Signals - UniMAP Portal CHAPTER 1
Signals and Systems
EKT 232
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Signals and Systems.
1.1 What is a Signal ?
1.2 Classification of a Signals.
1.2.1 Continuous-Time and Discrete-Time Signals
1.2.2 Even and Odd Signals.
1.2.3 Periodic and Non-periodic Signals.
1.2.4 Deterministic and Random Signals.
1.2.5 Energy and Power Signals.
1.3 Basic Operation of the Signal.
1.4 Elementary Signals.
1.4.1 Exponential Signals.
1.4.2 Sinusoidal Signal.
1.4.3 Sinusoidal and Complex Exponential Signals.
1.4.4 Exponential Damped Sinusoidal Signals.
1.4.5 Step Function.
1.4.6 Impulse Function.
1.4.7 Ramped Function.
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Cont’d…
1.5 What is a System ?
1.5.1 System Block Diagram.
1.6 Properties of the System.
1.6.1 Stability.
1.6.2 Memory.
1.6.3 Causality.
1.6.4 Invertability.
1.6.5 Time Invariance.
1.6.6 Linearity.
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1.1 What is a Signal ?
 A common form of human communication;
(i) use of speech signal, face to face or telephone channel.
(ii) use of visual, signal taking the form of images of people or
objects around us.
 Real life example of signals;
(i) Doctor listening to the heartbeat, blood pressure and temperature
of the patient. These indicate the state of health of the patient.
(ii) Daily fluctuations in the price of stock market will convey an
information on the how the share for a company is doing.
(iii) Weather forecast provides information on the temperature,
humidity, and the speed and direction of the prevailing wind.
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Cont’d…
 By definition, signal is a function of one or more variable,
which conveys information on the nature of a physical
phenomenon.
 A function of time representing a physical or mathematical
quantities.
e.g. : Velocity, e.g. : Velocity, acceleration of a car, voltage/current
of a circuit.
An example of signal; the electrical activity of the heart recorded
with electrodes on the surface of the chest — the
electrocardiogram (ECG or EKG) in the figure below.
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1.2 Classifications of a Signal.
 There are five types of signals;
(i) Continuous-Time and Discrete-Time Signals
(ii) Even and Odd Signals.
(iii) Periodic and Non-periodic Signals.
(iv) Deterministic and Random Signals.
(v) Energy and Power Signals.
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1.2.1 Continuous-Time and
Discrete-Time Signals.
Continuous-Time (CT) Signals
 Continuous-Time (CT) Signals are functions whose amplitude
or value varies continuously with time, x(t).
 The symbol t denotes time for continuous-time signal and ( ) used
to denote continuous-time value quantities.
 Example, speed of car, converting acoustic or light wave into
electrical signal and microphone converts variation in sound
pressure into correspond variation in voltage and current.
Figure 1.1: Continuous-Time Signal.
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Cont’d…
Discrete-Time Signals
 Discrete-Time Signals are function of discrete variable, i.e. they
are defined only at discrete instants of time.
xn  xnTs ,
n  0,1,2,....
 It is often derived from continuous-time signal by sampling at
uniform rate. Ts denotes sampling period and n denotes integer.
 The symbol n denotes time for discrete time signal and [. ] is
used to denote discrete-value quantities.
 Example: the value of stock at the end of the month.
Figure 1.3: Discrete-Time Signal.
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1.2.2 Even and Odd Signals.
 A continuous-time signal x(t) is said to be an even signal if
xt   xt 
for all t
 The signal x(t) is said to be an odd signal if
xt   xt 
for
all t
 In summary, an even signal are symmetric about the vertical axis
(time origin) whereas an odd signal are antisymetric about the
origin.
Figure 1.4: Even Signal
Figure 1.5: Odd Signal.
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Cont’d…
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Example 1.1: Even and Odd Signals.
Find the even and odd components of each of the following
signals:
(a) x(t) = 4cos(3πt)
ge(t) = 4cos(3πt)
go(t) = 0
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1.2.3 Periodic and Non-Periodic
Signals.
Periodic Signal.
 A periodic signal x(t) is a function of time that satisfies the
condition
xt   xt  T 
for all t,
where T is a positive constant.
 The smallest value of T that satisfy the definition is called a period.
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Figure 1.6: Aperiodic Signal.
Figure 1.7: Periodic Signal.
1.2.4 Deterministic and Random
Signals.
Deterministic Signal.
 A deterministic signal is a signal that has no uncertainty with
respect to its value at any time.
 The deterministic signal can be modeled as completely specified
function of time.
Figure 1.8: Deterministic Signal; Square Wave.
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Cont’d…
Random Signal.
 A random signal is a signal about which there is uncertainty
before it occurs. The signal may be viewed as belonging to an
ensemble or a group of signals which each signal in the ensemble
having a different waveform.
 The signal amplitude fluctuates between positive and negative in a
randomly fashion.
 Example; noise generated by amplifier of a radio or television.
Figure 1.9: Random Signal
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Energy Signal and Power Signals.
A signal with finite signal energy is
called an energy signal.
A signal with infinite signal energy
and finite average signal power is
called a power signal.
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1.2.5 Energy Signal and Power
Signals.
Energy Signal.
E

