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

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## Book Description

**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.

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, continuous- and 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.

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.

## Table of Contents

**Continuous-Time and Discrete-Time Signals and Systems**Introduction

Continuous-Time Signals

Periodic Functions

Unit Step Function

Graphical Representation of Functions

Even and Odd Parts of a Function

Dirac-Delta Impulse

Basic Properties of the Dirac-Delta Impulse

Other Important Properties of the Impulse

Continuous-Time Systems

Causality, Stability

Examples of Electrical Continuous-Time Systems

Mechanical Systems

Transfer Function and Frequency Response

Convolution and Correlation

A Right-Sided and a Left-Sided Function

Convolution with an Impulse and Its Derivatives

Additional Convolution Properties

Correlation Function

Properties of the Correlation Function

Graphical Interpretation

Correlation of Periodic Functions

Average, Energy and Power of Continuous-Time Signals

Discrete-Time Signals

Periodicity

Difference Equations

Even/Odd Decomposition

Average Value, Energy and Power Sequences

Causality, Stability

Problems

Answers to Selected Problems

**Trigonometric Fourier Series**

Fourier Series Expansion

Fourier Series Expansion

Exponential Fourier Series

Exponential versus Trigonometric Series

Periodicity of Fourier Series

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

Differentiation of Discontinuous Functions

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

**Introduction**

Laplace Transform

Laplace Transform

Bilateral Laplace Transform

Conditions of Existence of Laplace Transform

Basic Laplace Transforms

Notes on the ROC of Laplace Transform

Properties of Laplace Transform

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

Gamma Function

Table of Additional Laplace Transforms

Problems

Answers to Selected Problems

**Definition of the Fourier Transform**

Fourier Transform

Fourier Transform

Fourier Transform as a Function of f

From Fourier Series to Fourier Transform

Conditions of Existence of the Fourier Transform

Table of Properties of the Fourier Transform

System Frequency Response

Even–Odd Decomposition of a Real Function

Causal Real Functions

Transform of the Dirac-Delta Impulse

Transform of a Complex Exponential and Sinusoid

Sign Function

Unit Step Function

Causal Sinusoid

Table of Fourier Transforms of Basic Functions

Relation between Fourier and Laplace Transforms

Relation to Laplace Transform with Poles on Imaginary Axis

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

Ideal Sampling of a Bandpass Signal

Sampling an Arbitrary Signal

Sampling the Fourier Transform

Problems

Answers to Selected Problems

**Transfer Function**

System Modeling, Time and Frequency Response

System Modeling, Time and Frequency Response

Block Diagram Reduction

Galvanometer

DC Motor

A Speed-Control System

Homology

Transient and Steady-State Response

Step Response of Linear Systems

First Order System

Second Order System Model

Settling Time

Second Order System Frequency Response

Case of a Double Pole

The Over-Damped Case

Evaluation of the Overshoot

Causal System Response to an Arbitrary Input

System Response to a Causal Periodic Input

Response to a Causal Sinusoidal Input

Frequency Response Plots

Decibels, Octaves, Decades

Asymptotic Frequency Response

Bode Plot of a Composite Linear System

Graphical Representation of a System Function

Vectorial Evaluation of Residues

Vectorial Evaluation of the Frequency Response

A First Order All-Pass System

Filtering Properties of Basic Circuits

Lowpass First Order Filter

Minimum Phase Systems

General Order All-Pass Systems

Signal Generation

Application of Laplace Transform to Differential Equations

Transformation of Partial Differential Equations

Problems

Answers to Selected Problems

**Introduction**

Discrete-Time Signals and Systems

Discrete-Time Signals and Systems

Linear Time-Invariant Systems

Linear Constant-Coefficient Difference Equations

The z-Transform

Convergence of the z-Transform

Inverse z-Transform

Inverse z-Transform by Partial Fraction Expansion

Inversion by Long Division

Inversion by a Power Series Expansion

Inversion by Geometric Series Summation

Table of Basic z-Transforms

Properties of the z-Transform

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

Problems

Answers to Selected Problems

**Laplace, Fourier and z-Transform Relations**

Discrete-Time Fourier Transform

Discrete-Time Fourier Transform

Discrete-Time Processing of Continuous-Time Signals

