JAYPEE INSTITUTE OF INFORMATION TECHNOLOGY
Electronics and Communication Engineering
Signals and Systems (18B11EC214)
Tutorial Sheet: 1 (b)
Q.1 [CO1]
Determine the values of Px and Ex for each of the following signals:
(a) x1 ( t )  e 2t u( t )
(b) x2 ( t )  e j ( 2t  / 4 )
(c) x3 ( t )  cos( t )
n
1
(d) x1 n    u( n )
2
(e) x2 n  e j (  / 2 n / 8 )
 
(f) x3 n  cos n 
4 
Q.2 [CO1]
Determine the fundamental period of the signal
x( t )  2 cos10t  1  sin4t  1
Q.3 [CO1]
Determine the fundamental period of the signal
xn  1  e j 4n / 7  e j 2n / 5
Q.4 [CO1] Determine whether or not each of the following continuous-time signals is
periodic. If the signal is periodic, determine its fundamental period.
(a) x( t )  3 cos4t   / 3
(b) x( t )  e j ( t 1 )
(c) x( t )  cos2t   / 3
2
JAYPEE INSTITUTE OF INFORMATION TECHNOLOGY
Electronics and Communication Engineering
Signals and Systems (18B11EC214)
Tutorial Sheet: 1 (a)
Q.1 [CO1]
A continuous-time signal x(t) is shown in Figure. Sketch and label carefully
each of the following signals:
(a) x( t  1 )
(b) x( 2  t )
(c) x( 2t  1 )
t

(d) x 4  
2

(e) x( t )  x( t )u( t )
  3   3 
(f) x( t )  t      t  
  2   2 
Q.2 [CO1]
A discrete-time signal is shown in Figure. Sketch and label carefully each of
the following signals:
(a) xn  4
(b) x3  n
(c) x3n
(d) x3n  1
(e) xnu3  n
(f) xn  2 n  2
(g)
1
1
n
xn   1 xn
2
2

(h) x n  1
2

Q.3 [CO1]
Determine whether or not each of the following signals is periodic:
(a) x1 ( t )  2e j ( t  / 4 )u( t )
(b) x2 n  un  u n
Q.4 [CO1]
Determine whether or not each of the following signals is periodic. If a
signal is periodic, specify its fundamental period.
(a) x1 ( t )  e j10t
(b) x2 ( t )  e( 1 j )t
(c) x3 n  e j 7n
(d) x4 n  3e j 3 ( n1 / 2 ) / 5
(e) x5 n  3e j 3 / 5( n1 / 2 )
JAYPEE INSTITUTE OF INFORMATION TECHNOLOGY
Electronics and Communication Engineering
Signals and Systems (18B11EC214)
TUTORIAL SHEET: 2
1. [CO1] Determine the properties which hold and which do not hold for each of the following
Continuous-time systems. Justify your answers. In each example y (t) denotes the system
output and x (t) is the system input.
a. y(t)  x(t  3)  x(3  t)
b. y(t)  cos[ x(t )]
t
c. y(t )   x ()d
d. y(t )  x(t / 3)

e. y( t ) 
0,
 x ( t )  x ( t  2),
dx ( t )
dt
f. y(t )  
x(t)  0
x(t )  0
2. [CO1] Determine the properties which hold and which do not hold for each of the following
Discrete-time systems. Justify your answers. In each example y [n] denotes the system output
and x [n] is the system input.
a. y[n]  nx[n]
b. y[n]  x[n]
c. y[n]  x[n]x[n  1]
d. y[n ]   x[k]
(1) x[n],

