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 |  موضوع: كتاب Signals and Systems with MATLAB الأحد 17 مارس 2024, 4:03 am | |
| 
أخواني في الله
أحضرت لكم كتاب
Signals and Systems with MATLAB
Won Y. Yang · Tae G. Chang · Ik H. Song · Yong S. Cho · Jun Heo · Won G. Jeon · Jeong W. Lee · Jae K. Kim
و المحتوى كما يلي :
Contents
1 Signals and Systems . 1
1.1 Signals . 2
1.1.1 Various Types of Signal 2
1.1.2 Continuous/Discrete-Time Signals . 2
1.1.3 Analog Frequency and Digital Frequency 6
1.1.4 Properties of the Unit Impulse Function
and Unit Sample Sequence . 8
1.1.5 Several Models for the Unit Impulse Function 11
1.2 Systems 12
1.2.1 Linear System and Superposition Principle 13
1.2.2 Time/Shift-Invariant System . 14
1.2.3 Input-Output Relationship of Linear
Time-Invariant (LTI) System 15
1.2.4 Impulse Response and System (Transfer) Function 17
1.2.5 Step Response, Pulse Response, and Impulse Response 18
1.2.6 Sinusoidal Steady-State Response
and Frequency Response . 19
1.2.7 Continuous/Discrete-Time Convolution . 22
1.2.8 Bounded-Input Bounded-Output (BIBO) Stability 29
1.2.9 Causality . 30
1.2.10 Invertibility . 30
1.3 Systems Described by Differential/Difference Equations . 31
1.3.1 Differential/Difference Equation and System Function . 31
1.3.2 Block Diagrams and Signal Flow Graphs 32
1.3.3 General Gain Formula – Mason’s Formula . 34
1.3.4 State Diagrams 35
1.4 Deconvolution and Correlation . 38
1.4.1 Discrete-Time Deconvolution 38
1.4.2 Continuous/Discrete-Time Correlation 39
1.5 Summary . 45
Problems . 45
ixx Contents
2 Continuous-Time Fourier Analysis . 61
2.1 Continuous-Time Fourier Series (CTFS) of Periodic Signals 62
2.1.1 Definition and Convergence Conditions
of CTFS Representation . 62
2.1.2 Examples of CTFS Representation . 65
2.1.3 Physical Meaning of CTFS Coefficients – Spectrum . 70
2.2 Continuous-Time Fourier Transform of Aperiodic Signals 73
2.3 (Generalized) Fourier Transform of Periodic Signals . 77
2.4 Examples of the Continuous-Time Fourier Transform 78
2.5 Properties of the Continuous-Time Fourier Transform 86
2.5.1 Linearity . 86
2.5.2 (Conjugate) Symmetry . 86
2.5.3 Time/Frequency Shifting (Real/Complex Translation) . 88
2.5.4 Duality 88
2.5.5 Real Convolution 89
2.5.6 Complex Convolution (Modulation/Windowing) 90
2.5.7 Time Differential/Integration – Frequency
Multiplication/Division 94
2.5.8 Frequency Differentiation – Time Multiplication 95
2.5.9 Time and Frequency Scaling 95
2.5.10 Parseval’s Relation (Rayleigh Theorem) . 96
2.6 Polar Representation and Graphical Plot of CTFT . 96
2.6.1 Linear Phase 97
2.6.2 Bode Plot 97
2.7 Summary . 98
Problems . 99
3 Discrete-Time Fourier Analysis . 129
3.1 Discrete-Time Fourier Transform (DTFT) 130
3.1.1 Definition and Convergence Conditions of DTFT
Representation . 130
3.1.2 Examples of DTFT Analysis 132
3.1.3 DTFT of Periodic Sequences 136
3.2 Properties of the Discrete-Time Fourier Transform 138
3.2.1 Periodicity . 138
3.2.2 Linearity . 138
3.2.3 (Conjugate) Symmetry . 138
3.2.4 Time/Frequency Shifting (Real/Complex Translation) . 139
3.2.5 Real Convolution 139
3.2.6 Complex Convolution (Modulation/Windowing) 139
3.2.7 Differencing and Summation in Time . 143
3.2.8 Frequency Differentiation . 143
3.2.9 Time and Frequency Scaling 143
3.2.10 Parseval’s Relation (Rayleigh Theorem) . 144Contents xi
3.3 Polar Representation and Graphical Plot of DTFT . 144
3.4 Discrete Fourier Transform (DFT) 147
3.4.1 Properties of the DFT 149
3.4.2 Linear Convolution with DFT . 152
3.4.3 DFT for Noncausal or Infinite-Duration Sequence 155
3.5 Relationship Among CTFS, CTFT, DTFT, and DFT . 160
3.5.1 Relationship Between CTFS and DFT/DFS 160
3.5.2 Relationship Between CTFT and DTFT . 161
3.5.3 Relationship Among CTFS, CTFT, DTFT, and DFT/DFS 162
3.6 Fast Fourier Transform (FFT) 164
3.6.1 Decimation-in-Time (DIT) FFT 165
3.6.2 Decimation-in-Frequency (DIF) FFT . 168
3.6.3 Computation of IDFT Using FFT Algorithm . 169
3.7 Interpretation of DFT Results 170
3.8 Effects of Signal Operations on DFT Spectrum . 178
3.9 Short-Time Fourier Transform – Spectrogram 180
3.10 Summary . 182
Problems . 182
4 The z-Transform 207
4.1 Definition of the z-Transform 208
4.2 Properties of the z-Transform 213
4.2.1 Linearity . 213
4.2.2 Time Shifting – Real Translation . 214
4.2.3 Frequency Shifting – Complex Translation 215
