Signals and Systems: A Practical Approach

Preface to the Second Edition
Preface to the First Edition
Contents
Abbreviations
1 Discrete Signals
1.1 Introduction
1.2 Basic Signals
1.2.1 Unit-Impulse Signal
1.2.2 Unit-Step Signal
1.2.3 Unit-Ramp Signal
1.2.4 Sinusoids and Exponentials
1.2.4.1 The Polar Form of Sinusoids
1.2.4.2 The Rectangular Form of Sinusoids
1.2.4.3 The Sum of Sinusoids of the Same Frequency
1.2.4.4 Exponentials
1.2.4.5 The Complex Sinusoids
1.2.4.6 Exponentially Varying Amplitude Sinusoids
1.2.4.7 The Sampling Theorem and the Aliasing Effect
1.2.4.8 Frequency-Sampling Theorem
1.3 Classification of Signals
1.3.1 Continuous, Discrete, and Digital Signals
1.3.2 Periodic and Aperiodic Signals
1.3.3 Energy and Power Signals
1.3.4 Even- and Odd-Symmetric Signals
1.3.5 Causal and Noncausal Signals
1.3.6 Deterministic and Random Signals
1.4 Signal Operations
1.4.1 Time Shifting
1.4.1.1 Circular Shifting
1.4.2 Time Reversal
1.4.2.1 Circular Time Reversal
1.4.3 Time Scaling
1.4.4 Zero Padding
1.5 Numerical Integration
1.6 The Organization of this Book
1.7 Summary
Exercises
2 Continuous Signals
2.1 Basic Signals
2.1.1 The Unit-Step Signal
2.1.2 The Unit-Impulse Signal
2.1.2.1 The Impulse Representation of Signals
2.1.2.2 The Unit-Impulse as the Derivative of the Unit-Step
2.1.2.3 The Scaling Property of the Impulse
2.1.3 The Unit-Ramp Signal
2.1.4 Sinusoids
2.1.4.1 The Polar Form of Sinusoids
2.1.4.2 The Rectangular Form of Sinusoids
2.1.4.3 The Sum of Sinusoids of the Same Frequency
2.1.4.4 The Complex Sinusoids
2.1.4.5 Exponentially Varying Amplitude Sinusoids
2.2 Classification of Signals
2.2.1 Continuous Signals
2.2.2 Periodic and Aperiodic Signals
2.2.3 Energy and Power Signals
2.2.4 Even- and Odd-Symmetric Signals
2.2.5 Causal and Noncausal Signals
2.3 Signal Operations
2.3.1 Time Shifting
2.3.2 Time Reversal
2.3.3 Time Scaling
2.4 Summary
Exercises
3 Time-Domain Analysis of Discrete Systems
3.1 Difference Equation Model
3.1.1 System Response
3.1.1.1 Zero-State Response
3.1.1.2 Zero-Input Response
3.1.1.3 Complete Response
3.1.1.4 Transient and Steady-State Responses
3.1.1.5 Coding and Simulation
3.1.1.6 Zero-Input Response by Solving the Difference Equation
3.1.2 Impulse Response
3.1.3 Characterization of Systems by Their Responses to Impulse and Unit-Step Signals
3.2 Classification of Systems
3.2.1 Linear and Nonlinear Systems
3.2.2 Time-Invariant and Time-Varying Systems
3.2.3 Causal and Noncausal Systems
3.2.4 Instantaneous and Dynamic Systems
3.2.5 Inverse Systems
3.2.6 Continuous and Discrete Systems
3.3 Convolution-Summation Model
3.3.1 Properties of Convolution-Summation
3.3.2 The Difference Equation and the Convolution-Summation
3.3.3 Response to Complex Exponential Input
3.4 System Stability
3.5 Realization of Discrete Systems
3.5.1 Decomposition of Higher-Order Systems
3.5.2 Feedback Systems
3.6 Summary
Exercises
4 Time-Domain Analysis of Continuous Systems
4.1 Classification of Systems
4.1.1 Linear and Nonlinear Systems
4.1.2 Time-Invariant and Time-Varying Systems
4.1.3 Causal and Noncausal Systems
4.1.4 Instantaneous and Dynamic Systems
4.1.5 Lumped-Parameter and Distributed-Parameter Systems
4.1.6 Inverse Systems
4.2 Differential Equation Model
4.3 Convolution-Integral Model
4.3.1 Properties of Convolution-Integral
4.3.2 Convolution of a Function with a Narrow Unit Area Pulse
4.4 System Response
4.4.1 Impulse Response
4.4.2 Response to Unit-Step Input
4.4.3 Characterization of Systems by Their Responses to Impulse and Unit-Step Signals
4.4.4 Response to Complex Exponential Input
4.5 System Stability
4.6 Realization of Continuous Systems
4.6.1 Decomposition of Higher-Order Systems
4.6.2 Feedback Systems
