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River Publishers Series in Signal, Image and Speech Processing
Muhammad N. Khan, Syed K. Hasnain and Mohsin Jamil
Digital Signal Processing
A Breadth-first Approach

Digital Signal Processing:
A Breadth-First Approach

River Publishers Series in Signal, Image and Speech Processing
Volume 1
Series Editors
MONCEF GABBOUJ THANOS STOURAITIS
Tampere University of Technology University of Patras
Finland Greece
The “River Publishers Series in Signal, Image and Speech Processing” is a series
of comprehensive academic and professional books which focus on all aspects of
the theory and practice of signal processing. Books published in the series include
research monographs, edited volumes, handbooks and textbooks. The books provide
professionals, researchers, educators, and advanced students in the field with an
invaluable insight into the latest research and developments.
Topics covered in the series include, but are by no means restricted to the
following:
• Signal Processing Systems
• Digital Signal Processing
• Image Processing
• Signal Theory
• Stochastic Processes
• Detection and Estimation
• Pattern Recognition
• Optical Signal Processing
• Multi-dimensional Signal Processing
• Communication Signal Processing
• Biomedical Signal Processing
• Acoustic and Vibration Signal Processing
• Data Processing
• Remote Sensing
• Signal Processing Technology
• Speech Processing
• Radar Signal Processing
For a list of other books in this series, visit www.riverpublishers.com

Digital Signal Processing:
A Breadth-First Approach
Muhammad N. Khan
The University of Lahore
Pakistan
Syed K. Hasnain
Swedish College of Engineering and Technology
Pakistan
Mohsin Jamil
National University of Sciences and Technology
Pakistan

Published, sold and distributed by:
River Publishers
Niels Jernes Vej 10
9220 Aalborg Ø
Denmark
River Publishers
Lange Geer 44
2611 PW Delft
The Netherlands
Tel.: +45369953197
www.riverpublishers.com
ISBN: 978-87-93379-40-4 (Hardback)
978-87-93379-39-8 (Ebook)
©2016 River Publishers
All rights reserved. No part of this publication may be reproduced, stored in
a retrieval system, or transmitted in any form or by any means, mechanical,
photocopying, recording or otherwise, without prior written permission of
the publishers.

I dedicate this book to my loving family: Atika, M. Hamdaan Khan,
Ayaan Ahmad Khan, and Imaan Khan and to my parents:
Habibullah Khan and Saira Khan.

Contents
Preface xix
Acknowledgments xxiii
List of Figures xxv
List of Tables xxxi
List of Abbreviations xxxiii
1 Introduction 1
1.1 Concept of Signal Processing . . . . . . . . . . . . . . . . . 1
1.1.1 Analog Signal Processing . . . . . . . . . . . . . . 1
1.1.2 Digital Signal Processing . . . . . . . . . . . . . . . 1
1.2 Roots of DSP . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Advantages of DSP . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Basic Blocks of Signal Processing System . . . . . . . . . . 3
1.5 DSP Key Operations . . . . . . . . . . . . . . . . . . . . . 4
1.6 Classification of Signals . . . . . . . . . . . . . . . . . . . 4
1.6.1 Continuous-Time versus Discrete-Time Signals . . . 4
1.6.2 Continuous-Valued versus Discrete-Valued
Signals . . . . . . . . . . . . . . . . . . . . . . . . 4
1.6.3 Deterministic versus Random Signals . . . . . . . . 5
1.6.4 Multi-Channel and Multi-Dimensional Signals . . . 5
1.7 Application of DSP . . . . . . . . . . . . . . . . . . . . . . 6
1.7.1 Telecommunications . . . . . . . . . . . . . . . . . 6
1.7.1.1 Multiplexing . . . . . . . . . . . . . . . . 6
1.7.1.2 Compression . . . . . . . . . . . . . . . . 7
1.7.1.3 Echo control . . . . . . . . . . . . . . . . 7
1.7.2 Audio Signal Processing . . . . . . . . . . . . . . . 7
1.7.2.1 Speech generation . . . . . . . . . . . . . 7
1.7.2.2 Speech recognition . . . . . . . . . . . . 7
vii

viii Contents
1.7.3 Echo Location . . . . . . . . . . . . . . . . . . . . 7
1.7.3.1 Radar (RAdio Detection And Ranging) . . 8
1.7.3.2 Sonar (SOund Navigation
And Ranging) . . . . . . . . . . . . . . . 8
1.7.3.3 Reflection seismology . . . . . . . . . . . 9
1.7.4 Image Processing . . . . . . . . . . . . . . . . . . . 9
1.7.4.1 Medical . . . . . . . . . . . . . . . . . . 9
1.7.4.2 Space . . . . . . . . . . . . . . . . . . . . 10
2 Signals and Systems (Continuous and Discrete) 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 CT Signals . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Unit Impulse Function . . . . . . . . . . . . . . . . 12
2.2.2 Step Function . . . . . . . . . . . . . . . . . . . . . 12
2.2.2.1 Properties of unit step function . . . . . . 13
2.2.3 Ramp Function . . . . . . . . . . . . . . . . . . . . 13
2.2.4 Parabolic Function . . . . . . . . . . . . . . . . . . 13
2.2.5 Exponential Function . . . . . . . . . . . . . . . . . 13
2.2.6 Sinusoidal Function . . . . . . . . . . . . . . . . . 14
2.3 Concept of Frequency: Continuous Time Signals . . . . . . 14
2.3.1 Periodic and Aperiodic Signals . . . . . . . . . . . . 14
2.4 DT Signals . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.1 CT versus DT Signals . . . . . . . . . . . . . . . . 17
2.4.2 Unit Impulse . . . . . . . . . . . . . . . . . . . . . 17
2.4.3 Unit Step Function . . . . . . . . . . . . . . . . . . 17
2.4.4 Ramp Function . . . . . . . . . . . . . . . . . . . . 17
2.4.5 Parabolic Function . . . . . . . . . . . . . . . . . . 18
2.4.6 Exponential Function . . . . . . . . . . . . . . . . . 18
2.4.7 Sinusoidal Function . . . . . . . . . . . . . . . . . 18
2.4.8 Concept of Frequency: DT Signals . . . . . . . . . . 18
2.5 Time-Domain and Frequency-Domain . . . . . . . . . . . . 20
2.6 A/D and D/A Conversion . . . . . . . . . . . . . . . . . . . 21
2.6.1 Processing Steps for A/D Conversion . . . . . . . . 21
2.6.1.1 Sample and hold . . . . . . . . . . . . . . 21
2.6.1.2 Quantization . . . . . . . . . . . . . . . . 21
2.6.1.3 Coding . . . . . . . . . . . . . . . . . . . 22
2.6.2 Sampling of Analog Signals . . . . . . . . . . . . . 22
2.7 The Sampling Theorem . . . . . . . . . . . . . . . . . . . . 25
2.8 Quantization Error . . . . . . . . . . . . . . . . . . . . . . 29

Contents ix
2.9 Further about DT Signals . . . . . . . . . . . . . . . . . . . 31
2.9.1 Representing DT Signal . . . . . . . . . . . . . . . 31
2.9.1.1 Graphical representation . . . . . . . . . . 31
2.9.1.2 Functional representation . . . . . . . . . 32
2.9.1.3 Sequence representation . . . . . . . . . . 32
2.9.1.4 Tabular representation . . . . . . . . . . . 32
2.10 Simple Manipulations . . . . . . . . . . . . . . . . . . . . 32
2.10.1 Reflection/Folding/Flipping . . . . . . . . . . . . . 32
2.10.2 Shifting (Advance and Delayed) . . . . . . . . . . . 32
2.10.3 Scaling (Time and Magnitude) . . . . . . . . . . . . 33
2.10.4 Addition and Multiplication . . . . . . . . . . . . . 34
2.10.5 Even and Odd Signals . . . . . . . . . . . . . . . . 38
2.11 Energy and Power Signals . . . . . . . . . . . . . . . . . . 39
2.12 Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.12.1 DT Systems . . . . . . . . . . . . . . . . . . . . . . 42
2.13 System’s Representation . . . . . . . . . . . . . . . . . . . 42
2.13.1 Symbol used for DT Systems . . . . . . . . . . . . 42
2.13.2 An Adder . . . . . . . . . . . . . . . . . . . . . . . 43
2.13.3 A Constant Multiplier . . . . . . . . . . . . . . . . . 43
2.13.4 A Signalmultiplier . . . . . . . . . . . . . . . . . . 43
2.13.5 Unit Delay Element . . . . . . . . . . . . . . . . . . 43
2.13.6 Unit Advanced Element . . . . . . . . . . . . . . . 44
2.14 System’s Classification . . . . . . . . . . . . . . . . . . . . 46
2.14.1 Static versus Dynamic Systems . . . . . . . . . . . 46
2.14.2 Time-Invariant versus Time-Variant System . . . . . 46
2.14.2.1 Method to workout for time-invariant
and time-variant system . . . . . . . . . . 47
2.14.3 Linear versus Non-linear System . . . . . . . . . . . 50
2.14.3.1 Linear system . . . . . . . . . . . . . . . 50
2.14.3.2 Non-linear system . . . . . . . . . . . . . 50
2.14.4 Causal versus Non-Causal System . . . . . . . . . . 54
2.14.5 Stable versus Un-Stable System . . . . . . . . . . . 55
2.15 Problems and Solutions . . . . . . . . . . . . . . . . . . . . 56
3 Convolution and Correlation 77
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.2 The Convolution Sum . . . . . . . . . . . . . . . . . . . . . 78
3.3 Properties of Convolution . . . . . . . . . . . . . . . . . . . 80
3.3.1 Commutative Law . . . . . . . . . . . . . . . . . . 80

x Contents
3.3.2 Associative Law . . . . . . . . . . . . . . . . . . . 80
3.3.3 Distributive Law . . . . . . . . . . . . . . . . . . . 80
3.4 Application of Convolution . . . . . . . . . . . . . . . . . . 81
3.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.5 Methods of Calculating Convolution . . . . . . . . . . . . . 83
3.5.1 Convolution of Delta Function with Delta
Function . . . . . . . . . . . . . . . . . . . . . . . 83
3.5.2 Convolution of Delta Function with Step
Function . . . . . . . . . . . . . . . . . . . . . . . 84
3.5.3 Convolution of Step Function with Step
Function . . . . . . . . . . . . . . . . . . . . . . . 84
3.5.4 Linear Convolution: Function Format . . . . . . . . 85
3.5.5 Linear Convolution: Sequence Format . . . . . . . . 90
3.5.5.1 Linear convolution by graphical
method . . . . . . . . . . . . . . . . . . . 90
3.5.5.2 Linear convolution by analytical
method . . . . . . . . . . . . . . . . . . . 94
3.5.5.3 Linear convolution by matrix method . . . 97
3.5.5.4 Linear convolution by overlap and add
method . . . . . . . . . . . . . . . . . . . 98
3.5.6 Circular Convolution . . . . . . . . . . . . . . . . . 101
3.6 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.7 Properties of Correlation . . . . . . . . . . . . . . . . . . . 103
3.8 Application of Correlation . . . . . . . . . . . . . . . . . . 103
3.9 Types of Correlation . . . . . . . . . . . . . . . . . . . . . 105
3.9.1 Cross-Correlation . . . . . . . . . . . . . . . . . . . 105
3.9.2 Auto-Correlation . . . . . . . . . . . . . . . . . . . 106
3.10 Further Analysis of Cross-Correlation . . . . . . . . . . . . 108
3.11 Cross-Correlation Coefficient . . . . . . . . . . . . . . . . . 112
3.12 Correlation Methods . . . . . . . . . . . . . . . . . . . . . 116
3.12.1 Correlation by Graphical Method . . . . . . . . . . 117
3.12.2 Correlation by Analytical Method . . . . . . . . . . 119
3.12.3 Correlation by Tabular Shifting Method . . . . . . . 120
3.12.4 Correlation by Convolution Property
Method . . . . . . . . . . . . . . . . . . . . . . . . 121
3.13 Cyclic Correlation . . . . . . . . . . . . . . . . . . . . . . . 123
3.14 Further Applications of Correlation . . . . . . . . . . . . . . 125

Contents xi
4 Z-Transform 155
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.2 Z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.3 Inverse Z-Transform . . . . . . . . . . . . . . . . . . . . . 161
4.3.1 Using the Property of Z-Transform . . . . . . . . . 161
4.3.2 Using the Long Division . . . . . . . . . . . . . . . 162
4.3.3 Using Residue Method . . . . . . . . . . . . . . . . 163
4.3.3.1 When the poles are real and non-
repeated . . . . . . . . . . . . . . . . . . 163
4.3.3.2 When the poles are real and repeated . . . 164
4.3.3.3 When the poles are complex . . . . . . . . 165
4.4 Theorems and Properties of Z-Transform . . . . . . . . . . 166
4.4.1 Multiplication Property . . . . . . . . . . . . . . . . 167
4.4.2 Linearity Property . . . . . . . . . . . . . . . . . . 168
4.4.3 Time Shifting Property . . . . . . . . . . . . . . . . 169
4.4.4 Scaling Property . . . . . . . . . . . . . . . . . . . 170
4.4.5 Time Reversal Property . . . . . . . . . . . . . . . 172
4.4.6 Differentiation Property . . . . . . . . . . . . . . . 172
4.4.7 Convolution Property . . . . . . . . . . . . . . . . 173
4.4.8 Correlation Property . . . . . . . . . . . . . . . . . 175
4.4.9 Initial Value Theorem . . . . . . . . . . . . . . . . . 176
4.4.10 Final Value Theorem . . . . . . . . . . . . . . . . . 176
4.4.11 Time Delay Property (One-Sided z-Transform) . . . 177
4.4.12 Time Advance Property . . . . . . . . . . . . . . . 177
5 Solution of Difference Equation 199
5.1 Constant-Coefficient Difference Equation . . . . . . . . . . 199
5.2 Solution of Difference Equation . . . . . . . . . . . . . . . 200
5.2.1 Using Sequential Procedure . . . . . . . . . . . . . 200
5.2.2 Using Z-Transform . . . . . . . . . . . . . . . . . . 201
5.2.3 Classical Technique of Difference Equation . . . . . 201
5.2.4 The Homogeneous Solution . . . . . . . . . . . . . 202
5.2.4.1 When the auxiliary polynomial roots
are real and distinct . . . . . . . . . . . . 202
5.2.4.2 When the characteristics polynomial roots
are real and repeated . . . . . . . . . . . . 203
5.2.5 The Particular Solution of Difference
Equation . . . . . . . . . . . . . . . . . . . . . . . 204

xii Contents
5.2.6 Rules for Choosing Particular Solutions . . . . . . . 204
5.2.6.1 When the forcing function is having term
different from the value of the roots
of the auxiliary equation . . . . . . . . . . 204
5.2.6.2 When the forcing function is having same
term as in the roots of the auxiliary
equation . . . . . . . . . . . . . . . . . . 207
5.2.6.3 When the forcing function is having
sinusoidal forcing function . . . . . . . . 210
5.3 Problems and Solutions . . . . . . . . . . . . . . . . . . . 211
6 Discrete-Time Fourier Transform Discrete Fourier Transform 239
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 239
6.2 Periodic Function and Fourier Synthesis . . . . . . . . . . . 239
6.2.1 Constructing a Waveform with Sine Waves . . . . . 240
6.2.2 Constructing a Waveform with Cosine Waves . . . . 241
6.2.3 Constructing a Waveform with Cosine and Sine
Waves . . . . . . . . . . . . . . . . . . . . . . . . . 242
6.2.4 Constructing a Waveform with Sine, Cosine,
and a DC . . . . . . . . . . . . . . . . . . . . . . . 243
6.2.5 Gibbs’ Phenomenon . . . . . . . . . . . . . . . . . 244
6.3 Introduction to Fourier Transforms . . . . . . . . . . . . . 247
6.4 DT Fourier Transform . . . . . . . . . . . . . . . . . . . . . 248
6.5 Properties of the DTFT . . . . . . . . . . . . . . . . . . . . 250
6.5.1 Periodicity . . . . . . . . . . . . . . . . . . . . . . 251
6.5.2 Linearity . . . . . . . . . . . . . . . . . . . . . . . 252
6.5.3 Time Shifting . . . . . . . . . . . . . . . . . . . . . 252
6.5.4 Frequency Shifting . . . . . . . . . . . . . . . . . . 254
6.5.5 Scaling . . . . . . . . . . . . . . . . . . . . . . . . 254
6.5.6 Multiplication by n (Frequency Differentiation) . . . 254
6.5.7 Time Reversal . . . . . . . . . . . . . . . . . . . . 254
6.5.8 Convolution . . . . . . . . . . . . . . . . . . . . . 255
6.5.9 Multiplication in Time Domain . . . . . . . . . . . 256
6.5.10 Complex Conjugation and Conjugate
Symmetry . . . . . . . . . . . . . . . . . . . . . . . 256
6.5.11 Parseval’s Theorem . . . . . . . . . . . . . . . . . . 256
6.5.12 Energy Density Spectrum . . . . . . . . . . . . . . 257
6.6 Why the DFT? . . . . . . . . . . . . . . . . . . . . . . . . 260
6.6.1 Window . . . . . . . . . . . . . . . . . . . . . . . . 260

