Digital Signal Processing

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1 Digital Signal Processing Fourth Edition John G. Proakis Department of Electrical and Computer Engineering Northeastern University Boston, Massachusetts Dimitris G. Manolakis MIT Lincoln Laboratory Lexington, Massachusetts PEARSON Prentice Hall Upper Saddle River, New Jersey 07458

2 Contents Preface xvii 1 Introduction Signals, Systems, and Signal Processing Basic Elements of a Digital Signal Processing System Advantages of Digital over Analog Signal Processing Classification of Signals Multichannel and Multidimensional Signals Continuous-Time Versus Discrete-Time Signals Continuous-Valued Versus Discrete-Valued Signals Deterministic Versus Random Signals The Concept of Frequency in Continuous-Time and Discrete-Time Signals Continuous-Time Sinusoidal Signals Discrete-Time Sinusoidal Signals Harmonically Related Complex Exponentials Analog-to-Digital and Digital-to-Analog Conversion Sampling of Analog Signals The Sampling Theorem Quantization of Continuous-Amplitude Signals Quantization of Sinusoidal Signals Coding of Quantized Samples Digital-to-Analog Conversion Analysis of Digital Signals and Systems Versus Discrete-Time Signals 36 and Systems 1.5 Summary and References 37 Problems 37

3 vi Contents 2 Discrete-Time Signals and Systems Discrete-Time Signals Some Elementary Discrete-Time Signals Classification of Discrete-Time Signals Simple Manipulations of Discrete-Time Signals Discrete-Time Systems Input-Output Description of Systems Block Diagram Representation of Discrete-Time Systems Classification of Discrete-Time Systems Interconnection of Discrete-Time Systems Analysis of Discrete-Time Linear Time-Invariant Systems Techniques for the Analysis of Linear Systems Resolution of a Discrete-Time Signal into Impulses Response of LTI Systems to Arbitrary Inputs: The Convolution Sum Properties of Convolution and the Interconnection of LTI Systems Causal Linear Time-Invariant Systems Stability of Linear Time-Invariant Systems Systems with Finite-Duration and Infinite-Duration Impulse 88 Response 2.4 Discrete-Time Systems Described by Difference Equations Recursive and Nonrecursive Discrete-Time Systems Linear Time-Invariant Systems Characterized by 93 Constant-Coefficient Difference Equations Solution of Linear Constant-Coefficient Difference Equations The Impulse Response of a Linear Time-Invariant Recursive System Implementation of Discrete-Time Systems Structures for the Realization of Linear Time-Invariant Systems Recursive and Nonrecursive Realizations of FIR Systems Correlation of Discrete-Time Signals Crosscorrelation and Autocorrelation Sequences Properties of the Autocorrelation and Crosscorrelation Sequences Correlation of Periodic Sequences Input-Output Correlation Sequences Summary and References 128 Problems 129

4 Contents VII The z -Transform and Its Application to the Analysis of LTI 147 Systems 3.1 The z-transform The Direct z-transform The Inverse z -Transform Properties of the z-transform Rational z-transforms Poles and Zeros Pole Location and Time-Domain Behavior for Causal Signals The System Function of a Linear Time-Invariant System Inversion of the z-transform The Inverse z-transform by Contour Integration The Inverse z-transform by Power Series Expansion The Inverse z-transform by Partial-Fraction Expansion Decomposition of Rational z-transforms Analysis of Linear Time-Invariant Systems in the z-domain Response of Systems with Rational System Functions Transient and Steady-State Responses Causality and Stability Pole-Zero Cancellations Multiple-Order Poles and Stability Stability of Second-Order Systems The One-sided z-transform Definition and Properties Solution of Difference Equations Response of Pole-Zero Systems with Nonzero Initial Conditions Summary and References 214 Problems 214 Frequency Analysis of Signals Frequency Analysis of Continuous-Time Signals The Fourier Series for Continuous-Time Periodic Signals Power Density Spectrum of Periodic Signals The Fourier Transform for Continuous-Time Aperiodic Signals Energy Density Spectrum of Aperiodic Signals 238