4.2.4 Time Reversal 215
4.2.5 Real Convolution 215
4.2.6 Complex Convolution 216
4.2.7 Complex Differentiation 216
4.2.8 Partial Differentiation 217
4.2.9 Initial Value Theorem 217
4.2.10 Final Value Theorem . 218
4.3 The Inverse z-Transform 218
4.3.1 Inverse z-Transform by Partial Fraction Expansion 219
4.3.2 Inverse z-Transform by Long Division 223
4.4 Analysis of LTI Systems Using the z-Transform . 224
4.5 Geometric Evaluation of the z-Transform 231
4.6 The z-Transform of Symmetric Sequences . 236
4.6.1 Symmetric Sequences 236
4.6.2 Anti-Symmetric Sequences . 237
4.7 Summary . 240
Problems . 240xii Contents
5 Sampling and Reconstruction 249
5.1 Digital-to-Analog (DA) Conversion[J-1] . 250
5.2 Analog-to-Digital (AD) Conversion[G-1, J-2, W-2] 251
5.2.1 Counter (Stair-Step) Ramp ADC . 251
5.2.2 Tracking ADC 252
5.2.3 Successive Approximation ADC . 253
5.2.4 Dual-Ramp ADC 254
5.2.5 Parallel (Flash) ADC . 256
5.3 Sampling . 257
5.3.1 Sampling Theorem 257
5.3.2 Anti-Aliasing and Anti-Imaging Filters . 262
5.4 Reconstruction and Interpolation . 263
5.4.1 Shannon Reconstruction 263
5.4.2 DFS Reconstruction . 265
5.4.3 Practical Reconstruction 267
5.4.4 Discrete-Time Interpolation . 269
5.5 Sample-and-Hold (S/H) Operation 272
5.6 Summary . 272
Problems . 273
6 Continuous-Time Systems and Discrete-Time Systems 277
6.1 Concept of Discrete-Time Equivalent 277
6.2 Input-Invariant Transformation . 280
6.2.1 Impulse-Invariant Transformation 281
6.2.2 Step-Invariant Transformation . 282
6.3 Various Discretization Methods [P-1] 284
6.3.1 Backward Difference Rule on Numerical Differentiation . 284
6.3.2 Forward Difference Rule on Numerical Differentiation 286
6.3.3 Left-Side (Rectangular) Rule on Numerical Integration 287
6.3.4 Right-Side (Rectangular) Rule on Numerical Integration . 288
6.3.5 Bilinear Transformation (BLT) – Trapezoidal Rule on
Numerical Integration . 288
6.3.6 Pole-Zero Mapping – Matched z-Transform [F-1] . 292
6.3.7 Transport Delay – Dead Time . 293
6.4 Time and Frequency Responses of Discrete-Time Equivalents . 293
6.5 Relationship Between s-Plane Poles and z-Plane Poles . 295
6.6 The Starred Transform and Pulse Transfer Function 297
6.6.1 The Starred Transform . 297
6.6.2 The Pulse Transfer Function . 298
6.6.3 Transfer Function of Cascaded Sampled-Data System . 299
6.6.4 Transfer Function of System in A/D-G[z]-D/A Structure . 300
Problems . 301Contents xiii
7 Analog and Digital Filters . 307
7.1 Analog Filter Design . 307
7.2 Digital Filter Design 320
7.2.1 IIR Filter Design 321
7.2.2 FIR Filter Design 331
7.2.3 Filter Structure and System Model Available in MATLAB . 345
7.2.4 Importing/Exporting a Filter Design 348
7.3 How to Use SPTool 350
Problems . 357
8 State Space Analysis of LTI Systems 361
8.1 State Space Description – State and Output Equations 362
8.2 Solution of LTI State Equation . 364
8.2.1 State Transition Matrix . 364
8.2.2 Transformed Solution 365
8.2.3 Recursive Solution . 368
8.3 Transfer Function and Characteristic Equation 368
8.3.1 Transfer Function 368
8.3.2 Characteristic Equation and Roots 369
8.4 Discretization of Continuous-Time State Equation . 370
8.4.1 State Equation Without Time Delay 370
8.4.2 State Equation with Time Delay . 374
8.5 Various State Space Description – Similarity Transformation 376
8.6 Summary . 379
Problems . 379
A The Laplace Transform . 385
A.1 Definition of the Laplace Transform . 385
A.2 Examples of the Laplace Transform . 385
A.2.1 Laplace Transform of the Unit Step Function . 385
A.2.2 Laplace Transform of the Unit Impulse Function . 386
A.2.3 Laplace Transform of the Ramp Function 387
A.2.4 Laplace Transform of the Exponential Function 387
A.2.5 Laplace Transform of the Complex Exponential Function 387
A.3 Properties of the Laplace Transform . 387
A.3.1 Linearity . 388
A.3.2 Time Differentiation . 388
A.3.3 Time Integration 388
A.3.4 Time Shifting – Real Translation . 389
A.3.5 Frequency Shifting – Complex Translation 389
A.3.6 Real Convolution 389
A.3.7 Partial Differentiation 390
A.3.8 Complex Differentiation 390
A.3.9 Initial Value Theorem 391xiv Contents
A.3.10 Final Value Theorem . 391
A.4 Inverse Laplace Transform . 392
A.5 Using the Laplace Transform to Solve Differential Equations 394
B Tables of Various Transforms 399
C Operations on Complex Numbers, Vectors, and Matrices 409
