DISCRETE-TIME SIGNAL PROCESSING

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1 THIRD EDITION DISCRETE-TIME SIGNAL PROCESSING ALAN V. OPPENHEIM MASSACHUSETTS INSTITUTE OF TECHNOLOGY RONALD W. SCHÄFER HEWLETT-PACKARD LABORATORIES Upper Saddle River Boston Columbus San Francisco New York Indianapolis London Toronto Sydney Singapore Tokyo Montreal Dubai Madrid Hong Kong Mexico City Munich Paris Amsterdam Cape Town

CONTENTS Preface 15 The Companion Website 22 Acknowledgments 25 1 Introduction 29 2 Discrete-Time Signals and Systems 37 2.0 Introduction 37 2.1 Discrete-Time Signals 38 2.

2 CONTENTS Preface 15 The Companion Website 22 Acknowledgments 25 1 Introduction 29 2 Discrete-Time Signals and Systems Introduction Discrete-Time Signals Discrete-Time Systems Memoryless Systems Linear Systems Time-Invariant Systems Causality Stability LTI Systems Properties of Linear Time-Invariant Systems Linear Constant-Coefficient Difference Equations Frequency-Domain Representation of Discrete-Time Signals and Systems Eigenfunctions for Linear Time-Invariant Systems Suddenly Applied Complex Exponential Inputs Representation of Sequences by Fourier Transforms Symmetry Properties of the Fourier Transform Fourier Transform Theorems Linearity of the Fourier Transform Time Shifting and Frequency Shifting Theorem Time Reversal Theorem 87 5

6 Contents 2.9.4 Differentiation in Frequency Theorem 87 2.9.5 Parseval's Theorem 88 2.9.6 The Convolution Theorem 88 2.9.7 The Modulation or Windowing Theorem 89 2.

3 6 Contents Differentiation in Frequency Theorem Parseval's Theorem The Convolution Theorem The Modulation or Windowing Theorem Discrete-Time Random Signals Summary 98 Problems 98 3 The z -Transform Introduction z-transform Properties of the ROC for the z-transform The Inverse z-transform Inspection Method Partial Fraction Expansion Power Series Expansion z-transform Properties Linearity Time Shifting Multiplication by an Exponential Sequence Differentiation of X(z) Conjugation of a Complex Sequence Time Reversal Convolution of Sequences Summary of Some z-transform Properties z-transforms and LTI Systems The Unilateral z-transform Summary 166 Problems Sampling of Continuous-Time Signals Introduction Periodic Sampling Frequency-Domain Representation of Sampling Reconstruction of a Bandlimited Signal from Its Samples Discrete-Time Processing of Continuous-Time Signals Discrete-Time LTI Processing of Continuous-Time Signals Impulse Invariance Continuous-Time Processing of Discrete-Time Signals Changing the Sampling Rate Using Discrete-Time Processing Sampling Rate Reduction by an Integer Factor Increasing the Sampling Rate by an Integer Factor Simple and Practical Interpolation Filters Changing the Sampling Rate by a Noninteger Factor Multirate Signal Processing Interchange of Filtering with Compressor/Expander Multistage Decimation and Interpolation 224

Contents 7 4.7.3 Polyphase Decompositions 226 4.7.4 Polyphase Implementation of Decimation Filters 228 4.7.5 Polyphase Implementation of Interpolation Filters 229 4.7.6 Multirate Filter Banks 230 4.

4 Contents Polyphase Decompositions Polyphase Implementation of Decimation Filters Polyphase Implementation of Interpolation Filters Multirate Filter Banks Digital Processing of Analog Signals Prefiltering to Avoid Aliasing A/D Conversion Analysis of Quantization Errors D/A Conversion Oversampling and Noise Shaping in A/D and D/A Conversion Oversampled A/D Conversion with Direct Quantization Oversampled A/D Conversion with Noise Shaping Oversampling and Noise Shaping in D/A Conversion Summary 265 Problems Transform Analysis of Linear Time-Invariant Systems Introduction The Frequency Response of LTI Systems Frequency Response Phase and Group Delay Illustration of Effects of Group Delay and Attenuation System Functions Linear Constant-Coefficient Difference Equations Stability and Causality Inverse Systems Impulse Response for Rational System Functions Frequency Response for Rational System Functions Frequency Response of l st -Order Systems Examples with Multiple Poles and Zeros Relationship between Magnitude and Phase All-Pass Systems Minimum-Phase Systems Minimum-Phase and All-Pass Decomposition Frequency-Response Compensation of Non-Minimum-Phase Systems Properties of Minimum-Phase Systems Linear Systems with Generalized Linear Phase Systems with Linear Phase Generalized Linear Phase Causal Generalized Linear-Phase Systems Relation of FIR Linear-Phase Systems to Minimum-Phase Systems Summary 368 Problems 369

8 Contents 6 Structures for Discrete-Time Systems 402 6.0 Introduction 402 6.1 Block Diagram Representation of Linear Constant-Coefficient Difference Equations 403 6.

