Signals, systems, and transforms

Cover
Title Page
Copyright Page
CONTENTS
PREFACE
1 INTRODUCTION
	1.1 Modeling
	1.2 Continuous-Time Physical Systems
		Electric Circuits
		Operational Amplifier Circuits
		Simple Pendulum
		DC Power Supplies
		Analogous Systems
	1.3 Samplers and Discrete-Time Physical Systems
		Analog-to-Digital Converter
		Numerical Integration
		Picture in a Picture
		Compact Discs
		Sampling in Telephone Systems
		Data-Acquisition System
	1.4 MATLAB and SIMULINK
2 CONTINUOUS-TIME SIGNALS AND SYSTEMS
	2.1 Transformations of Continuous-Time Signals
		Time Transformations
		Amplitude Transformations
	2.2 Signal Characteristics
		Even and Odd Signals
		Periodic Signals
	2.3 Common Signals in Engineering
	2.4 Singularity Functions
		Unit Step Function
		Unit Impulse Function
	2.5 Mathematical Functions for Signals
	2.6 Continuous-Time Systems
		Interconnecting Systems
		Feedback System
	2.7 Properties of Continuous-Time Systems
		Stability
		Linearity
	Summary
	Problems
3 CONTINUOUS-TIME LINEAR TIME-INVARIANT SYSTEMS
	3.1 Impulse Representation of Continuous-Time Signals
	3.2 Convolution for Continuous-Time LTI Systems
	3.3 Properties of Convolution
	3.4 Properties of Continuous-Time LTI Systems
		Memoryless Systems
		Invertibility
		Causality
		Stability
		Unit Step Response
	3.5 Differential-Equation Models
		Solution of Differential Equations
		General Case
		Relation to Physical Systems
	3.6 Terms in the Natural Response
		Stability
	3.7 System Response for Complex-Exponential Inputs
		Linearity
		Complex Inputs for LTI Systems
		Impulse Response
	3.8 Block Diagrams
		Direct Form I
		Direct Form II
		nth-Order Realizations
		Practical Considerations
	Summary
	Problems
4 FOURIER SERIES
	4.1 Approximating Periodic Functions
		Periodic Functions
		Approximating Periodic Functions
	4.2 Fourier Series
		Fourier Series
		Fourier Coefficients
	4.3 Fourier Series and Frequency Spectra
		Frequency spectra
	4.4 Properties of Fourier series
	4.5 System Analysis
	4.6 Fourier Series Transformations
		Amplitude Transformations
		Time Transformations
	Summary
	Problems
5 THE FOURIER TRANSFORM
	5.1 Definition of the Fourier Transform
	5.2 Properties of the Fourier Transform
		Linearity
		Time Scaling
		Time Shifting
		Time Reversal
		Time Transformation
		Duality
		Convolution
		Frequency Shifting
		Time Integration
		Time Differentiation
		Frequency Differentiation
		Symmetry
		Summary
	5.3 Fourier Transforms of Time Functions
		DC Level
		Unit Step Function
		Switched Cosine
		Pulsed Cosine
		Exponential Pulse
		Fourier Transforms of Periodic Functions
		Summary
	5.4 Application of the Fourier Transform
		Frequency Response of Linear Systems
		Frequency Spectra of Signals
		Summary
	5.5 Energy and Power Density Spectra
		Energy Density Spectrum
		Power Density Spectrum
		Power and Energy Transmission
		Summary
	Summary
	Problems
6 APPLICATIONS OF THE FOURIER TRANSFORM
	6.1 Ideal Filters
	6.2 Real Filters
		RC Low-Pass Filter
		Butterworth Filter
		Bandpass Filters
		Active Filters
		Summary
	6.3 Bandwidth Relationships
	6.4 Sampling Continuous-Time Signals
		Impulse Sampling
		Shannon’s Sampling Theorem
		Practical Sampling
	6.5 Reconstruction of Signals from Sample Data
		Interpolating Function
		Digital-to-Analog Conversion
		Quantization Error
	6.6 Sinusoidal Amplitude Modulation
		Frequency-Division Multiplexing
	6.7 Pulse-Amplitude Modulation
		Time-Division Multiplexing
		Flat-Top PAM
	Summary
	Problems
7 THE LAPLACE TRANSFORM
	7.1 Definitions of Laplace Transforms
	7.2 Examples
	7.3 Laplace Transforms of Functions
	7.4 Laplace Transform Properties
		Real Shifting
		Differentiation
		Integration
	7.5 Additional Properties
		Multiplication by t
		Initial Value
		Final Value
		Time Transformation
	7.6 Response of LTI systems
		Initial Conditions
		Transfer Functions
		Convolution
		Transforms with Complex Poles
		Functions with Repeated Poles
	7.7 LTI Systems Characteristics
		Causality
		Stability
		Invertibility
		Frequency Response
		Step Response
	7.8 Bilateral Laplace Transform
		Region of Convergence
		Bilateral Transform from Unilateral Tables
		Inverse Bilateral Laplace Transform