Contents xiii
6.6.2 Orthogonal Signals . . . . . . . . . . . . . . . . . . 261
6.6.3 Inside the DFT . . . . . . . . . . . . . . . . . . . . 262
6.6.4 DFT Frequencies and Frequency Resolution . . . . . 264
6.6.4.1 Spectral leakage due to correlation . . . . 265
6.6.4.2 Spectral leakage due to discontinuities . . 266
6.7 Discrete Fourier Transform . . . . . . . . . . . . . . . . . . 267
6.7.1 Inverse Discrete Fourier Transform . . . . . . . . . 271
6.7.2 DFT: Matrix Method . . . . . . . . . . . . . . . . . 272
6.7.3 IDFT: Matrix Method . . . . . . . . . . . . . . . . 274
6.8 Properties of the DFT . . . . . . . . . . . . . . . . . . . . . 275
6.8.1 Periodicity . . . . . . . . . . . . . . . . . . . . . . 275
6.8.2 Linearity . . . . . . . . . . . . . . . . . . . . . . . 276
6.8.3 Time Reversal . . . . . . . . . . . . . . . . . . . . 276
6.8.4 Circular Time Shift . . . . . . . . . . . . . . . . . . 276
6.8.5 Circular Frequency Shift . . . . . . . . . . . . . . . 276
6.8.6 Circular Convolution . . . . . . . . . . . . . . . . . 277
6.8.7 Circular Correlation . . . . . . . . . . . . . . . . . 277
6.8.8 Multiplication of Two Sequences . . . . . . . . . . 277
6.8.9 Even Functions . . . . . . . . . . . . . . . . . . . . 277
6.8.10 Odd Functions . . . . . . . . . . . . . . . . . . . . 277
6.8.11 Parseval’s Theorem . . . . . . . . . . . . . . . . . . 278
6.9 Comparison between DTFT and DFT . . . . . . . . . . . . 278
6.10 Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . 282
6.10.1 Decomposition-in-Time (DIT) FFT Algorithm . . . 283
6.10.1.1 Two-point FFT . . . . . . . . . . . . . . . 283
6.10.1.2 Four-point FFT . . . . . . . . . . . . . . 284
6.10.1.3 Eight-point FFT . . . . . . . . . . . . . . 286
6.11 Decomposition-in-Frequency (DIF) FFT Algorithm . . . . . 290
6.11.1 Two-point DFT . . . . . . . . . . . . . . . . . . . . 290
6.11.2 Four-point DFT . . . . . . . . . . . . . . . . . . . . 291
7 Structure for FIR and IIR Filters 305
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 305
7.2 Structure Form of FIR Filters . . . . . . . . . . . . . . . . . 306
7.2.1 Direct Form (Transversal) . . . . . . . . . . . . . . 306
7.2.2 Lattice Structure . . . . . . . . . . . . . . . . . . . 306
7.2.2.1 Direct form filter-to-lattice coefficients . . 310
7.2.2.2 Lattice-to-direct form coefficients . . . . . 312

xiv Contents
7.2.3 Frequency Sampling Form . . . . . . . . . . . . . . 315
7.2.4 Fast Convolution Form . . . . . . . . . . . . . . . . 316
7.3 Realization Form of IIR Filters . . . . . . . . . . . . . . . . 316
7.3.1 Direct Form I . . . . . . . . . . . . . . . . . . . . . 316
7.3.2 Direct Form II . . . . . . . . . . . . . . . . . . . . 318
7.3.3 Cascade (Series) Form . . . . . . . . . . . . . . . . 319
7.3.4 Parallel Form . . . . . . . . . . . . . . . . . . . . . 321
7.3.5 Lattice Structure for IIR Filter . . . . . . . . . . . . 324
7.3.5.1 Gray–Markel method of IIR lattice
structure for ladder coefficients . . . . . . 328
7.3.5.2 Calculation of ladder coefficients using
Gray–Markel method . . . . . . . . . . . 329
7.4 Implementation of Filters . . . . . . . . . . . . . . . . . . . 330
8 Introduction to Digital Filters 359
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 359
8.1.1 Types of Filters . . . . . . . . . . . . . . . . . . . . 359
8.1.2 Classification of Filters Development Wise . . . . . 360
8.1.3 Analog Filters . . . . . . . . . . . . . . . . . . . . . 360
8.1.4 Types of Analog Filter . . . . . . . . . . . . . . . . 360
8.2 Digital Filters . . . . . . . . . . . . . . . . . . . . . . . . . 360
8.3 Importance and Advantages . . . . . . . . . . . . . . . . . . 361
8.4 Disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . 361
8.4.1 Speed Limitation . . . . . . . . . . . . . . . . . . . 361
8.4.2 Finite Word-Length Effects . . . . . . . . . . . . . . 362
8.4.3 Limit Cycles . . . . . . . . . . . . . . . . . . . . . 362
8.4.4 Long Design and Development Times . . . . . . . . 363
8.5 Types of Digital Filters . . . . . . . . . . . . . . . . . . . . 363
8.5.1 FIR (Finite Impulse Response) Filters . . . . . . . . 363
8.5.2 IIR (Infinite Impulse Response) Filters . . . . . . . . 364
8.6 Choosing between FIR and IIR Filters . . . . . . . . . . . . 364
8.7 Tolerance Scheme of FIR and IIR Filters . . . . . . . . . . . 365
8.7.1 FIR Filters . . . . . . . . . . . . . . . . . . . . . . 365
8.7.2 IIR Filters . . . . . . . . . . . . . . . . . . . . . . . 367
8.8 FIR and IIR Filter Design Stages . . . . . . . . . . . . . . . 368
8.8.1 Filter Specification . . . . . . . . . . . . . . . . . . 368
8.8.2 Coefficient Calculation . . . . . . . . . . . . . . . . 368
8.8.3 Realization . . . . . . . . . . . . . . . . . . . . . . 369

Contents xv
8.8.4 Analysis of Finite Word-Length Effects . . . . . . . 369
8.8.5 Implementation . . . . . . . . . . . . . . . . . . . . 369
8.9 Filters Coefficient Calculation Methods . . . . . . . . . . . 369
8.9.1 FIR Filters Coefficient Calculation Methods . . . . . 369
8.9.2 IIR Filters Coefficient Calculation Methods . . . . . 370
8.9.3 Summary of Filters Coefficient Calculation
Methods . . . . . . . . . . . . . . . . . . . . . . . 370
9 Step-by-Step Design of Digital FIR Filters 373
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 373
9.2 Why is it Called the Finite Impulse Response Filter? . . . . 373
9.2.1 Features of FIR Filters . . . . . . . . . . . . . . . . 374
9.2.2 Linear-Phase Implications . . . . . . . . . . . . . . 377
9.2.2.1 Effect of phase distortion on signals . . . . 377
9.2.2.2 Phase delay . . . . . . . . . . . . . . . . 377
9.2.2.3 Group delay . . . . . . . . . . . . . . . . 378
9.3 Type of FIR Filters . . . . . . . . . . . . . . . . . . . . . . 381
9.3.1 Type-1 FIR Filter (Length of the filter N is odd) . . . 381
9.3.2 Type-II FIR Filter (Length of the filter N is even) . . 383
9.3.3 Type III-FIR Filter (Length of the filter N is odd) . . 384
9.3.4 Type-IVFIR Filter (Length of the filter N is even) . . 386
9.4 Basic Principle of FIR Filter Design . . . . . . . . . . . . . 396
9.4.1 Windows Used in FIR Filters . . . . . . . . . . . . . 397
9.4.1.1 Windowing a signal . . . . . . . . . . . . 398
9.4.1.2 Rectangular window . . . . . . . . . . . . 398
9.4.1.3 Hanning window . . . . . . . . . . . . . 398
9.4.1.4 Hamming window . . . . . . . . . . . . . 400
9.5 Design of FIR Filter using the Window Method . . . . . . . 402
9.5.1 To Find the Filter Coefficients using Window . . . . 405
9.5.2 Filter Design Steps for Non-causal Filters . . . . . . 405
9.5.3 Filter Design Steps for Causal Filters . . . . . . . . 406
9.5.4 Designing Other Types of FIR Filters . . . . . . . . 409
9.5.5 Smearing Effect . . . . . . . . . . . . . . . . . . . 415
9.5.6 Kaiser Window . . . . . . . . . . . . . . . . . . . . 419
9.5.6.1 Procedure to FIR filter design using Kaiser
windows . . . . . . . . . . . . . . . . . . 420
9.5.7 Comparison of Window Methods . . . . . . . . . . 421
9.5.8 Analysis of Different FIR Filter Types . . . . . . . . 421
9.5.9 Conclusion for the Types of FIR Filter . . . . . . . . 422

xvi Contents
9.5.10 Advantages/Disadvantages of the Window
Method . . . . . . . . . . . . . . . . . . . . . . . 422
10 Step-by-Step Design of IIR Filters 457
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 457
10.2 Analog Prototype Filters . . . . . . . . . . . . . . . . . . . 457
10.2.1 Preview of Butterworth Filter . . . . . . . . . . . . 458
10.2.2 Preview of Chebyshev Filter . . . . . . . . . . . . 460
10.2.3 Preview of Elliptic filter . . . . . . . . . . . . . . . 462
10.3 Basic Structure of IIR Filters . . . . . . . . . . . . . . . . 462
10.4 Bilinear z-Transform (BZT) Method . . . . . . . . . . . . 463
10.5 Frequency Transformation . . . . . . . . . . . . . . . . . . 466
10.6 Design of Filters for Known Transfer Function . . . . . . . 467
10.7 Design of Filters for Unknown Transfer Function . . . . . 472
10.7.1 Low-Pass Filter – Basic Concepts . . . . . . . . . 473
10.7.2 The Order (N) of the Butterworth
Approximation . . . . . . . . . . . . . . . . . . . 474
10.7.3 High-Pass Filter – Basic Concepts . . . . . . . . . 478
10.7.4 Band-Pass Filters – Basic Concepts . . . . . . . . . 482
10.7.5 Band-Stop Filters – Basic Concepts . . . . . . . . 486
10.8 Pole-Zero Placement Method . . . . . . . . . . . . . . . . 491
10.9 Impulse Invariant Method . . . . . . . . . . . . . . . . . . 495
10.9.1 Limitation of this Method . . . . . . . . . . . . . . 499
10.9.2 Illustration: Impulse Invariant Method . . . . . . . 499
11 Finite Word-Length Effects in Digital Filters 529
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 529
11.2 Methods of Representation of Binary Numbers . . . . . . . 529
11.3 Fixed-Point Arithmetic for Binary Number . . . . . . . . . 531
11.3.1 Representation of Fixed-Point Number . . . . . . . 532
11.3.1.1 Sign-magnitude form . . . . . . . . . . . 532
11.3.1.2 One’s complement form . . . . . . . . . 533
11.3.1.3 Two’s complement form . . . . . . . . . 534
11.3.2 Addition . . . . . . . . . . . . . . . . . . . . . . . 535
11.3.3 Multiplication . . . . . . . . . . . . . . . . . . . . 537

