Practical Signals Theory with MATLAB Applications

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.

• PrefacePedagogyOrganizationChapter 1. Practical MATLAB with Signals TheoryChapter 2. Introduction to Signals and SystemsChapter 3. Classification of SignalsChapter 4. Linear SystemsChapter 5. The Fourier SeriesChapter 6. The Fourier TransformChapter 7. Practical Fourier TransformsChapter 8. The Laplace TransformChapter 9. Discrete SignalsChapter 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 Series0.1.4 Orthogonality0.1.5 Signal Inner Product0.1.6 Convolution0.1.7 Fourier Series0.1.8 Complex Fourier Series0.1.9 Fourier Transform0.1.10 Laplace Transform0.1.11 z-Transform0.2 List of Acronyms0.2.1 Communications Acronyms  1 Practical MATLAB with Signals Theory 1Learning Objectives1.1 Introduction1.1.1 Accessing MATLAB1.1.2 Learning MATLAB1.1.3 The MATLAB Desktop1.1.4 Help with MATLAB1.1.5 Numeric Variables for Signals Theory1.1.6 MATLAB Arrays, Matrices, Vectors1.1.7 Recording a MATLAB session1.2 Visualizing Functions1.2.1 Making a Rough Sketch of a Function1.2.2 Plotting a Function by Hand1.2.3 Plotting a Function with MATLAB1.2.4 Enhanced Plotting Functions1.3 MATLAB M-Files1.3.1 Creating a MATLAB Function1.3.2 Anonymous Functions1.4 Numerical Integration1.4.1 Generalized Numerical Integration1.5 The for loop1.6 Conditional and Logical Expressions1.7 Piecewise Continuous Signals1.8 Complex Numbers in MATLAB1.8.1 Representation of Complex Numbers1.8.2 Euler's Formula1.8.3 The Complex PlaneViewing a Function from Different Perspectives1.9 Conclusions1.10 Worked Problems1.11 End of Chapter ExercisesBibliography2 Introduction to Signals and SystemsLearning Objectives2.1 Introduction2.1.1 What is a Signal?2.1.2 What is a System?2.2 Introduction to Signal Manipulation2.2.1 Amplification2.2.2 Shifting2.2.3 Scaling2.2.4 Linear Combination2.2.5 Addition and Multiplication of Signals2.2.6 Visualizing Signals - An Important Skill2.3 Basic Signals2.3.1 The Unit Rectangle : rect(t)2.3.2 The Unit Step u(t)2.3.3 The Exponential ekt 2.3.4 The Unit Impulse δ(t)2.3.5 Plotting the Impulse Aδ(t-x)2.4 The Sinusoidal Signal2.4.1 The One-Sided Cosine Representation2.4.2 Phase Change -Phase Change vs. Time Shift2.4.3 Sine vs. Cosine2.5 Conclusions2.6 Worked Problems2.7 End of Chapter ExercisesBibliography3 Classification of SignalsLearning Objectives3.1 Introduction3.2 Odd and Even Signals3.2.1 Combining Odd and Even signals3.2.2 The constant value s(t) = k3.3 Periodic Signals3.3.1 DC Component in Periodic Signals3.3.2 Sinusoids and Rectifiers3.3.3 Square Wave3.3.4 Sawtooth Wave3.3.5 Triangle wave3.3.6 Pulse Train3.3.7 Rectangular Pulse Train3.3.8 Impulse Train3.3.9 Trigonometric Identities3.3.10 Sinusoidal MultiplicationModulation PropertyDial Tone GeneratorSquaring the Sinusoid3.4 Energy and Power Signals3.4.1 Periodic Signals = Power SignalsVrms is not always A/√23.4.2 Comparing Signal Power: The Decibel (dB)3.5 Complex Signals3.6 Discrete Time Signals3.7 Random Signals3.8 Conclusions3.9 Worked Problems3.10 End of Chapter ExercisesBibliography4 Linear SystemsLearning Objectives4.1 Introduction4.2 Definition of a Linear System4.2.1 Superposition4.2.2 Example 1: Zero-State Response4.2.3 Example 2: Operating in a linear region4.2.4 Example 3: Mixer4.2.5 Linear Time-Invariant (LTI) Systems4.2.6 Bounded Input, Bounded Output4.2.7 System Behavior as a Black Box4.3 LTI System Response Function h(t)4.4 Convolution4.4.1 The Convolution Integral4.4.2 Convolution is Commutative4.4.3 