	7.9 Relationship of the Laplace Transform to the Fourier Transform
	Summary
	Problems
8 STATE VARIABLES FOR CONTINUOUS-TIME SYSTEMS
	8.1 State-Variable Modeling
	8.2 Simulation Diagrams
	8.3 Solution of State Equations
		Laplace-Transform Solution
		Convolution Solution
		Infinite Series Solution
	8.4 Properties of the State-Transition Matrix
	8.5 Transfer Functions
		Stability
	8.6 Similarity Transformations
		Transformations
		Properties
	Summary
	Problems
9 DISCRETE-TIME SIGNALS AND SYSTEMS
	9.1 Discrete-Time Signals and Systems
		Unit Step and Unit Impulse Functions
		Equivalent Operations
	9.2 Transformations of Discrete-Time signals
		Time Transformations
		Amplitude Transformations
	9.3 Characteristics of Discrete-Time signals
		Even and Odd Signals
		Signals Periodic in n
		Signals Periodic in Ω
	9.4 Common Discrete-Time Signals
	9.5 Discrete-Time Systems
		Interconnecting Systems
	9.6 Properties of Discrete-Time systems
		Systems with Memory
		Invertibility
		Inverse of a System
		Causality
		Stability
		Time Invariance
		Linearity
		Summary
		Problems
10 DISCRETE-TIME LINEAR TIME-INVARIANT SYSTEMS
	10.1 Impulse Representation of Discrete-Time Signals
	10.2 Convolution for Discrete-Time systems
		Properties of Convolution
	10.3 Properties of Discrete-Time LTI systems
		Memory
		Invertibility
		Causality
		Stability
		Unit Step Response
	10.4 Difference-Equation Models
		Difference-Equation Models
		Classical Method
		Solution by Iteration
	10.5 Terms in the Natural Response
		Stability
	10.6 Block Diagrams
		Two Standard Forms
	10.7 System Response for Complex-Exponential Inputs
		Linearity
		Complex Inputs for LTI Systems
		Stability
		Sampled Signals
		Impulse Response
	Summary
	Problems
11 THE Z-TRANSFORM
	11.1 Definitions of z-Transforms
	11.2 Examples
		Two z-Transforms
		Digital-Filter Example
	11.3 z-Transforms of Functions
		Sinusoids
	11.4 z-Transform Properties
		Real shifting
		Initial and Final Values
	11.5 Additional Properties
		Time Scaling
		Convolution in Time
	11.6 LTI System Applications
		Transfer Functions
		Inverse z-Transform
		Complex Poles
		Causality
		Stability
		Invertibility
		Frequency Response
	11.7 Bilateral z-Transform
		Bilateral Transforms
		Regions of Convergence
		Inverse Bilateral Transforms
	Summary
	Problems
12 FOURIER TRANSFORMS OF DISCRETE-TIME SIGNALS
	12.1 Discrete-Time Fourier Transform
		z-Transform
	12.2 Properties of the Discrete-Time Fourier Transform
		Periodicity
		Linearity
		Time Shift
		Frequency Shift
		Symmetry
		Time Reversal
		Convolution in Time
		Convolution in Frequency
		Multiplication by n
		Parseval’s Theorem
	12.3 Discrete-Time Fourier Transform of Periodic Sequences
	12.4 Discrete Fourier Transform
		Shorthand Notation for the DFT
		Frequency Resolution of the DFT
		Validity of the DFT
		Summary
	12.5 Fast Fourier Transform
		Decomposition-in-Time Fast Fourier Transform Algorithm
		Decomposition-in-Frequency Fast Fourier Transform
		Summary
	12.6 Applications of the Discrete Fourier Transform
		Calculation of Fourier Transforms
		Convolution
		Filtering
		Correlation
		Energy Spectral Density Estimation
		Summary
	12.7 The Discrete Cosine Transform
	Summary
	Problems
13 STATE VARIABLES FOR DISCRETE-TIME SYSTEMS
	13.1 State-Variable Modeling
	13.2 Simulation Diagrams
	13.3 Solution of State Equations
		Recursive Solution
		z-Transform Solution
	13.4 Properties of the State Transition Matrix
	13.5 Transfer Functions
		Stability
	13.6 Similarity Transformations
		Properties
	Summary
	Problems
APPENDICES
	A. Integrals and Trigonometric Identities
		Integrals
		Trigonometric Identities
	B. Leibnitz’s and L’Hôpital’s Rules
		Leibnitz’s Rule
		L’Hôpital’s Rule
	C. Summation Formulas for Geometric Series
	D. Complex Numbers and Euler’s Relation
		Complex-Number Arithmetic
		Euler’s Relation
		Conversion Between Forms
	E. Solution of Differential Equations
		Complementary Function
		Particular Solution
		General Solution
		Repeated Roots
	F. Partial-Fraction Expansions
	G. Review of Matrices
		Algebra of Matrices
		Other Relationships
	H. Answers to Selected Problems
	I. Signals and Systems References
INDEX
	A
	B
	C
	D
	E
	F
	G
	H
	I
	J
	L
	M
	N
	O
	P
	Q
	R
	S
	T
	U
	V
	W
	Y
	Z