This book is a self-teaching reference focused on visualization of signals and systems with MATLAB . The book covers the 108 key topics including the thorough analysis of some discrepancies between the Fourier transform, the Laplace transform, the discrete-time Fourier transform, and the z transform , for example, of u(t)and u(n). Each topic consists of three steps: an introduction to the topic in depth, problems, and Matlab solutions to each problem. The accompanying CD-ROM includes all of the Matlab programs for 108 topics and the 14 new functions developed for the book.
Features:
- The Complete Matlab solutions for all of the 108 topics
- Touching signals and systems for oneself
- Over 400 Figures
- New Matlab functions for basic signals and systems
Important Topics:
- 3D surface mesh plot of 1/s close to Re(s)=0
- Dirac-delta impulse on the complex plane
- Generalized Fourier and Laplace transforms
- Generalized z transform
Contents
Preface 1
1 Background 9
1.1 Sinusoidal Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Complex Exponential Waveforms . . . . . . . . . . . . . . . . . . . . . . 12
1.3 3D Line Plot on the Unit Circle . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Pulse Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5 Sinc Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.6 Dirichlet Kernel Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.7 Convolution Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.8 Properties of Convolution Integral . . . . . . . . . . . . . . . . . . . . . . 26
1.9 Convolution Integral with Step Signal . . . . . . . . . . . . . . . . . . . . 28
1.10 Convolution between Finite Length Impulse Trains . . . . . . . . . . . . . 31
1.11 Hilbert Transform: Shifting the Phase by ¡90± . . . . . . . . . . . . . . . 33
1.12 Generalized Unit Impulse and Step Functions . . . . . . . . . . . . . . . 36
1.13 Orthogonal Signal Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.14 Partial Fraction Expansion of Irrational Polynomials . . . . . . . . . . . . 45
1.15 Partial Fraction Expansion of Laplace and z Transforms . . . . . . . . . . 49
1.16 Area under 1 s on a vertical line in the Complex Plane . . . . . . . . . . . 52
1.17 Contour Integral of 1 s in the Complex Plane . . . . . . . . . . . . . . . . 54
1.18 3D Line Plot of 1 s over the Right Half Complex Plane . . . . . . . . . . . 55
1.19 3D Line Plot of 1 s over the Entire Complex Plane . . . . . . . . . . . . . . 59
1.20 3D Surface Mesh Plot of 1 s close to Re(s) = 0 . . . . . . . . . . . . . . . . 64
2 Continuous-Time Signals and Systems 71
2.1 The Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.2 Fourier Coe±cients of Pulse Train . . . . . . . . . . . . . . . . . . . . . . 78
2.3 Fourier Coe±cients of Recti？ed Waveforms . . . . . . . . . . . . . . . . . 80
2.4 Fourier Coe±cients of Impulse Trains . . . . . . . . . . . . . . . . . . . . 83
2.5 Fourier Coe±cients of Dirichlet Kernel . . . . . . . . . . . . . . . . . . . 86
2.6 Fourier Series Expansion of Impulse Train . . . . . . . . . . . . . . . . . . 89
2.7 Fourier Series Expansion of Pulse Train . . . . . . . . . . . . . . . . . . . 92
2.8 Gibbs Phenomenon Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 96
2.9 Generalized Impulse Train (GIT) . . . . . . . . . . . . . . . . . . . . . . . 100
2.10 Fourier Series Expansion of Generalized Impulse Trains . . . . . . . . . . 103
2.11 Fourier Series Expansion of Generalized Pulse Train . . . . . . . . . . . . 106
2.12 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
2.13 Fourier Transform of |(t), sinc(t), and cos(2¼t)|(2t) . . . . . . . . . . . 113
2.14 Inverse Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
2.15 Fourier Transforms of Impulse Trains of Finite Duration . . . . . . . . . 118
