Linear Systems and Signals

by Lathi, B. P.

• ISBN13: 9780195158335
• ISBN10: 0195158334
• Edition: 2nd
• Format: Hardcover
• Copyright: 2004-07-01
• Publisher: Oxford University Press

Summary

Incorporating new problems and examples, the second edition of Linear Systems and Signals features MATLABRG material in each chapter and at the back of the book. It gives clear descriptions of linear systems and uses mathematics not only to prove axiomatic theory, but also to enhance physicaland intuitive understanding.

Author Biography

B. P. Lathi is Professor Emeritus of Electrical Engineering at California State University, Sacramento. He is the author of Signal Processing and Linear Systems (OUP, 2000) and Modern Digital and Analog Communications Systems, 3/e (OUP, 1998).

Table of Contents

PREFACE	xiii
B BACKGROUND
B.1 Complex Numbers
B.1-1 A Historical Note	1	(4)
B.1-2 Algebra of Complex Numbers	5	(11)
B.2 Sinusoids	16	(6)
B.2-1 Addition of Sinusoids	18	(3)
B.2-2 Sinusoids in Terms of Exponentials: Euler's Formula	21	(1)
B.3 Sketching Signals	22	(2)
B.3-1 Monotonic Exponentials	22	(1)
B.3-2 The Exponentially Varying Sinusoid	23	(1)
B.4 Cramer's Rule	24	(2)
B.5 Partial Fraction Expansion	26	(11)
B.5-1 Method of Clearing Fractions	27	(1)
B.5-2 The Heaviside "Cover-Up" Method	28	(4)
B.5-3 Repeated Factors of Q(x)	32	(2)
B.5-4 Mixture of the Heaviside "Cover-Up" and Clearing Fractions	34	(1)
B.5-5 Improper F(x) with m = n	35	(1)
B.5-6 Modified Partial Fractions	36	(1)
B.6 Vectors and Matrices	37	(11)
B.6-1 Some Definitions and Properties	38	(1)
B.6-2 Matrix Algebra	39	(4)
B.6-3 Derivatives and Integrals of a Matrix	43	(2)
B.6-4 The Characteristic Equation of a Matrix: The Cayley-Hamilton Theorem	45	(2)
B.6-5 Computation of an Exponential and a Power of a Matrix	47	(1)
B.7 Miscellaneous	48	(20)
B.7-1 L'Hôpital's Rule	48	(1)
B.7-2 The Taylor and Maclaurin Series	48	(1)
B.7-3 Power Series	48	(1)
B.7-4 Sums	49	(1)
B.7-5 Complex Numbers	49	(1)
B.7-6 Trigonometric Identities	49	(1)
B.7-7 Indefinite Integrals	50	(1)
B.7-8 Common Derivative Formulas	51	(1)
B.7-9 Some Useful Constants	52	(1)
B.7-10 Solution of Quadratic and Cubic Equations	52	(1)
References	53	(1)
MATLAB Session B: Elementary Operations	53	(11)
Problems	64	(4)
1 SIGNALS AND SYSTEMS
1.1 Size of a Signal	68	(83)
1.1-1 Signal Energy	69	(1)
1.1-2 Signal Power	69	(6)
1.2 Some Useful Signal Operations	75	(7)
1.2-1 Time Shifting	75	(2)
1.2-2 Time Scaling	77	(3)
1.2-3 Time Reversal	80	(1)
1.2-4 Combined Operations	81	(1)
1.3 Classification of Signals	82	(4)
1.3-1 Continuous-Time and Discrete-Time Signals	82	(1)
1.3-2 Analog and Digital Signals	82	(1)
1.3-3 Periodic and Aperiodic Signals	83	(3)
1.3-4 Energy and Power Signals	86	(1)
1.3-5 Deterministic and Random Signals	86	(1)
1.4 Some Useful Signal Models	86	(10)
1.4-1 Unit Step Function u(t)	87	(3)
1.4-2 The Unit Impulse Function δ(t)	90	(4)
1.4-3 The Exponential Function est	94	(2)
1.5 Even and Odd Functions	96	(3)
1.5-1 Some Properties of Even and Odd Functions	96	(1)
1.5-2 Even and Odd Components of a Signal	97	(2)
1.6 Systems	99	(2)
1.7 Classification of Systems	101	(12)
1.7-1 Linear and Nonlinear Systems	101	(5)
1.7-2 Time-Invariant and Time-Varying Systems	106	(1)
1.7-3 Instantaneous and Dynamic Systems	107	(1)
1.7-4 Causal and Noncausal Systems	108	(2)
1.7-5 Continuous-Time and Discrete-Time Systems	110	(1)
1.7-6 Analog and Digital Systems	111	(1)
1.7-7 Invertible and Noninvertible Systems	112	(1)
1.7-8 Stable and Unstable Systems	112	(1)
1.8 System Model: Input Output Description	113	(9)
1.8-1 Electrical Systems	113	(4)
