Signals, Systems, Transforms, and Digital Signal Processing with MATLAB

Cover......Page 1
Title Page......Page 4
16.1 Nonparametric Methods of Power Spectrum Estimation......Page 5
Contents......Page 8
Preface......Page 26
Acknowledgment......Page 28
1 Continuous-Time and Discrete-Time Signals and Systems......Page 30
1.2 Continuous-Time Signals......Page 31
1.3 Periodic Functions......Page 32
1.4 Unit Step Function......Page 33
1.5 Graphical Representation of Functions......Page 34
1.6 Even and Odd Parts of a Function......Page 35
1.7 Dirac-Delta Impulse......Page 36
1.8 Basic Properties of the Dirac-Delta Impulse......Page 37
1.10 Continuous-Time Systems......Page 40
1.12 Examples of Electrical Continuous-Time Systems......Page 41
1.13 Mechanical Systems......Page 42
1.14 Transfer Function and Frequency Response......Page 43
1.15 Convolution and Correlation......Page 44
1.16 A Right-Sided and a Left-Sided Function......Page 49
1.18 Additional Convolution Properties......Page 50
1.20 Properties of the Correlation Function......Page 51
1.21 Graphical Interpretation......Page 52
1.23 Average, Energy and Power of Continuous-Time Signals......Page 54
1.24 Discrete-Time Signals......Page 55
1.25 Periodicity......Page 56
1.27 Even/Odd Decomposition......Page 57
1.28 Average Value, Energy and Power Sequences......Page 58
1.30 Problems......Page 59
1.31 Answers to Selected Problems......Page 69
2.1 Trigonometric Fourier Series......Page 76
2.2 Exponential Fourier Series......Page 77
2.3 Exponential versus Trigonometric Series......Page 79
2.4 Periodicity of Fourier Series......Page 80
2.5 Dirichlet Conditions and Function Discontinuity......Page 82
2.7 Analysis Interval versus Function Period......Page 84
2.8 Fourier Series as a Discrete-Frequency Spectrum......Page 85
2.10.1 Linearity......Page 87
2.10.3 Frequency Shift......Page 89
2.10.5 Reflection......Page 90
2.10.6 Symmetry......Page 93
2.10.7 Half-Periodic Symmetry......Page 94
2.10.8 Double Symmetry......Page 96
2.10.9 Time Scaling......Page 99
2.10.10 Differentiation Property......Page 101
2.11.1 Multiplication in the Time Domain......Page 103
2.11.3 Integration......Page 104
2.12 Fourier Series of an Impulse Train......Page 106
2.13 Expansion into Cosine or Sine Fourier Series......Page 107
2.14 Deducing a Function Form from Its Expansion......Page 110
2.15 Truncated Sinusoid Spectral Leakage......Page 112
2.16 The Period of a Composite Sinusoidal Signal......Page 115
2.17 Passage through a Linear System......Page 117
2.18 Parseval’s Relations......Page 118
2.19 Use of Power Series Expansion......Page 119
2.20 Inverse Fourier Series......Page 120
2.21 Problems......Page 121
2.22 Answers to Selected Problems......Page 129
3.2 Bilateral Laplace Transform......Page 134
3.3 Conditions of Existence of Laplace Transform......Page 136
3.4 Basic Laplace Transforms......Page 139
3.5 Notes on the ROC of Laplace Transform......Page 141
3.6 Properties of Laplace Transform......Page 144
3.6.3 Multiplication by Powers of Time......Page 145
3.6.5 Integration in Time......Page 146
3.6.7 Time Scaling......Page 147
3.6.10 Final Value Theorem......Page 148
3.6.11 Laplace Transform of Anticausal Functions......Page 149
3.6.12 Shift in Time......Page 150
3.7 Applications of the Differentiation Property......Page 151
3.8 Transform of Right-Sided Periodic Functions......Page 152
3.9 Convolution in Laplace Domain......Page 153
3.10 Cauchy’s Residue Theorem......Page 154
3.11 Inverse Laplace Transform......Page 157
3.12 Case of Conjugate Poles......Page 158
3.13 The Expansion Theorem of Heaviside......Page 160
3.14 Application to Transfer Function and Impulse Response......Page 161
3.15 Inverse Transform by Differentiation and Integration......Page 162
3.16 Unilateral Laplace Transform......Page 163
3.16.1 Differentiation in Time......Page 164
3.16.4 Division by Time Property......Page 166
3.17 Gamma Function......Page 167
3.18 Table of Additional Laplace Transforms......Page 170
