Practical Signals Theory with MATLAB Applications

A hands-on resource designed to teach the mathematics of signals and systems with MATLAB

In this newly revised second edition of Practical Signals Theory with MATLAB Applications, Richard Tervo delivers an articulate presentation of the mathematics underlying real world engineering applications and everyday electronic devices. The new edition provides extended coverage of communication systems—including digital and spread spectrum communications—as well as a new introductory chapter on using MATLAB as a tool to visualize the mathematics of signals and systems.

The text contains numerous hands-on examples and expanded end-of-chapter exercises. It is a one-stop reference for signals and systems, explaining aspects of commonplace signal types, orthogonality and signal decomposition, transformations, and the graphical presentation of calculations and results. Readers will also find:

• A solid introduction to the mathematics of continuous and discrete signals represented in time and frequency domains
• Thorough coverage of the classic Fourier, Laplace and z-transforms, and their many applications
• New end-of-chapter worked exercises, a variety of in-line study questions with answers and easily reproducible MATLAB code demonstrations
• Bonus material on related applications in different fields of study and a companion website designed to support additional learning

Perfect for undergraduate and graduate students of signals and systems, signals theory, and related areas of electrical engineering,Practical Signals Theory with MATLAB Applications will also benefit researchers and professors in the field of system design and signal processing.

