Published by Pearson (November 1, 2010) © 2011
* Homework problems at the end of the chapters.
 * New material as indicated by the "(NEW)" marking in the below proposed TABLE OF CONTENTS.
* Improve the material in the current UDSP, 2/e material by way of additional examples, figures, and explanations.Â
* Expansion of the current Chapter 13 ("Digital Signal Process Tricks") by roughly 35%. (Chapter 13 of the current UDSP, 2/e book is arguably the most practical, and useful, single book chapter for working DSP engineers ever published.)Â
About the Author     xxiii
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Chapter 1: Discrete Sequences and Systems     1
1.1 Discrete Sequences and their Notation  2
1.2 Signal Amplitude, Magnitude, Power  8
1.3 Signal Processing Operational Symbols  10
1.4 Introduction to Discrete Linear Time-Invariant Systems  12
1.5 Discrete Linear Systems  12
1.6 Time-Invariant Systems  17
1.7 The Commutative Property of Linear Time-Invariant Systems  18
1.8 Analyzing Linear Time-Invariant Systems  19
References  21
Chapter 1 Problems  23
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Chapter 2: Periodic Sampling     33
2.1 Aliasing: Signal Ambiguity in the Frequency Domain  33
2.2 Sampling Lowpass Signals  38
2.3 Sampling Bandpass Signals  42
2.4 Practical Aspects of Bandpass Sampling  45
References  49
Chapter 2 Problems  50
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Chapter 3: The Discrete Fourier Transform    59
3.1 Understanding the DFT Equation  60
3.2 DFT Symmetry  73
3.3 DFT Linearity  75
3.4 DFT Magnitudes  75
3.5 DFT Frequency Axis  77
3.6 DFT Shifting Theorem  77
3.7 Inverse DFTÂ Â 80
3.8 DFT Leakage   81
3.9 Windows  89
3.10 DFT Scalloping Loss  96
3.11 DFT Resolution, Zero Padding, and Frequency-Domain Sampling  98
3.12 DFT Processing Gain  102
3.13 The DFT of Rectangular Functions  105
3.14 Interpreting the DFT Using the Discrete-Time Fourier Transform  120
References  124
Chapter 3 Problems  125
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Chapter 4: The Fast Fourier Transform     135
4.1 Relationship of the FFT to the DFT 136
4.2 Hints on Using FFTs in Practice 137
4.3 Derivation of the Radix-2 FFT Algorithm 141
4.4 FFT Input/Output Data Index Bit Reversal 149
4.5 Radix-2 FFT Butterfly Structures 151
4.6 Alternate Single-Butterfly Structures 154
References 158
Chapter 4 Problems 160
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Chapter 5: Finite Impulse Response Filters     169
5.1 An Introduction to Finite Impulse Response (FIR) Filters  170
5.2 Convolution in FIR Filters  175
5.3 Lowpass FIR Filter Design  186
5.4 Bandpass FIR Filter Design  201
5.5 Highpass FIR Filter Design  203
5.6 Parks-McClellan Exchange FIR Filter Design Method  204
5.7 Half-band FIR Filters  207
5.8 Phase Response of FIR Filters  209
5.9 A Generic Description of Discrete Convolution  214
5.10 Analyzing FIR Filters  226
References  235
Chapter 5 Problems  238
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Chapter 6: Infinite Impulse Response Filters     253
6.1 An Introduction to Infinite Impulse Response Filters  254
6.2 The Laplace Transform  257
6.3 The z-Transform  270
6.4 Using the z-Transform to Analyze IIR Filters  274
6.5 Using Poles and Zeros to Analyze IIR Filters  282
6.6 Alternate IIR Filter Structures  289
6.7 Pitfalls in Building IIR Filters  292
6.8 Improving IIR Filters with Cascaded Structures  295
6.9 Scaling the Gain of IIR Filters  300
6.10 Impulse Invariance IIR Filter Design Method  303
6.11 Bilinear Transform IIR Filter Design Method  319
6.12 Optimized IIR Filter Design Method  330
6.13 A Brief Comparison of IIR and FIR Filters  332
References  333