3.19 Problems......Page 172
3.20 Answers to Selected Problems......Page 178
4.1 Definition of the Fourier Transform......Page 182
4.2 Fourier Transform as a Function of f......Page 184
4.3 From Fourier Series to Fourier Transform......Page 185
4.4 Conditions of Existence of the Fourier Transform......Page 186
4.5 Table of Properties of the Fourier Transform......Page 187
4.5.1 Linearity......Page 188
4.5.2 Duality......Page 189
4.5.6 Frequency Shift......Page 190
4.5.7 Modulation Theorem......Page 191
4.5.9 Initial Frequency Value......Page 192
4.5.12 Integration in Time......Page 193
4.5.14 Real Functions......Page 194
4.6 System Frequency Response......Page 195
4.7 Even–Odd Decomposition of a Real Function......Page 196
4.8 Causal Real Functions......Page 197
4.10 Transform of a Complex Exponential and Sinusoid......Page 198
4.11 Sign Function......Page 200
4.14 Table of Fourier Transforms of Basic Functions......Page 201
4.15 Relation between Fourier and Laplace Transforms......Page 203
4.16 Relation to Laplace Transform with Poles on Imaginary Axis......Page 204
4.17 Convolution in Time......Page 205
4.18 Linear System Input–Output Relation......Page 206
4.20 Parseval’s Theorem......Page 207
4.21 Energy Spectral Density......Page 208
4.22 Average Value versus Fourier Transform......Page 209
4.23 Fourier Transform of a Periodic Function......Page 210
4.25 Fourier Transform of Powers of Time......Page 211
4.27 Stability of a Linear System......Page 212
4.29 Transform of a Train of Rectangles......Page 213
4.30 Fourier Transform of a Truncated Sinusoid......Page 214
4.31 Gaussian Function Laplace and Fourier Transform......Page 215
4.32 Inverse Transform by Series Expansion......Page 216
4.33 Fourier Transform in ω and f......Page 217
4.34 Fourier Transform of the Correlation Function......Page 218
4.35 Ideal Filters Impulse Response......Page 219
4.37 Ideal Sampling......Page 220
4.38 Reconstruction of a Signal from its Samples......Page 222
4.39.1 Natural Sampling......Page 224
4.39.2 Instantaneous Sampling......Page 226
4.40 Ideal Sampling of a Bandpass Signal......Page 229
4.41 Sampling an Arbitrary Signal......Page 230
4.42 Sampling the Fourier Transform......Page 232
4.43 Problems......Page 233
4.44 Answers to Selected Problems......Page 251
5.2 Block Diagram Reduction......Page 262
5.3 Galvanometer......Page 263
5.4 DC Motor......Page 266
5.5 A Speed-Control System......Page 268
5.6 Homology......Page 274
5.7 Transient and Steady-State Response......Page 276
5.9 First Order System......Page 277
5.10 Second Order System Model......Page 278
5.11 Settling Time......Page 279
5.12 Second Order System Frequency Response......Page 282
5.13 Case of a Double Pole......Page 283
5.15 Evaluation of the Overshoot......Page 284
5.16 Causal System Response to an Arbitrary Input......Page 285
5.17 System Response to a Causal Periodic Input......Page 286
5.18 Response to a Causal Sinusoidal Input......Page 288
5.20 Decibels, Octaves, Decades......Page 289
5.21.1 A Simple Zero at the Origin......Page 290
5.21.3 A Simple Zero in the Left Plane......Page 291
5.21.5 Second Order System......Page 293
5.22 Bode Plot of a Composite Linear System......Page 296
5.23 Graphical Representation of a System Function......Page 297
5.24 Vectorial Evaluation of Residues......Page 298
5.25 Vectorial Evaluation of the Frequency Response......Page 302
5.27 Filtering Properties of Basic Circuits......Page 304
5.28 Lowpass First Order Filter......Page 306
5.29 Minimum Phase Systems......Page 309
5.30 General Order All-Pass Systems......Page 310
5.31 Signal Generation......Page 312
5.32 Application of Laplace Transform to Differential Equations......Page 313
5.32.2 Linear First Order Differential Equation......Page 314
5.32.3 General Order Differential Equations with Constant Coeffcients......Page 315
5.32.4 Homogeneous Linear Differential Equations......Page 316