5 VIII Contents 4.2 Frequency Analysis of Discrete-Time Signals The Fourier Series for Discrete-Time Periodic Signals Power Density Spectrum of Periodic Signals The Fourier Transform of Discrete-Time Aperiodic Signals Convergence of the Fourier Transform Energy Density Spectrum of Aperiodic Signals Relationship of the Fourier Transform to the z-transform /The Cepstrum *The Fourier Transform of Signals with Poles on the Unit Circle Frequency-Domain Classification of Signals: The Concept of 265 Bandwidth The Frequency Ranges of Some Natural Signals Frequency-Domain and Time-Domain Signal Properties Properties of the Fourier Transform for Discrete-Time Signals Symmetry Properties of the Fourier Transform Fourier Transform Theorems and Properties Summary and References 291 Problems Frequency-Domain Analysis of LTI Systems Frequency-Domain Characteristics of Linear Time-Invariant Systems Response to Complex Exponential and Sinusoidal Signals: The 301 Frequency Response Function Steady-State and Transient Response to Sinusoidal Input Signals Steady-State Response to Periodic Input Signals Response to Aperiodic Input Signals Frequency Response of LTI Systems Frequency Response of a System with a Rational System Function Computation of the Frequency Response Function Correlation Functions and Spectra at the Output of LTI Systems Input-Output Correlation Functions and Spectra Correlation Functions and Power Spectra for Random Input Signals Linear Time-Invariant Systems as Frequency-Selective Filters Ideal Filter Characteristics Lowpass, Highpass, and Bandpass Filters Digital Resonators Notch Filters Comb Filters 341

6 Contents IX All-Pass Filters Digital Sinusoidal Oscillators Inverse Systems and Deconvolution Invertibility of Linear Time-Invariant Systems Minimum-Phase, Maximum-Phase, and Mixed-Phase Systems System Identification and Deconvolution Homomorphic Deconvolution Summary and References 362 Problems Sampling and Reconstruction of Signals Ideal Sampling and Reconstruction of Continuous-Time Signals Discrete-Time Processing of Continuous-Time Signals Analog-to-Digital and Digital-to-Analog Converters Analog-to-Digital Converters Quantization and Coding Analysis of Quantization Errors Digital-to-Analog Converters Sampling and Reconstruction of Continuous-Time Bandpass Signals Uniform or First-Order Sampling Interleaved or Nonuniform Second-Order Sampling Bandpass Signal Representations Sampling Using Bandpass Signal Representations Sampling of Discrete-Time Signals Sampling and Interpolation of Discrete-Time Signals Representation and Sampling of Bandpass Discrete-Time Signals Oversampling A/D and D/A Converters Oversampling A/D Converters Oversampling D/A Converters Summary and References 440 Problems 440

7 X Contents 7 The Discrete Fourier Transform: Its Properties and Applications Frequency-Domain Sampling: The Discrete Fourier Transform Frequency-Domain Sampling and Reconstruction of Discrete-Time 449 Signals The Discrete Fourier Transform (DFT) The DFT as a Linear Transformation ^Relationship of the DFT to Other Transforms Properties of the DFT Periodicity, Linearity, and Symmetry Properties Multiplication of Two DFTs and Circular Convolution Additional DFT Properties Linear Filtering Methods Based on the DFT Use of the DFT in Linear Filtering Filtering of Long Data Sequences Frequency Analysis of Signals Using the DFT The Discrete Cosine Transform Forward DCT Inverse DCT DCT as an Orthogonal Transform Summary and References 501 Problems Efficient Computation of the DFT: Fast Fourier Transform 511 Algorithms 8.1 Efficient Computation of the DFT: FFT Algorithms Direct Computation of the DFT Divide-and-Conquer Approach to Computation of the DFT Radix-2 FFT Algorithms Radix-4 FFT Algorithms Split-Radix FFT Algorithms Implementation of FFT Algorithms Applications of FFT Algorithms Efficient Computation of the DFT of Two Real Sequences Efficient Computation of the DFT of a 2 N -Point Real Sequence Use of the FFT Algorithm in Linear Filtering and Correlation 540

