Signals, Systems, Transforms, and Digital Signal Processing with MATLAB  book cover
1st Edition

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

ISBN 9781420090482
Published June 17, 2009 by CRC Press
1256 Pages 904 B/W Illustrations

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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
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
Difference Equations
Even/Odd Decomposition
Average Value, Energy and Power Sequences
Causality, Stability
Answers to Selected Problems

Fourier Series Expansion
Trigonometric Fourier Series
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
Answers to Selected Problems

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
Answers to Selected Problems

Fourier Transform
Definition of the 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
Answers to Selected Problems

System Modeling, Time and Frequency Response
Transfer Function
Block Diagram Reduction
DC Motor
A Speed-Control System
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
Answers to Selected Problems

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
Answers to Selected Problems

Discrete-Time Fourier Transform
Laplace, Fourier and z-Transform Relations
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
Answers to Selected Problems

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
Answers to Selected Problems

Filters of Continuous-Time Domain
Lowpass Approximation
Butterworth Approximation
Denormalization of Butterworth Filter Prototype
Denormalized Transfer Function
The Case ε 6= 1
Butterworth Filter Order Formula
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
Rectangular Window
Triangle (Bartlett) Window
Hanning Window
Hamming Window
Answers to Selected Problems

Passive and Active Filters
Design of Passive 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
Answers to Selected Problems

Digital Filters
Signal Flow Graphs
IIR Filter Models
First Canonical Form
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
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
Answers to Selected Problems

Energy and Power Spectral Densities
Energy Spectral Density
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
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
Answers to Selected Problems

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)
Answers to Selected Problems

Fourier-, Laplace- and z-Related Transforms
Walsh Transform
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
Answers to Selected Problems

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
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
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
Answers to Selected Problems

Random Signal Processing
Nonparametric Methods of Power Spectrum Estimation
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
FIR Filter Design
Power Spectrum Estimation Using MATLAB
Parametric Modeling Functions
A z-Domain Counterpart to Prony’s Method
Answers to Selected Problems

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
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
Answers to Selected Problems

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
Answers to Selected Problems

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

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