Instructor solution manual for Digital signal processing

Contents......Page 5
Preface......Page 17
12.3.2 Backward Linear Prediction 84......Page 21
1.1 Signals, Systems, and Signal Processing......Page 22
1.1.1 Basic Elements of a Digital Signal Processing System......Page 24
1.1.2 Advantages of Digital over Analog Signal Processing......Page 25
1.2.1 Multichannel and Multidimensional Signals......Page 26
1.2.2 Continuous-Time Versus Discrete-Time Signals......Page 29
1.2.3 Continuous-Valued Versus Discrete-Valued Signals......Page 30
1.2.4 Deterministic Versus Random Signals......Page 31
1.3.1 Continuous-Time Sinusoidal Signals......Page 32
1.3.2 Discrete-Time Sinusoidal Signals......Page 34
1.3.3 Harmonically Related Complex Exponentials......Page 37
1.4 Analog-to-Digital and Digital-to-Analog Conversion......Page 39
1.4.1 Sampling of Analog Signals......Page 41
1.4.2 The Sampling Theorem......Page 46
1.4.3 Quantization of Continuous-Amplitude Signals......Page 51
1.4.4 Quantization of Sinusoidal Signals......Page 54
1.4.5 Coding of Quantized Samples......Page 55
1.4.7 Analysis of Digital Signals and Systems Versus Discrete-Time Signals and Systems......Page 56
Problems......Page 57
2 Discrete-Time Signals and Systems......Page 61
2.1 Discrete-Time Signals......Page 62
2.1.1 Some Elementary Discrete-Time Signals......Page 63
2.1.2 Classification of Discrete-Time Signals......Page 65
2.1.3 Simple Manipulations of Discrete-Time Signals......Page 70
2.2 Discrete-Time Systems......Page 73
2.2.1 Input-Output Description of Systen1s......Page 74
2.2.2 Block Diagram Representation of Discrete-Time Systems......Page 77
2.2.3 Classification of Discrete-Time Systems......Page 79
2.2.4 Interconnection of Discrete-Time Systems......Page 87
2.3.1 Techniques for the Analysis of Linear Systems......Page 89
2.3.2 Resolution of a Discrete-Time Signal into Impulses......Page 91
2.3.3 Response of LTI Systems to Arbitrary Inputs: The Convolution Sum......Page 93
2.3.4 Properties of Convolution and the Interconnection of LTI Systems......Page 100
2.3.5 Causal Linear Time-Invariant Systems......Page 103
2.3.6 Stability of Linear Time-Invariant Systems......Page 105
2.3.7 Systems with Finite-Duration and Infinite-Duration Impulse Response......Page 108
2.4 Discrete-Time Systems Described by Difference Equations......Page 109
2.4.1 Recursive and Nonrecursive Discrete-Time Systems......Page 110
2.4.2 Linear Time-Invariant Systems Characterized by Constant-Coefficient Difference Equations......Page 113
2.4.3 Solution of Linear Constant-Coefficient Difference Equations......Page 118
2.4.4 The Impulse Response of a Linear Time-Invariant Recursive System......Page 126
2.5.1 Structures for the Realization of Linear Time-Invariant Systems......Page 129
2.5.2 Recursive and Nonrecursive Realizations of FIR Systems......Page 133
2.6 Correlation of Discrete-Time Signals......Page 136
2.6.1 Crosscorrelation and Autocorrelation Sequences......Page 138
2.6.2 Properties of the Autocorrelation and Crosscorrelation Sequences......Page 140
2.6.3 Correlation of Periodic Sequences......Page 143
2.6.4 Input-Output Correlation Sequences......Page 145
2.7 Summary and References......Page 148
Problems......Page 149
3.1.1 The Direct z-Transform......Page 167
3.1.2 The Inverse z-Transform......Page 176
3.2 Properties of the z-Transform......Page 177
3.3.1 Poles and Zeros......Page 190
3.3.2 Pole Location and Time-Domain Behavior for Causal Signals......Page 194
3.3.3 The System Function of a Linear Time-Invariant System......Page 197
3.4.1 The Inverse z-Transform by Contour Integration......Page 200
3.4.2 The Inverse z-Transform by Power Series Expansion......Page 202
3.4.3 The Inverse z-Transform by Partial-Fraction Expansion......Page 204
3.4.4 Decomposition of Rational z-Transforms......Page 212
3.5 Analysis of Linear Time-Invariant Systems in the z-Domain......Page 213
