.
Sampling rate is chosen significantly
greater than the Nyquist rate
List - II
2.
To convert the discrete time sequence
back to a continuous time signal.
2. Continuous, periodic
3.
Assign values between samples.
1. Discrete, periodic
3. Discrete, aperiodic
4. Continuous, aperiodic
Codes:
(2)
(A)
A
3
B
2
C
1
(B)
1
3
2
(C)
2
1
3
(D)
2
3
1
Codes:
AC [IES - EC - 2012]
The property of Fourier transforms which
states that the compression in time domain is
equivalent to expansion in the frequency
domain is
(5)
(A) duality
A
B
C
D
(A)
3
4
1
2
(B)
1
2
4
3
(C)
3
2
4
1
(D)
1
4
2
3
AC [IES-EC-2013]
Which of the following Derichlets conditions
are correct for convergence of Fourier
transform of the function x (t)?
1.
x (t) is square integral
(C) time scaling
2.
x(t) must be periodic
(D) frequency shifting
3.
AB [GATE – EC – 1992]
If G(f) represents the Fourier transform of a
signal g(t) which is real and odd symmetric
in time, then
x(t) should have finite number of
maxima and minima within any finite
interval
4.
x(t) should have finite number of
discontinuities within any finite interval
(A) G(f) is complex
(A) 1, 2, 3 and 4
(B) G(f) is imaginary
(B) 1, 2 and 4 only
(C) G(f) is real
(C) 1, 3 and 4 only
(D) G(f) is real and non-negative
(D) 2, 3 and 4 only
(B) scaling
(3)
AA [IES - EC - 2002]
Match List-I (Fourier series and fourier
transforms) with List-II (Their properties)
and select the correct answer using the codes
given below the lists :
www.targate.org
Page 45
SIGNAL & SYSTEM
(6)
AC [IES-EC-2013]
For certain sequence which are neither
absolutely summable nor square summable,
it is possible to have a Fourier Transform
(FT) representation if we
(A) Take short time FT
(B) Evaluate FT only the real part of the
sequence
(7)
(B) j 2f X ( f )
(C) jf X ( f )
(D)
X( f )
jf
AB [GATE – EC – 1998]
(11) The Fourier transform of a voltage signal
x(t ) X ( F ) , find the unit of | X ( f ) | .
(A) Volt
(B) Volt-sec
(D) Evaluate FT over a limited time span
(C) Volt/ sec
(D) Volt 2
AC [GATE – EC – 1996]
The Fourier transform of a real valued time
signal has
(B) even symmetry
(C) conjugate symmetry
(D) no symmetry
AB [GATE-EE1-2014]
Let f (t ) be a continuous time signal and let
F () be its Fourier Transform defined by
AC [GATE – EC – 2004]
(12) The Fourier transform of a conjugate
symmetric function is always
(A) imaginary
(B) conjugate anti-symmetric
(C) real
(D) conjugate symmetric
AC [GATE - IN - 2008]
(13) The Fourier transform of x (t ) e at u ( t ),
where u(t) is the unit step function,
(A) Exists for any real value of a
(B) Does not exist for any real value of a
(C) Exists if the real value of a is strictly
negative
F () f (t )e jt dt
Define g(t) by
g (t ) F (u )e jut du
What is the relationship between f(t) and
g(t) ?
(A) g(t) would always be proportional to
f(t)
(B) g(t) would be proportional to f(t) if f(t)
is an even function.
(C) g(t) would be proportional to f(t) only if
f(t) is a sinusoidal function.
(D) g(t) would never be proportional to f(t).
(9)
dX ( f )
dt
(C) Allow DTFT to contain impulse
(A) odd symmetry
(8)
(A)
AC [GATE – EC – 1998]
The amplitude spectrum of a Gaussian pulse
is
(A) uniform
(D) Exits if the real value of a is strictly
positive
AC [GATE - EE - 1997]
(14) A differentiator has transfer function whose
(A) Phase increases linearly with frequency
(B) Amplitude remains constant
(C) Amplitude
frequency
increases
linearly
with
(D) Amplitude
frequency
decreases
linearly
with
AD [IES - EC - 2002]
(15) Consider the following statements :
1.
Fourier transform is special case of
Laplace transform
2.
Region of convergence need not be
specified for Fourier transform
3.
Laplace transform is not unique unless
the region of convergence is specified
4.
Laplace transform is a special case of
Fourier transform
(B) a sine function
(C) Gaussian
(D) an impulse function
AB [GATE – EC – 1998]
(10) The Fourier transform of a function x(t) is
dx (t )
X(f). The Fourier transform of
will be
dt
Page 46
Which of these statements are correct ?
(A) 2 and 4
(B) 4 and 1
(C) 4, 3 and 2
(D) 1, 2 and 3
TARGATE EDUCATION GATE-(EE/EC)
Topic.4 - Fourier Transform
AA [IES - EC - 2002]
(16) Match List-I (Fourier series and Fourier
transforms) with List-II (Their properties)
and select the correct answer using the codes
given below the lists :
List - I
A. Fourier series
B. Fourier transform
C. Discrete time Fourier series
D. Discrete Fourier transform
List - II
1. Discrete, periodic
2. Continuous, periodic
3. Discrete, aperiodic
4. Continuous, aperiodic
1
f ()
2
1
f ( )
(B)
2
(C) 2f ( )
(A)
(D) none of the above
AB [GATE – EC – 1999]
(21) A signal x (t) has a Fourier transform X () .
If x(t) is a real and odd function of t, then
X () is
(A) A real and even function of
(B) an imaginary and odd function of
(C) an imaginary and even function of
(D) a real and odd function of
Codes:
A
B
C
D
(A)
3
4
1
2
(B)
1
2
4
3
(C)
3
2
4
1
(D)
1
4
2
3
**********
AD [IES - EC - 2010]
(17) For distortion less transmission through LTI
system phase of H( ) is
(A) Constant
(B) One
(C) Zero
(D) Linearly dependent on
AC [IES - EC - 2012]
(18) The Fourier transform of a rectangular pulse
is :
(A) Another rectangular pulse
(B) Triangular pulse
(C) Sinc function
(D) Impulse function
AB [IES - EC - 2012]
(19) The function which has its Fourier transform,
Laplace transform and Z- transform unity is
(A) Gaussian
(B) impulse
(C) sinc
(D) pulse
AAC [GATE – EC – 1997]
(20) The function f (t) has the Fourier Transform
g() .
The
Fourier
Transform
of
g
(t)
jt
g (t )e dt is
www.targate.org
Page 47
SIGNAL & SYSTEM
e j10t
x(t)
0
Numerical Problem
(1)
AC [GATE – EE – 2008]
1
Let x(t) = rect t (where rect(t) = 1 for
2
1
1
and zero otherwise). Then
x
2
2
sin x
sinc(x) =
, the Fourier transformer
x
of x(t) + x(– t) will be given by
Its Fourier Transform is
(A)
2sin( 10)
10
j10
(B) 2e
(C)
(A) sin c
2
(6)
G
2 21
2 9
AC [IES-EC-2013]
If f (t) is a real and odd function, then its
Fourier transform F(ω) will be
(A) t 2exp 3 t
(A) Real and even function of ω
(C) sin 3t 7cos t
(B) Real and odd function of ω
(D) sin 3 t 21exp 3 t
(B) cos 3 t 21exp 3t
(C) Imaginary and odd function of ω
(D) Imaginary function of ω
(3)
2sin
AA [GATE – IN – 2006]
The Fourier transform of a function g(t) is
given
(D) sin c sin
2
2
(2)
sin( 10)
10
2sin
10
j10
(D) e
(B) 2 sin c
2
(C) 2 sin c cos
2
2
for | t | 1
for | t | 1
(7)
AB [GATE-EE2-2014]
A differentiable non constant even function
x(t) has a derivative y(t), and their respective
Fourier Transforms are X () and Y () .
Which of the following statements is TRUE?
(A) X () and Y () are both real.
(B) X () is real and Y () is imaginary.
AD [GATE – IN – 2006]
The magnitude of Fourier transform X
of a function x(t) is shown below in figure
(A). The magnitude of Fourier transform
Y of another function y(t) is shown
below in figure (B). The phases of
X and Y are zero for all . The
magnitude and frequency units are identical
in both figures. The function y(t) cam
expressed in terms of x(t) as
(C) X () and Y () are both imaginary.
(D) X () is imaginary and Y () is real.
(4)
AB [GATE – EC – 1999]
A signal x (t) has a Fourier transform X () .
If x (t) is a real and odd function of t. then
X () is
(a)
(b)
(A)
2 t
3 2
(B)
2
2t
3
(C)
2
2t
3
(D)
3 t
2 2
(A) A real and even function of
(B) an imaginary and odd function of
(C) an imaginary and even function of
(D) a real and odd function of
(5)
AA [GATE – EE2 – 2015]
Consider a signal defined by
Page 48
(8)
AB [GATE – EC – 2014]
For a function g(t), it is given that
TARGATE EDUCATION GATE-(EE/EC)
Topic.4 - Fourier Transform
gte
jt
2
dt e 2 for any real value .
t
If y t
g d , then
y t dt
is -
________
(A) 0
(C)
(9)
(B) -j
j
2
(D)
j
2
(A) 1 sin
(B) 1 cos
A59.9to60.1 [GATE – EC2 – 2014]
In the figure, M(f) is the Fourier transform of
the message signal m (t) where A = 100 Hz
and B = 40 Hz. Given
(C)
2(1 cos )
2
(D)
2(1 cos
2
v(t) =cos 2 fct
and w(t) =cos 2 (f c +A)t , where fc > A.
The cut-off frequencies of both the filters fc.
AA [IES - EC - 1999]
(13) The signal (1 + M cos(4 t )) cos(2 103 t )
contains the frequency components (in Hz)
(A) 998, 1000 and 1002
(B) 1000 and 2000
(C) dc, 2 and 1000
(D) ....., 996, 998, 1000, 1002, 1004,......
The bandwidth of the signal at the output of
the modulator (in Hz) is -------.
AA [GATE – EC – 2000]
(14) The Hilbert transform of [cos 1t sin 2 t ]
is :
AD [GATE - IN - 2003]
(10) A real function f(t) has a Fourier transform
The
Fourier
transform
of
F ( ).
[ f (t ) f ( t )] is
(A) Zero
(C) real and odd
(A) sin 1t cos 2 t
(B) sin 1t cos 2 t
(C) cos 1t sin 2 t
(B) real
(D) imaginary
AC [GATE - IN - 2004]
j
(11) If the Fourier transform of x[ n ] is X ( e ),
n
then the Fourier transform of ( 1) x[ n ] is
(D) sin 1t sin 2 t
AC [GATE - EE/EC/IN - 2012]
(15) The Fourier transform of a signal h(t) is
H(j = (2 cos )(sin 2 ) / . The value
of h(0) is
(A) ( j ) X ( e j )
(A) 1/4
(B) 1/2
(B) ( 1) X ( e j )
(C) 1
(D) 2
AB [GATE - EE - 2008]
(C) X ( e j ( π ) )
(D)
(16) Let x(t) = rect t
d
X (e j )
d
for
AC [GATE - IN - 2006]
(12) If the waveform, shown in the following
figure, corresponds to the second derivative
of a given function f(t), then the Fourier
transform of f(t) is
www.targate.org
if
1
(where rect(x) = 1
2
1
1
x and zero otherwise). Then
2
2
sin(πx)
sin c( x)
,
the
Fourier
πx
Transform of x (t ) x ( t ) will be given by
Page 49
SIGNAL & SYSTEM
AA [IES-EC-2013]
(20) A unit impulse function δ(t) is defined by
2π
(A) sin c
1 δ(t) = 0 for all t except t = 0
2π
(B) 2sin c
2
(t ) dt 1
The Fourier transform F(ω) of δ(t) is
cos
2π
2
(C) 2sin
cos
2π 2
(D) sin c
AC [IES - EC - 1997]
(17) Which one of the following is the correct
Fourier transform of the unit step signal?
(21)
(A) 1
(B)
1
(C) 0
(D)
1
j
AD [GATE – EC4 – 2014]
A real-valued signal x (t) limited to the
frequency band f
u(t) = 1 for t ≥ 0
W
is passed through a
2
linear time invariant system whose frequency
= 0 for t ≤ 0
j 4 f
e
response is H f
0,
(A) ( )
1
(B)
j
W
2
W
f
2
f
.
The output of the system is
(C)
1
( )
j
1
(D)
2 ( )
j
(B) ( )
(D) ( )
d ( )
d
(C) x t 2
(D) x t 2
n
j 6 f
Ae
2
F .T
u n 3
2
3
1 e j 2 f
3
where u [n] denotes the unit step sequence.
The values of A is -----.
d ( )
d 2
(C) ( )
(B) x t 4
A3.36to3.39 [GATE – EC4 – 2014]
(22) A Fourier transform pair is given by
AA [IES - EC - 1995]
(18) The group delay function ( ) is related to
phase function ( ) as
(A) ( )
(A) x t 4
AD [GATE-EE2-2014]
(23) A continuous-time LTI system with system
function H() has the following pole-zero
plot. For this system, which of the
alternatives is TRUE?
d 2 ( )
d 2
d 2 ( )
d
AA [IES-EC-2014]
(19) The Fourier transform of a rectangular pulse
for a period
t
T
T
to t
2
2
is
(A) a sinc function
(A) | H (0) || H ( ) |;| | 0
(B) a sine function
(B) | H () | has multiple maxima, at 1 and
(C) a cosine function
2
(D) a sine-squared function
Page 50
TARGATE EDUCATION GATE-(EE/EC)
Topic.4 - Fourier Transform
(C) | H (0) || H () |;| | 0
(A) e f u (t )
(B) e f u ( t )
(D) | H () | = constant;
(C) e f u ( f )
(D) e f u ( f )
AC [GATE-EE2-2014]
(24) A function f(t) is shown in the figure.
AA [GATE – EC – 2006]
(28) Let x(t ) X ( j) be Fourier Transform
pair. The Fourier Transform of the signal
x(5t 3) in terms of X ( j is given as
(A)
1 j53 j
e
X
5
5
(B)
1 j 35 j
e X
5
5
(C)
1 j 3 j
e
X
5
5
(D)
1 j 3 j
e X
5
5
The Fourier transform F () of f(t) is
(A) real and even function of .
(B) real and odd function of .
AD [GATE - IN - 2005]
(C) imaginary and odd function of .
(29) The continuous – time signal x (t )
(D) imaginary and even function of .
AA [GATE-EE2-2014]
(25) A signal is represented by
π
exp ( a | |).
a
The signal x (t ) cos( bt ) has the Fourier
transform
has the Fourier transform
1 | t | 1
x (t )
0 | t | 1
The Fourier transform of the convolved
signal is y (t ) x (2t ) * x (t / 2) is
(A)
(A)
4
(B)
sin
2
2
4
sin (2)
(C)
2
4
sin 2
(D)
2
(C) A B | f |2
π
[exp( a | |) exp( a | |)]
2a
(C)
π
[exp( a | |) cos b]
2a
(D)
π
[exp( a | b |) exp
2a
(B) A e Bf
AC [GATE - IN - 2011]
(30) Consider the signal
2
(D) Ae Bf
AC [GATE – EC – 2002]
(27) The Fourier transform F {e t u ( t )} is equal
to
(B)
(a | b |)]
AD [GATE – EC – 2000]
(26) The Fourier Transform of the signal x (t) =
2
e3t is of the following form where A and B
are constants:
(A) Ae
π
[exp( a | b |) exp
2a
(a | b |)]
4
sin sin(2)
2
2
Bf 2
1
a t2
2
1
1
. Therefore, F
is
1 j 2 f
1 j 2t
www.targate.org
et ,
x(t )
0,
t0
t0
Let X ( denote the Fourier transform of
this signal. The integral
1
X ()d is
2π
(A) 0
(B) 1/2
(C) 1
(D)
Page 51
SIGNAL & SYSTEM
AB [GATE – EC – 2007]
(31) The frequency response of a linear, time –
invariant
system
is
given
by
5
. The step response of the
1 j10πf
system is
H( f )
(A) 5(1 e 5 t ) u ( t )
t
Match each of the items 1, 2 on the left with
the most appropriate item A, B, C or D on
the right.
AB [GATE – EC – 2004]
(35) Let x (t) and y (t) (with Fourier transforms X
(f) and
Y (f) respectively) be related as
shown in Fig. (1) & (2)
(B) 5 1 e 5 u (t )
(C)
1
(1 e5t )u(t )
5
(D)
1
1
5
1
e
u(t )
5
AA [GATE – EC – 2005]
(32) The output y (t) of a linear time invariant
system is related to its input x (t) by the
following
equation:
y(t)=0.5x
(t td T ) x(t td ) 0.5 x(t td T ). The
Fig. (1)
Fig. (2)
Then Y (f) is :
(A)
1
X ( f / 2)e j 2 f
2
filter transfer function H () of such a
system is given by
(B)
1
X ( f / 2)e j 2f
2
(A) (1 cos e j td
(C) X ( f / 2) e j 2 f
(B) (1 0.5 cos T ) e j td
(D) X ( f / 2) e j 2 f
(C) (1 cos T ) e j td
AD [GATE - EE - 2010]
(36) x(t) is a positive rectangular pulse from t = -1
to t + 1 with unit height as shown in the
(D) (1 0.5 cos T ) e j t d
AC [GATE – EC – 2003]
(33) Let x (t) be the input to a linear, timeinvariant system. The required output is 4x (t
– 2).
figure. The value of
X
2
d {where
X( ) is the Fourier transform of x(t)} is
The transfer function of the system should be
(A) 4 e j 4 f
(B) 2 e j 8 f
(C) 4 e j 4 f
(D) 2 e j 8 f
A(1-A)(2-C) [GATE – EC – 1997]
(34) If the Fourier Transform of a deterministic
signal g(t) is G(f) then
(1)
The
Fourier
Transform of g(t
– 2) is
(A) 2
(B) π
(C) 4
(D) 4 π
j (4f )
(A) G(f) e
(37) The
AC [IES - EC - 1993]
Fourier
transform
of
inverse
(2)
The
Fourier (B) G(2f)
Transform
of
g(t/2) is
(C) 2G(2f)
F ( j )
exp( jt ) f ( t )dt
(A) f ( t )
exp( jt ) F ( j )d
(D) G(f – 2)
(B) f (t )
Page 52
is
1
2
TARGATE EDUCATION GATE-(EE/EC)
exp( jt ) F ( j ) d
Topic.4 - Fourier Transform
(C) f (t )
(D) f (t )
1
2
1
2
exp( j t ) F ( j )d
exp( jt ) F ( j )d
AD [IES - EC - 1995]
(41) The two inputs to an analogue multiplier are
x(t) and y(t) with Fourier transforms X(f)
and Y(f) respectively. The output z(t) will
have a transform Z(f) given by
(A) X(f) . Y(f)
AB [IES - EC - 1994]
(38) Which one of the following is the Fourier
transform of the signal given in Fig. (B), if
the Fourier transform of the signal in Fig.(A)
sin T1
is given by 2
?
(B) X(f) + Y(f)
(C) X(f)/Y(f)
(D)
X ( )Y ( f )d
AD [IES - EC - 1997]
(42) The Fourier transform of v(t) = cos 0t is
given by
1
(A) V ( f ) ( f f 0 )
2
1
(B) V ( f ) ( f f 0 )
2
Fig. (A)
1
(C) V ( f ) [ ( f f 0 ) ( f f 0 )]
2
1
(D) V ( f ) [ ( f f 0 ) ( f f 0 )]
2
AA [IES - EC - 1998]
(43) Let F(ω) be the Fourier transform of a
function f(t), then F(0) is
Fig. (B)
sin T1 jT1
e
sin T1 jT1
e
(B) 2
sin T1 jT1
e
(C)
sin T1 j ( T1 2)
e
(D)
(A) 2
(A)
f ( t ) dt
(B)
2
2
tf ( t ) dt
AC [IES - EC - 1995]
(39) Fourier transform F ( j ) of an arbitrary real
signal has the property.
f (t )
(C)
(D)
tf (t )dt
AD [IES - EC - 1998]
(44) Given that the Fourier transform of f(t) is
F(jω), which of the following pairs of
functions of time and the corresponding
Fourier transforms are correctly matched?
1. f ( t 2) e j 2 F ( j )
(A) F ( j ) F ( j )
2. f ( 0.5t ) 2 F ( 2 j )
(B) F ( j ) F ( j )
t
3.
*
(C) F ( j ) F ( j )
1
f ( t ) dt F ( j )
( )
j
Select the correct answer using the codes
given below:
(D) F ( j ) F * ( j )
AD [IES - EC - 1995]
(40) The inverse Fourier transform of the function
1
F ( )
( ) is :
j
(A) sint
(B) cost
(C) sgn (t)
(D) u(t)
(A) 1 and 2
(B) 1 and 3
(C) 2 and 3
(D) 1, 2 and 3
AA [IES - EC - 1999]
(45) Match the list -I (Fourier transforms) with
List-II (Functions of time) and select the
correct answer using the codes given below
the lists :
www.targate.org
Page 53
SIGNAL & SYSTEM
List- I
sin k
(A)
(C)
(A)
1
j
(B) j
(C)
1
(1 j )
(D) ( )
(B) e jd
1
( j 2) 2
(D) k ( )
1
j
AB [IES - EC - 2002]
(49) Match List-I (Functions) with List-II (Fourier
transforms) and select the correct answer
using the codes given below the lists :
List -II
1. A constant
2. Exponential function
List – I
3. t - multiplied exponential function
A. exp(t)u(t), 0
4. Rectangular pulse
B. exp( | t |), 0
5. Impulse function
C. t exp( t)u(t), 0
Codes:
A
B
C
D
D. exp( j2 t / t 0 )
(A)
4
5
3
1
List - II
(B)
4
5
3
2
(C)
3
4
2
1
(D)
3
4
2
5
AC [IES - EC - 2000]
(46) A voltage signal v(t) has the following
Fourier transforms :
e jd for 1
V ( j )
0 for 1
1.
1
( j2f ) 2
2.
1
j2f
3. f
t0
4.
