Instrumentation Engineering : Signals & systems, THE GATE ACADEMY

Signals & Systems
for
EC / EE / IN
By
www.thegateacademy.com

Syllabus Signals & Systems
THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30
th
Cross, 10
th
Main, Jayanagar 4
th
Block, Bangalore-11
: 080-65700750,  info@thegateacademy.com © Copyright reserved. Web: www.thegateacademy.com
Syllabus for Signals & Systems
Definitions and properties of Laplace transform, continuous-time and discrete-time Fourier
series, continuous-time and discrete-time Fourier Transform, DFT and FFT, Z-transform.
Sampling theorem. Linear Time-Invariant (LTI) Systems: definitions and properties; causality,
stability, impulse response, convolution, poles and zeros, parallel and cascade structure,
frequency response, group delay, phase delay. Signal transmission through LTI systems.
Analysis of GATE Papers
(Signals & Systems)
Year ECE EE IN
2010 10.00 11.00 9.00
2011 12.00 7.00 9.00
2012 9.00 10.00 8.00
2013 11.00 7.00 6.00
Over All
Percentage
10.50% 8.75% 8.00

Contents Signals and Systems
THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30
th
Cross, 10
th
Main, Jayanagar 4
th
Block, Bangalore-11
: 080-65700750,  info@thegateacademy.com © Copyright reserved. Web: www.thegateacademy.com Page I
CC OO NN TT EE NN TT SS
Chapters Page No.
#1. Introduction to Signals & Systems 1-31
 Introduction 1-2
 Classification of Signals 2-9
 Systems 10- 11
 Solved Examples 12-17
 Assignment 1 18-21
 Answer Keys 25
 Explanations 25-31
#2. Linear Time Invariant (LTI) Systems 32 -53
 Introduction 32
 Properties of Convolution 33
 Properties/Characterization of LTI System 33 –35
 Solved Examples 36 -41
 Answer Keys 49
#3. Fourier Representation of Signals 54-77
 Introduction 54-55
 Fourier Series (FS) for Continuous –Time Periodic Signals 55-57
 Properties of Fourier Representation 57-59
 Differentiation and Integration 59-60
 Answer Keys 71
#4. Z-Transform 78-98
 Introduction 78
 Properties of ROC 79
 Properties of Z – Transform 79-81
 Characterization of LTI System from H(Z) and ROC 81-82
 Answer Keys 93
 Explanations 93 -98

Contents Signals and Systems
THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30
th
Cross, 10
th
Main, Jayanagar 4
th
Block, Bangalore-11
: 080-65700750,  info@thegateacademy.com © Copyright reserved. Web: www.thegateacademy.com Page II
#5. Laplace Transform 99 - 120
 Introduction 99
 Properties of Laplace Transform 99-101
 Laplace Transform of Standard Functions 101-102
 Assignment 1 106 - 109
 Answer Keys 114
#6. Frequency Response of LTI Systems and
Diversified Topics 121 - 148
 Frequency Response of a LTI System 121-122
 Standard LTI Systems 122-123
 Magnitude Transfer Function 123-124
 Sampling 124-125
 Discrete Fourier Transform (DFT) 125
 Fast Fourier Transform (FFT) 126
 FIR Filters 126-132
 Answer Keys 144
Module Test 149-161
 Test Questions 149 - 156
 Answer Keys 157
Reference Book 162

Chapter 1 Signals & Systems
th
Cross, 10
th
Main, Jayanagar 4
th
Block, Bangalore-11
: 080-65700750,  info@thegateacademy.com © Copyright reserved. Web: www.thegateacademy.com Page 1
CHAPTER 1
Introduction to Signals & Systems
Introduction
Signal is defined as a function that conveys useful information about the state or behaviour of a
physical phenomenon. Signal is typically the variation with respect to an independent quantity
like time as shown in figure below. Time is assumed as independent variable for remaining part
of the discussion, unless mentioned.
(1) Speech signal – plot of amplitude with respect to time [x(t)]
(2) Image –plot of intensity with respect to spatial co-ordinates [I(x, y)]
(3) Video – plot of intensity with respect to spatial co-ordinates and time [V(x, y, t)]
Fig 1.1 Continuous –time signal
System
System is defined as an entity which extracts useful information from the signal or processes the
signal as per a specific function.
Eg:- speech signal filtering
Classification of Signals
Depending on property under consideration, signals can be classified in the following ways.
Continuous-time vs Discrete-time Signals
Continuous-time signal is defined as a signal which is defined for all instants of time. Discrete
time signal is a signal which is defined at specific instants of time only and is obtained by
sampling a continuous – time signal. Hence, discrete-time signal is not defined for non-integer
instants and is often identified as sequence of numbers, denoted as x [n] where n is integer.
x[n] = x(t)| ∀n = 0, 1, 2, 3. . . .
x[n] = { x(0), x(T), x(2T). . . . . . }
Digital signal is defined as a signal which is defined at specific instants of time and also
dependent variables can take only specific values. Digital signal is obtained from discrete-time
signal by quantization.
X(t)
t

