Signals and systems

An academic textbook that is a systematic lecture on signals and systems. It presents: analog and discrete signals, simple signal transformations, even and odd signal components, generalized signals, sampling and convolution of signals, as well as types of systems (with memory, without memory, linear, stationary, fixed constants, SLS, block) . System equations in time domain, convolution in linear systems, Laplace transform, Z transformation, transmittance, and system stability are presented. The representation of analog periodic signals by means of series, Fourier transform of analog and discrete signals as well as their applications and discrete Fourier transforms are also described. All issues are illustrated with carefully selected examples, various methods, procedures and computing techniques are given, commented on to better understand the topics discussed, and many tasks are included for self-solution.
Recipients of the book: students of electronics, electrical engineering and automation at universities and all interested in expanding knowledge in this field.

Table of Contents:

Introduction 9
1. Signals 15
1.1. Introductory information 15
1.2. Mathematical representation of signals 17
1.3. Periodic signals 21
1.4. Examples of elementary signals 27
1.4.1. Analog signals with limited energy and finite duration 27
1.4.2. Analog signals with limited energy and infinite duration 28
1.4.3. Analog signals with limited power 31
1.4.4. Discrete signals with limited energy and finite duration 35
1.4.5. Discrete signals with limited energy and infinite duration 37
1.4.6. Discrete signals with limited power 39
1.4.7. Other signals 40
1.5. Simple transformations of signals 41
1.5.1. Time shift 41
1.5.2. Changing the time scale 42
1.5.3. Inversion of time 43
1.5.4. Multiplying signals 44
1.6. Even and odd components of signals 45
1.7. Generalized signals 46
1.7.1. Distribution Dirac 47
1.7.2. Basic distribution properties of Dirac 50
1.7.3. Dirac distribution relationship with unit jump 51
1.7.4. Derivative of distribution 53
1.7.5. Examples and applications 54
1.8. Sampling of signals 56
1.9. Weave of signals 58
1.9.1. Weave discrete signals 58
1.9.2. Weave of continuous time signals 61
1.10. Tasks 64
Literature 70

2. Systems 71
2.1. Introductory information 71
2.2. Systems with and without memory 73
2.3. State variables 75
2.4. Stability 76
2.5. Linear systems 78
2.6. Stationary systems 83
2.7. Fixed systems 86
2; 8. SLS 88 systems
2.9. Block diagrams 91
2.10. Tasks 99
Literature 104

3. Equations of time-domain systems 106
3.1. SLS systems of continuous time 106
3.1.1. Input-output equations 106
3.1.2. First order systems 107
3.1.3. Second-order systems 113
3.1.4. Discussion 118
3.2. SLS discrete time systems 118
3.2..1. Input-output equations 118
3.2.2. First-order systems 120
3.3. Block diagrams and system equations 123
3.4. Discretization of continuous time equations 126
3.5. Tasks 130
Literature 138

4. Weave in linear systems 139
4.1. SLS discrete time systems 139
4.1.1. The convolution relation of input and output signals 139
4.1.2. Comparison of convolution description and description by means of differential equations 142
4.2. SLS systems of continuous time 142
4.3. Linear part-time systems 147
4.4. Tasks 149
Literature 151

5. Laplace transformation 152
5.1. Mathematical foundations 152
5.2. Properties of the Laplace transform 155
5.3. Reverse Lap1ace transformation - distribution to fractions 163
5.3.1. Introduction 163
5.3.2. Distribution for linear fractions 165
5.3.3. Distribution using square fractions 169
5.4. Application of transforms for analysis of SLS 171 systems
5.4.1. Solving differential equations 171
5.4.2. Second order systems 175
5.5. Tasks 178
Literature 185