Convolution is Associative4.4.4 Convolution is Distributive over Addition4.4.5 Evaluation of the Convolution IntegralGraphical Convolution 1: Rectangle with Itself4.4.6 Convolution PropertiesGraphical Convolution 2: Two RectanglesGraphical Convolution 3: Rectangle and Exponential Decay4.4.7 Convolution in MATLAB4.5 Determining h(t) in an Unknown System4.5.1 The Unit Impulse δ(t) Test Signal4.5.2 Convolution and Signal DecompositionConvolution and Periodic Signals4.5.3 An Ideal Distortionless SystemDeconvolution4.6 Causality4.6.1 Causality and Zero Input Response4.7 Combined Systems4.8 Convolution and Random Numbers4.9 Useful Hints and Help with MATLAB4.10 Chapter Summary4.11 Conclusions4.12 Worked Problems4.13 End of Chapter ExercisesBibliography5 The Fourier SeriesLearning ObjectivesChapter Overview5.1 Introduction5.2 Expressing Signals by Components5.2.1 The Spectrum Analyzer5.2.2 Approximating a Signal s(t) by Another5.2.3 Estimating One Signal by Another5.3 Part One - Orthogonal Signals5.4 Orthogonality5.4.1 An Orthogonal Signal Space5.4.2 The Signal Inner Product Formulation5.4.3 Complete Set of Orthogonal Signals5.4.4 What if a Complete Set is not Present?5.4.5 An Orthogonal Set of Signals5.5 Part Two - The Fourier Series5.5.1 The Orthogonal Signals {sin(2ϖmƒot); cos(2ϖnƒot)}5.5.2 The Fourier Series - An Orthogonal Set?5.6 Computing Fourier Series Components5.6.1 Fourier Series Approximation to an Odd Square Wave5.6.2 Zero-Frequency (DC) Component5.6.3 Fundamental Frequency Component5.6.4 Higher Order Components5.6.5 Frequency Spectrum of the Square Wave s(t)5.7 Odd and Even Square Waves5.7.1 The Fourier Series Components of an Even Square Wave5.8 Gibb's Phenomenon5.9 Setting-Up the Fourier Series Calculation5.9.1 Appearance of Pulse Train Frequency Components5.10 Some Common Fourier Series5.11 Practical Harmonics5.11.1 Audio Ampli_er Specs - Total Harmonic Distortion5.11.2 The CB Radio Booster5.12 Part Three: The Complex Fourier Series5.12.1 Not all Signals are Even or Odd5.13 The Complex Fourier Series5.13.1 Complex Fourier Series - The Frequency Domain5.13.2 Comparing the Real and Complex Fourier Series5.13.3 Magnitude and Phase5.14 Complex Fourier Series Components5.14.1 Real Signals and the Complex Fourier Series5.14.2 Stretching and Squeezing: Time vs. Frequency5.14.3 Shift in Time5.14.4 Change in Amplitude5.14.5 Power in Periodic SignalsFind the Total Power in s(t) = Acos(t) + B sin(t)5.14.6 Parseval's Theorem for Periodic Signals5.15 Properties of the Complex Fourier Series5.16 Analysis of a DC Power Supply5.16.1 The DC Component5.16.2 An AC-DC Converter5.16.3 Vrms is always greater than or equal to Vdc 5.16.4 Fourier Series: The Full-wave Rectifier5.16.5 Complex Fourier series components Cn Power in the Fundamental Frequency 120 Hz5.17 The Fourier Series with MATLAB5.17.1 Finding Fourier Series ComponentsA full-wave rectified cosine (60 Hz)5.17.2 Effective use of the Fast Fourier Transform5.18 Conclusions5.19 Worked Problems5.20 End of Chapter ExercisesBibliography6 The Fourier TransformLearning Objectives6.1 Introduction6.1.1 A Fresh Look at the Fourier SeriesPeriodic and Non-Periodic Signals6.1.2 Approximating a Non-Periodic Signal Over All Time6.1.3 Definition of the Fourier Transform6.1.4 Existence of the Fourier Transform6.1.5 The Inverse Fourier Transform6.2 Properties of the Fourier Transform6.2.1 Linearity of the Fourier Transform6.2.2 Value of the Fourier transform at the Origin6.2.3 Odd and Even Functions and the Fourier Transform6.3 