1.8-2 Mechanical Systems	117	(4)
1.8-3 Electromechanical Systems	121	(1)
1.9 Internal and External Description of a System	122	(2)
1.10 Internal Description: The State-Space Description	124	(5)
1.11 Summary	129	(2)
References	131	(1)
MATLAB Session I: Working with Functions	131	(7)
Problems	138	(13)
2 TIME-DOMAIN ANALYSIS OF CONTINUOUS-TIME SYSTEMS
2.1 Introduction	151	(1)
2.2 System Response to Internal Conditions: The Zero-Input Response	152	(12)
2.2-1 Some Insights into the Zero-Input Behavior of a System	163	(1)
2.3 The Unit Impulse Response h(t)	164	(5)
2.4 System Response to External Input: Zero-State Response	169	(29)
2.4-1 The Convolution Integral	171	(8)
2.4-2 Graphical Understanding of Convolution Operation	179	(13)
2.4-3 Interconnected Systems	192	(3)
2.4-4 A Very Special Function for LTIC Systems: The Everlasting Exponential est	195	(2)
2.4-5 Total Response	197	(1)
2.5 Classical Solution of Differential Equations	198	(9)
2.5-1 Forced Response: The Method of Undetermined Coefficients	199	(8)
2.6 System Stability	207	(8)
2.6-1 Internal (Asymptotic) Stability	209	(2)
2.6-2 Relationship Between BIBO and Asymptotic Stability	211	(4)
2.7 Intuitive Insights into System Behavior	215	(8)
2.7-1 Dependence of System Behavior on Characteristic Modes	215	(1)
2.7-2 Response Time of a System: The System Time Constant	216	(2)
2.7-3 Time Constant and Rise Time of a System	218	(1)
2.7-4 Time Constant and Filtering	218	(2)
2.7-5 Time Constant and Pulse Dispersion (Spreading)	220	(1)
2.7-6 Time Constant and Rate of Information Transmission	220	(1)
2.7-7 The Resonance Phenomenon	221	(2)
2.8 Appendix 2.1: Determining the Impulse Response	223	(2)
2.9 Summary	225	(1)
References	226	(1)
MATLAB Session 2: M-Files	227	(8)
Problems	235	(10)
3 TIME-DOMAIN ANALYSIS OF DISCRETE-TIME SYSTEMS
3.1 Introduction	245	(3)
3.1-1 Size of a Discrete-Time Signal	246	(2)
3.2 Useful Signal Operations	248	(4)
3.3 Some Useful Discrete-Time Signal Models	252	(7)
3.3-1 Discrete-Time Impulse Function σ[η]	252	(1)
3.3-2 Discrete-Time Unit Step Function u[η]	253	(1)
3.3-3 Discrete-Time Exponential γη	254	(3)
3.3-4 Discrete-Time Sinusoid cos(Ωη + theta)	257	(2)
3.3-5 Discrete-Time Complex Exponential ejΩη	259	(1)
3.4 Examples of Discrete-Time Systems	259	(11)
3.4-1 Classification of Discrete-Time Systems	268	(2)
3.5 Discrete-Time System Equations	270	(6)
3.5-1 Recursive (Iterative) Solution of Difference Equation	271	(5)
3.6 System Response to Internal Conditions: The Zero-Input Response	276	(6)
3.7 The Unit Impulse Response h[η]	282	(4)
3.8 System Response to External Input: The Zero-State Response	286	(18)
3.8-1 Graphical Procedure for the Convolution Sum	294	(6)
3.8-2 Interconnected Systems	300	(2)
3.8-3 A Very Special Function for LTID Systems: The Everlasting Exponential zη	302	(1)
3.8-4 Total Response	303	(1)
3.9 Classical Solution of Linear Difference Equations	304	(7)
3.10 System Stability: The External (BIBO) Stability Criterion	311	(29)
3.10-1 Internal (Asymptotic) Stability	312	(2)
3.10-2 Relationship Between BIBO and Asymptotic Stability	314	(3)
3.11 Intuitive Insights into System Behavior	317	(1)
3.12 Appendix 3.1: Impulse Response for a Special Case	318	(1)
3.13 Summary	318	(2)
MATLAB Session 3: Discrete-Time Signals and Systems	320	(6)
Problems	326	(14)
4 CONTINUOUS-TIME SYSTEM ANALYSIS USING THE LAPLACE TRANSFORM
4.1 The Laplace Transform	340	(20)
4.1-1 Finding the Inverse Transform	348	(12)
4.2 Some Properties of the Laplace Transform	360	(11)
4.2-1 Time Shifting	360	(3)
4.2-2 Frequency Shifting	363	(1)
4.2-3 The Time-Differentiation Property	364	(2)
