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THE GATE ACADEMY's GATE Correspondence Materials consist of complete GATE syllabus in the form of booklets with theory, solved examples, model tests, formulae and questions in various levels of difficulty in all the topics of the syllabus. The material is designed in such a way that it has proven to be an ideal material in-terms of an accurate and efficient preparation for GATE. Quick Refresher Guide : is especially developed for the students, for their quick revision of concepts preparing for GATE examination. Also get 1 All India Mock Tests with results including Rank,Percentile,detailed performance analysis and with video solutions GATE QUESTION BANK : is a topic-wise and subject wise collection of previous year GATE questions ( 2001 – 2013). Also get 1 All India Mock Tests with results including Rank,Percentile,detailed performance analysis and with video solutions Bangalore Head Office: THE GATE ACADEMY # 74, Keshava Krupa(Third floor), 30th Cross, 10th Main, Jayanagar 4th block, Bangalore- 560011 E-Mail: [email protected] Ph: 080-61766222
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Syllabus Signals & Systems
THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th
Cross, 10th
Main, Jayanagar 4th
Block, Bangalore-11 : 080-65700750, [email protected] © 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), 30th
Cross, 10th
Main, Jayanagar 4th
Block, Bangalore-11 : 080-65700750, [email protected] © 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
Assignment 2 21-24
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
Assignment 1 42-46
Assignment 2 46-48
Answer Keys 49
Explanations 49-53
#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
Solved Examples 61-65
Assignment 1 66-69
Assignment 2 69-70
Answer Keys 71
Explanations 71-77
#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
Solved Examples 83-86
Assignment 1 87-90
Assignment 2 90-92
Answer Keys 93
Explanations 93 -98
Contents Signals and Systems
THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th
Cross, 10th
Main, Jayanagar 4th
Block, Bangalore-11 : 080-65700750, [email protected] © 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
Solved Examples 102-105
Assignment 1 106 - 109
Assignment 2 110-113
Answer Keys 114
Explanations 114-120
#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
Solved Examples 133-136
Assignment 1 137-140
Assignment 2 140-143
Answer Keys 144
Explanations 144-148
Module Test 149-161 Test Questions 149 - 156
Answer Keys 157
Explanations 157-161
Reference Book 162
Chapter 1 Signals & Systems
THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th
Cross, 10th
Main, Jayanagar 4th
Block, Bangalore-11 : 080-65700750, [email protected] © 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
Chapter 1 Signals & Systems
THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th
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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
Sampling Continuous-time signal Discrete- time signal
n
Chapter 1 Signals & Systems
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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)
Chapter 1 Signals & Systems
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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
Chapter 1 Signals & Systems
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(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 t 0.5 t
Chapter 1 Signals & Systems
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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 1 t 1 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