2.16 Fourier Transform of Generalized Impulse Functions . . . . . . . . . . . . 123
2.17 Generalized Fourier Transforms of u(t), 1, and sgn(t) . . . . . . . . . . . . 127
2.18 Generalized Fourier Transform of Complex Exponential . . . . . . . . . . 130
2.19 Generalized Fourier Transform of Impulse Train . . . . . . . . . . . . . . 132
2.20 Orthogonality of the sinc function . . . . . . . . . . . . . . . . . . . . . . 135
2.21 Ideal Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
2.22 Sampling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
2.23 Dual Sampling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
2.24 DC Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
2.25 Sampling of |(f) and Pn sinc(t ¡ nT) . . . . . . . . . . . . . . . . . . . 147
2.26 The Value of a Time-Limited Signal at time origin . . . . . . . . . . . . . 151
2.27 Ideal Lowpass ？lter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
2.28 Real Lowpass Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
2.29 Lowpass Filtering of Periodic Pulse Train . . . . . . . . . . . . . . . . . . 163
2.30 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
2.31 Amplitude Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
2.32 Single-Sideband Modulation . . . . . . . . . . . . . . . . . . . . . . . . . 172
2.33 Frequency Response of a Linear Time-Invariant System . . . . . . . . . . 174
2.34 Analysis of a Second Order LTI System . . . . . . . . . . . . . . . . . . . 177
2.35 Frequency Response of Butterworth Filter . . . . . . . . . . . . . . . . . 181
2.36 Transfer Function of Butterworth Filter . . . . . . . . . . . . . . . . . . . 184
2.37 Filter Conversion from Lowpass Filter . . . . . . . . . . . . . . . . . . . . 186
2.38 Butterworth Filter Conversion . . . . . . . . . . . . . . . . . . . . . . . . 189
2.39 3D Mesh Plot of Laplace Transform of u(t) . . . . . . . . . . . . . . . . . 192
2.40 Generalized Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . 194
2.41 3D Mesh Plot of the Second Order Transfer Functions . . . . . . . . . . . 200
2.42 The Limiting Function of 1¡e¡j!T j! as T ! 1 . . . . . . . . . . . . . . . . 204
2.43 Dirac-Delta Impulse on the Complex Plane . . . . . . . . . . . . . . . . . 207
3 Discrete-Time Signals and Systems 217
3.1 The Discrete-Time Fourier Transform (DTFT) . . . . . . . . . . . . . . . 219
3.2 DTFT of Windowed Complex Exponential . . . . . . . . . . . . . . . . . . 224
3.3 DTFT of Windowed Sinusoid . . . . . . . . . . . . . . . . . . . . . . . . . 230
3.4 DTFT of Rectangular Pulse and dc Sequences . . . . . . . . . . . . . . . 235
3.5 DTFT of Triangular Pulse Sequence . . . . . . . . . . . . . . . . . . . . . 239
3.6 Dirichlet and sinc functions . . . . . . . . . . . . . . . . . . . . . . . . . . 243
3.7 DTFTs of sgn[n] Based on the FT of sgn(t) . . . . . . . . . . . . . . . . . 245
3.8 DTFT of Exponential Sequence from the FT of Exponential Signal . . . . 248
3.9 Characteristics of Q(r; !) = 1+re¡j! 1¡re¡j! . . . . . . . . . . . . . . . . . . . . . 252
3.10 Sum of an In？nite Geometric Series . . . . . . . . . . . . . . . . . . . . . . 256
3.11 DTFTs of u(n) u[n], sgn[n], and 1 . . . . . . . . . . . . . . . . . . . . . . 261
3.12 DTFT of a Causal Rectangular Pulse Sequence . . . . . . . . . . . . . . . 264
3.13 Mesh Plot of the DTFT of rnu[n] . . . . . . . . . . . . . . . . . . . . . . . 268
3.14 Digital Di®erentiator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
3.15 Periodic Convolution in Frequency Domain . . . . . . . . . . . . . . . . . 274
3.16 Upsampling and Downsampling . . . . . . . . . . . . . . . . . . . . . . . . 277
3.17 Samplings in Both Time and Frequency Domains (I) . . . . . . . . . . . . 285