The Rectangle SignalAlternate Solution6.4 The Sinc Function6.4.1 Expressing a Function in Terms of sinc(t)6.4.2 The Fourier Transform of a General Rectangle6.5 Signal Manipulations: Time and Frequency6.5.1 Amplitude Variations6.5.2 Stretch and Squeeze: The Sinc Function6.5.3 The Scaling Theorem6.5.4 Testing the Limits6.5.5 A Shift in Time6.5.6 The Shifting Theorem6.5.7 The Fourier Transform of a Shifted RectangleMagnitude of G(ƒ)Phase of G(ƒ)6.5.8 Impulse Series - The Line Spectrum6.5.9 Shifted Impulse δ(ƒ – ƒo)6.5.10 Fourier Transform of a Periodic Signal6.6 Fourier Transform Pairs6.6.1 The Illustrated Fourier Transform6.7 Rapid Changes vs. High Frequencies6.7.1 Derivative Theorem6.7.2 Integration Theorem6.8 Conclusions6.9 Worked Problems6.10 End of Chapter ExercisesBibliography7 Practical Fourier Transforms 3497.1 IntroductionLearning Objectives7.2 Convolution: Time and FrequencyThe Logarithm Domain7.2.1 Simplifying the Convolution Integral7.3 Transfer Function of a Linear System7.3.1 Impulse Response: The Frequency Domain7.3.2 Frequency Response Curve7.4 Energy in Signals: Parseval's Theorem for the Fourier Transform7.4.1 Energy Spectral Density7.5 Data Smoothing and the Frequency Domain7.6 Ideal Filters7.6.1 The Ideal Low-Pass Filter is not Causal7.7 A Real Low-Pass FilterMATLAB Example 1: First Order Filter7.8 The Modulation Theorem7.8.1 A Voice Privacy SystemSpectral Inversion7.9 Periodic Signals and the Fourier Transform7.9.1 The Impulse Train7.9.2 General Appearance of Periodic Signals7.9.3 The Fourier Transform of a Square waveChanging the Pulse Train Appearance7.9.4 Other Periodic Waveforms7.10 The Analog Spectrum Analyzer7.11 Conclusions7.12 Worked Problems7.13 End of Chapter ExercisesBibliography8 The Laplace TransformLearning Objectives8.1 Introduction8.2 The Laplace Transform8.2.1 The Frequency Term ejwt 8.2.2 The Exponential Term eσt 8.2.3 The s-domain8.3 Exploring the s-domain8.3.1 Poles and Zeros8.3.2 A Pole at the origin8.3.3 Decaying Exponential8.3.4 A Sinusoid8.3.5 A Decaying Sinusoid8.3.6 An Unstable System8.4 Visualizing the Laplace Transform8.4.1 First Order Low-pass Filter8.4.2 Pole Position Determines Frequency Response8.4.3 Second Order Low-pass Filter8.4.4 Two-Sided Laplace Transform8.4.5 The Bode Diagram8.4.6 Calculating the Laplace Transform8.4.7 System Analysis in MATLAB8.5 Properties of the Laplace Transform8.6 Differential Equations8.6.1 Solving a Differential Equation8.6.2 Transfer Function as Differential Equations8.7 Laplace Transform Pairs8.7.1 The Illustrated Laplace Transform8.8 Circuit Analysis with the Laplace Transform8.8.1 Voltage Divider8.8.2 A First-Order Low-pass Filter8.8.3 A First-Order High-pass Filter8.8.4 A Second Order Filter8.9 State Variable Analysis8.9.1 State Variable Analysis - First Order System8.9.2 First Order State Space Analysis with MATLAB8.9.3 State Variable Analysis - Second Order System8.9.4 Matrix Form of the State Space Equations8.9.5 Second Order State Space Analysis with MATLAB8.9.6 Differential Equation8.9.7 State Space and Transfer Functions with MATLAB8.10 Conclusions8.11 Worked Problems8.12 End of Chapter ExercisesBibliography9 Discrete Signals9.1 IntroductionLearning Objectives9.2 Discrete Time vs. Continuous Time Signals9.3 A Discrete Time Signal9.3.1 Digital Signal Processing9.3.2 A Periodic Discrete Time Signal9.4 Data Collection and Sampling Rate9.4.1 The Selection of a Sampling Rate9.4.2 Bandlimited Signal9.4.3 Theory of Sampling9.4.4 The Sampling Function9.4.5 Recovering a Waveform