Continuous and Discrete Signals and Systems

by Soliman, Samir S.; Mandyam D. Srinath

• ISBN13: 9780135184738
• ISBN10: 0135184738
• Edition: 2nd
• Format: Paperback
• Copyright: 1997-12-22
• Publisher: Pearson

Summary

This complete introductory book assists readers in developing the ability to understand and analyze both continuous and discrete-time systems.The author presents the most widely used techniques of signal and system analysis in a highly readable and understandable fashion.For anyone interested in Signals & Systems, and Transform Theory.

Table of Contents

PREFACE	xiii
1 REPRESENTING SIGNALS	1	(40)
1.1 Introduction	1	(1)
1.2 Continuous-Time vs. Discrete-Time Signals	2	(2)
1.3 Periodic vs. Aperiodic Signals	4	(3)
1.4 Energy and Power Signals	7	(3)
1.5 Transformations of the Independent Variable	10	(9)
1.5.1 The Shifting Operation	10	(3)
1.5.2 The Reflection Operation	13	(4)
1.5.3 The Time-Scaling Operation	17	(2)
1.6 Elementary Signals	19	(13)
1.6.1 The Unit Step Function	19	(2)
1.6.2 The Ramp Function	21	(1)
1.6.3 The Sampling Function	22	(1)
1.6.4 The Unit Impulse Function	22	(8)
1.6.5 Derivatives of the Impulse Function	30	(2)
1.7 Other Types of Signals	32	(1)
1.8 Summary	33	(2)
1.9 Checklist of Important Terms	35	(1)
1.10 Problems	35	(6)
2 CONTINUOUS-TIME SYSTEMS	41	(65)
2.1 Introduction	41	(1)
2.2 Classification of Continuous-Time Systems	42	(10)
2.2.1 Linear and Nonlinear Systems	42	(4)
2.2.2 Time-Varying and Time-Invariant Systems	46	(1)
2.2.3 Systems with and without Memory	47	(1)
2.2.4 Causal Systems	48	(2)
2.2.5 Invertibility and Inverse Systems	50	(1)
2.2.6 Stable Systems	51	(1)
2.3 Linear Time-Invariant Systems	52	(12)
2.3.1 The Convolution Integral	52	(6)
2.3.2 Graphical Interpretation of Convolution	58	(6)
2.4 Properties of Linear, Time-Invariant Systems	64	(3)
2.4.1 Memoryless LTI Systems	64	(1)
2.4.2 Causal LTI Systems	64	(1)
2.4.3 Invertible LTI Systems	65	(1)
2.4.4 Stable LTI Systems	65	(2)
2.5 Systems Described by Differential Equations	67	(9)
2.5.1 Linear, Constant-Coefficient Differential Equations	67	(1)
2.5.2 Basic System Components	68	(2)
2.5.3 Simulation Diagrams for Continuous-Time Systems	70	(3)
2.5.4 Finding the Impulse Response	73	(3)
2.6 State-Variable Representation	76	(18)
2.6.1 State Equations	77	(1)
2.6.2 Time-Domain Solution of the State Equations	78	(8)
2.6.3 State Equations in First Canonical Form	86	(1)
2.6.4 State Equations in Second Canonical Form	87	(4)
2.6.5 Stability Considerations	91	(3)
2.7 Summary	94	(2)
2.8 Checklist of Important Terms	96	(1)
2.9 Problems	96	(10)
3 FOURIER SERIES	106	(56)
3.1 Introduction	106	(1)
3.2 Orthogonal Representations of Signals	107	(5)
3.3 The Exponential Fourier Series	112	(10)
3.4 Dirichlet Conditions	122	(3)
3.5 Properties of Fourier Series	125	(10)
3.5.1 Least Squares Approximation Property	125	(2)
3.5.2 Effects of Symmetry	127	(2)
3.5.3 Linearity	129	(1)
3.5.4 Product of Two Signals	130	(1)
3.5.5 Convolution of Two Signals	131	(1)
3.5.6 Parseval's Theorem	132	(1)
3.5.7 Shift in Time	133	(1)
3.5.8 Integration of Periodic Signals	134	(1)
3.6 Systems with Periodic Inputs	135	(7)
3.7 The Gibbs Phenomenon	142	(3)
3.8 Summary	145	(3)
3.9 Checklist of Important Terms	148	(1)
3.10 Problems	148	(12)
3.11 Computer Problems	160	(2)
4 THE FOURIER TRANSFORM	162	(62)
4.1 Introduction	162	(1)
4.2 The Continuous-Time Fourier Transform	163	(8)
4.2.1 Development of the Fourier Transform	163	(2)
4.2.2 Existence of the Fourier Transform	165	(1)
4.2.3 Examples of the Continuous-Time Fourier Transform	166	(5)
