Digital Signal Processing

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  • Edition: 4th
  • Format: Paperback
  • Copyright: 2006-03-28
  • Publisher: Pearson

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This fourth edition covers the fundamentals of discrete-time signals, systems, and modern digital signal processing. Appropriate for students of electrical engineering, computer engineering, and computer science, the book is suitable for undergraduate and graduate courses and provides balanced coverage of both theory and practical applications.

Table of Contents

Prefacep. xvii
Introductionp. 1
Signals, Systems, and Signal Processingp. 2
Basic Elements of a Digital Signal Processing Systemp. 4
Advantages of Digital over Analog Signal Processingp. 5
Classification of Signalsp. 6
Multichannel and Multidimensional Signalsp. 6
Continuous-Time Versus Discrete-Time Signalsp. 9
Continuous-Valued Versus Discrete-Valued Signalsp. 10
Deterministic Versus Random Signalsp. 11
The Concept of Frequency in Continuous-Time and Discrete-Time Signalsp. 12
Continuous-Time Sinusoidal Signalsp. 12
Discrete-Time Sinusoidal Signalsp. 14
Harmonically Related Complex Exponentialsp. 17
Analog-to-Digital and Digital-to-Analog Conversionp. 19
Sampling of Analog Signalsp. 21
The Sampling Theoremp. 26
Quantization of Continuous-Amplitude Signalsp. 31
Quantization of Sinusoidal Signalsp. 34
Coding of Quantized Samplesp. 35
Digital-to-Analog Conversionp. 36
Analysis of Digital Signals and Systems Versus Discrete-Time Signals and Systemsp. 36
Summary and Referencesp. 37
Problemsp. 37
Discrete-Time Signals and Systemsp. 41
Discrete-Time Signalsp. 42
Some Elementary Discrete-Time Signalsp. 43
Classification of Discrete-Time Signalsp. 45
Simple Manipulations of Discrete-Time Signalsp. 50
Discrete-Time Systemsp. 53
Input-Output Description of Systemsp. 54
Block Diagram Representation of Discrete-Time Systemsp. 57
Classification of Discrete-Time Systemsp. 59
Interconnection of Discrete-Time Systemsp. 67
Analysis of Discrete-Time Linear Time-Invariant Systemsp. 69
Techniques for the Analysis of Linear Systemsp. 69
Resolution of a Discrete-Time Signal into Impulsesp. 71
Response of LTI Systems to Arbitrary Inputs: The Convolution Sump. 73
Properties of Convolution and the Interconnection of LTI Systemsp. 80
Causal Linear Time-Invariant Systemsp. 83
Stability of Linear Time-Invariant Systemsp. 85
Systems with Finite-Duration and Infinite-Duration Impulse Responsep. 88
Discrete-Time Systems Described by Difference Equationsp. 89
Recursive and Nonrecursive Discrete-Time Systemsp. 90
Linear Time-Invariant Systems Characterized by Constant-Coefficient Difference Equationsp. 93
Solution of Linear Constant-Coefficient Difference Equationsp. 98
The Impulse Response of a Linear Time-Invariant Recursive Systemp. 106
Implementation of Discrete-Time Systemsp. 109
Structures for the Realization of Linear Time-Invariant Systemsp. 109
Recursive and Nonrecursive Realizations of FIR Systemsp. 113
Correlation of Discrete-Time Signalsp. 116
Crosscorrelation and Autocorrelation Sequencesp. 118
Properties of the Autocorrelation and Crosscorrelation Sequencesp. 120
Correlation of Periodic Sequencesp. 123
Input-Output Correlation Sequencesp. 125
Summary and Referencesp. 128
Problemsp. 129
The z-Transform and Its Application to the Analysis of LTI Systemsp. 147
The z-Transformp. 147
The Direct z-Transformp. 147
The Inverse z-Transformp. 156
Properties of the z-Transformp. 157
Rational z-Transformsp. 170
Poles and Zerosp. 170
