Signals & Systems for
EC / EE / IN By
www.thegateacademy.com
Syllabus
Signals & Systems
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
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Contents
Signals and Systems
CONTENTS
#1.
#2.
#3.
#4.
Chapters
Page No.
Introduction to Signals & Systems Introduction Classification of Signals Systems Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations
1-31
Linear Time Invariant (LTI) Systems Introduction Properties of Convolution Properties/Characterization of LTI System Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations
32 -53
Fourier Representation of Signals Introduction Fourier Series (FS) for Continuous –Time Periodic Signals Properties of Fourier Representation Differentiation and Integration Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations
54-77
Z-Transform Introduction Properties of ROC Properties of Z – Transform Characterization of LTI System from H(Z) and ROC Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations
78-98
1-2 2-9 10- 11 12-17 18-21 21-24 25 25-31 32 33 33 –35 36 -41 42-46 46-48 49 49-53
th
54-55 55-57 57-59 59-60 61-65 66-69 69-70 71 71-77
78 79 79-81 81-82 83-86 87-90 90-92 93 93 -98 th
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Contents
#5.
#6.
Signals and Systems
Laplace Transform Introduction Properties of Laplace Transform Laplace Transform of Standard Functions Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations
99 - 120 99 99-101 101-102 102-105 106 - 109 110-113 114 114-120
Frequency Response of LTI Systems and Diversified Topics Frequency Response of a LTI System Standard LTI Systems Magnitude Transfer Function Sampling Discrete Fourier Transform (DFT) Fast Fourier Transform (FFT) FIR Filters Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations
121 - 148 121-122 122-123 123-124 124-125 125 126 126-132 133-136 137-140 140-143 144 144-148
Module Test Test Questions Answer Keys Explanations
149-161
Reference Book
162
149 - 156 157 157-161
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Chapter 1
Signals & Systems
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)] X(t)
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.
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Chapter 1
Continuous-time signal
Signals & Systems
Discrete- time signal
Sampling
x[n]
X(t)
x(T) x(2T)
0 t
→
n ←
T
Fig. 1.2.Demonstration of sampling 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 =
(
)
(
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 =
)
=
and
=
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 1. 2. 3. 4. 5. 6. 7. 8.
Nature of signal Complex, continuous-time Complex, continuous-time Real, continuous –time Real, continuous –time Complex, discrete –time Complex, discrete-time Real, discrete-time Real, discrete-time
Property Conjugate symmetry Conjugate skew symmetry Even function Odd function Conjugate symmetry Conjugate skew symmetry Even function Odd function th
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Condition x(t) = x t x(t) = x t x(t) = x(-t) x(t) = -x(-t) x[n] = x n x[n] = x n x[n] = x[ n] x[n] = x[ n] th
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Chapter 1
Signals & Systems
Table 1.2 Decompostion Based on Nature of Signal S.No
Nature of signal
Decomposition x x x x
Properties t =x t t = x t t =x t t = x t
1.
Complex, continuous –time
x(t) = x t
x t
2.
Real, continuous time
x(t) = x t
x t
3.
Complex, discrete-time
x[n] = x n
x n
x n =x x n = x
4.
Real, discrete-time
x[n] = x n
x n
x n =x n x n = x n
n
n
In the figure shown below, x (t), x2[n] are even signals and y (t), y [n] are odd signals. y (t) x (t) A
T
t
T
-T
2
1
1
t
T -A y [n]
x n] 2 1
3
+ A
1 3 -2 2
3
-1
1
2
3
n
-1
n
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) (b) (c) (d)
x x x x
t x t is conjugate symmetric t x t is conjugate skew symmetric t . x t is conjugate symmetric 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}
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Chapter 1
Signals & Systems
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 x(t)
-2T
-T
0
t
T
2 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 ∫ = im → T
Continuous-time, non-periodic
= ∫ x t
2.
Continuous-time, periodic signal with period T
= ∫
3.
Discrete-time, nonperiodic
=
im ∑ |x n |
= im
4.
Discrete-time, periodic signal with period N
=
im ∑ |x n |
=
1.
dt
x t
dt
– /
x t
dt P = ∫–
→
→
/ /
x t
→
dt 1 2N 1
1 2N 1
∑ |x n |
∑ |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. th
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Chapter 1
Signals & Systems
(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. S. No 1 2 3 4 5
Table 1.4 Summary of basic operations on dependent variable of a signal Operation Continuous-time signal Discrete –time signal Amplitude scaling y(t) = c.x(t) y[n]= c.x[n] Addition y(t) = x t x t y[n] =x n x n Multiplication y(t) = x t x t y[n] =x n . x n Differentiation y[n]= x[n] x[n-1] y(t) = x t Integration
y(t) = ∫
y[n] = ∑
x t dt
xn
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: x(t)
1
1
1
-1
x( )
x(2t)
1
t
0.5
0.5 t
-2
t 2
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
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Chapter 1
Signals & Systems
Eg: 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
n
7
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: X(t – T0)
X(t)
X(t + T0)
t
t
0
t
T0
T0
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) x(t)
y(t) = x(-t)
1
-1
t
1
-1
t
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] .. } ↑ th
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