C.1 Complex Addition . 409
C.2 Complex Multiplication . 409
C.3 Complex Division 409
C.4 Conversion Between Rectangular Form and Polar/Exponential Form409
C.5 Operations on Complex Numbers Using MATLAB 410
C.6 Matrix Addition and Subtraction[Y-1] . 410
C.7 Matrix Multiplication . 411
C.8 Determinant . 411
C.9 Eigenvalues and Eigenvectors of a Matrix1 . 412
C.10 Inverse Matrix . 412
C.11 Symmetric/Hermitian Matrix 413
C.12 Orthogonal/Unitary Matrix 413
C.13 Permutation Matrix . 414
C.14 Rank . 414
C.15 Row Space and Null Space 414
C.16 Row Echelon Form . 414
C.17 Positive Definiteness 415
C.18 Scalar(Dot) Product and Vector(Cross) Product . 416
C.19 Matrix Inversion Lemma 416
C.20 Differentiation w.r.t. a Vector 416
D Useful Formulas 419
E MATLAB 421
E.1 Convolution and Deconvolution 423
E.2 Correlation 424
E.3 CTFS (Continuous-Time Fourier Series) . 425
E.4 DTFT (Discrete-Time Fourier Transform) 425
E.5 DFS/DFT (Discrete Fourier Series/Transform) 425
E.6 FFT (Fast Fourier Transform) 426
E.7 Windowing . 427
E.8 Spectrogram (FFT with Sliding Window) 427
E.9 Power Spectrum . 429
E.10 Impulse and Step Responses . 430
E.11 Frequency Response 433
E.12 Filtering 434
E.13 Filter Design 436Contents xv
E.13.1 Analog Filter Design . 436
E.13.2 Digital Filter Design – IIR (Infinite-duration Impulse
Response) Filter 437
E.13.3 Digital Filter Design – FIR (Finite-duration Impulse
Response) Filter 438
E.14 Filter Discretization 441
E.15 Construction of Filters in Various Structures Using dfilt() . 443
E.16 System Identification from Impulse/Frequency Response . 447
E.17 Partial Fraction Expansion and (Inverse) Laplace/z-Transform . 449
E.18 Decimation, Interpolation, and Resampling . 450
E.19 Waveform Generation 452
E.20 Input/Output through File . 452
F Simulink R . 453
Index . 461
Index for MATLAB routines . 467
Index for Examples . 471
Index for Remarks . 473
Index
A
ADC (Analog-to-Digital conversion), 251–256
A/D-G[z]-D/A structure, 259, 275, 283
Additivity, 13
Aliasing, 8
AM (amplitude modulation), 91, 113
Analog filter design, 307–320
Analog frequency, 6, 7
Analog signal, 2
Analytic signal, 109, 110
Anti-aliasing, 262–263
Anti-causal sequence, 208
Anticipatory, 30
Anti-imaging, 262–263
Anti-symmetric, 145, 232, 237–240
ASK (amplitude-shift keying), 115
B
Backward difference rule, 284
Bandpass filter (BPF), 309
Bandstop filter (BSF), 309
Bandwidth, 76
BIBO (bounded-input bounded-output), 29
BIBO stability condition, 29, 46
Bilateral (two-sided) z-transform, 208
Bilinear transformation, 288–292, 301, 302
Bit reversed order, 166
Block diagram, 32
Bode plot, 97
BPF realization, 123, 184
Butterfly computation, 167, 168
Butterworth filter, 313, 314, 329
C
Cascade form, 309
Causal, 30, 85, 208
Causality, 30, 225
Causal sequence, 208
Characteristic equation, 369
Characteristic root, 369
Chebyshev I filter, 311, 314, 324
Chebyshev II filter, 312, 314, 326
Circulant matrix, 382
Circular convolution, 90, 142, 151, 152, 403
Circular shift, 150
Complex convolution, 90, 139, 151, 184, 216
Complex envelope, 112
Complex exponential function, 4
Complex exponential sequence, 4
Complex number operation, 409
Complex sinusoidal function, 5
Complex sinusoidal sequence, 5
Complex translation, 88, 139, 150, 215, 389
Conjugate symmetry, 86, 138
Continuous-amplitude signal, 2
Continuous-time convolution, 7, 13, 19, 20
Continuous-time Fourier series, see CTFS
Continuous-time Fourier transform, see CTFT
Continuous-time frequency, 7
Continuous-time signal, 1
Continuous-time state equation, 50, 370
Continuous-time system, 13
Continuous-value signal, 2
Controllable canonical form, 36, 363
Convolution, 9–10, 16, 17, 183
Convolution property, 90, 109
Correlation, 38–39, 58, 60, 116, 424
Correlation coefficient, 43
CTFS, 62–73, 399, 401
CTFS and DFT/DFS, 160–164
CTFS coefficient, 70
CTFS of an impulse train, 70
CTFS of a rectangular wave, 65
CTFS of a triangular wave, 65
CTFS spectrum, 67
CTFT, 78–96, 400, 401
CTFT and DTFT, 161–162
CTFT of a cosine function, 85
461462 Index
CTFT of an impulse train, 85
CTFT of a periodic signal, 77
CTFT of a polygonal signal, 117
CTFT of a rectangular pulse, 75
CTFT of a sine function, 85
CTFT of a triangular pulse, 76
CTFT of the unit impulse function, 81
CTFT of the unit step function, 83
D
DAC (Digital-to-Analog conversion), 250–251