Pole Location and Time-Domain Behavior for Causal Signalsp. 174
The System Function of a Linear Time-Invariant Systemp. 177
Inversion of the z-Transformp. 180
The Inverse z-Transform by Contour Integrationp. 180
The Inverse z-Transform by Power Series Expansionp. 182
The Inverse z-Transform by Partial-Fraction Expansionp. 184
Decomposition of Rational z-Transformsp. 192
Analysis of Linear Time-Invariant Systems in the z-Domainp. 193
Response of Systems with Rational System Functionsp. 194
Transient and Steady-State Responsesp. 195
Causality and Stabilityp. 196
Pole-Zero Cancellationsp. 198
Multiple-Order Poles and Stabilityp. 200
Stability of Second-Order Systemsp. 201
The One-sided z-Transformp. 205
Definition and Propertiesp. 206
Solution of Difference Equationsp. 210
Response of Pole-Zero Systems with Nonzero Initial Conditionsp. 211
Summary and Referencesp. 214
Problemsp. 214
Frequency Analysis of Signalsp. 224
Frequency Analysis of Continuous-Time Signalsp. 225
The Fourier Series for Continuous-Time Periodic Signalsp. 226
Power Density Spectrum of Periodic Signalsp. 230
The Fourier Transform for Continuous-Time Aperiodic Signalsp. 234
Energy Density Spectrum of Aperiodic Signalsp. 238
Frequency Analysis of Discrete-Time Signalsp. 241
The Fourier Series for Discrete-Time Periodic Signalsp. 241
Power Density Spectrum of Periodic Signalsp. 245
The Fourier Transform of Discrete-Time Aperiodic Signalsp. 248
Convergence of the Fourier Transformp. 251
Energy Density Spectrum of Aperiodic Signalsp. 254
Relationship of the Fourier Transform to the z-Transformp. 259
The Cepstrump. 261
The Fourier Transform of Signals with Poles on the Unit Circlep. 262
Frequency-Domain Classification of Signals: The Concept of Bandwidthp. 265
The Frequency Ranges of Some Natural Signalsp. 267
Frequency-Domain and Time-Domain Signal Propertiesp. 268
Properties of the Fourier Transform for Discrete-Time Signalsp. 271
Symmetry Properties of the Fourier Transformp. 272
Fourier Transform Theorems and Propertiesp. 279
Summary and Referencesp. 291
Problemsp. 292
Frequency-Domain Analysis of LTI Systemsp. 300
Frequency-Domain Characteristics of Linear Time-Invariant Systemsp. 300
Response to Complex Exponential and Sinusoidal Signals: The Frequency Response Functionp. 301
Steady-State and Transient Response to Sinusoidal Input Signalsp. 310
Steady-State Response to Periodic Input Signalsp. 311
Response to Aperiodic Input Signalsp. 312
Frequency Response of LTI Systemsp. 314
Frequency Response of a System with a Rational System Functionp. 314
Computation of the Frequency Response Functionp. 317
Correlation Functions and Spectra at the Output of LTI Systemsp. 321
Input-Output Correlation Functions and Spectrap. 322
Correlation Functions and Power Spectra for Random Input Signalsp. 323
Linear Time-Invariant Systems as Frequency-Selective Filtersp. 326
Ideal Filter Characteristicsp. 327
Lowpass, Highpass, and Bandpass Filtersp. 329
Digital Resonatorsp. 335
Notch Filtersp. 339
Comb Filtersp. 341
All-Pass Filtersp. 345
Digital Sinusoidal Oscillatorsp. 347
Inverse Systems and Deconvolutionp. 349
Invertibility of Linear Time-Invariant Systemsp. 350
Minimum-Phase, Maximum-Phase, and Mixed-Phase Systemsp. 354
System Identification and Deconvolutionp. 358
Homomorphic Deconvolutionp. 360
Summary and Referencesp. 362
Problemsp. 363
Sampling and Reconstruction of Signalsp. 384
Ideal Sampling and Reconstruction of Continuous-Time Signalsp. 384
Discrete-Time Processing of Continuous-Time Signalsp. 395
Analog-to-Digital and Digital-to-Analog Convertersp. 401
