Cover......Page 1
Contents......Page 6
Preface......Page 12
1 DISCRETE SEQUENCES AND SYSTEMS......Page 20
1.1 Discrete Sequences and Their Notation......Page 21
1.2 Signal Amplitude, Magnitude, Power......Page 27
1.3 Signal Processing Operational Symbols......Page 28
1.5 Discrete Linear Systems......Page 31
1.6 Time-Invariant Systems......Page 36
1.7 The Commutative Property of Linear Time-Invariant Systems......Page 37
1.8 Analyzing Linear Time-Invariant Systems......Page 38
2.1 Aliasing: Signal Ambiquity in the Frequency Domain......Page 40
2.2 Sampling Low-Pass Signals......Page 45
2.3 Sampling Bandpass Signals......Page 49
2.4 Spectral Inversion in Bandpass Sampling......Page 58
3 THE DISCRETE FOURIER TRANSFORM......Page 64
3.1 Understanding the DFT Equation......Page 65
3.2 DFT Symmetry......Page 77
3.3 DFT Linearity......Page 79
3.4 DFT Magnitudes......Page 80
3.5 DFT Frequency Axis......Page 81
3.6 DFT Shifting Theorem......Page 82
3.7 Inverse DFT......Page 84
3.8 DFT Leakage......Page 85
3.9 Windows......Page 93
3.10 DFT Scalloping Loss......Page 101
3.11 DFT Resolution, Zero Padding, and Frequency-Domain Sampling......Page 102
3.12 DFT Processing Gain......Page 107
3.13 The DFT of Rectangular Functions......Page 110
3.14 The DFT Frequency Response to a Complex Input......Page 131
3.15 The DFT Frequency Response to a Real Cosine Input......Page 135
3.16 The DFT Single-Bin Frequency Response to a Real Cosine Input......Page 136
3.17 Interpreting the DFT......Page 139
4 THE FAST FOURIER TRANSFORM......Page 144
4.1 Relationship of the FFT to the DFT......Page 145
4.2 Hints on Using FFTs in Practice......Page 146
4.3 FFT Software Programs......Page 150
4.4 Derivation of the Radix-2 FFT Algorithm......Page 151
4.5 FFT Input/Output Data Index Bit Reversal......Page 158
4.6 Radix-2 FFT Butterfly Structures......Page 160
5 FINITE IMPULSE RESPONSE FILTERS......Page 170
5.1 An Introduction to Finite Impulse Response FIR Filters......Page 171
5.2 Convolution in FIR Filters......Page 176
5.3 Low-Pass FIR Filter Design......Page 186
5.4 Bandpass FIR Filter Design......Page 202
5.5 Highpass FIR Filter Design......Page 203
5.6 Remez Exchange FIR Filter Design Method......Page 205
5.7 Half-Band FIR Filters......Page 207
5.8 Phase Response of FIR Filters......Page 209
5.9 A Generic Description of Discrete Convolution......Page 214
6 INFINITE IMPULSE RESPONSE FILTERS......Page 230
6.1 An Introduction to Infinite Impulse Response Filters......Page 231
6.2 The Laplace Transform......Page 234
6.3 The z- Transform......Page 247
6.4 Impulse Invariance IIR Filter Design Method......Page 262
6.5 Bilinear Transform IIR Filter Design Method......Page 278
6.6 Optimized IIR Filter Design Method......Page 289
6.7 Pitfalls in Building IIR Digital Filters......Page 291
6.8 Improving IIR Filters with Cascaded Structures......Page 293
6.9 A Brief Comparison of IIR and FIR Filters......Page 298
7 SPECIALIZED LOWPASS FIR FILTERS......Page 302
7.1 Frequency Sampling Filters: The Lost Art......Page 303
7.2 Interpolated Lowpass FIR Filters......Page 338
8 QUADRATURE SIGNALS......Page 354
8.2 The Notation of Complex Numbers......Page 355
8.3 Representing Real Signals Using Complex Phasors......Page 361
8.4 A Few Thoughts on Negative Frequency......Page 365
8.5 Quadrature Signals in the Frequency Domain......Page 366
8.6 Bandpass Quadrature Signals in the Frequency Domain......Page 369
8.7 Complex Down-Conversion......Page 371
8.8 A Complex Down-Conversion Example......Page 373
8.9 An Alternate Down-Conversion Method......Page 377
9 THE DISCRETE HILBERT TRANSFORM......Page 380
9.1 Hilbert Transform Definition......Page 381
9.2 Why Care About the Hilbert Transform?......Page 383
9.3 Impulse Response of a Hilbert Transformer......Page 388
9.4 Designing a Discrete Hilbert Transformer......Page 390
