Applied Digital Signal Processing: Theory and Practice

Cover......Page 1
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 15
1 Introduction......Page 19
1.1 Signals......Page 20
1.1.1 Mathematical representation of signals......Page 21
1.1.2 Physical representation of signals......Page 24
1.1.3 Deterministic and random signals......Page 26
1.2.1 Continuous-time systems......Page 27
1.2.3 Interface systems......Page 28
1.3 Analog, digital, and mixed signal processing......Page 31
1.4 Applications of digital signal processing......Page 34
1.5 Book organization......Page 36
Further Reading......Page 38
Review questions......Page 39
2 Discrete-time signals and systems......Page 41
2.1 Discrete-time signals......Page 42
2.2 Signal generation and plotting in Matlab......Page 45
2.3 Discrete-time systems......Page 49
2.3.1 Causality and stability......Page 50
2.3.2 Linearity and time invariance......Page 51
2.3.3 Block diagrams, signal flow graphs, and practical realizability......Page 53
2.4 Convolution description of linear time-invariant systems......Page 55
2.5.1 Properties of convolution......Page 63
2.5.2 Causality and stability......Page 65
2.5.4 Response to simple test sequences......Page 67
2.6 Analytical evaluation of convolution......Page 68
2.7 Numerical computation of convolution......Page 73
2.8 Real-time implementation of FIR filters......Page 75
2.9 FIR spatial filters......Page 77
2.10 Systems described by linear constant-coefficientdifference equations......Page 79
2.11 Continuous-time LTI systems......Page 87
Problems......Page 92
3 The z-transform......Page 107
3.1 Motivation......Page 108
3.2 The z-transform......Page 109
3.3 The inverse z-transform......Page 117
3.4 Properties of the z-transform......Page 121
3.5 System function of LTI systems......Page 124
3.6 LTI systems characterized by linear constant-coefficientdifference equations......Page 128
3.7 Connections between pole-zero locations and time-domain behavior......Page 132
3.7.1 First-order systems......Page 134
3.7.2 Second-order systems......Page 135
3.8 The one-sided z-transform......Page 136
Learning summary......Page 138
Further Reading......Page 139
Problems......Page 141
4 Fourier representation of signals......Page 152
4.1.1 Continuous-time sinusoids......Page 153
4.1.2 Discrete-time sinusoids......Page 156
4.2 Fourier representation of continuous-time signals......Page 160
4.2.1 Fourier series for continuous-time periodic signals......Page 161
4.2.2 Fourier transforms for continuous-time aperiodic signals......Page 168
4.3.1 Fourier series for discrete-time periodic signals......Page 175
4.3.2 Fourier transforms for discrete-time aperiodic signals......Page 181
4.4 Summary of Fourier series and Fourier transforms......Page 187
4.5 Properties of the discrete-time Fourier transform......Page 189
4.5.1 Relationship to the z-transform and periodicity......Page 190
4.5.2 Symmetry properties......Page 191
4.5.3 Operational properties......Page 198
4.5.4 Correlation of signals......Page 204
Learning summary......Page 206
Problems......Page 207
5 Transform analysis of LTI systems......Page 219
5.1 Sinusoidal response of LTI systems......Page 220
5.2.1 Response to periodic inputs......Page 228
5.2.2 Response to aperiodic inputs......Page 230
5.2.3 Energy or power gain......Page 232
5.3 Distortion of signals passing through LTI systems......Page 233
5.4 Ideal and practical filters......Page 239
5.5 Frequency response for rational system functions......Page 242
5.6.1 Geometrical evaluation of H(ej omega)
 from poles and zeros......Page 249
5.6.2 Significance of poles and zeros......Page 251
5.7 Design of simple filters by pole-zero placement......Page 255
5.7.1 Discrete-time resonators......Page 256
5.7.2 Notch filters......Page 258
5.7.3 Comb filters......Page 260
5.7.4 Pole-zero pattern rotation -- frequency transformations......Page 261
5.8 Relationship between magnitude and phase responses......Page 265
5.9 Allpass systems......Page 267
5.10 Invertibility and minimum-phase systems......Page 272
5.11.1 System function and frequency response......Page 276
5.11.2 The Laplace transform......Page 278
5.11.3 Systems with rational system functions......Page 281
5.11.4 Frequency response from pole-zero location......Page 284
5.11.5 Minimum-phase and allpass systems......Page 288
Problems......Page 291
6 Sampling of continuous-time signals......Page 310
6.1 Ideal periodic sampling of continuous-time signals......Page 311
6.2 Reconstruction of a bandlimited signal from its samples......Page 315
6.3 The effect of undersampling: aliasing......Page 318
6.4 Discrete-time processing of continuous-time signals......Page 329
6.5.1 Analog-to-digital conversion......Page 336
6.5.2 Digital-to-analog conversion......Page 341
6.6 Sampling of bandpass signals......Page 345
6.6.1 Integer band positioning......Page 346
