Signals and Systems

by Oppenheim, Alan V.; Willsky, Alan S.; Hamid, with S.

• ISBN13: 9780138147570
• ISBN10: 0138147574
• Edition: 2nd
• Format: Hardcover
• Copyright: 1996-08-06
• Publisher: Pearson

Summary

This authoritative book, highly regarded for its intellectual quality and contributions provides a solid foundation and life-long reference for anyone studying the most important methods of modern signal and system analysis.The major changes of the revision are reorganization of chapter material and the addition of a much wider range of difficulties.

Table of Contents

PREFACE	XVII	(8)
ACKNOWLEDGMENTS	XXV	(2)
FOREWORD	XXVII
1 SIGNALS AND SYSTEMS	1	(73)
1.0 Introduction	1	(1)
1.1 Continuous-Time and Discrete-Time Signals	1	(6)
1.1.1 Examples and Mathematical Representation	1	(4)
1.1.2 Signal Energy and Power	5	(2)
1.2 Transformations of the Independent Variable	7	(7)
1.2.1 Examples of Transformations of the Independent Variable	8	(3)
1.2.2 Periodic Signals	11	(2)
1.2.3 Even and Odd Signals	13	(1)
1.3 Exponential and Sinusoidal Signals	14	(16)
1.3.1 Continuous-Time Complex Exponential and Sinusoidal Signals	15	(6)
1.3.2 Discrete-Time Complex Exponential and Sinusoidal Signals	21	(4)
1.3.3 Periodicity Properties of Discrete-Time Complex Exponentials	25	(5)
1.4 The Unit Impulse and Unit Step Functions	30	(8)
1.4.1 The Discrete-Time Unit Impulse and Unit Step Sequences	30	(2)
1.4.2 The Continuous-Time Unit Step and Unit Impulse Functions	32	(6)
1.5 Continuous-Time and Discrete-Time Systems	38	(6)
1.5.1 Simple Examples of Systems	39	(2)
1.5.2 Interconnections of Systems	41	(3)
1.6 Basic System Properties	44	(12)
1.6.1 Systems with and without Memory	44	(1)
1.6.2 Invertibility and Inverse Systems	45	(1)
1.6.3 Causality	46	(2)
1.6.4 Stability	48	(2)
1.6.5 Time Invariance	50	(3)
1.6.6 Linearity	53	(3)
1.7 Summary	56	(1)
Problems	57	(17)
2 LINEAR TIME-INVARIANT SYSTEMS	74	(103)
2.0 Introduction	74	(1)
2.1 Discrete-Time LTI Systems: The Convolution Sum	75	(15)
2.1.1 The Representation of Discrete-Time Signals in Terms of Impulses	75	(2)
2.1.2 The Discrete-Time Unit Impulse Response and the Convolution-Sum Representation of LTI Systems	77	(13)
2.2 Continuous-Time LTI Systems: The Convolution Integral	90	(13)
2.2.1 The Representation of Continuous-Time Signals in Terms of Impulses	90	(4)
2.2.2 The Continuous-Time Unit Impulse Response and the Convolution Integral Representation of LTI Systems	94	(9)
2.3 Properties of Linear Time-Invariant Systems	103	(13)
2.3.1 The Commutative Property	104	(1)
2.3.2 The Distributive Property	104	(3)
2.3.3 The Associative Property	107	(1)
2.3.4 LTI Systems with and without Memory	108	(1)
2.3.5 Invertibility of LTI Systems	109	(3)
2.3.6 Causality for LTI Systems	112	(1)
2.3.7 Stability for LTI Systems	113	(2)
2.3.8 The Unit Step Response of an LTI System	115	(1)
2.4 Causal LTI Systems Described by Differential and Difference Equations	116	(11)
2.4.1 Linear Constant-Coefficient Differential Equations	117	(4)
2.4.2 Linear Constant-Coefficient Difference Equations	121	(3)