Dead time, 293
Decimation-in-frequency (DIF) FFT, 168
Decimation-in-time (DIT) FFT, 165
Deconvolution, 38, 54, 423
Demodulation, 92
DFS, 151–152, 405–406
DFS reconstruction, 265
DFT, 147–164, 178–180, 403–404
DFT for an infinite-duration sequence,
155–160
DFT for a noncausal sequence, 156–157,
159, 187
DFT of a triangular sequence, 175
DFT size, 149, 164
Difference equation, 31, 207, 208, 213,
225, 226
Differential equation, 33
Differentiation w.r.t. a vector, 388
Differentiator, 184, 338
Digital filter design, 320–350
Digital frequency, 6–8, 147
Digital fundamental frequency, 147
Digital resolution frequency, 147
Digital signal, 2
Dirac delta function, 8
Direct form, 36, 323, 325
Dirichlet condition, 63
Discrete-amplitude signal, 2
Discrete Fourier series (DFS), 147
Discrete Fourier transform, see DFT
Discrete-time convolution, 8, 17, 22, 24
Discrete-time equivalence criterion, 278, 279
Discrete-time equivalent, 277–280, 293
Discrete-time Fourier series (DTFS), 149
Discrete-time Fourier transform, see DTFT
Discrete-time frequency, 6
Discrete-time interpolation, 269–271, 275
Discrete-time sampling, 189
Discrete-time signal, 2
Discrete-time state equation, 54
Discrete-time system, 13
Discrete-value signal, 2
Discretization, 284–293, 370–376
Discretization of continuous-time state
equation, 370–376, 381
DTFT, 130–138, 160–164, 402
DTFT of a cosine sequence, 137
DTFT of a discrete-time impulse train, 189
DTFT of a periodic sequence, 136, 188
DTFT of a rectangular pulse, 132
DTFT of a sine sequence, 137
DTFT of asymmetric sequences, 145
DTFT of symmetric sequences, 145
DTFT of the unit impulse sequence, 135
DTFT of the unit step sequence, 137
DTFT spectrum, 141, 186
Duality, 88
Dual-tone multi-frequency (DTMF), 202
E
Eigenfunction, 116
Eigenvalue, 369, 378, 412
Elliptic filter, 313, 315, 328
Envelope detector, 112
Exponential function, 4
Exponential sequence, 4, 208
Export (a filter), 348
F
Fast Fourier transform (FFT), 164
FDATool, 348, 350, 352, 354
FDM (frequency-division multiplexing), 124
Filter structure, 345–348
Final value theorem, 218, 391
Finite-duration impulse response, see FIR
Finite pulsewidth sampler, 92
FIR, 32
FIR filter design, 331–344
FIR lowpass filter (LPF), 140
Folding frequency, 262
Formulas, 419
Forward difference, 286
Forward substitution, 39
Fourier, 61
Fourier reconstruction, 69
Fourier series, see CTFS
Fourier series and Fourier ransform, 76
Fourier transform, see CTFT
Fourier transform and Laplace transform, 74
Frequency, 6–8
Frequency aliasing, 257, 258, 260
Frequency resolution, 178
Frequency response, 19–22, 74, 225, 226,
228, 237
Frequency shifting, 88, 139, 150, 215, 389
Frequency transformation, 282, 289–291Index 463
FSK (frequency-shift keying), 115
Fundamental frequency, 64, 161
Fundamental matrix, 364–365
Fundamental period, 62
G
General gain formula, 34
Generalized Fourier transform, 77
Gibbs phenomenon, 69
Goertzel algorithm, 246
Group delay, 332
H
Half-power frequency, 79
Half-size DFT computation, 196
Harmonic, 72
Highpass filter (HPF), 307
Hilbert transformer, 109–113, 184, 338–339
Homogeneity, 13
I
Ideal BPF (bandpass filter), 119
Ideal LPF frequency response, 74–75, 120
Ideal LPF impulse response, 74–75
Ideal sampler, 92
IDFT (inverse discrete Fourier transform),
148, 169
IIR, 32
IIR filter design, 321–331
Import (a filter), 348
Impulse-invariant transformation, 281–282
Impulse response, 15, 16, 17, 216, 368
Impulse signal, 81
Incrementally linear, 14
Infinite-duration impulse response, see IIR
Initial value theorem, 217, 391
In-place computation, 167
Interpolation, 263
Inverse discrete Fourier series (IDFS), 151
Inverse discrete Fourier transform (IDFT), 148
Inverse Laplace transform, 392
Inverse system, 48
Inverse z-transform, 218–223, 226, 230–231
Invertible, 30
J
Jordan canonical form, 378–379
K
Kronecker delta sequence, 10
L
Ladder, 345
Laplace transform, 384–397, 406, 407
Lattice, 345
Least squares error (LSE), 126
Left-sided sequence, 208
Left-side (rectangular) rule, 287
Linear, 13
Linear convolution, 152–155, 423
Linear convolution with DFT, 152–155, 192
Linear phase, 97, 133, 238
Linear system, 13
Linear time-invariant (LTI) system, 15
Long division, 223
Lowpass equivalent, 112
Lowpass filter (LPF), 270
M
Mason’s formula, 34
Matched filter, 40, 41, 42, 55
Matched z-transform, 292
MATLAB, 395
