Signals, systems, transforms, and digital signal processing with MATLAB (1)

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Signals, Systems, Transforms, and Digital
Signal Processing with MATLAB®

Electrical Engineering
Signals, Systems, Transforms, and Digital Signal Processing
with MATLAB® has as its principal objective simplification
without compromise of rigor. Graphics, called by the author “the
language of scientists and engineers”, physical interpretation
of subtle mathematical concepts, and a gradual transition from
basic to more advanced topics are meant to be among the
important contributions of this book. The text establishes a
solid background in Fourier, Laplace and z-transforms, before
extending them in later chapters. After illustrating the analysis
of a function through a step-by-step addition of harmonics, the
book deals with Fourier and Laplace transforms. It then covers
discrete time signals and systems, the z-transform, continuousand discrete-time filters, active and passive filters, lattice filters,
and continuous- and discrete-time state space models. The
author goes on to discuss the Fourier transform of sequences,
the discrete Fourier transform, and the fast Fourier transform,
followed by Fourier-, Laplace, and z-related transforms, including
Walsh–Hadamard, generalized Walsh, Hilbert, discrete cosine,
Hartley, Hankel, Mellin, fractional Fourier, and wavelet. He also
surveys the architecture and design of digital signal processors,
computer architecture, logic design of sequential circuits, and
random signals. He concludes with simplifying and demystifying
the vital subject of distribution theory.
Features
• Shows how the Fourier transform is a special case of the
Laplace transform
• Presents a unique matrix-equation-matrix sequence of
operations that dispels the mystique of the fast Fourier

transform
• Examines how parallel processing and wired-in design can
lead to optimal processor architecture
• Explores the application of digital signal processing
technology to real-time processing
• Introduces the author’s own generalization of the Dirac-delta
impulse and distribution theory
ã Offers extensive referencing to MATLABđ and Mathematicađ
for solving the examples
Drawing on much of the author’s own research work, this
book expands the domains of existence of the most important
transforms and thus opens the door to a new world of applications
using novel, powerful mathematical tools.

90488_Cover.indd 1

Corinthios

Signals,
Systems,
Transforms, and
Digital Signal
Processing with
®
MATLAB

Michael Corinthios

90488

4/12/10 10:23 AM

Signals,
Systems,
Transforms, and
Digital Signal
Processing
®
with MATLAB

Signals,
Systems,
Transforms, and
Digital Signal
Processing
®
with MATLAB

Michael Corinthios
École Polytechnique de Montréal
Montréal, Canada

MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the
accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products
does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular

use of the MATLAB® software.

CRC Press
Taylor & Francis Group
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© 2009 by Taylor & Francis Group, LLC
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Version Date: 20150206
International Standard Book Number-13: 978-1-4200-9049-9 (eBook - PDF)
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To Maria, Angela, Gis`ele, John.

v

Contents

Preface

xxv

Acknowledgment

xxvii

1 Continuous-Time and Discrete-Time Signals and Systems
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Continuous-Time Signals . . . . . . . . . . . . . . . . . . . .
1.3 Periodic Functions . . . . . . . . . . . . . . . . . . . . . . .
1.4 Unit Step Function . . . . . . . . . . . . . . . . . . . . . . .
1.5 Graphical Representation of Functions . . . . . . . . . . . .
1.6 Even and Odd Parts of a Function . . . . . . . . . . . . . .
1.7 Dirac-Delta Impulse . . . . . . . . . . . . . . . . . . . . . .
1.8 Basic Properties of the Dirac-Delta Impulse . . . . . . . . .
1.9 Other Important Properties of the Impulse . . . . . . . . .
1.10 Continuous-Time Systems . . . . . . . . . . . . . . . . . . .
1.11 Causality, Stability . . . . . . . . . . . . . . . . . . . . . . .
1.12 Examples of Electrical Continuous-Time Systems . . . . . .
1.13 Mechanical Systems . . . . . . . . . . . . . . . . . . . . . .

1.14 Transfer Function and Frequency Response . . . . . . . . .
1.15 Convolution and Correlation . . . . . . . . . . . . . . . . . .
1.16 A Right-Sided and a Left-Sided Function . . . . . . . . . . .
1.17 Convolution with an Impulse and Its Derivatives . . . . . .
1.18 Additional Convolution Properties . . . . . . . . . . . . . .
1.19 Correlation Function . . . . . . . . . . . . . . . . . . . . . .
1.20 Properties of the Correlation Function . . . . . . . . . . . .
1.21 Graphical Interpretation . . . . . . . . . . . . . . . . . . . .
1.22 Correlation of Periodic Functions . . . . . . . . . . . . . . .
1.23 Average, Energy and Power of Continuous-Time Signals . .
1.24 Discrete-Time Signals . . . . . . . . . . . . . . . . . . . . .
1.25 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.26 Difference Equations . . . . . . . . . . . . . . . . . . . . . .
1.27 Even/Odd Decomposition . . . . . . . . . . . . . . . . . . .
1.28 Average Value, Energy and Power Sequences . . . . . . . .
1.29 Causality, Stability . . . . . . . . . . . . . . . . . . . . . . .
1.30 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.31 Answers to Selected Problems . . . . . . . . . . . . . . . . .

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1
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15
20
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22
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27
28
28
29
30
30
40

2 Fourier Series Expansion
2.1 Trigonometric Fourier Series . . . . . . .
2.2 Exponential Fourier Series . . . . . . . .
2.3 Exponential versus Trigonometric Series
2.4 Periodicity of Fourier Series . . . . . . .

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47
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51

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vii

viii

Signals, Systems, Transforms and Digital Signal Processing with MATLAB
2.5
2.6
2.7
2.8
2.9
2.10

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3 Laplace Transform
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Bilateral Laplace Transform . . . . . . . . . . . . . . .
3.3 Conditions of Existence of Laplace Transform . . . . .
3.4 Basic Laplace Transforms . . . . . . . . . . . . . . . .
3.5 Notes on the ROC of Laplace Transform . . . . . . . .
3.6 Properties of Laplace Transform . . . . . . . . . . . .
3.6.1 Linearity . . . . . . . . . . . . . . . . . . . . . .
3.6.2 Differentiation in Time . . . . . . . . . . . . . .
3.6.3 Multiplication by Powers of Time . . . . . . . .
3.6.4 Convolution in Time . . . . . . . . . . . . . . .
3.6.5 Integration in Time . . . . . . . . . . . . . . . .
3.6.6 Multiplication by an Exponential (Modulation)
3.6.7 Time Scaling . . . . . . . . . . . . . . . . . . .
3.6.8 Reflection . . . . . . . . . . . . . . . . . . . . .
3.6.9 Initial Value Theorem . . . . . . . . . . . . . .
3.6.10 Final Value Theorem . . . . . . . . . . . . . . .
3.6.11 Laplace Transform of Anticausal Functions . .
3.6.12 Shift in Time . . . . . . . . . . . . . . . . . . .

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105
105
105
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116
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119
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121

2.11

2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19
2.20
2.21
2.22

Dirichlet Conditions and Function Discontinuity
Proof of the Exponential Series Expansion . . .
Analysis Interval versus Function Period . . . .

Fourier Series as a Discrete-Frequency Spectrum
Meaning of Negative Frequencies . . . . . . . .
Properties of Fourier Series . . . . . . . . . . .
2.10.1 Linearity . . . . . . . . . . . . . . . . . .
2.10.2 Time Shift . . . . . . . . . . . . . . . . .
2.10.3 Frequency Shift . . . . . . . . . . . . . .
2.10.4 Function Conjugate . . . . . . . . . . .
2.10.5 Reflection . . . . . . . . . . . . . . . . .
2.10.6 Symmetry . . . . . . . . . . . . . . . . .
2.10.7 Half-Periodic Symmetry . . . . . . . . .
2.10.8 Double Symmetry . . . . . . . . . . . .
2.10.9 Time Scaling . . . . . . . . . . . . . . .
2.10.10 Differentiation Property . . . . . . . . .
Differentiation of Discontinuous Functions . . .
2.11.1 Multiplication in the Time Domain . . .
2.11.2 Convolution in the Time Domain . . . .
2.11.3 Integration . . . . . . . . . . . . . . . .
Fourier Series of an Impulse Train . . . . . . .
Expansion into Cosine or Sine Fourier Series . .
Deducing a Function Form from Its Expansion
Truncated Sinusoid Spectral Leakage . . . . . .
The Period of a Composite Sinusoidal Signal .
Passage through a Linear System . . . . . . . .
Parseval’s Relations . . . . . . . . . . . . . . .
Use of Power Series Expansion . . . . . . . . .
Inverse Fourier Series . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . .
Answers to Selected Problems . . . . . . . . . .