th
Cross, 10
th
Main, Jayanagar 4
th
Block, Bangalore-11
In the figure shown above, x[n] is the discrete time signal obtained by uniform sampling of x(t)
with a sampling period T.
Classification of Signals
Conjugate Symmetric vs Skew Symmetric Signals.
A continuous time signal x(t) is conjugate symmetric if x(t) = x*(-t);∀t.Also, x(t) is conjugate
skew symmetric if x(t) = -x* (-t); ∀t
Any arbitrary signal x(t) can be considered to constitute 2 parts as below,
x(t) = x t + x t
Where x t = conjugate symmetric part of signal =
( )
= and
x t = conjugate skew symmetric part of signal =
( )
=
If signal x(t) is real, x(t) constitutes even and odd parts.
x(t) = x t x t
Where x t = and x t = t
x t = and x t = t
For real signals, conjugate symmetry property implies even function property and conjugate
skew symmetry property implies odd function property. Above properties can also be applied
for discrete time signals and are summarized in the following table.
Table 1.1 Symmetry Properties Based on Nature of Signal
S.NO Nature of signal Property Condition
1. Complex, continuous-time Conjugate symmetry x(t) = x t
2. Complex, continuous-time Conjugate skew symmetry x(t) = x t
3. Real, continuous –time Even function x(t) = x(-t)
4. Real, continuous –time Odd function x(t) = -x(-t)
5. Complex, discrete –time Conjugate symmetry x[n] = x n
6. Complex, discrete-time Conjugate skew symmetry x[n] = x n
7. Real, discrete-time Even function x[n] = x[ n]
8. Real, discrete-time Odd function x[n] = x[ n]
X(t)
t
x(T)
T
→ ←
x[n]
x(2T)
0
Fig. 1.2.Demonstration of sampling
SamplingContinuous-time signal Discrete- time signal
n

th
Cross, 10
th
Main, Jayanagar 4
th
Block, Bangalore-11
Table 1.2 Decompostion Based on Nature of Signal
S.No Nature of signal Decomposition Properties
1. Complex, continuous –time x(t) = x t x t
x t = x t
x t = x t
2. Real, continuous time x(t) = x t x t
x t = x t
x t = x t
3. Complex, discrete-time x[n] = x n x n
x n = x n
x n = x n
4. Real, discrete-time x[n] = x n x n
x n = x n
x n = x n
In the figure shown below, x (t), x2[n] are even signals and y (t), y [n] are odd signals.
Fig 1.3 Examples of even and odd signals
If x t and x (t) are complex conjugate symmetric signals and x (t) and x (t) are complex
conjugate skew – symmetric signals, then
(a) x t x t is conjugate symmetric
(b) x t x t is conjugate skew symmetric
(c) x t . x t is conjugate symmetric
(d) x t x t is conjugate symmetric
Above applies for complex discrete-time signals also and can be equivalently derived for real
signals, based on even-odd function properties.
Periodic vs Non-periodic Signals
A continuous –time signal is periodic if there exists T such that
x(t+T) = x(t), ∀ t ;T R – {0}
2
x n]
3 2 1 1 2 3
1
n
1 2 3
3 -2 -1
-1
1
y [n]
n
-
T T
-A
+
A
t
y (t)
A
-T T
x
t
(t)

th
Cross, 10
th
Main, Jayanagar 4
th
Block, Bangalore-11
The smallest positive value of T that satisfies above condition is called fundamental period of
x(t). Also, angular frequency of continuous-time signals is defined as, = 2π/T and is measured
in rad/sec.A discrete-time signal is periodic if there exists N such that
= , ∀ – 0
The smallest positive N that satisfies above condition is called fundamental period of x[n]. Here
N is always positive integer and angular frequency is defined as  = 2π/N and is measured in
radians/samples. If x t and x t are periodic signals with periods T and T respectively,
then x(t) = x t + x t is periodic iff (if and only if) is a rational number and period of x(t)
is least common multiple (LCM) of T and T .If x n is periodic with fundamental period N and
x n is periodic with fundamental period M than x[n] = x n + x n is always periodic with
fundamental period equal to the least common multiple (LCM) of M and N.
Figure below shows a signal, x(t) of period T
Fig.1.4 Example of a periodic signal
Energy & Power Signals
The formulas for calculation of energy, E and power, P of a continuous/discrete-time signal are
given in table below,
Table 1.3 Formulas for calculation of energy and power
S. NO Nature of the signal Formulas for energy & power calculation
1.
Continuous-time,
non-periodic
= ∫ x t dt = im
→
1
T
∫ x t dt
/
– /
2.
Continuous-time,
periodic signal with
period T
= ∫ x t dt P = ∫ x t dt
/
– /
3.
Discrete-time, non-
periodic
= im
→
∑ |x n | = im
→
1
2N 1
∑ |x n |
4.
Discrete-time,
periodic signal with
period N
= im
→
∑ |x n | =
1
2N 1
∑ |x n |
A signal is called energy signal if f 0< E < . So P = 0
A signal is called power signal if f 0< P < . So →
Note: (1) Energy signal has zero average power.
(2) Power signal has infinite energy.
(3) Usually periodic signals and random signals are power signals.
-2T -T T 2
T
0
x(t)
t