2.4.3 Block Diagram Representations of First-Order Systems Described by Differential and Difference Equations	124	(3)
2.5 Singularity Functions	127	(10)
2.5.1 The Unit Impulse as an Idealized Short Pulse	128	(3)
2.5.2 Defining the Unit Impulse through Convolution	131	(1)
2.5.3 Unit Doublets and Other Singularity Functions	132	(5)
2.6 Summary	137	(1)
Problems	137	(40)
3 FOURIER SERIES REPRESENTATION OF PERIODIC SIGNALS	177	(107)
3.0 Introduction	177	(1)
3.1 A Historical Perspective	178	(4)
3.2 The Response of LTI Systems to Complex Exponentials	182	(4)
3.3 Fourier Series Representation of Continuous-Time Periodic Signals	186	(9)
3.3.1 Linear Combinations of Harmonically Related Complex Exponentials	186	(4)
3.3.2 Determination of the Fourier Series Representation of a Continuous-Time Periodic Signal	190	(5)
3.4 Convergence of the Fourier Series	195	(7)
3.5 Properties of Continuous-Time Fourier Series	202	(9)
3.5.1 Linearity	202	(1)
3.5.2 Time Shifting	202	(1)
3.5.3 Time Reversal	203	(1)
3.5.4 Time Scaling	204	(1)
3.5.5 Multiplication	204	(1)
3.5.6 Conjugation and Conjugate Symmetry	204	(1)
3.5.7 Parseval's Relation for Continuous-Time Periodic Signals	205	(1)
3.5.8 Summary of Properties of the Continuous-Time Fourier Series	205	(1)
3.5.9 Examples	205	(6)
3.6 Fourier Series Representation of Discrete-Time Periodic Signals	211	(10)
3.6.1 Linear Combinations of Harmonically Related Complex Exponentials	211	(1)
3.6.2 Determination of the Fourier Series Representation of a Periodic Signal	212	(9)
3.7 Properties of Discrete-Time Fourier Series	221	(5)
3.7.1 Multiplication	222	(1)
3.7.2 First Difference	222	(1)
3.7.3 Parseval's Relation for Discrete-Time Periodic Signals	223	(1)
3.7.4 Examples	223	(3)
3.8 Fourier Series and LTI Systems	226	(5)
3.9 Filtering	231	(8)
3.9.1 Frequency-Shaping Filters	232	(4)
3.9.2 Frequency-Selective Filters	236	(3)
3.10 Examples of Continuous-Time Filters Described by Differential Equations	239	(5)
3.10.1 A Simple RC Lowpass Filter	239	(2)
3.10.2 A Simple RC Highpass Filter	241	(3)
3.11 Examples of Discrete-Time Filters Described by Difference Equations	244	(5)
3.11.1 First-Order Recursive Discrete-Time Filters	244	(1)
3.11.2 Nonrecursive Discrete-Time Filters	245	(4)
3.12 Summary	249	(1)
Problems	250	(34)
4 THE CONTINUOUS-TIME FOURIER TRANSFORM	284	(74)
4.0 Introduction	284	(1)
4.1 Representation of Aperiodic Signals: The Continuous-Time Fourier Transform	285	(11)
4.1.1 Development of the Fourier Transform Representation of an Aperiodic Signal	285	(4)
4.1.2 Convergence of Fourier Transforms	289	(1)
4.1.3 Examples of Continuous-Time Fourier Transforms	290	(6)
4.2 The Fourier Transform for Periodic Signals	296	(4)
4.3 Properties of the Continuous-Time Fourier Transform	300	(14)
4.3.1 Linearity	301	(1)
4.3.2 Time Shifting	301	(2)
4.3.3 Conjugation and Conjugate Symmetry	303	(3)
4.3.4 Differentiation and Integration	306	(2)
4.3.5 Time and Frequency Scaling	308	(1)
4.3.6 Duality	309	(3)
4.3.7 Parseval's Relation	312	(2)
4.4 The Convolution Property	314	(8)
4.4.1 Examples	317	(5)
4.5 The Multiplication Property	322	(6)