The energy that would be dissipated in a 1 Ω
resistor fed from v(t) is
2
(A) Joules
2
(2f ) 2
Codes:
A
B
C
D
(A)
3
1
4
2
(B)
2
4
1
3
(C)
3
4
1
2
(D)
2
1
4
3
2e 2 d
Joules
1
(C) Joules
(B)
1
Joules
(D)
2
2
AB [IES - EC - 2002]
2
AA [IES - EC - 2001]
(47) The Fourier transform of a double-sided
exponential signal x(t) = e
(A) is
2
the Fourier transform of et is.
b t
(B)
f
e
2
(C)
1
e j tan ( / b )
(B) is
(b 2 2 )
2 2
1
e f
(D)
e
f2
2
AC [IES - EC - 2002]
(51) The Fourier transform X(f) of the periodic
delta functions,
(C) does not exist
(D) exists only when it is single sided
AD [IES - EC - 2001]
(48) The Fourier transform of u(t) is
Page 54
2 2
2
1
(A) e f
2b
(b 2 )
2
(50) The Fourier transform of et is ef ; then
x (t )
(t kT )
k
TARGATE EDUCATION GATE-(EE/EC)
is :
Topic.4 - Fourier Transform
( f kT )
(A) T
k
(B) T
k
f T
(B)
k
(C)
(D)
1
T
1
T
k
f T
k
f kT
k
AD [IES - EC - 2004]
(52) What is the Nyquist rate for the signal
(C)
x (t ) cos 2000 πt 3sin 6000 πt ?
(A) 2 kHz
(B) 4 kHz
(C) 12 kHz
(D) 6 kHz
AD [IES - EC - 2004]
(53) Match List-I with List-II pertaining to
Fourier Representation Periodicity Properties
and select the correct answer using the codes
given the lists:
List I
List II
(Time
Domains
Property)
(Frequency
Domain
Property)
(A)
Continuous
1.
Periodic
(B)
Discrete
2.
Continuous
(C)
Periodic
3.
Non-periodic
(D)
Non-periodic
4.
Discrete
(D)
AB [IES - EC - 2005]
(55) Match List I (Time Function) with List II
(Fourier Spectrum/Fourier Transform) and
select the correct answer using the code
given below the Lists:
Codes:
A
B
C
D
(A)
3
4
1
2
(B)
2
4
1
3
(C)
2
1
4
3
(D)
3
1
4
2
List II
(Time
Function)
(Fourier
Spectrum/Fourier
Transform)
(A) Periodic
Function
(B)
AD [IES - EC - 2005]
(54) Which one of the following represents the
phase response of the function
s 2 20
H (s) 2
?
s ( 0 / Q ) s 2
List I
1.
A periodic 2.
Function
(C) Unit Impulse 3.
(t)
(D)
sin t
4.
Continuous
spectrum at
frequencies
all
()
Line
spectrum
discrete
1
Codes:
(A)
(B)
(C)
(D)
A
4
3
4
3
B
2
1
1
2
C
3
4
3
4
D
1
2
2
1
(A)
AA [IES - EC - 2005]
(56) A signal represented by
x(t ) 5cos 400πt
www.targate.org
Page 55
SIGNAL & SYSTEM
is sampled at a rate 300 sample/s. The
resulting samples are passed through an ideal
low pass filter of cut-off frequency 150 Hz.
Which of the following will be contained in
the output of the LPF?
(B) Magnitude of X ( f ) has odd symmetry
while phase of X ( f ) has even symmetry
(A) 100 Hz
(D) Both magnitude and phase of X ( f )
have odd symmetry
(B) 100 Hz, 150 Hz
AD [IES - EC - 2008]
(60) Which one of the following is the correct
relation?
(C) 50 Hz 100 Hz
(D) 50 Hz, 100 Hz, 150 Hz
AD [IES - EC - 2006]
(57) Match List-I with List-II and select the
correct answer using the code given
List I
List II
(CT Function)
(CT
Fourier
Transform)
(A)
e t u (t )
1.
1 2
(B)
1,| t | 1
x (t )
0,| t | 1
2.
j X ( j )
(C)
dx(t )
dt
3.
1
1 j
(D)
e
4.
2sin
t
Codes:
A
B
C
D
(A)
1
4
2
3
(B)
3
2
4
1
(C)
1
2
4
3
(D)
3
4
2
1
1
1
(t )
2
2πt
(B)
1
( t )
2
(B) F (at ) aF ( a)
(C) F (t / a) aF ( a)
(D) F (at ) (1 / a) F ( a)
AC [IES - EC - 2008]
(61) The Fourier transform of a function is equal
to its two-sided Laplace transform evaluated
(A) On the real axis of the s-plane
(B) On a line parallel to the real axis of the
s-plane
(C) On the imaginary axis of the s-plane
(D) On a line parallel to the imaginary axis
of the s-plane
AB [IES - EC - 2008]
(62) If the Fourier transform of f (t ) is f ( j),
then what is the Fourier transform of
f ( t ) ?
(B) F ( j)
(C) F ( j)
(D) Complex conjugate of F ( j)
AC [IES - EC - 2011]
(63) An electrical system transfer function has a
pole at s = –2 and a zero at s = –1 with
system gain 10. For sinusoidal current
excitation, voltage response of the system :
(A) Is zero
(B) Is in phase with the current
(C) Leads the current
1
(C) 2(t )
πt
(D) Lags behind the current
(D) 2(t ) sgn(t )
AA [IES - EC - 2007]
(59) A real signal x(t) has Fourier transform X(f).
Which one of the following is correct?
(A) Magnitude of X ( f ) has even symmetry
while phase of X ( f ) has odd symmetry
Page 56
(A) F (at ) aF ( / a)
(A) F ( j)
AA [IES - EC - 2006]
(58) What is the inverse Fourier transform of
u () ?
(A)
(C) Both magnitude and phase of X ( f )
have even symmetry
AB [IES - EC - 2008]
(64) If f (t ) is an even function, then what is its
Fourier transform F ( j)?
(A)
0
(B) 2
f (t )cos(2t )dt
0
f (t )cos(t )dt
TARGATE EDUCATION GATE-(EE/EC)
Topic.4 - Fourier Transform
(C) 2
(D)
0
0
f (t )sin(t )dt
(C)
x *[n]
3.
(D)
x[n 1]
4.
n
1
h[ n] u ( n) in response to the input
2
π
x[n] 3 cos πn ?
3
1
π
(A) y[ n] 3 cos πn
3
3
Codes:
A
B
C
D
(A)
1
3
2
4
(B)
2
4
1
3
(C)
1
4
2
3
(D)
2
3
1
4
2
π
(C) y[ n] 1 sin πn
3
3
List - I
2
π
(D) y[ n] 6 cos πn
3
3
B. Gate Function
A. Delta Function
C. Normalized Gaussian function
AD [IES - EC - 2006]
2
(66) If the Fourier transform of x(t) is
sin(π then what is the Fourier transform
D. Sinusoidal function
List-II
1. Delta function
of e j 5 t x (t ) ?
2. Gaussian function
2
sin(π)
(A)
5
(D)
d
X (e j )
d
AB [IES - EC - 2002]
(68) Match List-I (Functions in the time domain)
with List-Ii (Fourier transform of the
function) and select the correct answer using
the codes given below the lists :
2
π
(B) y[ n] 3 cos πn
3
3
(C)
j
f (t )sin(2t )dt
AD [IES - EC - 2004]
(65) What is the output of the system with
(B)
e j X ( e j )
3. Constant function
4. Sampling function
2
sin{π(
Codes:
2
sin{π ( 5)}
2
sin{π (
5
AB [IES - EC - 2007]
(67) A discrete-time signal x[n] has Fourier
transform X ( e j ).
Match List-I with List-II and select the
correct answer using the code given below
the lists:
List I
List II
(Signal)
(Fourier
Transform)
(A)
x[n]
1.
X * (e
(B)
nx[ n]
2.
X (e j )
j
A
B
C
D
(A)
1
2
4
3
(B)
3
4
2
1
(C)
1
4
2
1
(D)
3
2
4
1
AC [IES - EC - 1997]
(69) If g(t) G(f) represents a Fourier transform
pair, then according to the duality property of
Fourier transforms.
(A) G(t ) g ( f )
(B) G (t ) g * ( f )
(C) G(t ) g ( f )
)
(D) G ( t ) g * ( f )
AA, C [GATE – EC – 2008]
(70) The signal x (t) is described by
www.targate.org
Page 57
SIGNAL & SYSTEM
for 1 t 1
otherwise
1
x(t )
0
Two of the angular frequencies at which its
Fourier transform becomes zero are
(A) , 2
(B) 0.5, 1.5
(C) 0.
(D) 2 , 2.5
AC [IES - EC - 1991]
(71) The Fourier transform of the function Sgn(t)
defined in the Fig. is
(A)
(C)
(72)
2
j
2
j
(B)
4
j
(D)
1
1
j
AB [IES - EC - 2009]
When y (t ) Y ( j );
FT
FT
x (t )
X ( j );
(A) cos
(B) cos
(C) sin
(D) sin
AA[GATE-IN-2007]
(76) Consider the periodic signal
x t 1 0.5cos 40 t cos 200t where t is
in seconds. The fundamental frequency in Hz
is
(A) 20
(B) 40
(C) 100
(D) 200
AMTA[GATE–S1–EE–2017]
t0
t t ,
(77) Consider g t
, where
t t , otherwise
t R . Here t represents the largest
integer less than or equal to t and t
denotes the smallest integer greater than or
equal to t. The coefficient of the second
harmonic component of the Fourier series
representing g(t) is _________.
AC [GATE – IN – 2018]
(78) The Fourier transform of a signal x(t),
denoted by X ( j) , is shown in the figure.
FT
h (t )
H ( j );
What is Y ( j ) ?
(A)
1
h(n) ( [n] [n 2])
2
The magnitude of the response can be
expressed as
X ( j )
H ( j )
(B) X ( j ) H ( j )
(C) X ( j ) H ( j )
(D) X ( j ) H ( j )
AC [IES - EC - 2010]
(73) The Fourier transform of unit step sequence
is
(A) π()
(C) π ( )
(B)
1
1 e j
1
1 e j
(D) 1 e j
AB [IES - EC - 2011]
(74) What are the gain and phase angle of a
system having the transfer function G(s) = (s
+ 1) at a frequency of 1 rad/sec ?
(A) 0.41 and 00
(B) 1.41 and 450
(C) 1.41 and –450
(D) 2.41 and 900
AB [IES - EC - 2012]
(75) The impulse response of a discrete time
system is given by
Page 58
Let y (t ) x(t ) e jt x(t ) . The value of Fourier
transform of y (t ) evaluated at the angular
frequency 0.5 rad/s is
(A) 0.5
(B) 1
(C) 1.5
(D) 2.5
A9 [GATE-IN-2019]
(79) The output of a continuous-time system y(t)
is related to its input x(t) as
1
y (t ) x(t ) x(t 1) . If the Fourier
2
transforms of x(t) and y(t) are X ( ) and
Y () respectively and | X (0) |2 4 , the
value of | Y (0) |2 is _____.
-------0000-------
TARGATE EDUCATION GATE-(EE/EC)
05
Laplace Transform
II. There is no causal and BIBO stable
system with a pole in the right half of
the complex plane.
Theoretical Problem
(1)
(2)
AB [IES - EC - 1998]
The impulse response of a single-pole system
would approach a non-zero constant as
t if and only if the pole is located in the
s-plane
(A) on the negative real axis
(B) at the origin
(C) on the positive real axis
(D) on the imaginary axis
AA [GATE – EC – 1995]
The final value theorem is used to find the
Which one among the following is correct?
(A) Both I and II are true
(B) Both I and II are not true
(C) Only I is true
(D) Only II is true
(5)
(A) steady state value of the system output
(B) initial value of the system output
(C) transient behaviour of the system output
(D) none of these
(3)
AB [IES - EC - 2010]
Consider the following statements:
1.
The Laplace transform of the unit
impulse function is s Laplace
transform of the unit ramp function.
2.
The impulse function is a
derivative of the ramp function.
time
3.
The Laplace transform of the unit
sLaplace
impulse
function
is
transform of the unit step function.
4.
The impulse function is a time
derivative of the unit step function.
AD [GATE–S2–EC–2017]
A second order LTI system is described by
the following state equations
d
x1 (t ) x2 (t ) 0
dt
d
x2 (t ) 2 x1 (t ) 3x2 (t ) r (t )
dt
where x1 (t ) and x2 (t ) are the two state
variables and r(t) denotes the input. The
output c (t ) x1 (t ) . The system is :
(A) undamped (oscillatory)
(B) underdamped
(C) critically damped
(D) overdamped
***********
Which of the above statements are correct?
(4)
(A) 1 and 2 only
(B) 3 and 4 only
(C) 2 and 3 only
(D) 1, 2, 3 and 4
AD [GATE–S1–EC–2017]
Consider the following statements for
continuous-time linear time invariant (LTI)
systems.
I.
There is no bounded input bounded
output (BIBO) stable system with a pole
in the right half of the complex plane.
www.targate.org
Page 59
SIGNAL & SYSTEM
(A) 0
(C) 1
Numerical Problem
(1)
AB [IES - EC - 1993]
Given the Laplace transform, V(s) =
(6)
e
st
v ( t ) dt . The inverse transform v(t) is
0
(B) 0.5
(D) 2
AC [GATE - EE - 2002]
Let Y(s) be the Laplace transformation of the
function y(t), then the final value of the
function is
j
(A)
estV ( s )ds
(A) lim Y ( s )
(B) lim Y ( s )
(C) lim sY ( s )
(D) lim sY ( s )
s 0
s
j
j
(B)
1
e stV ( s )ds
2 j j
s 0
(7)
(C)
1
e stV ( s )ds
2 j 0
j
1
e stV ( s)ds
(D)
2 j j
(2)
(3)
(C) s – 6
(D) s + 1
(4)
(5)
(8)
1
s 3s 2
The steady state value of the output of this
system for a unit impulse input applied at
time instant t = 1 will be
G ( s)
Page 60
2
f (t )e
st
dt
σ has a value less than zero
(A) 1, 2 and 3
(B) 1 and 2
(C) 2 and 3
(D) 1 and 3
AD [GATE – EE – 2004]
5
Consider the function, F(s) =
2
s s 3s 2
where F(s) is the Laplace transform of the
function f(t). The initial value of f(t) is equal
to
(A) 5
(C)
(9)
AA [GATE - EE - 2008]
The transfer function of a linear time
invariant system is given as
σ is responsible for convergence of
Select the correct answer using the codes
given below:
AC [GATE – EC – 1995]
The transfer function of a linear system is the
(A) ratio of the output. v0 (t ) and input
(B) ratio of the derivatives of the output and
the input
(C) ratio of the Laplace transform of the
output and that of the input with all
initial conditions zeros
(D) none of these
2.
3.
1
is always a stable transfer function
G (s)
v1 (t )
σ has a damping effect.
0
AFALSE [GATE – EC – 1994]
Indicate whether the following statement is
TRUE/FALSE. Give reason for our answer.
If G(s) is a stable transfer function, then T(s)
=
1.
integral
H1 s among the following options is
(B) s – 2
AB [IES - EC - 1994]
In Laplace transform, the variable 's' equals
( j ) . Which of the following represents
the true nature of σ ?
AB [GATE – EC – 2014]
A stable linear time invariant(LTI) system
1
has a transfer function H(s) = 2
. To
s s6
make this system causal it needs to be
cascaded with another LTI system having a
transfer function H1 s . A correct choice for
(A) s + 3
s
5
3
(B)
5
2
(D) 0
AB [GATE – EE/IN – 2013]
Assuming zero initial condition, the response
y(t) of the system given below to a unit step
input u(t) is
(A) u(t)
(C)
t2
u t
2
(B) t u(t)
(D) e t u t
AC [GATE – EE/IN – 2013]
(10) Which one of the following statements is
NOT TRUE for a continuous time causal and
stable LTI system?
TARGATE EDUCATION GATE-(EE/EC)
Topic.5 - Laplace Transform
(A) All the poles of the system must lie on
the left side of the j axis
(A)
1
30
(B)
1
15
(B) Zeros of the system can lie anywhere in
the s-plane.
(C)
3
4
(D)
4
3
(C) All the poles must lie within s 1
(D) All the roots of the characteristic
equation must be located on the left side
of the j axis.
AC [GATE – EC – 1987]
(11) Laplace transforms of the functions tu(t) and
u(t)sin(t) are respectively:
1
s
(A) 2 , 2
s s 1
1 1
(B) , 2
s s 1
1
1
, 2
2
s s 1
(C)
(D) s,
s
s 1
2
AB [GATE – EE – 2014]
(15) Consider an LTI system with impulse
response h t e 5 t u t . If the output of the
system is y t e 3t u t e 5 t u t then the
input, x(t) is given by
(B) 2e t e3t
(C) e t 2e 3t
(D) e t 2e3t
A0.99to1.01 [GATE – EC – 2014]
(13) A causal LTI system has zero initial
conditions and impulse response h(t). Its
input y(t) and output x(t) are related thorough
the linear constant-coefficient differential
equation.
dt
2
dy t
dt
(C) e 5 t u t
(D) 2 e 5 t u t
(A) 0
(B) 1
(C) 2
(D) 3
A-0.01to0.01 [GATE – EC – 2014]
(17) The input 3e 2 t u t , where u(t) is the unit
step function , is applied to a system with
s2
transfer function
. If the initial value of
s3
the output is -2, then the value of the output
at steady state is _______.
(A) 2e t e 3t
d2 y t
(B) 2 e 3t u t
AB [GATE – EC/IN – 2013]
(16) The impulse response of a continuous time
system is given by h t t 1 t 3 .
The value of the step response at t = 2 is
AA [GATE – EC – 1996]
(12) The inverse Laplace transform of the
function
s5
is
s 1s 3
(A) e 3t u t
AA [GATE – EC – 1997]
(18) The Laplace transform of e t cos t is
equal
(A)
(B)
y t x t
2
Let another signal g(t) be defined as
t
g t 2 h d
0
dh t
dt
(C)
h t
s
2
2
s
s
2
2
1
s 2
(D) none of these
If G(s) is the Laplace transform of g(t), then
the number of poles of G(s) is _______
AB [GATE – EC – 1999]
(19) If £[f(t)] = F(s), then £[f(t – T)] is equal to
(A) e sT F s
AB [GATE – EE – 2014]
(14) Consider an LTI system with transfer
function
H s
s
1
s s 4
If the input to the system is cos(3t) and the
steady state output is A sin 3t , then the
value of A is
(C)
F s
1 e
sT
(B) e sT F s
(D)
Fs
1 esT
AC [GATE – EC – 2003]
(20) The Laplace transform of i(t) is given by
2
I s
s 1 s
www.targate.org
As t , the value of i(t) would be:
Page 61
SIGNAL & SYSTEM
(A) 0
(B) 1
(C) 1 f 1
(D)
AA [GATE – EC – 2011]
(21) If the unit step response of a network is
1 et , then its unit impulse response is
(A) et
(B)
1
t
(25) Let the signal f(t) = 0 outside the interval
T1 , T2 , where T1 and T2 are finite.
e
1 0 t 2
x t
0 otherwise
dy
Assuming that y(0)=0 and
0 at t = 0,
dt
the Laplace transform of y(t) is
2s
e
s s 2 s 3
2s
(D)
f t . The region of
convergence (ROC) of the signal’s bilateral
Laplace transform F(s) is
(A) a parallel strip containing the j axis
Furthermore,
AB [GATE – EC – 2013]
(22) A system is described by the differential
d2 y
dy
equation
5 dy t x t . Let x(t)
2
dt
dt
be a rectangular pulse given by
(C)
(C) 50 49e 0.2 t
AC [GATE – EC – 2015]
e
(D) 1 e t
(B)
(B) 2 e 0.2 t
1 t
(C) 1
(A)
(A) 2 e 0.2 t
(D) 50 49e0.2t
1
dx
10 0.2x with initial conduction x(0) =
dt
1. The response x(t) for t > 0 is
1 e
s s 2 s 3
(B) a parallel strip not containing the j
axis
(C) the entire s-plane
(D) a half plane containing the j axis
AA [GATE – IN – 2005]
(26) Identify the transfer function corresponding
to an all-pass filter from the following:
(A)
1 s
1 s
(B)
1 s1
1 s2
(C)
1
1 s
(D)
s
1 s
AA [GATE – IN – 2007]
(27) Let the signal x(t) have the Fourier transform
e2s
s 2 s 3
Consider the signal
1 e2s
s 2 s 3
d
x t t d
dt
(Where t d is an arbitrary delay)
AA [GATE – EC – 2014]
(23) A system is described by the following
differential equation, where u(t) is the input
to the system and y(t) is the output of the
system
y t 5y t u t
When y(0) = 1 and u(t) is a unit step
function, y(t) is
(A) 0.2 0.8e 5t
(B) 0.2 0.2e 5t
The magnitude of the fourier transform of
y(t) is given by the expression
(A) X
(B) X .