3.5.1 Response of Systems with Rational System Functions......Page 214
3.5.2 Transient and Steady-State Responses......Page 215
3.5.3 Causality and Stability......Page 216
3.5.4 Pole-Zero Cancellations......Page 218
3.5.5 Multiple-Order Poles and Stability......Page 220
3.5.6 Stability of Second-Order Systems......Page 221
3.6 The One-sided z-Transform......Page 225
3.6.1 Definition and Properties......Page 226
3.6.2 Solution of Difference Equations......Page 230
3.6.3 Response of Pole-Zero Systems with Nonzero Initial Conditions......Page 231
Problems......Page 234
4 Frequency Analysis of Signals......Page 244
4.1 Frequency Analysis of Continuous-Time Signals......Page 245
4.1.1 The Fourier Series for Continuous-Time Periodic Signals......Page 246
4.1.2 Power Density Spectrum of Periodic Signals......Page 250
4.1.3 The Fourier Transform for Continuous-Time Aperiodic Signals......Page 254
4.1.4 Energy Density Spectrum of Aperiodic Signals......Page 258
4.2.1 The Fourier Series for Discrete-Time Periodic Signals......Page 261
4.2.2 Power Density Spectrum of Periodic Signals......Page 265
4.2.3 1he fourier Transform of Discrete-Time Aperiodic Signals......Page 268
4.2.4 Convergence of the Fourier Transform......Page 271
4.2.5 Energy Density Spectrum of AperiodiG Signals......Page 274
4.2.6 Relationship of the Fourier Transform to the z-Transform......Page 279
4.2.7 The Cepstrum......Page 281
4.2.8 The Fourier Transform of Signals with Poles on the Unit Circle......Page 282
4.2.9 Frequency-Domain Classification of Signals: The Concept of Bandwidth......Page 285
4.2.10 1he Frequency Ranges of Some Natural Signals......Page 287
4.3 Frequency-Domain and Time-Domain Signal Properties......Page 288
4.4 Properties of the Fourier Transform for Discrete-Time Signals......Page 291
4.4.1 Symmetry Properties of the Fourier Transform......Page 292
4.4.2 Fourier Transform Theorems and Properties......Page 299
4.5 Summary and References......Page 311
Problems......Page 312
5.1 Frequency-Domain Characteristics of linear Time-Invariant Systems......Page 320
5.1.1 Response to Complex Exponential and Sinusoidal Signals: The Frequency Response Function......Page 321
5.1.2 Steady-State and Transient Response to Sinusoidal Input Signals......Page 330
5.1.3 Steady-State Response to Periodic Input Signals......Page 331
5.1.4 Response to Aperiodic Input Signals......Page 332
5.2.1 Frequency Response of a System with a Rational System Function......Page 334
5.2.2 Computation of the Frequency Response Function......Page 337
5.3 Correlation Functions and Spectra at the Output of l Tl Systems......Page 341
5.3.1 Input-Output Correlation Functions and Spectra......Page 342
5.3.2 Correlation Functions and Power Spectra for Random Input Signals......Page 343
5.4 linear Time-Invariant Systems as Frequency-Selective Filters......Page 346
5.4.1 Ideal Filter Characteristics......Page 347
5.4.2 Lowpass, Highpass, and Bandpass Filters......Page 349
5.4.3 Digital Resonators......Page 355
5.4.4 Notch Filters......Page 359
5.4.5 Comb Filters......Page 361
5.4.6 All-Pass Filters......Page 365
5.4.7 Digital Sinusoidal Oscillators......Page 367
5.5 Inverse Systems and Deconvolution......Page 369
5.5.1 Invertibility of Linear Time-Invariant Systems......Page 370
5.5.2 Minimum-Phase, Maximum-Phase, and Mixed-Phase Systems......Page 374
5.5.3 System Identification and Deconvolution......Page 378
5.5.4 Homomorphic Deconvolution......Page 380
5.6 Summary and References......Page 382
Problems......Page 383
6.1 Ideal Sampling and Reconstruction of Continuous-Time Signals......Page 404
6.2 Discrete-Time Processing of Continuous-Time Signals......Page 415
6.3.1 Analog-to-Digital Converters......Page 421
6.3.2 Quantization and Coding......Page 423
6.3.3 Analysis of Quantization Errors......Page 426
6.3.4 Digital-to-Analog Converters......Page 428
6.4 Sampling and Reconstruction of Continuous-Time Bandpass Signals......Page 430