Contents xvii
11.4 Floating-Point Number Representation . . . . . . . . . . . 537
11.5 Comparison of Fixed- and Floating-Point
Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . 539
11.6 Block Floating-Point Numbers . . . . . . . . . . . . . . . 539
11.7 The Quantization Noise . . . . . . . . . . . . . . . . . . . 539
11.7.1 Quantization Error Due to Truncation
and Rounding . . . . . . . . . . . . . . . . . . . . 540
11.7.2 Truncation . . . . . . . . . . . . . . . . . . . . . . 540
11.7.2.1 Truncation using two’s complement
representation . . . . . . . . . . . . . . 541
11.7.2.2 Truncation using one’s complement
representation . . . . . . . . . . . . . . 543
11.7.3 Rounding . . . . . . . . . . . . . . . . . . . . . . 545
11.8 The Input Quantization Error . . . . . . . . . . . . . . . . 546
11.9 The Coefficient Quantization Error . . . . . . . . . . . . . 547
11.10 Effects in FIR Digital Filters . . . . . . . . . . . . . . . . 549
12 Review Question with Answers and Multiple Choice Questions 563
12.1 Review Questions with Answers . . . . . . . . . . . . . . 563
12.2 Multiple Choice Questions . . . . . . . . . . . . . . . . . 597
13 Examination Question Papers 615
13.1 Practice Question Paper 1 . . . . . . . . . . . . . . . . . . 615
13.10 Practice Question Paper 10 . . . . . . . . . . . . . . . . . 640

xviii Contents
References 655
Index 667
About the Authors 669

Preface
With the rate at which technology is advancing and the level of research being
conducted in various fields, particularly Digital Signal Processing (DSP), it is
vital to keep up with the times.
By the grace of almighty Allah, We are able to complete my book in
the form of Fourth edition, which we feel it is complete course book for
undergraduate students. It has been our prime aim to streamline the flow of
the book by connecting the numerical problems with the theory in a manner
which will be most beneficial to the student. We wish to thank all our students
and colleagues in suggesting improvements for this book. The organization
of the book is as follows:
Chapter 1 includes the introduction of Digital Signal Processing, with a brief
history of DSP, classification of signals and application of DSP signal.
Chapter 2 is devoted entirely to the characterization and analysis of linear
timeinvariantcontinuousanddiscretetimesignalsandsystems,blockdiagram
representation of discrete time systems, which can be left if the students
have already gone through the course of Signals and Systems in previous
semesters.
Chapter 3 is devoted to impulse response, convolution sum, application of
convolution, properties of convolution, and different methods and techniques
of finding convolution, correlation and its properties, application, and corre-
lation coefficients.
Chapter 4 plays an important role in signal processing applications which
brushes up the Z-transform, its properties which has been introduced are used
to solve the problem, three cases of inverse Z-Transform, classical method for
solution of difference equations with different forcing functions has also been
included.
xix

xx Preface
Chapter 5 describes the classical method of solution of difference equation.
Classical method for solution of difference equations with different forcing
functions has also been included.
Chapter 6 covers Fourier synthesis, discrete time Fourier transform (DTFT),
discrete Fourier transform (DFT), and its properties and finally fast Fourier
transform (FFT) of radix-2 by two methods, decimation in time fast Fourier
transform (DIT-FFT) and decimation in frequency fast Fourier transform
(DIF-FFT).
Chapter 7 focuses on the realization structures of the FIR and IIR digital filter
using interconnection of basic building block, few basic structures of the filter
has been included to give basic concepts of it such as direct form-1, direct
form-2, cascade, parallel and lattice, and lattice-ladder form.
Chapter 8 gives a basic knowledge about filter types used in DSP, criteria
for selecting digital filters, design steps, advantage and disadvantage of Finite
Impulse Response (FIR) and Infinite Impulse Response (IIR) filters.
Chapter 9 is devoted to FIR filter design consideration. A simple approach
of designing filter coefficients has been described including FIR low pass,
high pass, band pass, and band stop filters using window methods have been
analyzed in detail for all type of filters non-causal and causal results of filter
coefficient calculation using Matlab has also been included.
Chapter 10 is developed for IIR Filter design consideration of analog filters;
step by step method of designing the digital IIR filter employing Butterworth
and Chebyshev approximation. Bilinear transformation, Impulse invariance,
and Pole placement methods for calculation of coefficients has been described
using the help of MATLAB.
Chapter 11 is finite word length effects in digital filters, which we think
should be inducted at undergraduate level, covering fixed point and floating
point numbers, quantization noise, etc. Although this chapter is not included
in the course at undergraduate level, it has been included here to provide basic
knowledge regarding this chapter.
Chapter 12 provides a 245 review questions and 85 multiple choice questions
to test the knowledge of students.

Preface xxi
Chapter 13 includes sample test papers for practicing by the students; these
questions can be given to students as assignment work.
Audience
This textbook is for a first course on DSP. It can be used in both computer
science and electrical engineering departments. In terms of programming
languages, the book assumes only that the student may have basic experience
with MATLAB or C language. Although this book is more precise and
analytical than many other introductory DSP texts, it uses mathematical
concepts that are taught in higher secondary school.We have made a deliberate
effort to avoid using most advanced calculus, probability, or stochastic process
concepts (although we’ve included some basic and homework problems for
students with this advanced background). The book is, therefore, appropriate
for undergraduate courses. It should also be useful to practitioners in the
telecommunications industry.
Unique about This Textbook
The subject of DSP is enormously complex, involving many concepts, proba-
bilities, and signal processing that are woven together in an intricate manner.
To cope with this scope and complexity, many DSP texts are often organized
around the “numerical examples” of a communication system. With such
organization, students can see through the complexity of DSP they learn about
the distinct concepts and protocols in one part of the communication system
while seeing the big picture of how all parts fit together. From a pedagogical
perspective, our personal experience has been that such approach indeed
works well.
Special Features for Students and Instructors
MATLAB includes several Signal Processing features and is an important tool
for illustrating many of the field’s applications. The use of MATLAB has been
linked to some aspects of this book to assist students in their understanding
and to give them confidence in the subject.
MATLAB is not a pre-requisite for this book. Its working is described
in sections where it is utilized. For further specifics the help documentation
is available online from Mathworks (http://www.mathworks.com), which is
easy to use and contains many examples. Our experience has shown that
signal processing students completely unfamiliar with MATLAB are able to
use MATLAB within a week or two of exposure to tutorial exercises.

xxii Preface
Every attempt has been made to ensure the accuracy of all material of the
book. Readers are highly welcomed for a positive criticism and comments.
Any suggestions or error reporting can be sent to dr.nasirkhan@ucp.edu.pk
One Final Note: We’d Love to Hear from You
We encourage students and instructors to e-mail us with any comments they
might have about our book. It’s been wonderful for us to hear from so many
instructors and students from around the world about our first international
edition. We also encourage instructors to send us new homework problems
(and solutions) that would complement the current homework problems. We
also encourage instructors and students to create new MATLAB programs
that illustrate the concepts in this book. If you have any topic that you think
would be appropriate for this text, please submit it to us. So, as the saying
goes, “Keep those cards and letters coming!” Seriously, please do continue to
send us interesting URLs, point out typos, disagree with any of our claims,
and tell us what works and what doesn’t work. Tell us what you think should
or shouldn’t be included in the next edition.

Acknowledgments
Since we began writing this book in 2013, many people have given us
invaluable help and have been influential in shaping our thoughts on how to
best organize. We want to say A BIG THANKS to everyone who has helped
us in drafting this book. Our special thanks to:
• Dr. Hammad Omer from COMSATS, Islamabad who has helped me to
in thorough checking of the manuscript of the book for the International
edition.
• Dr. Mohsin Jamil, Deputy Head of Department, School of Manufactur-
ing and Mechanical Engineering, National University of Science and
Technology, Islamabad for keeping my moral up to keep writing and
compiling such engineering books for undergraduate students.
• Prof. Dr. Mansoor-U-Zafar Dawood from King Abdul Aziz University,
Kingdom of Saudi Arabia for giving the manuscript a final reading for
the International edition.
• Prof. Dr. Jameel Ahmad, Chairman, Department of Electrical Engineer-
ing, HITEC University Islamabad for checking the manuscript for this
edition.
• Dr. Ishtiaq Ahmad, Assistant Professor, Department of Electrical Engi-
neering, The University of Lahore, Lahore for checking the manuscript
for this edition.
• Ms Tarbia Iftikhar, my student of the University of Lahore at graduate
level, for helping me in writing and editing my book.
• Dr. Kamran Ezdi, Assistant Professor, Electrical Engineering Depart-
ment, University of Central Punjab, Lahore for technical support during
preparation of this book.
• Dr. Ghulam Abbas, Assistant Professor, Electrical Engineering Depart-
ment, The University of Lahore, Lahore for technical support during
preparation of this book.
• Mr. Farhan Abbas Jaffery from OGDCL and Mr. Tajammul Ahsan
Rasheed from SNGPL for designing the cover of this edition of the book.
xxiii

xxiv Acknowledgments
• Mr. Qaiser Mahmood for typing this book for first and second edition
with great zeal and enthusiasm.
We also want to thank the entire publisher team, who has done an absolutely
outstanding job on this International edition. Finally, most special thanks go
to the editor. This book would not be what it is (and may well not have been
at all) without their graceful management, constant encouragement, nearly
infinite patience, good humor, and perseverance.
Muhammad N. Khan
S. K. Hasnain

List of Figures
Figure1.1 Fuzzy and overlapping boundaries of DSP. . . . . . 3
Figure1.2 Digital system. . . . . . . . . . . . . . . . . . . . 4
Figure2.1 Basic blocks of an A/D converter. . . . . . . . . . 22
Figure2.2 (a) Sampler and (b) Periodic sampling of an analog
signal. . . . . . . . . . . . . . . . . . . . . . . . . 23
Figure2.3 Time (or sample) number: (a) Analog Frequency =
0.0 (i.e., DC). (b) Analog frequency = 0.09 of
sample rate. (c) Analog Frequency = 0.31
of sampling rate. (d) Analog Frequency = 0.95
of sampling rate. . . . . . . . . . . . . . . . . . . 26
Figure2.4 (a) Sampling with frequency of 1 Hz. . . . . . . . 30
Figure2.4 (b) Illustration of quantization by rounding off. . . 30
Figure2.5 Graphical representation of signal. . . . . . . . . . 31
Figure2.6 (a) Signal x(n) and (b) Reflected signal x(−n). . . 33
Figure2.7 Original signal x(n), delayed by 3 units
and advanced by 1 unit. . . . . . . . . . . . . . . . 33
Figure2.8 (a) Odd signal; (b) Even signal. . . . . . . . . . . . 38
Figure2.9 Representation of a discrete time system. . . . . . 42
Figure2.10 (a) Graphical representation of adder. . . . . . . . 43
Figure2.10 (b) Graphical representation of a constant
multiplier. . . . . . . . . . . . . . . . . . . . . . . 43
Figure2.10 (c) Graphical representation of a multiplier. . . . . 43
Figure2.10 (d) Unit delay element. . . . . . . . . . . . . . . . 44
Figure2.10 (e) Unit delay element. . . . . . . . . . . . . . . . 44
Figure2.11 Time-invariant (a) and some time-variant
systems (b)–(d). . . . . . . . . . . . . . . . . . . . 47
Figure2.12 Graphical representation of the superposition
principle, H is linear if and only if y(n) = y(n). . 50
Figure3.1 A LTI system with and without delay. . . . . . . . 78
Figure3.2 Discrete time signals for convolution. . . . . . . . 79
Figure3.3 Commutative law. . . . . . . . . . . . . . . . . . . 80
xxv

xxvi List of Figures
Figure3.4 Associative law. . . . . . . . . . . . . . . . . . . . 80
Figure3.5 Distributive law. . . . . . . . . . . . . . . . . . . 80
Figure3.6 (a, b) Sample of a sine wave and sample of a sine
wave with noise added to it, and (c, d) Coefficients
used for averaging the noisy sine-wave signal
and result of convolution between the
coefficients and the noisy sine wave. . . . . . . . . 83
Figure3.8 A sine wave and its auto-correlation. The auto-
correlation of a sine wave is a sinc function. . . . . 104
Figure3.9 Random noise and its auto-correlation: the auto-
correlation of random noise is an impulse. . . . . . 105
Figure3.10 Sine wave buried in noise and its auto-correlation.
The autocorrelation clearly shows the presence of
noise and a periodic signal. . . . . . . . . . . . . . 105
Figure3.11 Auto-correlation function of a random waveform. . 107
Figure3.12 Waveform x2 = x1 + j shifted j lags to the left of
waveform x1. . . . . . . . . . . . . . . . . . . . . 108
Figure3.13 The effect of the end-effect on the cross-correlation
r12( j). . . . . . . . . . . . . . . . . . . . . . . . . 111
Figure3.14 Pairs of waveforms {x1(n), x2(n), x3(n), x4(n)}
of different magnitudes but equal cross-correlations. 113
Figure4.1 (a) The exponential signal x(n) = αnu(n) and
(b) the ROC of its z-transform. . . . . . . . . . . 160
Figure4.2 (a) Anticausal signal x(n) = −αnu(−n − 1),
(b) ROC of its z-transform . . . . . . . . . . . . . 187
Figure6.1 (a) Sine wave 1 with f = 2 Hz, amplitude = 7;
(b) Sine wave 2 with f = 4 Hz, amplitude = 2;
(c) Sine wave 3 with f = 6 Hz, amplitude = 4
and (d) Addition of sine waves 1, 2, 3. . . . . . . . 240
Figure6.2 (a) Cosine wave 1 with f = 2 Hz, amplitude = 7;
(b) Cosine wave 2 with f = 4 Hz, amplitude = 2;
(c) Cosine wave 3 with f = 6 Hz, amplitude = 4
and (d) Addition of cosine waves in (a), (b),
and (c). The maximum value occurs at t = 0. . . . . 241
Figure6.3 Combination of sine waves from 6.1(a)–(c) and the
cosine waves from 6.2(a)–(c). The value at t = 0
is neither 0 nor maximum, but somewhere
in between. . . . . . . . . . . . . . . . . . . . . . 242

List of Figures xxvii
Figure6.4 (a) Square wave with fundamental harmonic;
(b) Square wave with fundamental and third
harmonic; (c) Square wave with fundamental, third,
fifth harmonic; (d) Square wave with fundamental,
third, fifth and seventh harmonic and (e) Square
wave with fundamental, third, fifth, seventh
and ninth harmonic. . . . . . . . . . . . . . . . . . 246
Figure6.5 (a) Sample of one cycle of a sine wave; (b) Sample
of two cycles of a sine wave; (c) Sample of one cycle
of a sine wave; (d) Sample of three cycles of a sine
wave and (e) Inside DFT. . . . . . . . . . . . . . . 261
Figure6.6 Samples of the windowed sine wave and samples of
one cycle of a sine wave (f = 1 Hz) used for
correlation. . . . . . . . . . . . . . . . . . . . . . 262
Figure6.7 Samples of the windowed sine wave and samples
of two cycles of a sine wave (f = 2 Hz) used
for correlation. . . . . . . . . . . . . . . . . . . . 263
of three cycles of a sine wave (f = 3 Hz) used
for correlation. . . . . . . . . . . . . . . . . . . . 263
of one cycle of a cosine wave (f = 1 Hz) used
for correlation. . . . . . . . . . . . . . . . . . . . 264
of two cycles of a cosine wave (f = 2 Hz) used
for correlation. . . . . . . . . . . . . . . . . . . . 264
of three cycles of a cosine wave (f = 3 Hz) used
for correlation. . . . . . . . . . . . . . . . . . . . 264
Figure6.12 (a) x(nT) versus t. . . . . . . . . . . . . . . . . . . 269
Figure6.12 (b) |X(k)| versus k. . . . . . . . . . . . . . . . . . . 270
Figure6.12 (c) φ(k). . . . . . . . . . . . . . . . . . . . . . . . 270
Figure6.13 Butterfly diagram for a 2-point DFT. . . . . . . . . 284
Figure6.14 Signal flow graph for a four-point DFT. . . . . . . 285
Figure6.15 Decomposition-in-time fast Fourier transforms. . . 289
Figure6.16 A general decomposition-in-frequency FFT. . . . . 293
Figure6.17 A 4-point Decomposition-in-frequency FFT. . . . . 293
Figure7.1 The direct form of FIR filter. . . . . . . . . . . . . 306