9.5 Time-Domain Analytic Signal Generation......Page 396
9.6 Comparing Analytical Signal Generation Methods......Page 398
10 SAMPLE RATE CONVERSION......Page 400
10.1 Decimation......Page 401
10.2 Interpolation......Page 406
10.3 Combining Decimation and Interpolation......Page 408
10.4 Polyphase Filters......Page 410
10.5 Cascaded Integrator-Comb Filters......Page 416
11 SIGNAL AVERAGING......Page 430
11.1 Coherent Averaging......Page 431
11.2 Incoherent Averaging......Page 438
11.3 Averaging Multiple Fast Fourier Transforms......Page 441
11.4 Filtering Aspects of Time-Domain Averaging......Page 449
11.5 Exponential Averaging......Page 451
12.1 Fixed-Point Binary Formats......Page 458
12.2 Binary Number Precision and Dynamic Range......Page 464
12.3 Effects of Finite Fixed-Point Binary Word Length......Page 465
12.4 Floating-Point Binary Formats......Page 481
12.5 Block Floating-Point Binary Format......Page 487
13.1 Frequency Translation without Multiplication......Page 490
13.2 High-Speed Vector-Magnitude Approximation......Page 498
13.3 Frequency-Domain Windowing......Page 503
13.4 Fast Multiplication of Complex Numbers......Page 506
13.5 Efficiently Performing the FFT of Real Sequences......Page 507
13.6 Computing the Inverse FFT Using the Forward FFT......Page 519
13.8 Reducing A/D Converter Quantization Noise......Page 522
13.9 A/D Converter Testing Techniques......Page 529
13.10 Fast FIR Filtering Using the FFT......Page 534
13.11 Generating Normally Distributed Random Data......Page 535
13.12 Zero-Phase Filtering......Page 537
13.13 Sharpened FIR Filters......Page 538
13.14 Interpolating a Bandpass Signal......Page 540
13.15 Spectral Peak Location Algorithm......Page 542
13.16 Computing FFT Twiddle Factors......Page 544
13.17 Single Tone Detection......Page 547
13.18 The Sliding DFT......Page 551
13.19 The Zoom FFT......Page 560
13.20 A Practical Spectrum Analyzer......Page 563
13.21 An Efficient Arctangent Approximation......Page 566
13.22 Frequency Demodulation Algorithms......Page 568
13.23 DC Removal......Page 571
13.24 Improving Traditional CIC Filters......Page 575
13.25 Smoothing Impulsive Noise......Page 580
13.26 Efficient Polynomial Evaluation......Page 582
13.27 Designing Very High-Order FIR Filters......Page 583
13.28 Time-Domain Interpolation Using the FFT......Page 587
13.30 Automatic Gain Control (AGC)......Page 590
13.31 Approximate Envelope Detection......Page 593
13.32 A Quadrature Oscillator......Page 595
13.33 Dual-Mode Averaging......Page 597
A.1 Graphical Representation of Real and Complex Numbers......Page 604
A.2 Arithmetic Representation of Complex Numbers......Page 605
A.3 Arithmetic Operations of Complex Numbers......Page 607
A.4 Some Practical Implications of Using Complex Numbers......Page 612
APPENDIX B. CLOSED FORM OF A GEOMETRIC SERIES......Page 614
APPENDIX C. TIME REVERSAL AND THE DFT......Page 618
D.1 Statistical Measures......Page 622
D.2 Standard Deviation, or RMS, of a Continuous Sinewave......Page 625
D.3 The Mean and Variance of Random Functions......Page 626
D.4 The Normal Probability Density Function......Page 629
E.1 Using Logarithms to Determine Relative Signal Power......Page 632
E.2 Some Useful Decibel Numbers......Page 636
E.3 Absolute Power Using Decibels......Page 638
APPENDIX F. DIGITAL FILTER TERMINOLOGY......Page 640
G.1 Frequency Response of a Comb Filter......Page 652
G.2 Single Complex FSF Frequency Response......Page 653
G.3 Multisection Complex FSF Phase......Page 654
G.4 Multisection Complex FSF Frequency Response......Page 655
G.5 Real FSF Transfer Function......Page 657
G.6 Type-IV FSF Frequency Response......Page 659
APPENDIX H. FREQUENCY SAMPLING FILTER DESIGN TABLES......Page 662
B......Page 676
D......Page 677
F......Page 678
I......Page 680
O......Page 681
S......Page 682
T......Page 683
Z......Page 684
ABOUT THE AUTHOR......Page 686