2 x[n],
e. y[n]  
n

k n
x[n]  0
x[n]  0
f. y[n]  8y[n  1]  x(n]
3. [CO2] The following are impulse responses of either DT or CT LTI Systems. Determine
whether the corresponding system is memoryless, causal and/or stable. Justify your answer.
a. h(t )  e 4 t ,
b. h(t)  u(1  t)  12 e  t u(t)
c. h(t )  e15t [u(t  1)  u(t  100)]
d. h[n]  3 n u[2  n]
e. h[n]  [1  (0.99) n ]u[n]
f. h[n]  n13  u[n  3]
n
JAYPEE INSTITUTE OF INFORMATION TECHNOLOGY
Electronics and Communication Engineering
Signals and Systems (18B11EC214) - 2020 ODD-SEM
TUTORIAL-4
Q.1 Compute and sketch the output y(t) for a continuous time LTI system whose impulse response
h(t) and input x(t) are given by
a) h(t) = 𝑒
u(t) & x(t) = 𝑒 u(-t)
b) h(t) = cos(t)u(t) & x(t) = u(t)
Q.2 Consider an input x[n] and a unit impulse response h[n] is given by
x[n] =𝛼 u[n]
h[n] = u[n]
where 0<α<1
Compute the output y[n]
Q.3 x[n] = δ[n] + 2δ[n-1] - δ[n-3] & h[n] = 2 δ[n+1] + 2δ[n-1]
Compute and plot each of the following convolutions
a) y1[n] = x[n] * h[n]
b) y2[n] = x[n] * h[n+2]
c) y3[n] = x[n+2] * h[n]
Q.4 Let h(t) = 𝑒 u(-t+4) + 𝑒
u(t-5)
Determine A and B such that
(
𝑒
h(t-𝜏) =
)
0
𝑒
(
)
, 𝜏<𝐴
, 𝐴<𝜏<𝐵
,𝐵 < 𝜏
Q.5 x(t) = u(t-3) – u(t-5) & h(t) = 𝑒
u(t)
a) Compute y(t) = x(t)*h(t)
b)Compute g(t) = differentiation of x(t) *h(t)
Q.6 For the given system compute and sketch the output response y(t)
x1(t)
u(t)
+
x2(t)
-u (t-3)
x(t)
=
h(t)
y(t)
h(t)= u(t)-u(t-2)
JAYPEE INSTITUTE OF INFORMATION TECHNOLOGY
Electronics and Communication Engineering
Signals and Systems (18B11EC214)
Tutorial Sheet: 4
Q1. [CO3] A CT periodic signal x(t) is real-valued and has fundamental period T = 8. The nonzero
Fourier series coefficients for x(t) are specified as
a1 = a  −1 = j , a3 = a− 3 = e j / 4 , a5 = a− 5 = 2
Express x(t) in the form

