Linear Systems and Signals

by Lathi, B. P.

• ISBN13: 9780195151299
• ISBN10: 0195151291
• Format: Hardcover
• Copyright: 2001-10-04
• Publisher: Oxford University Press

Summary

This introductory level book gives comprehensive treatment to signals andlinear systems. It emphasizes the physical appreciation of concepts rather thanthe mere mathematical manipulation of symbols. Mathematics is used to enhancephysical and intuitive understanding, instead of to prove axiomatic theory. Thisconveniently organized book is divided into five parts and allows for theflexible teaching of discrete-time and continuous-time systems. Whereverpossible, theoretical results are interpreted heuristically and are supported bycarefully chosen examples and analogies.

Table of Contents

Preface	xi
PART I: INTRODUCTION
B. Background	3	(646)
Complex Numbers	3	(13)
A Historical Note	3	(3)
Algebra of Complex Numbers	6	(10)
Sinusoids	16	(9)
Addition of Sinusoids	19	(3)
Sinusoids in Terms of Exponentials: Euler's Formula	22	(1)
Generalized Sinusoids: Complex Frequency	23	(2)
Sketching Signals	25	(4)
Monotonic Exponentials	25	(1)
Exponentially Varying Sinusoid	26	(2)
Amplitude Modulated Sinusoid	28	(1)
Some Useful Signal Operations	29	(5)
Time-Shifting	29	(1)
Time-Scaling	30	(2)
Time-Inversion	32	(1)
Combined Operations	33	(1)
Mathematical Description of a Signal from Its Sketch	34	(3)
Even and Odd Functions	37	(2)
Some Properties of Even and Odd Functions	38	(1)
Even and Odd Components of a Signal	38	(1)
Cramer's Rule	39	(2)
Partial fraction expansion	41	(10)
Method of Clearing Fractions	42	(1)
Heaviside ``Cover-up'' Method	43	(5)
Mixture of Clearing Fractions and Heaviside ``Cover-up''	48	(2)
Partial Fraction Expansion of Improper F(x) with m = n	50	(1)
Modified Partial Fractions	51	(1)
Vectors and Matrices	51	(13)
Some Definitions and Properties	53	(1)
Matrix Algebra	54	(5)
Derivatives and Integrals of a Matrix	59	(1)
Characteristic Equation of a Matrix: Cayley-Hamilton Theorem	60	(2)
Computation of an Exponential and a Power of a Matrix	62	(2)
Miscellaneous	64	(5)
L'Hopital's Rule	64	(1)
Taylor and Maclaurin Series	64	(1)
Power Series Expansion	64	(1)
Sums	64	(1)
Complex Numbers	65	(1)
Trigonometric Identities	65	(1)
Indefinite Integrals	66	(1)
Differentiation Table	67	(1)
Some Useful Constants	67	(1)
Solution of Quadratic and Cubic Equations	67	(2)
Introduction to Systems	69	(36)
Signals and Systems	69	(3)
Classification of Systems	72	(10)
Linear and Nonlinear Systems	72	(4)
Time-Invariant and Time-varying Parameter Systems	76	(1)
Instantaneous and Dynamic Systems	77	(1)
Causal and Noncausal Systems	77	(3)
Lumped-Parameter and Distributed Parameter Systems	80	(1)
Continuous-Time and Discrete-Time Systems	80	(1)
Analog and Digital Systems	81	(1)
System Model: Input-output Description	82	(8)
Electrical Systems	83	(2)
Mechanical Systems	85	(4)
Electromechanical Systems	89	(1)
Simultaneous Differential Equations	90	(2)
Internal and External Description of a System	92	(4)
Large Systems	95	(1)
State-Space Description of a System	96	(2)
Summary	98	(7)
PART II: TIME-DOMAIN ANALYSIS OF LTI SYSTEMS
Time-Domain Analysis: Continuous-Time Systems	105	(84)
Introduction	105	(2)
System Response to Internal Conditions: Zero-Input response	107	(13)
Some Insights Into the Zero-Input Behavior of a System	118	(2)
Unit Impulse Function	120	(7)
The Unit Impulse response h(t)	123	(4)