Matrix operation, 409–417
Modal matrix, 378–379, 382
Modulation property, 91, 108, 139
Multi-band FIR filter design, 240, 335–336
N
Non-anticipatory, 30
Non-causal, 30
Nyquist frequency, 258
Nyquist rate, 258, 261
O
Observable canonical form, 36, 363
Order, 32
Orthogonal, 127, 413
Output equation, 362
P
Parallel computation of two DFTs, 194
Parallel form, 309
Parseval’s relation, 96, 116, 144
Partial fraction expansion, 219–223, 393
Passband edge frequency, 307–308
Passband ripple, 307–308
Period, 6, 7, 62
Periodic, 6, 62
Periodic convolution, 139
Periodic extension, 149
Phase delay, 332
Phase jump, 133
Physical realizability, 84
Picket fence effect, 172, 177
Plane impulse train, 106
Poisson sum formula, 123464 Index
Pole, 29, 210, 224
Pole location, 295–296, 303, 305
Pole-zero mapping, 292
Pole-zero pattern, 210, 232
Power theorem, 116
Practical reconstruction, 267
Pre-envelope signal, 112
Prewarping, 290–291, 302
Pulse response, 18
Pulse transfer function, 297
PWM (pulse-width modulated), 382
Q
Quadrature multiplexing, 124, 184
R
Rayleigh theorem, 96, 144
Real convolution, 89, 139, 149, 215, 389
Real translation, 88, 139, 150, 214, 389
Reconstruction, 263–271, 274
Rectangular function, 5
Rectangular sequence, 5
Rectangular windowing, 140
Recursive, 32, 321
Recursive computation of DFT, 198
Region of convergence (ROC), 208, 209, 213,
223
Resolution frequency, 147
Right-sided sequence, 210
Right-side (rectangular) rule, 288
S
Sample-and-hold (S/H), 272
Sampler, 92–93
Sampling, 92–93, 186, 249
Sampling interval, 178
Sampling period, 178, 259
Sampling property, 10, 11
Sampling theorem, 249, 257
Scaling, 95, 143, 189
Second-order active filter, 318–319
Shannon reconstruction, 263
Shift-invariant, 14
Short-time Fourier transform, 180, 200
Sifting property, 10, 11
Signal, 2
Signal bandwidth, 79
Signal flow graph, 32–34
Similarity transformation, 376–379
Simulink, 453
Sinc function, 11, 45, 76
Sinusoidal function, 5
Sinusoidal sequence, 5
Sinusoidal steady-state response, 19–20, 234
Spectral leakage, 140, 164, 171, 193
Spectrogram, 180, 201, 427
Spectrum, 64, 67, 70–73
Spectrum blurring, 176, 194
SPTool, 350–359
Stability, 29, 242
Stability condition, 29, 74, 397
Stability of discrete-time systems, 47, 225
Stable, 29
Starred transform, 268, 297
State, 362
State diagram, 32, 35, 37, 51, 53
State equation, 50, 53, 362–363, 364, 370–376
State space description, 362
State transition matrix, 364–365
State variable, 362
State vector, 362
Step-invariant transformation, 282–283
Step response, 18
Stopband attenuation, 307–308
Stopband edge frequency, 307–308
Stroboscopic effect, 8, 258
Superheterodyne receiver, 120
Superposition principle, 13, 15, 16
Symmetric sequence, 145, 146, 236, 238
System, 17
System bandwidth, 79
System function, 18, 31, 225
T
Tapped delay lines, 26–28
TDM (Time-Division multiplexing), 124
Time-aliasing, 154, 262
Time constant, 78
Time-invariant, 14
Time resolution, 182
Time reversal, 215
Time sampling method, 279
Time shifting, 88, 139, 148, 214, 389
Transfer function, 17, 31, 368
Transmission matrix, 38
Transportation delay, 293
Transposed direct form, 37, 434
Trapezoidal rule, 288
Tustin’s method, 288–292
Two-dimensional DFT, 199
Two-dimensional Fourier transform, 106
U
Uncertainty principle, 67, 155
Unilateral (one-sided) z-transform, 208
Unit impulse function, 3, 8, 10, 11, 45, 386
Unit impulse sequence, 3, 10, 11Index 465
Unit sample response, 16
Unit sample sequence, 4, 10, 11
Unit step function, 3, 362
Unit step sequence, 3, 362
W
Wagon-wheel effect, 8, 258
White spectrum, 81
Whittaker’s cardinal interpolation, 264
Windowing, 193
Windowing method (for FIR filter design), 333
Windowing property, 90, 139, 185
Z
Zero, 31, 212
Zero-insertion, 176
Zero-order-hold equivalent, 283, 301, 374
Zero-padding, 152, 156, 164, 174
z-transform, 208, 213, 406, 407, 408
z-transform and DTFT, 211Index for MATLAB routines
MATLAB Page
routine name Description number
bilinear() bilinear transformation (optionally with prewarping) 442
butter() designs Butterworth filter with an order and cutoff
frequency
310, 436
buttord() the order and cutoff frequency of Butterworth filter 310, 436
cfirpm() designs a (possibly complex) FIR filter 332, 344
cheby1() designs Chebyshev I filter with an order and cutoff
frequency
311, 436
cheby1order() the order and cutoff frequency of Chebyshev I filter 311