Preface

Pedagogy

Organization

Chapter 1. Practical MATLAB with Signals Theory

Chapter 2. Introduction to Signals and Systems

Chapter 3. Classification of Signals

Chapter 4. Linear Systems

Chapter 5. The Fourier Series

Chapter 6. The Fourier Transform

Chapter 7. Practical Fourier Transforms

Chapter 8. The Laplace Transform

Chapter 9. Discrete Signals

Chapter 10. The z-Transform

Chapter 11. Communications Systems

0.1 Useful Information (inside cover / endpaper)

0.1.1 Identities

0.1.2 Definite Integrals

0.1.3 Infinite Series

0.1.4 Orthogonality

0.1.5 Signal Inner Product

0.1.6 Convolution

0.1.7 Fourier Series

0.1.8 Complex Fourier Series

0.1.9 Fourier Transform

0.1.10 Laplace Transform

0.1.11 z-Transform

0.2 List of Acronyms

0.2.1 Communications Acronyms

1 Practical MATLAB with Signals Theory 1

Learning Objectives

1.1 Introduction

1.1.1 Accessing MATLAB

1.1.2 Learning MATLAB

1.1.3 The MATLAB Desktop

1.1.4 Help with MATLAB

1.1.5 Numeric Variables for Signals Theory

1.1.6 MATLAB Arrays, Matrices, Vectors

1.1.7 Recording a MATLAB session

1.2 Visualizing Functions

1.2.1 Making a Rough Sketch of a Function

1.2.2 Plotting a Function by Hand

1.2.3 Plotting a Function with MATLAB

1.2.4 Enhanced Plotting Functions

1.3 MATLAB M-Files

1.3.1 Creating a MATLAB Function

1.3.2 Anonymous Functions

1.4 Numerical Integration

1.4.1 Generalized Numerical Integration

1.5 The for loop

1.6 Conditional and Logical Expressions

1.7 Piecewise Continuous Signals

1.8 Complex Numbers in MATLAB

1.8.1 Representation of Complex Numbers

1.8.2 Euler's Formula

1.8.3 The Complex Plane

Viewing a Function from Different Perspectives

1.9 Conclusions

1.10 Worked Problems

1.11 End of Chapter Exercises

Bibliography

2 Introduction to Signals and Systems

Learning Objectives

2.1 Introduction

2.1.1 What is a Signal?

2.1.2 What is a System?

2.2 Introduction to Signal Manipulation

2.2.1 Amplification

2.2.2 Shifting

2.2.3 Scaling

2.2.4 Linear Combination

2.2.5 Addition and Multiplication of Signals

2.2.6 Visualizing Signals - An Important Skill

2.3 Basic Signals

2.3.1 The Unit Rectangle : rect(t)

2.3.2 The Unit Step u(t)

2.3.3 The Exponential e^kt

2.3.4 The Unit Impulse δ(t)

2.3.5 Plotting the Impulse Aδ(t-x)

2.4 The Sinusoidal Signal

2.4.1 The One-Sided Cosine Representation

2.4.2 Phase Change -

Phase Change vs. Time Shift

2.4.3 Sine vs. Cosine

2.5 Conclusions

2.6 Worked Problems

2.7 End of Chapter Exercises

Bibliography

3 Classification of Signals

Learning Objectives

3.1 Introduction

3.2 Odd and Even Signals

3.2.1 Combining Odd and Even signals

3.2.2 The constant value s(t) = k

3.3 Periodic Signals

3.3.1 DC Component in Periodic Signals

3.3.2 Sinusoids and Rectifiers

3.3.3 Square Wave

3.3.4 Sawtooth Wave

3.3.5 Triangle wave

3.3.6 Pulse Train

3.3.7 Rectangular Pulse Train

3.3.8 Impulse Train

3.3.9 Trigonometric Identities

3.3.10 Sinusoidal Multiplication

Modulation Property

Dial Tone Generator

Squaring the Sinusoid

3.4 Energy and Power Signals

3.4.1 Periodic Signals = Power Signals

V_rms is not always A/√2

3.4.2 Comparing Signal Power: The Decibel (dB)

3.5 Complex Signals

3.6 Discrete Time Signals

3.7 Random Signals

3.8 Conclusions

3.9 Worked Problems

3.10 End of Chapter Exercises

Bibliography

4 Linear Systems

Learning Objectives

4.1 Introduction

4.2 Definition of a Linear System

4.2.1 Superposition

4.2.2 Example 1: Zero-State Response

4.2.3 Example 2: Operating in a linear region

4.2.4 Example 3: Mixer

4.2.5 Linear Time-Invariant (LTI) Systems

4.2.6 Bounded Input, Bounded Output

4.2.7 System Behavior as a Black Box

4.3 LTI System Response Function h(t)

4.4 Convolution

4.4.1 The Convolution Integral

4.4.2 Convolution is Commutative

4.4.3 Convolution is Associative

4.4.4 Convolution is Distributive over Addition

4.4.5 Evaluation of the Convolution Integral

Graphical Convolution 1: Rectangle with Itself

4.4.6 Convolution Properties