5 8 Contents 6 Structures for Discrete-Time Systems Introduction Block Diagram Representation of Linear Constant-Coefficient Difference Equations Signal Flow Graph Representation Basic Structures for IIR Systems Direct Forms Cascade Form Parallel Form Feedback in IIR Systems Transposed Forms Basic Network Structures for FIR Systems Direct Form Cascade Form Structures for Linear-Phase FIR Systems Lattice Filters FIR Lattice Filters All-Pole Lattice Structure Generalization of Lattice Systems Overview of Finite-Precision Numerical Effects Number Representations Quantization in Implementing Systems The Effects of Coefficient Quantization Effects of Coefficient Quantization in IIR Systems Example of Coefficient Quantization in an Elliptic Filter Poles of Quantized 2 nd -Order Sections Effects of Coefficient Quantization in FIR Systems Example of Quantization of an Optimum FIR Filter Maintaining Linear Phase Effects of Round-off Noise in Digital Filters Analysis of the Direct Form IIR Structures Scaling in Fixed-Point Implementations of IIR Systems Example of Analysis of a Cascade IIR Structure Analysis of Direct-Form FIR Systems Floating-Point Realizations of Discrete-Time Systems Zero-Input Limit Cycles in Fixed-Point Realizations of IIR Digital Filters Limit Cycles Owing to Round-off and Truncation Limit Cycles Owing to Overflow Avoiding Limit Cycles Summary 491 Problems Filter Design Techniques Introduction Filter Specifications 523

Contents 9 7.2 Design of Discrete-Time IIR Filters from Continuous-Time Filters... 525 7.2.1 Filter Design by Impulse Invariance 526 7.2.2 Bilinear Transformation 533 7.

6 Contents Design of Discrete-Time IIR Filters from Continuous-Time Filters Filter Design by Impulse Invariance Bilinear Transformation Discrete-Time Butterworth, Chebyshev and Elliptic Filters Examples of IIR Filter Design Frequency Transformations of Lowpass IIR Filters Design of FIR Filters by Windowing Properties of Commonly Used Windows Incorporation of Generalized Linear Phase The Kaiser Window Filter Design Method Examples of FIR Filter Design by the Kaiser Window Method Lowpass Filter Highpass Filter Discrete-Time Differentiators Optimum Approximations of FIR Filters Optimal Type I Lowpass Filters Optimal Type II Lowpass Filters The Parks-McClellan Algorithm Characteristics of Optimum FIR Filters Examples of FIR Equiripple Approximation Lowpass Filter Compensation for Zero-Order Hold Bandpass Filter Comments on IIR and FIR Discrete-Time Filters Design of an Upsampling Filter Summary 611 Problems The Discrete Fourier Transform Introduction Representation of Periodic Sequences: The Discrete Fourier Series Properties of the DFS Linearity Shift of a Sequence Duality Symmetry Properties Periodic Convolution Summary of Properties of the DFS Representation of Periodic Sequences The Fourier Transform of Periodic Signals Sampling the Fourier Transform Fourier Representation of Finite-Duration Sequences Properties of the DFT Linearity Circular Shift of a Sequence Duality Symmetry Properties 678

10 Contents 8.6.5 Circular Convolution 679 8.6.6 Summary of Properties of the DFT 684 8.7 Linear Convolution Using the DFT 685 8.7.1 Linear Convolution of Two Finite-Length Sequences 686 8.7.2 Circular Convolution as Linear Convolution with Aliasing.

7 10 Contents Circular Convolution Summary of Properties of the DFT Linear Convolution Using the DFT Linear Convolution of Two Finite-Length Sequences Circular Convolution as Linear Convolution with Aliasing Implementing Linear Time-Invariant Systems Using the DFT The Discrete Cosine Transform (DCT) Definitions of the DCT Definition of the DCT-1 and DCT Relationship between the DFT and the DCT Relationship between the DFT and the DCT Energy Compaction Property of the DCT Applications of the DCT Summary 708 Problems Computation of the Discrete Fourier Transform Introduction Direct Computation of the Discrete Fourier Transform Direct Evaluation of the Definition of the DFT The Goertzel Algorithm Exploiting both Symmetry and Periodicity Decimation-in-Time FFT Algorithms Generalization and Programming the FFT In-Place Computations Alternative Forms Decimation-in-Frequency FFT Algorithms In-Place Computation Alternative Forms Practical Considerations Indexing Coefficients More General FFT Algorithms Algorithms for Composite Values of N Optimized FFT Algorithms Implementation of the DFT Using Convolution Overview of the Winograd Fourier Transform Algorithm The Chirp Transform Algorithm Effects of Finite Register Length Summary 788 Problems Fourier Analysis of Signals Using the Discrete Fourier Transform Introduction Fourier Analysis of Signals Using the DFT 818

Contents 11 10.2 DFT Analysis of Sinusoidal Signals 822 10.2.1 The Effect of Windowing 822 10.2.2 Properties of the Windows 825 10.2.3 The Effect of Spectral Sampling 826 10.