cheby2() designs Chebyshev II filter with an order and cutoff
frequency
312, 438
cheby2order() the order and cutoff frequency of Chebyshev II filter 312
chirp() swept-frequency cosine generator 452
conv() (linear) convolution 154, 423
conv circular() circular convolution 424
cpsd() cross power spectral density 429
c2d() discretization (continuous-to-discrete conversion) 442
CTFS exponential() find the CTFS coefficients in exponential form 425
CTFT poly() CTFT of a polygonal signal 118
decimate() Reduces the sampling rate to produce a decimated
sequence
450
deconv() deconvolution 424
dimpulse() impulse response of a discrete-time system 431
detrend() remove the best straight-line fit or the mean value 451
dfilt digital filter structure conversion 443
DFS discrete Fourier series 425
DFT discrete Fourier transform 425
dlsim() time response of a discrete-time system to a given
input
432
dstep() step response of a discrete-time system 432
DTFT discrete-time Fourier transform 425
d2c() discrete-to-continuous conversion 442
dtmf decoder() DTMF (dual-tone multi-frequency) signal decoder 205
dtmf generator() DTMF (dual-tone multi-frequency) signal generator 202
ellip() designs elliptic filter with an order and cutoff
frequency
437
fft() fast Fourier transform (FFT) 426
fftshift() swaps the first and second halves 427
467468 Index for MATLAB routines
MATLAB Page
routine name Description number
filter() the output of a digital filter (with an initial state) to an
input
434
filter cas() filtering in a cascade form 445
filter latc nr() filtering in a nonrecursive lattice form 447
filter latc r() filtering in a recursive lattice form 447
filter par() filtering in a parallel form 445
fir1(), fir2() designs a FIR filter using windowing 332, 439
fircls(), fircls1() designs a FIR filter using constrained least squares 332, 441
firls(), firpm() designs a FIR filter using equiripple or least squares 332, 440
firrcos() designs a FIR filter using raised cosine 332
Fourier analysis() CTFT analysis of an LTI system with a transfer
function
105
freqs() frequency response of a continuous-time system 433
freqz() frequency response of a discrete-time system 433
hilbert() analytic signal with Hilbert transform on the
imaginary part
111
ifft() inverse (fast) Fourier transform 426
ilaplace() inverse Laplace transform 394, 449
impinv() impulse-invariant discretiztion of a continuous-time
system
441
impulse() impulse response of a continuous-time system 431
impz() impulse response of a discrete-time system B[z]/A[z] 431
interp() increase the sampling rate to produce an interpolated
sequence
450-451
interpolation discrete() discrete-time interpolation (Sec. 5.4.4) 271
invfreqs() identifies continuous-time system from its frequency
response
448
invfreqz identifies discrete-time system from its frequency
response
448
iztrans() inverse z-transform 221, 449
jordan() Jordan canonical form of state equation 379
laplace() Laplace transform 449
latc2tf() lattice structure to transfer function 347
load load (read) a file 452
lsim() time response of a continuous-time system to a given
input
432
music wave() melody generator 200
par2tf() parallel form to transfer function 347
prony() identifies a discrete-time system based on its impulse
response
447
pulstran() generates a pulse train 452
rectpuls generates a rectangular pulse 452
resample() change the sampling rate 451
residue() partial fraction expansion of a Laplace transform
expression
394, 449
residuez() partial fraction expansion of a z-transform expression 220, 449
save save (write) a file 452
sos2ss() second-order sections to state-space description 347
sos2tf() second-order sections to transfer function 347
sos2zp() second-order sections to zero-pole form 347
specgram() spectrogram (old version) 427
spectrogram() spectrogram 427Index for MATLAB routines 469
MATLAB Page
routine name Description number
ss2sos() state-space description to second-order sections 347
ss2tf() state-space description to transfer function 347
ss2zp() state-space description to zero-pole form 347
step() step response of a continuous-time system 432
stmcb() identifies a discrete-time system 448
tfe (discrete-time) transfer function estimation 448
tf2latc() transfer function to lattice form 347, 443
tf2latc my() transfer function to lattice form 446