xxviii List of Figures
Figure7.2 The lattice structure for FIR filter. . . . . . . . . . 307
Figure7.3 An N-stage FIR lattice structure. . . . . . . . . . . 310
Figure7.4 Realization structures for FIR filters: frequency
sampling structure. . . . . . . . . . . . . . . . . . 315
Figure7.5 Realization structures for FIR filters: fast
convolution. . . . . . . . . . . . . . . . . . . . . . 316
Figure7.6 Direct form-I realization: k-th order H(z). . . . . . 317
Figure7.7 Direct form-I realization: second-order H(z). . . . 317
Figure7.8 Direct form II realization: (a) kth-order H(z);
(b) second-order H(z). . . . . . . . . . . . . . . . 319
Figure7.9 Cascade or series realization. . . . . . . . . . . . . 319
Figure7.10 Direct form I realization of fourth order IIR filter. . 320
Figure7.11 Cascade realization of a fourth order IIR filter. . . . 320
Figure7.12 Parallel realization of a fourth-order IIR filter. . . . 321
Figure7.13 Single pole lattice filter structure. . . . . . . . . . . 326
Figure7.14 Two pole lattice filter structure. . . . . . . . . . . . 326
Figure7.15 Lattice ladder structure of a pole-zero system. . . . 328
Figure7.16 Lattice ladder structure of a pole-zero system. . . . 329
Figure8.1 Tolerance scheme for a low-pass FIR filter. . . . . 366
Figure8.2 Tolerance scheme for an IIR band-pass filter. . . . 367
Figure9.1 Type-I FIR filter: Length of the filter N is odd,
positive symmetry. . . . . . . . . . . . . . . . . . 382
Figure9.2 Type-II FIR filter: Length of the filter N is even,
positive symmetry. . . . . . . . . . . . . . . . . . 384
Figure9.3 Type-III FIR filter: Length of the filter N is odd,
negative symmetry. . . . . . . . . . . . . . . . . . 386
Figure9.4 Type-IV FIR filter: Length of the filter N is even,
negative symmetry. . . . . . . . . . . . . . . . . . 388
Figure9.5 Hanning expression non-causal ω(n) = 0.5
(1 + cos 2nπ/N − 1). . . . . . . . . . . . . . . . 399
Figure9.6 Hanning expression causal ω(n) = 0.5
(1 − cos 2nπ/N − 1). . . . . . . . . . . . . . . . 399
Figure9.7 Hamming expression non-causal ω(n) = 0.54 + 0.46
cos 2πn/(N − 1). . . . . . . . . . . . . . . . . . . 401
Figure9.8 Hamming expression causal ω(n) = 0.54 − 0.46 cos
2πn/(N − 1). . . . . . . . . . . . . . . . . . . . . 401
Figure9.9 (a, b, c, d) Ideal frequency response of low-pass,
high-pass, band-pass and bandstop filters.
(e) Impulse response of an ideal low-pass filter. . . 403

List of Figures xxix
Figure9.10 (a) Coefficients of a low-pass FIR filter, non-causal.
(b) Magnitude response of a low-pass FIR filter. . . 406
Figure10.1 Sketches of frequency responses of some classical
analog filters (a) Butterworth; (b) Chebyshev
type I; (c) Chebyshev type II; and (d) Elliptic. . . . 458
Figure10.2 (a) An illustration of the s-plane (s = σ + jΩ) to z-
plane mapping using the bilinear Z-transformation
(z = σ + jω). Note that +jΩ axis maps to the
upper half and the negative jΩ axis maps to the
lower half unit circle in the z-plane and (b) Relation-
ship between analog and digital frequencies shows
the warping effect. Notice that the equally spaced
analog pass bands are pushed together at the high
frequency and, after transformation, in the digital
domain. . . . . . . . . . . . . . . . . . . . . . . . 465
Figure10.3 Relationships between frequencies in the
denormalized LP and prototype LP filters. . . . . . 473
Figure10.4 Relationships between frequencies in the
denormalized HP and prototype LP Filters. . . . . . 478
Figure10.5 Mapping of the prototype LP to BPF. . . . . . . . . 483
Figure10.6 Relationship between the frequencies in the
denormalized BS and prototype LP filters. . . . . . 487
Figure11.1 Block diagram of an A/D converter. . . . . . . . . 540
Figure11.2 Probability density functions P(e) for truncation
for fixed point and floating point. . . . . . . . . . . 544
Figure11.3 Probability density functions P(ε) for rounding. . . 546
Figure11.4 (a) Quantizer characteristics with rounding.
(b) Probability density functions for roundoff
error. . . . . . . . . . . . . . . . . . . . . . . . . . 547
Figure11.5 (a) Quantizer characteristics with two’s
complement truncation. (b) Probability
density function of roundoff error. . . . . . . . . . 548
Figure11.6 Direct form realization of an FIR system (a) Ideal
(b) Fixed-point roundoff noise mode. . . . . . . . . 550
Figure11.7 Cascade realization of an FIR system (a) Ideal
(b) Fixed-point roundoff noise mode. . . . . . . . . 551
Figure12.1 A typical DSP system. . . . . . . . . . . . . . . . 599
Figure12.2 Two sine waves of frequencies 2 and 6 kHz, each
of peak amplitude 1 V. . . . . . . . . . . . . . . . 599

xxx List of Figures
Figure12.3 A waveform in one part of a typical DSP system. . 599
Figure12.4 Two sine waves of frequencies 2 and 6 kHz, each
of peak amplitude 0.5 V. . . . . . . . . . . . . . . 600
Figure12.5 Samplers of a sine wave. . . . . . . . . . . . . . . 600
Figure12.6 Sine wave of frequency 2 kHz. . . . . . . . . . . . 600
Figure12.7 Pressure wave with Alternate compressions
and rarefactions. . . . . . . . . . . . . . . . . . . . 600
Figure12.8 A filter with one delay, two adders, and three
multipliers. . . . . . . . . . . . . . . . . . . . . . 611

List of Tables
Table2.1 A comparison of energy and power signal . . . . . 40
Table3.1 Closed form expression for some commonly used
series in convolution . . . . . . . . . . . . . . . . 86
Table4.1 Z-transform . . . . . . . . . . . . . . . . . . . . . 161
Table4.2 Properties of Z-transform . . . . . . . . . . . . . . 167
Table5.1 Rules for choosing particular solutions . . . . . . . 205
Table6.1 Used for selection of the Fourier series coefficients 245
Table6.2 Properties of DTFT . . . . . . . . . . . . . . . . . 251
Table6.3 Computational cost . . . . . . . . . . . . . . . . . 289
Table6.4 DFT and FFT comparison (number of complex
multiplications required) . . . . . . . . . . . . . . 289
Table8.1 A comparison of the characteristics of FIR and IIR
filters . . . . . . . . . . . . . . . . . . . . . . . . 365
Table9.1 Asummary of the key points of the four types of FIR
filters . . . . . . . . . . . . . . . . . . . . . . . . 393
Table9.2 Summary of ideal responses for standard frequency
selective filters . . . . . . . . . . . . . . . . . . . 404
Table9.3 Summary of important features of common window
functions . . . . . . . . . . . . . . . . . . . . . . 416
Table11.1 Comparison of fixed- and floating-point arithmetic 539
Table11.2 Quantization error ranges due to truncation and
rounding . . . . . . . . . . . . . . . . . . . . . . . 545
xxxi

List of Abbreviations
ACF Auto-Correlation Function
ADC A/D (Analog to Digital Converter)
BZT Bilinear Z-Transform
CAD Computer Aided Design
CCF Cross Correlation Function
CT Continues Time
CTFT CT Fourier Transforms
DAC D/C (Digital to Analog Converter)
DFS Discrete Fourier Series
DIF Decomposition in Frequency
DIT Decomposition in-Time
DSP Digital Signal Processing
DT Discrete Time
DTFS DT Fourier Series
DTFT Discrete Time Fourier Transforms
ECG Electrocardiogram
FFT Fast Fourier Transform
FIR Finite Impulse Response
IC Integrated Circuit
IDTFT Inverse DTFT
IIR Infinite Impulse Response
LTI Linear Time Invariant
MRI Magnetic Resonance Imaging
PDF Probability Density Function
RADAR RAdio Detection And Ranging
RC Resistance Capacitance
RMS Root Mean Square
ROC Region of Convergence
SNR Signal-to-Noise Ratio
SONAR SOund Navigation And Ranging
VLSI Very Large Scale Integration
xxxiii

1
Introduction
This chapter covers fundamental concepts of analog signal processing and
digital signal processing, history of digital signal processing, basic definition,
advantages of the signal processing in basic systems, basic blocks, key
operation, and classification of signals along with applications of digital signal
processing.
1.1 Concept of Signal Processing
Signal Processing is basically the analysis, interpretation, and manipulation of
signals. It is the manipulation of the basic nature of a signal to get the desired
shaping of the signal at the output. It is concerned with the representation,
transformation, and manipulation of signals and the information they contain.
Signal processing can be grouped into two classes:
• Analog Signal Processing
• Digital Signal Processing
1.1.1 Analog Signal Processing
It is the analysis of analog signals through analog means. In analog signal
processing, continuous time signals are processed. Different types of analog
signals are processed through low-pass filters, high-pass filters, band pass
filters, and band stop filters to get the desired shaping of the input signal.
1.1.2 Digital Signal Processing
Digital signal processing is the numerical processing of signals on a digital
computer or some other data processing machine. Digital signal processors
(DSPs) take real-world signals like voice, audio, video, temperature, pressure,
or position that have been digitized and then mathematically manipulate
1

2 Introduction
them. For example, a digital system such as a digital computer takes a signal
in discrete-time sequence form and converts it into a discrete time output
sequence.
The environment is full of signals that we sense; examples including the
sound, temperature and light.
In case of sound we use our ears to convert into electrical signal in to our
brain. We then analyze properties such as frequency, amplitude, and phase
to categorize the sound and determine its direction. We may recognize it as
music, speech or noise of a machine.
In case of temperature our nerves are exposed through skin will send
signals to the brain. The example includes in this case is witch on of a heating
or opening a window.
In case of light, our eyes focus the image into the retina, which converts it
into electrical signal to send to the brain. Our brain analyses the color, shape,
intensity, etc.
The processing which apply to the signals is carried out by the digital
computer and is thus called digital signal processing (DSP).
Digital signal processing involves the extraction of information from
signals which in turn, depends upon the type of signal and the nature of
information it carries. In the case of a calculator, the application of different
operators on a set of values comes under DSP.
It covers the mathematics, the algorithms, and the techniques used to
influence and control signals after they have been converted into digital form.
Visual images, recognition and generation of speech, compression of data for
storage and transmission are some of its applications.
1.2 Roots of DSP
Owing to the high level of research conducted, the subject of DSP has
developed very rapidly over the last few decades. This rapid development
has been a result of significant advances in digital computer technology and
IC fabrication techniques.
Since the advent of computers in the 1960s, time and money have been
invested in incorporating DSP in all the sections of engineering. The efforts
were first made in four key areas, namely
1. Radar and Sonar (where national security was at risk)
2. Oil Exploration (where large amounts of money could be made)
3. Space Exploration (where the data are irreplaceable)
4. Medical Imaging (where lives could be saved)

1.4 Basic Blocks of Signal Processing System 3
Figure 1.1 Fuzzy and overlapping boundaries of DSP.
Digital Signal Processing has ties to many other areas of science and
engineering as shown in Figure 1.1. Hence, for a thorough knowledge of the
subject, it is vital to have some level of exposure to these other fields.
1.3 Advantages of DSP
What is it about DSP that makes it so popular? The answer to it is not short
and clear. There are many advantages in using digital technique for general
purpose signal processing.
1. Digital programmable systems allow flexibility. DSP programs can be
configured by simply making alterations in our program. Reconfiguration
of an analog system usually implies a redesign of the hardware.
2. Digital signal processing systems exhibit high accuracy.
3. DSP programs can be stored on magnetic media (disk) without any loss
in signal. As a consequence, the signals become portable and can be
processed off-line in a remote laboratory.
4. Processing in DSP reduces the cost by time-sharing of the processor
among a number of signals.
5. Digital circuits are less sensitive to tolerance of a component value.
6. The implementation of highly sophisticated signal processing algorithms
is made possible with DSP. It is very difficult to perform precise
mathematical operations on signals in the analog form.
1.4 Basic Blocks of Signal Processing System
Most signals are analog in nature. For us to apply DSP on these signals, it is
vital to efficiently bring these signals to the digital realm. Thus, there is a need
for an interface between the analog and the digital signal processor as shown
in Figure 1.2.

4 Introduction
Figure 1.2 Digital system.
The interface that performs this operation is called the analog-to-digital
(A/D) converter. The digital output is inputted to a digital processor as per our
requirements.
1.5 DSP Key Operations
There are five main principles in DSP operations that need to be studied to
familiarize with the field of DSP, which are as under:
• Convolution,
• Correlation,
• Filtering,
• Transformation, and
• Modulation.
1.6 Classification of Signals
The characteristic attributes of a specific signal determine which methods to
employ in processing a signal or analyzing the response of a system to a signal.
There are techniques that apply only to specific families of signals.
1.6.1 Continuous-Time versus Discrete-Time Signals
Continuous-time (or analog) signals exist for the continuous interval (a, b)
where a can be –∞ and b can be ∞. Discrete-time signals exist only for
certain specific instances of time. These time instants may not be equidistant,
but in practice, they are usually taken at equally spaced intervals for ease of
calculations and mathematical tractability.
1.6.2 Continuous-Valued versus Discrete-Valued Signals
Continuous-time or discrete-time signals may give values that are continuous
or discrete. If a signal takes on all possible values on a finite or an infinite range,

1.6 Classification of Signals 5
it is said to be a continuous-valued signal. A discrete-time signal having a set
of discrete values is called a digital signal.
1.6.3 Deterministic versus Random Signals
A deterministic signal is one that can be uniquely described by an explicit
mathematical expression, a table of data, or a well-defined rule. Signals that
cannot be described to any reasonable degree of accuracy by explicit math-
ematical formulas are of little practical use. Some examples of deterministic
signals are:
x(t) = bt. (1.1)
The above equation represents a ramp signal whose amplitude increases
linearly with time and the slope is b.
x(n) = A sin ωn. (1.2)
In the above case, the amplitude varies sinusoidally with time and has
maximum amplitude of A. For such signals, it can be seen that the amplitude
can be predicted in advance for any time instant. Hence, the signals represented
above are deterministic signals.
A non-deterministic signal, on the other hand, is one whose occurrence is
always random in nature. The absence of a relationship in these signals implies
that they evolve over time in an unpredictable manner; such signals are also
called random signals.
A typical example of non-deterministic signals is thermal noise generated
in an electric circuit. Such a signal has probabilistic behavior.
1.6.4 Multi-Channel and Multi-Dimensional Signals
A signal is expressed by a function of one or more independent variables.
In the case of electrocardiograms, 3 lead and 12 lead ECG are often used.
This results in 3 channel and 12 channel signals. If the signal is a function
of a single independent variable, the signal is called a one-dimensional
signal.
One example of a two-dimensional signal is a picture, since the intensity or
the brightness I(x, y) at each point is a function of two-independent variables.
Furthermore, since the brightness is a function of time, it may be represented as
I(x, y, t). Hence, the TV picture may be treated as a three-dimensional signal.
The color TV picture may be described by three intensity function I(x, y, t),
I(x, y, t), and I(x, y, t) corresponding to the brightness of the three principle

6 Introduction
colors (red, green, and blue) as a function of time. The color TV picture is,
thus, a three channel, three-dimensional signal, which can be represented by
the vector.
I(x, y, t) =
⎡
⎣
Ir(x, y, t)
Ig(x, y, t)
Ib(x, y, t)
⎤
⎦
1.7 Application of DSP
High-resolution spectral analysis has created various application areas in
DSP. It requires a high-speed processor to implement the Fast Fourier Trans-
form (FFT).
New applications are being added to DSP all the time. The sound pro-
duction in home theatre systems employs DSP. Digital Computers are used
to process the images of Mars sent back to Earth by the Mars pathfinder at
the National Aeronautics and Space Administration (NASA). Any area where
information is handled in digital form or controlled by a digital processor is
working on the principles of DSP.
Even the special effects in movies such as Pan’s Labyrinth, Harry Potter,
and the Lord of the rings are created using special purpose digital computers
and softwares. The generation of the cartoon characters and the lighting and
shading effects in computer animation movies such as Shrek and Wall-E have
all been carried out digitally.
Signaling tone generation and detection, frequency band shifting, filter-
ing to remove power line hum etc., are all implemented by DSP. Further
application areas are discussed below.
1.7.1 Telecommunications
Let us consider the three examples of multiplexing, compression, and echo
control present in the telephone network.
1.7.1.1 Multiplexing
Here, audio signals are converted into a stream of serial digital data by the
process of DSP.The property of bits to be easily intertwined and later separated
allows telephone conversations to be transmitted on a single channel.
The financial advantage of transmitting digitally is enormous. Digital logic
gates are far cheaper than wires and analog switches.