(C) 2 . X
jt
(D) X e d
AD [GATE – IN – 2013]
(28) The Laplace Transform representation of the
triangular pulse shown below is
(C) 0.8 0.2e5t
(D) 0.8 0.8e 5 t
AC [GATE – EC – 2015]
(24) Consider the differential equation
Page 62
yt
TARGATE EDUCATION GATE-(EE/EC)
Topic.5 - Laplace Transform
(A)
1
1 e 2s
s2
(B)
1
1 e s e 2s
2
s
(C)
1
1 e s 2e 2s
s2
(D)
F(s) =
1
(2 s 1)
System 2 : G(s) =
1
(5 s 1)
(C) 1 f () 1
(D)
1
. The unilateral Laplace transform
s s 1
of t f(t) is
(A)
(B)
e (a b)
s
(D)
e (a b)
s
s
s
(B)
AA [GATE - IN - 2004]
(30) Consider the following systems:
System 1 : G(s) =
(B) 1
2
Z
e as e bs
s
(A) 0
AD [GATE - EE/EC/IN - 2012]
(34) The unilateral Laplace transform of f(t) is
AC [GATE – EC2 – 2015]
(29) The bilateral Laplace transform of a function
1 if a t b
is
f (t)
0 otherwise
(C)
2
s 1
2
2s 1
( s s 1)2
2
(C)
s
( s s 1)2
(D)
2s 1
( s s 1)2
2
2
AA [GATE - EE - 2005]
(35) The Laplace transform of a function f(t) is
The true statement regarding the system is
F(s) =
(B) Bandwidth of system 1 is lower than the
bandwidth of system 2
(C) Bandwidth of both the systems are the
same
5s 2 23s 6
. As t , f(t)
s ( s 2 2 s 2)
approaches
(A) Bandwidth of system 1 is greater than the
bandwidth of system 2
(A) 3
(B) 5
(C) 17/2
(D)
AC [GATE - EE - 2011]
(36) The Laplace transform of g(t) is
(D) Bandwidth of both the systems are
infinite
(A)
1 3s 5 s
(e e )
s
AA [GATE – EC – 1998]
(31) If L[ f ( t )] s 2 ) , then the value of
(B)
1 5 s 3s
(e e )
s
(C)
e 3 s
(1 e 2 s )
s
(D)
1 5 s 3s
(e e )
s
2
lim f (t )
t
(A) cannot be determined
(B) is zero
(C) is unity
(D) is infinite
AD [GATE– EC – 2007]
(32) If the Laplace transform of a signal y (t) is
Y(s) =
Re[s] > 0
The final value of f (t) would be:
1
1 2e s e 2s
s2
ab
(A)
s
0
s 02
2
1
, then its final value is
s ( s 1)
(A) -1
(B) 0
(C) 1
(D) Unbounded
AC [GATE - EE - 2000]
(37) A linear time-invariant system initially at
rest, when subjected to a unit-step input,
gives a response y(t) = te t , t 0. The
transfer function of the system is
AC [GATE – EC – 2006]
(33) Consider the function f (t) having Laplace
transform
www.targate.org
(A)
1
( s 1) 2
(B)
1
s ( s 1) 2
Page 63
SIGNAL & SYSTEM
(C)
s
( s 1) 2
(D)
1
s ( s 1)
AA [GATE - EE - 2002]
(38) The transfer function of the system described
d 2 y dy du
2u with u as input and
dt 2 dt dt
y as output is
by
(A)
( s 2)
( s 2 s)
(B)
( s 1)
( s 2 s)
(C)
2
( s s)
(D)
2s
( s s)
2
2
AA [GATE – EC – 2005]
(39) In what range should Re(s) remain so that the
Laplace transform of the function of the
( a 2) t 5
function e
AC [GATE - EE - 1998]
(42) The Laplace transform of (t 2 2t )u (t 1) is
(A)
s 2 s
e 2e
s3
s
2 2 s 2 s
e 2e
s3
s
2 s 1 s
(C) 3 e e
s
s
(B)
(D) None of the above
AB [IES - EC - 1997]
(43) If δ(t) denotes a unit impulse, then the
d 2 ( t )
Laplace transform of
will be
dt 2
(A) 1
(B) s 2
(C) s
(D) s 2
exists?
AB [IES - EC - 1994]
(44) Which one of the following is the correct
Laplace transform of the signal in the given
Fig.?
(A) Re(s) > a + 2
(B) Re(s) > a + 7
(C) Re(s) < 2
(D) Re(s) > a + 5
Common Data for the Next two Questions:
Given f(t) and g(t) as shown below:
1
[1 e2 s (1 2s )
2s2
1 2s
(B) 2 [e 1 2s]
2s
1 2s
(C) 2 [e 1 2s]
2s
1
2 s
(D) 2 [1 e 2s]
2s
(A)
AD [GATE - EE - 2010]
(40) g(t) can be expressed as
(A) g(t) = f (2t 3)
t
2
3
(C) g (t ) f 2t
2
(B) g (t ) f 3
AD [IES - EC - 1999]
(45) Laplace transform of sin (t ) u(t) is
t 3
2 2
(A)
exp( s / )
s 2
(B)
exp( s / )
s 2
(C)
s
exp( s / )
s 2
(D) g (t ) f
AC [GATE - EE - 2010]
(41) The Laplace transform of g(t) is
1 3s
e e5s
s
1
(B) e 5s e 3s
s
e 3s
(C)
1 e 2s
s
1
(D) e5s e3s
s
(A)
Page 64
2
2
2
(D) None
AA [IES - EC - 2000]
(46) Which one of the following transfer
functions represents the critically damped
system?
(A) H1 ( s )
TARGATE EDUCATION GATE-(EE/EC)
1
s 4s 4
2
Topic.5 - Laplace Transform
(B) H 2 ( s )
AB [GATE – EE2 – 2015]
(51) The Laplace transform of f (t) 2 t / is
1
s 3s 4
2
s3/ 2 . The Laplace transform of g(t) = 1/ t
1
(C) H 3 ( s ) 2
s 2s 4
(D) H 4 ( s )
(A)
1
s s4
3s –5/2
2
(B) s–1/2
(C) s1/2
2
AA [IES - EC - 2005]
(47) What is the Laplace transform of the
waveform shown below?
(D) s3/2
AC [GATE – EC – 1994]
(52) The Laplace transform of a unit ramp
function starting at t = a, is
(A)
1
( s a )2
(B)
e as
( s a )2
(C)
e as
s2
(D)
a
s2
AB [GATE – EC – 1995]
(53) If L [f (t)] =
1 1 s 2 2 s
(A) F (s ) e e
s s
s
2( s 1)
, then f (0+) and
s 2s 5
2
f () are given by
1 1 s 2 2 s
(B) F (s ) e e
s s
s
(A) 0, 2 respectively
1 1 s 2 2s
(C) F (s) e e
s s
s
(C) 0, 1 respectively
1 1 2 s 2 s
(D) F (s ) e e
s s
s
Note: ‘L’ stands for [‘Laplace transform of ’]
(B) 2, 0 respectively
(D) 2/5, 0 respectively
AD [IES - EC - 2007]
(48) What is the output as t for a system that
2
has a transfer function G ( s) 2
;
s s2
when subjected to a step input?
(A) -1
(B) 1
(C) 2
(D) Unbounded
AA [GATE – EC – 1997]
(54) The Laplace Transform of e t cos( t ) is
equal to
(A)
(s )
(s )2 2
(B)
(s )
(s )2 2
1
( s )2
(D) none of the above
(C)
AB [IES - EC - 2011]
(49) What is the unit impulse response of the
system shown in figure for t 0 ?
AB [GATE – EC – 1999]
(55) If L[f (t)] = F(s), then L [f (t – T)] is equal to
(A) e sT F ( s )
(A) 1 e
(C) e
t
t
(B) 1 e
(D) e
(C)
t
A–2 [GATE – EC2 – 2015]
(50) Let x(t) = s(t) + s(–t) with s(t) = e-4t
u(t), where u(t) is unit step function . If the
bilateral Laplace transform of x(t) is
X(S)
(B) e sT F ( s )
t
F (s)
1 esT
(D)
F (s)
1 e sT
AB [GATE – EC – 1998]
(56) The transfer function of a zero – order – hold
system is
(A) (1 / s )(1 e sT )
16
4 Re{s} 4
2
S 16
(B) (1 / s )(1 e sT )
(C) 1 (1 / s ) e sT
Then the value of is _____.
(D) 1 (1 / s ) e sT
www.targate.org
Page 65
SIGNAL & SYSTEM
AB [GATE – EC – 2000]
(57) Given that
L [f (t)] =
s2
,
s2 1
AD [GATE - EE - 2001]
(61) Given the relationship between the input u(t)
and the output y(t) to be
t
y (t ) =
2
L [g (t)] =
s 1
( s 3)( s 2)
(2 t τ )e
3(t τ )
u( τ )dτ
0
the transfer function Y(s)/U(s) is :
(A)
2e 2 s
s3
(B)
s2
( s 3) 2
(C)
2s 5
s3
(D)
2s 7
( s 3) 2
t
h (t) = f ( g (t ) d ,
0
L [h (t)] is
s2 1
(A)
s3
AC [GATE - IN - 2010]
(62) u(t) represents the unit step function. The
Laplace transform of u ( t τ ) is
1
(B)
s 3
(C)
s2 1
s2
2
( s 3)( s 2) s 1
(D) None of the above
AA [GATE - IN - 2003]
(58) The transfer function of a second order band
– pass filter, having a centre frequency of
1000 rad/s, selectivity of 100 and a gain of 0
dB at the centre frequency, is
(A)
1
sτ
(B)
(C)
e sτ
s
(D) esτ
AB [GATE – EC – 1988]
(63) The Laplace transform of a function f (t) u
(t). Where f (t) is periodic with period T, is
A(s) times the Laplace transform of its first
period. Then
(A)
10 s
s 10 s 10 6
(B)
s
s s 10 6
(B) A(s) = 1/ (1 exp(Ts))
100 s
s 100 s 107
(D) A(s) = exp(Ts )
(C)
2
1
sτ
(A) A(s) = s
2
(C) A(s) = 1/ (1 exp(Ts))
2
100 s
(D) 2
s 100 s 106
AC [GATE – EC – 2003]
(59) The Laplace transform of i (t) is given by
I(s) =
2
s (1 s )
A* [GATE – EC – 1993]
(64) The Laplace transform of the periodic
function f(t) described by the curve below,
i.e.
sin t if (2n 1) t 2n(n 1, 2,3...)
f (t )
otherwise
0
is_______ (fill in the blank)
As t , the value of i (t) tends to
(A) 0
(B) 1
(C) 2
(D)
AA [GATE – EC – 1988]
(60) The transfer function of a zero-order hold is
(A)
1 exp(Ts)
s
(B) 1/s
(C) 1
(D) 1/ [-exp (-Ts)]
Page 66
AD [GATE – EC – 1993]
(65) If F(s) = L [f (t)] =
lim f (t ) is given by
t
TARGATE EDUCATION GATE-(EE/EC)
K
then
( s 1)( s 2 4)
Topic.5 - Laplace Transform
(A) K/4
(B) zero
(C) infinite
(D) undefined
AD [GATE - EE - 1999]
(71) A rectangular current pulse of duration T and
magnitude 1 has the Laplace transform
AD [GATE – EC – 2002]
(66) The Laplace transform of a continuous-time
signal x (t) is X(s) =
(A) 1/s
(B) (1/s) exp(-Ts)
(C) (1/s)exp(Ts)
(D) (1/s)[1 – exp(-Ts)]
5 s
. If the
s s2
2
Fourier transform of this signal exists, then x
(t) is
(A) e 2 t u (t ) 2e t u (t )
(B) e 2 t u ( t ) 2 e t u (t )
AB [GATE - EE - 2005]
(72) For the equation
x ( t ) 3 x (t ) 2 x (t ) 5,
the solution x(T) approaches which of the
following values as t ?
(C) e 2 t u ( t ) 2 e t u (t )
(A) 0
(B) 5/2
(D) e 2 t u ( t ) 2 e t u (t )
(C) 5
(D) 10
AB [GATE – EC – 2009]
(67) Given that F(s) is the one-sided Laplace
transform of
f (t), the Laplace transform
AD [GATE - EE - 2006]
(73) The running integrator, given by y(t) =
t
t
of
x ( ) d
f ( ) d is
0
(A) sF(s) – f (0)
(A) Has no finite singularities in its double
sided Laplace Transform Y(s)
(B)
1
F(s)
s
(B) Produces a bounded output for every
causal bounded input
(C)
(D)
1
[F(s) – f (0)]
s
s
0
(C) Produces a bounded output for every
anticausal bounded input
F ( d
(D) Has no finite zeros in its double sided
Laplace Transform Y(s)
AD [GATE – EC – 2010]
3s 1
1
(68) Given f (t) = L 3
. If
2
s 4 s ( K 3) s
lim f (t ) 1, then the value of K is
t
AB [GATE - EE - 2011]
(74) Let the Laplace transform of a function f(t)
which exists for t>0 be F1(s) and the Laplace
transform of its delayed version f(t - τ ) be
F2 ( s ). F1 * ( s ) be the complex conjugate of
(A) 1
(B) 2
F1 (s) with the Laplace variable set as
(C) 3
(D) 4
F2 ( s).F1 * ( s )
, then
| F1 ( s ) |2
the inverse Laplace transform of G(s) is
s j. If G(s) =
AB [GATE – EC – 2011]
(69) If F(s) = L[f(t)] =
2( s 1)
then the initial
s 4s 7
(A) An ideal impulse (t )
2
and final values of f (t) are respectively
(A) 0, 2
(B) 2, 0
(C) 0, 2/7
(D) 2/7, 0
(B) An ideal delayed impulse ( t τ )
(C) An ideal step function u(t)
AD [GATE - EE - 1995]
(70) The Laplace transformation of f(t) is F(s).
Given F(s) = 2
, the final value of f(t)
s 2
is
(D) An ideal delayed step function u ( t τ )
AD [IES - EC - 1991]
(75) The Laplace transform of the waveform
shown in the Fig. is
(A) infinity
(B) zero
(C) one
(D) none of these
www.targate.org
Page 67
SIGNAL & SYSTEM
1
(1 e as )
(A) V ( s )
(C) Lim
s
as
(B) V ( s ) 1 e2
2s
(C) V ( s ) 1 e
as
2
s 0
X ( s)
s
t
f (t )
h (t ) e( )d
is given by
0
AA [IES - EC - 1992]
(76) Assertion (A): The Laplace transform of
e at sin t is
( s a )2 2
Reason (R) : If the Laplace transform of f(t)
= F(s), then Laplace transform of eat f(t) is
F(s + a)
Codes:
(A) Both A and R are true and R is the
correct explanation of A
(B) Both A and R are true and R is NOT the
correct explanation of A
(C) A is true but R is false
(D) A is false but R is true
AA [IES - EC - 1995]
(77) The following table gives some time
functions and their Laplace transforms:
f(t)
1. ( t ) .......................
F(s)
s
2. u(t) ...................
1/s
3.t u(t) ...................
2 / s2
4. t 2u(t ) ....................
2 / s3
Of these , the correctly matched pair are
(A) 2 and 4
(B) 1 and 4
(C) 3 and 4
(D) 1 and 2
AC [IES - EC - 1995]
2s 1
1
(78) The final value of L 4
is
s 8s3 16s 2 s
(A) infinity
(B) 2
(C) 1
(D) zero
AA [IES - EC - 1997]
(79) If x(t) and its first derivative are Laplace
transformable and the Laplace transform of
x(t) is X(s), then Lim x ( t ) is given by
t 0
Page 68
(D) Lim
AC [IES - EC - 1997]
(80) Given that h(t) = 10 e 10 t u( t ) , and e(t) = sin
10t u(t), the Laplace transform of the signal
e as
s
1 e as e as
1
(D) V ( s )
2
s (1 e as )
as
(A) Lim sX ( s )
s
X ( s)
s
(B) Lim sX ( s )
s 0
(A)
10
( s 10)( s 2 100)
(B)
10( s 10)
( s 2 100)
(C)
100
( s 10)( s 2 100)
(D)
1
( s 10)( s 2 100)
AD [IES - EC - 1998]
(81) Of the following transfer functions of second
order liner time-invariant systems, the
underdamped system is represented by
(A) H ( s )
1
s 4s 4
(B) H ( s )
1
s 5s 4
(C) H ( s )
1
s 4.5s 4
(D) H ( s )
1
s 3s 4
2
2
2
2
AB [IES - EC - 1998]
dx(t)
(82) If x(t) and
are Laplace transformable
dt
and Lim x ( t ) exists, then Lim x ( t ) is equal
t
t
to
(A) Lim sX ( s )
s
(C) Lim
s
X ( s)
s
(B) Lim sX ( s )
s 0
(D) Lim
s 0
X ( s)
s
AD [IES - EC - 1998]
(83) The transfer function of an active network
with gain 'K' is given by:
V2 ( s)
K
2 2 2
V1 ( s ) s C R sCR (3 K ) 1
Assertion (A): The network is unstable for
all values of K.
Reason (R): The poles of the network
function depend on the parameter K.
TARGATE EDUCATION GATE-(EE/EC)
Topic.5 - Laplace Transform
Codes:
(A) Both A and R are true and R is the
correct explanation of A
(B) Both A and R are true but R is NOT a
correct explanation of A
A.
es
s 1
1.
B.
1
s s 1
2.
C.
2
( s 1) 2
3.
D.
1
s s
4.
(C) A is true but R is false
(D) A is false but R is true
AD [IES - EC - 1999]
(84) The function f(t) shown in the given Fig. will
have Laplace transform as
(A)
1 1 s 1 2 s
e 2e
s2 s
s
(B)
1
(1 e s e 2 s )
s2
(C)
1
(1 e s e 2 s )
s
(D)
1
(1 e s se2 s )
2
s
2
2
5.
Codes:
AA [IES - EC - 1999]
(85) Inverse Laplace transform of the function
2s 5
is :
2
s 5s 6
A
B
C
D
(A) 3
4
1
2
(B) 5
2
3
4
(C) 3
2
1
4
(D) 5
4
3
2
(A) 2exp( 2.5t ).cosh(0.5t )
(B) exp( 2 t ).cos( 3t )
(C) 2 exp( 2.5t )sinh(0.5t )
(D) 2exp( 2.5t )cos0.5t
AD [IES - EC - 2000]
(86) The output of a linear system to a unit step
input u(t) is t 2e2t .
AD [IES - EC - 2001]
(88) Which one of the following transfer
functions does correspond to a non-minimum
phase system?
(A)
s
s 2s 1
(B)
s 1
s 2s 1
(C)
s 1
s 2s 1
(D)
s 1
s 2s 1
The system function H(s) is
(A)
2
s ( s 2)
(B)
s
( s 2) 2
(C)
2
( s 2) 3
(D)
2s
( s 2) 3
2
2
2
2
2
AB [IES - EC - 2003]
(89) The relationship between the input x(t) and
the output y(t) of a system is
AC [IES - EC - 2000]
(87) Match List - I (System function) with List II (Impulse response) and select the correct
answer using the codes given the Lists:
d2y
d2x
x
(
t
2)
u
(
t
2)
dt 2
dt 2
The transfer function for system is
www.targate.org
Page 69
SIGNAL & SYSTEM
(A) 1
s2
e2s
(B) 1
e 2 s
s
2.
3.
e2s
(C) 1 2
s
s2
(D) 1 2 s
e
AC [IES - EC - 2003]
27 s 97
(90) If 2
is the Laplace transform of
s 33s
f (t ), then f (0 ) is
(A) Zero
97
(B)
33
(C) 27
(D) infinity
AD [IES - EC - 2003]
(91) Match List I with List II (and select the
correct answer using the codes given
below the Lists:
List I
List II
(F(s))
( f (t ) )
dF ( s )
ds
L(t a) f (t ) asF (s )
Lf (t )
dF (t )
sF ( s ) f (0 )
dt
Select the correct answer using the codes
given below:
(A) 1, 2 and 3
(B) 1, 2 and 4
(D) 2, 3 and 4
(D) 1, 3 and 4
4.
L
AB [IES - EC - 2004]
(94) What is the inverse Laplace transform of
e as
?
s
(A) e at
(B) u (t a)
(C) (t a )
(D) (t a )u (t a )
(A)
10
s ( s 10)
1.
10(t )
(B)
10
s( s 10)
2.
( e 10 t cos10 t ).u (t )
( s 10)
( s 10) 2 100
3.
AD [IES - EC - 2004]
(95) Laplace transforms of f (t ) and g (t ) are F(s)
and G(s), respectively. Which one of the
following expressions gives the inverse
Laplace transform of F(s) G(s)?
2
(C)
(D) 10
(B)
(C) f ( t ) g (t )
(D) f (t ) * g (t )
(sin10t ).u (t )
4.
(1 e 10 t ).u (t )
f (t )
g (t )
(A) f (t ) g (t )
AB [IES - EC - 2005]
(96) For the signal shown below:
Codes:
A
B
C
D
(A)
3
4
1
2
(B)
4
3
1
2
(C)
3
4
2
1
(D)
4
3
2
1
(A) Only Fourier transform exists
AD [IES - EC - 2003]
(92) The Laplace transform of sin t is
(B) Only Laplace transform exists
(C) Both Laplace and Fourier transforms
exist
(A)
s
2
s 2
(B)
2
s 2 2
(D) Neither Laplace transform nor Fourier
transform exists
(C)
s2
s 2 2
(D)
s 2
AB [IES - EC - 2006]
(97) What is the Laplace transform of
2
x ( t ) e 2 t u ( t ) * (tu (t )) ?
AB [IES - EC - 2003]
(93) Given : Lf (t ) F ( s ) f (t )e st dt
0
Which of the following expression are
correct?
1. L[ f (t a )u (t a )] F ( s )e sa
Page 70
(A)
1
s ( s 2)
(B)
1
s ( s 2)
(C)
1
s ( s 2)
(D)
1
s ( s 2)
2
2
TARGATE EDUCATION GATE-(EE/EC)
2
Topic.5 - Laplace Transform
AA [IES - EC - 2006]
(98) The Laplace transform of the waveform
shown in the figure. Is
(B)
(C)
1
(1 Ae s Be 4s Ce 6s De8s )
2
s
What is the value of D ?
(A) – 0.5
(B) – 1.5
(C) 0.5
(D) 2.0
(D)
AB [IES - EC - 2006]
8s 10
(99) What is F ( s )
is equal to
( s 1)( s 2)3
(A)
2
4
4
2
3
2
s 1 ( s 2) ( s 2)
s2
AC [IES - EC - 2007]
(102) What is the Laplace transform of a delayed
unit impulse function (t 1) ?
2
6
2
2
(B)
3
2
s 1 ( s 2) ( s 2)
( s 2)
(C)
2
6
2
2
3
2
s 1 ( s 2) ( s 2)
s2
(D)
4
6
2
4
3
2
s 1 ( s 2) ( s 2)
s2
(B) zero
(C) exp ( s)
(D) s
AB [IES - EC - 2008]
(103) What is the Laplace transform of cos 0 t ?
AB [IES - EC - 2006]
(100) For the function x (t ), x ( s ) is given by
2
x(s ) e s
s ( s 2)
Then, what are the initial and final and final
values of x(t ), respectively?