System Response to External Input: Zero-State Response	127	(23)
The Convolution Integral	130	(8)
Graphical Understanding of Convolution	138	(11)
Why Convolution? An Intuitive Explanation of System response	149	(1)
Some Reflections on the Use of Impulse Function	150	(1)
Numerical Convolution	150	(6)
System Stability	156	(5)
System Response to Bounded Inputs	159	(2)
Intuitive Insights In System Behavior	161	(9)
Dependence of System Behavior on Characteristic Modes	161	(2)
Response-Time of a System: The System Time-Constant	163	(1)
Time-Constant and Rise-Time of a System	164	(1)
Time-Constant and Filtering	165	(1)
Time-Constant and Pulse Dispersion (Spreading)	166	(1)
Time-Constant and Rate of Information transmission	167	(1)
Resonance Phenomenon	167	(3)
Classical Solution of Differential equations	170	(7)
Forced response: Method of Undetermined Coefficients	170	(7)
Determining The Impulse Response	177	(2)
Summary	179	(10)
Time-Domain Analysis: Discrete-Time Systems	189	(60)
Discrete-Time Systems	189	(6)
Discrete-Time System Equations	195	(5)
Initial Conditions and Interative solution of Difference Equations	196	(4)
System Response to Internal Conditions: Zero-Input Response	200	(8)
Unit Impulse Response h[k]	208	(5)
Determining Impulse Response h[k]	209	(4)
System Response to External Input: Zero-State Response	213	(10)
Graphical Procedure for Convolution Sum	220	(3)
System Stability	223	(7)
System Response to Bounded Inputs	229	(1)
Intuitive insights In System Behavior	230	(1)
Classical Solution of Linear Difference Equations	230	(5)
Determining Impulse Response	235	(2)
Summary	237	(12)
PART III: FREQUENCY-DOMAIN (TRANSFORM) ANALYSIS OF LTI SYSTEMS
Continuous-Time Systems: Laplace Transform Analysis	249	(104)
The Laplace Transform	249	(16)
Finding Inverse Transform	255	(10)
Some Properties of the Laplace Transform	265	(9)
Transform Solution of Differential and Integro-Differential Equations	274	(11)
The Zero-State Response of LTIC Systems: The Transfer Function	278	(3)
Relationship Between h(t) and H(s)	281	(1)
Frequency-Domain Interpretation of the Transfer function	282	(2)
Physical Interpretation of the Laplace Transform	284	(1)
Analysis of Electrical Networks: Transformed Network Method	285	(13)
Analysis of Active Circuits	294	(3)
Initial and Final Values	297	(1)
Block Diagrams	298	(5)
System Realization	303	(12)
Canonical or Direct-Form Realization	304	(5)
Cascade and Parallel Realizations	309	(3)
System Realization Using Operational Amplifiers	312	(3)
Frequency-Response of an LTIC System	315	(14)
Dependence of Frequency Response on Poles and Zeros of H(s)	322	(7)
Bilateral Laplace Transform	329	(8)
Linear System Analysis Using Bilateral Transform	335	(2)
Second Canonical Realization	337	(2)
Summary	339	(14)
Discrete-Time Systems: Z-Transform Analysis	353	(62)
The Z-Transform	353	(10)
Finding the Inverse Transform	357	(6)
Some Properties of the Z-Transform	363	(5)
Z-Transform solution of Difference equations	368	(9)
Zero-State Response of LTID Systems: The Transfer Function	372	(3)
Relationship Between h[k] and H[z]	375	(1)
Frequency-Domain Interpretation of the Transfer Function	376	(1)
Physical Interpretation of the Z-Transform	376	(1)
System Realization	377	(3)
Frequency Response of Discrete-Time Systems	380	(16)
Periodic Nature of Frequency Response	385	(1)
Aliasing and Sampling Rate	386	(2)