Chapter 6 Problems  336
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Chapter 7: Specialized Digital Networks and Filters     361
7.1 Differentiators  361
7.2 Integrators  370
7.3 Matched Filters  376
7.4 Interpolated Lowpass FIR Filters  381
7.5 Frequency Sampling Filters: The Lost Art  392
References  426
Chapter 7 Problems  429
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Chapter 8: Quadrature Signals      439
8.1 Why Care about Quadrature Signals?  440
8.2 The Notation of Complex Numbers  440
8.3 Representing Real Signals Using Complex Phasors  446
8.4 A Few Thoughts on Negative Frequency  450
8.5 Quadrature Signals in the Frequency Domain  451
8.6 Bandpass Quadrature Signals in the Frequency Domain  454
8.7 Complex Down-Conversion  456
8.8 A Complex Down-Conversion Example  458
8.9 An Alternate Down-Conversion Method  462
References  464
Chapter 8 Problems  465
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Chapter 9: The Discrete Hilbert Transform      479
9.1 Hilbert Transform Definition  480
9.2 Why Care about the Hilbert Transform?  482
9.3 Impulse Response of a Hilbert Transformer  487
9.4 Designing a Discrete Hilbert Transformer  489
9.5 Time-Domain Analytic Signal Generation  495
9.6 Comparing Analytical Signal Generation Methods  497
References  498
Chapter 9 Problems  499
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Chapter 10: Sample Rate Conversion      507
10.1 Decimation  508
10.2 Two-Stage Decimation  510
10.3 Properties of Downsampling  514
10.4 Interpolation  516
10.5 Properties of Interpolation  518
10.6 Combining Decimation and Interpolation  521
10.7 Polyphase Filters  522
10.8 Two-Stage Interpolation  528
10.9 z-Transform Analysis of Multirate Systems  533
10.10 Polyphase Filter Implementations  535
10.11 Sample Rate Conversion by Rational Factors  540
10.12 Sample Rate Conversion with Half-band Filters  543
10.13 Sample Rate Conversion with IFIR Filters  548
10.14 Cascaded Integrator-Comb Filters  550
References  566
Chapter 10 Problems  568
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Chapter 11: Signal Averaging     589
11.1 Coherent Averaging  590
11.2 Incoherent Averaging  597
11.3 Averaging Multiple Fast Fourier Transforms  600
11.4 Averaging Phase Angles  603
11.5 Filtering Aspects of Time-Domain Averaging  604
11.6 Exponential Averaging  608
References  615
Chapter 11 Problems  617
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Chapter 12: Digital Data Formats and their Effects     623
12.1 Fixed-Point Binary Formats  623
12.2 Binary Number Precision and Dynamic Range  632
12.3 Effects of Finite Fixed-Point Binary Word Length  634
12.4 Floating-Point Binary Formats  652
12.5 Block Floating-Point Binary Format  658
References  658
Chapter 12 Problems  661
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Chapter 13: Digital Signal Processing Tricks       671
13.1 Frequency Translation without Multiplication  671
13.2 High-Speed Vector Magnitude Approximation  679
13.3 Frequency-Domain Windowing  683
13.4 Fast Multiplication of Complex Numbers  686
13.5 Efficiently Performing the FFT of Real Sequences  687
13.6 Computing the Inverse FFT Using the Forward FFTÂ Â 699
13.7 Simplified FIR Filter Structure  702
13.8 Reducing A/D Converter Quantization Noise  704
13.9 A/D Converter Testing Techniques  709
13.10 Fast FIR Filtering Using the FFTÂ Â 716
13.11 Generating Normally Distributed Random Data  722
13.12 Zero-Phase Filtering  725
13.13 Sharpened FIR Filters  726
13.14 Interpolating a Bandpass Signal  728
13.15 Spectral Peak Location Algorithm  730
13.16 Computing FFT Twiddle Factors  734
13.17 Single Tone Detection  737
13.18 The Sliding DFTÂ Â 741
13.19 The Zoom FFTÂ Â 749