4.7 Summary
Exercises
5 The Discrete Fourier Transform
5.1 The Time-Domain and the Frequency-Domain
5.2 The Fourier Analysis
5.2.1 The Four Versions of Fourier Analysis
5.3 The Discrete Fourier Transform
5.3.1 The Approximation of Arbitrary Waveforms with Finite Number of Samples
5.3.2 The DFT and the IDFT
5.3.2.1 Center-Zero Format of the DFT and IDFT
5.3.3 DFT of Some Basic Signals
5.4 Properties of the Discrete Fourier Transform
5.4.1 Linearity
5.4.2 Periodicity
5.4.3 Circular Time Reversal
5.4.4 Duality
5.4.5 Sum and Difference of Sequences
5.4.6 Upsampling of a Sequence
5.4.7 Zero Padding the Data
5.4.8 Circular Shift of a Sequence
5.4.9 Circular Shift of a Spectrum
5.4.10 Symmetry
5.4.11 Circular Convolution of Time-Domain Sequences
5.4.12 Circular Convolution of Frequency-Domain Sequences
5.4.13 Parseval's Theorem
5.5 Applications of the Discrete Fourier Transform
5.5.1 Computation of the Linear Convolution Using the DFT
5.5.2 Interpolation and Decimation
5.5.2.1 Interpolation
5.5.2.2 Decimation
5.5.2.3 Interpolation and Decimation
5.5.3 Image Boundary Representation
5.6 Summary
Exercises
6 Fourier Series
6.1 Fourier Series
6.1.1 FS as the Limiting Case of the DFT
6.1.2 The Compact Trigonometric Form of the FS
6.1.3 The Trigonometric Form of the FS
6.1.4 Periodicity of the FS
6.1.5 Existence of the FS
6.1.6 Gibbs Phenomenon
6.2 Properties of the Fourier Series
6.2.1 Linearity
6.2.2 Symmetry
6.2.2.1 Even Symmetry
6.2.2.2 Odd Symmetry
6.2.2.3 Half-Wave Symmetry
6.2.3 Time Shifting
6.2.4 Frequency Shifting
6.2.5 Time Reversal
6.2.6 Convolution in the Time-Domain
6.2.7 Convolution in the Frequency-Domain
6.2.8 Duality
6.2.9 Time Scaling
6.2.10 Time-Differentiation
6.2.11 Time-Integration
6.2.11.1 Rate of Convergence of the Fourier Series
6.2.12 Parseval's Theorem
6.3 Approximation of the Fourier Series
6.3.1 Aliasing Effect
6.4 Applications of the Fourier Series
6.4.1 Analysis of Rectified Power Supply
6.5 Summary
Exercises
7 The Discrete-Time Fourier Transform
7.1 The Discrete-Time Fourier Transform
7.1.1 The DTFT as the Limiting Case of the DFT
7.1.2 The Dual Relationship between the DTFT and the FS
7.1.3 The DTFT of a Discrete Periodic Signal
7.1.4 Determination of the DFT from the DTFT
7.2 Properties of the Discrete-Time Fourier Transform
7.2.1 Linearity
7.2.2 Time Shifting
7.2.3 Frequency Shifting
7.2.4 Convolution in the Time-Domain
7.2.5 Convolution in the Frequency-Domain
7.2.6 Symmetry
7.2.7 Time Reversal
7.2.8 Time Expansion
7.2.9 Frequency Differentiation
7.2.10 Difference
7.2.11 Summation
7.2.12 Parseval's Theorem and the Energy Transfer Function
7.3 Approximation of the Discrete-Time Fourier Transform
7.3.1 Approximation of the Inverse DTFT by the IDFT
7.4 Applications of the Discrete-Time Fourier Transform
7.4.1 Transfer Function and the System Response
7.4.2 Digital Filter Design Using DTFT
7.4.2.1 Rectangular Window
7.4.2.2 Hamming Window
7.4.3 Digital Differentiator
7.4.4 Hilbert Transform
7.4.5 Downsampling
7.5 Summary
Exercises
8 The Fourier Transform
8.1 The Fourier Transform
8.1.1 The FT as a Limiting Case of the DTFT
8.1.2 Existence of the FT
8.2 Properties of the Fourier Transform
8.2.1 Linearity
8.2.2 Duality
8.2.3 Symmetry
8.2.4 Time Shifting
8.2.5 Frequency Shifting
8.2.6 Convolution in the Time Domain
8.2.7 Convolution in the Frequency Domain
8.2.8 Conjugation
8.2.9 Time Reversal
8.2.10 Time Scaling
8.2.11 Time Differentiation
8.2.12 Time Integration
8.2.13 Frequency Differentiation
8.2.14 Parseval's Theorem and the Energy Transfer Function
8.3 Fourier Transform of Mixed Class of Signals
8.3.1 The FT of a Continuous Periodic Signal
8.3.2 Determination of the FS from the FT