tf2par s() transfer function (in Laplace transform) to parallel
form
444
tf2par z() transfer function (in z-transform) to parallel form 347, 443
tf2sos() transfer function to second-order sections 347
tf2ss() transfer function to state-space description 347
tf2zp() transfer function to zero-pole form 347
tripuls() generates a triangular pulse 452
upfirdn() upsamples, applies a FIR filter, and downsamples 451
windowing() various windowing techniques 427
xcorr() correlation 42, 423
xcorr circular() circular correlation 425
zp2sos() zero-pole form to second-order sections 347
zp2ss() zero-pole form to state-space description 347
ztrans() z-transform 449Index for Examples
Example no. Description Page number
Example 1.1 Convolution of Two Rectangular Pulses 22
Example 1.2 Approximation of a Continuous-Time Convolution 25
Example 1.3 Tapped Delay Lines 26
Example 1.4a Differential Equation and Continuous-Time State Diagram 36
Example 1.4b Difference Equation and Discrete-Time State Diagram 36
Example 1.5 Correlation and Matched Filter 41
Example 1.6 Correlation for Periodic Signals with Random Noise 43
Example 2.1 Fourier Spectra of a Rectangular Wave and a Triangular Wave 65
Example 2.2 Fourier Spectrum of an Impulse Train 70
Example 2.3 CTFT Spectra of Rectangular Pulse and a Triangular Pulse 75
Example 2.4 Fourier Transform of an Exponential Function 78
Example 2.5 Fourier Transform of an Even-Symmetric Exponential Function 80
Example 2.6 Fourier Transform of the Unit Impulse Function 81
Example 2.7 Fourier Transform of a Constant Function 82
Example 2.8 Fourier Transform of the Unit Step Function 83
Example 2.9 Inverse Fourier Transform of an ideal LPF Frequency Response 84
Example 2.10 Fourier Transform of an Impulse Train 85
Example 2.11 Fourier Transform of Cosine/Sine Functions 85
Example 2.12 Sinusoidal Amplitude Modulation and Demodulation 91
Example 2.13 Ideal (Impulse or Instant) Sampler and Finite Pulsewidth
Sampler
92
Example 3.1 DTFT of a Rectangular Pulse Sequence 132
Example 3.2 DTFT of an Exponential Sequence 133
Example 3.3 DTFT of a Symmetrical Exponential Sequence 134
Example 3.4 DTFT of the Unit Sample (Impulse) Sequence 135
Example 3.5 IDTFT of an Ideal Lowpass Filter Frequency Response 136
Example 3.6 DTFT of a Constant Sequence 137
Example 3.7 DTFT of Cosine/Sine Sequences 137
Example 3.8 DTFT of the Unit Step Sequence 137
Example 3.9 Effect of Rectangular Windowing on the DTFT of a Cosine
Wave
140
Example 3.10 Impulse Response and Frequency Response of a FIR LPF 140
Example 3.11 DTFT of an Odd Sequence 145
Example 3.12 DTFT of an Anti-Symmetric Sequence 146
Example 3.13 Linear Convolution Using the DFT 152
Example 3.14 DFT of a Noncausal Pulse Sequence 156
Example 3.15 DFT of an Infinite-Duration Sequence 157
Example 3.16 DFT Spectrum of a Single-Tone Sinusoidal Wave 170
471472 Index for Examples
Example no. Description Page number
Example 3.17 DFT Spectrum of a Multi-Tone Sinusoidal Wave 173
Example 3.18 DFT Spectrum of a Triangular Wave 175
Example 4.1 The z-Transform of Exponential Sequences 208
Example 4.2 A Causal Sequence Having a Multiple-Pole z-Transform 210
Example 4.3 The z-Transform of a Complex Exponential Sequence 211
Example 4.4 The z-Transform of an Exponentially Decreasing Sinusoidal
Sequence
212
Example 4.5 Applying Linearity and Time Shifting Properties of the
z-Transform
214
Example 4.6 Complex Differentiation and Partial Differentiation 217
Example 4.7 The Inverse z-Transform by Partial Fraction Expansion 220
Example 4.8 The Inverse z-Transform by Partial Fraction Expansion 222
Example 4.9 Difference Equation, System Function, and Impulse Response 227
Example 4.10 Different Difference Equations Describing the Same System 229
Example 4.11 Pole-Zero Pattern and Frequency Response 233
Example 4.12 Pole-Zero Pattern of Symmetric or Anti-Symmetric
Sequences
238
Example 5.1 Discrete-Time Interpolation 270
Example 6.1 Impulse-Invariant Transformation–Time-Sampling Method 282
Example 6.2 Step-Invariant Transformation (Zero-Order-Hole Equivalent) 283
Example 6.3 Backward Difference Rule 285
Example 6.4 Forward Difference Rule 286
Example 6.5 Bilinear Transformation 289
Example 6.6 Bilinear Transformation with Prewarping 291
Example 6.7 Pole-Zero Mapping 292
Example 7.1 Analog Filter Design Using the MATLAB Functions 309
