book

Discrete Systems and Digital Signal Processing with MATLAB, 2nd Edition

Name: Discrete Systems and Digital Signal Processing with MATLAB, 2nd Edition
Author: Taan S. ElAli
ISBN: 9781000755688

by Taan S. ElAli

April 2016

Intermediate to advanced

620 pages

9h 55m

English

CRC Press

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Cover
Half Title
Title Page
Copyright Page
Dedication
Table of Contents
Preface
Acknowledgments
Author
1. Signal Representation
1.1 Introduction1.2 Why Do We Discretize Continuous Systems?1.3 Periodic and Nonperiodic Discrete Signals1.4 Unit Step Discrete Signal1.5 Impulse Discrete Signal1.6 Ramp Discrete Signal1.7 Real Exponential Discrete Signal1.8 Sinusoidal Discrete Signal1.9 Exponentially Modulated Sinusoidal Signal1.10 Complex Periodic Discrete Signal1.11 Shifting Operation1.12 Representing a Discrete Signal Using Impulses1.13 Reflection Operation1.14 Time Scaling1.15 Amplitude Scaling1.16 Even and Odd Discrete Signal1.17 Does a Discrete Signal Have a Time Constant?1.18 Basic Operations on Discrete Signals1.18.1 Modulation1.18.2 Addition and Subtraction1.18.3 Scalar Multiplication1.18.4 Combined Operations1.19 Energy and Power Discrete Signals1.20 Bounded and Unbounded Discrete Signals1.21 Some Insights: Signals in the Real World1.21.1 Step Signal1.21.2 Impulse Signal1.21.3 Sinusoidal Signal1.21.4 Ramp Signal1.21.5 Other SignalsEnd of Chapter ExamplesEnd of Chapter Problems