5.32.5 The General Solution of a Linear Differential Equation......Page 317
5.32.6 Partial Differential Equations......Page 320
5.33 Transformation of Partial Differential Equations......Page 322
5.34 Problems......Page 326
5.35 Answers to Selected Problems......Page 343
6.1 Introduction......Page 352
6.3 Linear Constant-Coeffcient Difference Equations......Page 353
6.4 The z-Transform......Page 354
6.5 Convergence of the z-Transform......Page 356
6.6 Inverse z-Transform......Page 359
6.7 Inverse z-Transform by Partial Fraction Expansion......Page 365
6.8 Inversion by Long Division......Page 366
6.9 Inversion by a Power Series Expansion......Page 367
6.10 Inversion by Geometric Series Summation......Page 368
6.12.3 Conjugate Sequence......Page 369
6.12.4 Initial Value......Page 370
6.12.6 Convolution in Frequency......Page 373
6.12.8 Final Value Theorem......Page 376
6.12.10 Frequency Translation......Page 377
6.13 Geometric Evaluation of Frequency Response......Page 378
6.14 Comb Filters......Page 380
6.15 Causality and Stability......Page 382
6.16 Delayed Response and Group Delay......Page 383
6.17 Discrete-Time Convolution and Correlation......Page 384
6.18 Discrete-Time Correlation in One Dimension......Page 386
6.19 Convolution and Correlation as Multiplications......Page 389
6.21 Notes on the Cross-Correlation of Sequences......Page 390
6.22 LTI System Input/Output Correlation Sequences......Page 391
6.24 Two-Dimensional Signals......Page 392
6.25 Linear Systems, Convolution and Correlation......Page 395
6.26 Correlation of Two-Dimensional Signals......Page 399
6.27 IIR and FIR Digital Filters......Page 403
6.28 Discrete-Time All-Pass Systems......Page 404
6.29 Minimum-Phase and Inverse System......Page 407
6.30 Unilateral z-Transform......Page 410
6.30.1 Time Shift Property of Unilateral z-Transform......Page 412
6.31 Problems......Page 413
6.32 Answers to Selected Problems......Page 419
7.1 Laplace, Fourier and z-Transform Relations......Page 424
7.3 A/D Conversion......Page 429
7.4 Quantization Error......Page 432
7.5 D/A Conversion......Page 433
7.6 Continuous versus Discrete Signal Processing......Page 435
7.7 Interlacing with Zeros......Page 436
7.8 Sampling Rate Conversion......Page 438
7.8.1 Sampling Rate Reduction......Page 439
7.8.2 Sampling Rate Increase: Interpolation......Page 443
7.8.3 Rational Factor Sample Rate Alteration......Page 446
7.9 Fourier Transform of a Periodic Sequence......Page 448
7.10 Table of Discrete-Time Fourier Transforms......Page 449
7.11 Reconstruction of the Continuous-Time Signal......Page 453
7.14 Parseval’s Theorem......Page 454
7.15 Fourier Series and Transform Duality......Page 455
7.16 Discrete Fourier Transform......Page 458
7.17 Discrete Fourier Series......Page 462
7.18 DFT of a Sinusoidal Signal......Page 463
7.19 Deducing the z-Transform from the DFT......Page 465
7.20 DFT versus DFS......Page 467
7.21 Properties of DFS and DFT......Page 468
7.21.1 Periodic Convolution......Page 470
7.22 Circular Convolution......Page 472
7.23 Circular Convolution Using the DFT......Page 474
7.24 Sampling the Spectrum......Page 475
7.25 Table of Properties of DFS......Page 476
7.26 Shift in Time and Circular Shift......Page 477
7.27 Table of DFT Properties......Page 478
7.28 Zero Padding......Page 479
7.29 Discrete z-Transform......Page 482
7.30 Fast Fourier Transform......Page 484
7.31 An Algorithm for a Wired-In Radix-2 Processor......Page 491
7.31.1 Post-Permutation Algorithm......Page 493
7.31.2 Ordered Input/Ordered Output (OIOO) Algorithm......Page 494
7.32 Factorization of the FFT to a Higher Radix......Page 495
7.32.1 Ordered Input/Ordered Output General Radix FFT Algorithm......Page 498
7.33 Feedback Elimination for High-Speed Signal Processing......Page 499
7.34 Problems......Page 501
7.35 Answers to Selected Problems......Page 507
8.2 Note on Notation......Page 512
8.3 State Space Model......Page 513
8.4 System Transfer Function......Page 517
