Instrumentation Engineering : Signals & systems, THE GATE ACADEMY

Signals & Systems

for

EC / EE / IN

By

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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


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


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


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




Answer Keys 144


Module Test 149-161 Test Questions 149 - 156

Answer Keys 157


Reference Book 162

Chapter 1 Signals & Systems


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



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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



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Main, Jayanagar 4th


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


(d) x t x


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)



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Main, Jayanagar 4th


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



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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



Cross, 10th

Main, Jayanagar 4th


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

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Instrumentation Engineering : Signals & systems, THE GATE ACADEMY