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Table of Contents
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16

3.17
3.18
3.19
3.20

Applications of the Differentiation Property . . . . . .
Transform of Right-Sided Periodic Functions . . . . .
Convolution in Laplace Domain . . . . . . . . . . . . .
Cauchy’s Residue Theorem . . . . . . . . . . . . . . .
Inverse Laplace Transform . . . . . . . . . . . . . . . .
Case of Conjugate Poles . . . . . . . . . . . . . . . . .
The Expansion Theorem of Heaviside . . . . . . . . . .
Application to Transfer Function and Impulse Response
Inverse Transform by Differentiation and Integration .
Unilateral Laplace Transform . . . . . . . . . . . . . .
3.16.1 Differentiation in Time . . . . . . . . . . . . . .

3.16.2 Initial and Final Value Theorem . . . . . . . .
3.16.3 Integration in Time Property . . . . . . . . . .
3.16.4 Division by Time Property . . . . . . . . . . .
Gamma Function . . . . . . . . . . . . . . . . . . . . .
Table of Additional Laplace Transforms . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . .
Answers to Selected Problems . . . . . . . . . . . . . .

4 Fourier Transform
4.1 Definition of the Fourier Transform . . . . . . . . . . .
4.2 Fourier Transform as a Function of f . . . . . . . . .
4.3 From Fourier Series to Fourier Transform . . . . . . .
4.4 Conditions of Existence of the Fourier Transform . . .
4.5 Table of Properties of the Fourier Transform . . . . . .
4.5.1 Linearity . . . . . . . . . . . . . . . . . . . . . .
4.5.2 Duality . . . . . . . . . . . . . . . . . . . . . .
4.5.3 Time Scaling . . . . . . . . . . . . . . . . . . .
4.5.4 Reflection . . . . . . . . . . . . . . . . . . . . .
4.5.5 Time Shift . . . . . . . . . . . . . . . . . . . . .
4.5.6 Frequency Shift . . . . . . . . . . . . . . . . . .
4.5.7 Modulation Theorem . . . . . . . . . . . . . . .
4.5.8 Initial Time Value . . . . . . . . . . . . . . . .
4.5.9 Initial Frequency Value . . . . . . . . . . . . .
4.5.10 Differentiation in Time . . . . . . . . . . . . . .
4.5.11 Differentiation in Frequency . . . . . . . . . . .
4.5.12 Integration in Time . . . . . . . . . . . . . . . .
4.5.13 Conjugate Function . . . . . . . . . . . . . . .
4.5.14 Real Functions . . . . . . . . . . . . . . . . . .
4.5.15 Symmetry . . . . . . . . . . . . . . . . . . . . .
4.6 System Frequency Response . . . . . . . . . . . . . . .

4.7 Even–Odd Decomposition of a Real Function . . . . .
4.8 Causal Real Functions . . . . . . . . . . . . . . . . . .
4.9 Transform of the Dirac-Delta Impulse . . . . . . . . .
4.10 Transform of a Complex Exponential and Sinusoid . .
4.11 Sign Function . . . . . . . . . . . . . . . . . . . . . . .
4.12 Unit Step Function . . . . . . . . . . . . . . . . . . . .
4.13 Causal Sinusoid . . . . . . . . . . . . . . . . . . . . . .
4.14 Table of Fourier Transforms of Basic Functions . . . .
4.15 Relation between Fourier and Laplace Transforms . . .
4.16 Relation to Laplace Transform with Poles on Imaginary

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Signals, Systems, Transforms and Digital Signal Processing with MATLAB
4.17
4.18
4.19
4.20
4.21
4.22

4.23
4.24
4.25
4.26
4.27
4.28
4.29
4.30
4.31
4.32
4.33
4.34
4.35
4.36
4.37
4.38
4.39

4.40
4.41
4.42
4.43
4.44

Convolution in Time . . . . . . . . . . . . . . . . . .
Linear System Input–Output Relation . . . . . . . .
Convolution in Frequency . . . . . . . . . . . . . . .
Parseval’s Theorem . . . . . . . . . . . . . . . . . . .
Energy Spectral Density . . . . . . . . . . . . . . . .
Average Value versus Fourier Transform . . . . . . .

Fourier Transform of a Periodic Function . . . . . .
Impulse Train . . . . . . . . . . . . . . . . . . . . . .
Fourier Transform of Powers of Time . . . . . . . . .
System Response to a Sinusoidal Input . . . . . . . .
Stability of a Linear System . . . . . . . . . . . . . .
Fourier Series versus Transform of Periodic Functions
Transform of a Train of Rectangles . . . . . . . . . .
Fourier Transform of a Truncated Sinusoid . . . . . .
Gaussian Function Laplace and Fourier Transform .
Inverse Transform by Series Expansion . . . . . . . .
Fourier Transform in ω and f . . . . . . . . . . . . .
Fourier Transform of the Correlation Function . . . .
Ideal Filters Impulse Response . . . . . . . . . . . .
Time and Frequency Domain Sampling . . . . . . . .
Ideal Sampling . . . . . . . . . . . . . . . . . . . . .
Reconstruction of a Signal from its Samples . . . . .
Other Sampling Systems . . . . . . . . . . . . . . . .
4.39.1 Natural Sampling . . . . . . . . . . . . . . . .
4.39.2 Instantaneous Sampling . . . . . . . . . . . .
Ideal Sampling of a Bandpass Signal . . . . . . . . .
Sampling an Arbitrary Signal . . . . . . . . . . . . .
Sampling the Fourier Transform . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . .
Answers to Selected Problems . . . . . . . . . . . . .

5 System Modeling, Time and Frequency Response
5.1 Transfer Function . . . . . . . . . . . . . . . . . . .
5.2 Block Diagram Reduction . . . . . . . . . . . . . .
5.3 Galvanometer . . . . . . . . . . . . . . . . . . . . .
5.4 DC Motor . . . . . . . . . . . . . . . . . . . . . . .

5.5 A Speed-Control System . . . . . . . . . . . . . . .
5.6 Homology . . . . . . . . . . . . . . . . . . . . . . .
5.7 Transient and Steady-State Response . . . . . . . .
5.8 Step Response of Linear Systems . . . . . . . . . .
5.9 First Order System . . . . . . . . . . . . . . . . . .
5.10 Second Order System Model . . . . . . . . . . . . .
5.11 Settling Time . . . . . . . . . . . . . . . . . . . . .
5.12 Second Order System Frequency Response . . . . .
5.13 Case of a Double Pole . . . . . . . . . . . . . . . .
5.14 The Over-Damped Case . . . . . . . . . . . . . . .
5.15 Evaluation of the Overshoot . . . . . . . . . . . . .
5.16 Causal System Response to an Arbitrary Input . .
5.17 System Response to a Causal Periodic Input . . . .
5.18 Response to a Causal Sinusoidal Input . . . . . . .
5.19 Frequency Response Plots . . . . . . . . . . . . . .