Graphical Convolution 2: Two Rectangles

Graphical Convolution 3: Rectangle and Exponential Decay

4.4.7 Convolution in MATLAB

4.5 Determining h(t) in an Unknown System

4.5.1 The Unit Impulse δ(t) Test Signal

4.5.2 Convolution and Signal Decomposition

Convolution and Periodic Signals

4.5.3 An Ideal Distortionless System

Deconvolution

4.6 Causality

4.6.1 Causality and Zero Input Response

4.7 Combined Systems

4.8 Convolution and Random Numbers

4.9 Useful Hints and Help with MATLAB

4.10 Chapter Summary

4.11 Conclusions

4.12 Worked Problems

4.13 End of Chapter Exercises

Bibliography

5 The Fourier Series

Learning Objectives

Chapter Overview

5.1 Introduction

5.2 Expressing Signals by Components

5.2.1 The Spectrum Analyzer

5.2.2 Approximating a Signal s(t) by Another

5.2.3 Estimating One Signal by Another

5.3 Part One - Orthogonal Signals

5.4 Orthogonality

5.4.1 An Orthogonal Signal Space

5.4.2 The Signal Inner Product Formulation

5.4.3 Complete Set of Orthogonal Signals

5.4.4 What if a Complete Set is not Present?

5.4.5 An Orthogonal Set of Signals

5.5 Part Two - The Fourier Series

5.5.1 The Orthogonal Signals {sin(2ϖmƒ_ot); cos(2ϖnƒ_ot)}

5.5.2 The Fourier Series - An Orthogonal Set?

5.6 Computing Fourier Series Components

5.6.1 Fourier Series Approximation to an Odd Square Wave

5.6.2 Zero-Frequency (DC) Component

5.6.3 Fundamental Frequency Component

5.6.4 Higher Order Components

5.6.5 Frequency Spectrum of the Square Wave s(t)

5.7 Odd and Even Square Waves

5.7.1 The Fourier Series Components of an Even Square Wave

5.8 Gibb's Phenomenon

5.9 Setting-Up the Fourier Series Calculation

5.9.1 Appearance of Pulse Train Frequency Components

5.10 Some Common Fourier Series

5.11 Practical Harmonics

5.11.1 Audio Ampli_er Specs - Total Harmonic Distortion

5.11.2 The CB Radio Booster

5.12 Part Three: The Complex Fourier Series

5.12.1 Not all Signals are Even or Odd

5.13 The Complex Fourier Series

5.13.1 Complex Fourier Series - The Frequency Domain

5.13.2 Comparing the Real and Complex Fourier Series

5.13.3 Magnitude and Phase

5.14 Complex Fourier Series Components

5.14.1 Real Signals and the Complex Fourier Series

5.14.2 Stretching and Squeezing: Time vs. Frequency

5.14.3 Shift in Time

5.14.4 Change in Amplitude

5.14.5 Power in Periodic Signals

Find the Total Power in s(t) = Acos(t) + B sin(t)

5.14.6 Parseval's Theorem for Periodic Signals

5.15 Properties of the Complex Fourier Series

5.16 Analysis of a DC Power Supply

5.16.1 The DC Component

5.16.2 An AC-DC Converter

5.16.3 Vrms is always greater than or equal to V_dc

5.16.4 Fourier Series: The Full-wave Rectifier

5.16.5 Complex Fourier series components C_n

Power in the Fundamental Frequency 120 Hz

5.17 The Fourier Series with MATLAB

5.17.1 Finding Fourier Series Components

A full-wave rectified cosine (60 Hz)

5.17.2 Effective use of the Fast Fourier Transform

5.18 Conclusions

5.19 Worked Problems

5.20 End of Chapter Exercises

Bibliography

6 The Fourier Transform

Learning Objectives

6.1 Introduction

6.1.1 A Fresh Look at the Fourier Series

Periodic and Non-Periodic Signals

6.1.2 Approximating a Non-Periodic Signal Over All Time

6.1.3 Definition of the Fourier Transform

6.1.4 Existence of the Fourier Transform

6.1.5 The Inverse Fourier Transform

6.2 Properties of the Fourier Transform

6.2.1 Linearity of the Fourier Transform

6.2.2 Value of the Fourier transform at the Origin

6.2.3 Odd and Even Functions and the Fourier Transform

6.3 The Rectangle Signal

Alternate Solution

6.4 The Sinc Function

6.4.1 Expressing a Function in Terms of sinc(t)

6.4.2 The Fourier Transform of a General Rectangle

6.5 Signal Manipulations: Time and Frequency

6.5.1 Amplitude Variations

6.5.2 Stretch and Squeeze: The Sinc Function

6.5.3 The Scaling Theorem

6.5.4 Testing the Limits

6.5.5 A Shift in Time

6.5.6 The Shifting Theorem

6.5.7 The Fourier Transform of a Shifted Rectangle

Magnitude of G(ƒ)

Phase of G(ƒ)