(A) 0 and 1
(B) 0 and - 1
(C) 1 and 1
(D) – 1 and 0
AC [IES - EC - 2006]
(101) The Laplace transform X(s) of a function x(t)
is :
1 e sT
X ( s)
s
(A) 1
(A)
0
s 20
(B)
s
s 20
(C)
0
( s 0 ) 2
(D)
s
( s 0 ) 2
2
2
B [IES - EC - 2008]
(104) A voltage having the Laplace transform
4s 2 3s 2
is applied across a 2H inductor
7s 2 6s 5
having zero initial current. What is the
current in the inductor at t ?
(A) Zero
(B) (1/5) A
(C) (2/7) A
(D) (2/5) A
AC [IES - EC - 2008]
(105) Match List-I (Function in time domain f(t)]
with List-II (Property of F(s)) and select the
correct answer using the code given below
the lists
List - I
Which is the wave shape of x(t )?
(A)
www.targate.org
A.
sin 0 t u(t t 0 )
B.
sin 0 (t t 0 ) u(t t 0 )
C.
sin 0 (t t 0 ) u(t)
D.
sin 0 t u (t)
Page 71
SIGNAL & SYSTEM
List - II
1.
AA [IES - EC - 2012]
(109) A system described by the following
differential equation is initially at rest and
then excited by the input
0
s 20
2
d2y
dy
4 3 y x (t )
2
dt
dt
The output y(t) is
2. 2 0 2 e t 0s
s 0
3.
4.
x (t ) 3u( t ) :
e t 0s
sin 0 t 0 tan 1 0
s
s
2
(A) 1 1.5e t 0.5e 3t
2
0
(B) 1 0.5e t 1.5e 3t
sin 0 t 0 tan 1 0
s
s
1
2
(C) 1 1.5e t 0.5e 3t
2
0
(D) 1 0.5e t 1.5e 3t
Codes:
(A)
(B)
(C)
(D)
A
3
4
3
4
B
1
2
2
1
C
4
3
4
3
AB [IES - EC - 2012]
(110) If F(s) and G(s) are the Laplace transforms of
f(t) and g(t) , then their product F(s).G(s) =
H(s), where H(s) is the Laplace transform of
h(t), is d
D
2
1
1
2
(A) (f.g)(t)
AB [IES - EC - 2010]
(106) The output of a linear system for step input is
t 2et , then the transfer function is
(A)
s
( s 1)2
(B)
2s
( s 1)3
(C)
s
s ( s 1)
(D)
1
( s 1)3
2
AB [IES - EC - 2011]
(107) Consider a second order all-pole transfer
function model, if the desired settling time
(5%) is 0.60 sec and the desired damping
ratio 0.707, where should the poles be
located in s-plane?
(A) 5 j4 2
(B) 5 j5
(C) 4 j5 2
(D) 4 j7
AA [IES - EC - 2011]
(108) Laplace transform of the function f(t) shown
in the below figure is
2
[1 e0.5s ]2
s2
2
(B) 2 [1 e0.5s ]2
s
2
(C) 2 [1 e 0.5s ]2
s
2
(D) 2 [1 e0.5s ]2
s
(B)
t
0
f ( ) g ( t ) d
(C) Both (A) and (B) are correct
(D) f(t).g(t)
AA [IES - EC - 2012]
(111) Consider a system with transfer function
3s 2 2
s 2 3s 2
The step response of the system is given by
H (s)
(A) C (t ) 5e2t e t 1
(B) C (t ) 3 (t ) 10e 2t e t
(C) C (t ) 4e t e 2t 1
(D) C (t ) 2(1 e 2 t )
AC [IES - EC - 2012]
(112) The unit step response y(t) of a linear system
is y(t) = u(t) For the system function, the
frequency at which the forced response
become zero is
1
rad / s
2
(B)
(C)
2rad / s
(D) 2 rad / s
A90[GATE-EE-2015]
(A)
Page 72
1
rad / s
2
(A)
(113) A moving average function is given by
1 t
y t u d . If the input u is a
T t T
1
sinusoidal signal of frequency
Hz , then
2T
in steady state, the output y will lag u (in
degree) by__________
TARGATE EDUCATION GATE-(EE/EC)
Topic.5 - Laplace Transform
AA [GATE-EE-2015]
(114) The following discrete-time equations result
from the numerical integration of the
differential equations of an un-damped single
harmonic oscillator with state variables x and
y. The integration time step is h.
S3A0.96-1.04 [GATE – EC – 2016]
(117) The
of
the
system
s2
to the unit step input ( )
G (s)
( s 1)( s 3)
is ( ).
The value of
x k 1 x k
yk
h
dy
dt
at = 0+ is _______
S6AA [GATE – EE – 2016]
y k 1 y k
x k
h
(118) The
Laplace
Trnasform
of
f ( t ) e sin(5t )u ( t ) is
2t
For this discrete-time system, which one of
the following statements is TRUE?
(A)
(A) The system is not stable for h > 0
(C)
1
(B) The system is stable for h >
(C) The system is stable for 0 h
(D) The system is stable for
1
2
1
1
h
2
(C) 8
(D) 6
5
s 5
2
s2
5
(D)
s 4s 29
s5
S8AD [GATE – EE – 2016]
2
x(t) 3e u(t) , where u(t) denotes the unit
discrete-time
sequence
j
x n 2, 0, 1, 3, 4,1, 1 , X e is
(B) 6
(B)
t
3
the
(A) 8
5
s 4s 29
2
(119) Consider a causal LTI system characterized
dy(t ) 1
y(t ) 3x(t )
by differential equation
dt
6
. The response of the system to the input
AD[GATE-IN-2005]
(115) Given
response
step function is
t
(A) 9e 3 u (t )
t
(B) 9 e 6 u (t )
S1AMTA [GATE – EC – 2016]
(116) The Laplace transform of the causal periodic
square wave of period T shown in the figure
below is :
t
t
(C) 9 e 3 u (t ) 6 e 6 u ( t )
t
t
(D) 54 e 6 u (t ) 54e 3 u (t )
S8AA [GATE – EE – 2016]
(120) Consider a linear time invariant system
x A x , with initial condition x(0) at t = 0.
Suppose and are eigenvectors of (2 x 2)
matrix A corresponding to distinct
eigenvalues and respectively. Then the
response x(t) of the system due to initial
condition x(0) = is
1
(A) F (s)
(B) F ( s)
1
1 e sT /2
(A) e 1t
(B) e2 t
1
(C) e2t
sT
s 1 e 2
(C) F ( s )
1
s (1 e sT )
(D) F (s)
1
1 esT
2
(D) e 1t e 2t
A0.45-0.55 [GATE–S2–EC–2017]
(121) The transfer function of a causal LTI system
is H(s) = 1/s. If the input to the system is
x(t ) [sin(t ) / t ] u (t ) , where u(t) is a unit
step function, the system output y(t) as
t is ____.
www.targate.org
Page 73
SIGNAL & SYSTEM
AB [GATE–S1–EE–2017]
(122) Let a causal LTI system be characterized by
the following differential equation, with
intial rest condition
dx t
d2 y
dy
7 10y t 4x t 5
2
dt
dt
dt
Where, x(t) and y(t) are the input and output
respectively. The impulse response of the
system is (u(t) is the unit step function)
(A) 2e2t u t 7e5t u t
A0.46 to 0.48 [GATE-IN-2019]
(127) The transfer function relating the input x(t) to
the output y(t) of a system is given by G(s) =
1/(s+3). A unit-step input is applied to the
system at time t = 0. Assuming that y(0) = 3,
the value of y(t) at time t = 1 is ______
(Answer should be rounded off to two
decimal places)
AC [GATE-IN-2019]
(128) The output y(t) of a system is related to its
input x (t ) as
(B) 2e2t u t 7e5t u t
t
y(t ) x( 2)d ,
(C) 7e2t u t 2e5t u t
0
(D) 7e2t u t 2e5t u t
where, x(t) = 0 and y(t) = 0 for t 0 . The
transfer function of the system is :
A0.550 TO 0.556 [GATE–S1–EE–2017]
(123) For a system having transfer function
s 1
, a unit step input is applied at
G s
s 1
time t = 0. The value of the response of the
system at t = 1.5 sec(rounded off to the three
decimal places) is __________.
A1.284 [GATE–S2–EE–2017]
(124) Consider the system described by the
following state space representation :
x1 (t ) 0 1 x1 (t ) 0
x (t ) 0 2 x (t ) 0 u (t )
2
2
x (t )
y (t ) [1 0] 1
x 2 (t )
x (0) 1
If u(t) is a unit step input and 1 ,
x 2 (0) 0
value of output y (t ) at t = 1 sec (rounded off
to three decimal places) is ______ .
AB [GATE – IN – 2017]
(125) A system is described by the following
differential
equation:
dy t
dx t
2y t
x t , x 0 y 0 0
dt
dt
Where x(t) and y(t) are the input and output
variables respectively. The transfer function
of the inverse system is
(A)
s 1
s2
(B)
s2
s 1
(C)
s 1
s2
(D)
s 1
s2
A-2.4 to -2.0 [GATE – IN – 2017]
(126) The Laplace transform of a causal signal y(t)
s2
is Y s
. The value of the signal y t
s6
at t = 0.1 s is ________units.
Page 74
(A)
1
s
(B)
(1 e2 s )
s
(C)
e 2 s
s
(D)
1 2 s
e
s
AD [GATE-EE-2019]
(129) The output response of a system is denoted
as y(t), and its Laplace transform is given by
10
Y ( s)
. The steady state
2
s( s s 100 2)
value of y(t) is :
(A) 100 2
(B)
(C) 10 2
(D)
1
100 2
1
10 2
AC [GATE-EE-2019]
inverse Laplace transform of
s3
H ( s) 2
for t 0 is
s 2s 1
(130) The
(A) 3te t e t
(B) 3e t
(C) 2tet et
(D) 4tet et
AA [GATE-EC-2019]
(131) Let Y(s) be the unit-step response of a causal
system having a transfer function
3 s
G (s)
G (s )
that is, Y ( s)
.
( s 1)(s 3)
s
The forced response of the system is
(A) u (t )
(B) 2u (t )
(C) 2u(t ) 2et u(t ) e3t u(t )
(D) u(t ) 2et u(t ) e3t u(t )
-------0000-------
TARGATE EDUCATION GATE-(EE/EC)
06
Sampling Theorem
(1)
(2)
(3)
2.99to3.01 [GATE – EC1 – 2014]
Consider two real valued signals x (t) band
limited to [-500HZ, 500HZ] and y (t) band
limited to [-1 kHz, 1 kHz]. For z (t) = x (t). Y
(t), the Nyquist sampling frequency (in kHz)
is ---------.
and
x (t ) 10 cos(8 x 10 3 )t
Ts 100 sec . When y(t) is passed through
an ideal low pass filter with a cot off
frequency of 5 KHz, the output of the filter is
A9.5to10.5 [GATE – EC3 – 2014]
A
modulated
signal
is
y t =m t cos 40000 t , where the
baseband signal m (t) has frequency
components less than 5 kHz, only. The
minimum required rate (in kHz), at which y
(t) should be sampled to recover m (t) is ----.
(B) 5 10 5 cos(8 10 3 )t
(A) 5 10 6 cos(8 10 3 )t
(C) 5 10 1 cos(8 10 3 )t
(D) 10 cos(8 10 3 )t
(6)
AA [GATE – EC – 1994]
Increased pulse-width in the flat-top
sampling, leads to
AB [GATE – EC – 2014]
For a given sample-and-hold circuit, if the
value of the hold capacitor is increased, then
(A) drop rate decreases and acquisition time
decreases
(B) drop rate decreases and acquisition time
increases
(C) drop rate increases and acquisition time
decreases
(D) drop rate increases and acquisition time
increases
(A) attenuation of high frequencies in
reproduction
(B) attenuation of
reproduction
low
frequencies
(7)
in
(C) greater aliasing errors in reproduction
x n x n 1 x n 2
Y n
. For a
x n 4 / 4
step input, the maximum time taken for the
output to reach the final value after the input
transition is
(D) no harmful effects in reproduction
(4)
AB [GATE – EC – 1999]
The Nyquist sampling frequency (in Hz) of a
signal given by
16 10 4 sin c 2 (400t ) *10 6 sin c 3 (100 t )
(5)
(A) 200
(B) 300
(C) 500
(D) 1000
Consider
a
AC [GATE – EC – 2002]
sampled signal y(t) =
AB [GATE – IN – 2006]
A digital measuring instrument employs a
sampling rate of 100 samples/second. The
sampled input x(n) is averaged using the
difference equation
(8)
(A) 20 ms
(B) 40 ms
(C) 80 ms
(D)
AC [GATE – IN – 2007]
Let x(t) be a continuous-time, real-valued
signal band-limited to F Hz.
5 106 x(t ) (t nTs ) where
The Nyquist sampling rate in Hz.
n
For y t x 0.5t x t x 2t is
www.targate.org
Page 75
SIGNAL & SYSTEM
(9)
(A) F
(B) 2F
(C) 4F
(D) 8F
AC [GATE – IN – 2015]
The highest frequency present in the signal
x(t) is f max . The highest frequency present in
the signal y t x 2 t is
(A)
(B) f max
(D) 4f max
AC [GATE – EC – 2004]
(10) A 1 kHz sinusoidal signal is ideally sampled
at 1500 samples/sec and the sampled signal
is passed through an ideal low-pass filter
with cut-off frequency 800 Hz. The output
signal has the frequency
(A) zero Hz
(B) 0.75 kHz
(C) 0.5 kHz
(D) 0.25 kHz
AD [GATE - IN - 2004]
(11) A signal x(t) = 5 cos(150 πt 60) is sampled
at 200 Hz. The fundamental period of the
sampled sequence x[n] is
1
200
(B)
(C) 4
2
200
(D) 8
AB [GATE - IN - 2011]
(12) The continuous time signal x(t) =
cos(100πt ) sin(300πt ) is sampled at the
rate 100 Hz to get the signal
xs ( t )
(A) 5 Hz and 15Hz only
(B) 10 Hz and15 Hz only
(C) 5 Hz, 10 Hz and 15 Hz only
(D) 5 Hz only
1
f max
2
(C) 2f max
(A)
The frequency / frequencies present in the
reconstructed signal is / are
x ( nTS ) (t nTs ),
n
Ts sampling period
The signal xs (t ) is passed through an ideal
low pass filter with cut-off frequency 100
Hz.
A14 [GATE-EE2-2014]
(14) For the signal f (t ) 3sin 8t 6sin12t
sin14 t ,
the
minimum
sampling
frequency (in Hz) satisfying the Nyquist
criterion is _________.
AC [GATE-EE2-2014]
(15) A sinusoid x(t) of unknown frequency is
sampled by an impulse train of period 20 ms.
The resulting sample train is next applied to
an ideal low pass filter with a cutoff at 25
Hz. The filter output is seen to be a sinusoid
of frequency 20 Hz. This means that x(t) has
a frequency of
(A) 10 Hz
(B) 60 Hz
(C) 30 Hz
(D) 90 Hz
AA [GATE-EC/EE/IN-2013]
(16) A band-limited signal with a maximum
frequency of 5 kHz is to be sampled.
According to the sampling theorem, the
sampling frequency which is not valid is
(A) 5 kHz
(B) 12 kHz
(C) 15 kHz
(D) 20 kHz
AC [GATE – EC – 1995]
(17) A 1.0 kHz signal is flat-top sampled at the
rate of 1800 samples/sec and the samples are
applied to an ideal rectangular LPF with cutoff frequency of 1100 Hz, then the output of
the filter contains
(A) only 800 Hz component
The output of the filter is proportional to
(B) 800 Hz and 900 Hz components
(A) cos(100 πt )
(C) 800 Hz and 1000 Hz components
(B) cos(100πt ) sin(100πt )
(D) 800 Hz, 900
components
(C) cos(100πt ) sin(100πt )
and 1000
Hz
AA [GATE – EC – 1998]
(18) Flat top sampling of low pass signals
(D) sin(100 πt )
AA [GATE – EC3 – 2014]
(13) Let x (t) cos 10 t c os 3 0 t
be
sampled at 20 Hz and reconstructed using an
ideal low pass filter with cut off frequency of
20 Hz.
Page 76
Hz
(A) gives rise to aperture effect
(B) implies oversampling
(C) leads to aliasing
(D) introduces delay distortion
TARGATE EDUCATION GATE-(EE/EC)
Topic.6 - Sampling Theorem
AD [GATE – EC – 2001]
(19) A band limited signal is sampled at the
Nyquist rate. The signal can be recovered by
passing the samples through
AC [GATE – EC – 2001]
(24) The Nyquist sampling interval, for the signal
Sinc(700t) + Sinc(500t) is
(A) an RC filter
(A)
1
sec
350
(B)
sec
350
(C)
1
sec
700
(D)
sec
175
(B) an envelope detector
(C) a PLL
(D) an ideal low-pass filter
appropriate bandwidth
with the
AD [GATE – EC – 1988]
(20) A signal containing only two frequency
components (3kHz and 6 kHz) is sampled at
the rate of 8 kHz, and then passed through a
low pass filter with a cut-off frequency of 8
kHz.
AD [GATE – EC – 2002]
(25) A signal x(t) = 100 cos(24 10 3 ) t is
ideally sampled with a sampling period of 50
sec and then passed through an ideal low
pass filter with cut-off frequency of 15 KHz.
Which of the following frequencies is/are
present at the filter output?
(A) 12 KHz only
The filter output
(B) 8 KHz only
(A) is an undistorted version of the original
signal
(C) 12 KHz and 9 KHz
(D) 12 KHz and 8 KHz
(B) contains only the 3 kHz component
(C) contains the 3kHz component and a
spurious component of 2 kHz
(D) contains both the components of the
original signal and two spurious
components of 2 kHz and 5 kHz.
AA [GATE - IN - 2012]
(21) The transfer function of a Zero- order – Hold
system with sampling interval T is
(A)
1
(1 e Ts )
s
(B)
2
1
1 e Ts
s
(C)
1 Ts
e
s
(D)
1 Ts
e
s2
AB [GATE – EC – 1990]
(22) A 4 GHz carrier is DSB-SC modulated by a
low pass message signal with maximum
frequency of 2 MHz. The resultant signal is
to be ideally sampled. The minimum
frequency of the sampling impulse train
should be:
(A) 4 MHz
(B) 8 MHz
(C) 8 GHz
(D) 8.004 GHz
AC [GATE – EC – 2003]
(26) Let x(t) = 2cos (80t ) + cos (1400t ) . x(t)
is sampled with the rectangular pulse train
shown in figure. The only spectral
components (in kHz) present in the sampled
signal in the frequency range 2.5 kHz to 3.5
kHz are
(A) 2.7,3.4
(B) 3.3,3.6
(C) 2.6, 2.7, 3.3, 3.4, 3.6
(D) 2.7, 3.3
AB [GATE – EC – 2006]
(27) A signal m(t) with bandwidth 500 Hz is first
multiplied by a signal g(t) where
A3.6kHz [GATE – EC – 1991]
(23) A signal has frequency components from 300
Hz to 1.8 KHz. The minimum possible rate at
which the signal has to be sampled is _____
(fill in the blank).
www.targate.org
(1)
g (t )
k
(t 0.5 10 4 k )
k
The resulting signal is then passed through
an ideal low pass filter with bandwidth 1
kHz.
The output of the low pass filter would be
(A) (t)
(B) m(t )
(C) 0
(D) m(t )(t )
Page 77
SIGNAL & SYSTEM
AC [GATE – EC – 2006]
(28) The minimum sampling frequency (in
samples/sec) required to reconstruct the
following signal from its samples without
distortion
3
sin 21000t
sin 2 1000t
x (t ) 5
7
t
t
2
would be
3
(B) 4 10
3
(D) 8 10
(A) 2 10
(C) 6 10
3
3
AC [GATE – EC – 2010]
(29) The Nyquist sampling rate for the signal s(t)
sin(500t ) sin(700t )
=
is given by
t
t
(A) 400 Hz
(B) 600 Hz
(C) 1200 Hz
(D) 1400 Hz
AB [GATE - EE - 2007]
(30) The frequency spectrum of a signal is shown
in the figure. If this signal is ideally sampled
at intervals of 1 ms, then the frequency
spectrum of the sampled signal will be
5000 samples/s. For a signal x(t) = [ s (t )]2
the corresponding Nyquist sampling rate in
samples/s is
(A) 2500
(B) 5000
(C) 10000
(D) 25000
AA [GATE - IN - 2010]
(33) A signal with frequency components 50 Hz,
100 Hz and 200 Hz only is sampled at 150
samples/s. The ideally reconstructed signal
will have frequency component (s) of
(A) 50 Hz only
(B) 75 Hz only
(C) 50 Hz and 75 Hz
(D) 50 Hz, 75 Hz and 100 Hz
AC [GATE - IN - 2006]
(34) The spectrum of a band limited signal after
sampling is shown below. The value of the
sampling interval is
(A)
(A) 1 ms
(B) 2 ms
(C) 4 ms
(D) 8 ms
(B)
AD[GATE-EE-2012]
(35) Consider
the
differential
2
d y t
dy t
2
y t t
2
dt
dt
dy
y t t 0 2 and
is
dt t 0
(C)
(D)
AA [GATE - EE/EC/IN - 2012]
(31) Let y[n] denote the convolution of h[n] and
g[n], where h[n] = (1/ 2) n u[n] and g[n] is a
causal sequence. If y[0] = 1 and y[1] = ½,
then g[1] equals
(A) 0
(B) 1/2
(C) 1
(D) 3/2
AC [GATE - IN - 2004]
(32) The Nyquist rate of sampling of an analog
signal s(t) for alias free reconstruction is
Page 78
(A) -2
(B) -1
(C) 0
(D) 1
equation
with
S1A12-14 [GATE – EC – 2016]
(36) A continuous-time sinusoid of frequency 33
Hz is multiplied with a periodic Dirac
impulse train of frequency 46 Hz. The
resulting signal is passed through an ideal
analog low-pass filter with a cutoff frequency
of 23 Hz. The fundamental frequency (in Hz)
of the output is _____.