1.7 Application of DSP 7
1.7.1.2 Compression
A voice signal is digitized at 8000 samples/s. Much of this digital information
is superfluous. For this reason, several DSP algorithms have been developed
to convert digitized voice signals into data streams that require fewer bits
per second. These are called data compression algorithms. Matching of un-
compressed algorithms is used to restore the signal to its original form.
1.7.1.3 Echo control
Echoes are a common problem faced in long distance telephone connections.
With increasing distances, the echo becomes even more noticeable and
irritating. It is particularly objectionable for intercontinental communication,
where the delay can be several hundred milliseconds.
Digital signal process tackles this problem by measuring the returned
signal and generating an appropriate anti-signal to cancel the offending echo.
This technique allows speakerphone users to hear and speak at the same time
without fighting audio feedback (squealing). In this way, environmental noise
can also be reduced by cancelling it with digitally generated anti-noise signal.
1.7.2 Audio Signal Processing
The area of Speech Signal Processing has been revolutionized by the
introduction of DSP.
1.7.2.1 Speech generation
Speech generation and recognition are used for the communication between
humans and machines. Generated speech systems provide digital recording
and vocal tract simulation.
1.7.2.2 Speech recognition
Digital signal processing approaches the problem of voice recognition in two
steps: Feature extraction followed by feature matching.
1.7.3 Echo Location
Acommon method of obtaining information about a remote object is to bounce
a wave of it. For example, radar operates by transmitting pulses of radio waves
and examining the received signal for echoes from aircraft.
In sonar applications, submarines, and other submerged objects are
detected by transmitting sound waves through the water. Geophysicists have

8 Introduction
long probed the earth by setting off explosions and listening for the echoes
from deeply buried layers of rock. While these applications have a common
thread, each has its own specific problems and needs. DSP has produced
revolutionary changes in all three areas.
1.7.3.1 Radar (RAdio Detection And Ranging)
Forafewmicroseconds,aradiotransmitterproducesapulseofradiofrequency
energy. This pulse is inputted to a highly directional antenna, where the
resulting radio wave propagates away at the speed of light. The aircraft in
the path of this wave will reflect a small portion of the energy back toward
a receiving antenna, situated near the transmission site. The elapsed time
between the transmitted pulse and the received echo are used in the distance
calculation. The direction to the object is known, as we know which direction
the directional antenna was facing when the echo was received.
Digital signal processing has revolutionized radar in three areas, all of
which relate to this basic problem. First, pulse compression can be carried
out by DSP after it is received, providing better distance determination
without reducing the operating range. Second, DSP can filter the received
signal to decrease the noise. This increases the range, without degrading the
distance determination. Third, DSP enables the rapid selection and generation
of different pulse shapes and lengths. Among other things, this allows the
pulse to be optimized for a particular detection problem. Much of this is
done at a sampling rate comparable to the radio frequency used, as high as
several hundred megahertz. When it comes to the radar, DSP is as much about
high-speed hardware design as it is about algorithms.
1.7.3.2 Sonar (SOund Navigation And Ranging)
In active sonar, sound pulses transmitted into the water are between 2 and
40kHz,andtheresultingechoesaredetectedandanalyzed.Someusesofactive
sonar are: the detection and localization of undersea bodies, for navigation,
communication, and mapping the sea floor, with a maximum operating range
of 10–100 km.
Passive sonar covers listening to underwater sounds, such as: natural
turbulence, marine life, and mechanical sounds from submarines and surface
vessels.
No energy is emitted in passive sonar and is, therefore, ideal for covert
operations. You want to detect the enemy, without him detecting you.
The most significant application of passive sonar is in military surveillance
systems that detect and track submarines. The frequencies utilized by passive

1.7 Application of DSP 9
sonar are typically lower than those of active sonar because less absorption
occurs as they propagate through water. Detection ranges can be thousands of
kilometers.
Rather than just a single channel, sonar systems usually employ extensive
arrays of transmitting and receiving elements. The sonar system can steer the
emitted pulse to the desired location by properly controlling and mixing the
signals in these elements, and determine the direction the echoes are received
from. To handle these multiple channels, sonar systems require the same
massive DSP computing power as those employed in radars.
1.7.3.3 Reflection seismology
Today, the primary method for locating petroleum and mineral deposits is the
reflection seismic method. Ideally, a sound pulse sent into the ground produces
a single echo for each boundary layer the pulse passes through.
Each echo returning to the surface must pass through all the other boundary
layers above from where it is originated. This can result in the echo bouncing
between layers, giving rise to echoes that is being detected at the surface.
These secondary echoes can make the detected signal very complicated and
difficult to interpret.
Since the 1960s, DSP has been widely used to isolate the primary
echoes from the secondary echoes in reflection seismograms. DSP allows
oil exploration at difficult locations, such as under the ocean.
1.7.4 Image Processing
Images are signals with special characteristics. While most signals are a
measure of a parameter over time, images are a measure of a parameter over
space (distance). They contain a great deal of information. More than 10 MB
can be required to store one second of television video. This is more than a
thousand times greater than for a similar length voice signal. The final judge of
quality is often a subjective human evaluation, rather than being an objective
criterion. These special characteristics have made image processing a distinct
subgroup within DSP.
1.7.4.1 Medical
Since19thcentury,medicalX-rayimagingwaslimitedbyfourproblems.First,
overlapping structures in the body hide behind each other creating problems
with visibility. For example, portions of the heart might not be visible behind
the ribs. Second, it would not always be possible to distinguish between similar

10 Introduction
tissues. For example, it may be possible to discern bone from soft tissue, but
distinguishing a tumor from the liver would not be as straightforward. Third,
X-ray images show anatomy, the body’s structure, and not physiology, the
body’s operation. The X-ray image of a living person would look exactly like
the X-ray image of a dead person. Finally, X-ray exposure can also cause
cancer, requiring it to be used sparingly and only with proper justification.
The last three X-ray problems have been solved by the use of penetrating
energy other than X-rays, such as radio and sound waves. Magnetic resonance
imaging (MRI) uses magnetic fields in conjunction with radio waves to probe
the interior of the human body. This resonance results in the emission of
a secondary radio wave, detected with an antenna placed near the body.
Information about the localized region in resonance can be obtained from
the strength and other characteristics of this detected signal.
With the adjustment of the magnetic field, the resonance region can scan
throughout the body, mapping the internal structure. Just as in computed
tomography, this information is usually presented as images. Besides provid-
ing excellent discrimination between different types of soft tissue, MRI can
provide information about physiology, such as blood flow through arteries. It
relies totally on DSP techniques and could not be implemented without them.
1.7.4.2 Space
With images taken from unmanned satellites and space exploration vehicles,
the feed received is frequently of the lowest quality. DSP can improve the
quality of images taken under extremely unfavorable conditions in several
ways: brightness and contrast adjustment, edge detection, noise reduction,
focus adjustment, motion blur reduction, etc. It is due to these merits that DSP
proves ideal in this application as well.

2
Signals and Systems
(Continuous and Discrete)
This chapter presents the basic foundation of signal and systems in discrete-
time (DT). We introduced important types of signals with their properties
and operations. Major topics of signals and systems have been introduced
in this chapter. This chapter covers: Different continuous-time (CT) signals,
concepts of frequency in CT signals, processing of analog-to-digital (A/D)
and digital-to-analog (D/A) conversion, sampling theorem, quantization error,
DT signals, concepts of frequency in DT signals, simple manipulation of DT
signals, classification of DT signals, energy and power signals, DT systems,
block diagram representation of DT systems, classification of DT systems,
and Problems and solutions.
2.1 Introduction
In this modern age of microelectronics, signals, and systems play vital
roles. A function of one or more independent variables which contains some
information is called a signal. In other words, a signal can be defined as a
varying phenomenon, which can be measured. Signals could be varied with
respect to time or space. More suitable examples of signals include sounds,
temperature, a voltage, and an image of video camera. Signals can be thought
as either CT or DT. Signals normally occurring in nature (e.g., speech) are
continuous in time as well as amplitude. Such signals are called CT signals.
DT signals have values defined only at discrete instants of time. These time
instants need not be equidistant, but in practice, they are usually taken at
equally spaced intervals for computational convenience and mathematical
tractability. If amplitude of DT signal is also made discrete through process
of quantization or rounding off, then this becomes a Digital Signal. Digital
signal processing (DSP) is concerned with digital processing of signals.
11

12 Signals and Systems
2.2 CT Signals
An analog signal has infinite variety of values with the varying time and
continuous changes (e.g., smoothly) over time. CT signals are often denoted
by x(t). Such a signal is often called an analog signal, but a better term is
continuous signal. The following are few CT signals for positive values of
time (i.e., t ≥ 0). The values of these signals are given below for t ≥ 0 for a
causal input
x1(t) = 1 (2.1)
x2(t) = t (2.2)
x3(t) = t2
(2.3)
x4(t) = e−t
(2.4)
x5(t) = cos(ωt + θ). (2.5)
2.2.1 Unit Impulse Function
The first specific signal we discuss is the unit impulse as is given in
Equation (2.1). The unit impulse is a building block signal used for creating
more complex signals as well as an effective signal for determining the time
and frequency domain characteristics of certain classes of systems. The unit
impulse has a magnitude of ∞, pulse width or time duration of 0 and area of
1. Following are the simple examples of such signals:
δ(t) =

∞, t = 0
0, t = 0
, (2.6)
and Equation (2.6) is also constrained to satisfy the identity as
+∞

−∞
δ(t)dt = 1 (2.7)
2.2.2 Step Function
The step function is commonly used to test the response time of a system. The
unit step response is desirable because the signal varies from zero magnitude
value to a finite value theoretically at zero time. The most often used unit step
function is described as a function having magnitude of 1 occurring at time
equal to and greater than zero.

2.2 CT Signals 13
u(t) =

0, t 0
1, t ≥ 0
. (2.8)
2.2.2.1 Properties of unit step function
Properties describe the continuous as well as sampled impulse and step
function. Following are the important properties of the Unit step function
1. u(t)δ (t − a) = δ(t − a)
2. du(t)
dt = δ(t) and then, u(t) =
∞
t=−∞ δ(t)dt
3. The step and impulse response are related by derivatives; the imp-
ulse represents the instantaneous rate of change of the step function
and accordingly the step function is equal to the integral of impulse
function.
2.2.3 Ramp Function
The ramp function is uniformly increasing time domain signal of a constant
slopek.Therampfunctioniscommonlyusedasatestsignalafterstepfunction.
The signal is designate as r(t). The ramp function is described as a function
having a magnitude of t at t ≥ 0.
r(t) =

0, t 0
t, t ≥ 0
. (2.9)
2.2.4 Parabolic Function
The parabolic function is not uniformly increasing with respect to time and
having a slope k. The signal is designated as t2.
t2
=

0, t 0
t2, t ≥ 0
. (2.10)
2.2.5 Exponential Function
The exponential function is increasing or decreasing exponentially. Because
of stability issue, we use most frequently the decreasing exponential function.
The signal is designate as e−t. The unit exponential function is described as
having a magnitude of 1 at zero time and exponentially decaying or rising for
time greater than zero.

14 Signals and Systems
e−t
=

1, t 0
e−t, t 0
. (2.11)
2.2.6 Sinusoidal Function
The sinusoidal function is time domain signal. The signal is designated
as sin ωt. This signal is used to find out the steady-state response of a
system.
2.3 Concept of Frequency: Continuous Time Signals
A simple harmonic oscillation is given by
x(t) = A cos (ωt + θ) for −∞ t ∞, (2.12)
Where x(t) is a CT analog signal, A is the amplitude, ω is the frequency in
radian per second, and θ is the phase.
2.3.1 Periodic and Aperiodic Signals
A periodic signal is that type of signal which has a finite pattern and repeat
with a repetition period of T. In other words, a CT signal is called periodic if
it exhibits. The smallest value of period T, which satisfies Equation (2.12) is
called fundamental period and is denoted by To.
The CT sinusoids are characterized by the following properties:
(1) Periodic functions are assumed to exist for all time. In Equation (2.12),
we can eliminate the limit of t.
(2) Aperiodic function can be written with period nT, where n is an integer.
Hence for a periodic function, x(t) = x (t + T) = x(t + nT) with n be
any integer.
(3) We define the fundamental period To as the minimum value of the period
T 0 that satisfies x(t) = x (t + T).
Example 2.1
Determine the fundamental period and periodicity of the following sinusoids.
(a) x(t) = sin π t
(b) x(t) = sin
√
2 π t.

2.3 Concept of Frequency: Continuous Time Signals 15
Solution
(a) The fundamental period T is given as T = 2 π
ω = 2 π
π = 2, x(t) = sin πt
is a periodic signal.
x(t + T) = sin π(t + T) = sin π( t + 2 )
= sin πt cos 2π + cos π t sin 2π = sin πt.
(b) The fundamental period T is given as T = 2π
Ω = 2π
π
√
2
= 2
√
2
=
√
2 =
1.414 , x(t) = sin π
√
2t is a periodic signal
x(t + T) = sin
√
2πt = sin
√
2π( t +
√
2 )
= sin
√
2πt cos 2π + cos
√
2πt sin 2 π = sin
√
2πt.
Example 2.2
Assume x1(t) and x2(t) are periodic signals with period T1 and T2, respec-
tively. Under what conditions the sum x(t) = x1(t) + x2(t) is periodic. What
will be period of x(t), if it is periodic?
Solution
Given that x1(t) and x2(t) are periodic signals with period T1 and T2,
respectively.
Thus x1(t) and x2(t) may be written as
x1(t) = x1 (t + T1) = x1 (t + mT1) , where m is an integer;
x2(t) = x2 (t + T2) = x2 (t + nT2) , where m is an integer.
Now, if T1 and T2 are such that mT1 = nT2 = T. Then,
x (t + T) = x1 (t + T1) + x2 (t + T2);
x (t + T) = x1(t) + x2(t), i.e., x(t) is periodic in this case.
Therefore, condition of x(t) to be periodic is T1
T2
= n
m is a rational number.
Example 2.3
The sinusoidal signal x(t) = 10 cos(200t + π/2) is passed through a square-
law device defined by the input–output relation. Using the trigonometric
identity cos2 θ = 1
2 (cos 2θ + 1).
(a) Specify the DC component.
(b) Specify the amplitude and fundamental frequency of the sinusoidal
component in the output y(t).