2
x
 n
n  
 A signal is refer to energy signal if and only if the total energy
satisfy the condition;
0 E 
Power Signal.
1
P
N
N 1
 x n
2
n 0
 A signal is refer to power signal if and only if the average power
of signal satisfy the condition;
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0 P
1.3 Basic Operation of the Signals.
1.3.1 Time Scaling.
1.3.2 Reflection and Folding.
1.3.3 Time Shifting.
1.3.4 Precedence Rule for Time Shifting and Time Scaling.
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1.3.1 Time Scaling.
 Time scaling refers to the multiplication of the variable by a real
positive constant.
yt   xat 
 If a &gt; 1 the signal y(t) is a compressed version of x(t).
 If 0 &lt; a &lt; 1 the signal y(t) is an expanded version of x(t).
 Example:
Figure 1.11: Time-scaling operation; continuous-time signal x(t),
(b) version of x(t) compressed by a factor of 2, and
(c) version of x(t) expanded by a factor of 2.
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Cont’d…
 In the discrete time,
yn  xkn,
 It is defined for integer value of k, k &gt; 1. Figure below for k = 2,
sample for n = +-1,
Figure 1.12: Effect of time scaling on a discrete-time signal:
(a) discrete-time signal x[n] and (b) version of x[n] compressed by a factor of 2, with
some values of the original x[n] lost as a result of the compression.
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1.3.2 Reflection and Folding.
 Let x(t) denote a continuous-time signal and y(t) is the signal
obtained by replacing time t with –t;
yt   x t 
 y(t) is the signal represents a refracted version of x(t) about t = 0.
 Two special cases for continuous and discrete-time signal;
(i) Even signal; x(-t) = x(t) an even signal is same as reflected
version.
(ii) Odd signal; x(-t) = -x(t) an odd signal is the negative of its
reflected version.
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Example 1.2: Reflection.
Given the triangular pulse x(t), find the reflected version of x(t) about
the amplitude axis (origin).
Solution:
Replace the variable t with –t, so we get y(t) = x(-t) as in figure below.
Figure 1.13: Operation of reflection: (a) continuous-time signal x(t) and
(b) reflected version of x(t) about the origin
x(t) = 0 for t &lt; -T1 and t &gt; T2.
y(t) = 0 for t &gt; T1 and t &lt; -T2.
.
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1.3.3 Time Shifting.
 A time shift delay or advances the signal in time by a time interval
+t0 or –t0, without changing its shape.
y(t) = x(t - t0)
 If t0 &gt; 0 the waveform of y(t) is obtained by shifting x(t)
toward the right, relative to the tie axis.
 If t0 &lt; 0, x(t) is shifted to the left.
Example:
Figure 1.14: Shift to the Left.
Q: How does the x(t) signal looks like?
Figure 1.15: Shift to the Right.
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Example 1.3: Time Shifting.
Given the rectangular pulse x(t) of unit amplitude and unit duration.
Find y(t)=x (t - 2)
Solution:
t0 is equal to 2 time units. Shift x(t) to the right by 2 time units.
(a)
.
Figure 1.16: Time-shifting operation:
continuous-time signal in the form of a rectangular pulse of amplitude 1.0 and
duration 1.0, symmetric about the origin; and
(b) time-shifted version of x(t) by 2 time shifts.
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1.3.4 Precedence Rule for Time
Shifting and Time Scaling.
 Time shifting operation is performed first on x(t), which results in
 Time shift has replace t in x(t) by t - b.
 Time scaling operation is performed on v(t), replacing t by at and
resulting in,
y t   vat 
y t   xat  b 
 Example in real-life: Voice signal recorded on a tape recorder;
 (a &gt; 1) tape is played faster than the recording rate, resulted in
compression.
 (a &lt; 1) tape is played slower than the recording rate, resulted
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in expansion.
Example 1.4: Continuous Signal.
A CT signal is shown in Figure 1.17 below, sketch and label each of
this signal;
a) x(t -1)
b) x(2t)
c) x(-t)
x(t)
2
t
-1
3
Figure 1.17
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Solution:
(a) x(t -1)
x(t)
(b) x(2t)
x(t-1)
2
2
t
t
0
4
-1/2
3/2
(c) x(-t)
x(-t)
2
t
-3
1
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Example 1.5: Continuous Signal.
A continuous signal x(t) is shown in Figure 1.17a. Sketch and label each
of the following signals.
a) x(t)= u(t -1)
b) x(t)= [u(t)-u(t-1)]
c) x(t)= d(t - 3/2)
Solution:
Figure 1.17a
(a) x(t)= u(t -1)
(b) x(t)= [u(t)-u(t-1)]
(c) x(t)=d(t - 3/2)
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Example 1.5: Discrete Time Signal.
A discrete-time signal x[n] is shown below,
Sketch and label each of the following signal.
(a) x[n – 2]
(c.) x[-n+2]
(b) x[2n]
(d) x[-n]
x[n]
4
2
0
1
2
3
n
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Cont’d…
(a) A discrete-time signal, x[n-2].
 A delay by 2
x[n-2]
4
2
0
1
2
3
4
5
n
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Cont’d…
(b) A discrete-time signal, x[2n].
Down-sampling by a factor of 2.
x(2n)
4
2
0
1
2
3
n
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Cont’d…
(c) A discrete-time signal, x[-n+2].
Time reversal and shifting
x(-n+2)
4
2
-1
0
1
2
n
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Cont’d…
(d) A discrete-time signal, x[-n].
 Time reversal
x(-n)
4
2
-3
-2
-1
0
1
n
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In Class Exercises .
A continuous-time signal x(t) is shown below, Sketch and label each
of the following signal
(a) x(t – 2)
(b) x(2t)
(c.) x(t/2)
(d) x(-t)
x(t)
4
0
4
t
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1.4.2 Sinusoidal Signals.
 A general form of sinusoidal signal is
xt   A coswot  q 
 where A is the amplitude, wo is the frequency in radian per
second, and q is the phase angle in radians.
Figure 1.20: Continuous-Time Sinusoidal signal A cos(ωt + Φ).
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1.4.5 Step Function.
 The discrete-time version of the unit-step function is defined
by,