Example 7.2 IIR Filter Design 321
Example 7.3 Standard Band FIR Filter Design 334
Example 7.4 Multi-Band FIR Filter Design 336
Example 7.5 Anti-Symmetric Filters–Hilbert Transformer and
Differentiator
338
Example 7.6 Multi-Band CLS FIR Filter Design 340
Example 7.7 CLS (Constrained Least-Squares) FIR LPF/HPF Design 341
Example 7.8 Complex-Coefficient, Arbitrary Magnitude Response FIR
Filter Design
343
Example 8.1 Solving a State Equation 366
Example 8.2 Transfer Function 369
Example 8.3 Discretization of a Continuous-Time State Equation 371
Example 8.4 Discretization of a Double Integrator 374
Example 8.5 Discretization of a Double Integrator with Time Delay 376
Example 8.6 Diagonal/Jordan Canonical Form of State Equation 378Index for Remarks
Remark no. Description Page number
Remark 1.1 Analog Frequency and Digital Frequency 7
Remark 1.2a Properties of the Unit Impulse Function 9
Remark 1.2b Properties of the Unit Impulse Sequence 10
Remark 1.3 Linearity and Incremental Linearity 14
Remark 1.4 Frequency Response and Sinusoidal Steady-State Response 21
Remark 1.5 Convolution of Two Rectangular Pulses 24
Remark 1.6 Stability of LTI systems with System Function G(s)/G[z] 29
Remark 1.7 Properties of Autocorrelation 40
Remark 1.8 Convolution vs. Correlation and Matched Filter 40
Remark 1.9 xcorr()–MATLAB function for Correlation 42
Remark 2.1 Convergence of Fourier Series Reconstruction 69
Remark 2.2 Physical Meaning of Complex Exponential Fourier Series
Coefficients
71
Remark 2.3 Effects of Smoothness and Period on Spectrum 72
Remark 2.4 Physical Meaning of Fourier Transform 74
Remark 2.5 Frequency Response Existence Condition and Stability
Condition
74
Remark 2.6 Fourier Transform and Laplace Transform 74
Remark 2.7 Fourier Series and Fourier Transform 74
Remark 2.8 Fourier Transform of a Periodic Signal 76
Remark 2.9 Signal Bandwidth and System Bandwidth–Uncertainty
Principle
79
Remark 2.10 An Impulse Signal and Its (White/Flat) Spectrum 82
Remark 2.11 Physical Realizability and Causality Condition 84
Remark 3.1 Physical Meaning of DTFT 130
Remark 3.2 Frequency Response Existence Condition and Stability
Condition
131
Remark 3.3 Phase Jumps in DTFT Phase Spectrum 144
Remark 3.4 The DTFT Magnitude/Phase Spectra of a Symmetric
Sequence
144
Remark 3.5 How to Choose the DFT Size N in Connection with Zero
Padding
155
Remark 3.6 The DFT got Noncausal/Infinite-Duration Sequences 159
Remark 3.7 Relationship among the CTFS, CTFT, DTFT, and DTFS
(DFT/DFS)
162
Remark 3.8 Data Arrangement in Bit Reversed Order 166
Remark 3.9 Simplified Butterfly Computation 166
473474 Index for Remarks
Remark no. Description Page number
Remark 3.10 DFS/DFT (Discrete Fourier Series/Transform) and Spectral
Leakage
177
Remark 3.11 The Effects of Sampling Interval T and DFT Size N on DFT 178
Remark 4.1 Region of Convergence (ROC) 209
Remark 4.2 z-Transform and DTFT (Discrete-Time Fourier Transform) 211
Remark 4.3 Poles and Zeros 212
Remark 4.4 System Function, Pole Location, ROC, Causality, and
Stability
213
Remark 4.5 Simplified Butterfly Computation 225
Remark 4.6 Computational Method for Inverse z-Transform 228
Remark 4.7 Frequency Response and Pole-Zero Pattern 232
Remark 4.8 Pole-Zero Pattern, Linear Phase of (Anti-)Symmetric
Sequences
238
Remark 5.1 z-Transform and DTFT (Discrete-Time Fourier Transform) 261
Remark 5.2 Poles and Zeros 262
Remark 5.3 Discrete-Time Interpolation, Zero Insertion, and Lowpass
Filtering
270
Remark 6.1 Equivalence Criterion and Band-Limitedness Condition 279
Remark 6.2 Time-Sampling Method–Impulse-Invariant Transformation 280
Remark 6.3 Frequency Response Aspect of Impulse-Invariant
Transformation
280
Remark 6.4 Mapping of Stability Region by Impulse-Invariant
Transformation
281
Remark 6.5 Frequency Transformation by Impulse-Invariant
Transformation
282
Remark 6.6 Mapping of Stability Region and Frequency Transformation 285
Remark 6.7 Mapping of Stability Region by Forward Difference Rule 287
Remark 6.8 Mapping of Stability Region and Frequency Transformation
by BLT
289
Remark 6.9 Prewarping 291
Remark 6.10 DC Gain Adjustment 293
Remark 8.1 Discretized State Equation and Zero-Order-Hold Equivalent 374
Remark 8.2 Similarity Transformation–Equivalence Transformation 377
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