8.5 System Response with Initial Conditions......Page 518
8.6 Jordan Canonical Form of State Space Model......Page 519
8.7 Eigenvalues and Eigenvectors......Page 526
8.8 Matrix Diagonalization......Page 527
8.9 Similarity Transformation of a State Space Model......Page 528
8.10 Solution of the State Equations......Page 530
8.11 General Jordan Canonical Form......Page 536
8.12 Circuit Analysis by Laplace Transform and State Variables......Page 538
8.13 Trajectories of a Second Order System......Page 542
8.14 Second Order System Modeling......Page 544
8.15 Transformation of Trajectories between Planes......Page 548
8.16 Discrete-Time Systems......Page 551
8.18 Transfer Function......Page 557
8.19 Change of Variables......Page 558
8.20 Second Canonical Form State Space Model......Page 560
8.21 Problems......Page 562
8.22 Answers to Selected Problems......Page 567
9.1 Lowpass Approximation......Page 572
9.2 Butterworth Approximation......Page 573
9.3 Denormalization of Butterworth Filter Prototype......Page 576
9.4 Denormalized Transfer Function......Page 579
9.5 The Case ε ≠ 1......Page 581
9.6 Butterworth Filter Order Formula......Page 582
9.7 Nomographs......Page 583
9.8 Chebyshev Approximation......Page 585
9.10 Transfer Function of the Chebyshev Filter......Page 589
9.11 Maxima and Minima of Chebyshev Filter Response......Page 592
9.13 Evaluation of Chebyshev Filter Gain......Page 593
9.14 Chebyshev Filter Tables......Page 594
9.15 Chebyshev Filter Order......Page 596
9.16 Denormalization of Chebyshev Filter Prototype......Page 597
9.17 Chebyshev’s Approximation: Second Form......Page 600
9.18 Response Decay of Butterworth and Chebyshev Filters......Page 601
9.19 Chebyshev Filter Nomograph......Page 604
9.20.1 Elliptic Integral......Page 605
9.21 Properties, Poles and Zeros of the sn Function......Page 606
9.21.1 Elliptic Filter Approximation......Page 609
9.22 Pole Zero Alignment and Mapping of Elliptic Filter......Page 613
9.23 Poles of H(s)......Page 618
9.26 Points of Maxima/Minima......Page 620
9.27 Elliptic Filter Nomograph......Page 621
9.28 N = 9 Example......Page 626
9.29 Tables of Elliptic Filters......Page 628
9.30 Bessel’s Constant Delay Filters......Page 640
9.31 A Note on Continued Fraction Expansion......Page 641
9.32 Evaluating the Filter Delay......Page 646
9.33 Bessel Filter Quality Factor and Natural Frequency......Page 647
9.34 Maximal Flatness of Bessel and Butterworth Response......Page 648
9.36 Denormalization and Deviation from Ideal Response......Page 651
9.38 Bessel Filter’s Butterworth Asymptotic Form......Page 655
9.39 Delay of Bessel–Butterworth Asymptotic Form Filter......Page 657
9.40 Delay Plots of Butterworth Asymptotic Form Bessel Filter......Page 658
9.41 Bessel Filters Frequency Normalized Form......Page 662
9.43 Response and Delay of Normalized Form Bessel Filter......Page 663
9.44 Bessel Frequency Normalized Form Attenuation Setting......Page 664
9.46 Frequency Transformations......Page 668
9.47 Lowpass to Bandpass Transformation......Page 670
9.48 Lowpass to Band-Stop Transformation......Page 680
9.49 Lowpass to Highpass Transformation......Page 682
9.50 Note on Lowpass to Normalized Band-Stop Transformation......Page 686
9.51 Windows......Page 690
9.52 Rectangular Window......Page 691
9.54 Hanning Window......Page 692
9.55 Hamming Window......Page 693
9.56 Problems......Page 694
9.57 Answers to Selected Problems......Page 700
10.2 Design of Passive Ladder Lowpass Filters......Page 706
10.3 Analysis of a General Order Passive Ladder Network......Page 709
10.4 Input Impedance of a Single-Resistance Terminated Network......Page 712
10.5 Evaluation of the Ladder Network Components......Page 713
10.6 Matrix Evaluation of Input Impedance......Page 718
10.7 Bessel Filter Passive Ladder Networks......Page 722
10.8 Tables of Single-Resistance Ladder Network Components......Page 723
10.9.1 Input Impedance Evaluation......Page 724