6.5.8 Impulse Series - The Line Spectrum

6.5.9 Shifted Impulse δ(ƒ – ƒ_o)

6.5.10 Fourier Transform of a Periodic Signal

6.6 Fourier Transform Pairs

6.6.1 The Illustrated Fourier Transform

6.7 Rapid Changes vs. High Frequencies

6.7.1 Derivative Theorem

6.7.2 Integration Theorem

6.8 Conclusions

6.9 Worked Problems

6.10 End of Chapter Exercises

Bibliography

7 Practical Fourier Transforms 349

7.1 Introduction

Learning Objectives

7.2 Convolution: Time and Frequency

The Logarithm Domain

7.2.1 Simplifying the Convolution Integral

7.3 Transfer Function of a Linear System

7.3.1 Impulse Response: The Frequency Domain

7.3.2 Frequency Response Curve

7.4 Energy in Signals: Parseval's Theorem for the Fourier Transform

7.4.1 Energy Spectral Density

7.5 Data Smoothing and the Frequency Domain

7.6 Ideal Filters

7.6.1 The Ideal Low-Pass Filter is not Causal

7.7 A Real Low-Pass Filter

MATLAB Example 1: First Order Filter

7.8 The Modulation Theorem

7.8.1 A Voice Privacy System

Spectral Inversion

7.9 Periodic Signals and the Fourier Transform

7.9.1 The Impulse Train

7.9.2 General Appearance of Periodic Signals

7.9.3 The Fourier Transform of a Square wave

Changing the Pulse Train Appearance

7.9.4 Other Periodic Waveforms

7.10 The Analog Spectrum Analyzer

7.11 Conclusions

7.12 Worked Problems

7.13 End of Chapter Exercises

Bibliography

8 The Laplace Transform

Learning Objectives

8.1 Introduction

8.2 The Laplace Transform

8.2.1 The Frequency Term e^jwt

8.2.2 The Exponential Term e^σt

8.2.3 The s-domain

8.3 Exploring the s-domain

8.3.1 Poles and Zeros

8.3.2 A Pole at the origin

8.3.3 Decaying Exponential

8.3.4 A Sinusoid

8.3.5 A Decaying Sinusoid

8.3.6 An Unstable System

8.4 Visualizing the Laplace Transform

8.4.1 First Order Low-pass Filter

8.4.2 Pole Position Determines Frequency Response

8.4.3 Second Order Low-pass Filter

8.4.4 Two-Sided Laplace Transform

8.4.5 The Bode Diagram

8.4.6 Calculating the Laplace Transform

8.4.7 System Analysis in MATLAB

8.5 Properties of the Laplace Transform

8.6 Differential Equations

8.6.1 Solving a Differential Equation

8.6.2 Transfer Function as Differential Equations

8.7 Laplace Transform Pairs

8.7.1 The Illustrated Laplace Transform

8.8 Circuit Analysis with the Laplace Transform

8.8.1 Voltage Divider

8.8.2 A First-Order Low-pass Filter

8.8.3 A First-Order High-pass Filter

8.8.4 A Second Order Filter

8.9 State Variable Analysis

8.9.1 State Variable Analysis - First Order System

8.9.2 First Order State Space Analysis with MATLAB

8.9.3 State Variable Analysis - Second Order System

8.9.4 Matrix Form of the State Space Equations

8.9.5 Second Order State Space Analysis with MATLAB

8.9.6 Differential Equation

8.9.7 State Space and Transfer Functions with MATLAB

8.10 Conclusions

8.11 Worked Problems

8.12 End of Chapter Exercises

Bibliography

9 Discrete Signals

9.1 Introduction

Learning Objectives

9.2 Discrete Time vs. Continuous Time Signals

9.3 A Discrete Time Signal

9.3.1 Digital Signal Processing

9.3.2 A Periodic Discrete Time Signal

9.4 Data Collection and Sampling Rate

9.4.1 The Selection of a Sampling Rate

9.4.2 Bandlimited Signal

9.4.3 Theory of Sampling

9.4.4 The Sampling Function

9.4.5 Recovering a Waveform from Samples

9.4.6 A Practical Sampling Signal

9.4.7 Minimum Sampling Rate

9.4.8 Nyquist Sampling Rate

9.4.9 The Nyquist Sampling Rate is a Theoretical Minimum

9.4.10 Sampling Rate and Alias Frequency

9.4.11 Practical Aliasing

9.4.12 Analysis of Aliasing

9.4.13 Anti-Alias Filter

9.5 Introduction to Digital Filtering

9.5.1 Impulse Response Function

9.5.2 A Discrete Response Function

9.5.3 Delay Blocks are a Natural Consequence of Sampling

9.5.4 General Digital Filtering

9.5.5 The Fourier Transform of Sampled Signals

9.5.6 The Discrete Fourier Transform (DFT)

9.5.7 A Discrete Fourier Series

9.5.8 Computing the Discrete Fourier Transform (DFT)

9.5.9 The Fast Fourier Transform (FFT)

9.6 Illustrative Examples

The FFT (fft) and Inverse FFT (ifft)