8 Contents XI 8.3 A Linear Filtering Approach to Computation of the DFT The Goertzel Algorithm The Chirp-z Transform Algorithm Quantization Effects in the Computation of the DFT Quantization Errors in the Direct Computation of the DFT Quantization Errors in FFT Algorithms Summary and References 555 Problems Implementation of Discrete-Time Systems Structures for the Realization of Discrete-Time Systems Structures for FIR Systems Direct-Form Structure Cascade-Form Structures Frequency-Sampling Structures Lattice Structure Structures for MR Systems Direct-Form Structures Signal Flow Graphs and Transposed Structures Cascade-Form Structures Parallel-Form Structures Lattice and Lattice-Ladder Structures for IIR Systems Representation of Numbers Fixed-Point Representation of Numbers Binary Floating-Point Representation of Numbers Errors Resulting from Rounding and Truncation Quantization of Filter Coefficients Analysis of Sensitivity to Quantization of Filter Coefficients Quantization of Coefficients in FIR Filters Round-Off Effects in Digital Filters Limit-Cycle Oscillations in Recursive Systems Scaling to Prevent Overflow Statistical Characterization of Quantization Effects in Fixed-Point 631 Realizations of Digital Filters 9.7 Summary and References 640 Problems 641

9 XII Contents 1 0 Design of Digital Filters General Considerations Causality and Its Implications Characteristics of Practical Frequency-Selective Filters Design of FIR Filters Symmetric and Antisymmetric FIR Filters /Design of Linear-Phase FIR Filters Using Windows Design of Linear-Phase FIR Filters by the Frequency-Sampling 671 Method Design of Optimum Equiripple Linear-Phase FIR Filters Design of FIR Differentiators Design of Hilbert Transformers Comparison of Design Methods for Linear-Phase FIR Filters Design of MR Filters From Analog Filters IIR Filter Design by Approximation of Derivatives IIR Filter Design by Impulse Invariance IIR Filter Design by the Bilinear Transformation Characteristics of Commonly Used Analog Filters Some Examples of Digital Filter Designs Based on the Bilinear 727 Transformation 10.4 Frequency Transformations Frequency Transformations in the Analog Domain Frequency Transformations in the Digital Domain Summary and References 734 Problems Multirate Digital Signal Processing Introduction Decimation by a Factor D Interpolation by a Factor / Sampling Rate Conversion by a Rational Factor I / D Implementation of Sampling Rate Conversion Polyphase Filter Structures Interchange of Filters and Downsamplers/Upsamplers Sampling Rate Conversion with Cascaded Integrator Comb Filters Polyphase Structures for Decimation and Interpolation Filters Structures for Rational Sampling Rate Conversion 774

10 Contents XIII 11.6 Multistage Implementation of Sampling Rate Conversion Sampling Rate Conversion of Bandpass Signals Sampling Rate Conversion by an Arbitrary Factor Arbitrary Resampling with Polyphase Interpolators Arbitrary Resampling with Farrow Filter Structures Applications of Multirate Signal Processing Design of Phase Shifters Interfacing of Digital Systems with Different Sampling Rates Implementation of Narrowband Lowpass Filters Subband Coding of Speech Signals Digital Filter Banks Polyphase Structures of Uniform Filter Banks Transmultiplexers Two-Channel Quadrature Mirror Filter Bank Elimination of Aliasing Condition for Perfect Reconstruction Polyphase Form of the QMF Bank Linear Phase FIR QMF Bank IIR QMF Bank Perfect Reconstruction Two-Channel FIR QMF Bank Two-Channel QMF Banks in Subband Coding M-Channel QMF Bank Alias-Free and Perfect Reconstruction Condition Polyphase Form of the M -Channel QMF Bank Summary and References 813 Problems Linear Prediction and Optimum Linear Filters Random Signals, Correlation Functions, and Power Spectra Random Processes Stationary Random Processes Statistical (Ensemble) Averages Statistical Averages for Joint Random Processes Power Density Spectrum Discrete-Time Random Signals Time Averages for a Discrete-Time Random Process Mean-Ergodic Process Correlation-Ergodic Processes 832

