Understanding Digital Signal Processing, Second Edition

Understanding Digital Signal Processing ......Page 1
Contents ......Page 5
Preface ......Page 11
1 DISCRETE SEQUENCES AND SYSTEMS ......Page 19
1.1 Discrete Sequences and Their Notation ......Page 20
1.2 Signal Amplitude, Magnitude, Power ......Page 26
1.3 Signal Processing Operational Symbols ......Page 27
1.5 Discrete Linear Systems ......Page 30
1.6 Time-Invariant Systems ......Page 35
1.7 The Commutative Property of Linear TIme-Invariant Systems ......Page 36
1.8 Analyzing Linear Time-Invariant Systems ......Page 37
2.1 Aliasing: Signal Ambiquity in the Frequency Domain ......Page 39
2.2 Sampling Low-PClss Signals ......Page 44
2.3 Sampling Bandpass Signals ......Page 48
2.4 Spectral Inversion in Bandpass Sampling ......Page 57
3 THE DISCRETE FOURIER TRANSFORM ......Page 62
3.1 Understanding the Off Equation ......Page 63
3.2 Off Symmetry ......Page 75
3.3 DFT Linearity ......Page 77
3.4 DFT Magnitudes ......Page 78
3.5 DFT Frequency Axis ......Page 79
3.6 DFT Shifting Theorem ......Page 80
3.7 Inverse DFT ......Page 82
3.8 DFT Leakage ......Page 83
3.9 Windows ......Page 91
3.10 DFT Scalloping Loss ......Page 99
3.11 DFT Resolution, Zero Padding, and Frequency-Domain Sampling ......Page 100
3.12 DFT Processing Gain ......Page 105
3.13 The DFT of Rectangular Functions ......Page 108
3.14 The OFT Frequency Response to a Complex Input ......Page 129
3.15 The OFT Frequency Response to a Real Cosine Input ......Page 131
3.16 The OFT Single-Bin Frequency Response to a Real Cosine Input ......Page 132
3.17 Interpreting the DFT ......Page 135
4 THE FAST FOURIER TRANSFORM ......Page 139
4.1 Relationship of the FFT to the DFT ......Page 140
4.2 Hints on Using FFTs in Practice ......Page 141
4.3 FFT Software Programs ......Page 145
4.4 Derivation of the Radix-2 FFT Algorithm ......Page 146
4.5 FFT Input/Output Data Index Bit Reversal ......Page 153
4.6 Radix-2 FFT Butterfly Structures ......Page 155
5 FINITE IMPULSE RESPONSE FILTERS ......Page 165
5.1 An Introduction to Finite Impulse Response FIR Filters ......Page 166
5.2 Convolution in FIR Filters ......Page 171
5.3 Low-Pass FIR Filter Design ......Page 181
5.4 Bandpass FIR Filter Design ......Page 197
5.5 Highpass FIR Filter Design ......Page 198
5.6 Remez Exchange FIR Filter Design Method ......Page 200
5.7 Half-Band FIR Filters ......Page 202
5.8 Phase Response of FIR Filters ......Page 204
5.9 A Generic Description of Discrete Convolution ......Page 209
6 INFINITE IMPULSE RESPONSE FILTERS ......Page 224
6.1 An Introduction to Infinite Impulse Response Filters ......Page 225
6.2 The Laplace Transform ......Page 228
6.3 The z- Transform ......Page 241
6.4 Impulse Invariance IIR Filter Design Method ......Page 256
6.5 Bilinear Transform IIR Filter Design Method ......Page 272
6.6 Optimized IIR Filter Design Method ......Page 283
6.7 Pitfalls in Building IIR Digital Filters ......Page 285
6.8 Improving I1R Filters with Cascaded Structures ......Page 287
6.9 A Brief Comparison of I1R and FIR Filters ......Page 292
7 SPECIALIZED LOWPASS FIR FILTERS ......Page 296
7.1 Frequency Sampling Filters: The Lost Art ......Page 297
7.2 Interpolated Lowpass FIR Filters ......Page 332
8 QUADRATURE SIGNALS ......Page 348
8.2 The Notation of Complex Numbers ......Page 349
8.3 Representing Real Signals Using Complex Phasors ......Page 355
8.4 A Few Thoughl\'i on Negative Frequency ......Page 359
8.5 Quadrature Signals in the Frequency Domain ......Page 360
8.6 Bandpass Quadrature Signals in the Frequency Domain ......Page 363
8.7 Complex Down-Conversion ......Page 365
8.8 A Complex Down-Conversion Example ......Page 367
8.9 An Alternate Down-Conversion Method ......Page 371
9 THE DISCRETE HILBERT TRANSFORM ......Page 374
9.1 Hilbert Transform Definition ......Page 375
9.2 Why Care About the Hilbert Transform ......Page 377
9.3 Impulse Response of a Hilbert Transformer ......Page 382
