Transforms and applications primer for engineers with examples and MATLAB

Title Page\r......Page 3
Contents......Page 5
Preface......Page 9
Author......Page 10
1.2 Signals......Page 11
1.3 Circuit Elements and Equation......Page 23
1.4.1 Linear Mechanical Systems......Page 31
1.4.2 Rotational Mechanical Systems......Page 32
1.5 Discrete Equations and Systems......Page 33
1.7 Convolution of Analog Signals......Page 36
1.8 Convolution of Discrete Signals......Page 39
2.2 Fourier Series in a Complex Exponential Form......Page 51
2.3.1 Differentiation of the Fourier Series......Page 52
2.5.1 Power Content: Parseval\'s Theorem......Page 53
2.5.3 Transmission without Distortion......Page 54
2.5.6 Product of Two Functions......Page 55
2.5.8 Gibbs\' Phenomenon......Page 56
2.5.9 Fourier Series of the Comb Function......Page 57
3.2.1 f(t) Is a Complex Function......Page 73
3.2.3 Imaginary Time Functions......Page 74
3.2.6 Odd and Even Representations......Page 75
3.2.7 Causal-Time Functions......Page 76
3.3 Fourier Transform Examples......Page 77
3.5 Examples on Fourier Properties......Page 80
3.6 FT Examples of Singular Functions......Page 84
3.7 Duration of a Signal and the Uncertainty Principle......Page 109
3.8 Applications to Linear-Time Invariant Systems......Page 110
3.9 Applications to Communication Signals......Page 119
3.10 Signals, Noise, and Correlation......Page 122
3.11 Average Power Spectra, Random Signals, Input–Output Relations......Page 123
3.12 FT in Probability Theory......Page 125
3.12.2 Joint Cumulative Distribution Function......Page 127
3.12.3 Characteristic Function of Two Variables......Page 128
4.2 Infinite Fourier Cosine Transform......Page 129
4.3 Applications to Boundary-Value Problems......Page 137
4.4 Finite Sine Fourier Transform and Finite Cosine Fourier Transform......Page 143
4.5 Two-Dimensional Fourier Transform......Page 146
4.5.2 Two-Dimensional Correlation......Page 149
4.5.3 Theorems of Two-Dimensional Functions......Page 150
5.1 Fundamentals of Sampling......Page 151
5.2 The Sampling Theorem......Page 156
6.1.1 Approximating the Fourier Transform......Page 164
6.2 Summary of DTFT Properties......Page 168
6.3 DTFT of Finite Time Sequences......Page 170
6.3.1 Windowing......Page 172
6.4 Frequency Response of LTI Discrete Systems......Page 174
6.5 Discrete Fourier Transform......Page 176
6.6 Summary of DFT Properties......Page 178
6.7 Multirate Digital Signal Processing and Spectra......Page 190
6.7.1 Down Sampling (or Decimation)......Page 191
6.7.2 Frequency Domain of Down-Sampled Signals......Page 193
6.7.3 Interpolation (Up-Sampling) by a Factor U......Page 197
6.7.4 Frequency Domain Characterization of Up-Sampled Signals......Page 198
6.A.1 Proofs of DTFT Properties......Page 201
6.A.2 Proofs of DFT Properties......Page 203
6.A.3 Fast Fourier Transform......Page 206
7.1 One-Sided Laplace Transform......Page 210
7.2 Summary of the Laplace Transform Properties......Page 213
7.3 Systems Analysis: Transfer Functions of LTI Systems......Page 217
7.4 Inverse Laplace Transform......Page 228
7.5.1 Ordinary Differential Equations......Page 235
7.5.2 Partial Differential Equations......Page 248
7.6 Frequency Response of LTI Systems......Page 258
7.7 Pole Location and the Stability of LTI Systems......Page 266
7.8 Feedback for Linear Systems......Page 269
7.9 Bode Plots......Page 280
7.10 *Inversion Integral......Page 284
7.11 *Complex Integration and the Bilateral Laplace Transform......Page 295
7.12 *State Space and State Equations......Page 297
7.12.1 State Equations in Phase Variable Form......Page 299
7.12.2 Time Response Using State Space Representation......Page 307
7.12.3 Solution Using the Laplace Transform......Page 311
7.12.4 State Space Transfer Function......Page 314
7.12.5 Impulse and Step Response......Page 315
8.1 The z-Transform......Page 336
8.2 Convergence of the z-Transform......Page 340
8.3 Properties of the z-Transform......Page 346
8.4 z-Transform Pairs......Page 355
8.5.1 Partial Fraction Expansion......Page 356
8.5.4 *Residues for Multiple Poles......Page 363
8.5.5 *Residues for Simple Poles Not Factorable......Page 364
8.6 Transfer Function......Page 366
8.6.1 Higher Order Transfer Functions......Page 372
8.7 Frequency Response of First-Order Discrete Systems......Page 374
8.7.1 Phase Shift in Discrete Systems......Page 380
8.8 Frequency Response of Higher Order Digital Systems......Page 381
8.9 z-Transform Solution of First-Order Difference Equations......Page 384
8.10 Higher Order Difference Equations......Page 388
8.10.1 Method of Undetermined Coefficients......Page 394
8.11 *LTI Discrete-Time Dynamical Systems......Page 399
8.12.1 Power Spectral Densities......Page 404
8.12.2 Linear Discrete-Time Filters......Page 406
8.12.3 Optimum Linear Filtering......Page 407
8.13 *Relationship between the Laplace and z-Transforms......Page 409
8.14 *Relationship to the Fourier Transform......Page 413
Appendix......Page 414
9.1 Definition......Page 424
9.2 Hilbert Transforms, Properties, and the Analytic Signal......Page 425
9.3 Hilbert Transform Properties and Hilbert Pairs......Page 438
Appendix A: Functions of a Complex Variable......Page 441
Appendix B: Series and Summations......Page 512
Appendix C: Definite Integrals......Page 520
Appendix D: Suggestions and Explanations for MATLAB® Use......Page 537