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Table of Contents

xi

5.20 Decibels, Octaves, Decades . . . . . . . . . . . . . . . . . . . . . . . . .
5.21 Asymptotic Frequency Response . . . . . . . . . . . . . . . . . . . . . .
5.21.1 A Simple Zero at the Origin . . . . . . . . . . . . . . . . . . . . .
5.21.2 A Simple Pole . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.21.3 A Simple Zero in the Left Plane . . . . . . . . . . . . . . . . . .
5.21.4 First Order System . . . . . . . . . . . . . . . . . . . . . . . . . .
5.21.5 Second Order System . . . . . . . . . . . . . . . . . . . . . . . .
5.22 Bode Plot of a Composite Linear System . . . . . . . . . . . . . . . . . .
5.23 Graphical Representation of a System Function . . . . . . . . . . . . . .
5.24 Vectorial Evaluation of Residues . . . . . . . . . . . . . . . . . . . . . .
5.25 Vectorial Evaluation of the Frequency Response . . . . . . . . . . . . . .
5.26 A First Order All-Pass System . . . . . . . . . . . . . . . . . . . . . . .
5.27 Filtering Properties of Basic Circuits . . . . . . . . . . . . . . . . . . . .
5.28 Lowpass First Order Filter . . . . . . . . . . . . . . . . . . . . . . . . . .
5.29 Minimum Phase Systems . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.30 General Order All-Pass Systems . . . . . . . . . . . . . . . . . . . . . . .
5.31 Signal Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.32 Application of Laplace Transform to Differential Equations . . . . . . .
5.32.1 Linear Differential Equations with Constant Coefficients . . . . .
5.32.2 Linear First Order Differential Equation . . . . . . . . . . . . . .
5.32.3 General Order Differential Equations with Constant Coefficients
5.32.4 Homogeneous Linear Differential Equations . . . . . . . . . . . .
5.32.5 The General Solution of a Linear Differential Equation . . . . . .
5.32.6 Partial Differential Equations . . . . . . . . . . . . . . . . . . . .
5.33 Transformation of Partial Differential Equations . . . . . . . . . . . . . .
5.34 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.35 Answers to Selected Problems . . . . . . . . . . . . . . . . . . . . . . . .
6 Discrete-Time Signals and Systems
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
6.2 Linear Time-Invariant Systems . . . . . . . . . . .
6.3 Linear Constant-Coefficient Difference Equations .
6.4 The z-Transform . . . . . . . . . . . . . . . . . . .
6.5 Convergence of the z-Transform . . . . . . . . . . .
6.6 Inverse z-Transform . . . . . . . . . . . . . . . . . .
6.7 Inverse z-Transform by Partial Fraction Expansion
6.8 Inversion by Long Division . . . . . . . . . . . . . .
6.9 Inversion by a Power Series Expansion . . . . . . .
6.10 Inversion by Geometric Series Summation . . . . .
6.11 Table of Basic z-Transforms . . . . . . . . . . . . .
6.12 Properties of the z-Transform . . . . . . . . . . . .
6.12.1 Linearity . . . . . . . . . . . . . . . . . . . .
6.12.2 Time Shift . . . . . . . . . . . . . . . . . . .
6.12.3 Conjugate Sequence . . . . . . . . . . . . .
6.12.4 Initial Value . . . . . . . . . . . . . . . . . .
6.12.5 Convolution in Time . . . . . . . . . . . . .

6.12.6 Convolution in Frequency . . . . . . . . . .
6.12.7 Parseval’s Relation . . . . . . . . . . . . . .
6.12.8 Final Value Theorem . . . . . . . . . . . . .
6.12.9 Multiplication by an Exponential . . . . . .
6.12.10 Frequency Translation . . . . . . . . . . . .

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xii

Signals, Systems, Transforms and Digital Signal Processing with MATLAB

6.13
6.14
6.15
6.16
6.17
6.18
6.19
6.20
6.21
6.22
6.23
6.24
6.25
6.26
6.27
6.28
6.29
6.30
6.31
6.32

6.12.11 Reflection Property . . . . . . . . . . . . . . . .
6.12.12 Multiplication by n . . . . . . . . . . . . . . . .
Geometric Evaluation of Frequency Response . . . . .
Comb Filters . . . . . . . . . . . . . . . . . . . . . . .
Causality and Stability . . . . . . . . . . . . . . . . . .
Delayed Response and Group Delay . . . . . . . . . .
Discrete-Time Convolution and Correlation . . . . . .
Discrete-Time Correlation in One Dimension . . . . .
Convolution and Correlation as Multiplications . . . .

Response of a Linear System to a Sinusoid . . . . . . .
Notes on the Cross-Correlation of Sequences . . . . . .
LTI System Input/Output Correlation Sequences . . .
Energy and Power Spectral Density . . . . . . . . . . .
Two-Dimensional Signals . . . . . . . . . . . . . . . .
Linear Systems, Convolution and Correlation . . . . .
Correlation of Two-Dimensional Signals . . . . . . . .
IIR and FIR Digital Filters . . . . . . . . . . . . . . .
Discrete-Time All-Pass Systems . . . . . . . . . . . . .
Minimum-Phase and Inverse System . . . . . . . . . .
Unilateral z-Transform . . . . . . . . . . . . . . . . . .
6.30.1 Time Shift Property of Unilateral z-Transform
Problems . . . . . . . . . . . . . . . . . . . . . . . . .
Answers to Selected Problems . . . . . . . . . . . . . .

7 Discrete-Time Fourier Transform
7.1 Laplace, Fourier and z-Transform Relations . . . . .
7.2 Discrete-Time Processing of Continuous-Time Signals
7.3 A/D Conversion . . . . . . . . . . . . . . . . . . . .
7.4 Quantization Error . . . . . . . . . . . . . . . . . . .
7.5 D/A Conversion . . . . . . . . . . . . . . . . . . . .
7.6 Continuous versus Discrete Signal Processing . . . .
7.7 Interlacing with Zeros . . . . . . . . . . . . . . . . .
7.8 Sampling Rate Conversion . . . . . . . . . . . . . . .
7.8.1 Sampling Rate Reduction . . . . . . . . . . .
7.8.2 Sampling Rate Increase: Interpolation . . . .
7.8.3 Rational Factor Sample Rate Alteration . . .
7.9 Fourier Transform of a Periodic Sequence . . . . . .
7.10 Table of Discrete-Time Fourier Transforms . . . . . .
7.11 Reconstruction of the Continuous-Time Signal . . . .

7.12 Stability of a Linear System . . . . . . . . . . . . . .
7.13 Table of Discrete-Time Fourier Transform Properties
7.14 Parseval’s Theorem . . . . . . . . . . . . . . . . . . .
7.15 Fourier Series and Transform Duality . . . . . . . . .
7.16 Discrete Fourier Transform . . . . . . . . . . . . . .
7.17 Discrete Fourier Series . . . . . . . . . . . . . . . . .
7.18 DFT of a Sinusoidal Signal . . . . . . . . . . . . . .
7.19 Deducing the z-Transform from the DFT . . . . . . .
7.20 DFT versus DFS . . . . . . . . . . . . . . . . . . . .
7.21 Properties of DFS and DFT . . . . . . . . . . . . . .
7.21.1 Periodic Convolution . . . . . . . . . . . . . .
7.22 Circular Convolution . . . . . . . . . . . . . . . . . .

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400
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406
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409
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414
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419
420
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425
425
425
426
429
433
434
436
438
439
441
443

Table of Contents
7.23
7.24

7.25
7.26
7.27
7.28
7.29
7.30
7.31

7.32
7.33
7.34
7.35

xiii

Circular Convolution Using the DFT . . . . . . . . . . . . . . . . . . .
Sampling the Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . .
Table of Properties of DFS . . . . . . . . . . . . . . . . . . . . . . . .
Shift in Time and Circular Shift . . . . . . . . . . . . . . . . . . . . . .
Table of DFT Properties . . . . . . . . . . . . . . . . . . . . . . . . . .
Zero Padding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Discrete z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . .
An Algorithm for a Wired-In Radix-2 Processor . . . . . . . . . . . . .
7.31.1 Post-Permutation Algorithm . . . . . . . . . . . . . . . . . . .
7.31.2 Ordered Input/Ordered Output (OIOO) Algorithm . . . . . . .
Factorization of the FFT to a Higher Radix . . . . . . . . . . . . . . .
7.32.1 Ordered Input/Ordered Output General Radix FFT Algorithm
Feedback Elimination for High-Speed Signal Processing . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Answers to Selected Problems . . . . . . . . . . . . . . . . . . . . . . .

