1st Edition

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

By Michael Corinthios Copyright 2009
    1344 Pages 904 B/W Illustrations
    by CRC Press

    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.

    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

    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
    Problems
    Answers to Selected Problems

    Laplace Transform
    Introduction
    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

    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
    Problems
    Answers to Selected Problems

    System Modeling, Time and Frequency Response
    Transfer Function
    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

    Discrete-Time Signals and Systems
    Introduction
    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

    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
    Problems
    Answers to Selected Problems

    State Space Modeling
    Introduction
    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

    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
    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

    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
    Problems
    Answers to Selected Problems

    Digital Filters
    Introduction
    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 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
    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 to Communication Systems
    Introduction
    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

    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
    Problems
    Answers to Selected Problems

    Digital Signal Processors: Architecture, Logic Design
    Introduction
    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

    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
    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

    Distributions
    Introduction
    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

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

    Appendix
    Symbols
    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

    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.