About the Author xxiii

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

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

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 DFT 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

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

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

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

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

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

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

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

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

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

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 FFT 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 FFT 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 DFT 741

13.19 The Zoom FFT 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 FFT 778

13.29 Frequency Translation Using Decimation 781

13.30 Automatic Gain Control (AGC) 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 FFT 831

13.50 Improved Narrowband Lowpass IIR Filters 833

13.51 A Stable Goertzel Algorithm 838

References 840

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

Appendix B: Closed Form of a Geometric Series 859

Appendix C: Time Reversal and the DFT 863

Appendix D: Mean, Variance, and Standard Deviation 867

D.1...