6.4.1 Uniform or First-Order Sampling......Page 431
6.4.2 Interleaved or Nonuniform Second-Order Sampling......Page 436
6.4.3 Bandpass Signal Representations......Page 442
6.4.4 Sampling Using Bandpass Signal Representations......Page 446
6.5.1 Sampling and Interpolation of Discrete-Time Signals......Page 447
6.5.2 Representation and Sampling of Bandpass Discrete-Time Signals......Page 450
6.6.1 Oversampling AID Converters......Page 453
6.6.2 Oversampling D/ A Converters......Page 459
Problems......Page 460
7.1.1 Frequency-Domain Sampling and Reconstruction of Discrete-Time Signals......Page 469
7.1.2 The Discrete Fourier Transform (DFT)......Page 474
7.1.3 The DFT as a Linear Transformation......Page 479
7.1.4 Relationship of the DFT to Other Transforms......Page 481
7.2 Properties of the OFT......Page 484
7.2.1 Periodicity, Linearity, and Symmetry Properties......Page 485
7.2.2 Multiplication of Two DFfs and Circular Convolution......Page 491
7.2.3 Additional DFT Properties......Page 496
7.3 linear Filtering Methods Based on the DFT......Page 500
7.3.1 Use of the DFT in Linear Filtering......Page 501
7.3.2 Filtering of Long Data Sequences......Page 505
7.4 Frequency Analysis of Signals Using the DFT......Page 508
7.5.1 Forward DCT......Page 515
7.5.2 Inverse DCT......Page 517
7.5.3 DCT as an Orthogonal Transform......Page 518
7.6 Summary and References......Page 521
Problems......Page 522
8.1 Efficient Computation of the OFT: FFT Algorithms......Page 531
8.1.1 Direct Computation of the DFT......Page 532
8.1.2 Divide-and-Conquer Approach to Computation of the DFT......Page 533
8.1.3 Radix-2 FFT Algorithms......Page 539
8.1.4 Radix-4 FFT Algorithms......Page 547
8.1.5 Split-Radix FFT Algorithms......Page 552
8.1.6 Implementation ofFFT Algorithms......Page 556
8.2.1 Efficient Computation of the DFT of Two Real Sequences......Page 558
8.2.2 Efficient Computation of the DFT of a 2N -Point Real Sequence......Page 559
8.2.3 Use of the FFf Algorithm in Linear Filtering and Correlation......Page 560
8.3.1 The Goertzel Algorithm......Page 562
8.3.2 The Chirp-z Transform Algorithm......Page 564
8.4.1 Quantization Errors in the Direct Computation of the DFT......Page 569
8.4.2 Quantization Errors in FFT Algorithms......Page 572
8.5 Summary and References......Page 575
Problems......Page 576
9.1 Structures for the Realization of Discrete-Time Systems......Page 583
9.2 Structures for FIR Systems......Page 585
9.2.1 Direct-Form Structure......Page 586
9.2.2 Cascade-Form Structures......Page 587
9.2.3 Frequency-Sampling Structures......Page 589
9.2.4 Lattice Structure......Page 594
9.3.1 Direct-Form Structures......Page 602
9.3.2 Signal Flow Graphs and Transposed Structures......Page 605
9.3.3 Cascade-Form Structures......Page 609
9.3.4 Parallel-Form Structures......Page 611
9.3.5 Lattice and Lattice-Ladder Structures for IIR Systems......Page 614
9.4.1 Fixed-Point Representation of Numbers......Page 621
9.4.2 Binary Floating-Point Representation of Numbers......Page 625
9.4.3 Errors Resulting from Rounding and Truncation......Page 628
9.5.1 Analysis of Sensitivity to Quantization of Filter Coefficients......Page 633
9.5.2 Quantization of Coefficients in FIR Filters......Page 640
9.6.1 Limit-Cycle Oscillations in Recursive Systems......Page 644
9.6.2 Scaling to Prevent Overflow......Page 649
9.6.3 Statistical Characterization of Quantization Effects in Fixed-Point Realizations of Digital Filters......Page 651
9.7 Summary and References......Page 660
Problems......Page 661
10.1 General Considerations......Page 674
10.1.1 Causality and Its Implications......Page 675
10.1.2 Characteristics of Practical Frequency-Selective Filters......Page 679
10.2.1 Symmetric and Antisymmetric FIR Filters......Page 680
10.2.2 Design of Linear-Phase FIR Filters Using Windows......Page 684