6.6.2 Arbitrary band positioning......Page 348
6.6.3 Creating integer band positioning with guard bands......Page 350
6.7 Image sampling and reconstruction......Page 351
Further Reading......Page 356
Problems......Page 359
7 The Discrete Fourier Transform......Page 371
7.1 Computational Fourier analysis......Page 372
7.2 The Discrete Fourier Transform (DFT)......Page 375
7.2.1 Algebraic formulation of DFT......Page 376
7.2.2 Matrix formulation of DFT......Page 378
7.2.3 Inherent periodicity of DFT and IDFT......Page 380
7.3 Sampling the Discrete-Time Fourier Transform......Page 381
7.3.1 Frequency-domain sampling......Page 382
7.3.2 Time-domain aliasing......Page 385
7.3.3 Reconstruction of DTFT tilde X (ej omega)
......Page 387
7.3.4 Relationships between CTFT, DTFT, and DFT......Page 390
7.4.1 Linearity......Page 392
7.4.2 Periodic, circular, and modulo-N operations......Page 393
7.4.3 Symmetry properties of the DFT......Page 396
7.4.4 Circular shift of a sequence......Page 401
7.4.5 Circular convolution......Page 403
7.4.6 Circular correlation......Page 407
7.4.7 DFT of stretched and sampled sequences......Page 408
7.4.8 Summary of properties of the DFT......Page 409
7.5.1 Linear convolution using circular convolution......Page 410
7.5.2 Implementation of FIR filters using the DFT......Page 412
7.6 Fourier analysis of signals using the DFT......Page 414
7.6.1 Effects of time-windowing on sinusoidal signals......Page 415
7.6.2 Effects of time-windowing on signals with continuous spectra......Page 419
7.6.3 ``Good'' windows and the uncertainty principle......Page 421
7.6.4 Effects of frequency-domain sampling......Page 426
7.6.5 The spectrogram......Page 430
Further Reading......Page 434
Problems......Page 438
8 Computation of the Discrete FourierTransform......Page 451
8.1 Direct computation of the Discrete Fourier Transform......Page 452
8.2 The FFT idea using a matrix approach......Page 453
8.3.1 Algebraic derivation......Page 457
8.3.2 Practical programming considerations......Page 461
8.3.3 Alternative forms......Page 465
8.4 Decimation-in-frequency FFT algorithms......Page 467
8.5 Generalizations and additional FFT algorithms......Page 471
8.6 Practical considerations......Page 473
8.7 Computation of DFT for special applications......Page 476
8.7.1 Goertzel's algorithm......Page 477
8.7.2 Chirp transform algorithm (CTA)......Page 479
8.7.3 The zoom-FFT......Page 482
8.7.4 A quick Fourier transform (QFT)......Page 484
8.7.5 Sliding DFT (SDFT)......Page 485
Further Reading......Page 486
Problems......Page 489
9 Structures for discrete-time systems......Page 502
9.1 Block diagrams and signal flow graphs......Page 503
9.2.1 Direct form structures......Page 505
9.2.2 Cascade form structures......Page 511
9.2.3 Parallel form structures......Page 514
9.3.1 Direct form......Page 518
9.3.2 Cascade form......Page 519
9.3.3 Direct form for linear-phase FIR systems......Page 520
9.3.4 Frequency-sampling form......Page 525
9.4.1 All-zero lattice structure......Page 528
9.4.2 All-pole lattice structure......Page 533
9.4.3 Further discussion......Page 535
9.5 Structure conversion, simulation, and verification......Page 536
Further Reading......Page 538
Problems......Page 541
10 Design of FIR filters......Page 554
10.1.1 Filter specifications......Page 555
10.1.2 Filter approximation......Page 558
10.1.3 Optimality criteria for filter design......Page 559
10.2 FIR filters with linear phase......Page 561
10.2.1 Type-I FIR linear-phase filters......Page 563
10.2.2 Type-II FIR linear-phase filters......Page 564
10.2.3 Type-III FIR linear-phase filters......Page 565
10.2.5 Amplitude response function of FIR filters with linear phase......Page 566
10.2.6 Zero locations of FIR filters with linear phase......Page 569
10.3.1 Direct truncation of an ideal impulse response......Page 573
10.3.2 Smoothing the frequency response using fixed windows......Page 577
10.3.3 Filter design using the adjustable Kaiser window......Page 583
10.4 Design of FIR filters by frequency sampling......Page 590
10.5.1 Definition and properties......Page 599
10.5.2 Minimax approximation optimality......Page 601
10.6.1 Problem formulation......Page 603
10.6.2 Specifying the optimum Chebyshev approximation......Page 605
10.6.3 Finding the optimum Chebyshev approximation......Page 607
10.6.4 Design examples using Matlab......Page 611
10.7.1 Discrete-time differentiators......Page 618
10.7.2 Discrete-time Hilbert transformers......Page 620
10.7.3 Ideal raised-cosine pulse-shaping lowpass filters......Page 622
Further Reading......Page 624
Problems......Page 627
11 Design of IIR filters......Page 641
11.1 Introduction to IIR filter design......Page 642
11.2 Design of continuous-time lowpass filters......Page 644
11.2.1 The Butterworth approximation......Page 646
11.2.2 The Chebyshev approximation......Page 651