Analog-to-Digital Convertersp. 401
Quantization and Codingp. 403
Analysis of Quantization Errorsp. 406
Digital-to-Analog Convertersp. 408
Sampling and Reconstruction of Continuous-Time Bandpass Signalsp. 410
Uniform or First-Order Samplingp. 411
Interleaved or Nonuniform Second-Order Samplingp. 416
Bandpass Signal Representationsp. 422
Sampling Using Bandpass Signal Representationsp. 426
Sampling of Discrete-Time Signalsp. 427
Sampling and Interpolation of Discrete-Time Signalsp. 427
Representation and Sampling of Bandpass Discrete-Time Signalsp. 430
Oversampling A/D and D/A Convertersp. 433
Oversampling A/D Convertersp. 433
Oversampling D/A Convertersp. 439
Summary and Referencesp. 440
Problemsp. 440
The Discrete Fourier Transform: Its Properties and Applicationsp. 449
Frequency-Domain Sampling: The Discrete Fourier Transformp. 449
Frequency-Domain Sampling and Reconstruction of Discrete-Time Signalsp. 449
The Discrete Fourier Transform (DFT)p. 454
The DFT as a Linear Transformationp. 459
Relationship of the DFT to Other Transformsp. 461
Properties of the DFTp. 464
Periodicity, Linearity, and Symmetry Propertiesp. 465
Multiplication of Two DFTs and Circular Convolutionp. 471
Additional DFT Propertiesp. 476
Linear Filtering Methods Based on the DFTp. 480
Use of the DFT in Linear Filteringp. 481
Filtering of Long Data Sequencesp. 485
Frequency Analysis of Signals Using the DFTp. 488
The Discrete Cosine Transformp. 495
Forward DCTp. 495
Inverse DCTp. 497
DCT as an Orthogonal Transformp. 498
Summary and Referencesp. 501
Problemsp. 502
Efficient Computation of the DFT: Fast Fourier Transform Algorithmsp. 511
Efficient Computation of the DFT: FFT Algorithmsp. 511
Direct Computation of the DFTp. 512
Divide-and-Conquer Approach to Computation of the DFTp. 513
Radix-2 FFT Algorithmsp. 519
Radix-4 FFT Algorithmsp. 527
Split-Radix FFT Algorithmsp. 532
Implementation of FFT Algorithmsp. 536
Applications of FFT Algorithmsp. 538
Efficient Computation of the DFT of Two Real Sequencesp. 538
Efficient Computation of the DFT of a 2N-Point Real Sequencep. 539
Use of the FFT Algorithm in Linear Filtering and Correlationp. 540
A Linear Filtering Approach to Computation of the DFTp. 542
The Goertzel Algorithmp. 542
The Chirp-z Transform Algorithmp. 544
Quantization Effects in the Computation of the DFTp. 549
Quantization Errors in the Direct Computation of the DFTp. 549
Quantization Errors in FFT Algorithmsp. 552
Summary and Referencesp. 555
Problemsp. 556
Implementation of Discrete-Time Systemsp. 563
Structures for the Realization of Discrete-Time Systemsp. 563
Structures for FIR Systemsp. 565
Direct-Form Structurep. 566
Cascade-Form Structuresp. 567
Frequency-Sampling Structuresp. 569
Lattice Structurep. 574
Structures for IIR Systemsp. 582
Direct-Form Structuresp. 582
Signal Flow Graphs and Transposed Structuresp. 585
Cascade-Form Structuresp. 589
Parallel-Form Structuresp. 591
Lattice and Lattice-Ladder Structures for IIR Systemsp. 594
Representation of Numbersp. 601
Fixed-Point Representation of Numbersp. 601
Binary Floating-Point Representation of Numbersp. 605
Errors Resulting from Rounding and Truncationp. 608
Quantization of Filter Coefficientsp. 613
Analysis of Sensitivity to Quantization of Filter Coefficientsp. 613
Quantization of Coefficients in FIR Filtersp. 620
Round-Off Effects in Digital Filtersp. 624
Limit-Cycle Oscillations in Recursive Systemsp. 624
Scaling to Prevent Overflowp. 629
Statistical Characterization of Quantization Effects in Fixed-Point Realizations of Digital Filtersp. 631