10.2.3 Design of Linear-Phase FIR Filters by the Frequency-Sampling  Method......Page 691
10.2.4 Design of Optimum Equiripple Linear-Phase FIR Filters......Page 698
10.2.5 Design of FIR Differentia tors......Page 711
10.2.6 Design of Hilbert Transformers......Page 713
10.2.7 Comparison of Design Methods for Linear-Phase FIR Filters......Page 720
10.3 Design of IIR Filters From Analog Filters......Page 721
10.3.1 IIR Filter Design by Approximation of Derivatives......Page 723
10.3.2 IIR Filter Design by Impulse Invariance......Page 727
10.3.3 IIR Filter Design by the Bilinear Transformation......Page 732
10.3.4 Characteristics of Comtnonly lTsed Analog Filters......Page 737
10.3.5 Some Examples of Digital Filter Designs Based on the Bilinear Transformation......Page 747
10.4.1 Frequency Transformations in the Analog Domain......Page 750
10.4.2 Frequency Transformations in the Digital Domain......Page 752
10.5 Summary and References......Page 754
Problems......Page 755
11 Multirate Digital Signal Processing......Page 770
11.1 Introduction......Page 771
11.2 Decimation by a Factor D......Page 775
11.3 Interpolation by a Factor I......Page 780
11.4 Sampling Rate Conversion by a Rational Factor I I D......Page 782
11.5.1 Polyphase Filter Structures......Page 786
11.5.2 Interchange of Filters and Downsamplers/Upsamplers......Page 787
11.5.3 Sampling Rate Conversion with Cascaded Integrator Comb Filters......Page 789
11.5.4 Polyphase Structures for Decimation and Interpolation Filters......Page 791
11.5.5 Structures for Rational Sampling Rate Conversion......Page 794
11.6 Multistage Implementation of Sampling Rate Conversion......Page 795
11.7 Sampling Rate Conversion of Bandpass Signals......Page 799
11.8 Sampling Rate Conversion by an Arbitrary Factor......Page 801
11.8.2 Arbitrary Resampling with Farrow Filter Structures......Page 802
11.9.1 Design of Phase Shifters......Page 804
11.9.2 Interfacing of Digital Systems with Different Sampling Rates......Page 805
11.9.3 Implementation of Narrowband Lowpass Filters......Page 806
11.9.4 Subband Coding of Speech Signals......Page 807
11.10 Digital Filter Banks......Page 810
11.10.1 Polyphase Structures of Uniform Filter Banks......Page 814
11.10.2 Transmultiplexers......Page 816
11.11 Two-Channel Quadrature Mirror Filter Bank......Page 818
11.11.1 Elimination of Aliasing......Page 819
11.11.3 Polyphase Form of the QMF Bank......Page 821
11.11.4 Linear Phase FIR QMF Bank......Page 822
11.11.6 Perfect Reconstruction Two-Channel FIR QMF Bank......Page 823
11.11.7 Two-Channel QMF Banks in Subband Coding......Page 826
11.12 M-Channel QMF Bank......Page 827
11.12.2 Polyphase Form of the M -Channel QMF Bank......Page 828
Problems......Page 833
12.1 Random Signals, Correlation Functions, and Power Spectra......Page 843
12.1.1 Random Processes......Page 844
12.1.3 Statistical (Ensemble) Averages......Page 845
12.1.4 Statistical Averages for Joint Random Processes......Page 846
12.1.5 Power Density Spectrum......Page 848
12.1.6 Discrete-Time Random Signals......Page 849
12.1.7 Time Averages for a Discrete-Time Random Process......Page 850
12.1.8 Mean-Ergodic Process......Page 851
12.1.9 Correlation-Ergodic Processes......Page 852
12.2 Innovations Representation of a Stationary Random Process......Page 854
12.2.1 Rational Power Spectra......Page 856
12.2.2 Relationships Between the Filter Parameters and the Autocorrelation Sequence......Page 857
12.3 Forward and Backward linear Prediction......Page 858
12.3.1 Forward Linear Prediction......Page 859
12.3.3 The Optimum Reflection Coefficients for the Lattice Forward and Backward Predictors......Page 865
12.4 Solution of the Normal Equations......Page 866
12.4.1 The Levinson-Durbin Algorithm......Page 867
12.4.2 The Schur Algorithm......Page 870
12.5 Properties of the linear Prediction-Error Filters......Page 875
12.6.1 AR Lattice Structure......Page 878
12.6.2 ARMA Processes and Lattice-Ladder Filters......Page 880