4.2-4 The Time-Integration Property	366	(2)
4.2-5 Time Convolution and Frequency Convolution	368	(3)
4.3 Solution of Differential and Integro-Differential Equations	371	(13)
4.3-1 Zero-State Response	376	(5)
4.3-2 Stability	381	(3)
4.3-3 Inverse Systems	384	(1)
4.4 Analysis of Electrical Networks: The Transformed Network	384	(13)
4.4-1 Analysis of Active Circuits	393	(4)
4.5 Block Diagrams	397	(2)
4.6 System Realization	399	(16)
4.6-1 Direct Form I Realization	400	(2)
4.6-2 Direct Form II Realization	402	(2)
4.6-3 Cascade and Parallel Realizations	404	(3)
4.6-4 Transposed Realization	407	(3)
4.6-5 Using Operational Amplifiers for System Realization	410	(5)
4.7 Application to Feedback and Controls	415	(8)
4.7-1 Analysis of a Simple Control System	417	(6)
4.8 Frequency Response of an LTIC System	423	(7)
4.8-1 Steady-State Response to Causal Sinusoidal Inputs	429	(1)
4.9 Bode Plots	430	(64)
4.9-1 Constant kappaα1α2/b1b3	432	(1)
4.9-2 Pole (or Zero) at the Origin	433	(1)
4.9-3 First-Order Pole (or Zero)	434	(3)
4.9-4 Second-Order Pole (or Zero)	437	(9)
4.9-5 The Transfer Function from the Frequency Response	446	(1)
4.10 Filter Design by Placement of Poles and Zeros of H(s)	446	(32)
4.10-1 Dependence of Frequency Response on Poles and Zeros of H(s)	447	(3)
4.10-2 Lowpass Filters	450	(1)
4.10-3 Bandpass Filters	451	(1)
4.10-4 Notch (Bandstop) Filters	451	(3)
4.10-5 Practical Filters and Their Specifications	454	(2)
4.11 The Bilateral Laplace Transform	456	(6)
4.11-1 Properties of Bilateral Laplace Transform	462	(1)
4.11-2 Using the Bilateral Transform for Linear System Analysis	463	(4)
4.12 Summary	467	(1)
References	468	(1)
MATLAB Session 4: Continuous-Time Filters	468	(10)
Problems	478	(16)
5 DISCRETE-TIME SYSTEM ANALYSIS USING THE z-TRANSFORM
5.1 The z-Transform	494	(12)
5.1-1 Finding the Inverse Transform	501	(5)
5.2 Some Properties of the z-Transform	506	(9)
5.3 z-Transform Solution of Linear Difference Equations	515	(10)
5.3-1 Zero-State Response of LTID Systems: The Transfer Function	519	(4)
5.3-2 Stability	523	(1)
5.3-3 Inverse Systems	524	(1)
5.4 System Realization	525	(6)
5.5 Frequency Response of Discrete-Time Systems	531	(13)
5.5-1 The Periodic Nature of the Frequency Response	537	(4)
5.5-2 Aliasing and Sampling Rate	541	(3)
5.6 Frequency Response from Pole-Zero Location	544	(9)
5.7 Digital Processing of Analog Signals	553	(7)
5.8 Connection Between the Laplace Transform and the z-Transform	560	(2)
5.9 The Bilateral z-Transform	562	(116)
5.9-1 Properties of the Bilateral z.-Transform	568	(1)
5.9-2 Using the Bilateral z-Transform for Analysis of LTID Systems	569	(2)
5.10 Summary	571	(1)
References	572	(1)
MATLAB Session 5: Discrete-Time IIR Filters	572	(9)
Problems	581	(13)
6 CONTINUOUS-TIME SIGNAL ANALYSIS: THE FOURIER SERIES
6.1 Periodic Signal Representation by Trigonometric Fourier Series	594	(84)
6.1-1 The Fourier Spectrum	600	(9)
6.1-2 The Effect of Symmetry	609	(2)
6.1-3 Determining the Fundamental Frequency and Period	611	(3)
6.2 Existence and Convergence of the Fourier Series	614	(1)
6.2-1 Convergence of a Series	615	(2)
6.2-2 The Role of Amplitude and Phase Spectra in Waveshaping	617	(6)
6.3 Exponential Fourier Series	623	(4)
6.3-1 Exponential Fourier Spectra	627	(7)
6.3-2 Parseval's Theorem	634	(3)
6.4 LTIC System Response to Periodic Inputs	637	(4)
6.5 Generalized Fourier Series: Signals as Vectors	641	(1)
6.5-1 Component of a Vector	641	(1)
6.5-2 Signal Comparison and Component of a Signal	642	(3)
6.5-3 Extension to Complex Signals	645	(1)
6.5-4 Signal Representation by an Orthogonal Signal Set	646	(12)