S4AC [GATE – EC – 2016]
(37) Consider
the
signal
x(t ) cos(6t ) sin(8t ) , where t is in
seconds. The Nyquist sampling rate (in
samples/second) for the signal y(t) = x(2t +
5) is :
TARGATE EDUCATION GATE-(EE/EC)
Topic.6 - Sampling Theorem
(A) 8
(B) 12
(C) 16
(D) 32
(A) -0.707
(C) 0
S8AB [GATE – EE – 2016]
(38) Let x1 (t ) X 1 () and x2 (t ) X 2 () be
two signals whose Fourier Transforms are as
shown in the figure below. In the figure, h(f)
= e–2|t| denotes the impulse response.
(B) -1
(D) 1
A13 [GATE – EC – 2018]
(42) A band limited low-pass signal x(t ) of
bandwidth 5 kHz is sampled at a sampling
rate f s . The signal ( ) is reconstructed
using the reconstruction filter ( ) whose
magnitude response is shown below :
The minimum sampling rate f s (in kHz) for
perfect reconstruction of x(t ) is _____.
AA [GATE – EE – 2018]
(43) Consider the two continuous-time signals
defined below :
For the system above, the minimum sampling
rate required to sample y(t), so that y(t) can
be uniquely reconstructed its samples, is
| t |, 1 t 1
x1 (t )
,
0, otherwise
(A) 2B1
(B) 2(B1+B2)
(C) 4(B1+B2)
(D) ∞
1 | t |, 1 t 1
x2 (t )
otherwise
0,
These signals are sampled with a sampling
period of T = 0.25 seconds to obtain
discretetime signals x1 [ n ] and x2 [ n ] ,
respectively. Which one of the following
statements is true?
(A) The energy of x1 [ n ] is greater than the
energy of x2 [ n ] .
S8A6.0 [GATE – EE – 2016]
(39) Suppose the maximum frequency in a bandlimited signal x(t) is 5 kHz. Then, the
maximum frequency in x(t) cos(2000πt), in
kHz, is _____.
AB [GATE–S2–EE–2017]
(40) The output y(t) of the following system is to
be sampled, so as to reconstruct it from its
samples uniquely. The required minimum
sampling rate is :
(B) The energy of x2 [ n ] is greater than the
energy of x1 [ n ] .
(C) x1 [ n ] and x2 [ n ] have equal energies.
(D) Neither x1 [ n ] nor
energy signal.
x2 [ n ] is a finite-
A8.0 [GATE-IN-2019]
(44) The frequency response of a digital filter
H ( ) has the following characteristics
Passband: 0.95 | H () | 1.05 for
0 0.3 and
(A) 1000 samples/s
(B) 1500 samples/s
(C) 2000 samples/s
(D) 3000 samples/s
AC [GATE – IN – 2017]
(41) If a continuous time signal x(t) = cos 2 t is
sampled at 4Hz, the value of the discrete
time sequence x n 5 is
www.targate.org
Stopband: 0 | H () | 0.005 for
0.4 ,
where is the normalized angular
frequency in rad/sample. If the analog upper
cut off frequency for the passband of the
above digital filter is to be 1.2 kHz, then the
sampling frequency should be ______ kHz.
-------0000------Page 79
07
Z- Transform
(1)
A0 [GATE – EC2 – 2015]
Two casual discrete-time signals x[n] and
n
y[n] are related as y[n] x[m] . If the zm 0
2
transform of y[n]
, the value of
z(z 1) 2
x[2] is ________.
(2)
AA [GATE – EE – 2008]
H(z) is a transfer function of a real system
when a signal x[n] = (1 + j)n is the input to
such a system, the output is zero. Further, the
Region of Convergence (ROC) of
1 1
1 z H z is the entire Z-plane(except
2
z = 0). It can then be inferred that H(z) can
have minimum of
(5)
1 az1
(A)
1 bz1
1 bz1
(B)
1 az1
1 az1
(C)
1 bz1
1 bz1
(D)
1 az1
A* [GATE – EC – 1998]
The z-transform of the time function
n k is
k 0
(A) one pole and one zero
(A)
z
z aT
(B)
z
z aT
(C)
z
z a T
(D)
z
z a T
(B) one pole and two zeros
(C) two poles and one zero
(D) two poles and two zeros
(3)
AC [GATE – EE – 2014]
Consider a discrete time signal given by
n
(6)
n
x n 0.25 u n 0.5 u n 1 . The
region of convergence of its Z-transform
would be
(A) the region inside the circle of radius 0.5
and centered at origin
(B) the region outside the circle of radius
0.25 and centered at origin
(C) the annular region between the two
circles, both centered at origin and
having radii 0.25 and 0.5
(7)
AA [GATE – EC – 1999]
The z-transform f(z) of the time function
f nT a nT is
(A)
z
z aT
(B)
z
z aT
(C)
z
z a T
(D)
z
z a T
AB [GATE – EC – 2003]
A sequence x(n) with the z-transform
x z z 4 z 2 2z 2 3z 4 is applied as
an input to a linear, time-invariant system
with the impulse response h (n) 2 n 3
where
n0
1,
n
0, otherwise
(D) the entire Z plane
(4)
AA [GATE – EC – 1988]
Consider the system shown in the figure
below. The transfer function Y(z)/X(z) of the
system is
Page 80
The output at n = 4 is
(A) – 6
(B) zero
(C) 2
(D) – 4
TARGATE EDUCATION GATE-(EE/EC)
Topic.7 - Z-Transform
(8)
AC [GATE – EC – 1999]
The z-transform of a signal is given by
1
4
1 z 1 z
. Its final value is
Cz
4 1 z 1 2
(9)
AC [GATE – EC – 2014]
(12) C is a closed path in the z-plane given by
z 3 . The value of the integral
z2 z 4
C z 2j dz is
(A) 1/4
(B) zero
(C) 1.0
(D) infinity
AD [GATE – EC – 2006]
If the region of convergence of
1
2
x 1 n x 2 n is z , then the region
3
3
of convergence of x1 n x 2 n includes
(A)
1
z 3
3
(B)
2
z 3
3
(C)
3
z 3
2
(D)
1
2
z
3
3
(A) 4 1 j2
(B) 4 3 j2
(C) 4 3 j2
(D) 4 1 j2
AC [GATE – EC – 2015]
(13) For the discrete-time system shown in the
figure, the poles of the system transfer
function are located at
AA [GATE – EC – 2010]
the
z-transform
The
x z 5z 2 4z 1 3; 0 z .
inverse z-transform x[n] is
(10) Consider
(A) 5 n 2 3 n 4 n 1
(C)
(B) 5 n 2 3 n 4 n 1
(C) 5u n 2 3 n 4u n 1
(D) 5u n 2 3u n 4u n 1
AB [GATE – EC – 2011]
(11) Two
systems
are
H1 z and H 2 z
connected in cascaded as shown below. The
overall output y(n) is the same as the input
x(n) with a one unit delay. The transfer
function of the second system H 2 z is
(A)
(B)
(C)
(D)
1 0.6z
z 1 0.4z
(B)
(C)
1
1
z 1 1 0.6z 1
1 0.4z
1
1 1
,
2 3
h[n] is real for all n
h[n] is purely imaginary for all n
h[n] is real for only even n
h[n] is purely imaginary for only odd n.
A0 [GATE – EC – 2015]
(15) Two causal discrete-time signals x[n] and
n
z 1 1 0.4z 1
y[n] are related as y n x m . If the z-
1 0.6z
1
m 0
transform of y[n] is
1 0.4z
z 1 0.6z
1
(D)
1
(B)
AA [GATE – EC – 2015]
(14) The pole-zero diagram of a causal and stable
discrete-time system is shown in the figure.
The zero at the origin has multiplicity 4. The
impulse response of the system is h[n]. If
h[0] = 1, we can conclude
1
(A)
1
,3
2
1
(D) 2,
3
(A) 2, 3
2
z z 1
2
, the value of
x[2] is ______ .
1
www.targate.org
Page 81
SIGNAL & SYSTEM
A2 [GATE – EC – 2015]
(16) The value of
1
n
n 2
is ________
n 0
AC [GATE – EC3 – 2015]
(17) Suppose x[n] is an absolutely summable
discrete-time signal. Its z-transform is a
rational function with two poles and two
zeroes. The poles are at z 2j . Which one
of the following statements is TRUE for the
signal x[n]?
(A) It is a finite duration signal.
H z 91 az1 , a is real and a 1 . The
impulse response of a stable system that
exactly compensates the magnitude of the
direction is
n
1
(A) u n
a
n
1
(B) u n 1
a
n
(C) a u n
(D) a n u n 1
(B) It is a causal signal.
(C) It is a non-causal signal.
(D) It is a periodic
AA [IES-EC-2013]
(18) The final value theorem is
AB [GATE – IN – 2004]
(22) In the IIR filter shown below, a is a variable
gain. For which of the following cases, the
system will transit from stable to unstable
condition
(A) lim x (k ) lim z 1 X ( z )
k
z 1
(B) lim x ( k ) lim X ( z )
k
z 1
(C) lim x ( k ) lim( z 1 ) X ( z )
k
(A) 0.1 a 0.5
z0
(B) 0.5 a 1.5
(D) lim x (k ) lim( z 1) 1 X ( z 1 )
k
(C) 1.5 a 2.5
z0
AD [IES-EC-2013]
(19) For the discrete signal the z-transform is
z a
(B)
z
z
(A)
z a
(C)
z
a
(D)
z
z a
(20) Let
n
n
.
The Region of convergence (ROC) of the ztransform of x [n]
(A) is z
1
9
(B) is z
1
3
(C) is
(C) z
1
2
(B) z 2
(D) z
1
2
AB [GATE – IN – 2010]
(24) H(z) is the discrete rational transfer function.
To ensure both H(z) and its inverse are stable
its
(A) Poles must be inside the unit circle and
zeros must be outside the unit circle
(B) Poles and zeros must be inside the unit
circle
(C) Poles and zeros must outside the unit
circle
1
1
z
3
9
(D) Poles must be outside the unit circle and
the zeros should be inside the unit circle.
(D) does not exist.
AD [GATE – IN – 2004]
(21) A discrete-time signal, x[n] suffered a
distortion modeled by an LTI system with
Page 82
AA [GATE – IN – 2008]
(23) The region of convergence of the Ztransform of the discrete-time signal
x n 2 n u n will be
(A) z 2
AC [GATE – EC1 – 2014]
1
1
x n u n u n 1
9
3
(D) 2 a
AA [GATE – EC/IN – 2015]
n
(25) The z-transform of x n , 0 1 , is
X(z). The region of convergence of x(z) is
TARGATE EDUCATION GATE-(EE/EC)
Topic.7 - Z-Transform
(A) z
AB [GATE – EC – 2007]
(31) The z-transform X (z) of a sequence x[n] is
1
(B) z
(C) z
given by X[z] =
1
region of convergence of X[z] includes the
unit circle. The value of x [0] is
1
(D) z min ,
AA [IES-EC-2013]
(26) If the z-transform of
z(8z 7)
, then the
4z 2 7z 3
X(n) is x(z) =
n
(A) 1
(C) ∞
(B) 2
(D) 0
H
A-0.6to-0.4 [GATE – EC3 – 2014]
(Z) = (1 – pz -1)-1, H
1
H2 z 1 qz
1 1
The quantities p, q, r are real number.
of H (Z) lies on the unit circle, then r = --------.
AC [GATE-IN-2014]
(28) The transfer function of a digital system is
given by:
b0
; where a 2 is real.
1
1 z a2 z 2
The transfer function is BIBO stable if the
value of 2 is:
(A) −1.5
(B) −0.75
(C) 0.5
(D) 1.5
1
| z | 3
3
(B)
1
(C) | z | 3
3
1
1
| z |
3
2
1
(D) | z |
3
(A) a
n1
(C) n a n
z
1
u[ n]
n!
(B)
1
u[n]
n !
1
u[n]
n!
1
u[ n 1]
(n 1)!
(A) u[n - m]
(B) δ(n - m)
(C) δ[m]
(D) δ[m - n]
the
sequence
5
6
n
u
(n)
6
5
n
u (n 1) must be
(A) | z |
(C)
5
6
5
6
| z |
6
5
(B) | z |
(D)
6
5
6
| z |
5
AC [IES - EC - 1998]
(35) If the function H1 ( z) (1 1.5z 1 z 2 ) and
at z = a for n 0 will
(B) a
(A)
AC [GATE – EC – 2005]
(34) The region of convergence of z-transform of
z
(30) Given X(z) =
with | z | a, the
( z a) 2
residue of X(z)
be
(D) 0.5
AB [IES - EC - 1997]
(33) Which one of the following represents the
impulse response of a system defined by
H(z) = z m ?
AD [GATE - EE - 2008]
n1
(C) 0.25
(D)
AC [GATE-EC/EE/IN-2012]
(29) If x[ n ] (1 / 3)|n| (1 / 2) n u[ n ], then the
region of convergence (ROC) of its Ztransform in the Z-plane will be
(A)
(B) 0
(C) (1)n
, H (Z) = H1(Z) + rH2(Z).
1
1
Consider P , q , r 1 If the zero
2
4
(A) 0.5
AA [GATE - IN - 2003]
(32) The sequence x[n] whose z – transform is
1/ z
X(z) = e is
lim x (n ) is
(27) Let
0.5
is given that the
1 2z 1
H 2 ( z ) z 2 1.5z 1 ,then
(A) the poles and zeros of the functions will
be the same
(B) the poles of the functions will be
identical but not zeros
(C) the zeros of the functions will be
identical but not poles
n
(D) n a n 1
www.targate.org
(D) neither the poles nor the zeros of the two
functions will be identical
Page 83
SIGNAL & SYSTEM
AD [IES - EC - 1994]
(36) Which one of the following is the region of
convergence (ROC) for the sequence x[n] =
bn u(n) b nu(n 1) ; b < 1?
(C) Y ( z)
X ( z)
1 z 1
(D) Y ( z )
dX ( z )
dz
(A) Region z 1
(B) Annular
AB [IES - EC - 2006]
strip
in
the
region
(40)
b z (1/ b)
(C) Region z 1
(D) Annualar
For the system shown,
strip
in
the
region
b z (1/ b)
x[ n] k [n],
and y[n ] is related to x[ n] as
AB [IES - EC - 2006]
(37) Which one of the following is the correct
statement ?
The region of convergence of z transform
x | n | consists of the value of z for which
x | n | r n is
(A) absolutely integrable
y[ n]
What is y[n ] equal to?
(A) k
(B) (1 / 2) n k
(C) nk
(D) 2n
AA [IES - EC - 2006]
(41) What is the inverse z transform of X(z)?
(B) absolutely summable
(C) unity
(D) < 1
AA [IES - EC - 2000]
(38)
Two linear time- invariant discrete time
systems s1 and s2 are cascaded as shown in
Fig. Each system is modeled by a second
order difference equation.
The difference equation of the overall
cascaded system can be the order of
(A) 0, 1, 2, 3 or 4
(B) either 2 or 4
(C) 2
(D) 4
AC [IES - EC - 2005]
(39) The output y[ n] of a discrete time LTI
system is related to the input x[ n] as given
below:
y[n]
x[k ]
k 0
Which one of the following correctly relates
the z-transforms of the input and output
denoted by X(z) and Y(z), respectively?
(A) Y(z) = (1 z 1 ) X ( z )
(B) Y ( z ) z 1 X ( z )
Page 84
1
y[ n 1] x[n ]
2
(A)
1
2πj
X ( z)z
n 1
(B)
1
2πj
X ( z)z
n 1
(C)
1
X z z1 n dz
2j
dz
dz
(D) 2nj X ( z ) z ( n 1) dz
AB [IES - EC - 2011]
(42) Assertion (A) : The system function
z3 2z2 z
is not causal.
1
1
z2 z
4
8
Reason (R) : If the numerator of H(z) is of
lower order than the denominator, the system
may be causal.
Codes :
(A) Both A and R are true and R is the
correct explanation of A
(B) Both A and R are true but R is not a
correct explanation of A
(C) A is true but R is false
(D) A is false but R is true
H(z)
AD [IES - EC - 2010]
(43) Frequency scaling [relationship between
discrete time frequency ( ) and continuous
time frequency () ] is defined as
TARGATE EDUCATION GATE-(EE/EC)
Topic.7 - Z-Transform
(A) 2
(B) 2TS /
(C) (0.2) n u[ n]
(C) 2 / TS
(D) TS
(D) (0.2) n u[ n 1]
A11.9to12.1 [GATE – EC3 – 2014]
(44) The z-transform of the sequence x [n] is
given by X (Z) =
1
1 2z 1
2
, with the
AA [GATE – EC – 2009]
(49) The ROC of Z-transform of the discrete time
sequence
n
region of convergence | z | > 2. Then, x [2] is
-------------.
AC [GATE-IN-2014]
(45) The system function of an LTI system is
given by
(A)
1
1
| z |
3
2
(C) | z |
1
1 Z 1
3
H ( z)
1 1
1 Z
4
1
2
(D) 2 | z | 3
AB [GATE – EC – 1990]
(50) The Z-transform of the following real
exponential sequence:
x ( nT ) a n .
1
(A) | z |
4
is given by
(D) | z |
nT 0
0 . nT 0, a 0
1
(B) | z |
12
1
4
(B) | z |
1
3
The above system can have stable inverse if
the region of convergence of H(z)is defined
as
(C) | z |
n
1
1
x( n) u ( n) u ( n 1) is
3
2
1
3
AC [GATE – EC – 1998]
(46) The z-transform of the time function
(A)
1
. | z | 1
1 z 1
(B)
1
:| z | a
1 az 1
( n k ) is
(C) 1 for all z
k 0
(A) ( z 1) / z
(B) z / ( z 1) 2
(C) z / ( z 1)
(D) ( z 1) 2 / z
AA [GATE – EC – 2001]
(47) The region of convergence of the z-transform
of a unit step function is
(D)
1
:| z | a
1 az 1
AA [GATE – EC – 1999]
(51) The z-transform of a signal is given by
1 z 1 (1 z 1 )
C( z )
4 (1 z 1 )2
(A) |z| > 1
Its final value is
(B) |z| < 1
(C) (Real part of z) > 0
(D) (Real part of z) < 0
AD [GATE – EC – 2004]
(48) The z-transform of a system is
H ( z)
z
z 0.2
(A) 1/4
(B) zero
(C) 1.0
(D) infinity
Statement for Linked Question for the Next Two
Questions :
In the following network (Fig.1). the switch is
closed at t = 0 and the sampling starts from t = 0.
The sampling frequency is 10 Hz.
If the ROC is |z| < 0.2, then the impulse
response of the system is
(A) (0.2) n u [ n ]
(B) (0.2) n u [ n 1]
www.targate.org
Page 85
SIGNAL & SYSTEM
AB [GATE – EC – 2008]
(52) The samples x (n) (n = 0, 1, 2 ...) are given
by
(A) 5(1 e 0.05 n )
(B) 5e
(C) 5(1 e 5 n )
(D) 5e
0.05 n
5n
AC [GATE – EC – 2003]
(53) The expression and the region of
convergence of the z-transform of the
sampled signal are
(A)
5z
,| z | e5
5
ze
5z
(B)
,| z | e0.05
z e0.05
(C)
5z
, z e 0.05
z e 0.05
(D)
5z
,| z | e5
z e5
AB [GATE - EE - 2005]
(54) If u(t) is the unit step and (t ) is the unit
impulse function, the inverse z-transform of
F (z)
H[n] = x[n – 1]* y[n]
Where*denotes discrete time convolution.
Then the output of the system for the input
[ n 1]
(A) Has Z-transform z 1 X ( z )Y ( z )
(B) Equals [ n 2] 3[ n 3] 2[ n 4]
6[ n 5]
(C) Has Z-transform 1 3 z 1 2 z 2 6 z 3
(D) Does not satisfy any of the above three.
AC [GATE - EE - 2007]
(57) A signal is processed by a causal filter with
transfer function G(s). For a distortion free
output signal waveform, G(s) must
(A) Provide zero phase shift for all
frequencies
(B) Provide constant phase shift for all
frequencies
(C) Provide linear phase shift that is
proportional to frequency
(D) Provide a phase shift that is inversely
proportional to frequency
AA [GATE - EE - 2007]
(58) G(z) = αz βz is allow pass digital filter
with a phase characteristic same as that of the
above question if
(A) α β
(B) α β
1
for k > 0 is
z 1
1
(A) ( 1) k ( k )
(B) ( k ) ( 1) k u(k)
3
(C) α β (1/3)
k
(C) ( 1) u ( k )
(D) u ( k ) ( 1) k
(55) The
has the impulse response h[n] defined by
these two signals as
AB [GATE - EE - 2006]
discrete-time
signal
3n 2 n
z , where
2n
denotes a transform-pair relationship, is
orthogonal to the signal
x[n] X ( z ) n 0
AA [GATE - EE - 2009]
(59) The z-transform of a signal x[n] is given by
4 z 3 3z 1 2 6 z 2 2 z 3 . It is applied to
a system, with a transfer function H(z) =
3 z 1 2. Let the output be y(n). Which of
the following is true?
(A) y(n) is non causal with finite support
(B) y(n) is causal with infinite support
(C) y(n) = 0; | n | 3
n
2 n
(A) y1[n] Y1 ( z ) n0 z
3
(B) y 2 n Y2 z n 0 5n n z
(D) α β (1/3)
(D) Re[Y ( z)]z e j = Re[Y ( z)]Z e j ;
2n 1
Im[Y ( z )]Z e j = Im[Y ( z )]Z e j ;
π π
(C) y 3 n Y3 z n 2 n z n
(D) y4 [n] Y4 ( z ) 2 z
4
3z 2 1
AB [GATE - EE - 2007]
(56) X(z) = 1 – 3z , Y(z) = 1 2z 2 are Ztransforms of two signals x[n], y[n]
respectively. A linear time invariant system
1
Page 86
AC [GATE - IN - 2004]
1
1
2 3 ,| a |, and
(60) Given X(z) =
1 az 1 1 bz 1
| b | 1 with the ROC specified as
| a || z || b |, x[0] of the corresponding
sequence is given by
TARGATE EDUCATION GATE-(EE/EC)
Topic.7 - Z-Transform
(A)
(C)
1
3
(B)
1
2
(D)
Codes:
5
6
(A) Both A and R are true and R is the
correct explanation of A
1
6
(B) Both A and R are true but R is NOT a
correct explanation of A
AB [IES - EC - 1997]
(61) Given that F(z) and G(z) are the one-sided Z
transforms of discrete time functions f(nT)
and
g(nT),
the
z
transform
of
f (kT ) g (nT kT ) is
k
(C) A is true but R is false
(D) A is false but R is true
AC [IES - EC - 1999]
(64) Consider the following statements regarding
a linear discrete - time system
Given by
H ( z)
n
(A)
f (nT ) g (nT ) z
(B)
f (nT )) z g (nT ) z
(C)
f (kT ) g (nT kT ) z n
(D)
f (nT kT ) g (nT ) z
n
n
n
AC [IES - EC - 1997]
(62) Match List-I (x[n]) with List-II (X(z)) and
select the correct answer using the codes
given below the Lists;
List-I
z2 1
( z 0.5)( z 0.5)
1.