3.18 Sampling in Both Time and Frequency Domains (II) . . . . . . . . . . . . 292
3.19 The Discrete Fourier Transform (DFT) . . . . . . . . . . . . . . . . . . . . 297
3.20 Relationship between the DFT and the DTFT . . . . . . . . . . . . . . . 301
3.21 Circular Shift in DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
3.22 DTFT and DFT of Complex Exponential Sequence . . . . . . . . . . . . 310
3.23 Periodic Convolution via Circular Convolution . . . . . . . . . . . . . . . 314
3.24 The z Transform and Its Properties . . . . . . . . . . . . . . . . . . . . . 319
3.25 z Transform of a Pulse Sequence . . . . . . . . . . . . . . . . . . . . . . . 324
3.26 z Transform of rjnj for jnj · N . . . . . . . . . . . . . . . . . . . . . . . . 328
3.27 Generalized z Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
3.28 Sampling the z Transform on the Unit Circle . . . . . . . . . . . . . . . . 342
3.29 Reconstruction of the z Transform on jzj = 1 . . . . . . . . . . . . . . . . 348
3.30 Reconstruction of the z Transform on jzj = r . . . . . . . . . . . . . . . . 352
3.31 Frequency Responses of Exponentials . . . . . . . . . . . . . . . . . . . . 359
3.32 Impulse Invariance Transformation (IIT) . . . . . . . . . . . . . . . . . . . 364
3.33 Digital Elliptic Filter Design Using the IIT . . . . . . . . . . . . . . . . . 371
3.34 Discrete-Time Linear Time-Invariant (DLTI) System . . . . . . . . . . . . 374
3.35 DC Blocker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
3.36 Analysis of a Di®erence Equation y[n] + r2y[n ¡ 2] = x[n] . . . . . . . . . 383
3.37 DLTI System of the Second Order Lowpass Filter . . . . . . . . . . . . . . 389
3.38 DLTI System of the Second Order Highpass Filter . . . . . . . . . . . . . 393
3.39 Bilinear Transformation (BT) and Frequency Warping . . . . . . . . . . . 396
3.40 Digital Filter Designs Using the IIT and the BT . . . . . . . . . . . . . . 401
3.41 Pole and Zero Distributions of Digital Filters . . . . . . . . . . . . . . . . 405
3.42 2D and 3D Visualizations of 1D Digital Filter (I) . . . . . . . . . . . . . . 409
3.43 2D and 3D Visualizations of 1D Digital Filter (II) . . . . . . . . . . . . . 413
3.44 Single Tone Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
3.45 Deconvolution (Inverse Filtering) . . . . . . . . . . . . . . . . . . . . . . . 423
Appendix 427
A Mathematical Formulas 427
B Evaluation of the area under sincN(t) 431
C Matlab Functions 433
Index 450
This book is primary intended for solving problems in signals and systems with use of the MATLABr and thereby understanding the solutions clearly. This MATLAB based book has resulted from my teaching experience of undergradu-ate and graduate-level courses of signals and systems, digital image processing over the past decade. Today most students prefer watching a screen to taking notes even in class of signals and systems really needed great e®ort, so it is liable to neglect their studies. Fortunately most students are familiar with the computers and the internet and also have some expe-rience of the MATLAB software. The MATLAB is easily accessible to even a beginner,e®ective in visualization of signals, has an abundance of mathematical and engineering functions and a great °exibility for user to make new commands and functions for his or her own purpose. For the course achievement, it requires as much direct and actual experience as pos-sible since touching signals and systems for oneself is worth much more than listening,seeing, or writing down. For this purpose, we ？rst select a set of 108 topics which are fundamentals in signals and systems, second introduce and discuss about each selected topic, and at last, set problems and solve the problems with MATLAB.
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