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For a moment several thought that Frank Simpson would collapse,
so surprised was he. Then he braced himself by a strong effort, and
tried to speak. For a second or two no words would come, and then,
in a husky voice he said:
“Part of that is true, and part is not. I did take part in those
games, but it was strictly as an amateur. I can prove that. I have
never been a professional.”
“Isn’t it true that you won the mile run?” asked Wallace.
“Yes, I did.”
“And wasn’t the first prize for that contest fifty dollars in gold?”
“It was, but——”
“Didn’t you win, and get the prize?”
“I won, but I did not get the money!” fairly shouted Frank. “I
never had a cent of it. I did win the race. The prize was fifty dollars,
but I never got it. I turned it over, without even taking it into my
possession, to the charitable committee. If that’s professionalism,
make the most of it!”
He sat down, and every lad in the room was on his feet in an
instant.
“Of course that’s not professionalism!”
“Never heard of such a thing!”
“That’s a silly charge!”
“The A. A. U. rules don’t make that professionalism!”
“Not by a long shot!”
Everyone seemed to be shouting something, and Holly managed
to hear the above expressions, amid the babble of others.
“Silence! Silence!” he cried.
“That’s our case,” Wallace managed to say.

Once more came hisses, that were not so easy to silence.
“We claim that is professionalism, and we won’t compete if Frank
Simpson represents Randall,” said Pendleton, who stood beside
Wallace.
“It seems like splitting hairs,” spoke Kindlings, “but——”
“Perhaps it does,” admitted Wallace calmly. “But we claim that
Simpson is a professional under the rules. It’s up to you fellows, but
——”
“Mr. Chairman, I move that the athletic committee of Randall go
into executive session at once, consider this matter, and let Exter
have our answer as soon as possible,” shouted Tom above the din.
“Second the motion!” cried Sid.
It was put and carried at once.
“Will you make yourselves comfortable until after our session?”
asked Holly of the Exter committee. “I’ll have you taken to our
chapter house,” and he called some lads, who were not members of
the committee, to act as the hosts of the visitors.

CHAPTER XXVI
FRANK WITHDRAWS
Tense and anxious faces looked into those of Holly and Kindlings
as the athletic committee drew closer to the platform in the
gymnasium. The doors were closed. The Exter lads had been taken
in charge by some Randall fraternity members, but it could not be
said that there was a spirit of gaiety observable. Only those of whom
it was absolutely required attended lectures. The others, not charged
with the extending of courtesies to the Exter lads, hung about the
gymnasium, waiting for any news that might leak out.
“Well, boys, what’s to be done?” asked Holly, rather helplessly, as
he faced his committee. Tom, Sid, Phil and Frank, of course, were
present.
“Who’s got anything to suggest?” asked Dan Woodhouse.
It seemed that the two trainers and managers were all at sea, as,
indeed, were most of the others.
“I suggest that Frank tells us all he knows about this case,” said
Tom, finally. “We’re with him to the last. I guess I needn’t say that,
though,” he added.
“That’s right,” chimed in several others.
Frank arose, all eyes turned toward him.
“Fellows,” he began, “I can’t tell you how sorry I am that this thing
has come to you. It’s like a bolt out of a clear sky to me, and I
needn’t say that I never dreamed of such a charge being brought.”
“We know it,” said someone.

“If I was surprised when the charge was made against Shambler—
and he admitted it was true,” went on the Big Californian. “I was
completely astounded when they named me as the second man. I
hardly know what to say.”
“Did you really take part in those games?” asked Holly.
“I did, but there was not the least hint of professionalism. No one
dreamed of such a thing. As I recollect it, a number of college
fellows were asked to compete. I was at Stanford University at the
time. I entered. It was for some charity. I’ve forgotten just what
now, but a hospital, I think. A business men’s committee was
formed, and I was told there were to be several prizes offered for
contestants. We didn’t care about them, for we only thought of
doing our best and winning. We all supposed the prizes would be
medals, cups, or something like that.
“Then there was some talk of money prizes being offered. But I
don’t believe any of us thought anything about it being professional
to compete for money, but I know we college fellows held a
meeting.
“We decided unanimously that whatever prizes we won we would
donate to the charity for which the contests were run off. None of us
wanted them. Then came the meet.
“I don’t know just how many events I took part in. I think I won
the pole vault, as well as the mile run, but I’m not sure. Anyhow, I
know that after the games a man came up to me, and some of the
other winners, with envelopes. I realize now that they must have
contained money—the prize money.
“Everyone of us waved him aside, and the general order was:
‘Give it to the hospital,’ if it was a hospital for which the meet was
held. I know I never accepted a cent, and none of the other college
fellows did. That’s all there is to it.”
There was a short period of silence following the statement by the
Big Californian. Then Tom arose in his seat.

“Mr. Chairman,” he said, “I move you that we take a vote of
confidence in Frank, first of all, and then that we send word to Exter
and Boxer Hall and Fairview, that the charges of professionalism are
groundless in this case, and that Frank will take part in the games.”
“Second it!” yelled Joe Jackson.
“One minute,” began Holly calmly. “I appreciate the spirit in which
that motion was made, and I’ll put it at the proper time. But, before
I do, I’d like to know if anyone here has a copy of the A. A. U. rules
bearing on professionalism. If he has will he see if they bear on this
case?”
“I’ve got a copy!” said Dan Woodhouse, “and I know ’em pretty
much by heart. I don’t believe that Frank would be barred under the
rules. They make the ‘acceptance’ of money a bar, I think, and by his
own evidence Frank didn’t accept it.”
“Not that I want to seem to believe for a moment this charge, but
because I think we ought to be very sure of our ground, I make this
suggestion,” spoke Phil Clinton. “Of course Frank didn’t take, or
accept, the money. But might it not be said that by tacitly turning it
over to the charity after winning it, that he had it? I’m afraid they’ll
say—the committee I mean—that when he competed for a money
prize he became a professional.”
“No! No!” cried several.
“Well, that’s one way of looking at it,” said Holly Cross. “That’s
what we’re here to decide. Shall we fight this case, and have it
threshed out in a general meeting, or——”
“Fight! Fight!” cried a number.
“Frank isn’t a professional, and never was,” declared Sid
Henderson, jumping up and excitedly waving his arms. “I say let’s
defy Exter and all the rest.”
“And maybe break up the meet?” asked Dan.

“Fellows, let me speak once more,” begged Frank. “I have thought
this matter over carefully in the last few minutes, and, while I don’t
retreat one point from my position, perhaps a compromise would be
better than a contest.”
“No! No! Contest it!” was the general cry.
“Wait!” begged the lad who had most at stake. “This comes at an
unfortunate moment. Shambler confessed that he was a
professional. Fortunately it came in time to save the honor of
Randall. Now, what I propose to do is for the further honor of our
college.”
“What’s the matter with Frank Simpson?” demanded Bean Perkins.
“He’s—all—right!” was thundered out.
“Thank you, boys,” responded the Big Californian, when quiet had
been restored. “I appreciate all that, but we must face the facts. As
soon as it becomes known that Shambler has confessed, there will
be a lot of talk. Fortunately Randall can’t be scorned. We have done
our duty. Now there’s this charge against me. There are some
complications in it. I believe——”
“A fair committee would never bar you,” broke in Tom.
“Perhaps not,” admitted Frank. “But we don’t want any question
raised. Boys,” he went on, and his voice was solemn, “we have to
think of the honor of Randall before we think of ourselves. It’s the
college and not the contestants who will be exalted, or dragged
down, as the case may be.
“I fully believe that I am in the right, and that no charge of
professionalism would stand against me. But, for the honor of
Randall I want you to let me withdraw. I——”
“No! No!” came a storm of protests.
“Stick it out!” urged Joe Jackson.
“We’re with you to the end,” added Phil.

Frank raised his hand for silence.
“It’s very good of you to say that,” he went on, when he could be
heard, “but I know how these things sometimes turn out. There is
talk afterward. You don’t want the success of Randall questioned, in
case she should win this meet.”
“But can we win with you and Shambler out?” someone asked.
“Boys, you’ve got to—for the honor of Randall,” said Frank quietly.
“You’ve just got to! You’ve got to let me drop out, and someone
must take my place. It can be done, easily. Someone must run for
Shambler, too. I know it’s going to be hard to get someone with his
record, but we’ll do it. Boys, I’m not going to take part in the games.
That’s final!”
In spite of the fact that they all expected this as a climax to what
Frank had started to say, it came as a shock. There was a tense
silence, and then someone asked:
“Isn’t there a way out? We need you, Simpson.”
“There is no way out, except my resignation,” answered Frank,
“and I hereby tender it now, formally, and ask that it be accepted at
once. Then you can go into the games with a clean slate, and—win!”

CHAPTER XXVII
“WHAT’S TO BE DONE?”
For perhaps five seconds no one spoke after Frank had announced
his decision, a decision that meant more to him than anyone
suspected. Then there came a spontaneous cheer—a cheer for the
lad who could sacrifice himself for the honor of his college.
“What’s the matter with Frank Simpson?” again demanded Bean
Perkins.
Instantly came the answer:
“He’s all right!”
“Tiger!” yelled the irrepressible Bean, and the yellow-striped cheer
was given with a will.
“Well, I suppose there’s nothing else to be done,” spoke Holly,
regretfully.
“Nothing,” replied Frank, and the wonder of it was that he could
smile. “Nothing but to accept my withdrawal, and so inform the
committee from Exter.”
“And then we’ve got to get busy and see who we can put in your
place, and Shambler’s,” added Kindlings.
The resignation was formally accepted, and word was sent to
Wallace and his friends. They expressed their regret at the necessity,
and even admitted that perhaps a ruling from the A. A. U. might
bear out Frank’s contention that he was not a professional.
“But we haven’t time for it,” said Holly. “We’ll take our medicine,
though it’s a bitter pill to swallow.”

“I hope you don’t think we did this because of any fear on our
part that we couldn’t win against your two men,” spoke the Exter
manager.
“Not at all,” Holly assured him. “I appreciate your position, but it’s
tough on us, to lose two good men. I can’t get over that cad
Shambler.”
“He certainly played a mean part,” agreed Wallace. “This
Simpson’s case is altogether different. I’m sorry for him.”
“We all are,” put in Kindlings. “Well, we’ve got a little time left in
which to make good. I’m glad we don’t have to go into the games
to-morrow.”
“Not wishing you any bad luck,” spoke the Exter lad, with a frank
laugh, “I hope we beat you.”
“Randall is hard to beat,” spoke Holly grimly. “You’ll find us on the
job when the time comes.”
But when the protesting committee had left the boys of Randall
looked at each other with troubled eyes.
“What’s to be done?” was the general question.
No one could answer.
“Of course we’ve got to go on and play the game,” declared Holly
Cross. “We’ve a few days in which to select some lads to take the
places of Shambler and Frank. Oh, why couldn’t it have been
someone else? This leaves the mile run and the broad jump open,
and we were counting on those two contests especially. Of the
others I’m not so much afraid. But who are we going to enter for
those contests?”
“We’re going to lose, I think,” said Jerry Jackson mournfully.
“That’s right—lose,” echoed his twin.
“Say, you fellows make me tired!” exploded Kindlings. “We’re not
going to lose!”

“That’s the way to talk, but how do you figure it out?” asked Holly.
“Who’ll substitute for Shambler and Frank?”
“Sid Henderson will have to make the jump, and Tom Parsons,
we’ll depend on you for the mile run!” answered Dan quickly.
“Who, me? I can never beat the Exter man in the jump,” asserted
Sid.
“Say, don’t you talk back to me!” retorted Kindlings, and there was
a new note in his voice. “I tell you you’re going to do it! Where’s
Parsons?”
“Here,” answered Tom meekly.
“You get into practice quick for that mile run,” ordered Dan.
“You’ve got to do it. Sid, get into your togs at once. Holly, come on
out and hold the watch on Tom. I’ll see Moses and make it all right
about lectures. We’re in a hole and we’ve got to pull ourselves out.”
At once it seemed as if new spirit had settled down over Randall.
There had been gloom, following the withdrawal of Shambler and
Frank, but with the manly way in which Kindlings met the situation
the skies seemed to clear.
It was the only way out of the dilemma. But everyone knew that,
at best, it was but a slim chance. Neither Tom nor Sid were brilliant
performers, though that is not saying they were to be despised, by
any means. Their talents simply lay in other directions than track
athletics. Yet they were not far behind Frank and Shambler in the
two events. They needed hard training, however, and the question
was, could they get in form in the short time left?
“They’ve got to!” declared Kindlings grimly. “It’s going to be train
—train—train! from now to the minute of the games. It means a lot
of practice—hard practice. Oh, if we only had a week more! Why
didn’t this come a little sooner?”
“Is there any chance of getting a postponement?” asked Phil. “I
think under the circumstances we’re entitled to it.”

“Entitled to it, yes, maybe,” assented Dan, “but we won’t crawl by
asking for it. We’ll take our medicine, and take it like men, and,
what’s more, we’ll turn the trick, too!”
The squad of athletes was ordered out soon after the momentous
meeting. Dr. Churchill met the situation squarely. He gave the boys
all the leeway needed in the matter of attending lectures, and wrote
a personal letter to the heads of Exter, Boxer Hall and Fairview,
expressing regret at the turn of affairs.
And then Randall grimly set to work on her uphill climb.
That it was to be an uphill climb was soon made very evident.
Whether it was because of nervousness, or real inability to make
good, or because they were so suddenly called on without adequate
preparation, was not made evident, but certain it was that neither
Tom nor Sid gave brilliant performances in the trials that followed.
Tom’s time was far behind that of Shambler in the mile run, and,
though it was only a matter of seconds, everyone knew that seconds
would count.
Sid, too, seemed to have lost his natural ability to cover ground in
the big jump, though he was by far the best man available after
Frank’s disbarment.
“This won’t do,” declared Holly, and though his heart was sinking,
he kept up a bold front. “Get at it, boys,” he urged the two on whom
so much depended. “You can make good yet! All you need is to think
so.”
“It’s easy enough to say,” complained Tom, who was tired from
many trials.
“Say, if you don’t win, I’ll roll you in the mud so your best girl
won’t speak to you for a month,” threatened Kindlings. “And, as for
you, Sid, I’ll have you run out of Randall on a rail. So make good—
both of you!”
“Um!” grunted Tom, disconsolately, and Sid looked at him with
despair in his eyes. They were both in a bad way.

