# Signals and Systems PDF by Rao K. Deergha

Download Signals and Systems PDF by Rao, K. Deergha – This textbook ( Signals and Systems ) covers the fundamental theories of signals and systems analysis, while incorporating recent developments from integrated circuits technology into its examples. Buy from Amazon

## Signals and Systems PDF

Starting with basic definitions in signal theory, the text explains the properties of continuous-time and discrete-time systems and their representation by differential equations and state space. From those tools, explanations for the processes of Fourier analysis, the Laplace transform, and the z-Transform provide new ways of experimenting with different kinds of time systems. The text also covers the separate classes of analog filters and their uses in signal processing applications.Signals and Systems PDF

Intended for undergraduate electrical engineering students, chapter sections include exercise for review and practice for the systems concepts of each chapter. Along with exercises, the text includes MATLAB-based examples to allow readers to experiment with signals and systems code on their own. An online repository of the MATLAB code from this textbook can be found at github.com/springer-math/signals-and-systems.

## Table of Content

Contents
1 Introduction â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦. 1
1.1 What is a Signal? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 What is a System? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Elementary Operations on Signals . . . . . . . . . . . . . . . . . . . . . . . 1
1.3.1 Time Shifting â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦. 2
1.3.2 Time Scaling â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.. 2
1.3.3 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Classification of Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4.1 Continuous-Time and Discrete-Time Signals . . . . . . . . . 5
1.4.2 Analog and Digital Signals . . . . â€¦â€¦â€¦â€¦â€¦â€¦ 5
1.4.3 Periodic and Aperiodic Signals . . â€¦â€¦â€¦â€¦â€¦.. 6
1.4.4 Even and Odd Signals â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦. 9
1.4.5 Causal, Noncausal, and Anticausal Signal . . . . . . . . . . . 12
1.4.6 Energy and Power Signals . . . . . . . . . . . . . . . . . . . . . . 13
1.4.7 Deterministic and Random Signals . . . . . . . . . . . . . . . . 20
1.5 Basic Continuous-Time Signals . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5.1 The Unit Step Function . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5.2 The Unit Impulse Function . . . . . . . . . . . . . . . . . . . . . . 21
1.5.3 The Ramp Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5.4 The Rectangular Pulse Function . . . . . . . . . . . . . . . . . . 22
1.5.5 The Signum Function . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5.6 The Real Exponential Function . . . . . . . . . . . . . . . . . . . 23
1.5.7 The Complex Exponential Function . . . . . . . . . . . . . . . 24
1.5.8 The Sinc Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.6 Generation of Continuous-Time Signals
Using MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.7 Typical Signal Processing Operations . . . . . . . . . . . . . . . . . . . . . 30
1.7.1 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.7.2 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.7.3 Modulation and Demodulation . . . . . . . . . . . . . . . . . . . 31
ix
1.7.4 Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.7.5 Multiplexing and Demultiplexing . . . . . . . . . . . . . . . . . 32
1.8 Some Examples of Real-World Signals and Systems . . . . . . . . . . 32
1.8.1 Audio Recording System . . . . . . . . . . . . . . . . . . . . . . . 32
1.8.2 Global Positioning System . . . . . . . . . . . . . . . . . . . . . . 33
1.8.3 Location-Based Mobile Emergency
Services System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.8.4 Heart Monitoring System . . . . . . . . . . . . . . . . . . . . . . . 34
1.8.5 Human Visual System . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.8.6 Magnetic Resonance Imaging . . . . . . . . . . . . . . . . . . . . 36
1.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.10 MATLAB Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2 Continuous-Time Signals and Systems . . . . . . . . . . . . . . . . . . . . . . . 41
2.1 The Representation of Signals in Terms of Impulses . . . . . . . . . . 41
2.2 Continuous-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.1 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.2 Time-Invariant System . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2.3 Causal System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.2.4 Stable System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2.5 Memory and Memoryless System . . . . . . . . . . . . . . . . . 49