10.10 Tables of Double-Resistance Terminated Ladder Network Components......Page 730
10.11 Closed Forms for Circuit Element Values......Page 732
10.12 Elliptic Filter Realization as a Passive Ladder Network......Page 735
10.12.1 Evaluating the Elliptic LC Ladder Circuit Elements......Page 736
10.14 Element Replacement for Frequency Transformation......Page 738
10.14.1 Lowpass to Bandpass Transformation......Page 739
10.14.3 Lowpass to Band-Stop Transformation......Page 740
10.16 Inverting Integrator......Page 742
10.17 Biquadratic Transfer Functions......Page 743
10.18 General Biquad Realization......Page 745
10.19 First Order Filter Realization......Page 750
10.20 A Biquadratic Transfer Function Realization......Page 752
10.21 Sallen–Key Circuit......Page 754
10.22 Problems......Page 757
10.23 Answers to Selected Problems......Page 758
11.2 Signal Flow Graphs......Page 762
11.5 Transposition......Page 763
11.6 Second Canonical Form......Page 765
11.7 Transposition of the Second Canonical Form......Page 766
11.9 Cascaded Form......Page 767
11.11 Matrix Representation......Page 768
11.12 Finite Impulse Response (FIR) Filters......Page 769
11.13 Linear Phase FIR Filters......Page 770
11.15 Impulse Invariance Approach......Page 772
11.16 Shortcut Impulse Invariance Design......Page 775
11.17 Backward-Rectangular Approximation......Page 776
11.18 Forward Rectangular and Trapezoidal Approximations......Page 778
11.19 Bilinear Transform......Page 780
11.21 Finite Impulse Response All-Zero Lattice Structures......Page 789
11.22 One-Zero FIR Filter......Page 790
11.23 Two-Zeros FIR Filter......Page 791
11.24 General Order All-Zero FIR Filter......Page 793
11.25 All-Pole Filter......Page 798
11.26 First Order One-Pole Filter......Page 799
11.27 Second Order All-Pole Filter......Page 800
11.28 General Order All-Pole Filter......Page 801
11.29 Pole-Zero IIR Lattice Filter......Page 804
11.30 All-Pass Filter Realization......Page 810
11.31 Schur–Cohn Stability Criterion......Page 811
11.32 Frequency Transformations......Page 812
11.34 Padé Approximation......Page 815
11.35 Error Minimization in Prony’s Method......Page 819
11.36 FIR Inverse Filter Design......Page 823
11.37 Impulse Response of Ideal Filters......Page 827
11.38 Spectral Leakage......Page 829
11.40 Ideal Digital Filters Rectangular Window......Page 830
11.41 Hanning Window......Page 831
11.42 Hamming Window......Page 832
11.43 Triangular Window......Page 833
11.44 Comparison of Windows Spectral Parameters......Page 834
11.45 Linear-Phase FIR Filter Design Using Windows......Page 836
11.46 Even- and Odd-Symmetric FIR Filter Design......Page 837
11.48 Sampling the Unit Circle......Page 839
11.49.1 Case I-1: Odd Order, Even Symmetry, µ = 0......Page 843
11.49.4 Case II-2: Even Order, Even Symmetry, µ = 1/2......Page 844
11.49.8 Case IV-2: Even Order, Odd Symmetry, µ = 1/2......Page 845
11.50 Problems......Page 846
11.51 Answers to Selected Problems......Page 857
12.1 Energy Spectral Density......Page 864
12.2 Average, Energy and Power of Continuous-Time Signals......Page 867
12.3 Discrete-Time Signals......Page 868
12.5 Autocorrelation of Energy Signals......Page 869
12.6 Energy Signal through Linear System......Page 871
12.7 Impulsive and Discrete-Time Energy Signals......Page 872
12.9 Cross-Correlation......Page 877
12.9.1 Power Spectral Density......Page 878
12.10 Power Spectrum Conversion of a Linear System......Page 879
12.11 Impulsive and Discrete-Time Power Signals......Page 881
12.12 Periodic Signals......Page 883
12.12.1 Response of an LTI System to a Sinusoidal Input......Page 884
12.13 Power Spectral Density of an Impulse Train......Page 885
12.14 Average, Energy and Power of a Sequence......Page 888
12.17 Power Density of a Sequence......Page 889
12.19 Problems......Page 890
12.20 Answers to Selected Problems......Page 898
13.1 Introduction......Page 904
13.2.1 Double Side-Band (DSB) Modulation......Page 905
13.2.2 Double Side-Band Suppressed Carrier (DSB-SC) Modulation......Page 906