x (t ) =  Ak cos( k t +  k )
k =0
Q2. [CO3] Determine the Fourier series representations of the following signals and plot the
magnitude and phase of these coefficients:
(a)
(b)
(c) x (t) periodic with period 2, and x(t ) = e − t ,−1  t  1
Q3. [CO3] Let x(t) be a periodic signal with fundamental period T and Fourier series coefficients
ak. Derive the Fourier series coefficients of (a) x(t) = x(T/4 − t) and (b) Odd{x(t − T/2)} in terms
of ak.
Q4. [CO3] Consider the following CT periodic signals, x(t), y(t), and z(t).
(a) Determine the fundamental frequency, period, and Fourier series coefficients, ak, for x (t).
(b) Determine the fundamental frequency, period, and Fourier series coefficients, bk, for y (t).
(c) Determine the fundamental frequency and period for z(t). Also, using the results of parts (a) and (b),
determine the Fourier series coefficients, ck for z(t).
Q5. [CO3] Let x1 (t) be a continuous-time periodic signal with fundamental frequency w 1 and
Fourier coefficients ak. Given that
x2(t) = x1(1- t) + x1(t- 1),
how is the fundamental frequency w2 of x2(t) related to w1? Also, find a relationship between the
Fourier series coefficients bk of x2(t) and the coefficients ak.
Q6. [CO3] Compute the CT Fourier transform of each of the following signals:
Q7. [CO3] Determine the continuous-time signal corresponding to each of the following transforms:
Q8. [CO3] Consider the signal x(t) with spectrum depicted in Figure given below.
Sketch the spectrum of
Q9. [CO3] Consider the system depicted below:
Q10. [CO3] The output y(t) of a causal LTI system is related to the input x(t) by the equation
(a) Find the frequency response H(jω) of this system.
(b) Determine the impulse response of the system.
(c) Find the differential equation relating the input and output of this system.
JAYPEE INSTITUTE OF INFORMATION TECHNOLOGY
Electronics and Communication Engineering
Signals and Systems (18B11EC214) - 2020 ODD SEM
TUTORIAL 6: Discrete Time Fourier series
1. A DT periodic signal x[n] is real valued and has fundamental period N = 9. The nonzero
Fourier series coefficients for x[n] are specified as
a0 = 2, a1 = a ∗ −1 = j , a4 = a−∗ 4 = e jπ / 4
Express x[n] in the form
∞
x[ n] = A0 + ∑ Ak sin(ω k n + φ k )
k =1
2. Determine the Fourier series coefficients for the periodic signals x[n] depicted below. Plot
the magnitude and phase of these coefficients.
(a)
 14π 
 10π 
n  + 2 sin 
n +1
 19 
 19 
(b) x[n] = cos
(c) x [n] periodic with period 4, and x[ n] =
∞
∑ {4δ[ n − 4m] + 8δ[ n − 1 − 4m]}
m = −∞
3. Let
be a periodic signal with fundamental period N = 10 and Fourier series coefficients ak. Also,
let g[n] = x[n]- x[n-1]
(a) Show that g[n] has a fundamental period of 10.
(b) Determine the Fourier series coefficients of g[n].
(c) Using the Fourier series coefficients of g[n] and property, determine ak for k ≠ 0.
4. Let x[n] be a periodic signal with fundamental period N and Fourier series coefficients ak.
Derive the Fourier series coefficients of (a) x(N/4 − n) and (b) Od{x(n − N/2)} in terms of
ak.
5. Consider the following DT periodic signals, x[n], y[n], and z[n] with fundamental period
π
 2π 
 2π
x[ n] = 1 + cos
n , y[ n] = 1 + sin 
n + , and z[ n] = x[ n] y[ n]
4
 6 
 6
(a) Determine the Fourier series coefficients, ak, for x[n].
(b) Determine the Fourier series coefficients, bk, for y[n].
(c) Also, using the results of parts (a) and (b) and property, determine the Fourier series
coefficients, ck for z[n].
JAYPEE INSTITUTE OF INFORMATION TECHNOLOGY
Electronics and Communication Engineering
Signals and Systems (18B11EC214) - 2020 ODD SEM
TUTORIAL 7: Discrete Time Fourier Transform
1. Compute the DT Fourier transform of each of the following signals:
n
1
(a ) x[n] =   u[− n − 2]
 3
2.
π 
(b) x[n] = sin  n  + cos n
2 
The following are Fourier transforms of discrete-time signals. Determine the
signal corresponding to each transform.
3. Let X(ejω) denote the Fourier transform of the signal x[n] depicted below.
4. Let x[n] be a DT signal with Fourier Transform X(ejω) as given below.
Sketch the Fourier Transform of w[n] = x[n]p[n].
5.
π 