8 Contents DFT Analysis of Sinusoidal Signals The Effect of Windowing Properties of the Windows The Effect of Spectral Sampling The Time-Dependent Fourier Transform Invertibility of X[n,) Filter Bank Interpretation of X[n,) The Effect of the Window Sampling in Time and Frequency The Overlap-Add Method of Reconstruction Signal Processing Based on the Time-Dependent Fourier Transform Filter Bank Interpretation of the Time-Dependent Fourier Transform Examples of Fourier Analysis of Nonstationary Signals Time-Dependent Fourier Analysis of Speech Signals Time-Dependent Fourier Analysis of Radar Signals Fourier Analysis of Stationary Random Signals: the Periodogram The Periodogram Properties of the Periodogram Periodogram Averaging Computation of Average Periodograms Using the DFT An Example of Periodogram Analysis Spectrum Analysis of Random Signals Computing Correlation and Power Spectrum Estimates Using the DFT Estimating the Power Spectrum of Quantization Noise Estimating the Power Spectrum of Speech Summary 887 Problems Parametric Signal Modeling Introduction All-Pole Modeling of Signals Least-Squares Approximation Least-Squares Inverse Model Linear Prediction Formulation of All-Pole Modeling Deterministic and Random Signal Models All-Pole Modeling of Finite-Energy Deterministic Signals Modeling of Random Signals Minimum Mean-Squared Error Autocorrelation Matching Property Determination of the Gain Parameter G Estimation of the Correlation Functions The Autocorrelation Method The Covariance Method Comparison of Methods 928

12 Contents 11.4 Model Order 929 11.5 All-Pole Spectrum Analysis 931 11.5.1 All-Pole Analysis of Speech Signals 932 11.5.2 Pole Locations 935 11.5.3 All-Pole Modeling of Sinusoidal Signals 937 11.

9 12 Contents 11.4 Model Order All-Pole Spectrum Analysis All-Pole Analysis of Speech Signals Pole Locations All-Pole Modeling of Sinusoidal Signals Solution of the Autocorrelation Normal Equations The Levinson-Durbin Recursion Derivation of the Levinson-Durbin Algorithm Lattice Filters Prediction Error Lattice Network All-Pole Model Lattice Network Direct Computation of the ^-Parameters Summary 950 Problems Discrete Hilbert Transforms Introduction Real-and Imaginary-Part Sufficiency of the Fourier Transform Sufficiency Theorems for Finite-Length Sequences Relationships Between Magnitude and Phase Hilbert Transform Relations for Complex Sequences Design of Hilbert Transformers Representation of Bandpass Signals Bandpass Sampling Summary 993 Problems Cepstrum Analysis and Homomorphic Deconvolution Introduction Definition of the Cepstrum Definition of the Complex Cepstrum Properties of the Complex Logarithm Alternative Expressions for the Complex Cepstrum Properties of the Complex Cepstrum Exponential Sequences Minimum-Phase and Maximum-Phase Sequences Relationship Between the Real Cepstrum and the Complex Cepstrum Computation of the Complex Cepstrum Phase Unwrapping Computation of the Complex Cepstrum Using the Logarithmic Derivative Minimum-Phase Realizations for Minimum-Phase Sequences Recursive Computation of the Complex Cepstrum for Minimumand Maximum-Phase Sequences The Use of Exponential Weighting Computation of the Complex Cepstrum Using Polynomial Roots

Contents 13 13.8 Deconvolution Using the Complex Cepstrum 1026 13.8.1 Minimum-Phase/Allpass Homomorphic Deconvolution... 1027 13.8.2 Minimum-Phase/Maximum-Phase Homomorphic Deconvolution 1028 13.

10 Contents Deconvolution Using the Complex Cepstrum Minimum-Phase/Allpass Homomorphic Deconvolution Minimum-Phase/Maximum-Phase Homomorphic Deconvolution The Complex Cepstrum for a Simple Multipath Model Computation of the Complex Cepstrum by z-transform Analysis Computation of the Cepstrum Using the DFT Homomorphic Deconvolution for the Multipath Model Minimum-Phase Decomposition Generalizations Applications to Speech Processing The Speech Model Example of Homomorphic Deconvolution of Speech Estimating the Parameters of the Speech Model Applications Summary 1056 Problems 1058 A Random Signals 1067 В Continuous-Time Filters 1080 С Answers to Selected Basic Problems 1085 Bibliography 1106 Index 1115

R13 SET - 1

R13 SET - 1 III B. Tech II Semester Regular Examinations, April - 2016 DIGITAL SIGNAL PROCESSING (Electronics and Communication Engineering) Time: 3 hours Maximum Marks: 70 Note: 1. Question Paper consists