9.6.1 FFT and Sample Rate

9.6.2 Practical DFT Issues

9.7 Filtering Application with MATLAB

9.7.1 Fourier Analysis

9.7.2 System Response

9.7.3 Check Calculation

9.8 Conclusions

9.9 Worked Problems

9.10 End of Chapter Exercises

Bibliography

10 The z-Transform 581

10.1 Introduction

Learning Objectives

10.2 The z-Transform

10.2.1 Fourier Transform, Laplace Transform, z-transform

10.2.2 Defnition of the z-Transform

10.2.3 The z-Plane and the Fourier Transform

10.3 Calculating the z-Transform

10.3.1 Unit Step u[n]

10.3.2 Exponential aⁿ u[n]

10.3.3 Sinusoid cos(nω_o) u[n] and sin(nω_o) u[n]

10.3.4 Differentiation

10.3.5 The Effect of Sampling Rate

10.4 A Discrete Time Laplace Transform

10.5 Properties of the z-Transform

10.6 z-Transform Pairs

10.7 Transfer Function of a Discrete Linear System

10.8 MATLAB Analysis with the z-transform

10.8.1 First Order Low-pass Filter

10.8.2 Pole-zero Plot

10.8.3 Bode diagram

10.8.4 Impulse Response

10.8.5 Calculating Frequency Response

10.8.6 Pole Position Determines Frequency Response

10.9 Digital Filtering - FIR Filter

10.9.1 A One Pole FIR Filter

10.9.2 A Two Pole FIR Filter

10.9.3 Higher Order FIR Filters

10.10Digital Filtering - IIR Filter

10.10.1A One Pole IIR Filter

10.10.2 IIR vs. FIR

10.10.3 Higher Order IIR Filters

10.10.4 Combining FIR and IIR Filters

10.11Conclusions

10.12Worked Problems

10.13End of Chapter Exercises

11 Communication Systems

Learning Objectives

11.1 Introduction

11.1.1 A Baseband Signal m(t)

11.1.2 The need for a Carrier Signal

11.1.3 A Carrier Signal c(t)

11.1.4 Modulation Techniques

11.1.5 The Radio Spectrum

11.2 Amplitude Modulation

11.2.1 Double Sideband Transmitted Carrier - (DSB-TC)

11.2.2 Demodulation of AM DSB-TC Signals

11.2.3 Graphical Analysis

11.2.4 AM Demodulation - Diode Detector

11.2.5 Examples of Diode Detection

11.3 Suppressed Carrier Transmission

11.3.1 Demodulation of Single Sideband Signals

11.3.2 Percent Modulation and Overmodulation

11.4 Superheterodyne Receiver

11.4.1 An Experiment with Intermediate Frequency

11.4.2 When Receivers become Transmitters

11.4.3 Image Frequency

11.4.4 Beat Frequency Oscillator

11.5 Digital Communications

11.5.1 Modulation Methods

11.5.2 Morse Code

11.5.3 Amplitude Shift Keying (ASK)

11.5.4 Frequency Shift Keying (FSK)

11.6 Phase Shift Keying (PSK)

11.6.1 Differential Coding

11.6.2 Quadrature Amplitude Modulation (QAM)

11.7 Spread Spectrum Systems

11.7.1 Introduction

11.7.2 Pseudorandom Noise

11.7.3 Encoding Bits in DSSS

11.7.4 Spectral Properties of a Pseudo-Random Sequence

11.7.5 Code Division Multiple Access (CDMA)

11.8 Conclusions

11.9 Worked Problems

11.10End of Chapter Exercises

Bibliography

A Reference Tables

A.1 Fourier Transform

A.1.1 Fourier Transform Theorems

A.2 Laplace Transform

A.2.1 Laplace Transform Theorems

A.3 z-Transform

A.3.1 z-Transform Theorems

B The Illustrated Fourier Transform

C The Illustrated Laplace Transform

D The Illustrated z-Transform

E MATLAB Reference Guide

E.1 Defining Signals

E.1.1 MATLAB Variables

E.1.2 The Time Axis

E.1.3 Common Signals

E.2 Complex Numbers

E.3 Plot Commands

E.4 Signal Operations

E.5 Defining Systems

E.5.1 System Definition

E.5.2 System Analysis

Richard Tervo, PhD, is a retired Professor of Electrical and Computer Engineering at the University of New Brunswick, Canada. For over 30 years, he taught signals and communications courses at the undergraduate and graduate levels. He is an expert in teaching the mathematical foundations of signal behavior.