6.6 Numerical Computation of Dn	658	(2)
6.7 Summary	660	(1)
References	661	(1)
MATLAB Session 6: Ferrier Series Applications	662	(7)
Problems	669	(9)
7 CONTINUOUS-TIME SIGNAL ANALYSIS: THE FOURIER 'IRANSFORM
7.1 Aperiodic Signal Representation by Fourier Integral	678	(9)
7.1-1 Physical Appreciation of the Fourier Transform	685	(2)
7.2 Transforms of Some Useful Functions	687	(11)
7.2-1 Connection Between the Fourier and Laplace Transforms	697	(1)
7.3 Some Properties of the Fourier Transform	698	(19)
7.4 Signal Transmission Through LTIC Systems	717	(9)
7.4-1 Signal Distortion During Transmission	719	(3)
7.4-2 Bandpass Systems and Group Delay	722	(4)
7.5 Ideal and Practical Filters	726	(3)
7.6 Signal Energy	729	(3)
7.7 Application to Communications: Amplitude Modulation	732	(14)
7.7-1 Double-Sideband, Suppressed-Carrier (DSB-SC) Modulation	732	(5)
7.7-2 Amplitude Modulation (AM)	737	(5)
7.7-3 Single-Sideband Modulation (SSB)	742	(3)
7.7-4 Frequency-Division Multiplexing	745	(1)
7.8 Data Truncation: Window Functions	746	(7)
7.8-1 Using Windows in Filter Design	751	(2)
7.9 Summary	753	(1)
References	754	(1)
MATLAB Session 7: Fourier Transform Topics	754	(6)
Problems	760	(10)
8 SAMPLING: THE BRIDGE FROM CONTINUOUS TO DISCRETE
8.1 The Sampling Theorem	770	(8)
8.1-1 Practical Sampling	775	(3)
8.2 Signal Reconstruction	778	(14)
8.2-1 Practical Difficulties in Signal Reconstruction	781	(9)
8.2-2 Some Applications of the Sampling Theorem	790	(2)
8.3 Analog-to-Digital (A/D) Conversion	792	(3)
8.4 Dual of Time Sampling: Spectral Sampling	795	(3)
8.5 Numerical Computation of the Fourier Transform: The Discrete Fourier Transform (DFT)	798	(19)
8.5-1 Some Properties of the DFT	811	(2)
8.5-2 Some Applications of the DFT	813	(4)
8.6 The Fast Fourier Transform (FFT)	817	(4)
8.7 Summary	821	(1)
References	822	(1)
MATLAB Session 8: The Discrete Fourier Transform	822	(7)
Problems	829	(8)
9 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS .
9.1 Discrete-Time Fourier Series (DTFS)	837	(10)
9.1-1 Periodic Signal Representation by Discrete-Time Fourier Series	838	(2)
9.1-2 Fourier Spectra of a Periodic Signal x[η]	840	(7)
9.2 Aperiodic Signal Representation by Fourier Integral	847	(12)
9.2-1 Nature of Fourier Spectra	850	(8)
9.2-2 Connection Between the DTFT and the z-Transform	858	(1)
9.3 Properties of the DTFT	859	(11)
9.4 LTI Discrete-Time System Analysis by DTFT	870	(5)
9.4-1 Distortionless Transmission	872	(2)
9.4-2 Ideal and Practical Filters	874	(1)
9.5 DTFT Connection with the CTFT	875	(3)
9.5-1 Use of DFT and FFT for Numerical Computation of DTFT	876	(2)
9.6 Generalization of the DTFT to the z-transform	878	(2)
9.7 Summary	880	(1)
Reference	881	(1)
MATLAB Session 9: Working with the DTFS and the DTFT	881	(9)
Problems	890
10 STATE-SPACE ANALYSIS
10.1 Introduction	899	(2)
10.2 A Systematic Procedure for Determining State Equations	901	(62)
10.2-1 Electrical Circuits	902	(2)
10.2-2 State Equations from a Transfer Function	904	(8)
10.3 Solution of State Equations	912	(1)
10.3-1 Laplace Transform Solution of State Equations	912	(7)
10.3-2 Time-Domain Solution of State Equations	919	(7)
10.4 Linear Transformation of State Vector	926	(4)
10.4-1 Diagonalization of Matrix Α	930	(4)
10.5 Controllability and Observability	934	(5)
10.5-1 Inadequacy of the Transfer Function Description of a System	939	(1)
10.6 State-Space Analysis of Discrete-Time Systems	940	(2)
10.6-1 Solution in State-Space	942	(5)
10.6-2 The z-Transform Solution	947	(1)
10.7 Summary	948	(1)
References	949	(1)
MATLAB Session 10: Toolboxes and State-Space Analysis	949	(8)
Problems	957	(6)
INDEX	963