13.2.3 Single Side-Band (SSB) Modulation......Page 908
13.2.5 Frequency Multiplexing......Page 911
13.3 Frequency Modulation......Page 912
13.4.1 Pulse Modulation Systems......Page 916
13.5.1 Pulse Code Modulation......Page 917
13.5.2 Pulse Duration Modulation......Page 919
13.5.3 Pulse Position Modulation......Page 921
13.7 Frequency Division Multiplexing (FDM)......Page 922
13.8 Problems......Page 923
13.9 Answers to Selected Problems......Page 933
14.2 Rademacher and Haar Functions......Page 940
14.3 Walsh Functions......Page 941
14.4 The Walsh (Sequency) Order......Page 942
14.6 Natural (Hadamard) Order......Page 943
14.7 Discrete Walsh Transform......Page 945
14.9.1 Natural (Hadamard) Order......Page 946
14.9.2 Dyadic or Paley Order......Page 947
14.10 Natural (Hadamard) Order Fast Walsh–Hadamard Transform......Page 948
14.11 Dyadic (Paley) Order Fast Walsh–Hadamard Transform......Page 949
14.12 Sequency Ordered Fast Walsh–Hadamard Transform......Page 950
14.14 Natural Order......Page 951
14.17 Walsh–Kaczmarz Transform......Page 952
14.19 Generalized Walsh Natural Order GWN Matrix......Page 953
14.20 Generalized Walsh–Paley GWP Transformation Matrix......Page 954
14.23 GWN Optimal Factorization......Page 955
14.25 GWK Optimal Factorization......Page 956
14.26 Karhunen Loève Transform......Page 957
14.27 Hilbert Transform......Page 960
14.28 Hilbert Transformer......Page 963
14.29 Discrete Hilbert Transform......Page 964
14.30 Hartley Transform......Page 965
14.31 Discrete Hartley Transform......Page 967
14.32 Mellin Transform......Page 968
14.33 Mellin Transform of e[sup(jx)]......Page 970
14.34 Hankel Transform......Page 972
14.35 Fourier Cosine Transform......Page 974
14.36 Discrete Cosine Transform (DCT)......Page 975
14.37 Fractional Fourier Transform......Page 977
14.39 Two-Dimensional Transforms......Page 979
14.40 Two-Dimensional Fourier Transform......Page 980
14.42 H[sub(I)] (jω) versus H[sub(R)](jω) with No Poles on Axis......Page 982
14.43 Case of Poles on the Imaginary Axis......Page 986
14.44 Hilbert Transform Closed Forms......Page 987
14.45 Wiener–Lee Transforms......Page 988
14.46 Discrete-Time Domain Hilbert Transform Relations......Page 990
14.47 Problems......Page 993
14.48 Answers to Selected Problems......Page 996
15.2 Systems for the Representation of Numbers......Page 1002
15.4 Integers, Fractions and the Binary Point......Page 1003
15.5.1 Sign and Magnitude Notation......Page 1004
15.5.2 1’s and 2’s Complement Notation......Page 1005
15.6 Integer and Fractional Representation of Signed Numbers......Page 1007
15.6.1 1’s and 2’s Complement of Signed Numbers......Page 1008
15.7.1 Addition in Sign and Magnitude Notation......Page 1011
15.7.2 Addition in 1’s Complement Notation......Page 1013
15.7.3 Addition in 2’s Complement Notation......Page 1014
15.8 Subtraction......Page 1015
15.8.1 Subtraction in Sign and Magnitude Notation......Page 1016
15.8.2 Numbers in 1’s Complement Notation......Page 1017
15.8.3 Subtraction in 2’s Complement Notation......Page 1018
15.9 Full Adder Cell......Page 1019
15.10 Addition/Subtraction Implementation in 2’s Complement......Page 1020
15.12 Multiplication of Unsigned Numbers......Page 1021
15.13 Multiplier Implementation......Page 1022
15.14 3-D Multiplier......Page 1024
15.14.2 Multiplication in 1’s Complement Notation......Page 1026
15.14.3 Numbers in 2’s Complement Notation......Page 1027
15.15 A Direct Approach to 2’s Complement Multiplication......Page 1029
15.16 Division......Page 1031
15.16.1 Division of Positive Numbers:......Page 1032
15.16.3 Division in 1’s Complement......Page 1033
15.16.4 Division in 2’s Complement......Page 1034
15.16.5 Nonrestoring Division......Page 1035
15.17 Cellular Array for Nonrestoring Division......Page 1038
15.18 Carry Look Ahead (CLA) Cell......Page 1040
15.19 2’s Complement Nonrestoring Division......Page 1043
15.20 Convergence Division......Page 1045