11 XIV Contents 12.2 Innovations Representation of a Stationary Random Process Rational Power Spectra Relationships Between the Filter Parameters and the 837 Autocorrelation Sequence 12.3 Forward and Backward Linear Prediction Forward Linear Prediction Backward Linear Prediction «The Optimum Reflection Coefficients for the Lattice Forward and 845 * Backward Predictors Relationship of an AR Process to Linear Prediction Solution of the Normal Equations The Levinson-Durbin Algorithm The Schur Algorithm Properties of the Linear Prediction-Error Filters AR Lattice and ARMA Lattice-Ladder Filters AR Lattice Structure ARMA Processes and Lattice-Ladder Filters Wiener Filters for Filtering and Prediction FIR Wiener Filter Orthogonality Principle in Linear Mean-Square Estimation IIR Wiener Filter Noncausal Wiener Filter Summary and References 873 Problems Adaptive Filters Applications of Adaptive Filters System Identification or System Modeling Adaptive Channel Equalization Echo Cancellation in Data Transmission over Telephone Channels Suppression of Narrowband Interference in a Wideband Signal Adaptive Line Enhancer Adaptive Noise Cancelling Linear Predictive Coding of Speech Signals Adaptive Arrays Adaptive Direct-Form FIR Filters The LMS Algorithm Minimum Mean-Square-Error Criterion The LMS Algorithm 905

12 Contents XV Related Stochastic Gradient Algorithms Properties of the LMS Algorithm Adaptive Direct-Form Filters RLS Algorithms RLS Algorithm The LDU Factorization and Square-Root Algorithms Fast RLS Algorithms Properties of the Direct-Form RLS Algorithms Adaptive Lattice-Ladder Filters Recursive Least-Squares Lattice-Ladder Algorithms Other Lattice Algorithms Properties of Lattice-Ladder Algorithms Summary and References 954 Problems Power Spectrum Estimation Estimation of Spectra from Finite-Duration Observations of Signals Computation of the Energy Density Spectrum Estimation of the Autocorrelation and Power Spectrum of Random 966 Signals: The Periodogram The Use of the DFT in Power Spectrum Estimation Nonparametric Methods for Power Spectrum Estimation The Bartlett Method: Averaging Periodograms The Welch Method: Averaging Modified Periodograms The Blackman and Tukey Method: Smoothing the Periodogram Performance Characteristics of Nonparametric Power Spectrum 981 Estimators Computational Requirements of Nonparametric Power Spectrum 984 Estimates 14.3 Parametric Methods for Power Spectrum Estimation Relationships Between the Autocorrelation and the Model 988 Parameters The Yule-Walker Method for the AR Model Parameters The Burg Method for the AR Model Parameters Unconstrained Least-Squares Method for the AR Model 994 Parameters Sequential Estimation Methods for the AR Model Parameters Selection of AR Model Order MA Model for Power Spectrum Estimation ARMA Model for Power Spectrum Estimation Some Experimental Results 1001

13 XVI Contents 14.4 Filter Bank Methods Filter Bank Realization of the Periodogram Minimum Variance Spectral Estimates Eigenanalysis Algorithms for Spectrum Estimation Pisarenko Harmonic Decomposition Method Eigen-decomposition of the Autocorrelation Matrix for Sinusoids in 1019 White Noise MUSIC Algorithm " ESPRIT Algorithm Order Selection Criteria Experimental Results Summary and References 1029 Problems 1030 A Random Number Generators 1041 B Tables of Transition Coefficients for the Design of Linear-Phase 1047 FIR Filters References and Bibliography 1053 Answers to Selected Problems 1067 Index 1077

Digital Signal Processing

Digital Signal Processing System Analysis and Design Paulo S. R. Diniz Eduardo A. B. da Silva and Sergio L. Netto Federal University of Rio de Janeiro CAMBRIDGE UNIVERSITY PRESS Preface page xv Introduction