2.2.6 Invertible System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2.7 Step and Impulse Responses . . . . . . . . . . . . . . . . . . . . . 49
2.3 The Convolution Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3.1 Some Properties of the Convolution Integral . . . . . . . . . 50
2.3.2 Graphical Convolution . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.3.3 Computation of Convolution Integral
Using MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.3.4 Interconnected Systems . . . . . . . . . . . . . . . . . . . . . . . . 74
2.3.5 Periodic Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.4 Properties of Linear Time-Invariant Continuous-Time
System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.4.1 LTI Systems With and Without Memory . . . . . . . . . . . . 77
2.4.2 Causality for LTI Systems . . . . . . . . . . . . . . . . . . . . . . 77
2.4.3 Stability for LTI Systems . . . . . . . . . . . . . . . . . . . . . . . 77
2.4.4 Invertible LTI System . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.5 Systems Described by Differential Equations . . . . . . . . . . . . . . . 82
2.5.1 Linear Constant-Coefficient Differential Equations . . . . . 82
2.5.2 The General Solution of Differential Equation . . . . . . . . 85
2.5.3 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.5.4 Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.5.5 Time-Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.5.6 Impulse Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
2.5.7 Solution of Differential Equations Using
MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
x Contents
2.5.8 Determining Impulse Response and Step
Response for a Linear System Described by
a Differential Equation Using MATLAB . . . . . . . . . . . . 92
2.6 Block-Diagram Representations of LTI Systems
Described by Differential Equations . . . . . . . . . . . . . . . . . . . . . . 93
2.7 Singularity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
2.8 State-Space Representation of Continuous-Time
LTI Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
2.8.1 State and State Variables . . . . . . . . . . . . . . . . . . . . . . . 98
2.8.2 State-Space Representation of Single-Input
Single-Output Continuous-Time LTI Systems . . . . . . . . 99
2.8.3 State-Space Representation of Multi-input
Multi-output Continuous-Time LTI Systems . . . . . . . . . 104
2.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
2.10 MATLAB Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3 Frequency Domain Analysis of Continuous-Time
Signals and Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.1 Complex Exponential Fourier Series Representation
of the Continuous-Time Periodic Signals . . . . . . . . . . . . . . . . . . . 111
3.1.1 Convergence of Fourier Series . . . . . . . . . . . . . . . . . . . . 113
3.1.2 Properties of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . 113
3.2 Trigonometric Fourier Series Representation . . . . . . . . . . . . . . . . 128
3.2.1 Symmetry Conditions in Trigonometric
Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
3.3 The Continuous Fourier Transform for Nonperiodic
Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3.3.1 Convergence of Fourier Transforms . . . . . . . . . . . . . . . . 135
3.3.2 Fourier Transforms of Some Commonly Used
Continuous-Time Signals . . . . . . . . . . . . . . . . . . . . . . . . 136
3.3.3 Properties of the Continuous-Time Fourier
Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
3.4 The Frequency Response of Continuous-Time Systems . . . . . . . . . 159
3.4.1 Distortion During Transmission . . . . . . . . . . . . . . . . . . . 160
3.5 Some Communication Application Examples . . . . . . . . . . . . . . . . 162
3.5.1 Amplitude Modulation (AM) and Demodulation
Amplitude Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 162
3.5.2 Single-Sideband (SSB) AM . . . . . . . . . . . . . . . . . . . . . . 164
3.5.3 Frequency Division Multiplexing (FDM) . . . . . . . . . . . . . 164
3.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
4 Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
4.1 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
4.1.1 Definition of Laplace Transform . . . . . . . . . . . . . . . . . . 171
Contents xi
4.1.2 The Unilateral Laplace Transform . . . . . . . . . . . . . . . . . 172
4.1.3 Existence of Laplace Transforms . . . . . . . . . . . . . . . . . . 172
4.1.4 Relationship Between Laplace Transform
and Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . 172
4.1.5 Representation of Laplace Transform in the S-Plane . . . . 173
4.2 Properties of the Region of Convergence . . . . . . . . . . . . . . . . . . 174
4.3 The Inverse Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . 176
4.4 Properties of the Laplace Transform . . . . . . . . . . . . . . . . . . . . . 178
4.4.1 Laplace Transform Properties of Even and
Odd Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
4.4.2 Differentiation Property of the Unilateral
Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
4.4.3 Initial Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 186
4.4.4 Final Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 187
4.5 Laplace Transforms of Elementary Functions . . . . . . . . . . . . . . . 187
4.6 Computation of Inverse Laplace Transform Using
Partial Fraction Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
4.6.1 Partial Fraction Expansion of X(s) with Simple
Poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
4.6.2 Partial Fraction Expansion of X(s) with Multiple
Poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
4.7 Inverse Laplace Transform by Partial Fraction
Expansion Using MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
4.8 Analysis of Continuous-Time LTI Systems Using
the Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
4.8.1 Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
4.8.2 Stability and Causality . . . . . . . . . . . . . . . . . . . . . . . . . 204
4.8.3 LTI Systems Characterized by Linear Constant
Coefficient Differential Equations . . . . . . . . . . . . . . . . . 207
4.8.4 Solution of linear Differential Equations Using
Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
4.8.5 Solution of Linear Differential Equations Using
Laplace Transform and MATLAB . . . . . . . . . . . . . . . . . 216
4.8.6 System Function for Interconnections
of LTI Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
4.9 Block-Diagram Representation of System Functions
in the S-Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
4.10 Solution of State-Space Equations Using Laplace
Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
4.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
4.12 MATLAB Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
xii Contents
5 Analog Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
5.1 Ideal Analog Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
5.2 Practical Analog Low-Pass Filter Design . . . . . . . . . . . . . . . . . . . 232
5.2.1 Filter Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
5.2.2 Butterworth Analog Low-Pass Filter . . . . . . . . . . . . . . . . 233
5.2.3 Chebyshev Analog Low-Pass Filter . . . . . . . . . . . . . . . . . 237
5.2.4 Elliptic Analog Low-Pass Filter . . . . . . . . . . . . . . . . . . . 245
5.2.5 Bessel Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
5.2.6 Comparison of Various Types of Analog Filters . . . . . . . . 249
5.2.7 Design of Analog High-Pass, Band-Pass,
and Band-Stop Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
5.3 Effect of Poles and Zeros on Frequency Response . . . . . . . . . . . . 264
5.3.1 Effect of Two Complex System Poles on
the Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . 264
5.3.2 Effect of Two Complex System Zeros on
the Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . 264
5.4 Design of Specialized Analog Filters by Pole-Zero
Placement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
5.4.1 Notch Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
5.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
6 Discrete-Time Signals and Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 271
6.1 The Sampling Process of Analog Signals . . . . . . . . . . . . . . . . . . 271
6.1.1 Impulse-Train Sampling . . . . . . . . . . . . . . . . . . . . . . . . 271
6.1.2 Sampling with a Zero-Order Hold . . . . . . . . . . . . . . . . . 272
6.1.3 Quantization and Coding . . . . . . . . . . . . . . . . . . . . . . . 274
6.2 Classification of Discrete-Time Signals . . . . . . . . . . . . . . . . . . . 276
6.2.1 Symmetric and Anti-symmetric Signals . . . . . . . . . . . . . 276
6.2.2 Finite and Infinite Length Sequences . . . . . . . . . . . . . . . 276
6.2.3 Right-Sided and Left-Sided Sequences . . . . . . . . . . . . . 277
6.2.4 Periodic and Aperiodic Signals . . . . . . . . . . . . . . . . . . . 277
6.2.5 Energy and Power Signals . . . . . . . . . . . . . . . . . . . . . . 279
6.3 Discrete-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
6.3.1 Classification of Discrete-Time Systems . . . . . . . . . . . . 282
6.3.2 Impulse and Step Responses . . . . . . . . . . . . . . . . . . . . . 286
6.4 Linear Time-Invariant Discrete-Time Systems . . . . . . . . . . . . . . . 286
6.4.1 Input-Output Relationship . . . . . . . . . . . . . . . . . . . . . . . 286
6.4.2 Computation of Linear Convolution . . . . . . . . . . . . . . . 288
6.4.3 Computation of Convolution Sum
Using MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
6.4.4 Some Properties of the Convolution Sum . . . . . . . . . . . . 291
6.4.5 Stability and Causality of LTI Systems
in Terms of the Impulse Response . . . . . . . . . . . . . . . . . 295
Contents xiii
6.5 Characterization of Discrete-Time Systems . . . . . . . . . . . . . . . . . 297
6.5.1 Non-Recursive Difference Equation . . . . . . . . . . . . . . . 298