8 State Space Modeling
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Note on Notation . . . . . . . . . . . . . . . . . . . . . . .
8.3 State Space Model . . . . . . . . . . . . . . . . . . . . . .
8.4 System Transfer Function . . . . . . . . . . . . . . . . . .
8.5 System Response with Initial Conditions . . . . . . . . . .
8.6 Jordan Canonical Form of State Space Model . . . . . . .
8.7 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . .
8.8 Matrix Diagonalization . . . . . . . . . . . . . . . . . . . .
8.9 Similarity Transformation of a State Space Model . . . . .
8.10 Solution of the State Equations . . . . . . . . . . . . . . .
8.11 General Jordan Canonical Form . . . . . . . . . . . . . . .
8.12 Circuit Analysis by Laplace Transform and State Variables
8.13 Trajectories of a Second Order System . . . . . . . . . . .
8.14 Second Order System Modeling . . . . . . . . . . . . . . .
8.15 Transformation of Trajectories between Planes . . . . . .
8.16 Discrete-Time Systems . . . . . . . . . . . . . . . . . . . .
8.17 Solution of the State Equations . . . . . . . . . . . . . . .
8.18 Transfer Function . . . . . . . . . . . . . . . . . . . . . . .
8.19 Change of Variables . . . . . . . . . . . . . . . . . . . . .
8.20 Second Canonical Form State Space Model . . . . . . . .
8.21 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.22 Answers to Selected Problems . . . . . . . . . . . . . . . .
9 Filters of Continuous-Time Domain
9.1 Lowpass Approximation . . . . . . . . . . . . . .
9.2 Butterworth Approximation . . . . . . . . . . . .
9.3 Denormalization of Butterworth Filter Prototype
9.4 Denormalized Transfer Function . . . . . . . . . .

9.5 The Case ε = 1 . . . . . . . . . . . . . . . . . . .
9.6 Butterworth Filter Order Formula . . . . . . . .
9.7 Nomographs . . . . . . . . . . . . . . . . . . . . .
9.8 Chebyshev Approximation . . . . . . . . . . . . .
9.9 Pass-Band Ripple . . . . . . . . . . . . . . . . . .

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445
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462
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472
478

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483
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543
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550
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xiv

Signals, Systems, Transforms and Digital Signal Processing with MATLAB
9.10
9.11
9.12
9.13
9.14
9.15
9.16

9.17
9.18
9.19
9.20
9.21
9.22
9.23
9.24
9.25
9.26
9.27
9.28
9.29
9.30
9.31
9.32
9.33
9.34
9.35
9.36
9.37
9.38
9.39
9.40
9.41
9.42
9.43
9.44
9.45
9.46

9.47
9.48
9.49
9.50
9.51
9.52
9.53
9.54
9.55
9.56
9.57

Transfer Function of the Chebyshev Filter . . . . . . . . . .
Maxima and Minima of Chebyshev Filter Response . . . . .
The Value of ε as a Function of Pass-Band Ripple . . . . .
Evaluation of Chebyshev Filter Gain . . . . . . . . . . . . .
Chebyshev Filter Tables . . . . . . . . . . . . . . . . . . . .
Chebyshev Filter Order . . . . . . . . . . . . . . . . . . . .
Denormalization of Chebyshev Filter Prototype . . . . . . .
Chebyshev’s Approximation: Second Form . . . . . . . . . .
Response Decay of Butterworth and Chebyshev Filters . . .
Chebyshev Filter Nomograph . . . . . . . . . . . . . . . . .
Elliptic Filters . . . . . . . . . . . . . . . . . . . . . . . . . .
9.20.1 Elliptic Integral . . . . . . . . . . . . . . . . . . . . .
Properties, Poles and Zeros of the sn Function . . . . . . .
9.21.1 Elliptic Filter Approximation . . . . . . . . . . . . .
Pole Zero Alignment and Mapping of Elliptic Filter . . . . .
Poles of H (s) . . . . . . . . . . . . . . . . . . . . . . . . . .
Zeros and Poles of G(ω) . . . . . . . . . . . . . . . . . . . .
Zeros, Maxima and Minima of the Magnitude Spectrum . .

Points of Maxima/Minima . . . . . . . . . . . . . . . . . . .
Elliptic Filter Nomograph . . . . . . . . . . . . . . . . . . .
N = 9 Example . . . . . . . . . . . . . . . . . . . . . . . . .
Tables of Elliptic Filters . . . . . . . . . . . . . . . . . . . .
Bessel’s Constant Delay Filters . . . . . . . . . . . . . . . .
A Note on Continued Fraction Expansion . . . . . . . . . .
Evaluating the Filter Delay . . . . . . . . . . . . . . . . . .
Bessel Filter Quality Factor and Natural Frequency . . . . .
Maximal Flatness of Bessel and Butterworth Response . . .
Bessel Filter’s Delay and Magnitude Response . . . . . . . .
Denormalization and Deviation from Ideal Response . . . .
Bessel Filter’s Magnitude and Delay . . . . . . . . . . . . .
Bessel Filter’s Butterworth Asymptotic Form . . . . . . . .
Delay of Bessel–Butterworth Asymptotic Form Filter . . . .
Delay Plots of Butterworth Asymptotic Form Bessel Filter .
Bessel Filters Frequency Normalized Form . . . . . . . . . .
Poles and Zeros of Asymptotic and Frequency Normalized
Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Response and Delay of Normalized Form Bessel Filter . . .
Bessel Frequency Normalized Form Attenuation Setting . .
Bessel Filter Nomograph . . . . . . . . . . . . . . . . . . . .
Frequency Transformations . . . . . . . . . . . . . . . . . .
Lowpass to Bandpass Transformation . . . . . . . . . . . . .
Lowpass to Band-Stop Transformation . . . . . . . . . . . .
Lowpass to Highpass Transformation . . . . . . . . . . . . .
Note on Lowpass to Normalized Band-Stop Transformation
Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rectangular Window . . . . . . . . . . . . . . . . . . . . . .
Triangle (Bartlett) Window . . . . . . . . . . . . . . . . . .
Hanning Window . . . . . . . . . . . . . . . . . . . . . . . .

Hamming Window . . . . . . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Answers to Selected Problems . . . . . . . . . . . . . . . . .

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629
633
634
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639
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651
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657
661
662
663
663
664
665
671

Table of Contents

xv

10 Passive and Active Filters
10.1 Design of Passive Filters . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Design of Passive Ladder Lowpass Filters . . . . . . . . . . . . . . . .
10.3 Analysis of a General Order Passive Ladder Network . . . . . . . . . .
10.4 Input Impedance of a Single-Resistance Terminated Network . . . . .
10.5 Evaluation of the Ladder Network Components . . . . . . . . . . . . .
10.6 Matrix Evaluation of Input Impedance . . . . . . . . . . . . . . . . . .
10.7 Bessel Filter Passive Ladder Networks . . . . . . . . . . . . . . . . . .
10.8 Tables of Single-Resistance Ladder Network Components . . . . . . . .
10.9 Design of Doubly Terminated Passive LC Ladder Networks . . . . . .
10.9.1 Input Impedance Evaluation . . . . . . . . . . . . . . . . . . . .
10.10 Tables of Double-Resistance Terminated Ladder Network Components
10.11 Closed Forms for Circuit Element Values . . . . . . . . . . . . . . . . .
10.12 Elliptic Filter Realization as a Passive Ladder Network . . . . . . . . .
10.12.1 Evaluating the Elliptic LC Ladder Circuit Elements . . . . . .
10.13 Table of Elliptic Filter Passive Network Components . . . . . . . . . .

10.14 Element Replacement for Frequency Transformation . . . . . . . . . .
10.14.1 Lowpass to Bandpass Transformation . . . . . . . . . . . . . .
10.14.2 Lowpass to Highpass Transformation . . . . . . . . . . . . . . .
10.14.3 Lowpass to Band-Stop Transformation . . . . . . . . . . . . . .
10.15 Realization of a General Order Active Filter . . . . . . . . . . . . . . .
10.16 Inverting Integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.17 Biquadratic Transfer Functions . . . . . . . . . . . . . . . . . . . . . .
10.18 General Biquad Realization . . . . . . . . . . . . . . . . . . . . . . . .
10.19 First Order Filter Realization . . . . . . . . . . . . . . . . . . . . . . .
10.20 A Biquadratic Transfer Function Realization . . . . . . . . . . . . . .
10.21 Sallen–Key Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.22 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.23 Answers to Selected Problems . . . . . . . . . . . . . . . . . . . . . . .