4.5.1 Frequency-Selective Filtering with Variable Center Frequency	325	(3)
4.6 Tables of Fourier Properties and of Basic Fourier Transform Pairs	328	(2)
4.7 Systems Characterized by Linear Constant-Coefficient Differential Equations	330	(3)
4.8 Summary	333	(1)
Problems	334	(24)
5 THE DISCRETE-TIME FOURIER TRANSFORM	358	(65)
5.0 Introduction	358	(1)
5.1 Representation of Aperiodic Signals: The Discrete-Time Fourier Transform	359	(8)
5.1.1 Development of the Discrete-Time Fourier Transform	359	(3)
5.1.2 Examples of Discrete-Time Fourier Transforms	362	(4)
5.1.3 Convergence Issues Associated with the Discrete-Time Fourier Transform	366	(1)
5.2 The Fourier Transform for Periodic Signals	367	(5)
5.3 Properties of the Discrete-Time Fourier Transform	372	(10)
5.3.1 Periodicity of the Discrete-Time Fourier Transform	373	(1)
5.3.2 Linearity of the Fourier Transform	373	(1)
5.3.3 Time Shifting and Frequency Shifting	373	(2)
5.3.4 Conjugation and Conjugate Symmetry	375	(1)
5.3.5 Differencing and Accumulation	375	(1)
5.3.6 Time Reversal	376	(1)
5.3.7 Time Expansion	377	(3)
5.3.8 Differentiation in Frequency	380	(1)
5.3.9 Parseval's Relation	380	(2)
5.4 The Convolution Property	382	(6)
5.4.1 Examples	383	(5)
5.5 The Multiplication Property	388	(2)
5.6 Tables of Fourier Transform Properties and Basic Fourier Transform Pairs	390	(1)
5.7 Duality	390	(6)
5.7.1 Duality in the Discrete-Time Fourier Series	391	(4)
5.7.2 Duality between the Discrete-Time Fourier Transform and the Continuous-Time Fourier Series	395	(1)
5.8 Systems Characterized by Linear Constant-Coefficient Difference Equations	396	(3)
5.9 Summary	399	(1)
Problems	400	(23)
6 TIME AND FREQUENCY CHARACTERIZATION OF SIGNALS AND SYSTEMS	423	(91)
6.0 Introduction	423	(1)
6.1 The Magnitude-Phase Representation of the Fourier Transform	423	(4)
6.2 The Magnitude-Phase Representation of the Frequency Response of LTI Systems	427	(12)
6.2.1 Linear and Nonlinear Phase	428	(2)
6.2.2 Group Delay	430	(6)
6.2.3 Log-Magnitude and Bode Plots	436	(3)
6.3 Time-Domain Properties of Ideal Frequency-Selective Filters	439	(5)
6.4 Time-Domain and Frequency-Domain Aspects of Nonideal Filters	444	(4)
6.5 First-Order and Second-Order Continuous-Time Systems	448	(13)
6.5.1 First-Order Continuous-Time Systems	448	(3)
6.5.2 Second-Order Continuous-Time Systems	451	(5)
6.5.3 Bode Plots for Rational Frequency Responses	456	(5)
6.6 First-Order and Second-Order Discrete-Time Systems	461	(11)
6.6.1 First-Order Discrete-Time Systems	461	(4)
6.6.2 Second-Order Discrete-Time Systems	465	(7)
6.7 Examples of Time- and Frequency-Domain Analysis of Systems	472	(10)
6.7.1 Analysis of an Automobile Suspension System	473	(3)
6.7.2 Examples of Discrete-Time Nonrecursive Filters	476	(6)
6.8 Summary	482	(1)
Problems	483	(31)
7 SAMPLING	514	(68)
7.0 Introduction	514	(1)
7.1 Representation of a Continuous-Time Signal by Its Samples: The Sampling Theorem	515	(7)
7.1.1 Impulse-Train Sampling	516	(4)
7.1.2 Sampling with a Zero-Order Hold	520	(2)
7.2 Reconstruction of a Signal from Its Samples Using Interpolation	522	(5)