6. Transformation From 186
6.1. Mathematical basis 186
6.2 .. Compound Z transform and Laplace transform 190
6.3. Transformation properties Z 192
6.4. Calculation of reverse transform Z 199
6.4.1. Introduction 199
6.4.2. Direct division method 200
6.4.3. Distribution for fractions 201
6.4.4. Distribution of X ( from ^-1 ) to linear fractions 201
6.4.5. Distribution of X ( from ^-1 ) using square fractions 204
6.4.6. An example of calculating the inverse transform of an irrational function 206
6..4.7. The distribution of functions X ( z ) 207
6.5. The use of Z transforms for analysis of SLS 209 systems
6.6. Tasks 215
Literature 220

7. Transmission 221
7.1. Transmission properties of continuous time systems 221
7.1.1. Transmittance of a continuous time system 221
7.1.2. Transmittance properties 225
7.1.3. Frequency characteristics 227
7.1.4. Transmittances of complex systems 232
7.1.5. Charts 241 charts
7.1.6. Asymptotic charts of Bode 245
7.2. Transmission properties of discrete time systems 249
7.2.1. Transmission of the discrete time system 249
7.2.2. Frequency response 253
7.2.3. Transmission of complex systems 259
7.3. Tasks 259
Literature 271

8. Stability 272
8.1. Introductory information 272
8.2. Stability of the continuous time system 273
8.2.1. Definition of stability 273
8.2.2. Stability conditions of the SLS 275 system
8.2.3. Hurwitz criterion 278
8.2.4. The Routh Criterion 283
8.2.5. The Nyquist criterion 286
8.3. Stability of discrete time systems 294
8.3.1. Definition of stability 294
8.3.2. Conditions for stability of the SLS 296 system
8.3.3. Criterion of the Jury 299
8.3.4. Supplement 303
8.3.5. The Nyquist 304 criterion
8.4. Tasks 306
Literature 311

9. Representation of analog periodic signals by means of series 313
9.1. Mathematical foundations 313
9.1.1. Signal spaces 313
9.1.2. Orthogonal signals 315
9.1,3. Approximation of signals 320
9.2. Fourier series 323
9.2.1. Trigonometric form 323
9.2.2. Exponential form 329
9.2.3. Properties of the Fourier series 335
9.3. Analysis of SLS systems with periodic excitations 340
9.4. Tasks 342
Literature 346

10. Fourier transform of analog signals and its applications 347
10.1. Mathematical foundations 347
10.2. Fourier transform in the limit sense 350
10.3. CTFT 352 properties
10.4. Frequency transmission 366
10.5. Frequency analysis of SLS 370 systems
10.6. Filtration 374
10.6.1. Basic concepts 374
10.6.2. Frequency transformation 378
10.6.3. Phase linearity 380
10.6.4. Perfect filter 381
10.7 Tasks 383
Literature 390

11. Sampling of signals 391
11.1. Pulse sampling 391
11.1.1. The sampling theorem 391
11.1.2. Recording of signal from samples 396
11.1.3. The spectrum of impulse-sampled signal - direct form 397
11.2. Reconstruction of the signal using interpolation 398
11.2.1. Sample-remembering system 398
11.2.2. A semi-linear approximation of the 400 signal
11.3. Frequency alignment 402
11.4. Spectrum sampling 406
11.5. Tasks 407
Literature 411

12. Fourier transform of discrete signals and its applications 412
12.1. Mathematical basis 412
12.2. Fourier transform of signals with finite power 416
12.3. DTFT 417 properties
12.4. Frequency transmission 425
12.5. Frequency analysis of SLS 427 systems
12.6. Discrete processing of analog signals 431
12.6.1. The principle of processing 431
12.6.2. Processing of discrete time 434 using SLS
12.6.3. Invariance impulse response 437
12.6.4. Examples 438
12.7. Tasks 442
Literature 477

13. Discrete Fourier transform 448
13.1. Mathematical basis of transformation 448
13.2. Spectrum estimation of analog signals 453
13.3. Discrete Fourier series 460
13.4. Features DFT 463
13.5. Circular weave 468
13.5.1. Weave in time domain 468
13.5.2. Weave in the frequency domain 471
13.6. Parseval's theorem 472
13.7. Tasks 473
Literature 476
Index 477