The system is stable.
2.
The initial value h(0) of the impulse
response is - 4 .
3.
The steady - state output is zero for a
sinusoidal discrete time input of
frequency equal to one - fourth the
sampling frequency.
Which of these statements are correct?
List-II
(A) 1, 2 and 3
(B) 1 and 2
(C) 1 and 3
(D) 2 and 3
A. a u[n]
az
1.
( z a )2
B. a n2u[n 2]
2.
ze j
ze j a
C. e jn a n
3.
z
z a
1
(B) y (n) [3x(n) 2 x(n 1) x(n 2)]
6
D. na nu[n]
4.
z 1
za
1
(C) y (n ) [ x( n ) 2 x( n 1) 3x(n 2)]
6
n
AD [IES - EC - 2001]
(65) Which one of the following digital filters
does have a linear phase response?
(A) y ( n ) y ( n 1) x ( n ) x ( n 1)
Codes:
A
B
C
D
(A) 3
2
4
1
(B) 2
3
4
1
(C) 3
4
2
1
(D) 1
4
2
3
1
(D) y (n) [ x(n) 2 x(n 1) x(n 2)]
4
AA [IES - EC – 2010/2011]
(66) Unit step response of the system described
by the equation y ( n) y (n 1) x (n) , is
AC [IES - EC - 1998]
(63) Assertion (A) : A linear time-invariant
discrete-time system having the system
function
H(z) =
z
z
1
2
is a stable system.
Reason (R) : The pole of H(z) is in the lefthalf plane for a stable system.
www.targate.org
Z2
(A)
(Z 1)(Z 1)
(B)
Z
( Z 1)( Z 1)
(C)
Z 1
Z 1
(D)
Z ( Z 1)
( Z 1)
Page 87
SIGNAL & SYSTEM
AC [IES - EC - 2008]
1
(67) If X(z) is
with | z | 1, then what is
1 z 1
the corresponding x ( n) ?
then the transfer function of the second
system would be
(A) H2 ( z)
z2 z 3
1 0.5z 1
AA [IES - EC - 2012]
(B) H 2 ( z )
z 2 0.8 z 3
1 0.5 z 1
Z-transform approach is used to analyze the
discrete time systems and is also called as
pulse transfer function approach.
(C) H 2 ( z )
z 1 0.2 z 3
1 0.4 z 1
Statement (II) :
(D) H 2 ( z )
z 2 0.8 z 3
1 0.5 z 1
(A) e n
(B) en
(C) u( n)
(D) (n)
(68) Statement (I) :
The sampled signal is assumed to be a train
of impulses whose strengths, or areas, are
equal to the continuous time signal at the
sampling instants.
(A) Both statement (I) and Statement (II) are
individually true and Statement (II) is
the correct explanation of Statement (I)
(B) Both Statement (I) and Statement (II)
are individually true but Statement (II)
is not the correct explanation of
Statement (I)
AD [IES - EC - 2000]
(71) Match List - I with List -II and select the
correct answer using the codes given below
the Lists:
List - I
{x(n)}
A. n u( n)
B. nu(n 1)
(C) Statement (I) is true but Statement (II) is
false
C. n n u(n 1)
(D) Statement (I) is false but Statement (II)
is true
D. n n u(n)
AB [IES - EC - 2000]
(69) The impulse response of a discrete system
with a simple pole is shown in Fig.
The pole of the system must be located on
the
(A) real axis at z = -1
List - II
{X(z)}
1.
z 1
1 z1
2.
1
1 z1
ROC : z
3.
1
1 z1
ROC : z
4.
z 1
1 2
1 z
ROC : z
ROC : z
(B) real axis between z = 0 and z = 1
Codes:
(C) imaginary axis at z = j
A
(A) 2
B
4
C
3
D
1
(B) 1
3
4
2
(C) 1
4
3
2
(D) 2
3
4
1
(D) imaginary axis between z = 0 and z = j
AD [IES - EC - 2000]
(70) Consider the compound system shown in
Fig. Its output is equal to input with a delay
of two units. If the transfer function of the
z 0.5
first system is given by H1 ( z)
,
z 0.8
Page 88
AC [IES - EC - 2001]
(72) The discrete time system described by y(n) =
x(n2) is
TARGATE EDUCATION GATE-(EE/EC)
Topic.7 - Z-Transform
(A) causal, linear and time-varying
(B) causal, non-linear and time-varying
(C) non-causal, linear and time-variant
(D) non-causal, non-linear ad time-variant
AA [IES - EC - 2001]
(73) The poles of a digital filter with linear phase
response can lie
3.
| z |
1
1
and | z |
3
2
4.
| z |
1
2
Codes:
A
B
C
D
(A) only at z = 0
(A)
4
2
1
3
(B) only on the unit circle
(B)
1
3
4
2
(C) only inside the unit circle but not at z = 0
(C)
4
3
1
2
(D)
1
2
4
3
(D) on the left side of Real (z) = 0 line
AB [IES - EC - 2000]
(74) The minimum number of delay elements
required for realizing a digital filter with the
transfer
function
1 az 1 bz 2
is
H ( z)
1 cz 1 dz 2 ez 3
(A) 2
(B) 3
(C) 4
(D) 5
AA [IES - EC - 2002]
(76) Assertion(A): The signals a n u ( n ) and
a n u (n 1) have the same Z-transform,
Z
.
Z a
Reason(R): For Region of Convergence
(ROC) for a n u ( n ) is | Z || a | , whereas the
AC [IES - EC - 2002]
ROC for a n u (n 1) is | Z || a | .
(75) For a Z-transform
Codes:
5
z 2z
6
X ( z)
1
1
z z
2
3
(A) Both A and R are true and R is the
correct explanation of A
(B) Both A and R are true but R is not a
correct explanation of A
Match List-I with List-II and select the
correct answer using codes given below the
lists:
List I
1 n 1 n
u ( n)
2 3
n
B.
(D) A is false but R is true
AA [IES - EC - 2002]
(77) The range of values of a and b for which the
linear time invariant system with impulse
response
(The sequences)
A.
(C) A is true but R is false
h( n ) a n , n 0
n
1
1
u ( n ) u ( n 1)
2
3
n
bn , n 0
n
1
1
u ( n 1) u ( n)
2
3
n
n
1 1
D. u ( n 1)
2 3
List II
(The region of convergence)
Will be stable is
C.
1.
1
1
| z |
3
2
2.
1
| z |
3
(A) | a | 1,| b | 1
(B) | a | 1,| b | 1
(C) | a | 1,| b | 1
(D) | a | 1,| b | 1
AB [IES - EC - 2005]
(78) Match List I (Discrete Time Signal) with List
II (Transform) and select the correct answer
using code given below the Lists:
www.targate.org
Page 89
SIGNAL & SYSTEM
List I
List II
(Discrete
Time Signal)
(Transform)
(A) n 1 and n 7
(A)
Unit
step
function
(B)
Unit impulse 2.
function
(C)
1.
sin t , t = 0,
1
cos t , t = 0,
(B) n 4 and n 2
z cos
2
z 2 z cos T +1
3.
z
z 1
4.
z sin T
z 2 z cos
T, 2T
(D)
T, 2T
2
Codes:
A
B
C
D
(A)
2
4
1
3
(B)
3
1
4
2
(C)
2
1
4
3
(D)
3
4
1
2
AA [IES - EC - 2005]
(79) Which one of the following is the inverse zz
transform of X(z) =
| z | 2?
( z 2)( z 3)
n
(D) n 2 and n 4
AC [IES - EC - 2007]
(82) What is the Z-transform of the signal
x[n ] α n u (n)?
1
z 1
1
(B) X ( z )
1 z
z
(C) X ( z )
zα
1
(D) X ( z )
zα
(A) X ( z )
AB [IES - EC - 2007]
(83) Algebraic expression for Z transform of x[n]
is X(z). What is the algebraic expression for
Z transform of e j0 n x[n]
(B) X e
(A) X ( z z0 )
j0
(D) X z e j
z
0
AB [IES - EC - 2010]
(84) Z and Laplace transform are related by
(B) 3 n 2 n u ( n 1)
(C) 2 n 3n u ( n 1)
(D) 2 n 3 n u ( n )
AB [IES - EC - 2006]
(80) Which one of the following is the correct
statement ?
The region of convergence of z transform
x | n | consists of the value of z for which
x|n|r
(C) n 6 and n 0
j
(C) X e 0 z
n
(A) 2 3 u ( n 1)
n
Determine the value of n for which
x[n 2] is guaranteed to be zero.
is
(A) s ln z
(B) s
ln z
T
(C) s z
(D) s
T
ln z
AB [IES - EC - 2010]
(85) Convolution of two sequences X1[n] and
X2[n] is represented as
(A) X1(z)*X2(z)
(B) X1(z).X2(z)
(C) X1(z) + X2(z)
(D) X1(z)/X2(z)
(A) absolutely integrable
AA [IES - EC - 2012]
(86) The step response of a discrete time system
with transfer function
(B) absolutely summable
(C) unity
(D) < 1
AC [IES - EC - 2006]
(81)
H ( z)
x[n] is defined as
0, for n 2 or n 4
x[n]
1, otherwise
Page 90
(A)
10
is given by
( Z 1)( Z 2)
10 10 10
n (2)n
9
3
9
(B) 5
n
( 2)n
2
TARGATE EDUCATION GATE-(EE/EC)
Topic.7 - Z-Transform
(C)
The region of convergence (ROC) of the ztransform of x[n] is
(A) | z | | a |
(B) | z | | b |
7 5
n (3)n
9 3
(D) 2 5(1 2n )
(C) | z | | a |
AB [IES - EC - 2012]
(87) The Z-transform corresponding to the
Laplace transform function
G ( s)
10
is :
s ( s 5)
(A)
2 Ze 5 z
( Z 1)( Z e T )
(B)
2(1 e ST ) z
( z 1)( z e ST )
(C)
e 5T
( Z 1) 2
(D)
e T
Z ( Z e 3T )
Z 0.32
Z 2 Z 0.16
(B)
1
Z 2 Z 0.16
| a | | z | | b |
S4AD [GATE – EC – 2016]
(91) The ROC (region of convergence) of the ztransform of a discrete-time signal is
represented by the shaded region in the zplane.
If
the
signal
x[ n ] (2.0) |n | , n , then the ROC of
its z-transform is represented by
(A)
AA [IES - EC - 2012]
(88) The difference equation for a system is given
by y(n+2) + y(n+1) + 0.16 y(n) = x(n+1)
0.32 x(n). The transfer function of the system
is
(A)
(D)
(B)
(C) Z2 0.32
Z 0.16
(D)
Z 0.32
( Z 1)( Z 2 Z 0.16)
AA[GATE-IN-2014]
(89) The impulse response of an LTI system is
given as
(C)
c
n0
h n
sin c n n n 0
n
It represents an ideal
(A) non-causal, low-pass filter
(B) causal, low-pass filter
(D)
(C) non-causal high-pass filter
(D) causal, high-pass filter
S1AB [GATE – EC – 2016]
the
sequence
x [ n ] a n u [ n ] b n u [ n ] , where u[ n ] denotes
the unit-step sequence and 0 < |a| < |b| < 1.
(90) Consider
www.targate.org
Page 91
SIGNAL & SYSTEM
S4AC [GATE – EC – 2016]
(92) The direct form structure of an FIR (finite
impulse response) filter is shown in the
figure.
The filter can be used to approximate a
(A) low-pass filter
(B) high-pass filter
The Z-transform of the convoluted sequence
x 1 n * x 2 n is
(A) 1 2z 1 3z 2
(B) Z2 3Z 2
(C) 1 3Z1 2Z2
(D) z 2 3z 3 2z 4
AA [GATE-EC-2019]
(97) Let H(z) be the z-transform of a real-valued
discrete-time
signal
h[n].
If
1 1
1
P( z ) H ( z ) H has a zero at z j
2 2
z
, and P ( z ) has a total of four zeros, which
one of the following plots represents all the
zeros correctly?
(A)
(C) band-pass filter
(D) band-stop filter
S4AC [GATE – EC – 2016]
(93) A discrete-time signal
x[ n] [n 3] 2[ n 5] has z-transform
X(z). If Y ( z) X ( z) is the z-transform of
another signal y[n], then
(A) y[n] = x[n]
(B) y ( n) x ( n)
(C) y ( n ) x[ n ]
(D) y (n) x[ n]
A0.09 TO 0.1 [GATE–S1–EE–2017]
(94) Consider a causal and stable LTI system with
rational transfer function H(z), whose
corresponding impulse response begins at n
5
= 0. Furthermore, H(1) . The poles of
4
2k 1
1
H(z) are pk
exp j
for k =
4
2
1,2,3,4. The zeros of H(z) are all at z = 0. Let
g n jn h n . The value of g[8]
equals_________. (Give the answer up to
three decimal places.)
(B)
AC [GATE – IN – 2017]
(95) The region of Convergenec(ROC) of the Ztransform of a causal unit step discrete-time
sequence is
(A) z 1
(B) z 1
(C) z 1
(D) z 1
AC [GATE – IN – 2017]
(96) Consider two discrete-time signals:
x 1 n 1,1 and x 2 n 1, 2 , for n = 0, 1
Page 92
TARGATE EDUCATION GATE-(EE/EC)
Topic.7 - Z-Transform
(C)
(D)
-------0000-------
www.targate.org
Page 93
08
DFS/DTFT/DFT/FFT
(1)
AA [GATE – EC2 – 2015]
The magnitude and phase of the complex
Fourier series coefficients ak of a periodic
signal x(t) are shown in the figure. Choose
the correct statement from the four choices
given. Notation C is the set of complex
numbers, R is the set of purely real numbers,
and P is the set of purely imaginary numbers.
(4)
AB [GATE – EC4 – 2014]
DFT X of a sequence
x n ,0 n N 1
is
given
by
The N-point
X k
N 1
1
x n e
N
j
2
nk
N
, 0 k N 1 .
n 0
Denote this relation as X = DFT(x). For N =
4, which one of the following sequences
satisfies
DFT (DFT (x)) = x?
(A) x 1 2 3 4
(B) x 1 2 3 2
(C) x 1 3 2 2
(D) x 1 2 2 3
AD [GATE – EC – 2005]
n
(5)
(A) x(t) R
(B) x(t) P
(3)
(D) the information given is not sufficient to
draw any conclusion about x(t)
Then Y (e j 0 ) is
AA [GATE-IN-2014]
( )is the Discrete Fourier Transform of a 6point real sequence ( ).
(A)
If (0)= 9 + 0, (2)= 2 + 2, (3)= 3 – 0,
(5)= 1 – 1, (0) is
(C) 4
(A) 3
(B) 9
(C)15
(D)18
A9.99to10.01 [GATE – EC1 – 2014]
Consider a discrete time periodic signal x[n]
n
. Let ak be the complex Fourier
5
(6)
1
4
(B) 2
(D)
4
3
AD [GATE – EC – 2009]
The 4-point Discrete Fourier Transform
(DFT) of a discrete time sequence {1, 0, 2,
3} is :
(A) [0, -2+2j, 2, -2, -2j]
= sin
(B) [2, 2+2j, 6, 2-2j]
series coefficients of x[n]. The coefficients
{ak} are non-zero when k = Bm ± 1, where m
is any integer. The value of B is --------
(C) [6, 1-3j, 2, 1+3j]
Page 94
y ( n) x 2 ( n).
And Y (e j ) be transform of y (n).
(C) x(t) (C R)
(2)
1
u (n) &
2
Let x ( n)
(D) [6, -1+3j, 0, -1-3j]
TARGATE EDUCATION GATE-(EE/EC)
Topic.8 – DFS/DTFT/DFT/FFT
Common Data for next two Questions
(D)
A sequence x[n] has non-zero values as shown in
figure
(8)
AC [GATE – EC – 2005]
The Fourier transform of y (2n) will be
AA [GATE – EC – 2005]
(7)
(A) e
The sequence
n
x 1
y (n) 2
0
[cos 4 2cos 2 2]
(B) [cos2 2cos 2]
for n even
(C) e j[cos 2 2cos 2]
for n odd
(D) e
will be :
(A)
j 2
(9)
j
2
cos 2 2 cos 2
AB [GATE – EC – 2007]
A 5-point sequence x[n] is given as
x[–3] = 1, x[–2] = 1, x[–1] = 0, x[0] = 5, x[1]
= 1.
Let X (e j ) denote the discrete – time
Fourier transform of x[n]. The value of
X e d is
j
(B)
(A) 5
(B) 10
(C) 16
(D) 5 + j 10
AD [GATE – EC – 2010]
(10) For an N-point FFT algorithm with N = 2m ,
which one of the following statements is
TRUE?
(A) It is not possible to construct a signal
flow graph with both input and output in
normal order.
(B) The number of butterflies in the mth
stage is N/m
(C)
(C) In – place computation requires storage
of only 2N node data
(D) Computation of a butterfly requires only
one complex multiplication
AA [GATE – EC – 2008]
(11) x (n)} is a real-valued periodic sequence with
a period N x (n) and X (k) form N-point
Discrete Fourier Transform (DFT) pairs. The
DFT Y (k) of the sequence
www.targate.org
Page 95
SIGNAL & SYSTEM
y (n) =
1
N
N 1
x (r ) x (n r )
is
r 0
(A) | X ( k ) |2
(B)
(C)
1
N
1
N
A2.05-2.15 [GATE–S1–EC–2017]
(16) Let h[n] be the impulse response of a
discrete-time linear time invariant(LTI) filter.
The impulse response is given by
1
1
1
h 0 ;h 1 ;h 2 ; and h n 0
3
3
3
for n 0 and n 2
N 1
X (r ) X * (k r )
r 0
N 1
X (r ) X (k r )
r 0
(D) 0
AB [GATE – EC – 2011]
(12) The first six points of the 8-point DFT of a
real valued sequence are 5, 1-j3, 0, 3-j4, 0
and 3 + j4. The last two points of the DFT
are respectively
(A) 0, 1 – j3
(B) 0, 1 + j3
(C) 1 + j3, 5
(D) 1 – j3, 5
S1A7.9-8.1 [GATE – EC – 2016]
(13) Consider the signal
Let H be the discrete-time Fourier
transform (DTFT) of h[n], where is the
normalized angular frequency in radians.
Given that H 0 0 and 0 0 , the
value of 0 (in radians) is equal to _______.
AB [GATE – IN – 2017]
(17) Three DFT coefficients, out of the DFT
coefficients of a five-point real sequence are
given as:
and
X 0 4, X 1 1 j1
X 3 2 j2 . The zeroth value of the
sequence x(n), x(0)
x[ n] 6[ n 2] 3[ n 1] 8[n] 7 [ n 1]
(A) 1
(B) 2
4[ n 2]
(C) 3
(D) 4
If X ( e j ) is the discrete-time Fourier
transform of x[n] ,
1
X (e j )sin 2 (2) d is equal to
_________.
then
S4A4096 [GATE – EC – 2016]
(14) A continuous-time speech signal xa(t) is
sampled at a rate of 8 kHz and the samples
are subsequently grouped in blocks, each of
size N. The DFT of each block is to be
computed in real time using the radix-2
decimation-in-frequency FFT algorithm. If
the processor performs all operations
sequentially, and takes 20 µs for computing
each complex multiplication (including
multiplications by 1 and −1) and the time
required
for
addition/subtraction
is
negligible, then the maximum value of N is
_______
S3A5.9-6.1 [GATE – EC – 2016]
(15) The Discrete Fourier Transform (DFT) of the
4-point sequence
x[n] = { x[0], x[1], x[2], x[3]} = {3, 2, 3, 4} is
A2.90-3.10 [GATE – EC – 2018]
(18) Let X [ k ] k 1, 0 k 7 be 8-point DFT of
a sequence x[ n] ,
N 1
where X [ k ] n 0 x[n ] e j 2 nk / N .
The value (correct to two decimal places) of
3
n0 x[2n] is ______.
AC [GATE – IN – 2018]
(19) For the sequence x[ n] {1, 1,1, 1} , with
n 0,1, 2,3 the DFT is computed as
3
j
2
nk
X (k ) n0 x[n]e 4 , for k 0,1, 2,3 .
The value of k for which X(k) is not zero is
(A) 0
(B) 1
(C) 2
(D) 3
A–27.01 to –26.99 [GATE-EC-2019]
(20) Let h[n] be a length-7 discrete-time finite
impulse response filter, given by
h[0] =4, h[1] = 3,
h[2] = 2,
h[3] = 1,
h[-1] = -3, h[-2] = -2, h[-3] = -1,
X[k] = {X[0], X[1], X[2], X[3]} = {12, 2j, 0,
−2j}.
and h[n] is zero for | n | 4 . A length-3 finite
impulse response approximation g[n] of h[n]
has to be obtained such that
If X1[k] is the DFT of the 12-point sequence
x1[n] = {3, 0, 0, 2, 0, 0, 3, 0, 0, 4, 0, 0}, the
E ( h, g ) H (e j ) G e j d
X1[8]
value of
is ________
X1[11]
is minimized, where H (e j ) and G(e j ) are
the discrete-time Fourier transforms of h[n]
Page 96
2
TARGATE EDUCATION GATE-(EE/EC)
Topic.8 – DFS/DTFT/DFT/FFT
and g[n], respectively. For the filter that
minimizes E(h, g), the value of 10g[-1] +
g[1], rounded off to 2 decimal places, is
______.
AA [GATE-EC-2019]
(21) Consider a six-point decimation-in-time Fast
Fourier Transform (FFT) algorithm, for
which the signal-flow graph corresponding to
X[1] is shown in the figure. Let
j2π
W6 = exp
. In the figure, what should
6
be the values of the coefficients a1,a2,a3 in
terms of W6 so that X[1] is obtained
correctly?