7.3 The Effect of Undersampling: Aliasing	527	(7)
7.4 Discrete-Time Processing of Continuous-Time Signals	534	(11)
7.4.1 Digital Differentiator	541	(2)
7.4.2 Half-Sample Delay	543	(2)
7.5 Sampling of Discrete-Time Signals	545	(10)
7.5.1 Impulse-Train Sampling	545	(4)
7.5.2 Discrete-Time Decimation and Interpolation	549	(6)
7.6 Summary	555	(1)
Problems	556	(26)
8 COMMUNICATION SYSTEMS	582	(72)
8.0 Introduction	582	(1)
8.1 Complex Exponential and Sinusoidal Amplitude Modulation	583	(4)
8.1.1 Amplitude Modulation with a Complex Exponential Carrier	583	(2)
8.1.2 Amplitude Modulation with a Sinusoidal Carrier	585	(2)
8.2 Demodulation for Sinusoidal AM	587	(7)
8.2.1 Synchronous Demodulation	587	(3)
8.2.2 Asynchronous Demodulation	590	(4)
8.3 Frequency-Division Multiplexing	594	(3)
8.4 Single-Sideband Sinusoidal Amplitude Modulation	597	(4)
8.5 Amplitude Modulation with a Pulse-Train Carrier	601	(3)
8.5.1 Modulation of a Pulse-Train Carrier	601	(3)
8.5.2 Time-Division Multiplexing	604	(1)
8.6 Pulse-Amplitude Modulation	604	(7)
8.6.1 Pulse-Amplitude Modulated Signals	604	(3)
8.6.2 Intersymbol Interference in PAM Systems	607	(3)
8.6.3 Digital Pulse-Amplitude and Pulse-Code Modulation	610	(1)
8.7 Sinusoidal Frequency Modulation	611	(8)
8.7.1 Narrowband Frequency Modulation	613	(2)
8.7.2 Wideband Frequency Modulation	615	(2)
8.7.3 Periodic Square-Wave Modulating Signal	617	(2)
8.8 Discrete-Time Modulation	619	(4)
8.8.1 Discrete-Time Sinusoidal Amplitude Modulation	619	(4)
8.8.2 Discrete-Time Transmodulation	623	(1)
8.9 Summary	623	(2)
Problems	625	(29)
9 THE LAPLACE TRANSFORM	654	(87)
9.0 Introduction	654	(1)
9.1 The Laplace Transform	655	(7)
9.2 The Region of Convergence for Laplace Transforms	662	(8)
9.3 The Inverse Laplace Transform	670	(4)
9.4 Geometric Evaluation of the Fourier Transform from the Pole-Zero Plot	674	(8)
9.4.1 First-Order Systems	676	(1)
9.4.2 Second-Order Systems	677	(4)
9.4.3 All-Pass Systems	681	(1)
9.5 Properties of the Laplace Transform	682	(10)
9.5.1 Linearity of the Laplace Transform	683	(1)
9.5.2 Time Shifting	684	(1)
9.5.3 Shifting in the s-Domain	685	(1)
9.5.4 Time Scaling	685	(2)
9.5.5 Conjugation	687	(1)
9.5.6 Convolution Property	687	(1)
9.5.7 Differentiation in the Time Domain	688	(1)
9.5.8 Differentiation in the s-Domain	688	(2)
9.5.9 Integration in the Time Domain	690	(1)
9.5.10 The Initial- and Final-Value Theorems	690	(1)
9.5.11 Table of Properties	691	(1)
9.6 Some Laplace Transform Pairs	692	(1)
9.7 Analysis and Characterization of LTI Systems Using the Laplace Transform	693	(13)
9.7.1 Causality	693	(2)
9.7.2 Stability	695	(3)
9.7.3 LTI Systems Characterized by Linear Constant-Coefficient Differential Equations	698	(3)
9.7.4 Examples Relating System Behavior to the System Function	701	(2)
9.7.5 Butterworth Filters	703	(3)
9.8 System Function Algebra and Block Diagram Representations	706	(8)
9.8.1 System Functions for Interconnections of LTI Systems	707	(1)
9.8.2 Block Diagram Representations for Causal LTI Systems Described by Differential Equations and Rational System Functions	708	(6)