π 



Consider the signal , x[n] = sin  8 n  − 2 cos 4 n  . Suppose that this signal is input
to LTI systems with the following impulse responses. Determine the output in each
case.
6. Consider a system consisting of the cascade of two LTI systems as depicted below:
jω
(a) What is the frequency response H( e ) of the overall system ?
(b) Find the difference equation for the overall system.
(c) Find the impulse response of the overall system.
JAYPEE INSTITUTE OF INFORMATION TECHNOLOGY
Electronics and Communication Engineering
Signals and Systems (18B11EC214) - 2021 ODD SEM
TUTORIAL 8: Laplace Transform
1. [CO4] Determine the Laplace transform and the associated region of convergence and polezero plot for each of the following functions of time:
2. [CO4] Determine the function of time, x(t), for each of the following Laplace transforms
and associated region of convergence:
3. [CO4] Consider an LTI system for which the system function H(s) is rational and has the
pole-zero pattern shown below:
(a) Indicate all possible ROC’s that can be associated with this pole-zero pattern.
(b) For each ROC identified in Part (a), specify whether the associated system is stable and/or
causal.
4. [CO4] Consider a signal y(t) which is related to two signals x1(t) and x2(t) by
y(t ) = x1 (t − 2)  x 2 (−t + 3)
where
x1 (t ) = e −2t u(t )
and
x 2 (t ) = e −3t u(t )
Use properties of the Laplace transform to determine the Laplace transform Y(s) of y(t). Also,
determine if this system is stable and justify your answer.
5.
(a)
(b)
(c)
(d)
[CO4] Consider an LTI system with input x(t ) = e −t u(t ) and impulse response h(t ) = e −2t u(t ) .
Determine the Laplace transform of x(t) and h(t).
Using the convolution property, determine the Laplace transform Y(s) of the output y(t).
From Laplace transform Y(s) of the output y(t), détermine y(t).
Verify your result in part (c) by explicitly convolving x(t) and h(t).
6. [CO4] Draw a direct-form representation for the causal LTI system with system function
7. [CO4] Consider the cascade of two LTI systems as depicted below:
where we have the following:
(a) System A is causal with impulse response
•
System B is causal and is characterized by the following differential equation relating its
input, w(t), and output, y(t):
•
If the input x(t) = e , the output y(t) = 0.
−3t
(a) Find the system function H(s) = Y (s)/X(s), determine its ROC and sketch its pole-zero pattern.
(b) Determine the differential equation relating y(t) and x(t).
8. [CO4] Consider the system S characterized by the differential equation
d 3 y (t )
d 2 y (t )
dy (t )
+6
+ 11
+ 6 y (t ) = x(t )
3
2
dt
dt
dt
(a) Determine the zero – state response of this system for the input x(t ) = e −4t u(t ) .
(b) Determine the zero-input response of this system for t > 0-, given that
−
y (0 ) = 1,
dy(t )
dt
= −1,
t = 0−
d 2 y (t )
dt 2
= 1,
t = 0−
(c) Determine the output of S when the input is x(t ) = e −4t u(t ) and the initial conditions are
the same as specified in part (b).
JAYPEE INSTITUTE OF INFORMATION TECHNOLOGY
Electronics and Communication Engineering
Signals and Systems (18B11EC214) - 2021 ODD SEM
TUTORIAL 9: z - Transform
1. [CO4] Determine the z-transform for each of the following sequences. Sketch pole-zero plot
and indicate the region of convergence. Indicate whether or not the Fourier transform of the
sequence exists.
2. [CO4] For each of the following z-transforms, determine the inverse z-transform
3. [CO4] Consider a signal y[n] which is related to two signals x1[n] and x2[n] by
where
Determine the z-transform Y (z) of y[n], together with its ROC.
4. [CO4] Consider the LTI system with input
1,0  n  N − 1
x[n] = 
and impulse response h[n] =
0
,
otherwise

(a)
(b)
(c)
(d)

a n , n  0

.

0, n  0
Determine the z-transform of x[n] and h[n].
Using the convolution property, determine the z- transform Y (z) of the output y[n].
From z- transform Y (z) of the output y[n], détermine y[n].
Verify your result in part (c) by explicitly convolving x[n] and h[n].
5. [CO4] A causal LTI system is described by the difference equation
(a) Find the system function H(z) for this system. Plot the poles and zeros of H(z) and indicate
ROC.
(b) Find unit sample response of the system.
(c) Find system is stable. If not, find a stable and noncausal unit sample response that satisfies
the difference equation.
6. [CO4] Determine the zero-input and zero-state response:
7. [CO4] The input x[n] and output y[n] of a causal LTI system are related through block
diagram representation shown in figure below:
(a) Determine a difference equation relating y[n] and x[n].
(b) Is this system stable?