Table of Contents

Signals And Systems Cover Image
Chapter 1 - Signal Representation and Modeling
  • Chapter Objectives
  • 1.1 Introduction
  • 1.2 Mathematical Modeling of Signals
  • 1.3 Continuous-Time Signals
    • 1.3.1 Signal operations
    • 1.3.2 Basic building blocks for continuous-time signals
    • 1.3.3 Impulse decomposition for continuous-time signals
    • 1.3.4 Signal classifications
    • 1.3.5 Energy and power definitions
    • 1.3.6 Symmetry properties
    • 1.3.7 Graphical representation of sinusoidal signals using phasors
  • 1.4 Discrete-Time Signals
    • 1.4.1 Signal operations
    • 1.4.2 Basic building blocks for discrete-time signals
    • 1.4.3 Impulse decomposition for discrete-time signals
    • 1.4.4 Signal classifications
    • 1.4.5 Energy and power definitions
    • 1.4.6 Symmetry properties
  • 1.5 Further Reading
  • MATLAB Exercises
    • Computing and graphing continuous-time signals
    • Describing signals using piecewise linear segments
    • Signal operations for continuous-time signals
    • Creating periodic signals
    • Functions for basic building blocks
    • Computing and graphing discrete-time signals
    • Periodic extension of a discrete-time signal
  • Problems
  • MATLAB Problems
  • MATLAB Projects
Chapter 2 - Analyzing Continuous-Time Systems in the Time Domain
  • Chapter Objectives
  • 2.1 Introduction
  • 2.2 Linearity and Time Invariance
    • 2.2.1 Linearity in continuous-time systems
    • 2.2.2 Time invariance in continuous-time systems
    • 2.2.3 CTLTI systems
  • 2.3 Differential Equations for Continuous-Time Systems
  • 2.4 Constant-Coefficient Ordinary Differential Equations
  • 2.5 Solving Differential Equations
    • 2.5.1 Solution of the first-order differential equation
    • 2.5.2 Solution of the general differential equation
    • 2.5.3 Finding the natural response of a continuous-time system
    • 2.5.4 Finding the forced response of a continuous-time system
  • 2.6 Block Diagram Representation of Continuous-Time Systems
  • 2.7 Impulse Response and Convolution
    • 2.7.1 Finding impulse response of a CTLTI system
    • 2.7.2 Convolution operation for CTLTI systems
  • 2.8 Causality in Continuous-Time Systems
  • 2.9 Stability in Continuous-Time Systems
  • 2.10 Approximate Numerical Solution of a Differential Equation
  • 2.11 Further Reading
  • MATLAB Exercises
    • Testing linearity of continuous-time systems
    • Testing time invariance of continuous-time systems
    • Using linearity to determine the response of the RC circuit
    • Numerical solution of the RC circuit using Euler method
    • Improved numerical solution of the RC circuit
  • Problems
  • MATLAB Problems
  • MATLAB Projects
Chapter 3 - Analyzing Discrete-Time Systems in the Time Domain
  • Chapter Objectives
  • 3.1 Introduction
  • 3.2 Linearity and Time Invariance
    • 3.2.1 Linearity in discrete-time systems
    • 3.2.2 Time invariance in discrete-time systems
    • 3.2.3 DTLTI systems
  • 3.3 Difference Equations for Discrete-Time Systems
  • 3.4 Constant-Coefficient Linear Difference Equations
  • 3.5 Solving Difference Equations
    • 3.5.1 Finding the natural response of a discrete-time system
    • 3.5.2 Finding the forced response of a discrete-time system
  • 3.6 Block Diagram Representation of Discrete-Time Systems
  • 3.7 Impulse Response and Convolution
    • 3.7.1 Finding impulse response of a DTLTI system
    • 3.7.2 Convolution operation for DTLTI systems
  • 3.8 Causality in Discrete-Time Systems
  • 3.9 Stability in Discrete-Time Systems
  • 3.10 Further Reading
  • MATLAB Exercises
    • Writing functions for moving average filters
    • Testing the functions written in MATLAB Exercise 3.1
    • Writing and testing a function for the exponential smoother
    • Iteratively solving a difference equation
    • Implementing a discrete-time system from its block diagram
    • Discrete-time system of MATLAB Exercise 3.5 revisited
    • Convolution using MATLAB
    • Implementing a moving average filter through convolution
  • Problems
  • MATLAB Problems
  • MATLAB Projects
Chapter 4 - Fourier Analysis for Continuous-Time Signals and Systems
  • Chapter Objectives
  • 4.1 Introduction
  • 4.2 Analysis of Periodic Continuous-Time Signals
    • 4.2.1 Approximating a periodic signal with trigonometric functions
    • 4.2.2 Trigonometric Fourier series (TFS)
    • 4.2.3 Exponential Fourier series (EFS)