4.3 Properties of the Fourier Transform	171	(19)
4.3.1 Linearity	171	(2)
4.3.2 Symmetry	173	(2)
4.3.3 Time Shifting	175	(1)
4.3.4 Time Scaling	175	(2)
4.3.5 Differentiation	177	(2)
4.3.6 Energy of Aperiodic Signals	179	(2)
4.3.7 Convolution	181	(3)
4.3.8 Duality	184	(1)
4.3.9 Modulation	185	(5)
4.4 Applications of the Fourier Transform	190	(14)
4.4.1 Amplitude Modulation	190	(2)
4.4.2 Multiplexing	192	(2)
4.4.3 The Sampling Theorem	194	(6)
4.4.4 Signal Filtering	200	(4)
4.5 Duration-Bandwidth Relationships	204	(7)
4.5.1 Definitions of Duration and Bandwidth	204	(4)
4.5.2 The Uncertainty Principle	208	(3)
4.6 Summary	211	(1)
4.7 Checklist of Important Terms	212	(1)
4.8 Problems	212	(12)
5 THE LAPLACE TRANSFORM	224	(54)
5.1 Introduction	224	(1)
5.2 The Bilateral Laplace Transform	225	(3)
5.3 The Unilateral Laplace Transform	228	(1)
5.4 Bilateral Transforms Using Unilateral Transforms	229	(2)
5.5 Properties of the Unilateral Laplace Transform	231	(15)
5.5.1 Linearity	232	(1)
5.5.2 Time Shifting	232	(1)
5.5.3 Shifting in the s Domain	233	(1)
5.5.4 Time Scaling	234	(1)
5.5.5 Differentiation in the Time Domain	234	(3)
5.5.6 Integration in the Time Domain	237	(1)
5.5.7 Differentiation in the s Domain	238	(1)
5.5.8 Modulation	239	(1)
5.5.9 Convolution	240	(3)
5.5.10 Initial-Value Theorem	243	(1)
5.5.11 Final-Value Theorem	244	(2)
5.6 The Inverse Laplace Transform	246	(4)
5.7 Simulation Diagrams for Continuous-Time Systems	250	(7)
5.8 Applications of the Laplace Transform	257	(6)
5.8.1 Solution of Differential Equations	257	(1)
5.8.2 Application to RLC Circuit Analysis	258	(2)
5.8.3 Application to Control	260	(3)
5.9 State Equations and the Laplace Transform	263	(3)
5.10 Stability in the s Domain	266	(2)
5.11 Summary	268	(2)
5.12 Checklist of Important Terms	270	(1)
5.13 Problems	270	(8)
6 DISCRETE-TIME SYSTEMS	278	(51)
6.1 Introduction	278	(4)
6.1.1 Classification of Discrete-Time Signals	279	(2)
6.1.2 Transformations of the Independent Variable	281	(1)
6.2 Elementary Discrete-Time Signals	282	(5)
6.2.1 Discrete Impulse and Step Functions	283	(1)
6.2.2 Exponential Sequences	284	(3)
6.3 Discrete-Time Systems	287	(7)
6.4 Periodic Convolution	294	(4)
6.5 Difference-Equation Representation of Discrete-Time Systems	298	(8)
6.5.1 Homogeneous Solution of the Difference Equation	299	(3)
6.5.2 The Particular Solution	302	(3)
6.5.3 Determination of the Impulse Response	305	(1)
6.6 Simulation Diagrams for Discrete-Time Systems	306	(4)
6.7 State-Variable Representation of Discrete-Time Systems	310	(6)
6.7.1 Solution of State-Space Equations	313	(3)
6.7.2 Impulse Response of Systems Described by State Equation	316	(1)
6.8 Stability of Discrete-Time Systems	316	(2)
6.9 Summary	318	(2)
6.10 Checklist of Important Terms	320	(1)
6.11 Problems	320	(9)
7 FOURIER ANALYSIS OF DISCRETE-TIME SYSTEMS	329	(46)
7.1 Introduction	329	(2)
7.2 Fourier-Series Representation of Discrete-Time Periodic Signals	331	(9)
7.3 The Discrete-Time Fourier Transform	340	(5)
7.4 Properties of the Discrete-Time Fourier Transform	345	(6)
7.4.1 Periodicity	345	(1)
7.4.2 Linearity	345	(1)
7.4.3 Time and Frequency Shifting	345	(1)
7.4.4 Differentiation in Frequency	346	(1)
7.4.5 Convolution	346	(4)
7.4.6 Modulation	350	(1)
7.4.7 Fourier Transform of Discrete-Time Periodic Sequences	350	(1)
7.5 Fourier Transform of Sampled Continuous-Time Signals	351	(16)
7.5.1 Reconstruction of Sampled Signals	356	(3)
7.5.2 Sampling-Rate Conversion	359	(5)
7.5.3 A/D and D/A Conversion	364	(3)
7.6 Summary	367	(2)