Summary and Referencesp. 640
Problemsp. 641
Design of Digital Filtersp. 654
General Considerationsp. 654
Causality and Its Implicationsp. 655
Characteristics of Practical Frequency-Selective Filtersp. 659
Design of FIR Filtersp. 660
Symmetric and Antisymmetric FIR Filtersp. 660
Design of Linear-Phase FIR Filters Using Windowsp. 664
Design of Linear-Phase FIR Filters by the Frequency-Sampling Methodp. 671
Design of Optimum Equiripple Linear-Phase FIR Filtersp. 678
Design of FIR Differentiatorsp. 691
Design of Hilbert Transformersp. 693
Comparison of Design Methods for Linear-Phase FIR Filtersp. 700
Design of IIR Filters From Analog Filtersp. 701
IIR Filter Design by Approximation of Derivativesp. 703
IIR Filter Design by Impulse Invariancep. 707
IIR Filter Design by the Bilinear Transformationp. 712
Characteristics of Commonly Used Analog Filtersp. 717
Some Examples of Digital Filter Designs Based on the Bilinear Transformationp. 727
Frequency Transformationsp. 730
Frequency Transformations in the Analog Domainp. 730
Frequency Transformations in the Digital Domainp. 732
Summary and Referencesp. 734
Problemsp. 735
Multirate Digital Signal Processingp. 750
Introductionp. 751
Decimation by a Factor Dp. 755
Interpolation by a Factor Ip. 760
Sampling Rate Conversion by a Rational Factor I/Dp. 762
Implementation of Sampling Rate Conversionp. 766
Polyphase Filter Structuresp. 766
Interchange of Filters and Downsamplers/Upsamplersp. 767
Sampling Rate Conversion with Cascaded Integrator Comb Filtersp. 769
Polyphase Structures for Decimation and Interpolation Filtersp. 771
Structures for Rational Sampling Rate Conversionp. 774
Multistage Implementation of Sampling Rate Conversionp. 775
Sampling Rate Conversion of Bandpass Signalsp. 779
Sampling Rate Conversion by an Arbitrary Factorp. 781
Arbitrary Resampling with Polyphase Interpolatorsp. 782
Arbitrary Resampling with Farrow Filter Structuresp. 782
Applications of Multirate Signal Processingp. 784
Design of Phase Shiftersp. 784
Interfacing of Digital Systems with Different Sampling Ratesp. 785
Implementation of Narrowband Lowpass Filtersp. 786
Subband Coding of Speech Signalsp. 787
Digital Filter Banksp. 790
Polyphase Structures of Uniform Filter Banksp. 794
Transmultiplexersp. 796
Two-Channel Quadrature Mirror Filter Bankp. 798
Elimination of Aliasingp. 799
Condition for Perfect Reconstructionp. 801
Polyphase Form of the QMF Bankp. 801
Linear Phase FIR QMF Bankp. 802
IIR QMF Bankp. 803
Perfect Reconstruction Two-Channel FIR QMF Bankp. 803
Two-Channel QMF Banks in Subband Codingp. 806
M-Channel QMF Bankp. 807
Alias-Free and Perfect Reconstruction Conditionp. 808
Polyphase Form of the M-Channel QMF Bankp. 808
Summary and Referencesp. 813
Problemsp. 813
Linear Prediction and Optimum Linear Filtersp. 823
Random Signals, Correlation Functions, and Power Spectrap. 823
Random Processesp. 824
Stationary Random Processesp. 825
Statistical (Ensemble) Averagesp. 825
Statistical Averages for Joint Random Processesp. 826
Power Density Spectrump. 828
Discrete-Time Random Signalsp. 829
Time Averages for a Discrete-Time Random Processp. 830
Mean-Ergodic Processp. 831
Correlation-Ergodic Processesp. 832
Innovations Representation of a Stationary Random Processp. 834
Rational Power Spectrap. 836
Relationships Between the Filter Parameters and the Autocorrelation Sequencep. 837
Forward and Backward Linear Predictionp. 838
Forward Linear Predictionp. 839
Backward Linear Predictionp. 841
The Optimum Reflection Coefficients for the Lattice Forward and Backward Predictorsp. 845