(A) a1 =1,a 2 = W6 ,a 3 = W62
(B) a1 = 1,a 2 = W62 ,a 3 = W6
(C) a1 =1,a 2 = W62 ,a 3 = W6
(D) a1 = 1,a 2 = W6 ,a 3 = W62
-------0000-------
www.targate.org
Page 97
09
Random Variable
(1)
(D) Cos 2 t
a bx
f (x)
0
1
for 0 x 1
otherwise
(4)
If the expected value E[X] = 2/3, then Pr[X <
0.5] is ___________
(2)
2
AA [IES - EC - 1997]
The autocorrelation function Rx ( ) of the
signal x(t) = V sin ωt is given by
(A) (1/ 2)V 2 cos
any conclusion about x(t)
(B) V 2 cos
AD [GATE – EC1 – 2015]
1
A source emits bit 0 with probability
and
3
2
bit 1 with probability . The emitted bits are
3
communicated to the receiver. The receiver
decides for either 0 or 1 based on the
received value R. It is given that the
conditional density functions of R are as
(C) V 2 cos2
(D) 2V 2 cos2
(5)
AA [IES - EC - 2002]
Two independent signals X and Y are known
to be Gaussian with mean values x0 and y0
2
and variances x and y2 . A signal Z = X –
Y is obtained from them. The mean z0 ,
2
1
, 3 x 1,
and
f R|0 (r) 4
0 otherwise,
variance z and p.d.f p( z ) of the signal Z
are given by
1
, 1 x 5,
f R||0 (r) 6
0 otherwise,
(B) x0 y0 , x2 y2 , Rayleigh
(A) x 0 y0 , x2 2y , Gaussian
(C) y0 x0 , y2 x2 , uniform
(D) x0 y0 , x2 y2 , Gaussian
The minimum decision error probability is
(3)
t
(C) Sin 2 t1 t2
A0.25 [GATE – EE1 – 2015]
A random variable X has probability density
function f(x) as given below :
(A) 0
(B) 1/12
(C) 1/9
(D) 1/6
AD [GATE – EC1 – 2014]
Consider a random process X (t) =
(6)
AB [IES - EC - 2006]
For random variable x having the probability
density function (PDF) as shown in the figure
below, what are the values of the mean and
the variance, respectively?
2 sin(2 t ) where the random phase
is uniformly distributed in the interval
[0,2 ] . The auto correlation E [X (t1) X
(t2)] is
(A) Cos 2 t1 t2
(B) Sin 2 t1 t2
Page 98
1
2
and
2
3
2
(C) 1 and
3
(A)
TARGATE EDUCATION GATE-(EE/EC)
4
3
4
(D) 2 and
3
(B) 1 and
Topic.9 – Random Variable
(7)
AA [IES - EC - 2008]
What is the spectral density of white noise?
(A)
(A) A constant
(B) ( )
2
(C) [()]
(D) A step function in
(8)
(B)
AB [IES - EC - 2010]
If a random process X(t) is ergodic then,
statistical averages
(A) and time averages are different
(C)
(B) and time averages are same
(C) are greater than time averages
(D) are smaller than time averages
(9)
AB [IES - EC - 1997]
The autocorrelation function satisfies which
one of the following properties?
(D)
(A) Rx ( ) Rx ( )
(B) Rx ( ) Rx ( )
(C) Rx ( ) Rx (0)
(12)
(D) Rx ( ) 1
A3.9to4.1 [GATE – EC2 – 2014]
The power spectral density of a real
stationary random process X (t) is given by
1 f
, W
S X f W
.
0, f W
A0.25 [GATE – EC3 – 2015]
(10) A random binary wave y(t) is given by
y(t)
The value of the expectation
X p(t nT )
n
n
where p(t) = u(t) – u(t – T), u(t) is the unit
step function and is an independent
random variable with uniform distribution in
[0, T]. The sequence {Xn} consists of
independent and dentically distributed binary
valued random variables with P{Xn = +1}=
P{Xn = –1} = 0.5 for each n.
AA [IES-EC-2013]
the power spectral density is,
R ( ) e j df and the auto correlation
2
function is defined by
value
of
the
autocorrelation
3T
3T
equals ____.
R yy
E y(t)y t
4
4
The integral on the right represents the
Fourier transform of
(A) Delta Function
AB [GATE – EC2 – 2015]
n
Xn n is
(B) Step function
an independent and distributed
(C) Ramp function
(i.i.d) random process with Xn equally likely
(D) Sinusoidal function
n
to be +1 or bute Yn n is another random
process obtained as Yn Xn 0.5Xn1 . The
autocorrelation
function
of
1
is ----------.
4W
(13) If
The
(11)
E X t X t
(14)
n
Yn n ,
denoted by RY [K] , is :
www.targate.org
AB [GATE – EC1 – 2014]
Let X be a real – valued random variable
with E [X] and E [X2] denoting the mean
values of X and X2, respectively. The
relation which always holds true is
Page 99
SIGNAL & SYSTEM
(A) (E[X]) 2 > E[X2]
(A)
A
( j 1)
(B)
(C)
A
( 1)
(D)
(B) E [X2] > (E[X]) 2
(C) E [X2] = (E[X]) 2
(D) E[X2] > (E[X]) 2
AB [GATE-EC-2012]
(15) The power spectral density of a real process
X(t) for positive frequencies is shown below.
The values of E[ X 2 (t )] and E[ X (t )] ,
respectively, are
2
A
( j 1)2
A
( j 1)
AA [IES - EC - 2002]
(19) The units of the spectrum obtained by
Fourier transforming the covariance function
of a stationary stochastic process is
(A) Power per Hertz
(B) energy per Hertz
(C) Power per second
(D) Energy per second
AB [IES - EC - 2002]
(20) If the cumulative distribution function is
Fx ( x ), then the probability density function
f x ( x ) is given as
(A) 6000/ π , 0
(B) 6400 / π , 0
(C) 6400 / π, 20 / (π 2)
(D) 6400 / π, 20 / (π 2)
AA [IES - EC - 1998]
(16) The spectral density of a random signal is
given by [ ( 0 ) ( 0 )] . The
auto-correlation function of the signal is
(A) cos
0
Fx ( x ) dx
(B)
d
Fx ( x)
dx
(C)
Fx ( x) dx
(D)
d
Fx ( x)
dx
AB [IES - EC - 2004]
(21) Which one of the following gives the average
value or expectation of the function g ( X ) of
the random variable X?
{Given f ( X ) is the probability density
function)
(A) E g ( X )
(B) sin 0
(C) cos[( 0 ) ]
(D) sin[( 0 ) ]
AB [IES - EC - 1998]
(17) The auto correlation of a wide-sense
stationary random process is given by e
.The peak value of the spectral density is
(A) 2
(B) 1
(C) e 1/ 2
(D) e
2
AC [IES - EC - 2000]
(18) A linear system has the transfer function
1
.When it is subjected to an
H ( j )
( j 1)
input white noise process with a constant
spectral density 'A', the spectral density of
the output will be :
Page 100
(A)
(B) E g ( X )
(C) E g ( X )
(D) E g ( X )
g ( X )dX
g ( X ) f ( X )dX
g * ( X )dX
g(X )
f ( X ) dX
AB [IES - EC - 2005]
(22) The auto-correlation function R x ( τ ) of a
random process has the property that Rx (0)
is equal to
(A) Square of the mean value of the process
(B) Mean squared value of the process
(C) mean squared value of the process
(D)
1
R x R x
2
TARGATE EDUCATION GATE-(EE/EC)
Topic.9 – Random Variable
AD [IES - EC - 2007]
(23) If a linear time invariant system is excited by
a true random signal like white noise, the
output of the linear system will have which
of the following properties?
AC [IES - EC - 2012]
(28) A random variable is known to have a
cumulative
distribution
function
2
x
Fx ( x ) U ( x ) 1 its density function is
b
(A) Output will be a white noise
(B) Output will be periodic
(A) U ( x )
2
2x
(1 e x /b )
b
(B) U ( x )
2 x x2 / b
e
b
(C) Output will not be random
(D) Output will be correlated or coloured
noise
x2
U
(
x
)
1
( x)
(C)
b
AC [IES - EC - 2007]
(24) Which of the following is/are not a
property/properties of a power spectral
density functions S x ()?
x2
x2 /b
1
( x) e
(D)
b
(A) S x () is a real function of
(B) S x () is an even function of
(C) S x () is a non-positive function of i.e.
S x () 0 for all
AC[GATE-EE-2014]
(29) An input signal x(t) = 2+ 5 sin 100 t is
sampled with a sampling frequency of 400
Hz and applied to the system whose transfer
function is represented by
(D) All of these
Y z
AB [IES - EC - 2008]
(25) Let x (n) be a real-valued sequence that is a
sample sequence of a wide-sense stationary
discrete-time random process. The power
density function of this signal is
X z
1 1 zN
N 1 z 1
Where, N represents the number of samples
pre cycle. The output y(n) of the system
under steady state is
(A) Real, odd and non-negative
(A) 0
(B) 1
(B) Real, even and non-negative
(C) 2
(D) 5
(C) Purely imaginary, even and negative
(D) Purely imaginary, odd and negative
AA [IES - EC - 2008]
(26) A random variable X is defined by the
double exponential distribution
Px ( x) aeb|x| , x
AA [GATE–S1–EC–2017]
(30) Let X(t) be a wide sense stationary random
process with the power spectral density
SX f as shown in figure(a), where f is in
Hertz(Hz). The random process X(t) is input
to an ideal lowpass filter with the frequency
response
Where a and b are +ve constants.
What is the relation between a and b so that
px ( x ) is a probability density function?
(A) a b / 2
(B) b a / 2
(C) a b
(D) a 1 / b
1,
Hf
0,
1
f Hz
2
1
f Hz
2
As shown in figure (b). The output of the
lowpass filter is Y(t)
AA [IES - EC - 2010]
(27) If random process X(t) and Y(t) are
orthogonal, then
(A) S XY ( f ) 0
(B) S XY ( f ) S X ( f ) SY ( f )
(C) RXY ( τ ) h( τ )
(D) H ( f ) 0
(a)
www.targate.org
Page 101
SIGNAL & SYSTEM
(b)
Let E be the expectation operator and
consider the following statements:
I.
E X t E Y t
II.
E X2 t E Y2 t
III. E Y 2 t 2
Select the correct option:
(A) only I is true
(B) Only II and III are true
(C) only I and II are true
(D) Only I and III are true
A2 [GATE–S2–EC–2017]
(31) Consider the random process
X (t ) U Vt ,
where U is a zero-mean Gaussian random
variable and V is a random variable
uniformly distributed between 0 and 2.
Assume that U and V are statistically
independent. The mean value of the random
process at t = 2 is _____ .
AB [GATE – EC – 2018]
(32) Consider a white Gaussian noise process
N (t ) with two-sided power spectral density
S N ( f ) 0.5 W/Hz as input to a filter with
2
impulse response 0.5e t / 2 (where
is in
seconds) resulting in output ( ). The power
in ( ) in watts is
(A) 0.11
(B) 0.22
(C) 0.33
(D) 0.44
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Page 102
TARGATE EDUCATION GATE-(EE/EC)
10
MISCELLANEOUS
(1)
AA [GATE – EC2 – 2015]
The signal cos 10t is ideally sampled
4
at a sampling frequency of 15 Hz. The
sampled signal is passed through a filter with
sin( t)
impulse response
cos 40t .
2
t
The filter output is
(A)
15
cos 40 t
2
4
(B)
15 sin( t)
cos 10t
2 t
4
(C)
(D)
(2)
1
n
2
n
(C) 5 u[n] 5 u[n]
3
3
n
n
(D) 5 2 u[n] 5 1 u[n]
3
3
(3)
A1.5 [GATE – EC3 – 2015]
Two sequences x1[n] and x2[n] have the same
energy. Suppose x1[n] 0.5n u[n], where
is a positive real number and u[n] is the
unit
step
sequence.
Assume
1.5 for n 0,1
x 2 [n]
other wise.
0
Then the value of is ______.
15
cos 10t
2
4
(4)
AA [GATE – EC3 – 2015]
Consider a four-point moving average filter
3
defined by the equation y[n] i x[n i] .
15 sin( t)
cos 40t
2 t
2
i 0
The condition on the filter coefficients that
results in a null at zero frequency is
AC [GATE – EC3 – 2015]
A realization of a stable discrete time system
is shown in the figure. If the system is
excited by a unit step sequence input x[n],
the response y[n] is
(A) 1 2 0; 0 3
(B) 1 2 1; 0 3
(C) 0 3 0; 1 2
(D) 1 2 0; 0 3
(5)
AC [GATE – EC – 1999]
The input to a channel is a band pass signal.
It is obtained by linearly modulating a
sinusoidal carrier with a single – tone signal.
The output of the channel due to this input is
given by y(t) = (1/100) cos(100t 10 6 )
cos(106 t 1.56). The group delay (t g ) and
1
n
2
n
(A) 4 u[n] 5 u[n]
3
3
n
the phase delay ( t p ), in seconds, of the
channel are
(A) t g 10 6 , t p 1.56
n
2
1
(B) 5 u[n] 3 u[n]
3
3
(B) t g 1.56, t p 10 6
www.targate.org
Page 103
SIGNAL & SYSTEM
(C) t g 10 8 , t p 1.56 10 6
(D) t g 10 8 , t p 1.56
(C)
0
d ()
,
| 0
(0 )
d
0
(6)
(7)
AC [GATE – IN – 2015]
The filter whose transfer function is of the
s 2 bs c
form G(s) 2
is :
s bs c
(A) a high-pass filter
(B) a low-pass filter
(C) an all-pass filter
(D) a band-reject filter
AB [GATE – EC – 2004]
(10) Consider the signal x(t) shown in Fig. 1. Let
h(t) denote the impulse response of the filter
matched to x(t), with h(t) being non-zero
only in the interval 0 to 4 sec. The slope of
h(t) in the interval 3 < t < 4 sec is
AA [GATE – EC – 2006]
In the system shown below, x (t) = (sin t) u
(t). In steady-state, the response y (t) will be
1
sin t
2
4
1
(B)
sin t
4
2
1 1
(C)
e sin(t )
2
(D) sin(t ) cos(t )
(A)
(8)
(D) 0 (0 ), ( )d
(A) ½
sec1
(C) 1/ 2
(B) 1
sec1
(D) 1
sec1
sec1
AD [GATE – EC – 1998]
(11) The ACF of a rectangular pulse of duration T
is
(A) a rectangular pulse of duration T
AC [GATE – EC – 1991]
The pole-zero pattern of a certain filter is
shown in Fig. 1. The filter must be of the
following type.
(B) a rectangular pulse of duration 2 T
(C) a triangular pulse of duration T
(D) a triangular pulse of duration 2T
AA [GATE – EC – 2004]
(12) Consider the sequence
x n 4 j5 1 j2 4
The conjugate anti-symmetric part of the
sequence is
(A) low-pass
(C) all-pass
(9)
(B) high-pass
(D) band-pass
AA [GATE – EC – 2000]
A system has a phase response given by
() where is the angular frequency.
The phase delay and group delay at 0
are respectively given by
(A)
( 0 ) d ( )
,
| 0
0
d
(B) ( 0 ),
Page 104
d 2 ( )
| 0
d 2
(A) [4 j 2.5
j2 4 j2.5]
(B) [ j 2.5 1
j 2.5]
(C) [ j5
j 2 0]
(D) [4 1 4]
AC [GATE – EC3 – 2015]
(13) The complex envelope of the bandpass signal
sin(t / 5)
x(t) 2
sin t ,
4
t / 5
1
centered about f Hz , is
2
sin( t / 5) j 4
(A)
e
t / 5
TARGATE EDUCATION GATE-(EE/EC)
Topic.10 - Miscellaneous
AB [GATE – EC2 – 2014]
(18) Let x [n] = x [-n]. Let X (z) is the z-transform
of x [n]. If 0.5 + j 0.25 is a zero of X (z)
which one of the following must also be a
zero of X (z).
sin( t / 5) j 4
(B)
e
t / 5
(C)
sin( t / 5) j 4
2
e
t / 5
(D)
sin(t / 5) j 4
2
e
t / 5
(A) 0.5 – j 0.25
(B) 1/ (0.5 + j0.25)
AD [GATE – EE1 – 2015]
(14) The transfer function of a second order real
system with a perfectly flat magnitude
response of unity has a pole at (2 input x[n],
the response y[n] isis p
(C) 1 / (0.5 – j0.25)
(D) 2 + j4
AB [GATE – EC3 – 2014]
1
(19)
For all pass system H
(A) Poles at (2 j3), no zeroes
1
H e j 1, for all . If Re
(B) Poles at ( 2 – j3), one zero at origin
where
(C) Poles at (2 –j3), (–2 + j3), zeroes at (–2–
j3), (2 + j3)
a 0, Im a 0,
(D) Poles at (2 j3), zeroes at (–2 j3)
(A) A
(B) a*
(C) 1/a*
(D) 1/a
AA [GATE – EE2 – 2015]
(15) The z-Transform of a sequence x[n] is given
as
X z 2z 4 4 / z 3 / z 2 . If y[n] is the
first difference of x[n], then Y(z) is given by
(A) rectangular pulse of duration T
(B) rectangular pulse of duration 2T
(C) triangular pulse
6 1 3
(B) 2z 2 2 3
z z z
(C) 2z 2
then b equals
AC [GATE – EC – 1996]
(20) A rectangular pulse of duration T is applied
to a filter matched to this input. The output of
the filter is a
8 7 3
(A) 2z 2 2 3
z z z
(D) sine function
AA [GATE – EC – 2002]
(21) A linear phase channel with phase delay T p
8 7 3
z z 2 z3
and group delay Tg must have
8 1 3
(D) 4z 2 2 3
z z z
(A) Tp Tg constant
AC [GATE – EC2 – 2014]
(16) An FIR system is described by the system
function.
The
system
is
(B) Tp F and Tg f
(C) Tp constant and Tg f
7
3
H Z 1 Z 1 Z 2
2
2
(D) Tp f and Tg constant
(A) Maximum phase
(
(B) minimum phase
(C) Mixed phase
(D) zero phase
(17)
z b ,
z
1 az
A44to46 [GATE – EC1 – 2014]
A continuous linear time-invariant filter has
an impulse response h (t) described by
3 for 0 t 3
ht
0 otherwise
f denotes frequency)
AB [GATE – EC – 2006]
(22) A low-pass filter having a frequency
response H ( j) = A()e j ( does not
produce any phase distortion if
When a constant input of value 5 is applied
to this filter the steady state output is -------.
www.targate.org
(A) A() C2 , () k 3
(B) A( ) C 2 , ( ) k
(C) A() C , () k 2
(D) A( ) C , () k 1
Page 105
SIGNAL & SYSTEM
(23) A
system
AB [GATE – EC – 2010]
with the transfer function
Y ( s)
s
X (s ) s p
has
an
output
y (t ) cos 2t for the input signal
3
x (t ) p cos 2t . Then, the system
2
parameter ‘p’ is
(A)
(C) 1
3
(B)
2
3
(D)
3
2
AC [GATE – EC – 2010]
(24) Consider the pulse shape s (t) as shown. The
impulse response h (t) of the filter matched to
this pulse is
H ( s)
K (s 2 20 )
s 2 (0 / Q)s 20
(A) all pass filter
(B) low pass filter
(C) band pass filter
(D) notch filter
AC [GATE – EC – 1990]
(26) The magnitude and phase functions for a
distortionless filter should respectively be:
(Magnitude)
(Phase)
(A) Linear
Constant
(B) Constant
Constant
(C) Constant
Linear
(D) Linear
Linear
AD [GATE – EC – 1999]
(27) The input to a matched filter is given by
10sin(2106 t ),
s (t )
0,
0 t | 104 sec
otherwise
The peak amplitude of the filter output is
(A) 10 volts
(B) 5 volts
(C) 10 millivolts
(D) 5 millivolts
(A)
AC [GATE – EC – 2002]
(28) In
Fig.
1
s(t) cos200t
m(t)
and
2sin 2t
,
t
sin199t
n(t )
.
t
=
The output y(t) will be
(B)
(C)
(A)
sin 2t
t
(B)
sin 2t sin t
cos3t
t
t
(C)
sin 2t sin 0.5t
cos1.5t
t
t
(D)
sin 2t sin t
cos 0.75t
t
t
Common Data Questions for the Next Two
Questions :
(D)
The system under consideration is an RC low-pass
filter (RC-LPF) with R = 1.0 k and C = 1.0 F.
AD [GATE – EC – 1988]
(25) Specify the filter type if its voltage transfer
function H(s) is given by
Page 106
AC [GATE – EC – 2003]
(29) Let H(f) denote the frequency response of the
RC-LPF. Let f1 be the highest frequency
TARGATE EDUCATION GATE-(EE/EC)
Topic.10 - Miscellaneous
such that 0 | f | f1 ,
| H ( f1 ) |
0.95 . Then
H (0)
(A) 0
(B) 20.25 cos(2t 0.125
f1 (in Hz) is
(A) 327.8
(C) 52.2
(C) 20.5 cos(2t 0.125)
(B) 163.9
(D) 104.4
(D) 20.5 cos(2t 0.25)
AA [GATE – EC – 2003]
(30) Let t g f be the group delay function of the
given RC-LPF and f 2 100 Hz.