13.20 A Practical Spectrum Analyzer  753
13.21 An Efficient Arctangent Approximation  756
13.22 Frequency Demodulation Algorithms  758
13.23 DC Removal  761
13.24 Improving Traditional CIC Filters  765
13.25 Smoothing Impulsive Noise  770
13.26 Efficient Polynomial Evaluation  772
13.27 Designing Very High-Order FIR Filters  775
13.28 Time-Domain Interpolation Using the FFTÂ Â 778
13.29 Frequency Translation Using Decimation  781
13.30 Automatic Gain Control (AGC)Â Â 783
13.31 Approximate Envelope Detection  784
13.32 AQuadrature Oscillator  786
13.33 Specialized Exponential Averaging  789
13.34 Filtering Narrowband Noise Using Filter Nulls  792
13.35 Efficient Computation of Signal Variance  797
13.36 Real-time Computation of Signal Averages and Variances  799
13.37 Building Hilbert Transformers from Half-band Filters  802
13.38 Complex Vector Rotation with Arctangents  805
13.39 An Efficient Differentiating Network  810
13.40 Linear-Phase DC-Removal Filter  812
13.41 Avoiding Overflow in Magnitude Computations  815
13.42 Efficient Linear Interpolation  815
13.43 Alternate Complex Down-conversion Schemes  816
13.44 Signal Transition Detection  820
13.45 Spectral Flipping around Signal Center Frequency  821
13.46 Computing Missing Signal Samples  823
13.47 Computing Large DFTs Using Small FFTs  826
13.48 Computing Filter Group Delay without Arctangents  830
13.49 Computing a Forward and Inverse FFT Using a Single FFTÂ Â 831
13.50 Improved Narrowband Lowpass IIR Filters  833
13.51 A Stable Goertzel Algorithm  838
References  840
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Appendix A: The Arithmetic of Complex Numbers      847
A.1 Graphical Representation of Real and Complex Numbers  847
A.2 Arithmetic Representation of Complex Numbers  848
A.3 Arithmetic Operations of Complex Numbers  850
A.4 Some Practical Implications of Using Complex Numbers  856
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Appendix B: Closed Form of a Geometric Series      859
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Appendix C: Time Reversal and the DFTÂ Â Â Â Â Â 863
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Appendix D: Mean, Variance, and Standard Deviation      867
D.1 Statistical Measures  867
D.2 Statistics of Short Sequences   870
D.3 Statistics of Summed Sequences  872
D.4 Standard Deviation (RMS) of a Continuous Sinewave  874
D.5 Estimating Signal-to-Noise Ratios  875
D.6 The Mean and Variance of Random Functions  879
D.7 The Normal Probability Density Function  882
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Appendix E: Decibels (DB and DBM)Â Â Â Â Â Â 885
E.1 Using Logarithms to Determine Relative Signal Power  885
E.2 Some Useful Decibel Numbers  889
E.3 Absolute Power Using Decibels  891
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Appendix F: Digital Filter Terminology      893
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Appendix G: Frequency Sampling Filter Derivations      903
G.1 Frequency Response of a Comb Filter  903
G.2 Single Complex FSF Frequency Response  904
G.3 Multisection Complex FSF Phase  905
G.4 Multisection Complex FSF Frequency Response  906
G.5 Real FSF Transfer Function  908
G.6 Type-IV FSF Frequency Response  910
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Appendix H: Frequency Sampling Filter Design Tables     913
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Appendix I: Computing Chebyshev Window Sequences       927
I.1 Chebyshev Windows for FIR Filter Design  927
I.2 Chebyshev Windows for Spectrum Analysis  929
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Index       931
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