Dependence of Frequency Response on Poles and Zeros of H[z]	388	(8)
Connection Between the Z-Transform and Laplace Transform	396	(3)
Bilateral Z-Transform	399	(5)
Analysis of LTID Systems Using Bilateral Z-Transform	402	(2)
Summary	404	(11)
PART IV: SIGNAL ANALYSIS
Continuous-Time Signal Analysis: The Fourier Series	415	(59)
Representation of Periodic Signals by Trigonometric Fourier Series	415	(23)
The Fourier Spectrum	421	(8)
Effect of Symmetry	429	(2)
Determining the Fundamental Frequency and Period	431	(1)
Role of Amplitude and Phase Spectra in Wave Shaping	432	(6)
Exponential Fourier Series	438	(11)
Exponential Fourier Spectra	442	(6)
Why Use exponentials?	448	(1)
Iternate View of Fourier Representation: Signal-Vector Analogy	449	(18)
Vector Comparison	449	(2)
Signal Comparison	451	(2)
Orthogonal Vector Space	453	(2)
Orthogonal Signal Space	455	(6)
Gibbs Phenomenon	461	(3)
Some Examples of Orthogonal functions	464	(3)
Summary	467	(7)
Continuous-Time Signal Analysis: The Fourier Transform	474	(58)
Nonperiodic Signal Representation By Fourier Integral	474	(6)
Physical Appreciation of the Fourier Transform	480	(4)
Connection Between the Fourier and Laplace Transforms	482	(2)
Transform of Some Useful Functions	484	(6)
Some Properties of the Fourier Transform	490	(16)
Symmetry of the Direct and Inverse Transform Operation: Time-Frequency Duality	492	(1)
Duality Property	493	(1)
Scaling Property	494	(2)
Time-Shifting Property	496	(2)
Frequency-shifting Property	498	(4)
Convolution	502	(1)
Time-Differentiation and Integration	503	(3)
LTI System Analysis by Fourier Transform	506	(2)
Signal Distortion During Transmission	508	(3)
Ideal and Practical Filters	511	(3)
Thinking in Time-Domain and Frequency-Domain: Two-Dimentional View of Signals and Systems	514	(1)
Signal Energy	515	(3)
Data Truncation: Window Functions	518	(4)
Summary	522	(10)
Sampling	532	(33)
The Sampling Theorem	532	(10)
Practical Difficulties in Signal Reconstruction	534	(4)
Some Applications of the Sampling Theorem	538	(4)
Dual of the Sampling Theorem: Spectral Sampling	542	(1)
Numerical Computation of Fourier Transform: The Discrete Fourier Transform (DFT)	543	(16)
Some Properties of DFT	552	(5)
Fast Fourier Transform (FFT)	557	(2)
Summary	559	(6)
Analysis of Discrate-Time Signals	565	(30)
Discrete-Time Periodic Signals	565	(9)
Representation of Periodic Signals: Discrete-Time Fourier Series	566	(8)
Nonperiodic Signals: The Discrete-Time Fourier Transform (DTFT)	574	(8)
Relationship of the DTFT to Z-transform	581	(1)
Properties of DTFT	582	(1)
LTID System analysis by DTFT	583	(2)
Relationships Among Various Transforms	585	(3)
Derivation of the Z-Transform Pair	588	(1)
Summary	589	(6)
PART V: STATE-SPACE ANALYSIS
State-Space Analysis	595	(54)
Introduction	595	(3)
Systematic Procedure for Determining State Equations	598	(12)
Electrical Circuits	599	(2)
State Equations From Transfer Function	601	(9)
Solution of State Equations	610	(12)
Laplace Transform Solution of State Equations	610	(5)
Time-Domain solution of State Equations	615	(7)
Linear Transformation of State Vector	622	(8)
Diagonalization of Matrix A	626	(4)
Controllability and Observability	630	(6)
Inadequacy of transfer function Description	635	(1)
State-Space Analysis of Discrete-Time Systems	636	(6)
Solution in State-Space	638	(3)
Z-Transform Solution	641	(1)
Summary	642	(7)
Supplementary Reading	649	(2)
Index	651