8.3.3 The FT of a Sampled Signal and the Aliasing Effect
8.3.4 The FT and the DTFT of Sampled Aperiodic Signals
8.3.5 The FT and the DFT of Sampled Periodic Signals
8.3.6 Approximation of the Continuous Signal from Its Sampled Version
8.4 Approximation of the Fourier Transform
8.5 Applications of the Fourier Transform
8.5.1 Transfer Function and the System Response
8.5.2 Ideal Filters and Their Unrealizability
8.5.3 Modulation and Demodulation
8.5.3.1 Double Sideband, Suppressed Carrier (DSB-SC), Amplitude Modulation
8.5.3.2 Double Sideband, with Carrier (DSB-WC), Amplitude Modulation
8.5.3.3 Pulse Amplitude Modulation (PAM)
8.6 Summary
Exercises
9 The z-Transform
9.1 Fourier Analysis and the z-Transform
9.2 The z-Transform
9.3 Properties of the z-Transform
9.3.1 Linearity
9.3.2 Left Shift of a Sequence
9.3.3 Right Shift of a Sequence
9.3.4 Convolution
9.3.5 Multiplication by n
9.3.6 Multiplication by an
9.3.7 Summation
9.3.8 Initial Value
9.3.9 Final Value
9.3.10 Transform of Semiperiodic Functions
9.4 The Inverse z-Transform
9.4.1 Finding the Inverse z-Transform
9.4.1.1 The Partial Fraction Method
9.4.1.2 The Long Division Method
9.5 Applications of the z-Transform
9.5.1 Transfer Function
9.5.2 Characterization of a System by Its Poles and Zeros
9.5.3 Frequency Response and the Locations of the Poles and Zeros
9.5.4 Design of Digital Filters
9.5.4.1 The Bilinear Transformation
9.5.4.2 Implementation of the Bilinear Transformation
9.5.5 System Response
9.5.5.1 Inverse Systems
9.5.6 System Stability
9.5.7 Realization of Systems
9.5.8 Feedback Systems
9.6 Summary
Exercises
10 The Laplace Transform
10.1 The Laplace Transform
10.1.1 Relationship Between the Laplace Transform and the z-Transform
10.2 Properties of the Laplace Transform
10.2.1 Linearity
10.2.2 Time Shifting
10.2.3 Frequency Shifting
10.2.4 Time Differentiation
10.2.5 Integration
10.2.6 Time Scaling
10.2.7 Convolution in Time
10.2.8 Multiplication by t
10.2.9 Initial Value
10.2.10 Final Value
10.2.11 Transform of Semiperiodic Functions
10.3 The Inverse Laplace Transform
10.3.1 Inverse Laplace Transform by Partial Fraction Expansion
10.4 Applications of the Laplace Transform
10.4.1 Transfer Function and the System Response
10.4.2 Characterization of a System by Its Poles and Zeros
10.4.3 Unit-Step Response and Transient-Response Specifications
10.4.4 System Stability
10.4.5 Realization of Systems
10.4.6 Frequency-Domain Representation of Circuits
10.4.7 Feedback Systems
10.4.8 Bode Diagram
10.4.9 The Nyquist Plot
10.4.9.1 Operational Amplifier Circuits
10.4.10 Analog Filters
10.4.10.1 Butterworth Filters
10.5 Summary
Exercises
11 State-Space Analysis of Discrete Systems
11.1 The State-Space Model
11.1.1 Parallel Realization
11.1.2 Cascade Realization
11.2 Time-Domain Solution of the State Equation
11.2.1 Iterative Solution
11.2.2 Closed-Form Solution
11.2.3 The Impulse Response
11.3 Frequency-Domain Solution of the State Equation
11.4 Linear Transformation of State Vectors
11.5 Summary
Exercises
12 State-Space Analysis of Continuous Systems
12.1 The State-Space Model
12.2 Time-Domain Solution of the State Equation
12.3 Frequency-Domain Solution of the State Equation
12.4 Linear Transformation of State Vectors
12.5 Diagonalization
12.6 Similarity Transformation
12.7 Controllability
12.8 Observability
12.9 Summary
Exercises
A Complex Numbers
B Transform Pairs and Properties
C Useful Mathematical Formulas
C.1 Trigonometric Identities
C.2 Series Expansions
C.3 Summation Formulas
C.4 Indefinite Integrals
C.5 Differentiation Formulas
C.6 L'Hôpital's Rule
C.7 Matrix Inversion
Answers to Selected Exercises
Answers to Selected Exercises
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Bibliography
Index