9.9 The Unilateral Laplace Transform	714	(6)
9.9.1 Examples of Unilateral Laplace Transforms	714	(2)
9.9.2 Properties of the Unilateral Laplace Transform	716	(3)
9.9.3 Solving Differential Equations Using the Unilateral Laplace Transform	719	(1)
9.10 Summary	720	(1)
Problems	721	(20)
10 THE Z-TRANSFORM	741	(75)
10.0 Introduction	741	(1)
10.1 The z-Transform	741	(7)
10.2 The Region of Convergence for the z-Transform	748	(9)
10.3 The Inverse z-Transform	757	(6)
10.4 Geometric Evaluation of the Fourier Transform from the Pole-Zero Plot	763	(4)
10.4.1 First-Order Systems	763	(2)
10.4.2 Second-Order Systems	765	(2)
10.5 Properties of the z-Transform	767	(7)
10.5.1 Linearity	767	(1)
10.5.2 Time Shifting	767	(1)
10.5.3 Scaling in the z-Domain	768	(1)
10.5.4 Time Reversal	769	(1)
10.5.5 Time Expansion	769	(1)
10.5.6 Conjugation	770	(1)
10.5.7 The Convolution Property	770	(2)
10.5.8 Differentiation in the z-Domain	772	(1)
10.5.9 The Initial-Value Theorem	773	(1)
10.5.10 Summary of Properties	774	(1)
10.6 Some Common z-Transform Pairs	774	(1)
10.7 Analysis and Characterization of LTI Systems Using z-Transforms	774	(9)
10.7.1 Causality	776	(1)
10.7.2 Stability	777	(2)
10.7.3 LTI Systems Characterized by Linear Constant-Coefficient Difference Equations	779	(2)
10.7.4 Examples Relating System Behavior to the System Function	781	(2)
10.8 System Function Algebra and Block Diagram Representations	783	(6)
10.8.1 System Functions for Interconnections of LTI Systems	784	(1)
10.8.2 Block Diagram Representations for Causal LTI Systems Described by Difference Equations and Rational System Functions	784	(5)
10.9 The Unilateral z-Transform	789	(7)
10.9.1 Examples of Unilateral z-Transforms and Inverse Transforms	790	(2)
10.9.2 Properties of the Unilateral z-Transform	792	(3)
10.9.3 Solving Difference Equations Using the Unilateral z-Transform	795	(1)
10.10 Summary	796	(1)
Problems	797	(19)
11 LINEAR FEEDBACK SYSTEMS	816	(93)
11.0 Introduction	816	(3)
11.1 Linear Feedback Systems	819	(1)
11.2 Some Applications and Consequences of Feedback	820	(12)
11.2.1 Inverse System Design	820	(1)
11.2.2 Compensation for Nonideal Elements	821	(2)
11.2.3 Stabilization of Unstable Systems	823	(3)
11.2.4 Sampled-Data Feedback Systems	826	(2)
11.2.5 Tracking Systems	828	(2)
11.2.6 Destabilization Caused by Feedback	830	(2)
11.3 Root-Locus Analysis of Linear Feedback Systems	832	(14)
11.3.1 An Introductory Example	833	(1)
11.3.2 Equation for the Closed-Loop Poles	834	(2)
11.3.3 The End Points of the Root Locus: The Closed-Loop Poles for K = 0 and ¦K¦ = +(XXX)	836	(1)
11.3.4 The Angle Criterion	836	(5)
11.3.5 Properties of the Root Locus	841	(5)
11.4 The Nyquist Stability Criterion	846	(12)
11.4.1 The Encirclement Property	847	(3)
11.4.2 The Nyquist Criterion for Continuous-Time LTI Feedback Systems	850	(6)
11.4.3 The Nyquist Criterion for Discrete-Time LTI Feedback Systems	856	(2)
11.5 Gain and Phase Margins	858	(8)
11.6 Summary	866	(1)
Problems	867	(42)
APPENDIX PARTIAL-FRACTION EXPANSION	909	(12)
BIBLIOGRAPHY	921	(10)
ANSWERS	931	(10)
INDEX	941