1,
un  

0,
n0
n0
Figure 1.24: Discrete–time of Step Function of Unit Amplitude.
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The Unit Step Function
1 , t  0

u  t   1/ 2 , t  0
0 , t  0

Precise Graph
Commonly-Used Graph
The product signal g(t)u(t) can be thought of as the signal g(t)
“turned on” at time t = 0.
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Dr. Abid Yahya
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The Unit Ramp Function
t , t  0  t
ramp  t   
   u    d   t u t 
0 , t  0  
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1.4.6 Impulse Function.
 The discrete-time version of the unit impulse is defined by,

1, n  0
d n  

0, n  0
Figure 1.26: Discrete-Time form of Impulse.
 Figure 1.26 is a graphical description of the unit impulse d(t).
 The continuous-time version of the unit impulse is defined by the
following pair,
d n  0
for
t0


d t dt  1

 The d(t) is also refer as the Dirac Delta function.
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Graphical Representation of
the Impulse
The impulse is not a function in the ordinary sense because its
value at the time of its occurrence is not defined. It is represented
graphically by a vertical arrow. Its strength is either written beside
it or is represented by its length.
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The Unit Step and Unit Impulse
As a approaches zero, g  t  approaches a unit
step and g  t  approaches a unit impulse
The unit step is the integral of the unit impulse and
the unit impulse is the generalized derivative of
the unit step
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Properties of the Impulse
The Sampling Property

 g  t d  t  t  dt  g  t 
0
0

The sampling property “extracts” the value of a function at
a point.
The Scaling Property
1
d  a  t  t0    d  t  t0 
a
This property illustrates that the impulse is different from
ordinary mathematical functions.
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The Unit Periodic Impulse
The unit periodic impulse is defined by
dT  t  