A/D Conversion

Quantization Error

D/A Conversion

Continuous versus Discrete Signal Processing

Interlacing with Zeros

Sampling Rate Conversion

Fourier Transform of a Periodic Sequence

Table of Discrete-Time Fourier Transforms

Reconstruction of the Continuous-Time Signal

Stability of a Linear System

Table of Discrete-Time Fourier Transform Properties

Parseval’s Theorem

Fourier Series and Transform Duality

Discrete Fourier Transform

Discrete Fourier Series

DFT of a Sinusoidal Signal

Deducing the z-Transform from the DFT

DFT versus DFS

Properties of DFS and DFT

Circular Convolution

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

Factorization of the FFT to a Higher Radix

Feedback Elimination for High-Speed Signal Processing

Problems

Answers to Selected Problems

**Introduction**

State Space Modeling

State Space Modeling

Note on Notation

State Space Model

System Transfer Function

System Response with Initial Conditions

Jordan Canonical Form of State Space Model

Eigenvalues and Eigenvectors

Matrix Diagonalization

Similarity Transformation of a State Space Model

Solution of the State Equations

General Jordan Canonical Form

Circuit Analysis by Laplace Transform and State Variables

Trajectories of a Second Order System

Second Order System Modeling

Transformation of Trajectories between Planes

Discrete-Time Systems

Solution of the State Equations

Transfer Function

Change of Variables

Second Canonical Form State Space Model

Problems

Answers to Selected Problems

**Lowpass Approximation**

Filters of Continuous-Time Domain

Filters of Continuous-Time Domain

Butterworth Approximation

Denormalization of Butterworth Filter Prototype

Denormalized Transfer Function

The Case ε 6= 1

Butterworth Filter Order Formula

Nomographs

Chebyshev Approximation

Pass-Band Ripple

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

Properties, Poles and Zeros of the sn Function

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 Bessel Filter 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

**Design of Passive Filters**

Passive and Active Filters

Passive and Active Filters

Design of Passive Ladder Lowpass Filters

Analysis of a General Order Passive Ladder Network

Input Impedance of a Single-Resistance Terminated Network

Evaluation of the Ladder Network Components

Matrix Evaluation of Input Impedance

Bessel Filter Passive Ladder Networks

Tables of Single-Resistance Ladder Network Components

Design of Doubly Terminated Passive LC Ladder Networks

Tables of Double-Resistance Terminated Ladder Network Components

Closed Forms for Circuit Element Values

Elliptic Filter Realization as a Passive Ladder Network

Table of Elliptic Filter Passive Network Components

Element Replacement for Frequency Transformation

Realization of a General Order Active Filter

Inverting Integrator

Biquadratic Transfer Functions

General Biquad Realization

First Order Filter Realization

A Biquadratic Transfer Function Realization

Sallen–Key Circuit

Problems

Answers to Selected Problems

**Introduction**

Digital Filters

Digital Filters

Signal Flow Graphs

IIR Filter Models

First Canonical Form

Transposition

Second Canonical Form

Transposition of the Second Canonical Form

Structures Based on Poles and Zeros

Cascaded Form

Parallel Form

Matrix Representation

Finite Impulse Response (FIR) Filters

Linear Phase FIR Filters

Conversion of Continuous-Time to Discrete-Time Filter

Impulse Invariance Approach

Impulse Invariance Approach Corrected

Backward-Rectangular Approximation

Forward Rectangular and Trapezoidal Approximations

Bilinear Transform

Lattice Filters

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

Problems

Answers to Selected Problems

**Energy Spectral Density**

Energy and Power Spectral Densities

Energy and Power Spectral Densities

Average, Energy and Power of Continuous-Time Signals

Discrete-Time Signals

Energy Signals

Autocorrelation of Energy Signals

Energy Signal through a Linear System

Impulsive and Discrete-Time Energy Signals

Power Signals

Cross-Correlation

Power Spectrum Conversion of a Linear System

Impulsive and Discrete-Time Power Signals

Periodic Signals

Power Spectral Density of an Impulse Train

Average, Energy and Power of a Sequence

Energy Spectral Density of a Sequence

Autocorrelation of an Energy Sequence

Power Density of a Sequence

Passage through a Linear System

Problems

Answers to Selected Problems

**Introduction**

Introduction to Communication Systems

Introduction to Communication Systems

Amplitude Modulation (AM) of Continuous-Time Signals

Frequency Modulation

Discrete Signals

Digital Communication Systems

PCM-TDM Systems

Frequency Division Multiplexing (FDM)