There was but one more day before the games. It dawned—or
rather, to quote Holly Cross, “it clouded up beautifully” from the
start. There was a chill, in the air, too.
“Tumble out!” cried Kindlings, as he banged on the door of the
room where the inseparables were sleeping. “Tom—Sid, we need
you for some morning practice.”
“Oh, go on away,” begged Tom.
“Let me dream on,” requested Sid, drowsily.
“Tumble out!” shouted the inexorable Kindlings. “This is your last
chance. It’s a nice cool morning for a run or a jump, and you’ll be all
the better for it. Come on.”
So, perforce, the substitutes who were to fill in for Frank and
Shambler “tumbled out,” literally, for they were half asleep. But a
shower bath, a brisk rub, and the cheerful talk of Holly and Kindlings
put new life into them, and soon they were at vigorous practice.
They did better than on the previous day.
“If we only had another week, or even three days, I wouldn’t be a
bit worried,” declared Holly at the conclusion of the trials. “They’re
both doing fine, Kindlings.”
“I don’t s’pose we can get an extension?”
“I wouldn’t have the nerve to ask for it.”
“Then we’ll have to stand or fall as we are.”
“That’s it—hang together or hang separately as Patrick Henry, or
some of the ancients, said,” quoted Holly.
The excitement over the unexpected charges had somewhat died
away, and Randall was more like herself. The withdrawal of
Shambler had created a little flurry, but not much. No one seemed to
know where he had gone, and no word came as to what to do with
his effects.

As for Frank, he was saddened, but not downcast. He announced
his intention of taking up his case with the Amateur Athletic Union as
soon as the games were completed.
“I’m sure they’ll uphold my contention,” he declared. “I’m an
amateur, and I can prove it!”
“But it will be too late for any use,” spoke Tom mournfully.
Words of sympathy had come from the girls, and Tom and his
chums were duly grateful for them. It developed that neither Boxer
Hall nor Fairview were in favor of forcing the issue against Randall,
but that Exter, with perhaps exaggerated notions as to what
constituted “amateur” sport, had taken the initiative. Still Randall’s
lads did not complain.
It was the night before the big games. Gathered in the room of
the inseparables were our old friends, Holly, Kindlings, Dutch, and a
few other kindred spirits.
“Well, it’s all over but the shouting,” said Dutch, in mournful tones.
“To-morrow will tell the tale.”
“Get out, you old croaker!” cried Kindlings.
“We’re going to win! I’m sure of it!”
“If we had another week, I believe we would,” asserted Holly.
“Tom and Sid could pull up by then. I’m almost tempted to
telephone, even at this late day, and ask for a postponement. We’re
entitled to it, under the circumstances.”
“Oh, forget it,” advised Phil. “Be a sport! Play the game!”
“Just the same I wish something would happen to put things off
until next Saturday,” insisted Holly.
“It’s too late now,” declared Kindlings. “We’ve got to take part to-
morrow unless——”
He stopped suddenly, and held up his hand.
“What’s the matter?” asked Tom, curiously.

“Hark!” exclaimed Dan. “What’s that noise?”

CHAPTER XXVIII
A BOTTLE OF MEDICINE
They all listened intently, looking the while curiously at Kindlings.
He seemed to be hearing something inaudible to the others.
“I don’t ‘hark’ to anything,” remarked Tom, “unless you mean a
sort of pattering noise, and——”
“That’s it!” interrupted Dan with a glad cry. “It’s the pattering
noise I mean. Fellows, there’s a way out after all. It’s raining, and if
it keeps up long enough the games will have to be called off. Now, if
any of you have any sort of pull with the weather man have him
make it rain like the old scratch, and keep it up. It’s our only
salvation. A postponement means a week, and in that time Tom and
Sid will be fit as fiddles. Come on, oh you rain drops!”
For a moment or two the students all stared at Dan as though
they thought he had taken leave of his senses. Then, as the patter
on the window ledge outside became more pronounced, and as the
gentle shower became a veritable downpour, all understood Dan’s
elation. Postponement—delay—was the thing they needed most of
all, and it seemed likely to be their luck.
“Oh, if it only lasts!” half-whispered Tom. “If it isn’t just a little
shower, that will only lay the dust!”
Dan jumped up, and made his way to the window, shoving Phil to
one side so forcibly that he toppled into one of the armchairs, with
impact enough to almost wreck it.
“Hey! Look out what you’re doing!” cried Phil. “What are you up
to, anyhow?”

“I’m going to stick my head out, and get soaked, then maybe the
rain-god will take that as a sort of votive offering, and keep the
faucets turned on all night,” replied Dan.
As he spoke there came a downpour harder than ever, and as he
thrust forth his head he was drenched in an instant.
“I guess it’ll keep up all night,” he remarked. “It seems a mean
thing to wish, perhaps, for it will spoil a lot of people’s fun, and the
other colleges won’t like the postponement, but it’s Randall’s only
hope. Rain on! Rain on!”
And rain it did, with increasing violence.
“How’s the wind?” asked Tom, with a memory of the days spent
on the farm, when the weather was a fruitful source of talk, and
when much depended on reading the signs.
“I can’t see it,” replied Dan. “Besides, what difference does that
make?”
“Lots,” replied Tom shortly. “Let me take a look. If we’ve got a
good east wind it means a long rain.”
He thrust his head out of the open window, into the darkness and
storm, while his chums awaited his verdict.
“It’s all right,” he announced after a moment. “It’s in the east.
There’ll be no games to-morrow.”
“You’ve got good eyes, to see wind in the dark,” remarked Sid.
“I didn’t see it—I felt it, you amiable cow,” answered Tom.
For a time they listened to the patter of the drops that meant so
much to Randall, and then the gathering broke up, the visitors going
to their rooms, leaving the inseparables to themselves.
It rained all night, and was still at it when morning broke. Several
times during the night Tom, or some of his chums, got up to see if
the storm was still doing its duty, and when they found that it was,
they returned to rest with sighs of satisfaction.

Of course there was nothing to do but call the games off. Boxer
Hall and Fairview, to whom Holly telephoned early in the day, agreed
to this. Exter held off, her manager saying he thought it might clear.
Perhaps he realized what the delay meant to his rivals. But even he
had to give in finally, and formal announcement of the
postponement was made, it being stated that all tickets would be
good the following Saturday.
“And now, Tom and Sid, you’ve got to train your heads off and be
fit to the minute,” declared Holly. “Into the gym until it clears, and
you won’t have any rest as soon as it’s dry enough to get on the
track.”
“We’ll sacrifice ourselves on the altar of duty,” replied Tom,
mockly-heroic.
“And you ought to be glad of the chance,” retorted Phil. “I wish I
was in your place.”
“I can’t tell you how sorry I am that this trouble occurred,” said
Frank to his two friends and some of the others as they were
gathered in the room of the inseparables the afternoon of the day
when the games were to have been held, and while it was still
pouring. “I feel as if I ought to have spoken of the chance of the
professional charge being brought against me, and then I could have
kept out. But I never dreamed of it. There never would have been
any question of Randall’s honor then.”
“And there isn’t now,” declared Kindlings sturdily. “It’s all right for
those fellows to take the stand they did, but I don’t believe they
were right in your case, Frank, and I don’t propose to let the matter
rest there.”
“What are you going to do?” asked Phil, as he shook the alarm
clock to cure it of a spasm of stopping that had developed that day.
“Are you going to raise a row over it?”
“Not a row, but I’m going to write to the heads of the A. A. U. and
state the case. Then I’m going to ask if Frank can be regarded as a
professional. This can’t stop here. We need Frank for something else

besides these games. We may have a rowing crew this year, or next;
besides, there’s football and baseball to consider. I’m going to the
bottom of this thing.”
“And I’m glad of it,” declared the Big Californian. “I don’t want this
charge hanging over me, and if you hadn’t asked for a ruling I
would. But it’s better to come from you, I guess.”
“And to think that now, if something hadn’t happened, we might
be sitting here, trying to figure out how we lost, if the games had
been held,” remarked Sid, as he listened to the rain.
It rained all the next day—Sunday—which had the effect of
keeping the lads indoors, making them fret, for they were all lovers
of fresh air, and were seldom in their rooms except to study or sleep.
In the afternoon Tom and the other three, in their raincoats, braved
the downpour, which had suddenly increased, and paid a visit to the
girls at Fairview.
“I believe you boys did this on purpose,” challenged Madge, as
they talked about the rain and the postponement.
“Don’t tell anybody—but we did,” whispered Tom with a smile.
“The rain spells success for Randall.”
The girls denied it, of course, but in spite of the jokes of our
heroes there was more or less of a feeling that Tom was right. The
Fairview boys fretted over the delay, but were good-natured about it.
Toward evening the rain slacked up a little, and the girls granted
the entreaties of the boys to come out for a walk, Miss Philock
according the necessary permission rather grudgingly.
It was too wet on Monday for out-door work, and Tom, Sid, and
the others kept to the gymnasium. There was a grim spirit about the
work now, for the boys felt that chance had played into their hands
and if they did not take advantage of it that there would be no more
hope for them.
“Luck doesn’t strike twice in the same place, even if lightning
does, the proverb to the contrary,” said Holly Cross.

Tom had a letter from his father that day, announcing that the
final hearing in the lawsuit might come off any day now.

“And I wish I could know how it’s coming out,” Mr.
Parsons wrote to his son. “It has me bothered and
worried more than a little. I don’t want to take you out
of college, Tom, my boy, but I’ll have to if I lose all this
money. I may need you to testify in the case, but if I
do I suppose I can reach you by telegram. If you do
get a wire, don’t delay.”
“Wow!” mused Tom, as he read that. “I hope dad doesn’t send for
me before the games. Not that I’m such a muchness, but it would
sort of break up the combination if I had to leave suddenly. Well,
there’s no help for it. If I have to go, I’ll have to go. If I don’t, in
case dad should telegraph for me, he might lose the case, and I’d
have to leave Randall.
“And yet if I left we might lose this contest. I wonder what is
better to do? Delay, in case dad sends for me, and help Randall win,
which may mean that I’m down and out afterward, or take a chance
on Randall losing, so I can come back? Pshaw! Of course I’ve got to
help win, no matter if I can’t come back. And yet for dad to lose all
that money——”
“Hang it all! I don’t know what to do!” burst out Tom. “I’m not
going to think any more about it. I’ll wait until the time comes, and if
dad does telegraph, I’ll tell the boys about it, and see what they
say.”
Then Tom resolutely put the affair as much out of his thoughts as
he could, for he found it interfering with his practice and training,
and he knew that he must bend every energy to win the mile run.
The practice went on unceasingly. The weather cleared, being
finer than ever, and the candidates went out on the track and field.
Meanwhile Holly and Kindlings had composed a letter to the
proper authorities of the Amateur Athletic Union, asking a ruling on
Frank’s case. Nothing more had been heard from Shambler,
excepting that he had sent for his baggage, and it was surmised that
he had quietly taken himself to parts unknown.

It was Wednesday afternoon, and Tom, coming from the
gymnasium, after a refreshing shower, following a hard spell of
practice in all-around work, was met by Wallops.
“Oh, Mr. Parsons,” said the messenger, “there was a young man
looking for you, with a package a while ago. I couldn’t find you, so I
sent him to your room with it. I guess he left it.”
“Are you sure it wasn’t a telegram?” asked our hero anxiously,
thinking of his father’s lawsuit.
“No, it was a package. It came by express, he said.”
“All right, Wallops. I’ll look out for it. Did you pay anything on it?”
“No, it was prepaid. I say, Mr. Parsons, do you think we’re going to
win the championship?” and the diminutive messenger looked at the
runner anxiously.
“Of course we are, Wallops. Why? You aren’t betting, I hope.”
“No, but you see—well, er—yes, I am in a way. A friend of mine
bet a box of candy—I mean I bet the box of candy and——”
“And she wagered a necktie, I suppose,” interrupted Tom with a
laugh. “Well, Wallops, I hope the young lady bet on us, and that you
lose, though I’d buy her the candy, if I were you.”
“Thanks, Mr. Parsons, I guess I will,” answered the messenger
with a cheerful grin. “She’s an awful nice girl.”
“Humph!” mused Tom, as he walked on. “Every fellow thinks that I
suppose, about his own. But I wonder what that package is?”
He found it outside the door, which was locked. None of his chums
was in as Tom swung the portal, and soon he was unwrapping the
bundle.
“Ha! A bottle of medicine,” remarked Tom, as the last paper came
off, revealing a flask of some dark fluid. “I wonder who could have
sent it to me?”

He looked at the wrapper, but it bore no sender’s name, and his
own address was in typewriting.
“Hello! What you got?” demanded Sid, as he entered at that
moment, and saw Tom holding the bottle up to the light.
“Search me,” was the answer. “It’s a bottle of some kind of
training dope I guess, to judge by the label.”
Sid looked at it.
“That’s good stuff,” he announced. “It’s a sort of iron tonic. I’ve
used it. It’s a patent medicine, but lots of fellows use it in training.
Who sent it?”
“I don’t know.”
Sid looked at the wrapper.
“It came from Fairview,” he declared. “Tom, some of the girls
thought you were losing your nerve, and they sent this. Well, a dose
of it won’t hurt you. They meant all right, I guess. Going to take
any? It’s fine for the stomach.”
“No, I don’t feel the need of it,” and Tom set the bottle of
medicine on the shelf.

CHAPTER XXIX
AN ALARM IN THE NIGHT
“What are you doing, Sid?”
“Writing a letter.”
“Of course. I can see that without glasses. But who to, if it’s not a
personal question?” persisted Tom tantalizingly, as he stretched out
on the old couch, and watched his chum busy with pen and ink. Phil
and Frank were making more or less successful pretenses at study.
“Well—er—it is sort of personal,” replied Sid, and Tom noticed that
the writer got red back of the ears. That is always regarded as a
sure sign.
“My! You’ve got it bad,” persisted Tom.
“Got what bad—what do you mean?”
“As if you didn’t know! You saw her Sunday, and here it is only
Wednesday, and you’re writing. I say, that’s against the union rules
you know; how about it fellows?”
“That’s right,” agreed Frank.
“And the punishment is that you’ll have to read the letter to us,”
went on Tom. “Failing to do that we will read it for ourselves.”
He arose suddenly, and made as if to look over Sid’s shoulder.
“No, you don’t!” cried the writer, dodging away from the table.
“You let me alone, and I’ll let you alone.”
“By Jove! He’s writing verse!” cried Tom. “Well, if that isn’t the
limit, fellows! Say, he has got ’em bad!”