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677
677
677
680
683
684
689
693
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695
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701
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706
707
709
709
710

711
711
713
713
714
716
721
723
725
728
729

11 Digital Filters
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
11.2 Signal Flow Graphs . . . . . . . . . . . . . . . . . . . .
11.3 IIR Filter Models . . . . . . . . . . . . . . . . . . . . .
11.4 First Canonical Form . . . . . . . . . . . . . . . . . . .
11.5 Transposition . . . . . . . . . . . . . . . . . . . . . . .
11.6 Second Canonical Form . . . . . . . . . . . . . . . . .
11.7 Transposition of the Second Canonical Form . . . . . .
11.8 Structures Based on Poles and Zeros . . . . . . . . . .
11.9 Cascaded Form . . . . . . . . . . . . . . . . . . . . . .
11.10 Parallel Form . . . . . . . . . . . . . . . . . . . . . . .
11.11 Matrix Representation . . . . . . . . . . . . . . . . . .
11.12 Finite Impulse Response (FIR) Filters . . . . . . . . .
11.13 Linear Phase FIR Filters . . . . . . . . . . . . . . . . .
11.14 Conversion of Continuous-Time to Discrete-Time Filter
11.15 Impulse Invariance Approach . . . . . . . . . . . . . .
11.16 Impulse Invariance Approach Corrected . . . . . . . .
11.17 Backward-Rectangular Approximation . . . . . . . . .

11.18 Forward Rectangular and Trapezoidal Approximations
11.19 Bilinear Transform . . . . . . . . . . . . . . . . . . . .
11.20 Lattice Filters . . . . . . . . . . . . . . . . . . . . . . .

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733
733
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734
734
736
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738
738
739
739
740
741

743
743
745
747
749
751
760

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xvi

Signals, Systems, Transforms and Digital Signal Processing with MATLAB
11.21
11.22
11.23
11.24
11.25
11.26
11.27
11.28
11.29
11.30

11.31
11.32
11.33
11.34
11.35
11.36
11.37
11.38
11.39
11.40
11.41
11.42
11.43
11.44
11.45
11.46
11.47
11.48
11.49

Finite Impulse Response All-Zero Lattice Structures . .
One-Zero FIR Filter . . . . . . . . . . . . . . . . . . . .
Two-Zeros FIR Filter . . . . . . . . . . . . . . . . . . . .
General Order All-Zero FIR Filter . . . . . . . . . . . .
All-Pole Filter . . . . . . . . . . . . . . . . . . . . . . . .
First Order One-Pole Filter . . . . . . . . . . . . . . . .
Second Order All-Pole Filter . . . . . . . . . . . . . . . .
General Order All-Pole Filter . . . . . . . . . . . . . . .
Pole-Zero IIR Lattice Filter . . . . . . . . . . . . . . . .
All-Pass Filter Realization . . . . . . . . . . . . . . . . .

Schur–Cohn Stability Criterion . . . . . . . . . . . . . .
Frequency Transformations . . . . . . . . . . . . . . . .
Least Squares Digital Filter Design . . . . . . . . . . . .
Pad´e Approximation . . . . . . . . . . . . . . . . . . . .
Error Minimization in Prony’s Method . . . . . . . . . .
FIR Inverse Filter Design . . . . . . . . . . . . . . . . .
Impulse Response of Ideal Filters . . . . . . . . . . . . .
Spectral Leakage . . . . . . . . . . . . . . . . . . . . . .
Windows . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ideal Digital Filters Rectangular Window . . . . . . . .
Hanning Window . . . . . . . . . . . . . . . . . . . . . .
Hamming Window . . . . . . . . . . . . . . . . . . . . .
Triangular Window . . . . . . . . . . . . . . . . . . . . .
Comparison of Windows Spectral Parameters . . . . . .
Linear-Phase FIR Filter Design Using Windows . . . . .
Even- and Odd-Symmetric FIR Filter Design . . . . . .
Linear Phase FIR Filter Realization . . . . . . . . . . .
Sampling the Unit Circle . . . . . . . . . . . . . . . . . .
Impulse Response Evaluation from Unit Circle Samples
11.49.1 Case I-1: Odd Order, Even Symmetry, µ = 0 . .
11.49.2 Case I-2: Odd Order, Even Symmetry, µ = 1/2 .
11.49.3 Case II-1 . . . . . . . . . . . . . . . . . . . . . .
11.49.4 Case II-2: Even Order, Even Symmetry, µ = 1/2
11.49.5 Case III-1: Odd Order, Odd Symmetry, µ = 0 . .
11.49.6 Case III-2: Odd Order, Odd Symmetry, µ = 1/2 .
11.49.7 Case IV-1: Even Order, Odd Symmetry, µ = 0 .
11.49.8 Case IV-2: Even Order, Odd Symmetry, µ = 1/2
11.50 Problems . . . . . . . . . . . . . . . . . . . . . . . . . .
11.51 Answers to Selected Problems . . . . . . . . . . . . . . .
12 Energy and Power Spectral Densities

12.1 Energy Spectral Density . . . . . . . . . . . . .
12.2 Average, Energy and Power of Continuous-Time
12.3 Discrete-Time Signals . . . . . . . . . . . . . .
12.4 Energy Signals . . . . . . . . . . . . . . . . . .
12.5 Autocorrelation of Energy Signals . . . . . . . .
12.6 Energy Signal through a Linear System . . . .
12.7 Impulsive and Discrete-Time Energy Signals . .
12.8 Power Signals . . . . . . . . . . . . . . . . . . .
12.9 Cross-Correlation . . . . . . . . . . . . . . . . .
12.9.1 Power Spectral Density . . . . . . . . .

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760
761
762
764
769
770
771
772
775
781
782
783
786
786
790
794
798
800
801
801
802
803
804
805
807
808
810
810

814
814
815
815
815
816
816
816
816
817
828

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835
835
838
839
840
840
842
843
848
848
849

Table of Contents
12.10 Power Spectrum Conversion of a Linear System . . . . .
12.11 Impulsive and Discrete-Time Power Signals . . . . . . .
12.12 Periodic Signals . . . . . . . . . . . . . . . . . . . . . . .
12.12.1 Response of an LTI System to a Sinusoidal Input
12.13 Power Spectral Density of an Impulse Train . . . . . . .
12.14 Average, Energy and Power of a Sequence . . . . . . . .
12.15 Energy Spectral Density of a Sequence . . . . . . . . . .
12.16 Autocorrelation of an Energy Sequence . . . . . . . . . .
12.17 Power Density of a Sequence . . . . . . . . . . . . . . .
12.18 Passage through a Linear System . . . . . . . . . . . . .
12.19 Problems . . . . . . . . . . . . . . . . . . . . . . . . . .
12.20 Answers to Selected Problems . . . . . . . . . . . . . . .

xvii

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850
852
854
855
856
859
860
860
860
861
861
869

13 Introduction to Communication Systems
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2 Amplitude Modulation (AM) of Continuous-Time Signals . . . . . .

13.2.1 Double Side-Band (DSB) Modulation . . . . . . . . . . . . .
13.2.2 Double Side-Band Suppressed Carrier (DSB-SC) Modulation
13.2.3 Single Side-Band (SSB) Modulation . . . . . . . . . . . . . .
13.2.4 Vestigial Side-Band (VSB) Modulation . . . . . . . . . . . . .
13.2.5 Frequency Multiplexing . . . . . . . . . . . . . . . . . . . . .
13.3 Frequency Modulation . . . . . . . . . . . . . . . . . . . . . . . . . .
13.4 Discrete Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.4.1 Pulse Modulation Systems . . . . . . . . . . . . . . . . . . . .
13.5 Digital Communication Systems . . . . . . . . . . . . . . . . . . . . .
13.5.1 Pulse Code Modulation . . . . . . . . . . . . . . . . . . . . .
13.5.2 Pulse Duration Modulation . . . . . . . . . . . . . . . . . . .
13.5.3 Pulse Position Modulation . . . . . . . . . . . . . . . . . . . .
13.6 PCM-TDM Systems . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.7 Frequency Division Multiplexing (FDM) . . . . . . . . . . . . . . . .
13.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.9 Answers to Selected Problems . . . . . . . . . . . . . . . . . . . . . .