Problems

Answers to Selected Problems

**Walsh Transform**

Fourier-, Laplace- and z-Related Transforms

Fourier-, Laplace- and z-Related Transforms

Rademacher and Haar Functions

Walsh Functions

The Walsh (Sequency) Order

Dyadic (Paley) Order

Natural (Hadamard) Order

Discrete Walsh Transform

Discrete-Time Walsh Transform

Discrete-Time Walsh–Hadamard Transform

Natural (Hadamard) Order Fast Walsh–Hadamard Transform

Dyadic (Paley) Order Fast Walsh–Hadamard Transform

Sequency Ordered Fast Walsh–Hadamard Transform

Generalized Walsh Transform

Natural Order

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

**Introduction**

Digital Signal Processors: Architecture, Logic Design

Digital Signal Processors: Architecture, Logic Design

Systems for the Representation of Numbers

Conversion from Decimal to Binary

Integers, Fractions and the Binary Point

Representation of Negative Numbers

Integer and Fractional Representation of Signed Numbers

Addition

Subtraction

Full Adder Cell

Addition/Subtraction Implementation in 2’s Complement

Controlled Add/Subtract (CAS) Cell

Multiplication of Unsigned Numbers

Multiplier Implementation

3-D Multiplier

A Direct Approach to 2’s Complement Multiplication

Division

Cellular Array for Nonrestoring Division

Carry Look Ahead (CLA) Cell

2’s Complement Nonrestoring Division

Convergence Division

Evaluation of the nth Root

Function Generation by Chebyshev Series Expansion

An Alternative Approach to Chebyshev Series Expansion

Floating Point Number Representation

Square Root Evaluation

Cellular Array for Nonrestoring Square Root Extraction

Binary Coded Decimal (BCD) Representation

Memory Elements

Design of Synchronous Sequential Circuits

Realization of a Counter Using T Flip-Flops

State Minimization

Asynchronous Sequential Machines

State Reduction

Control Counter Design for Generator of Prime Numbers

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

Instruction Syntax

TMS320C6000 Control Register File

Addressing Mode Register (AMR)

Syntax for Load/Store Address Generation

Programming the T.I. DSP

A Simple C Program

The Generated Assembly Code

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

MOS FET Logic Circuit Realization

Problems

Answers to Selected Problems

**Nonparametric Methods of Power Spectrum Estimation**

Random Signal Processing

Random Signal Processing

Correlation of Continuous-Time Random Signals

Passage through an LTI System

Wiener Filtering in Continuous-Time Domain

Causal Wiener Filter

Random Sequences

From Statistical to Time Averages

Correlation and Covariance in z-Domain

Random Signal Passage through an LTI System

PSD Estimation of Discrete-Time Random Sequences

Fast Fourier Transform (FFT) Evaluation of the Periodogram

Parametric Methods for PSD Estimation

The Yule–Walker Equations

System Modeling for Linear Prediction, Adaptive Filtering and Spectrum Estimation

Wiener and Least-Squares Models

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

**Introduction**

Distributions

Distributions

Distributions as Generalizations of Functions

What is a Distribution?

The Impulse as the Limit of a Sequence

Properties of Distributions

Approximating the Impulse

Other Approximating Sequences and Functions of the Impulse

Test Functions

Convolution

Multiplication by an Impulse Derivative

The Dirac-Delta Impulse as a Limit of a Gaussian Function

Fourier Transform of Unity

The Impulse of a Function

Multiplication by t

Time Scaling

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

**Introduction**

Generalization of Distributions Theory, Extending Laplace-, z- and Fourier-Related Transforms

Generalization of Distributions Theory, Extending Laplace-, z- and Fourier-Related Transforms

An Anomaly

Generalized Distributions for Continuous-Time Functions

Properties of the Generalized Impulse in s Domain

Generalized Impulse as a Limit of a Three-Dimensional Sequence

Discrete-Time Domain

3-D Test Function as a Possible Generalization

Properties of the Generalized Impulse in z-Domain

Additional Generalized Impulse Properties

Generalized Impulse as Limit of a 3-D Sequence

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

**Symbols**

Appendix

Appendix

Frequently Needed Expansions

Important Trigonometric Relations

Orthogonality Relations

Frequently Encountered Functions

Mathematical Formulae

Frequently Encountered Series Sums

Biographies of Pioneering Scientists

Plato (428 BC–347 BC)

Euclid (circa 300 BC)

Ptolemy (circa 90–168 AD)

Abu Jafar Muhammad ibn Musa Al-Khwarizmi (780–850 AD)

Nicolaus Copernicus (1473–1543)

Galileo Galilei (1564–1642)

Sir Isaac Newton (1643–1727)

Guillaume-Fran¸cois-Antoine de L’Hˆopital (1661–1704)

Pierre-Simon Laplace (1749–1827)

Gaspard Clair Fran¸cois Marie, Baron Riche de Prony (1755–1839)

Jean Baptiste Joseph Fourier (1768–1830)

Johann Carl Friedrich Gauss (1777–1855)

Friedrich Wilhelm Bessel (1784–1846)

Augustin-Louis Cauchy (1789–1857)

Niels Henrik Abel (1802–1829)

Johann Peter Gustav Lejeune Dirichlet (1805–1859)

Pafnuty Lvovich Chebyshev (1821–1894)

Paul A.M. Dirac

## Author(s)

### Biography

**Michael Corinthios**, Ph.D., Fellow IEEE, FIET is a professor in the Department of Electrical Engineering at the École Polytechnique de Montréal, Quebec, Canada.