    • 4.2.4 Compact Fourier series (CFS)
    • 4.2.5 Existence of Fourier series
    • 4.2.6 Gibbs phenomenon
    • 4.2.7 Properties of Fourier series
  • 4.3 Analysis of Non-Periodic Continuous-Time Signals
    • 4.3.1 Fourier transform
    • 4.3.2 Existence of Fourier transform
    • 4.3.3 Developing further insight
    • 4.3.4 Fourier transforms of some signals
    • 4.3.5 Properties of the Fourier transform
    • 4.3.6 Applying Fourier transform to periodic signals
  • 4.4 Energy and Power in the Frequency Domain
    • 4.4.1 Parseval’s theorem
    • 4.4.2 Energy and power spectral density
    • 4.4.3 Autocorrelation
  • 4.5 System Function Concept
  • 4.6 CTLTI Systems with Periodic Input Signals
    • 4.6.1 Response of a CTLTI system to complex exponential signal
    • 4.6.2 Response of a CTLTI system to sinusoidal signal
    • 4.6.3 Response of a CTLTI system to periodic input signal
  • 4.7 CTLTI Systems with Non-Periodic Input Signals
  • 4.8 Further Reading
  • MATLAB Exercises
    • Computing finite-harmonic approximation to pulse train
    • Computing multiple approximations to pulse train
    • Graphing the line spectrum in Example 4.5
    • Line spectrum for Example 4.6
    • Graphing system function for RC circuit
  • Problems
  • MATLAB Problems
  • MATLAB Projects
Chapter 5 - Fourier Analysis for Discrete-Time Signals and Systems
  • Chapter Objectives
  • 5.1 Introduction
  • 5.2 Analysis of Periodic Discrete-Time Signals
    • 5.2.1 Discrete-Time Fourier Series (DTFS)
    • 5.2.2 Properties of the DTFS
  • 5.3 Analysis of Non-Periodic Discrete-Time Signals
    • 5.3.1 Discrete-time Fourier transform (DTFT)
    • 5.3.2 Developing further insight
    • 5.3.3 Existence of the DTFT
    • 5.3.4 DTFT of some signals
    • 5.3.5 Properties of the DTFT
    • 5.3.6 Applying DTFT to periodic signals
  • 5.4 Energy and Power in the Frequency Domain
    • 5.4.1 Parseval’s theorem
    • 5.4.2 Energy and power spectral density
    • 5.4.3 Autocorrelation
  • 5.5 System Function Concept
  • 5.6 DTLTI Systems with Periodic Input Signals
    • 5.6.1 Response of a DTLTI system to complex exponential signal
    • 5.6.2 Response of a DTLTI system to sinusoidal signal
    • 5.6.3 Response of a DTLTI system to periodic input signal
  • 5.7 DTLTI Systems with Non-Periodic Input Signals
  • 5.8 Discrete Fourier Transform
    • 5.8.1 Relationship of the DFT to the DTFT
    • 5.8.2 Zero padding
    • 5.8.3 Properties of the DFT
    • 5.8.4 Using the DFT to approximate the EFS coefficients
    • 5.8.5 Using the DFT to approximate the continuous Fourier transform
  • 5.9 Further Reading
  • MATLAB Exercises
    • Developing functions to implement DTFS analysis and synthesis
    • Testing DTFS functions
    • Developing and testing a function to implement periodic convolution
    • Steady-state response of DTLTI system to sinusoidal input
    • Exploring the relationship between the DFT and the DTFT
    • Using the DFT to approximate the DTFT
    • Developing functions for circular time shifting and time reversal
    • Computing conjugate symmetric and antisymmetric components
    • Using the symmetry properties of the DFT
    • Circular and linear convolution using the DFT
    • Developing a convolution function using the DFT
    • Exponential Fourier series approximation using the DFT
    • Testing the EFS approximation function
    • Fourier transform approximation using DFT
  • Problems
  • MATLAB Problems
  • MATLAB Projects
Chapter 6 - Sampling and Reconstruction
  • Chapter Objectives
  • 6.1 Introduction
  • 6.2 Sampling of a Continuous-Time Signal
    • 6.2.1 Nyquist sampling criterion
    • 6.2.2 DTFT of sampled signal
    • 6.2.3 Sampling of sinusoidal signals
    • 6.2.4 Practical issues in sampling
  • 6.3 Reconstruction of a Signal From Its Sampled Version
  • 6.4 Resampling Discrete-Time Signals
    • 6.4.1 Reducing the sampling rate by an integer factor
    • 6.4.2 Increasing the sampling rate by an integer factor
  • 6.5 Further Reading
  • MATLAB Exercises
    • Spectral relations in impulse sampling
    • DTFT of discrete-time signal obtained through sampling
    • Sampling a sinusoidal signal
    • Natural sampling
    • Zero-order hold sampling
    • Graphing signals for natural and zero-order hold sampling
    • Reconstruction of right-sided exponential
    • Frequency spectrum of reconstructed signal