6.5.2 Recursive Difference Equation . . . . . . . . . . . . . . . . . . . 298
6.5.3 Solution of Difference Equations . . . . . . . . . . . . . . . . . . 299
6.5.4 Computation of Impulse and Step Responses
Using MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
6.6 Sampling of Discrete-Time Signals . . . . . . . . . . . . . . . . . . . . . . 305
6.6.1 Discrete-Time Down Sampler . . . . . . . . . . . . . . . . . . . . 306
6.6.2 Discrete-Time Up-Sampler . . . . . . . . . . . . . . . . . . . . . . 306
6.7 State-Space Representation of Discrete-Time LTI Systems . . . . . 307
6.7.1 State-Space Representation of Single-Input
Single-Output Discrete-Time LTI Systems . . . . . . . . . . . 307
6.7.2 State-Space Representation of Multi-input
Multi-output Discrete-Time LTI Systems . . . . . . . . . . . . 309
6.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
6.9 MATLAB Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
7 Frequency Domain Analysis of Discrete-Time Signals
and Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
7.1 The Discrete-Time Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . 313
7.1.1 Periodic Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
7.2 Representation of Discrete-Time Signals and Systems
in Frequency Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
7.2.1 Fourier Transform of Discrete-Time Signals . . . . . . . . . . 316
7.2.2 Theorems on DTFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
7.2.3 Some Properties of the DTFT of a Complex
Sequence x(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
7.2.4 Some Properties of the DTFT of a Real
Sequence x(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
7.3 Frequency Response of Discrete-Time Systems . . . . . . . . . . . . . . 332
7.3.1 Frequency Response Computation Using MATLAB . . . . . 338
7.4 Representation of Sampling in Frequency Domain . . . . . . . . . . . . 344
7.4.1 Sampling of Low-Pass Signals . . . . . . . . . . . . . . . . . . . . 346
7.5 Reconstruction of a Band-Limited Signal from Its Samples . . . . . . 347
7.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
8 The z-Transform and Analysis of Discrete Time
LTI Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
8.1 Definition of the z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
8.2 Properties of the Region of Convergence for
the z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
8.3 Properties of the z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
8.4 z-Transforms of Some Commonly Used Sequences . . . . . . . . . . . 365
8.5 The Inverse z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
xiv Contents
8.5.1 Modulation Theorem in the z-Domain . . . . . . . . . . . . . . 372
8.5.2 Parsevalâ€™s Relation in the z-Domain . . . . . . . . . . . . . . . 372
8.6 Methods for Computation of the Inverse z-Transform . . . . . . . . . 374
8.6.1 Cauchyâ€™s Residue Theorem for Computation
of the Inverse z-Transform . . . . . . . . . . . . . . . . . . . . . . 374
8.6.2 Computation of the Inverse z-Transform
Using the Partial Fraction Expansion . . . . . . . . . . . . . . . 375
8.6.3 Inverse z-Transform by Partial Fraction Expansion
Using MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
8.6.4 Computation of the Inverse z-Transform
Using the Power Series Expansion . . . . . . . . . . . . . . . . 380
8.6.5 Inverse z-Transform via Power Series Expansion
Using MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
8.6.6 Solution of Difference Equations
Using the z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . 383
8.7 Analysis of Discrete-Time LTI Systems in the
z-Transform Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
8.7.1 Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
8.7.2 Poles and Zeros of a Transfer Function . . . . . . . . . . . . . 386
8.7.3 Frequency Response from Poles and Zeros . . . . . . . . . . 388
8.7.4 Stability and Causality . . . . . . . . . . . . . . . . . . . . . . . . . 389
8.7.5 Minimum-Phase, Maximum-Phase, and
Mixed-Phase Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 395
8.7.6 Inverse System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
8.7.7 All-Pass System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
8.7.8 All-Pass and Minimum-Phase Decomposition . . . . . . . . 399
8.8 One-Sided z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
8.8.1 Solution of Difference Equations with
Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
8.9 Solution of State-Space Equations Using z-Transform . . . . . . . . . 405
8.10 Transformations Between Continuous-Time Systems
and Discrete-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
8.10.1 Impulse Invariance Method . . . . . . . . . . . . . . . . . . . . . . 409
8.10.2 Bilinear Transformation . . . . . . . . . . . . . . . . . . . . . . . . 411
8.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
8.12 MATLAB Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