Relationship of an AR Process to Linear Predictionp. 846
Solution of the Normal Equationsp. 846
The Levinson-Durbin Algorithmp. 847
The Schur Algorithmp. 850
Properties of the Linear Prediction-Error Filtersp. 855
AR Lattice and ARMA Lattice-Ladder Filtersp. 858
AR Lattice Structurep. 858
ARMA Processes and Lattice-Ladder Filtersp. 860
Wiener Filters for Filtering and Predictionp. 863
FIR Wiener Filterp. 864
Orthogonality Principle in Linear Mean-Square Estimationp. 866
IIR Wiener Filterp. 867
Noncausal Wiener Filterp. 872
Summary and Referencesp. 873
Problemsp. 874
Adaptive Filtersp. 880
Applications of Adaptive Filtersp. 880
System Identification or System Modelingp. 882
Adaptive Channel Equalizationp. 883
Echo Cancellation in Data Transmission over Telephone Channelsp. 887
Suppression of Narrowband Interference in a Wideband Signalp. 891
Adaptive Line Enhancerp. 895
Adaptive Noise Cancellingp. 896
Linear Predictive Coding of Speech Signalsp. 897
Adaptive Arraysp. 900
Adaptive Direct-Form FIR Filters-The LMS Algorithmp. 902
Minimum Mean-Square-Error Criterionp. 903
The LMS Algorithmp. 905
Related Stochastic Gradient Algorithmsp. 907
Properties of the LMS Algorithmp. 909
Adaptive Direct-Form Filters-RLS Algorithmsp. 916
RLS Algorithmp. 916
The LDU Factorization and Square-Root Algorithmsp. 921
Fast RLS Algorithmsp. 923
Properties of the Direct-Form RLS Algorithmsp. 925
Adaptive Lattice-Ladder Filtersp. 927
Recursive Least-Squares Lattice-Ladder Algorithmsp. 928
Other Lattice Algorithmsp. 949
Properties of Lattice-Ladder Algorithmsp. 950
Summary and Referencesp. 954
Problemsp. 955
Power Spectrum Estimationp. 960
Estimation of Spectra from Finite-Duration Observations of Signalsp. 961
Computation of the Energy Density Spectrump. 961
Estimation of the Autocorrelation and Power Spectrum of Random Signals: The Periodogramp. 966
The Use of the DFT in Power Spectrum Estimationp. 971
Nonparametric Methods for Power Spectrum Estimationp. 974
The Bartlett Method: Averaging Periodogramsp. 974
The Welch Method: Averaging Modified Periodogramsp. 975
The Blackman and Tukey Method: Smoothing the Periodogramp. 978
Performance Characteristics of Nonparametric Power Spectrum Estimatorsp. 981
Computational Requirements of Nonparametric Power Spectrum Estimatesp. 984
Parametric Methods for Power Spectrum Estimationp. 986
Relationships Between the Autocorrelation and the Model Parametersp. 988
The Yule-Walker Method for the AR Model Parametersp. 990
The Burg Method for the AR Model Parametersp. 991
Unconstrained Least-Squares Method for the AR Model Parametersp. 994
Sequential Estimation Methods for the AR Model Parametersp. 995
Selection of AR Model Orderp. 996
MA Model for Power Spectrum Estimationp. 997
ARMA Model for Power Spectrum Estimationp. 999
Some Experimental Resultsp. 1001
Filter Bank Methodsp. 1009
Filter Bank Realization of the Periodogramp. 1010
Minimum Variance Spectral Estimatesp. 1012
Eigenanalysis Algorithms for Spectrum Estimationp. 1015
Pisarenko Harmonic Decomposition Methodp. 1017
Eigen-decomposition of the Autocorrelation Matrix for Sinusoids in White Noisep. 1019
MUSIC Algorithmp. 1021
ESPRIT Algorithmp. 1022
Order Selection Criteriap. 1025
Experimental Resultsp. 1026
Summary and Referencesp. 1029
Problemsp. 1030
Random Number Generatorsp. 1041
Tables of Transition Coefficients for the Design of Linear-Phase FIR Filtersp. 1047
References and Bibliographyp. 1053
Answers to Selected Problemsp. 1067
Indexp. 1077
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