th
Cross, 10
th
Main, Jayanagar 4
th
Block, Bangalore-11
(4) Usually deterministic and non periodic signals are energy signals.
Real vs Complex Signals
A signal x(t) is real signal if its value are only real numbers and the signal x(t) is complex signal
if its value are complex numbers.
Deterministic Vs Random Signals
A signal is said to be deterministic signal whose values can be predicted in advance
Eg : A
A signa is said to be random signa whose va ues are can’t be predicted in advance
Eg : Noise
Basic Operations on Signals
Depending on nature of operation, different basic operations can be applied on dependent and
independent variables of a signal. The table below summaries basic operations that can be
performed on dependent variable of a signal.
Table 1.4 Summary of basic operations on dependent variable of a signal
S. No Operation Continuous-time signal Discrete –time signal
1 Amplitude scaling y(t) = c.x(t) y[n]= c.x[n]
2 Addition y(t) = x t x t y[n] =x n x n
3 Multiplication y(t) = x t x t y[n] =x n . x n
4 Differentiation y(t) = x t y[n]= x[n] x[n-1]
5 Integration y(t) = ∫ x t dt y[n] = ∑ x n
Similarly, following operations can be performed on independent variable of a signal.
Time Scaling
For continuous –time signals, y(t) = x(at). If a > 1, y(t) is obtained by compressing signal x(t)
a ong time axis by ‘a’ and vice versa.
Eg:
Fig 1.5.Demonstration of time scaling
For discrete time signal, y[n] =x([Kn]) where [ ] is some operation leading to integer result. If K
> 1, y[n] is compressed version of x[n] and vice versa.y[n] is defined only for integer values of n.
Therefore, compressing a discrete-time signal may result in data loss and expansion leads to
some instants where values of y[n] are zero
x(t)
-1 1
1
x( )
1
-2 2
t
x(2t)
1
0.5 t0.5
t

th
Cross, 10
th
Main, Jayanagar 4
th
Block, Bangalore-11
Eg:
Fig. 1.6.Demonstration of down sampling
Time Shifting
For continuous time signal,y(t) = x(t – T0). If T0> 0, y(t) is delayed version of x(t). If T0< 0, y(t)
is advanced version of x(t)
Eg:
Fig. 1.7.Demonstration of delaying a signal
For discrete time signal, y[n] = x[n-m];
where m is an integer. If m > 0, signal x[n] gets shifted to right by m (delayed by m samples).If
m < 0, signal x[n] gets shifted to left by m (advanced by m samples).
Reflection or Transposing of Time Variable
For continuous –time signal x(t), reflection is expressed as y(t) = x(-t). For discrete-time signals,
y[n] = x[ n] is the reflection of x[n].
Eg:-(1)
Fig. 1.8.Demonstration of reflection
(2) x[n] = {.. x[-2], x[-1], x[0], x[1], x[2] .. }
↑
x[-n] = {..x[2], x[1], x[0],x[-1],x[-2] .. }
↑
y(t) = x(-t)x(t)
-1 1t1 t-1
X(t + T0)
T0
t
X(t – T0)
T0
t
X(t)
0
t
X[n]
0 1 2 3 4 5 6 7 8 9 10 n
Y[n] = x[2n]
0 1 2 3 4 5 6 7 n

Instrumentation Engineering : Signals & systems, THE GATE ACADEMY

Instrumentation Engineering : Signals & systems, THE GATE ACADEMY

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Viewers also liked

Viewers also liked (14)

Similar to Instrumentation Engineering : Signals & systems, THE GATE ACADEMY

Similar to Instrumentation Engineering : Signals & systems, THE GATE ACADEMY (20)

More from klirantga

More from klirantga (11)

Recently uploaded

Recently uploaded (20)

Instrumentation Engineering : Signals & systems, THE GATE ACADEMY