15.21 Evaluation of the n[sup(th)] Root......Page 1047
15.22 Function Generation by Chebyshev Series Expansion......Page 1049
15.23 An Alternative Approach to Chebyshev Series Expansion......Page 1055
15.24 Floating Point Number Representation......Page 1056
15.24.2 Multiplication......Page 1058
15.25.1 The Paper and Pencil Method......Page 1059
15.25.3 Comparison Approach......Page 1060
15.25.5 Nonrestoring Approach......Page 1061
15.27 Binary Coded Decimal (BCD) Representation......Page 1062
15.28 Memory Elements......Page 1066
15.28.1 Set-Reset (SR) Flip-Flop......Page 1067
15.28.3 The JK Flip-Flop......Page 1069
15.28.4 Master-Slave Flip-Flop......Page 1070
15.29 Design of Synchronous Sequential Circuits......Page 1071
15.29.1 Realization Using SR Flip-Flops......Page 1073
15.29.2 Realization Using JK Flip-Flops......Page 1074
15.30.1 Realization Using JK Flip-Flops......Page 1075
15.31 State Minimization......Page 1077
15.32 Asynchronous Sequential Machines......Page 1079
15.33 State Reduction......Page 1080
15.34 Control Counter Design for Generator of Prime Numbers......Page 1083
15.34.1 Micro-operations and States......Page 1084
15.35 Fast Transform Processors......Page 1088
15.36 Programmable Logic Arrays (PLAs)......Page 1091
15.37 Field Programmable Gate Arrays (FPGAs)......Page 1092
15.38 DSP with Xilinx FPGAs......Page 1094
15.39 Texas Instruments TMS320C6713B Floating-Point DSP......Page 1096
15.40 Central Processing Unit (CPU)......Page 1098
15.41.1 General-Purpose Register Files......Page 1100
15.41.3 Register File Cross Paths......Page 1101
15.41.5 Data Address Paths......Page 1102
15.43 TMS320C6000 Control Register File......Page 1103
15.44 Addressing Mode Register (AMR)......Page 1104
15.45 Syntax for Load/Store Address Generation......Page 1105
15.45.1 Linear Addressing Mode......Page 1106
15.46 Programming the T.I. DSP......Page 1107
15.47 A Simple C Program......Page 1108
15.48 The Generated Assembly Code......Page 1109
15.48.1 Calling an Assembly Language Function......Page 1112
15.50 Finite Impulse Response (FIR) Filter......Page 1116
15.51 Infinite Impulse Response (IIR) Filter on the DSP......Page 1117
15.52 Real-Time DSP Applications Using MATLAB–Simulink......Page 1121
15.53.1 Steps to Implement a C++ Program on the DSP Card......Page 1123
15.53.2 Steps to Implement a Simulink Program on the DSP Card......Page 1125
15.54 Problems......Page 1127
15.55 Answers to Selected Problems......Page 1130
16 Random Signal Processing......Page 1134
16.2 Correlation of Continuous-Time Random Signals......Page 1138
16.3 Passage through an LTI System......Page 1139
16.4 Wiener Filtering in Continuous-Time Domain......Page 1142
16.5 Causal Wiener Filter......Page 1145
16.6 Random Sequences......Page 1147
16.7 From Statistical to Time Averages......Page 1148
16.8 Correlation and Covariance in z-Domain......Page 1149
16.9 Random Signal Passage through an LTI System......Page 1150
16.10 PSD Estimation of Discrete-Time Random Sequences......Page 1153
16.11 Fast Fourier Transform (FFT) Evaluation of the Periodogram......Page 1157
16.12 Parametric Methods for PSD Estimation......Page 1160
16.13 The Yule–Walker Equations......Page 1161
16.15 Wiener and Least-Squares Models......Page 1163
16.16 Wiener Filtering......Page 1164
16.18 Forward Linear Prediction......Page 1167
16.19 Backward Linear Prediction......Page 1169
16.20 Lattice MA FIR Filter Realization......Page 1172
16.22 ARMA(p,q) Process......Page 1175
16.23 Power Spectrum Estimation......Page 1176
16.24 FIR Wiener Filtering of Noisy Signals......Page 1177
16.25 Two-Sided IIR Wiener Filtering......Page 1180
16.26 Causal IIR Wiener Filter......Page 1181
16.27 Wavelet Transform......Page 1183
16.28 Discrete Wavelet Transform......Page 1186
16.29 Important Signal Processing MATLAB Functions......Page 1193
16.30 lpc......Page 1196
16.31 Yulewalk......Page 1197
16.32 dfilt......Page 1198
16.34 FIR Filter Design......Page 1199