    • Resampling discrete-time signals
  • Problems
  • MATLAB Problems
  • MATLAB Projects
Chapter 7 - Laplace Transform for Continuous-Time Signals and Systems
  • Chapter Objectives
  • 7.1 Introduction
  • 7.2 Characteristics of the Region of Convergence
  • 7.3 Properties of the Laplace Transform
    • 7.3.1 Linearity
    • 7.3.2 Time shifting
    • 7.3.3 Shifting in the s-domain
    • 7.3.4 Scaling in time and s- domains
    • 7.3.5 Differentiation in the time domain
    • 7.3.6 Differentiation in the s-domain
    • 7.3.7 Convolution property
    • 7.3.8 Integration property
  • 7.4 Inverse Laplace Transform
    • 7.4.1 Partial fraction expansion with simple poles
    • 7.4.2 Partial fraction expansion with multiple poles
  • 7.5 Using the Laplace Transform with CTLTI Systems
    • 7.5.1 Relating the system function to the differential equation
    • 7.5.2 Response of a CTLTI system to a complex exponential signal
    • 7.5.3 Response of a CTLTI system to an exponentially damped sinusoid
    • 7.5.4 Pole-zero plot for a system function
    • 7.5.5 Graphical interpretation of the pole-zero plot
    • 7.5.6 System function and causality
    • 7.5.7 System function and stability
    • 7.5.8 All-pass systems
    • 7.5.9 Inverse systems
    • 7.5.10 Bode plots
  • 7.6 Simulation Structures for CTLTI Systems
    • 7.6.1 Direct-form implementation
    • 7.6.2 Cascade and parallel forms
  • 7.7 Unilateral Laplace Transform
    • 7.7.1 Time shifting
    • 7.7.2 Differentiation in time
    • 7.7.3 Initial and final value theorems
  • 7.8 Further Reading
  • MATLAB Exercises
    • Three dimensional plot of Laplace transform
    • Computing the Fourier transform from the Laplace transform
    • Graphing poles and zeros
    • Residue calculations
    • Symbolic calculations for Laplace transform
    • Computing frequency response of a system from pole-zero layout
    • Frequency response from pole-zero layout revisited
    • System objects
    • Bode plots
    • Solving a differential equation through Laplace transform
  • Problems
  • MATLAB Problems
  • MATLAB Projects
Chapter 8 - z-Transform for Discrete-Time Signals and Systems
  • Chapter Objectives
  • 8.1 Introduction
  • 8.2 Characteristics of the Region of Convergence
  • 8.3 Properties of the z-Transform
    • 8.3.1 Linearity
    • 8.3.2 Time shifting
    • 8.3.3 Time reversal
    • 8.3.4 Multiplication by an exponential signal
    • 8.3.5 Differentiation in the z-domain
    • 8.3.6 Convolution property
    • 8.3.7 Initial value
    • 8.3.8 Correlation property
    • 8.3.9 Summation property
  • 8.4 Inverse z-Transform
    • 8.4.1 Inversion integral
    • 8.4.2 Partial fraction expansion
    • 8.4.3 Long division
  • 8.5 Using the z-Transform with DTLTI Systems
    • 8.5.1 Relating the system function to the difference equation
    • 8.5.2 Response of a DTLTI system to complex exponential signal
    • 8.5.3 Response of a DTLTI system to exponentially damped sinusoid
    • 8.5.4 Graphical interpretation of the pole-zero plot
    • 8.5.5 System function and causality
    • 8.5.6 System function and stability
    • 8.5.7 Allpass systems
    • 8.5.8 Inverse systems
  • 8.6 Implementation Structures for DTLTI Systems
    • 8.6.1 Direct-form implementations
    • 8.6.2 Cascade and parallel forms
  • 8.7 Unilateral z-Transform
  • 8.8 Further Reading
  • MATLAB Exercises
    • Three-dimensional plot of z-transform
    • Computing the DTFT from the z-transform
    • Graphing poles and zeros
    • Using convolution function for polynomial multiplication
    • Partial fraction expansion with MATLAB
    • Developing a function for long division
    • Computing frequency response of a system from pole-zero layout
    • Frequency response from pole-zero layout revisited
    • Preliminary calculations for a cascade-form block diagram
    • Preliminary calculations for a cascade-form block diagram revisited
    • Preliminary calculations for a parallel-form block diagram
    • Implementing a system using second-order sections
    • Solving a difference equation through z-transform
  • Problems
  • MATLAB Problems
  • MATLAB Projects
Chapter 9 - State-Space Analysis of Systems
  • Chapter Objectives
  • 9.1 Introduction
  • 9.2 State-Space Modeling of Continuous-Time Systems
    • 9.2.1 State-space models for CTLTI systems
    • 9.2.2 Obtaining state-space model from physical description