2. Discrete System
2.1 Definition of a System2.2 Input and Output2.3 Linear Discrete Systems2.4 Time Invariance and Discrete Signals2.5 Systems with Memory2.6 Causal Systems2.7 Inverse of a System2.8 Stable System2.9 Convolution2.10 Difference Equations of Physical Systems2.11 Homogeneous Difference Equation and Its Solution2.11.1 Case When Roots Are All Distinct2.11.2 Case When Two Roots Are Real and Equal2.11.3 Case When Two Roots Are Complex
2.12 Nonhomogeneous Difference Equations and Their Solutions
2.12.1 How Do We Find the Particular Solution?2.13 Stability of Linear Discrete Systems: The Characteristic Equation2.13.1 Stability Depending on the Values of the Poles2.13.2 Stability from the Jury Test2.14 Block Diagram Representation of Linear Discrete Systems2.14.1 Delay Element2.14.2 Summing/Subtracting Junction2.14.3 Multiplier2.15 From the Block Diagram to the Difference Equation2.16 From the Difference Equation to the Block Diagram: A Formal Procedure2.17 Impulse Response2.18 Correlation2.18.1 Cross-Correlation2.18.2 Auto-Correlation2.19 Some Insights2.19.1 How Can We Find These Eigenvalues?2.19.2 Stability and EigenvaluesEnd of Chapter ExamplesEnd of Chapter Problems
3. Fourier Series and the Fourier Transform of Discrete Signals
3.1 Introduction3.2 Review of Complex Numbers3.2.1 Definition3.2.2 Addition3.2.3 Subtraction3.2.4 Multiplication3.2.5 Division3.2.6 From Rectangular to Polar3.2.7 From Polar to Rectangular3.3 Fourier Series of Discrete Periodic Signals3.4 Discrete System with Periodic Inputs: The Steady-State Response3.4.1 General Form for yss(n)3.5 Frequency Response of Discrete Systems3.5.1 Properties of the Frequency Response3.5.1.1 Periodicity Property3.5.1.2 Symmetry Property3.6 Fourier Transform of Discrete Signals3.7 Convergence Conditions3.8 Properties of the Fourier Transform of Discrete Signals3.8.1 Periodicity Property3.8.2 Linearity Property3.8.3 Discrete-Time Shifting Property3.8.4 Frequency Shifting Property3.8.5 Reflection Property3.8.6 Convolution Property3.9 Parseval’s Relation and Energy Calculations3.10 Numerical Evaluation of the Fourier Transform of Discrete Signals3.11 Some Insights: Why Is This Fourier Transform?3.11.1 Ease in Analysis and Design3.11.2 Sinusoidal AnalysisEnd of Chapter ExamplesEnd of Chapter Problems
4. z-Transform and Discrete Systems
4.1 Introduction4.2 Bilateral z-Transform4.3 Unilateral z-Transform4.4 Convergence Considerations4.5 Inverse z-Transform4.5.1 Partial Fraction Expansion4.5.2 Long Division4.6 Properties of the z-Transform4.6.1 Linearity Property4.6.2 Shifting Property4.6.3 Multiplication by e−an4.6.4 Convolution4.7 Representation of Transfer Functions as Block Diagrams
4.8 x(n), h(n), y(n), and the z-Transform
4.9 Solving Difference Equation Using the z-Transform4.10 Convergence Revisited4.11 Final-Value Theorem4.12 Initial-Value Theorem4.13 Some Insights: Poles and Zeroes4.13.1 Poles of the System4.13.2 Zeros of the System4.13.3 Stability of the SystemEnd of Chapter ExercisesEnd of Chapter Problems
5. State-Space and Discrete Systems
5.1 Introduction5.2 Review on Matrix Algebra5.2.1 Definition, General Terms, and Notations5.2.2 Identity Matrix5.2.3 Adding Two Matrices5.2.4 Subtracting Two Matrices5.2.5 Multiplying a Matrix by a Constant5.2.6 Determinant of a Two-by-Two Matrix5.2.7 Transpose of a Matrix5.2.8 Inverse of a Matrix5.2.9 Matrix Multiplication5.2.10 Eigenvalues of a Matrix5.2.11 Diagonal Form of a Matrix5.2.12 Eigenvectors of a Matrix5.3 General Representation of Systems in State Space5.3.1 Recursive Systems5.3.2 Nonrecursive Systems5.3.3 From the Block Diagram to State Space5.3.4 From the Transfer Function H(z) to State Space
5.4 Solution of the State-Space Equations in the z-Domain
5.5 General Solution of the State Equation in Real Time5.6 Properties of An and Its Evaluation5.7 Transformations for State-Space Representations5.8 Some Insights: Poles and StabilityEnd of Chapter ExamplesEnd of Chapter Problems
6. Block Diagrams and Review of Discrete System Representations
6.1 Introduction6.2 Basic Block Diagram Components6.2.1 Ideal Delay6.2.2 Adder6.2.3 Subtractor6.2.4 Multiplier6.3 Block Diagrams as Interconnected Subsystems6.3.1 General Transfer Function Representation6.3.2 Parallel Representation6.3.3 Series Representation6.3.4 Basic Feedback Representation6.4 Controllable Canonical Form Block Diagrams with Basic Blocks6.5 Observable Canonical Form Block Diagrams with Basic Blocks6.6 Diagonal Form Block Diagrams with Basic Blocks6.6.1 Distinct Roots Case6.6.2 Repeated Roots Case6.7 Parallel Block Diagrams with Subsystems6.7.1 Distinct Roots Case6.7.2 Repeated Roots Case6.8 Series Block Diagrams with Subsystems6.8.1 Distinct Real Roots Case6.8.2 Mixed Complex and Real Roots Case6.9 Block Diagram Reduction Rules6.9.1 Using the Reduction Rules6.9.2 Using Mason’s RuleEnd of Chapter ExamplesEnd of Chapter Problems
7. Discrete Fourier Transform and Discrete Systems
