Digital Signal Processing (Proakis) (4th Edition).pdf

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Contents Preface xvii 1 Introduction 1.1 Signals, Systems, and Signal Processing 1.1.1 Basic Elements of a Digital Signal Processing System 1.1.2 Advantages of Digital over Analog Signal Processing 1.2 Classification of Signals 1.2.1 Multichannel and Multidimensional Signals 1.2.2 Continuous-Time Versus Discrete-TIme Signals 1.2.3 Continuous-Valued Versus Discrete-Valued Signals 1.2.4 Deterministic Versus Random Signals 1 2 4 5 6 6 9 10 11 1.3 The Concept of Frequency in Continuous-Time and Discrete-Time Signals 1.4 Analog-to-Digital and Digital-to-Analog Conversion 1.3.1 Continuous-Time Sinusoidal Signals 1.3.2 Discrete-Time Sinusoidal Signals 1.3.3 Harmonically Related Complex Exponentials 12 12 14 17 19 1.4.1 Sampling of Analog Signals 21 1.4.2 The Sampling Theorem 26 1.4.3 Quantization of Continuous-Amplitude Signals 31 1.4.4 Quantization of Sinusoidal Signals 34 1.4.5 Coding of Quantized Samples 35 1.4.6 Digital-to-Analog Conversion 36 1.4.7 Analysis of Digital Signals and Systems Versus Discrete-Time Signals 36 and Systems 1.5 Summary and References Problems 37 37

vi Contents 2 Discrete-Time Signals and Systems 2.1 Discrete-Time Signals 2.1.1 Some Elementary Discrete-Time Signals 2.1.2 Classification of Discrete-Time Signals 2.1.3 Simple Manipulations of Discrete-Time Signals 2.2 Discrete-Time Systems Input-Output Description of Systems 2.2.1 2.2.2 Block Diagram Representation of Discrete-Time Systems 2.2.3 Classification of Discrete-Time Systems 2.2.4 Interconnection of Discrete-Time Systems 2.3 Analysis of Discrete-Time Linear Time-Invariant Systems 2.3.1 Techniques for the Analysis of Linear Systems 2.3.2 Resolution of a Discrete-Time Signal into Impulses 2.3.3 Response of LTI Systems to Arbitrary Inputs: The Convolution Sum 2.3.4 Properties of Convolution and the Interconnection of LTI Systems 2.3.5 Causal Linear Time-Invariant Systems 2.3.6 Stability of Linear Time-Invariant Systems 2.3.7 Systems with Finite-Duration and Infinite-Duration Impulse Response 2.4 Discrete-Time Systems Described by Difference Equations 2.4.1 Recursive and Nonrecursive Discrete-Time Systems 2.4.2 Linear Time-Invariant Systems Characterized by Constant-Coefficient Difference Equations 41 42 43 45 50 53 54 57 59 67 69 69 71 73 80 83 85 88 89 90 93 2.4.3 Solution of Linear Constant-Coefficient Difference Equations 98 2.4.4 The Impulse Response of a Linear Time-Invariant Recursive System 106 2.5 Implementation of Discrete-Time Systems 2.5.1 Structures for the Realization of Linear Time-Invariant Systems 2.5.2 Recursive and Nonrecursive Realizations of FIR Systems 2.6 Correlation of Discrete-Time Signals 2.6.1 Crosscorrelation and Autocorrelation Sequences 2.6.2 Properties of the Autocorrelation and Crosscorrelation Sequences 2.6.3 Correlation of Periodic Sequences 2.6.4 Input-Output Correlation Sequences 2.7 Summary and References Problems 109 109 113 116 118 120 123 125 128 129