## Preface – Signals and Systems PDF

The signals and systems course is not only an important element for undergraduate electrical engineering students but the fundamentals and techniques of the subject are essential in all the disciplines of engineering. Signals and systems analysis has a long history, with its techniques and fundamentals found in broad areas of applications. The signals and systems is continuously evolving and developing in response to new problems, such as the development of integrated circuits technology and its applications. In this book, many illustrative examples are included in each chapter for easy understanding of the fundamentals and methodologies of signals and systems. An attractive feature of this book is the inclusion of MATLAB-based examples with codes to encourage readers to implement exercises on their personal computers in order to become confident with the fundamentals and to gain more insight into signals and systems. Signals and Systems PDF

In addition to the problems that require analytical solutions, MATLAB exercises are introduced to the reader at the end of some chapters. This book is divided into 8 chapters. Chapter 1 presents an introduction to signals and systems with basic classification of signals, elementary operations on signals, and some real-world examples of signals and systems. Chapter 2 gives time-domain analysis of continuous time signals and systems, and state-space representation of continuous-time LTI systems. Fourier analysis of continuous-time signals and systems is covered in Chapter 3. Chapter 4 deals with the Laplace transform and analysis of continuous-time signals and systems, and solution of state-space equations of continuous-time LTI systems using Laplace transform. Ideal continuoustime (analog) filters, practical analog filter approximations and design methodologies, and design of special class filters based on pole-zero placement are discussed in Chapter 5. Chapter 6 discusses the time-domain representation of discrete-time signals and systems, linear time-invariant (LTI) discrete-time systems and their properties, characterization of discrete-time systems, and state-space representation of discrete-time LTI systems. Representation of discrete-time signals and systems in frequency domain, representation of sampling in frequency domain, reconstruction of a band-limited signal from its samples, and sampling of discrete-time signals are detailed in Chapter 7. Chapter 8 describes the z-transform and analysis of LTI discrete-time systems, the solution of state-space equations of discrete-time LTI systems using z-transform, and transformations between the continuous-time systems and discrete-time systems.Signals and Systems PDF

The salient features of this book are as follows: â€¢ Provides introductory and comprehensive exposure to all aspects of signal and systems with clarity and in an easy way to understand. â€¢ Provides an integrated treatment of continuous-time signals and systems and discrete-time signals and systems. â€¢ Several fully worked numerical examples are provided to help students understand the fundamentals of signals and systems. â€¢ PC-based MATLAB m-files for the illustrative examples are included in this book. This book is written at introductory level for undergraduate classes in electrical engineering and applied sciences that are the prerequisite for upper level courses, such as communication systems, digital signal processing, and control systems. Hyderabad, India K. Deergha Rao