9.4 Designing a Discrete Hilbert Transformer ......Page 384
9.5 Time-Domain Analytic Signal Generation ......Page 390
9.6 Comparing Analytical Signal Generation Methods ......Page 392
10 SAMPLE RATE CONVERSION ......Page 394
10.1 Decimation ......Page 395
10.2 Interpolation ......Page 400
10.3 Combining Decimation and Interpolation ......Page 402
10.4 Polyphase Filters ......Page 404
10.5 Cascaded Integrator-Comb Filters ......Page 410
11 SIGNAL AVERAGING ......Page 424
11.1 Coherent Averaging ......Page 425
11.2 Incoherent Averaging ......Page 432
11.3 Averaging Multiple Fast Fourier Transforms ......Page 435
11.4 Filtering Aspects of Time-Domain Averaging ......Page 443
11.5 Exponential Averaging ......Page 445
12.1 Fixed-Point Binary Formats ......Page 452
12.2 Binary Number Precision and Dynamic Range ......Page 458
12.3 Effects of Finite Fixed-Point Binary Word Length ......Page 459
12.4 Floating-Point Binary Formats ......Page 475
12.5 Block Floating-Point Binary Format ......Page 481
13.1 Frequency Translation without Multiplication ......Page 484
13.2 High-Speed Vector-Magnitude Approximation ......Page 492
13.3 Frequency-Domain Windowing ......Page 497
13.4 Fast Multiplication of Complex Numbers ......Page 500
13.5 Efficiently Performing the FFf of Real Sequences ......Page 501
13.6 Computing the Inverse FFT Using the Forward FFT ......Page 513
13.8 Reducing AID Converter Quantization Noise ......Page 516
13.9 AID Converter Testing Techniques ......Page 523
13.10 Fast FIR Filtering Using the FFT ......Page 528
13.11 Generating Normally Distributed Random Data ......Page 529
13.12 Zero-Phase Filtering ......Page 531
13.13 Sharpened FIR Filters ......Page 532
13.14 Interpolating a Bandpass Signal ......Page 534
13.15 Spectral Peak Location Algorithm ......Page 536
13.16 Computing FFT Twiddle Factors ......Page 538
13.17 Single Tone Detection ......Page 541
13.18 The Sliding DFT ......Page 545
13.19 The Zoom FFT ......Page 554
13.20 A Practical Spectrum Analyzer ......Page 557
13.21 An Efficient Arctangent Approximation ......Page 560
13.22 Frequency Demodulation Algorithms ......Page 562
13.23 DC Removal ......Page 565
13.24 Improving Traditional CIC Filters ......Page 569
13.25 Smoothing Impulsive Noise ......Page 574
13.26 Efficient Polynomial Evaluation ......Page 576
13.27 Designing Very High-Order FIR Filters ......Page 577
13.28 Time-Domain Interpolation Using the FFT ......Page 581
13.30 Automatic Gain Control (AGC) ......Page 584
13.31 Approximate Envelope Detection ......Page 587
13.32 A Quadrature Oscillator ......Page 589
13.33 Dual-Mode Averaging ......Page 591
A.1 Graphical Representation of Real and Complex Numbers ......Page 598
A.2 Arithmetic Representation of Complex Numbers ......Page 599
A.3 Arithmetic Operations of Complex Numbers ......Page 601
A.4 Some Practical Implications of Using Complex Numbers ......Page 606
APPENDIX B. CLOSED FORM OF A GEOMETRIC SERIES ......Page 608
APPENDIX C. TIME REVERSAL AND THE DFT ......Page 611
D.1 Statistical Measures ......Page 614
D.2 Standard Deviation, or RMS, of a Continuous Sinewave ......Page 617
D.3 The Mean and Variance of Random Functions ......Page 618
D.4 The Normal Probability Density Function ......Page 621
E.1 Using Logarithms to Determine Relative Signal Power ......Page 623
E.2 Some Useful Decibel Numbers ......Page 627
E.3 Absolute Power Using Decibels ......Page 629
APPENDIX F. DIGITAL FILTER TERMINOLOGY ......Page 631
G.1 Frequency Response of a Comb Filter ......Page 642
G.2 Single Complex FSF Frequency Response ......Page 643
G.3 Multisection Complex FSF Phase ......Page 644
G.4 Multisection Complex FSF Frequency Response ......Page 645
G.5 Real FSF Transfer Function ......Page 647
G.6 Type-IV FSF Frequency Response ......Page 649
APPENDIX H. FREQUENCY SAMPLING FILTER DESIGN TABLES ......Page 652
INDEX ......Page 666
ABOUT THE AUTHOR ......Page 675