3 The z -Transform and Its Application to the Analysis of L TI Systems Contents vii 147 3.1 The z-Transform 3.1.1 The Direct z-Transform 3.1.2 The Inverse z-Transform 3.2 Properties of the z-Transform 3.3 Rational z-Transforms 3.3.1 Poles and Zeros 3.3.2 Pole Location and TIme-Domain Behavior for Causal Signals 3.3.3 The System Function of a Linear Time-Invariant System 3.4 Inversion of the z-Transform 3.4.1 The Inverse z-Transform by Contour Integration 3.4.2 The Inverse z-Transform by Power Series Expansion 3.4.3 The Inverse z-Transform by Partial-Fraction Expansion 3.4.4 Decomposition of Rational z-Transforms 3.5 Analysis of Linear Time-Invariant Systems in the z-Domain 3.5.1 Response of Systems with Rational System Functions 3.5.2 Transient and Steady-State Responses 3.5.3 Causality and Stability 3.5.4 Pole-Zero Cancellations 3.5.5 Multiple-Order Poles and Stability 3.5.6 Stability of Second-Order Systems 3.6 The One-sided z-Transform 3.6.1 Definition and Properties 3.6.2 Solution of Difference Equations 3.6.3 Response of Pole-Zero Systems with Nonzero Initial Conditions 3.7 Summary and References Problems 4 Frequency Analysis of Signals 4.1 Frequency Analysis of Continuous-Time Signals 4.1.1 The Fourier Series for Continuous-TIme Periodic Signals 4.1.2 Power Density Spectrum of Periodic Signals 4.1.3 The Fourier Transform for Continuous-Time Aperiodic Signals 4.1.4 Energy Density Spectrum of Aperiodic Signals 147 147 156 157 170 170 174 177 180 180 182 184 192 193 194 195 196 198 200 201 205 206 210 211 214 214 224 225 226 230 234 238

viii Contents 4.2 Frequency Analysis of Discrete-Time Signals 4.2.1 The Fourier Series for Discrete-TIme Periodic Signals 4.2.2 Power Density Spectrum of Periodic Signals 4.2.3 The fourier Transform of Discrete-TIme Aperiodic Signals 4.2.4 Convergence of the Fourier Transform 4.2.5 Energy Density Spectrum of Aperiodk Signals 4.2.6 Relationship of the Fourier Transform to the z -Transform 4.2.7 The Cepstrum 4.2.8 The Fourier Transform of Signals with Poles on the Unit Circle 4.2.9 Frequency-Domain Classification of Signals: The Concept of Bandwidth 4.2.10 The Frequency Ranges of Some Natural Signals 4.3 Frequency-Domain and Time-Domain Signal Properties 4.4 Properties of the Fourier Transform for Discrete-Time Signals 4.4.1 Symmetry Properties of the Fourier Transform 4.4.2 Fourier Transform Theorems and Properties 4.5 Summary and References Problems 5 Frequency-Domain Analysis of LTI Systems 5.1 Frequency-Domain Characteristics of Linear TIme-Invariant Systems 5.1.1 Response to Complex Exponential and Sinusoidal Signals: The 241 241 245 248 251 254 259 261 262 265 267 268 271 272 279 291 292 300 300 301 5.3 Correlation Functions and Spectra at the Output of LTI Systems Input-Output Correlation Functions and Spectra Frequency Response Function 5.1.2 Steady-State and Transient Response to Sinusoidal Input Signals 5.1.3 Steady-State Response to Periodic Input Signals 5.1.4 Response to Aperiodic Input Signals 5.2.1 Frequency Response of a System with a Rational System Function 5.2.2 Computation of the Frequency Response Function 310 311 312 314 314 317 321 5.3.1 322 5.3.2 Correlation Functions and Power Spectra for Random Input Signals 323 326 327 329 335 339 341 5.4.1 5.4.2 Lowpass, Highpass, and Bandpass Filters 5.4.3 Digital Resonators 5.4.4 Notch Filters 5.4.5 Comb Filters 5.2 Frequency Response of LTI Systems 5.4 Linear Time-Invariant Systems as Frequency-Selective Filters Ideal Filter Characteristics

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