7.7 Checklist of Important Terms	369	(1)
7.8 Problems	369	(6)
8 THE Z-TRANSFORM	375	(44)
8.1 Introduction	375	(1)
8.2 The Z-Transform	376	(2)
8.3 Convergence of the Z-Transform	378	(5)
8.4 Properties of the Z-Transform	383	(9)
8.4.1 Linearity	385	(1)
8.4.2 Time Shifting	386	(1)
8.4.3 Frequency Scaling	387	(1)
8.4.4 Differentiation with Respect to z	388	(1)
8.4.5 Initial Value	389	(1)
8.4.6 Final Value	389	(1)
8.4.7 Convolution	390	(2)
8.5 The Inverse Z-Transform	392	(6)
8.5.1 Inversion by a Power-Series Expansion	394	(1)
8.5.2 Inversion by Partial-Fraction Expansion	395	(3)
8.6 Z-Transfer Functions of Causal Discrete-Time Systems	398	(4)
8.7 Z-Transform Analysis of State-Variable Systems	402	(8)
8.8 Relation Between the Z-Transform and the Laplace Transform	410	(1)
8.9 Summary	411	(3)
8.10 Checklist of Important Terms	414	(1)
8.11 Problems	414	(5)
9 THE DISCRETE FOURIER TRANSFORM	419	(33)
9.1 Introduction	419	(2)
9.2 The Discrete Fourier Transform and Its Inverse	421	(1)
9.3 Properties of the DFT	422	(4)
9.3.1 Linearity	422	(1)
9.3.2 Time Shifting	422	(1)
9.3.3 Alternative Inversion Formula	423	(1)
9.3.4 Time Convolution	423	(1)
9.3.5 Relation to the Discrete-Time Fourier and Z-Transforms	424	(1)
9.3.6 Matrix Interpretation of the DFT	425	(1)
9.4 Linear Convolution Using the DFT	426	(2)
9.5 Fast Fourier Transforms	428	(8)
9.5.1 The Decimation-in-Time Algorithm	429	(4)
9.5.2 The Decimation-in-Frequency Algorithm	433	(3)
9.6 Spectral Estimation of Analog Signals Using the DFT	436	(9)
9.7 Summary	445	(3)
9.8 Checklist of Important Terms	448	(1)
9.9 Problems	448	(4)
10 DESIGN OF ANALOG AND DIGITAL FILTERS	452	(33)
10.1 Introduction	452	(3)
10.2 Frequency Transformations	455	(2)
10.3 Design of Analog Filters	457	(11)
10.3.1 The Butterworth Filter	458	(4)
10.3.2 The Chebyshev Filter	462	(6)
10.4 Digital Filters	468	(14)
10.4.1 Design of IIR Digital Filters Using Impulse Invariance	469	(4)
10.4.2 IIR Design Using the Bilinear Transformation	473	(2)
10.4.3 FIR Filter Design	475	(6)
10.4.4 Computer-Aided Design of Digital Filters	481	(1)
10.5 Summary	482	(1)
10.6 Checklist of Important Terms	483	(1)
10.7 Problems	483	(2)
APPENDIX A COMPLEX NUMBERS	485	(6)
A.1 Definition	485	(2)
A.2 Arithmetic Operations	487	(2)
A.2.1 Addition and Subtraction	487	(1)
A.2.2 Multiplication	487	(1)
A.2.3 Division	488	(1)
A.3 Powers and Roots of Complex Numbers	489	(1)
A.4 Inequalities	490	(1)
APPENDIX B MATHEMATICAL RELATIONS	491	(11)
B.1 Trigonometric Identities	491	(1)
B.2 Exponential and Logarithmic Functions	492	(1)
B.3 Special Functions	493	(1)
B.3.1 Gamma Functions	493	(1)
B.3.2 Incomplete Gamma Functions	494	(1)
B.3.3 Beta Functions	494	(1)
B.4 Power-Series Expansion	494	(1)
B.5 Sums of Powers of Natural Numbers	495	(1)
B.5.1 Sums of Binomial Coefficients	496	(1)
B.5.2 Series of Exponentials	496	(1)
B.6 Definite Integrals	496	(2)
B.7 Indefinite Integrals	498	(4)
APPENDIX C ELEMENTARY MATRIX THEORY	502	(10)
C.1 Basic Definition	502	(1)
C.2 Basic Operations	503	(1)
C.2.1 Matrix Addition	503	(1)
C.2.2 Differentiation and Integration	503	(1)
C.2.3 Matrix Multiplication	503	(1)
C.3 Special Matrices	504	(2)
C.4 The Inverse of a Matrix	506	(1)
C.5 Eigenvalues and Eigenvectors	507	(1)
C.6 Functions of a Matrix	508	(4)
APPENDIX D PARTIAL FRACTIONS	512	(7)
D.1 Case I: Nonrepeated Linear Factors	513	(1)
D.2 Case II: Repeated Linear Factors	514	(2)
D.3 Case III: Nonrepeated Irreducible Second-Degree Factors	516	(1)
D.4 Case IV: Repeated Irreducible Second-Degree Factors	517	(2)
BIBLIOGRAPHY	519	(2)
INDEX	521