16.35 fir2......Page 1202
16.37 Parametric Modeling Functions......Page 1203
16.38 prony......Page 1204
16.39 Problems......Page 1205
16.40 Answers to Selected Problems......Page 1208
17.2 Distributions as Generalizations of Functions......Page 1210
17.3 What is a Distribution?......Page 1211
17.5.1 Linearity......Page 1213
17.5.3 Time Scaling......Page 1214
17.5.5 Symmetry......Page 1215
17.6 Approximating the Impulse......Page 1216
17.7 Other Approximating Sequences and Functions of the Impulse......Page 1219
17.8 Test Functions......Page 1220
17.9 Convolution......Page 1221
17.10 Multiplication by an Impulse Derivative......Page 1222
17.11 The Dirac-Delta Impulse as a Limit of a Gaussian Function......Page 1224
17.13 The Impulse of a Function......Page 1225
17.15 Time Scaling......Page 1228
17.16 Some Properties of the Dirac-Delta Impulse......Page 1229
17.18 Riemann–Lebesgue Lemma......Page 1230
17.19 Generalized Limits......Page 1231
17.21 The Distribution t[sup(-k)]......Page 1233
17.22 Initial Derivatives of the Transform......Page 1235
17.23 The Unit Step Function as a Limit......Page 1236
17.24 Inverse Fourier Transform and Gibbs Phenomenon......Page 1237
17.25 Ripple Elimination......Page 1241
17.26 Transforms of |t| and tu(t)......Page 1242
17.27 The Impulse Train as a Limit......Page 1243
17.28 Sequence of Distributions......Page 1245
17.29 Poisson’s Summation Formula......Page 1247
17.30 Moving Average......Page 1248
17.31 Problems......Page 1249
17.32 Answers to Selected Problems......Page 1251
18.2 An Anomaly......Page 1254
18.3.3 Shift in s......Page 1255
18.3.6 Differentiation......Page 1256
18.4.3 Convolution......Page 1257
18.4.4 Convolution with an Ordinary Function......Page 1258
18.5 Additional Generalized Impulse Properties......Page 1259
18.6 Generalized Impulse as a Limit of a Three-Dimensional Sequence......Page 1261
18.7 Discrete-Time Domain......Page 1263
18.8 3-D Test Function as a Possible Generalization......Page 1264
18.8.4 Differentiation......Page 1265
18.9.1 Differentiation......Page 1266
18.10 Generalized Impulse as Limit of a 3-D Sequence......Page 1267
18.10.1 Convolution of Generalized Impulses......Page 1269
18.10.2 Convolution with an Ordinary Function......Page 1270
18.12 Generalization of Fourier-, Laplace- and z-Related Transforms......Page 1271
18.13 Hilbert Transform Generalization......Page 1274
18.15 Generalized Hartley Transform......Page 1276
18.16 Generalized Discrete Hartley Transform......Page 1277
18.17 Generalization of the Mellin Transform......Page 1278
18.18 Multidimensional Signals and the Solution of Differential Equations......Page 1280
18.19 Problems......Page 1283
18.20 Answers to Selected Problems......Page 1284
A.2 Frequently Needed Expansions......Page 1286
A.4 Orthogonality Relations......Page 1288
A.6 Mathematical Formulae......Page 1289
A.7 Frequently Encountered Series Sums......Page 1290
A.9 Plato (428 BC–347 BC)......Page 1291
A.10 Ptolemy (circa 90–168 AD)......Page 1293
A.11 Euclid (circa 300 BC)......Page 1294
A.12 Abu Ja’far Muhammad ibn Musa Al-Khwarizmi (780–850 AD)......Page 1295
A.13 Nicolaus Copernicus (1473–1543)......Page 1298
A.14 Galileo Galilei (1564–1642)......Page 1301
A.15 Sir Isaac Newton (1643–1727)......Page 1303
A.16 Guillaume-François-Antoine de L’Hôpital (1661–1704)......Page 1307
A.17 Pierre-Simon Laplace (1749–1827)......Page 1308
A.18 Gaspard Clair François Marie, Baron Riche de Prony (1755–1839)......Page 1310
A.19 Jean Baptiste Joseph Fourier (1768–1830)......Page 1314
A.20 Johann Carl Friedrich Gauss (1777–1855)......Page 1318
A.21 Friedrich Wilhelm Bessel (1784–1846)......Page 1319
A.22 Augustin-Louis Cauchy (1789–1857)......Page 1321
A.23 Niels Henrik Abel (1802–1829)......Page 1324
A.24 Johann Peter Gustav Lejeune Dirichlet (1805–1859)......Page 1326
A.25 Pafnuty Lvovich Chebyshev (1821–1894)......Page 6
A.26 Paul A.M. Dirac......Page 1329
References......Page 1332
Index......Page 1336