12.7 Wiener Filters for Filtering and Prediction......Page 883
12.7.1 FIR Wiener Filter......Page 884
12.7.2 Orthogonality Principle in Linear Mean-Square Estimation......Page 886
12.7.3 IJR Wiener Filter......Page 887
12.7.4 Noncausal Wiener Filter......Page 892
12.8 Summary and References......Page 893
Problems......Page 894
13.1 Applications of Adaptive Filters......Page 900
13.1.1 System Identification or System Modeling......Page 902
13.1.2 Adaptive Channel Equalization......Page 903
13.1.3 Echo Cancellation in Data Transmission over Telephone Channels......Page 907
13.1.4 Suppression of Narrowband Interference in a Wideband Signal......Page 911
13.1.5 Adaptive Line Enhancer......Page 915
13.1.6 Adaptive Noise Cancelling......Page 916
13.1.7 Linear Predictive Coding of Speech Signals......Page 917
13.1.8 Adaptive Arrays......Page 920
13.2 Adaptive Direct-Form FIR Filters The lMS Algorithm......Page 922
13.2.1 Minimum Mean-Square-Error Criterion......Page 923
13.2.2 The LMS Algorithm......Page 925
13.2.3 Related Stochastic Gradient Algorithms......Page 927
13.2.4 Properties of the LMS Algorithm......Page 929
13.3.1 RLS Algorithm......Page 936
13.3.2 The LDU Factorization and Square-Root Algorithms......Page 941
13.3.3 Fast RLS Algorithms......Page 943
13.3.4 Properties of the Direct-Form RLS Algorithms......Page 945
13.4 Adaptive Lattice-Ladder Filters......Page 947
13.4.1 Recursive Least-Squares Lattice-Ladder Algorithms......Page 948
13.4.2 Other Lattice Algorithms......Page 969
13.4.3 Properties of Lattice-Ladder Algorithms......Page 970
13.5 Summary and References......Page 974
Problems......Page 975
14 Power Spectrum Estimation......Page 980
14.1.1 Computation of the Energy Density Spectrum......Page 981
14.1.2 Estimation of the Autocorrelation and Power Spectrum of Random Signals: The Periodogram......Page 986
14.1.3 The Use of the Df1 in Power Spectrum Estimation......Page 991
14.2.1 The Bartlett Method: Averaging Periodograms......Page 994
14.2.2 The Welch Method: Averaging Modified Periodograms......Page 995
14.2.3 The Blackman and Tukey Method: Smoothing the Periodogram......Page 998
14.2.4 Performance Characteristics of Nonparametric Power Spectrum Estimators......Page 1001
14.2.5 Computational Requirements of Nonparametric Power Spectrum Estitnates......Page 1004
14.3 Parametric Methods for Power Spectrum Estimation......Page 1006
14.3.1 Relationships Between the Autocorrelation and the Model Parameters......Page 1008
14.3.2 lbe Yule-Walker Method for the AR Model Parameters......Page 1010
14.3.3 The Burg Method for the AR Model Parameters......Page 1011
14.3.4 Unconstrained Least-Squares Method for the AR Model Parameters......Page 1014
14.3.5 Sequential Estimation Methods for the AR Model Parameters......Page 1015
14.3.6 Selection of AR Model Order......Page 1016
14.3.7 MA Model for Power Spectrum Estimation......Page 1017
14.3.8 ARMA Model for Power Spectrum Estimation......Page 1019
14.3.9 Some Experimental Results......Page 1021
14.4 Filter Bank Methods......Page 1029
14.4.1 Filter Bank Realization of the Periodogram......Page 1030
14.4.2 Minimum Variance Spectral Estimates......Page 1032
14.5 Eigenanalysis Algorithms for Spectrum Estimation......Page 1035
14.5.1 Pisarenko Harmonic Decomposition Method......Page 1037
14.5.2 Eigen-decomposition of the Autocorrelation Matrix for Sinusoids in White Noise......Page 1039
14.5.3 MUSIC Algorithm......Page 1041
14.5.4 ESPRIT Algorithm......Page 1042
14.5.5 Order Selection Criteria......Page 1045
14.5.6 Experimental Results......Page 1046
14.6 Summary and References......Page 1049
Problems......Page 1050
A Random Number Generators......Page 1061
B Tables of Transition Coefficients for the Design of Linear-Phase FIR Filters......Page 1067
References and Bibliography......Page 1073
Answers to Selected Problems......Page 1087
Index......Page 1097