7.1 Introduction7.2 Discrete Fourier Transform and the Finite-Duration Discrete Signals7.3 Properties of the DFT7.3.1 How Does the Defining Equation Work?7.3.2 DFT Symmetry7.3.3 DFT Linearity7.3.4 Magnitude of the DFT7.3.5 What Does k in X(k), the DFT, Mean?7.4 Relation the DFT Has with the Fourier Transform of Discrete Signals, the z-Transform, and the Continuous Fourier Transform7.4.1 DFT and the Fourier Transform of x(n)7.4.2 DFT and the z-Transform of x(n)7.4.3 DFT and the Continuous Fourier Transform of x(t)7.5 Numerical Computation of the DFT7.6 Fast Fourier Transform: A Faster Way of Computing the DFT7.7 Applications of the DFT7.7.1 Circular Convolution7.7.2 Linear Convolution7.7.3 Approximation to the Continuous Fourier Transform7.7.4 Approximation to the Coefficients of the Fourier Series and the Average Power of the Periodic Signal x(t)7.7.5 Total Energy in the Signal x(n) and x(t)7.7.6 Block Filtering7.7.7 Correlation7.8 Some Insights7.8.1 DFT Is the Same as the fft7.8.2 DFT Points Are the Samples of the Fourier Transform of x(n)7.8.3 How Can We Be Certain That Most of the Frequency Contents of x(t) Are in the DFT?7.8.4 Is the Circular Convolution the Same as the Linear Convolution?7.8.5 Is |X(w)| ≅ |X(k)|?7.8.6 Frequency Leakage and the DFTEnd of Chapter ExercisesEnd of Chapter Problems
8. Sampling and Transformations
8.1 Need for Converting a Continuous Signal to a Discrete Signal8.2 From the Continuous Signal to Its Binary Code Representation8.3 From the Binary Code to the Continuous Signal8.4 Sampling Operation8.4.1 Ambiguity in Real-Time Domain8.4.2 Ambiguity in the Frequency Domain8.4.3 Sampling Theorem8.4.4 Filtering before Sampling8.4.5 Sampling and Recovery of the Continuous Signal8.5 How Do We Discretize the Derivative Operation?8.6 Discretization of the State-Space Representation8.7 Bilinear Transformation and the Relationship between the Laplace-Domain and the z-Domain Representations8.8 Other Transformation Methods8.8.1 Impulse Invariance Method8.8.2 Step Invariance Method8.8.3 Forward Difference Method8.8.4 Backward Difference Method8.8.5 Bilinear Transformation8.9 Some Insights8.9.1 Choice of the Sampling Interval Ts8.9.2 Effect of Choosing Ts on the Dynamics of the System8.9.3 Does Sampling Introduce Additional Zeros to the Transfer Function H(z)?End of Chapter ExamplesEnd of Chapter Problems
9. Infinite Impulse Response Filter Design
9.1 Introduction9.2 Design Process9.2.1 Design Based on the Impulse Invariance Method9.2.2 Design Based on the Bilinear Transform Method9.3 IIR Filter Design Using MATLAB®9.3.1 From the Analogue Prototype to the IIR Digital Filter9.3.2 Direct Design9.4 Some Insights9.4.1 Difficulty in Designing IIR Digital Filters in the z-Domain9.4.2 Using the Impulse Invariance Method9.4.3 Choice of the Sampling Interval TsEnd of Chapter ExamplesEnd of Chapter Problems
10. Finite Impulse Response Digital Filters
10.1 Introduction10.1.1 What Is an FIR Digital Filter?10.1.2 Motivating Example10.2 FIR Filter Design10.2.1 Stability of FIR Filters10.2.2 Linear Phase of FIR Filters10.3 Design Based on the Fourier Series: The Windowing Method10.3.1 Ideal Lowpass FIR Filter Design10.3.2 Other Ideal Digital FIR Filters10.3.3 Windows Used in the Design of the Digital FIR Filter10.3.4 Which Window Does Give the Optimal h(n)?10.3.5 Design of a Digital FIR Differentiator10.3.6 Design of Comb FIR Filters10.3.7 Design of a Digital Shifter: The Hilbert Transform Filter10.4 From IIR to FIR Digital Filters: An Approximation10.5 Frequency Sampling and FIR Filter Design10.6 FIR Digital Design Using MATLAB®10.6.1 Design Using Windows10.6.2 Design Using Least-Squared Error10.6.3 Design Using the Equiripple Linear Phase10.6.4 How to Obtain the Frequency Response10.7 Some Insights10.7.1 Comparison with IIR Filters10.7.2 Different Methods Used in the FIR Filter DesignEnd of the Chapter ExamplesEnd of Chapter Problems
Bibliography

Content preview from Discrete Systems and Digital Signal Processing with MATLAB, 2nd Edition

4.8 x(n), h(n), y(n), and the z-Transform

Throughout this book, we have been using x(n) to represent the input, h(n) to represent the impulse response, and y(n) to represent the output. In discrete real time, the output is given using the convolution.

$eq786.jpg$

But as we saw earlier, convolution is multiplication is the transform domain. Thus, we write Y(z) = X(z) H(z). We then can use Tables 4.1 and 4.2 to get back to the real-time signal y(n).

Example 4.16

If x(n) = u(n) is an input to the system with h(n) = (.5)ⁿ u(n), what is the output y(n)?

Solution

The output in the z-domain is

$eq787.jpg$

The input in the z-domain is X(z) = z/(z − 1) and the impulse response is H(z) = z/(z −.5). Thus, using the convolution property of the z-transform, ...

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Discrete Systems and Digital Signal Processing with MATLAB, 2nd Edition

by Taan S. ElAli

4.8 x(n), h(n), y(n), and the z-Transform

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