    • 9.2.3 Obtaining state-space model from differential equation
    • 9.2.4 Obtaining state-space model from system function
    • 9.2.5 Alternative state-space models
    • 9.2.6 CTLTI systems with multiple inputs and/or outputs
    • 9.2.7 Solution of state-space model
    • 9.2.8 Computation of the state transition matrix
    • 9.2.9 Obtaining system function from state-space model
  • 9.3 State-Space Modeling of Discrete-Time Systems
    • 9.3.1 State-space models for DTLTI systems
    • 9.3.2 Obtaining state-space model from difference equation
    • 9.3.3 Obtaining state-space model from system function
    • 9.3.4 Solution of state-space model
    • 9.3.5 Obtaining system function from state-space model
  • 9.4 Discretization of Continuous-Time State-Space Model
  • 9.5 Further Reading
  • MATLAB Exercises
    • Obtaining state-space model from system function
    • Diagonalizing the state matrix
    • Computation of the state transition matrix
    • Solving the homogeneous state equation
    • Symbolic computation of the state transition matrix
    • Obtaining system function from continuous-time state-space model
    • Obtaining system function from discrete-time state-space model
    • Discretization of state-space model
    • Discretization using Euler method
  • Problems
  • MATLAB Problems
Chapter 10 - Analysis and Design of Filters
  • Chapter Objectives
  • 10.1 Introduction
  • 10.2 Distortionless Transmission
  • 10.3 Ideal Filters
  • 10.4 Design of Analog Filters
    • 10.4.1 Butterworth lowpass filters
    • 10.4.2 Chebyshev lowpass filters
    • 10.4.3 Inverse Chebyshev lowpass filters
    • 10.4.4 Analog filter transformations
  • 10.5 Design of Digital Filters
    • 10.5.1 Design of IIR filters
    • 10.5.2 Design of FIR filters
  • 10.6 Further Reading
  • MATLAB Exercises
    • Butterworth analog filter design
    • Chebyshev polynomials
    • Chebyshev type-I analog filter design
    • Determining Chebyshev analog filter parameters
    • Chebyshev type-II analog filter design
    • Lowpass to highpass filter transformation
    • Lowpass to bandpass and lowpass to band-reject transformations
    • Impulse-invariant design
    • IIR filter design using bilinear transformation
    • IIR filter design using bilinear transformation revisited
    • A complete IIR filter design example
    • FIR filter design using Fourier series method
    • FIR filter design using Parks-McClellan technique
  • Problems
  • MATLAB Problems
  • MATLAB Projects
Chapter 11 - Amplitude Modulation
  • Chapter Objectives
  • 11.1 Introduction
  • 11.2 The Need for Modulation
  • 11.3 Types of Modulation
  • 11.4 Amplitude Modulation
    • 11.4.1 Frequency spectrum of the AM signal
    • 11.4.2 Power balance and modulation efficiency
    • 11.4.3 Generation of AM signals
    • 11.4.4 Demodulation of AM signals
  • 11.5 Double-Sideband Suppressed Carrier Modulation
    • 11.5.1 Frequency spectrum of the DSB-SC signal
  • 11.6 Single-Sideband Modulation
  • 11.7 Further Reading
  • MATLAB Exercises
    • Compute and graph the AM signal
    • EFS spectrum of the tone-modulated AM signal
    • Function to simulate a switching modulator
    • Testing the switching modulator
    • Function to simulate a square-law modulator
    • Testing the square-law modulator
    • Function to simulate envelope detector
    • Testing the envelope detector function
  • Problems
  • MATLAB Problems
  • MATLAB Projects
Appendices
  • A Complex Numbers and Euler’s Formula
    • A.1 Introduction
    • A.2 Arithmetic with Complex Numbers
      • A.2.1 Addition and subtraction
      • A.2.2 Multiplication and division
    • A.3 Euler’s Formula
  • B Mathematical Relations
    • B.1 Trigonometric Identities
    • B.2 Indefinite Integrals
    • B.3 Laplace Transform Pairs
    • B.4 z-Transform Pairs
  • C Closed Forms for Sums of Geometric Series
    • C.1 Infinite-Length Geometric Series
    • C.2 Finite-Length Geometric Series
    • C.3 Finite-Length Geometric Series (Alternative Form)
  • D Orthogonality of Basis Functions
    • D.1 Orthogonality for Trigonometric Fourier Series
    • D.2 Orthogonality for Exponential Fourier Series
    • D.3 Orthogonality for Discrete-Time Fourier Series
  • E Partial Fraction Expansion
    • E.1 Partial Fraction Expansion for Continuous-Time Signals and Systems
    • E.2 Partial Fraction Expansion for Discrete-Time Signals and Systems
  • F Review of Matrix Algebra