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875
875
876
876
877
879
882
882
883
887
887
888
888
890
892
893

893
894
904

14 Fourier-, Laplace- and z-Related Transforms
14.1 Walsh Transform . . . . . . . . . . . . . . . . . . . . . . . . .
14.2 Rademacher and Haar Functions . . . . . . . . . . . . . . . .
14.3 Walsh Functions . . . . . . . . . . . . . . . . . . . . . . . . .
14.4 The Walsh (Sequency) Order . . . . . . . . . . . . . . . . . .
14.5 Dyadic (Paley) Order . . . . . . . . . . . . . . . . . . . . . . .
14.6 Natural (Hadamard) Order . . . . . . . . . . . . . . . . . . .
14.7 Discrete Walsh Transform . . . . . . . . . . . . . . . . . . . .
14.8 Discrete-Time Walsh Transform . . . . . . . . . . . . . . . . .
14.9 Discrete-Time Walsh–Hadamard Transform . . . . . . . . . .
14.9.1 Natural (Hadamard) Order . . . . . . . . . . . . . . .
14.9.2 Dyadic or Paley Order . . . . . . . . . . . . . . . . . .
14.9.3 Sequency or Walsh Order . . . . . . . . . . . . . . . .
14.10 Natural (Hadamard) Order Fast Walsh–Hadamard Transform
14.11 Dyadic (Paley) Order Fast Walsh–Hadamard Transform . . .
14.12 Sequency Ordered Fast Walsh–Hadamard Transform . . . . .
14.13 Generalized Walsh Transform . . . . . . . . . . . . . . . . . .
14.14 Natural Order . . . . . . . . . . . . . . . . . . . . . . . . . . .

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911
911
911
912
913
914
914
916
917
917
917
918
919

919
920
921
922
922

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xviii
14.15
14.16
14.17
14.18
14.19
14.20
14.21
14.22
14.23
14.24
14.25
14.26
14.27
14.28

14.29
14.30
14.31
14.32
14.33
14.34
14.35
14.36
14.37
14.38
14.39
14.40
14.41
14.42
14.43
14.44
14.45
14.46
14.47
14.48

Signals, Systems, Transforms and Digital Signal Processing with MATLAB
Generalized Sequency Order . . . . . . . . . . . . . . . .
Generalized Walsh–Paley (p-adic) Transform . . . . . .
Walsh–Kaczmarz Transform . . . . . . . . . . . . . . . .
Generalized Walsh Factorizations for Parallel Processing
Generalized Walsh Natural Order GWN Matrix . . . . .
Generalized Walsh–Paley GWP Transformation Matrix
GWK Transformation Matrix . . . . . . . . . . . . . . .
High Speed Optimal Generalized Walsh Factorizations .

GWN Optimal Factorization . . . . . . . . . . . . . . .
GWP Optimal Factorization . . . . . . . . . . . . . . . .
GWK Optimal Factorization . . . . . . . . . . . . . . .
Karhunen Lo`eve Transform . . . . . . . . . . . . . . . .
Hilbert Transform . . . . . . . . . . . . . . . . . . . . .
Hilbert Transformer . . . . . . . . . . . . . . . . . . . .
Discrete Hilbert Transform . . . . . . . . . . . . . . . .
Hartley Transform . . . . . . . . . . . . . . . . . . . . .
Discrete Hartley Transform . . . . . . . . . . . . . . . .
Mellin Transform . . . . . . . . . . . . . . . . . . . . . .
Mellin Transform of ejx . . . . . . . . . . . . . . . . . .
Hankel Transform . . . . . . . . . . . . . . . . . . . . . .
Fourier Cosine Transform . . . . . . . . . . . . . . . . .
Discrete Cosine Transform (DCT) . . . . . . . . . . . .
Fractional Fourier Transform . . . . . . . . . . . . . . .
Discrete Fractional Fourier Transform . . . . . . . . . .
Two-Dimensional Transforms . . . . . . . . . . . . . . .
Two-Dimensional Fourier Transform . . . . . . . . . . .
Continuous-Time Domain Hilbert Transform Relations .
HI (jω) versus HR (jω) with No Poles on Axis . . . . . .
Case of Poles on the Imaginary Axis . . . . . . . . . . .
Hilbert Transform Closed Forms . . . . . . . . . . . . .
Wiener–Lee Transforms . . . . . . . . . . . . . . . . . .
Discrete-Time Domain Hilbert Transform Relations . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . .
Answers to Selected Problems . . . . . . . . . . . . . . .

15 Digital Signal Processors: Architecture, Logic Design
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
15.2 Systems for the Representation of Numbers . . . . . . .

15.3 Conversion from Decimal to Binary . . . . . . . . . . . .
15.4 Integers, Fractions and the Binary Point . . . . . . . . .
15.5 Representation of Negative Numbers . . . . . . . . . . .
15.5.1 Sign and Magnitude Notation . . . . . . . . . . .
15.5.2 1’s and 2’s Complement Notation . . . . . . . . .
15.6 Integer and Fractional Representation of Signed Numbers
15.6.1 1’s and 2’s Complement of Signed Numbers . . .
15.7 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.7.1 Addition in Sign and Magnitude Notation . . . .
15.7.2 Addition in 1’s Complement Notation . . . . . .
15.7.3 Addition in 2’s Complement Notation . . . . . .
15.8 Subtraction . . . . . . . . . . . . . . . . . . . . . . . . .
15.8.1 Subtraction in Sign and Magnitude Notation . .

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923
923
923
924
924
925
926
926
926
927
927
928
931
934
935
936
938
939
941
943
945
946
948
950
950
951

953
953
957
958
959
961
964
967

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973
973
973
974
974
975
975
976
978
979
982
982
984
985
986
987

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Table of Contents

15.9
15.10
15.11
15.12
15.13
15.14

15.15
15.16

15.17
15.18
15.19
15.20
15.21
15.22
15.23
15.24

15.25

15.26
15.27
15.28

15.29

15.30
15.31
15.32
15.33
15.34

15.8.2 Numbers in 1’s Complement Notation . . . . . .
15.8.3 Subtraction in 2’s Complement Notation . . . . .
Full Adder Cell . . . . . . . . . . . . . . . . . . . . . . .
Addition/Subtraction Implementation in 2’s Complement
Controlled Add/Subtract (CAS) Cell
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Multiplication of Unsigned Numbers . . . . . . . . . . .
Multiplier Implementation . . . . . . . . . . . . . . . . .
3-D Multiplier . . . . . . . . . . . . . . . . . . . . . . . .
15.14.1 Multiplication in Sign and Magnitude Notation .
15.14.2 Multiplication in 1’s Complement Notation . . .
15.14.3 Numbers in 2’s Complement Notation . . . . . .
A Direct Approach to 2’s Complement Multiplication .
Division . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.16.1 Division of Positive Numbers: . . . . . . . . . . .
15.16.2 Division in Sign and Magnitude Notation . . . .
15.16.3 Division in 1’s Complement . . . . . . . . . . . .
15.16.4 Division in 2’s Complement . . . . . . . . . . . .

15.16.5 Nonrestoring Division . . . . . . . . . . . . . . .
Cellular Array for Nonrestoring Division . . . . . . . . .
Carry Look Ahead (CLA) Cell . . . . . . . . . . . . . .
2’s Complement Nonrestoring Division . . . . . . . . . .
Convergence Division . . . . . . . . . . . . . . . . . . . .
Evaluation of the n th Root . . . . . . . . . . . . . . . .
Function Generation by Chebyshev Series Expansion . .
An Alternative Approach to Chebyshev Series Expansion
Floating Point Number Representation . . . . . . . . . .
15.24.1 Addition and Subtraction . . . . . . . . . . . . .
15.24.2 Multiplication . . . . . . . . . . . . . . . . . . . .
15.24.3 Division . . . . . . . . . . . . . . . . . . . . . . .
Square Root Evaluation . . . . . . . . . . . . . . . . . .
15.25.1 The Paper and Pencil Method . . . . . . . . . . .
15.25.2 Binary Square Root Evaluation . . . . . . . . . .
15.25.3 Comparison Approach . . . . . . . . . . . . . . .
15.25.4 Restoring Approach . . . . . . . . . . . . . . . .
15.25.5 Nonrestoring Approach . . . . . . . . . . . . . .
Cellular Array for Nonrestoring Square Root Extraction
Binary Coded Decimal (BCD) Representation . . . . . .
Memory Elements . . . . . . . . . . . . . . . . . . . . .
15.28.1 Set-Reset (SR) Flip-Flop . . . . . . . . . . . . . .
15.28.2 The Trigger or T Flip-Flop . . . . . . . . . . . .
15.28.3 The JK Flip-Flop . . . . . . . . . . . . . . . . .
15.28.4 Master-Slave Flip-Flop . . . . . . . . . . . . . . .
Design of Synchronous Sequential Circuits . . . . . . . .
15.29.1 Realization Using SR Flip-Flops . . . . . . . . .
15.29.2 Realization Using JK Flip-Flops. . . . . . . . . .
Realization of a Counter Using T Flip-Flops . . . . . . .
15.30.1 Realization Using JK Flip-Flops . . . . . . . . .