11.2.3 The inverse Chebyshev or Chebyshev II approximation......Page 660
11.2.4 The elliptic or Cauer approximation......Page 665
11.3.1 Impulse-invariance transformation......Page 670
11.3.2 Bilinear transformation......Page 677
11.4 Design examples for lowpass IIR filters......Page 685
11.5 Frequency transformations of lowpass filters......Page 690
11.5.1 Continuous-time frequency transformations......Page 691
11.5.2 Discrete-time frequency transformations......Page 693
11.6 Design examples of IIR filters using Matlab......Page 697
Further Reading......Page 703
Problems......Page 706
12 Multirate signal processing......Page 722
12.1.1 Sampling rate decrease by an integer factor......Page 723
12.1.2 Sampling rate increase by an integer factor......Page 732
12.1.3 Sampling rate change by a noninteger factor......Page 742
12.2.1 Sampling rate compressors and expanders......Page 744
12.2.2 The multirate identities......Page 746
12.2.3 Polyphase filter structures......Page 747
12.2.4 Polyphase structures for decimation and interpolation......Page 748
12.3.1 Half-band and Kth-band (Nyquist) FIR filters......Page 753
12.3.2 Multistage decimation and interpolation......Page 756
12.3.3 Interpolated FIR (IFIR) filters......Page 759
12.4 Two-channel filter banks......Page 763
12.4.1 Input-output description......Page 764
12.4.2 Conditions for perfect reconstruction......Page 766
12.4.3 Perfect reconstruction orthogonal FIR filter banks......Page 768
12.4.4 FIR quadrature mirror filter (QMF) banks......Page 773
12.5 Multichannel filter banks......Page 776
12.5.1 Modulated filter banks......Page 777
12.5.2 Tree-structured filter banks......Page 779
Further Reading......Page 780
Problems......Page 783
13 Random signals......Page 794
13.1.1 Randomness and statistical regularity......Page 795
13.1.3 Probability distributions......Page 797
13.1.4 Statistical averages......Page 799
13.1.5 Two useful random variables......Page 801
13.2.1 Probability functions......Page 803
13.2.2 Covariance and correlation......Page 805
13.2.3 Linear combinations of random variables......Page 807
13.3 Covariance, correlation, and linear estimation......Page 809
13.4 Random processes......Page 813
13.4.1 Statistical specification of random processes......Page 814
13.4.2 Stationary random processes......Page 816
13.4.3 Response of linear time-invariant systems to random processes......Page 819
13.4.4 Power spectral densities......Page 823
13.5.1 White noise process......Page 826
13.5.3 Autoregressive moving average (ARMA) processes......Page 827
13.5.4 Harmonic process models......Page 831
Further Reading......Page 832
Problems......Page 835
14 Random signal processing......Page 846
14.1.1 Basic concepts and terminology......Page 847
14.1.2 Sample mean......Page 849
14.1.3 Sample variance......Page 850
14.2 Spectral analysis of stationary processes......Page 851
14.2.1 Estimation of mean, variance, and ACVS/ACRS......Page 852
14.2.2 The periodogram......Page 856
14.2.3 Statistical properties of the periodogram......Page 858
14.2.4 The modified periodogram......Page 862
14.2.5 The Blackman--Tukey method: smoothing a single periodogram......Page 866
14.2.6 The Bartlett--Welch method: averaging multiple periodograms......Page 872
14.3.1 Filters that maximize the output signal-to-noise ratio......Page 875
14.3.2 Filters that minimize the output mean square error......Page 878
14.4.1 Linear prediction and AR modeling......Page 883
14.4.2 The Levinson--Durbin algorithm......Page 885
14.4.3 Lattice structures for linear prediction......Page 887
14.4.4 Linear prediction in practice......Page 890
14.5 Optimum orthogonal transforms......Page 894
Further Reading......Page 899
Problems......Page 903
15 Finite wordlength effects......Page 919
15.1.1 Binary fixed-point number representation......Page 920
15.1.2 Quantization process......Page 922
15.2 Statistical analysis of quantization error......Page 926
15.2.1 Input A/D quantization noise through discrete-time systems......Page 933
15.3.1 Oversampled A/D conversion with direct quantization......Page 936
15.3.2 Oversampled A/D conversion with noise shaping......Page 940
15.3.3 Oversampled D/A conversion with noise shaping......Page 944
15.4 Quantization of filter coefficients......Page 945
15.4.1 Quantization of IIR filter coefficients......Page 946
15.4.2 Quantization of FIR filter coefficients......Page 950
15.5 Effects of finite wordlength on digital filters......Page 953
15.5.1 Effects of round-off noise in direct-form FIR filters......Page 954
15.5.2 Scaling to avoid overflows in direct-form FIR filters......Page 956
15.5.3 Round-off noise and scaling in IIR filters......Page 957
15.5.4 Limit cycle oscillations......Page 966
15.6 Finite wordlength effects in FFT algorithms......Page 967
Further Reading......Page 969
Problems......Page 971
References
......Page 985
Index......Page 983