AB [GATE – EC – 2005]
(35) A signal x n sin 0 n f is the input to
a linear time-invariant system having a
frequency response H (e j ) . If the output of
Then t g ( f 2 ) in ms, is
the system is Ax( n n0 ), then the most
(A) 0.717
(B) 7.17
general form of H (e j ) will be
(C) 71.7
(D) 4.505
(A) n0 0 for any arbitrary real
AC [GATE – EC – 2004]
(31) A system has poles at 0.01 Hz, 1 Hz and 80
Hz; zeros at 5 Hz. 100 Hz and 200 Hz. The
approximate phase of the system response at
20 Hz is
(A)
900
(B)
00
(C)
900
(D)
1800
(B) n0 0 2k for any arbitrary integer k
(C) n0 0 2k for any arbitrary integer k
(D) n0 0
AA [GATE – EC – 2009]
(36) A system with transfer function H (z) has
impulse response h(n) defined as h(2) 1,
h(3) = -1 and h(k ) 0 otherwise. Consider
the following statements.
AA [GATE – EC – 2007]
(32) The 3 dB bandwidth of the low-pass
signal e t u (t ), where u(t) is the unit step
function, is given by
(A)
1
Hz
2
(B)
1
2
S1 : H z is a low-pass filter.
S 2 : H z is an FIR filter.
Which of the following is correct?
(A) Only S 2 is true
2 1 Hz
(B) Both S1 and S 2 are false
(C)
(C) Both S1 and S 2 are true, and S 2 is a
reason for S1
(D) 1 Hz
Statement for Linked Question for the Next Two
Questions :
(D) Both S1 and S 2 are true, but S 2 is not
a reason for S1
The impulse response h(t) of a linear time-invariant
continuous time system is given by h(t) = exp( 2
t) u(t), where u(t) denotes the unit step function.
Statement for Linked Question for the Next Two
Questions :
AC [GATE – EC – 2008]
(33) The frequency response H () of this
system in terms of angular frequency , is
It is required to design an anti-aliasing filter for an
8 bit ADC. The filter is a first order RC filter with
R = 1 and C = 1F. The ADC is designed to span
a sinusoidal signal with peak to peak amplitude
equal to the full scale range of the ADC.
given by
H () =
(A)
1
1 j 2
(B)
sin()
(C)
1
2 j
(D)
j
2 j
AD [GATE – EC – 2008]
The
output
of
this
system, to the sinusoidal
(34)
input x(t) = 2cos(2t ) for all time t, is
AA [GATE - EE - 2006]
(37) The transfer function of the filter and its roll
of respectively are :
www.targate.org
Page 107
SIGNAL & SYSTEM
(A) 1/(1 + RCs), - 20 dB/decade
(B) (1 + RCs), - 40dB/decade
(C) 1/(1 + RCs), - 40dB/decade
(D) {RCs/(1+RCs)}, -20 dB/decade
AB [GATE - EE - 2006]
(38) What is the SNR (in dB) of the ADC? Also
find the frequency (in decades) at the filter
output at which the filter attenuation just
exceeds the SNR of the ADC.
(A) 50 dB, 2 decades
(B) 50 dB, 2.5 decades
(C) 60 dB, 2 decades
(D) 60 dB, 2.5 decades
AC [GATE – EC – 2005]
(42) The derivative of the symmetric function
drawn in Fig. 1 will look like
(A)
AA [GATE – EC – 2005]
(39) The function x(t) is shown in Fig. Even and
odd parts of a unit-step function u(t) are
respectively,
(B)
1 1
, x(t )
2 2
1 1
(C) , x(t )
2 2
(A)
1 1
x(t )
2 2
1 1
(D) , x(t )
2 2
(B) ,
(C)
AA [GATE – EC – 2006]
(40) A solution for the differential equation
x(t) 2x(t ) (t) with initial condition
x(0) 0 is
(A) e2t u (t )
(B) e 2 t u (t )
(C) e t u (t )
(D) et u (t )
(D)
A(A-1,B-3,C-4) [GATE – EC – 1995]
(41) Match each of the items, A, B and C, with an
appropriate item from 1, 2, 3, 4 and 5.
(A) Fourier transform (1) Gaussian
of
a
Gaussian
function
function
(B) Convolution of a (2) Rectangular
rectangular
pulse
pulse
with itself
(C) Current through an (3) Triangular
inductor for a step
pulse
input voltage
(4) Ramp
function
(5) Zero
Page 108
AC [GATE – EC – 2005]
(43) Match the following and choose the correct
combination.
E-
Continuous
1- Fourier
and a periodic
representation is
signal
continuous and
periodic
F-
Continuous
2- Fourier
and periodic
representation is
signal
discrete and a
periodic
G- Discrete and a 3- Fourier
periodic signal
representation is
TARGATE EDUCATION GATE-(EE/EC)
Topic.10 - Miscellaneous
(A) e j t u (t )
continuous and
periodic
H- Discrete and 4- Fourier
periodic signal
representation is
discrete and
periodic
(C) e
AA [GATE – EC – 2009]
(45) An LTI system having transfer function
s2 1
and input x(t) = sin(t 1) is in
s 2 2s 1
j0 t
(D) sin(0t )
S3A0.04-0.06 [GATE – EC – 2016]
(48) A
continuous-time filter with transfer
2s 6
function H ( s ) 2
is converted to
s 6s 8
a discrete time filter with transfer function
2 z 2 0.5032 z
G (z) 2
so that the impulse
z 0.5032 z k
response of the continuous-time filter,
sampled at 2 Hz, is identical at the sampling
instants to the impulse response of the
discrete time filter. The value of
is
________
(A) E-3, F-2, G-4, H-1
(B) E-1, F-3, G-2, H-4
(C) E-1, F-2, G-3, H-4
(D) E-2, F-1, G-4, H-3
AA [GATE – EC – 2007]
(44) A Hilbert transformer is a
(A) non-linear system
(B) non-causal system
(C) time-varying system
(D) low – pass system
(B) cos( 0 t )
0
S1AC [GATE – EC – 2016]
(49) A first-order low-pass filter of time constant
T is excited with different input signals (with
zero initial conditions up to t = 0). Match the
excitation signals X, Y, Z with the
corresponding time responses for t 0:
steady state. The output is sampled at a rate
s rad/s to obtain the final output {y(k )}.
which of the following is true?
(A) y is zero for all sampling frequencies
s
X: Impulse
P: 1 e t /T
Y: Unit step
Q: t T (1 e t / T )
Z: Ramp
R: e t / T
(A) X→R, Y→Q, Z→P
(B) y is nonzero for all sampling frequencies
(B) X→Q, Y→P, Z→R
s
(C) X→R, Y→P, Z→Q
(C) y is nonzero for s > 2, but zero for
(D) X→P, Y→R, Z→Q
s 2
S6AD [GATE – EE – 2016]
(D) y is zero for s 2, but nonzero for
s 2
AC [GATE – EC – 2014]
(46) Let X(t) be a wide sense stationary (WSS)
random with power spectral density Sx f .
If Y(t) is the process defined as Y(t) = X(2t –
1), the power spectral density S Y f
1 f
(A) SY f SX e jt
2 2
1 f
(B) SY f SX e jt / 2
2 2
(50) The output of a continuous-time, linear timeinvariant system is denoted by T{x(t)} where
x(t) is the input signal. A signal z(t) is called
eigen-signal of the system T , when T{z(t)} =
yz(t), where y is a complex number, in
general, and is called an eigenvalue of T.
Suppose the impulse response of the system
T is real and even. Which of the following
statements is TRUE?
(A) cos(t) is an eigen-signal but sin(t) is not
(B) cos(t) and sin(t) are both eigen-signals
but with different eigenvalues
(C) sin(t) is an eigen-signal but cos(t) is not
1 f
(C) SY f SX
2 2
(D) cos(t) and sin(t) are both eigen-signals
with identical eigenvalues
1 f
(D) SY f SX e j2 t
2 2
S1AC [GATE – EC – 2016]
(47) Which one of the following is an eigen
function of the class of all continuous-time,
linear, time-invariant systems (u(t) denotes
the unit-step function)?
A7.90-8.10 [GATE–S1–EC–2017]
(51) A continuous time signal x(t) = 4
cos 200 t 8cos 400 t , where t is in
seconds, is the input to a linear time invariant
(LTI) filter with the impulse response
www.targate.org
Page 109
SIGNAL & SYSTEM
2sin 300t
,
ht
t
600,
t0
t 0
Let y(t) be the output of this filter. The
maximum value of y t is _________
AC [GATE–S2–EC–2017]
(52) An LTI system with unit sample response
h[ n] 5[ n] 7[ n 1] 7[ n 3]
5[ n 4] is a
(A) low-pass filter
(B) high-pass filter
(C) band-pass filter
(D) band-stop filter
AB [GATE–S2–EC–2017]
(53) The signal x(t) = sin(14000 t ) , where t is in
seconds is sampled at a rate of 9000 samples
per second. The sampled signal is the input to
an ideal lowpass filter with frequency
response H(f) as follows :
1, | f | 12 kHz
H( f )
0, | f | 12 kHz
What is the number of sinusoids in the output
and their frequencies in kHz?
(A) Number = 1, frequency = 7
Which one of the following is TRUE about
the frequency selectivity of these systems?
(B) Number = 3, frequency = 2, 7, 11
(A) All three are high-pass filters.
(C) Number = 2, frequency = 2, 7
(B) All three are band-pass filters.
(D) Number = 2, frequency = 7, 11
(C) All three are low-pass filters.
AD [GATE–S1–EE–2017]
(54) The transfer function of a system is given by,
V0 s 1 s
. Let the output of the system
Vi s 1 s
be
v0 t vm sin t for the input,
vi t Vm sin t . Then the minimum and
maximum values of (in radians) are
respectively
(A)
(B)
and
and 0
2
2
2
(C) 0 and
2
(D) and 0
AB [GATE–S2–EE–2017]
(55) The pole-zero plots of three discrete-time
systems P, Q and R on the z-plane are shown
below.
(D) P is a low-pass filter, Q is a band-pass
filter and R is a high-pass filter.
AC [GATE-EE-2019]
(56) A system transfer function is
a s 2 b1 s c1
H (s) 1 2
. If a1 b1 0 , and all
a2 s b2 s c2
other coefficients are positive, the transfer
function represents a
(A) band pass filter
(B) high pass filter
(C) low pass filter
(D) notch filter
AC [GATE-EC-2019]
(57) It is desired to find three-tap causal filter
which gives zero signal as an output to an
input
of
the
form
jn
j n
x(n) c1 exp
c2 exp
2
2
Where c1 and c2 are arbitrary real numbers.
The desired three-tap filter is given by
h[0] 1, h[1] a, h[2] b
Page 110
TARGATE EDUCATION GATE-(EE/EC)
Topic.10 - Miscellaneous
And
LTI Systems Continuous And
Discrete (Time Domain)
h[n] 0, for n 0 or n > 2 .
What are the values of the filter taps a and b
if the output is y[n] = 0 for all n, when x[n] is
as given above?
(1)
AC [GATE – EC – 1994]
The 3-dB bandwidth of a typical secondorder system with the transfer function.
Cs
R s
(A) a = 1, b = 1
(B) a = –1, b = 1
(C) a = 0, b = 1
(D) a = 0, b = –1
2n
, is given by
s 2 2n s 2n
2
(A) n 1 2
**********
(2)
(B) n
1 2
4 2 1
(C) n
1 2
4 4 4 2 2
(D) n
1 2
4 4 4 2 2
2
2
2
AB [GATE – EC – 2005]
For a signal x(t) the Fourier transform is
X(f). Then the inverse Fourier transform of
X(3f + 2) is given by
(A)
1 1 j3 t
x e
2 2
(B)
1 t j4 t /3
x e
3 3
(C) 3x 3t e j4 t
(D) x(3t + 2)
(3)
AD [GATE – EC – 2013]
Let g(t) = e , and h(t) is a filter matched
to g(t). If g(t) is applied as input to h(t), then
the Fourier transform of the output is
t 2
(A) ef
2
(B) ef
2
(D) e2 f
(C) e f
/2
2
AD [GATE – EC – 2005]
n
(4)
1
x n u n , y n x 2 n
2
Let
and
Y e j be the Fourier transform of y(n).
j0
Then Y e is
(A)
1
4
(C) 4
(5)
www.targate.org
(B) 2
(D)
4
3
AA [GATE – EC – 2013]
The DFT of a vector [a b c d] is the vector
. Consider the product
Page 111
SIGNAL & SYSTEM
(A) 1-j, 1.875
a b c d
d a b c
p
q
r
s
a
b
c
d
c d a b
b c d a
The DFT of a vector [p q r s] is a scaled
version of
2
2
(A)
(B)
(C)
(D)
(6)
2 2
AC [GATE – EC – 2015]
Two sequences [a, b, c] and [A, B, C] are
1 a
A 1 1
B 1 W 1 W 2 b
where
3
3
2
4
C 1 W3
W3 c
W3 e
j
2
3
If another sequence [p, q, r] is derived as,
p 1 1
q 1 W1
3
r 1 W32
(C) 1+j, 1.875
(D) 0.1-j0.1, 1.500
(9)
AA [GATE – EE – 2014]
A discrete system is represented by the
difference equation
X1 k 1 a
a 1 X1 k
X 2 k 1 a 1 1 X 2 k
(B) 1-j, 1.500
1 1 0
W32 0 W32
W34 0 0
0 A / 3
0 B / 3
W34 C / 3
It has initial conditions X1 0 1; X 2 0 0
. The pole locations of the system for a = 1,
are
(A) 1 j0
(B) 1 j0
(C) 1 j0
(D) 0 j1
AB [GATE – EC – 2018]
(10) A discrete-time all-pass system has two of its
poles at 0.250 0 and 2300 . Which one of
the following statements about the system is
TRUE?
(A) It has two more poles at 0.530 0 and
40 0 .
(B) It is stable only when the impulse
response is two-sided.
then the relationship between the sequences
[p, q, r] and [a, b, c] is
(C) It has constant phase response over all
frequencies.
(A) [p, q, r] = [b, a, c]
(D) It has constant phase response over the
entire z-plane.
(B) [p, q, r] = [b, c, a]
(C) [p, q, r] = [c, a, b]
(D) [p, q, r] = [c, b, a]
(7)
-------0000-------
A11 [GATE – EC – 2015]
Consider two real sequences with time-origin
marked by the bold value,
x 1 n 1, 2, 3, 0 , x 2 n 1, 3, 2, 1
Let X1 k and X 2 k be 4-point DFTs of
x 1 n and x 2 n , respectively
Another sequence x 3 n is derived by taking
4-point
inverse
DFT
of
X 3 k X 1 k X 2 k . The value of x 3 2 is
__________.
(8)
AA[GATE – IN – 2010]
4-Point DFT of a real discrete –time signal
x[n] of length 4 is given by
It is given that
X k , n 0,1, 2,3 .
X 0 5, X 1 1 j1, X 2 0 , then X[3]
and X[0] respectively are
Page 112
TARGATE EDUCATION GATE-(EE/EC)
ANSWERS
Signal & System Answer Keys
01 – Continuous Time Signal & Sys.
System’s Classfication
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
A
C
D
D
D
D
C
B
B
D
D
D
B
B
B
D
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
D
A
A
D
D
B
B
B
C
C
B
D
D
D
A
C
33
34
35
36
37
38
39
40
41
42
43
44
45
46
A
D
D
D
B
A
A
A
*
B
C
B
C
D
41.
(1-D),(2-B)
Continuous Signal
01
02
03
04
05
06
07
08
09
10
11
A
B
C
A
D
D
A
C
B
C
B
Periods of Signal
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
A
C
D
B
A
C
A
B,D
A
B
A
D
*
D
6
A
17
18
19
20
21
22
23
24
0
B
A
8
1
D
6
*
24.
11.99 to 12.01
Convolution Theorem
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
D
C
D
C
D
A
B
A
*
B
*
C
A
D
A
B
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
A
A
D
B
B
C
*
C
C
D
A
D
B
C
*
C
33
34
35
36
37
C
A
31
A
D
9.
3.9 to 4.1
11.
0.4
23.
h (t )
31.
1 2 t 1 t
3 e 3 e u (t )
d
f (t )
dt
www.targate.org
Page 113
SIGNAL & SYSTEM
Delta Functions
01
02
03
04
05
06
07
08
09
10
11
12
13
14
14
A
C
B
B
A
*
D
B
B
C
D
A
C
A
A
e2
6.
Energy, Power & RMS
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
*
A
B
C
D
D
A
C
C
C
B
B
D
*
C
B
17
18
19
20
21
22
23
24
25
26
27
A
A
B
A
D
A
*
*
7
6
D
14.
0.408
23.
0.24 to 0.26
24.
7.95 to 8.05
Miscellaneous
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
B
C
D
*
*
A
B
A
D
B
A
C
B
D
C
D
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
C
D
C
B
B
D
A
D
D
B
A
C
C
D
B
A
33
34
B
B
4.
0.155
5.
0.19to0.21
02 – Discrete signal & systems
System’s Classification
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
A
A
B
A
A
A
A
A
D
B
C
D
B
D
B
C
17
18
19
20
21
22
23
24
25
26
27
C
A
D
C
C
D
D
D
A
D
A
Miscellaneous
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
B
A
C
C
B
D
C
A
D
*
C
A
D
A
B
D
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
D
C
A
A
C
B
A
A
D
A
A
D
A
A
A
D
33
34
35
36
37
38
39
A
A
B
*
D
2
D
10.
9.9to10.1
Page 114
TARGATE EDUCATION GATE-(EE/EC)
ANSWERS
03 – Fourier Series
Theoretical Problem
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
A
D
A
D
B
C
A
D
A
D
D
A
D
A
A
B
Numerical Problem
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
D
A
0.5
A
0.5
C
C
*
D
B
A
C
B
A
D
*
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
B
B
D
C
C
D
C
C
C
D
D
A
C
A
A
A
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
D
A
B
B
A
A
C
A
A
B
B
C
C
C
C
A
49
50
*
C
8.
2
8
16.
0.50to0.52
49.
9.5-10.5
04 – Fourier Transform
Theoretical Problem
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
C
C
B
A
C
C
C
B
C
B
B
C
C
C
D
A
17
18
19
20
21
D
C
B
C
B
Numerical Problem
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
C
C
B
B
A
A
D
B
*
D
C
C
A
A
C
B
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
C
A
A
A
D
*
D
C
A
D
C
A
D
C
B
A
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
C
*
B
D
C
B
C
D
D
D
A
D
A
C
A
D
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
B
B
C
D
D
D
B
A
D
A
A
D
C
B
C
B
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
D
D
B
B
C
A,C
C
B
C
B
B
A
MTA
C
9
9.
59.9to60.1
www.targate.org
Page 115
SIGNAL & SYSTEM
22.
3.36to3.39
34.
(1-A)(2-C)
05 – Laplace Transform
Theoretical Problem
01
02
03
04
05
B
A
B
D
D
Numerical Problem
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
B
B
*
C
A
C
B
D
B
C
C
A
*
B
B
B
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
*
A
B
C
A
B
A
C
C
A
A
D
C
A
A
D
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
C
D
A
C
C
A
A
D
C
C
B
B
D
A
A
D
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
B
-2
B
C
B
A
B
B
B
A
C
A
D
C
B
*
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
D
D
B
D
B
D
D
B
D
B
D
A
A
C
A
C
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
D
B
D
D
A
D
C
D
B
C
D
D
B
B
D
B
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
B
A
B
B
C
C
B
B
C
B
B
A
A
B
A
C
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
90
A
D
MTA
*
A
D
A
*
B
*
*
B
*
*
C
129
130
131
D
C
A
3.
FALSE
13.
0.99to1.01
17.
-0.01to0.01
117. 0.96 to 1.04
121. 0.45to0.55
123. 0.550to0.556
124. 1.284
126. -2.4to-2.0
127. 0.46 to 0.48
Page 116
TARGATE EDUCATION GATE-(EE/EC)
ANSWERS
06 – Sampling Theorem
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
*
*
A
B
C
B
B
C
C
C
D
B
A
*
C
A
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
C
A
D
D
A
B
*
C
D
C
B
C
C
B
A
C
33
34
35
36
37
38
39
40
41
42
43
44
A
C
D
*
C
B
6
B
C
13
A
8.0
1.
2.99to3.01
2.
9.5to10.5
14.
14
23.
3.6kHz
36.
12to14
07 – Z- Transform
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
0
A
C
A
*
A
B
C
D
A
B
C
C
A
0
2
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
C
A
D
C
D
B
A
B
A
A
*
C
C
D
B
A
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
B
C
C
D
B
A
C
B
A
B
D
*
C
C
A
D
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
A
B
A
B
C
B
B
B
C
A
A
C
B
C
C
C
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
D
A
C
A
B
D
D
C
A
B
C
A
A
B
A
B
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
C
C
B
B
B
A
B
A
A
B
D
C
C
*
C
C
97
A
27.
-0.6to-0.4
44.
11.9to12.1
94.
0.09to0.1
08 – DSF/DTFT/DFT/FFT
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
A
A
*
B
D
D
A
C
B
D
A
B
*
*
*
*
17
18
19
20
21
B
*
C
*
A
3.
9.99to10.01
13.
7.9to8.1
www.targate.org
Page 117
SIGNAL & SYSTEM
14.
4096
15.
5.9-6.1
16.
2.05to2.15
18.
2.90-3.10
20.
–27.01 to 26.99
09 – Random Variable
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
0.25
D
D
A
A
B
A
B
B
0.25
B
*
A
B
B
A
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
B
C
A
B
B
B
D
C
B
A
A
C
C
A
2
B
12.
3.9-4.1
10 – Miscellaneous
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
A
C
1.5
A
C
C
A
C
A
B
D
A
C
D
A
C
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
*
B
B
C
A
B
B
C
D
C
D
C
C
A
C
A
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
C
D
B
A
A
B
A
A
*
C
C
A
A
C
C
*
49
50
51
52
53
54
55
56
57
C
D
*
C
B
D
B
C
C
17.
44-46
41.
(A-1,B-3,C-4)
48.
0.04to0.06
51.
7.9to8.1
LTI Systems Continuous And Discrete (Time Domain)
01
02
03
04
05
06
07
08
09
10
C
B
D
D
A
C
11
A
A
B
****************
Page 118
TARGATE EDUCATION GATE-(EE/EC)