State Minimization . . . . . . . . . . . . . . . . . . . . .
Asynchronous Sequential Machines . . . . . . . . . . . .
State Reduction . . . . . . . . . . . . . . . . . . . . . . .
Control Counter Design for Generator of Prime Numbers

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988
989
990
991
992
992
993
995
997
997
998
1000
1002
1003
1004
1004
1005
1006
1009
1011

1014
1016
1018
1020
1026
1027
1029
1029
1030
1030
1030
1031
1031
1032
1032
1033
1033
1037
1038
1040
1040
1041
1042
1044
1045
1046
1046
1048
1050
1051

1054

xx

Signals, Systems, Transforms and Digital Signal Processing with MATLAB

15.35
15.36
15.37
15.38
15.39
15.40
15.41

15.42
15.43
15.44
15.45
15.46
15.47
15.48
15.49
15.50
15.51
15.52
15.53

15.54
15.55

15.56

15.34.1 Micro-operations and States . . . . . . . . . . . . . . . . .
Fast Transform Processors . . . . . . . . . . . . . . . . . . . . . .
Programmable Logic Arrays (PLAs) . . . . . . . . . . . . . . . .
Field Programmable Gate Arrays (FPGAs) . . . . . . . . . . . .
DSP with Xilinx FPGAs . . . . . . . . . . . . . . . . . . . . . . .
Texas Instruments TMS320C6713B Floating-Point DSP . . . . .
Central Processing Unit (CPU) . . . . . . . . . . . . . . . . . . .
CPU Data Paths and Control . . . . . . . . . . . . . . . . . . . .
15.41.1 General-Purpose Register Files . . . . . . . . . . . . . . .
15.41.2 Functional Units . . . . . . . . . . . . . . . . . . . . . . .
15.41.3 Register File Cross Paths . . . . . . . . . . . . . . . . . .
15.41.4 Memory, Load, and Store Paths . . . . . . . . . . . . . . .
15.41.5 Data Address Paths . . . . . . . . . . . . . . . . . . . . .
Instruction Syntax . . . . . . . . . . . . . . . . . . . . . . . . . .
TMS320C6000 Control Register File . . . . . . . . . . . . . . . .
Addressing Mode Register (AMR) . . . . . . . . . . . . . . . . .
15.44.1 Addressing Modes . . . . . . . . . . . . . . . . . . . . . .
Syntax for Load/Store Address Generation . . . . . . . . . . . .
15.45.1 Linear Addressing Mode . . . . . . . . . . . . . . . . . . .
Programming the T.I. DSP . . . . . . . . . . . . . . . . . . . . .
A Simple C Program . . . . . . . . . . . . . . . . . . . . . . . . .
The Generated Assembly Code . . . . . . . . . . . . . . . . . . .
15.48.1 Calling an Assembly Language Function . . . . . . . . . .
Fibonacci Series in C Calling Assembly-Language Function . . .
Finite Impulse Response (FIR) Filter . . . . . . . . . . . . . . . .
Infinite Impulse Response (IIR) Filter on the DSP . . . . . . . .
Real-Time DSP Applications Using MATLAB–Simulink . . . . .
Detailed Steps for DSP Programming in C++ and Simulink . . .

15.53.1 Steps to Implement a C++ Program on the DSP Card . .
15.53.2 Steps to Implement a Simulink Program on the DSP Card
MOS FET Logic Circuit Realization . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Answers to Selected Problems . . . . . . . . . . . . . . . . . . . .

16 Random Signal Processing
16.1 Nonparametric Methods of Power Spectrum Estimation . . . .
16.2 Correlation of Continuous-Time Random Signals . . . . . . . .
16.3 Passage through an LTI System . . . . . . . . . . . . . . . . . .
16.4 Wiener Filtering in Continuous-Time Domain . . . . . . . . . .
16.5 Causal Wiener Filter . . . . . . . . . . . . . . . . . . . . . . . .
16.6 Random Sequences . . . . . . . . . . . . . . . . . . . . . . . . .
16.7 From Statistical to Time Averages . . . . . . . . . . . . . . . .
16.8 Correlation and Covariance in z-Domain . . . . . . . . . . . . .
16.9 Random Signal Passage through an LTI System . . . . . . . . .
16.10 PSD Estimation of Discrete-Time Random Sequences . . . . .
16.11 Fast Fourier Transform (FFT) Evaluation of the Periodogram .
16.12 Parametric Methods for PSD Estimation . . . . . . . . . . . . .
16.13 The Yule–Walker Equations . . . . . . . . . . . . . . . . . . . .
16.14 System Modeling for Linear Prediction, Adaptive Filtering and
Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.15 Wiener and Least-Squares Models . . . . . . . . . . . . . . . .

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1055
1059
1062
1063
1065
1067
1069
1071
1071
1072
1072
1073
1073
1074
1074
1075
1076
1076
1077
1078
1079
1080
1083
1087
1087
1088
1092
1094
1094

1096
1098
1099
1102
1105
1108
1109
1110
1113
1116
1118
1119
1120
1121
1124
1128
1131
1132
1134
1134

Table of Contents
16.16
16.17
16.18
16.19
16.20
16.21
16.22

16.23
16.24
16.25
16.26
16.27
16.28
16.29
16.30
16.31
16.32
16.33
16.34
16.35
16.36
16.37
16.38
16.39
16.40
16.41

Wiener Filtering . . . . . . . . . . . . . . . . . .
Least-Squares Filtering . . . . . . . . . . . . . . .
Forward Linear Prediction . . . . . . . . . . . . .
Backward Linear Prediction . . . . . . . . . . . .
Lattice MA FIR Filter Realization . . . . . . . .
AR Lattice of Order p . . . . . . . . . . . . . . .
ARMA(p, q) Process . . . . . . . . . . . . . . . .
Power Spectrum Estimation . . . . . . . . . . . .
FIR Wiener Filtering of Noisy Signals . . . . . .
Two-Sided IIR Wiener Filtering . . . . . . . . . .

Causal IIR Wiener Filter . . . . . . . . . . . . . .
Wavelet Transform . . . . . . . . . . . . . . . . .
Discrete Wavelet Transform . . . . . . . . . . . .
Important Signal Processing MATLAB Functions
lpc . . . . . . . . . . . . . . . . . . . . . . . . . .
Yulewalk . . . . . . . . . . . . . . . . . . . . . . .
dfilt . . . . . . . . . . . . . . . . . . . . . . . . .
logspace . . . . . . . . . . . . . . . . . . . . . . .
FIR Filter Design . . . . . . . . . . . . . . . . . .
fir2 . . . . . . . . . . . . . . . . . . . . . . . . . .
Power Spectrum Estimation Using MATLAB . .
Parametric Modeling Functions . . . . . . . . . .
prony . . . . . . . . . . . . . . . . . . . . . . . .
A z-Domain Counterpart to Prony’s Method . . .
Problems . . . . . . . . . . . . . . . . . . . . . .
Answers to Selected Problems . . . . . . . . . . .

xxi
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1135
1138
1138
1140
1143
1146
1146
1147
1148
1151
1152
1154
1157
1164
1167
1168
1169
1170
1170
1173
1174
1174
1175
1176