“Oh, you make me tired!” snapped Sid, as he stuffed the paper,
over which he had been laboring, into his pocket. “Can’t a fellow
write a letter? I’m going down in the reading room.”
And before they could stop him he had slipped out.
“Sid certainly is going some,” remarked Phil. “The germ is working.
Well, I’m going to turn in. I’m dead tired and I expect I’ll sleep like a
top.”
“Dutch wanted us to come to his room to-night,” remarked Frank.
“He’s got some feed.”
“Not for me,” spoke Tom. “I’m not going to risk anything that
Dutch will set up, when the games are so near. He’d feed us on
Welsh rabbit and cocoanut macaroons if he had his way. Not that he
wouldn’t eat ’em himself, but they don’t go with training diet.”
“Well, I’m out of it, so I’ll take a chance,” remarked Frank.
“Don’t take Sid,” Tom called after the big Californian. “He’s on
training diet, too. Dutch has the digestion of an ostrich, and it won’t
hurt him.”
“All right,” Frank retorted, and then Tom, together with Phil,
prepared to turn in.
Tom was thinking of many things. Of his father’s troubles, of the
possible outcome of the contests, and of his own chances. For the
first time since he had begun to train extra hard, because of the
necessity of taking Shambler’s place, Tom felt a little less “up to the
mark” than usual. He was more tired than he had been in several
weeks, and his stomach did not feel just right.
“I mustn’t overtrain,” he thought. “I can’t afford to go stale.”
He did not know what time it was when he awoke, but it must
have been quite late, for Sid and Frank had been in some time. The
unpleasant feeling in Tom’s stomach had increased, and he did not
know whether it was hunger or indigestion.

“Guess I worked a little bit too hard to-day,” he reflected. “I’ll be
all right in the morning.”
But he could not get to sleep again. He tossed restlessly on his
pillow, first trying one side of the bed, and then the other.
“Hang it all, what’s the matter with me?” he asked himself. “Guess
I’ll get up and take a drink of water.”
He moved quietly, so as not to disturb any of his chums, but Sid,
who was a light sleeper, heard him.
“Who’s that? What’s the matter?” demanded Tom’s team-mate.
“Oh, I just woke up—can’t seem to get to sleep again. I don’t feel
very good,” answered Tom.
“Take some of that medicine the girls sent,” advised Sid. “It’s a
harmless enough tonic, and it may do you good—send you to sleep.
You don’t want to get knocked out of your rest.”
“Guess I will,” agreed Tom. There was light enough coming in
through the transom over the door to the hall, to enable him to see
the bottle of medicine on the shelf. He drew the cork, poured out a
dose and swallowed it with a little water. The taste was not very
pleasant, but he did not mind that.
“Count sheep jumping over a stone fence, and you’ll drop off in no
time,” advised Sid, as Tom went back to bed. Sid was soon
slumbering again.
But, somehow or other, neither the counting of sheep nor any of
the other time-honored methods of wooing Morpheus availed Tom.
His restlessness increased, and he was aware of a growing distress
in his stomach.
Suddenly a sharp pain wrenched him, and, in spite of himself, he
cried out.
“What’s the matter?” asked Phil.

“I—I don’t know,” faltered Tom. “I’m sick, I guess. Oh, say, this is
fierce!” he cried, as another spasm racked him.
Phil was out of bed at once, and switched on the light. One look at
Tom was enough for him.
“Boy, you’re sick!” he declared. “I’m going to call the doctor. You
need looking after!”
“Oh, I guess I’ll be all right in a little while. I took some of that
new medicine, and——”
Another spasm of pain prevented Tom from continuing, and
hastened Phil’s decision. He slipped on some garments, awakened
Sid and Frank, and was soon communicating with Proctor Zane, who
at once summoned Dr. Marshall, the physician connected with
Randall.
The medical man came in at once, stopping only to slip on a
bathrobe.
“What have you been eating—or taking?” he demanded of Tom, as
he felt of the youth’s pulse, and examined him.
“Nothing but some of that Smith, Brown Robinson’s Tonic,”
groaned Tom, motioning toward the medicine bottle. Sid quickly
explained about it, handing the phial to the physician. The latter
smelled of the mixture, tasted it gingerly and then exclaimed:
“No wonder you’re sick, if you took that stuff!”
“Why, I’ve often taken it,” asserted Sid. “It did me good.”
“Not ‘doped’ as this is,” declared Dr. Marshall. “I know this
preparation. It is very good, but this has been tampered with.
There’s enough ‘dope’ in there to make a score of you boys sick.
Throw the stuff away, or, no, hold on, let me have it. I’ll look into
this. There’s been underhand work somewhere. You say some girl
friends sent it to you?”
“We thought so,” spoke Sid, “but if it’s been meddled with, of
course, they didn’t. I begin to suspect something now.”

“Well, talk about it later,” advised the doctor crisply. “I’ve got a
sick lad to look after now. Some of you get me a lot of hot water.
I’ve got to use a stomach pump,” and he mixed Tom some medicine,
while Sid hurried to rouse the housekeeper.

CHAPTER XXX
JUST A CHANCE
“Who you suppose could have sent that stuff?”
“We’ll have to look into it.”
“Yes, we ought to tell Dr. Churchill, and have him help us.”
Phil, Sid and Frank thus expressed themselves in whispers, as they
sat in their room. Tom had been moved to the infirmary, and Dr.
Marshall was working over him with the assistance of Professor
Langley, who, as physics instructor, knew something of medicine.
The three chums had just received word that Tom was practically
out of danger, and would be all right in a day or so, but that he was
still quite ill, and suffered much discomfort.
“Well, I don’t know how you fellows feel about it,” spoke Sid, “but
I’ve got my own opinion as to how that stuff came to be fixed, so as
to make Tom ill.”
“How?” demanded Frank.
“You mean——” began Phil.
“I mean Shambler, and I don’t care who knows it,” went on Sid,
raising his voice. “He’s a cad—and he’ll never be anything else. He
and Tom were on the outs from the first, partly over Miss Tyler, and
for other reasons.
“Then came the charge against Shambler, and, though Tom had
nothing to do with that, Shambler has probably heard that Tom has
taken his place for the mile run. He hates Randall, and he wants to
see her lose after what happened to him, and, he wants to make
Tom, by slumping, bring it about. That’s why he tried to ‘dope’ him.

Oh, if I had Shambler here!” and Sid clenched his fists with fierce
energy.
“Do you really think Shambler did it?” asked Frank.
“I’m sure of it!” declared Sid. “He is the only one who would have
an object.”
“What about Exter—or some of our enemies from Boxer Hall—or
even Fairview?” asked Phil. “You know the bottle came from
Fairview.”
“It might have come from there, but no one from Fairview
Institute sent it,” declared Sid confidently. “I’m going to look into
this.”
“But we ought to keep it quiet,” suggested Frank. “I don’t see that
any good can come of raising a row about it.”
“Me either,” agreed Phil. “Let’s work it out ourselves, with Dr.
Marshall to help us.”
Sid finally agreed with this view. The night wore on, and Tom, by
energetic measures, was soon brought out of danger. In fact he
never really was in what could be called “danger,” the only effect of
the stuff that had been put in the tonic, Dr. Marshall said, being to
make him ill and weak. This, in all likelihood, was the object of the
person who had fixed the dose. He hoped that Tom would be
incapacitated for a week or more.
For it developed that the original bottle, of what was a standard
remedy, had been opened, and a certain chemical oil added, that
would neutralize the good effects, and make the stuff positively
harmful.
“Say, but it was a scare all right, though,” remarked Sid, as the
three sat talking about it, too engrossed to go to bed. And, in their
case the usual rule of “lights out,” was not enforced on this occasion.
“I sort of think it was ‘up to me,’ for recommending Tom to take the
stuff.”

“Nonsense,” exclaimed Phil. “You meant all right. It was that cad
Shambler who ought to be pummeled.”
“It’ll be hard to fix it on him,” was Frank’s opinion; and so it
proved.
The next morning the three friends arranged with Dr. Marshall and
the college authorities to keep the real reason of Tom’s illness secret
from the students. It was given out that he was overtired from
training. Then they set to work to unravel the mystery.
But it was hard work. In the first place they learned that the girls
at Fairview knew nothing about the matter. Then Wallops was
interviewed.
He gave a good description of the boy who had brought the
bottle, and this personage developed, later, into a young employee
of a local express company. The boy was sought out.
All that he knew was that the bottle had been given him at the
Fairview office to take to Randall, and at the office a clerk had only a
dim recollection of the person who brought it in to be dispatched.
Shambler was described to him, and he said that youth might
have been the one. But it was flimsy evidence, and though Phil and
his chums were well enough satisfied in their own minds that
Shambler was the guilty one, there was no way of proving it.
So the matter was dropped, as much “for the honor of Randall,” as
for any other reason. For, as Phil said:
“Fellows, we don’t want it to get out that any lad who once
attended here could be guilty of such a thing.”
And so the affair rested.
It was two days before Tom was on his feet again, and though he
had a wretched time he was, in a measure, even better off than
before he took the unfortunate dose. For the rest had done him
good, and when he got back to practice, rather pale and uncertain,
he soon picked up his speed.

Sid, meanwhile, had been doing hard work, and the other
candidates were up to the difficult standard set by Holly and
Kindlings.
It was two days before the postponed games. All the difficulties
caused by the change of date had been overcome, and there was
every prospect of a successful meet.
“Now, Tom, do you feel like letting yourself go?” asked Holly, as
the pitcher came out for a trial on the track.
“Yes, I’m all right again,” was the answer. “In fact I think I’m
better than I was. Shall I do the whole distance?”
“No, try a half at first. Then, after you warm up, go the limit. We’ll
‘clock’ you.”
As Tom sped over the cinder track for the half mile run, he felt
within himself a confidence that he had not been conscious of
before.
“I believe that fit of sickness did me good,” he reflected. “It rested
me up, at any rate.” When he had come to the finish mark, and the
time was announced, it was two seconds better than he had ever
done before.
“Now for the mile,” suggested Kindlings. “But take a little rest.”
“No, I’ll go at something else,” decided Tom. “I don’t want to get
stiff.” So he did a little work at putting the shot, jumped over a few
hurdles, tried some high and broad leaping, and then announced
that he was ready for the mile test.
Quite a throng gathered about the track to watch Tom at his
practice, and he felt not a little nervousness as he got on his mark.
“Go,” shouted Kindlings, as he fired the pistol, and Tom was off
with some of the other candidates, who were in more to fill up, and
make a showing for Randall than because they, or their friends,
hoped they would win. And yet there was always the one chance.

Tom got off in good shape on the half mile track, two circuits of
which were necessary to make the required distance.
“He certainly can go,” observed Holly Cross, who, with Kindlings,
and some other kindred spirits, was watching the test.
“Come on! Come on!” yelled Bean Perkins, who was getting his
voice in shape for the strain that would be put on it when the games
were called. “Oh you, Tom Parsons! Come on!”
And Tom came. Running freely and well, he covered yard after
yard, doing the half just a shade better than his other performance.
“Now for the real test,” murmured Kindlings, as our hero swung
around the track on the final lap.
There were many eager faces lining the rail, and hands that held
stop watches trembled a bit. On and on ran Tom, until he breasted
the tape at the finish.
“Time! Time! What’s the time?” shouted the eager students who
knew that fifths of seconds counted in a championship meet.
“Four minutes, forty-one and two-fifth seconds,” announced Holly.
“Tom, that’s the best yet!”
“We’ll win! We’ll win!” screamed Bean. “Come on, boys!” he called
to his crowd of shouters, “let’s practice that new song, ‘We’ll cross
the line a winner, or we’ll never cross at all.’ All on the job, now.”
“Tom, old man, you’re all right,” cried Phil, as his chum slipped a
sweater over his shoulders. “You’re going to win!”
“I hope—so,” was the panting answer.
There was a comparison of records, and it was found that while
Tom’s was a little behind some mile run performances, it was better
than that of a number of former champions.
“I think he can cut down a second or two when the games are run
off,” said Kindlings, discussing the matter with Holly. “There’ll be a

band then, and that always helps a lot, and big crowds, to say
nothing of Bean and his shouters.”
“And the girls,” added wise Holly. “Tom’s got a girl in Fairview, I
understand, and if she’s on hand he’ll run his head off.”
“Then we’ll have to have her on hand, if we’ve got to bribe her,”
declared Kindlings.
“Oh, I guess she won’t need any bribing,” went on his chum. “Now
let’s see what Sid can do.”
Sid, on whom the hopes of Randall rested to win the broad jump,
was on his mettle. He could easily cover twenty feet, without
straining himself, and to-day, in what all regarded as among the last
of the important practices, he had several times, gone an inch or
two over.
“I don’t hope to equal Bowers who, in 1899, did twenty-one feet,
eight and one-half inches,” said Sid, “but I do want to do twenty
foot, six, and I’m going to make it, too.”
“Sheran, in 1909, only made twenty feet, seven and a half inches,”
Phil reminded his chum.
“Don’t make me envious,” begged Sid. “If I do twenty feet, six, I’ll
be satisfied.”
“Don’t be satisfied with anything but the limit,” suggested
Kindlings. But then he always was a hard trainer.
And so the practice went on, until Holly and Kindlings, seeing the
danger of weariness, called a halt.
“I think we’re coming on all right,” was Holly’s opinion as he and
his fellow coach left the field. “I’d like to get a line, though, on what
Boxer Hall and the others are doing.”
“So would I, and I believe we ought to. Is there anything in the
papers?”

“Yes, a lot of surmises, and some stuff that I believe is faked on
purpose to deceive us.”
“Well, we’ll see if we can get a line on their form.”
Accordingly certain “spies” were sent out to see if they could get
any information. It was regarded as legitimate then, for no
underhand methods were used. It was “all in the game,” and there
was a sort of friendly rivalry among the colleges.
A day later some of the lads whom Kindlings had sent out made a
report. On the receipt of it the young coach did some figuring on the
back of an envelope. Holly came upon him engaged in this
occupation.
“What’s up?” he demanded.
“Well, I’m trying to ‘dope out,’ where we stand,” was the reply.
“Got any line?”
“Yes, if I can depend on it. The way I figure out is this. We’ve
fairly got ’em all on some things. But not the mile run and the broad
jump. Of course something might go wrong with the dash, or the
hammer and weight throws, but I don’t think so.”
“What’s the matter with the run and jump?”
“Well, if these figures from Exter are true, they’ve got Tom by
about three seconds, and Sid by two inches. But I think Exter has
been too optimistic in giving the ‘dope.’”
“Maybe they’ve gone under their records to get better odds in
betting.”
“No, I don’t think so. The only one I’m really afraid of is Exter. I
think we can clean up Boxer Hall and Fairview. They can’t come near
us on anything except the weight throw and pole vault, and I know
Phil will make good on the vault, and if Dutch doesn’t get the fifty-
six over the twenty-five foot mark I’ll punch his head.”

“Then the way you figure it out, we’ve got our work cut out for
us?”
“We always had, but I think now that we’ve got just a chance to
win. A chance, and nothing more, for the championship. If Shambler
and Frank had stayed in it would have been different, but as it is,
and not to disparage Tom or Sid, we’ve got a fair chance and
nothing more.”
“To quote the raven,” said Holly with a smile. “‘Nevermore,’ Mr.
Poe. But I think we’ll do it, Kindlings.”
“I’m sure I hope so,” was the grave answer. “I hope so.”

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