 d t  nT 
, n an integer
n
The periodic impulse is a sum of infinitely many
uniformly-spaced impulses.
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FEW Questions from Past
Exam Paper , for Practice
and Use as a Tutorial
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Q.1
44
A.1
45
Q.2
46
A.2
47
A.2
48
Q.3
49
A.3
50
A.3
51
Q.4
52
Q.5
53
A.5
54
A.5
55
Q.6
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Q.6
57
Q.6
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Q.6
59
Q.6
60
Q.7
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Q.7
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Q.7
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The Rectangle Function
1 , n  N w 
rect N w  n   
 , N w  0 , N w an integer
0 , n  N w 
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Q.9
Find the signal energy of
these signals.
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1.5 What is a System ?
 A system can be viewed as an interconnection of operation that
transfer an input signal into an output signal with properties
different from those of the input signal.
 y(t) is the impulse response of the continuous-time system and
y[t] is the impulse response of the discrete-time system.
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Cont’d…
 Real life example of system;
(i) In automatic speaker recognition system; the system is to extract
the information from an incoming speech signal for the purpose of
recognizing and identifying the speaker.
(ii) In communication system; the system will transport the the
information contained in the message over a communication
channel and deliver that information to the destination.
Figure 1.30: Elements of a communication system.
Figure 1.31: Block diagram representation of a system.
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Cont’d…
 By definition, a system is an entity that manipulates one or
more signals to accomplish a function, thereby yielding new
signals.
 A physical process or a mathematical model of the physical process
that relates a set of input signals to yield another set of output
signal.
 Process input signals to produce output signals
 System representation of the systems.
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1.6 Properties of Systems.
 The properties of a system describe the characteristics of the
operator H representing the system.
 Basic properties of the system;
1.6.1 Stability.
1.6.2 Memory.
1.6.3 Causality.
1.6.4 Invertibility.
1.6.5 Time Invariance.
1.6.6 Linearity.
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1.6.1 Stability.
 A system is said to be bounded-input bounded-output (BIBO)
stable if and only if all bounded inputs result in bounded outputs.
The output of the system does not diverge if the input does not
diverge.
 For the resistor, if i(t) is bounded then so is v(t), but for the
capacitance this is not true. Consider i(t) = u(t) then v(t) = tu(t)
which is unbounded.
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1.6.2 Memory.
 A system is said to possess memory if its output signal depend on
pass or future values of the input signal.
 Note that v(t) depends not just on i(t) at one point in time t.
Therefore, the system that relates v to i exhibits memory.
 The system is said to be memoryless if its output signal depends
only on the present value of the input signal.
 Example: The resistive divider network
 Therefore, vo(to) depends upon the value of vi(to) and not on vi(t)
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for t = to.
Example 1.6: Memory and Memoryless System.
Below is the moving-average system described by the input-output
relation. Does it has memory or not?
(a) yn  1 xn  xn  1  xn  2
3
(b) yn  x 2 n
Solution:
(a) It has memory, the value of the output signal y[n] at time n depends
on the present and two pass values of x[n].
(b) It is memoryless, because the value of the output signal y[n]
depends only on the present value of the input signal x[n].
 .
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1.6.3 Causality.
Causal.
 A system is said to be casual if the present value of the output
signal depends only on the present or the past values of the input
signal. The system cannot anticipate the input.
Noncausal.
 In contrast, the output signal of a noncausal system depends on
one or more future values of the input signal.
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Causality
 Any system for which the response
occurs only during or after the time in
which the excitation is applied is
called a causal system.
 All real physical systems are causal
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Example 1.7: Causal and Noncausal.
1
yn  xn  1  xn  xn  1
3
Causal or noncausal?
Solution:
Noncausal; the output signal y[n] depends on a future value of the
input signal, x[n+1]
yn 
1
xn  1  xn  xn  1
3
Causality is required for a system to be capable of operating in real
time.
.
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1.6.4 Invertibility.
 A system is said to be invertible if the input of the system can be
recovered from the output.
H invyt   H invH xt 
 H invH xt 
Figure 1.32: The notion of system inevitability. The second operator Hinv is the
inverse of the first operator H. Hence, the input x(t) is passed through the
cascade correction of H and H-1 completely unchanged.
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1.6.5 Time Invariance.
 A system is said to be time invariant if the time delay or time
advance of the input signal leads to an identical time shift in the
output signal.
 The Time invariance system responds identically no mater when
the input signal is applied.
HS t0  S t0 H
Figure 1.33: (a) Time-shift operator St0 preceding operator H. (b) Time-shift operator
St0 following operator H. These two situations are equivalent, provided that H is
time invariant
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1.6.6 Linearity.
 A system is said to be linear in term of the system input
(excitation) x(t) and the system output (response) y(t) if it satisfies
the following two properties.
1. Superposition
 The system is initially at rest. The input is x(t)=x1(t), the
output y(t)=y1(t). So x(t)=x1(t)+x2(t) the corresponding
output y(t)=y1(t)+y2(t).
2. Homogeneity/Scaling
 The system is initially at rest. Input x(t) result in y(t). The
system exhibit the property of homogeneity if x(t) scaled by
constant factor a result in output y(t) is scaled by exact
constant a.
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Example:
1. Determine whether the system
1. y[n] = nx[n-1]+2 is :
(i) memoryless

(ii) time invariant
(iii) linear
(iv) causal
(v) stable
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3. Determine whether the corresponding
system is linear, time invariant or
both.

 (a) y[n] = x2[n-2]
 (b) y[n] = Od{x(t)}
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