SCHAUM'S OUTLINES OF Theory and Problems
of Signals and Systems
Contents
Chapter
1. Signals and Systems 1
1.1
Introduction 1
1.2
Signals and Classification of Signals 1
1.3
Basic Continuous-Time Signals 6
1.4
Basic Discrete-Time Signals 12
1.5
Systems and Classification of Systems 16
Solved
Problems 19
Chapter
2. Linear Time-Invariant Systems 56
2.1
Introduction 56
2.2
Response of a Continuous-Time LTI System and the Convolution Integral 56
2.3
Properties of Continuous-Time LTI Systems 58
2.4
Eigenfunctions of Continuous-Time LTI Systems 59
2.5
Systems Described by Differential Equations 60
2.6
Response of a Discrete-Time LTI System and Convolution Sum 61
2.7
Properties of Discrete-Time LTI Systems 63
2.8
Eigenfunctions of Discrete-Time LTI Systems 64
2.9
Systems Described by Difference Equations 65
Solved
Problems 66
Chapter
3. Laplace Transform and Continuous-Time LTI Systems 110
3.1
Introduction 110
3.2
The Laplace Transform 110
3.3
Laplace Transforms of Some Common Signals 114
3.4
Properties of the Laplace Transform 114
3.5
The Inverse Laplace Transform 119
3.6
The System Function 121
3.7
The Unilateral Laplace Transform 124
Solved
Problems 127
Chapter
4. The z-Transform and Discrete-Time LTI Systems 165
4.1
Introduction 165
4.2
The z-Transform 165
4.3
z-Transforms of Some Common Sequences 169
4.4
Properties of the z-Transform 171
4.5
The Inverse z-Transform 173
4.6
The System Function of Discrete-Time LTI Systems 175
4.7
The Unilateral z-Transform 177
Solved
Problems 178
Chapter
5. Fourier Analysis of Continuous-Time Signals and Systems 211
5.1
Introduction 211
5.2
Fourier Series Representation of Periodic Signals 211
5.3
The Fourier Transform 214
5.4
Properties of the Continuous-Time Fourier Transform 219
vii
5.5
The Frequency Response of Continuous-Time LTI Systems 223
5.6
Filtering 227
5.7
Bandwidth 230
Solved
Problems 231
Chapter
6. Fourier Analysis of Discrete-Time Signals and Systems 288
6.1
Introduction 288
6.2
Discrete Fourier Series 288
6.3
The Fourier Transform 291
6.4
Properties of the Fourier Transform 295
6.5
The Frequency Response of Discrete-Time LTI Systems 300
6.6
System Response to Sampled Continuous-Time Sinusoids 302
6.7
Simulation 303
6.8
The Discrete Fourier Transform 305
Solved
Problems 308
Chapter
7. State Space Analysis 365
7.1
Introduction 365
7.2
The Concept of State 365
7.3
State Space Representation of Discrete-Time LTI Systems 366
7.4
State Space Representation of Continuous-Time LTI Systems 368
7.5
Solutions of State Equations for Discrete-Time LTI Systems 371
7.6
Solutions of State Equations for Continuous-Time LTI Systems 374
Solved
Problems 377
Appendix
A. Review of Matrix Theory 428
A.1
Matrix Notation and Operations 428
A.2
Transpose and Inverse 431
A.3
Linear Independence and Rank 432
A.4
Determinants 433
A.5
Eigenvalues and Eigenvectors 435
A.6
Diagonalization and Similarity Transformation 436
A.7
Functions of a Matrix 437
A.8
Differentiation and Integration of Matrices 444
Appendix
B. Properties of Linear Time-Invariant Systems and Various Transforms 445
B.1
Continuous-Time LTI Systems 445
B.2
The Laplace Transform 445
B.3
The Fourier Transform 447
B.4
Discrete-Time LTI Systems 449
B.5
The z-Transform 449
B.6
The Discrete-Time Fourier Transform 451
B.7
The Discrete Fourier Transform 452
B.8
Fourier Series 453
B.9
Discrete Fourier Series 454
Appendix
C. Review of Complex Numbers 455
C.1
Representation of Complex Numbers 455
C.2
Addition, Multiplication, and Division 456
C.3
The Complex Conjugate 456
C.4
Powers and Roots of Complex Numbers 456
Appendix
D. Useful Mathematical Formulas 458
D.1
Summation Formulas 458
D.2
Euler's Formulas 458
viii
D.3
Trigonometric Identities 458
D.4
Power Series Expansions 459
D.5
Exponential and Logarithmic Functions 459
D.6
Some Definite Integrals 460
Index
461
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