1176
1179

17 Distributions
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.2 Distributions as Generalizations of Functions . . . . . . . . .
17.3 What is a Distribution? . . . . . . . . . . . . . . . . . . . . .
17.4 The Impulse as the Limit of a Sequence . . . . . . . . . . . .
17.5 Properties of Distributions . . . . . . . . . . . . . . . . . . . .
17.5.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . .
17.5.2 Time Shift . . . . . . . . . . . . . . . . . . . . . . . . .
17.5.3 Time Scaling . . . . . . . . . . . . . . . . . . . . . . .
17.5.4 Product with an Ordinary Function . . . . . . . . . .
17.5.5 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . .
17.5.6 Differentiation . . . . . . . . . . . . . . . . . . . . . .
17.5.7 Multiplication Times an Ordinary Function . . . . . .
17.5.8 Sequence of Distributions . . . . . . . . . . . . . . . .
17.6 Approximating the Impulse . . . . . . . . . . . . . . . . . . .
17.7 Other Approximating Sequences and Functions of the Impulse
17.8 Test Functions . . . . . . . . . . . . . . . . . . . . . . . . . .
17.9 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.10 Multiplication by an Impulse Derivative . . . . . . . . . . . .
17.11 The Dirac-Delta Impulse as a Limit of a Gaussian Function .
17.12 Fourier Transform of Unity . . . . . . . . . . . . . . . . . . .
17.13 The Impulse of a Function . . . . . . . . . . . . . . . . . . . .
17.14 Multiplication by t . . . . . . . . . . . . . . . . . . . . . . . .
17.15 Time Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1181
1181
1181
1182
1184
1184
1184
1185
1185
1186
1186
1187
1187
1187

1187
1190
1191
1192
1193
1195
1196
1196
1199
1199

xxii
17.16
17.17
17.18
17.19
17.20
17.21
17.22
17.23
17.24
17.25
17.26
17.27
17.28
17.29
17.30
17.31
17.32

Signals, Systems, Transforms and Digital Signal Processing with MATLAB
Some Properties of the Dirac-Delta Impulse . . . .
Additional Fourier Transforms . . . . . . . . . . . .
Riemann–Lebesgue Lemma . . . . . . . . . . . . .
Generalized Limits . . . . . . . . . . . . . . . . . .
Fourier Transform of Higher Impulse Derivatives .
The Distribution t−k . . . . . . . . . . . . . . . . .
Initial Derivatives of the Transform . . . . . . . . .
The Unit Step Function as a Limit . . . . . . . . .
Inverse Fourier Transform and Gibbs Phenomenon
Ripple Elimination . . . . . . . . . . . . . . . . . .
Transforms of |t| and tu(t) . . . . . . . . . . . . . .
The Impulse Train as a Limit . . . . . . . . . . . .
Sequence of Distributions . . . . . . . . . . . . . .
Poisson’s Summation Formula . . . . . . . . . . . .
Moving Average . . . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . .
Answers to Selected Problems . . . . . . . . . . . .

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1200
1201
1201
1202
1204
1204
1206
1207
1208
1212
1213
1214
1216
1218
1219
1220

1222

18 Generalization of Distributions Theory, Extending Laplace-, z- and
Fourier-Related Transforms
18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.2 An Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.3 Generalized Distributions for Continuous-Time Functions . . . . . . .
18.3.1 Properties of Generalized Distributions in s Domain . . . . . .
18.3.2 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.3.3 Shift in s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.3.4 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.3.5 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.3.6 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.3.7 Multiplication of Derivative by an Ordinary Function . . . . . .
18.4 Properties of the Generalized Impulse in s Domain . . . . . . . . . . .
18.4.1 Shifted Generalized Impulse . . . . . . . . . . . . . . . . . . . .
18.4.2 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.4.3 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.4.4 Convolution with an Ordinary Function . . . . . . . . . . . . .
18.4.5 Multiplication of an Impulse Times an Ordinary Function . . .
18.4.6 Multiplication by Higher Derivatives of the Impulse . . . . . .
18.5 Generalized Impulse as a Limit of a Three-Dimensional Sequence . . .
18.6 Discrete-Time Domain . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.7 3-D Test Function as a Possible Generalization . . . . . . . . . . . . .
18.7.1 Properties of Generalized Distributions in z-Domain . . . . . .
18.7.2 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.7.3 Scaling in z-Domain . . . . . . . . . . . . . . . . . . . . . . . .
18.7.4 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.7.5 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.8 Properties of the Generalized Impulse in z-Domain . . . . . . . . . . .

18.8.1 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.9 Additional Generalized Impulse Properties . . . . . . . . . . . . . . . .
18.10 Generalized Impulse as Limit of a 3-D Sequence . . . . . . . . . . . . .
18.10.1 Convolution of Generalized Impulses . . . . . . . . . . . . . . .
18.10.2 Convolution with an Ordinary Function . . . . . . . . . . . . .

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1225
1225
1225
1226
1226
1226
1226
1227
1227
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1228
1228
1228
1228
1228
1229
1230
1230
1230
1233

1234
1234
1235
1235
1235
1236
1236
1236
1237
1238
1240
1241

Table of Contents
18.11
18.12
18.13
18.14
18.15
18.16
18.17
18.18
18.19
18.20

Extended Laplace and z-Transforms . . . . . . . . . . . . . . . . .
Generalization of Fourier-, Laplace- and z-Related Transforms . . .
Hilbert Transform Generalization . . . . . . . . . . . . . . . . . . .
Generalizing the Discrete Hilbert Transform . . . . . . . . . . . . .

Generalized Hartley Transform . . . . . . . . . . . . . . . . . . . .
Generalized Discrete Hartley Transform . . . . . . . . . . . . . . .
Generalization of the Mellin Transform . . . . . . . . . . . . . . . .
Multidimensional Signals and the Solution of Differential Equations
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Answers to Selected Problems . . . . . . . . . . . . . . . . . . . . .

A Appendix
A.1 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 Frequently Needed Expansions . . . . . . . . . . . . . . . . .
A.3 Important Trigonometric Relations . . . . . . . . . . . . . . .
A.4 Orthogonality Relations . . . . . . . . . . . . . . . . . . . . .
A.5 Frequently Encountered Functions . . . . . . . . . . . . . . .
A.6 Mathematical Formulae . . . . . . . . . . . . . . . . . . . . .
A.7 Frequently Encountered Series Sums . . . . . . . . . . . . . .
A.8 Biographies of Pioneering Scientists . . . . . . . . . . . . . . .
A.9 Plato (428 BC–347 BC) . . . . . . . . . . . . . . . . . . . . .
A.10 Euclid (circa 300 BC) . . . . . . . . . . . . . . . . . . . . . .
A.11 Ptolemy (circa 90–168 AD) . . . . . . . . . . . . . . . . . . .
A.12 Abu Ja’far Muhammad ibn Musa Al-Khwarizmi (780–850 AD)
A.13 Nicolaus Copernicus (1473–1543) . . . . . . . . . . . . . . . .
A.14 Galileo Galilei (1564–1642) . . . . . . . . . . . . . . . . . . .
A.15 Sir Isaac Newton (1643–1727) . . . . . . . . . . . . . . . . . .
A.16 Guillaume-Fran¸cois-Antoine de L’Hˆopital (1661–1704) . . . .
A.17 Pierre-Simon Laplace (1749–1827) . . . . . . . . . . . . . . .
A.18 Gaspard Clair Fran¸cois Marie, Baron Riche de Prony
(1755–1839) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.19 Jean Baptiste Joseph Fourier (1768–1830) . . . . . . . . . . .
A.20 Johann Carl Friedrich Gauss (1777–1855) . . . . . . . . . . .
A.21 Friedrich Wilhelm Bessel (1784–1846) . . . . . . . . . . . . .

A.22 Augustin-Louis Cauchy (1789–1857) . . . . . . . . . . . . . .
A.23 Niels Henrik Abel (1802–1829) . . . . . . . . . . . . . . . . .
A.24 Johann Peter Gustav Lejeune Dirichlet (1805–1859) . . . . .
A.25 Pafnuty Lvovich Chebyshev (1821–1894) . . . . . . . . . . . .
A.26 Paul A.M. Dirac . . . . . . . . . . . . . . . . . . . . . . . . .
Index

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