from Samples9.4.6 A Practical Sampling Signal9.4.7 Minimum Sampling Rate9.4.8 Nyquist Sampling Rate9.4.9 The Nyquist Sampling Rate is a Theoretical Minimum9.4.10 Sampling Rate and Alias Frequency9.4.11 Practical Aliasing9.4.12 Analysis of Aliasing9.4.13 Anti-Alias Filter9.5 Introduction to Digital Filtering9.5.1 Impulse Response Function9.5.2 A Discrete Response Function9.5.3 Delay Blocks are a Natural Consequence of Sampling9.5.4 General Digital Filtering9.5.5 The Fourier Transform of Sampled Signals9.5.6 The Discrete Fourier Transform (DFT)9.5.7 A Discrete Fourier Series9.5.8 Computing the Discrete Fourier Transform (DFT)9.5.9 The Fast Fourier Transform (FFT)9.6 Illustrative ExamplesThe FFT (fft) and Inverse FFT (ifft)9.6.1 FFT and Sample Rate9.6.2 Practical DFT Issues9.7 Filtering Application with MATLAB9.7.1 Fourier Analysis9.7.2 System Response9.7.3 Check Calculation9.8 Conclusions9.9 Worked Problems9.10 End of Chapter ExercisesBibliography10 The z-Transform 58110.1 IntroductionLearning Objectives10.2 The z-Transform10.2.1 Fourier Transform, Laplace Transform, z-transform10.2.2 Defnition of the z-Transform10.2.3 The z-Plane and the Fourier Transform10.3 Calculating the z-Transform10.3.1 Unit Step u[n]10.3.2 Exponential an u[n]10.3.3 Sinusoid cos(nωo) u[n] and sin(nωo) u[n]10.3.4 Differentiation10.3.5 The Effect of Sampling Rate10.4 A Discrete Time Laplace Transform10.5 Properties of the z-Transform10.6 z-Transform Pairs10.7 Transfer Function of a Discrete Linear System10.8 MATLAB Analysis with the z-transform10.8.1 First Order Low-pass Filter10.8.2 Pole-zero Plot10.8.3 Bode diagram10.8.4 Impulse Response10.8.5 Calculating Frequency Response10.8.6 Pole Position Determines Frequency Response10.9 Digital Filtering - FIR Filter10.9.1 A One Pole FIR Filter10.9.2 A Two Pole FIR Filter10.9.3 Higher Order FIR Filters10.10Digital Filtering - IIR Filter10.10.1A One Pole IIR Filter10.10.2 IIR vs. FIR10.10.3 Higher Order IIR Filters10.10.4 Combining FIR and IIR Filters10.11Conclusions10.12Worked Problems10.13End of Chapter Exercises11 Communication SystemsLearning Objectives11.1 Introduction11.1.1 A Baseband Signal m(t)11.1.2 The need for a Carrier Signal11.1.3 A Carrier Signal c(t)11.1.4 Modulation Techniques11.1.5 The Radio Spectrum11.2 Amplitude Modulation11.2.1 Double Sideband Transmitted Carrier - (DSB-TC)11.2.2 Demodulation of AM DSB-TC Signals11.2.3 Graphical Analysis11.2.4 AM Demodulation - Diode Detector11.2.5 Examples of Diode Detection11.3 Suppressed Carrier Transmission11.3.1 Demodulation of Single Sideband Signals11.3.2 Percent Modulation and Overmodulation11.4 Superheterodyne Receiver11.4.1 An Experiment with Intermediate Frequency11.4.2 When Receivers become Transmitters11.4.3 Image Frequency11.4.4 Beat Frequency Oscillator11.5 Digital Communications11.5.1 Modulation Methods11.5.2 Morse Code11.5.3 Amplitude Shift Keying (ASK)11.5.4 Frequency Shift Keying (FSK)11.6 Phase Shift Keying (PSK)11.6.1 Differential Coding11.6.2 Quadrature Amplitude Modulation (QAM)11.7 Spread Spectrum Systems11.7.1 Introduction11.7.2 Pseudorandom Noise11.7.3 Encoding Bits in DSSS11.7.4 Spectral Properties of a Pseudo-Random Sequence11.7.5 Code Division Multiple Access (CDMA)11.8 Conclusions11.9 Worked Problems11.10End of Chapter ExercisesBibliographyA Reference TablesA.1 Fourier TransformA.1.1 Fourier Transform TheoremsA.2 Laplace TransformA.2.1 Laplace Transform TheoremsA.3 z-TransformA.3.1 z-Transform TheoremsB The Illustrated Fourier TransformC The Illustrated Laplace TransformD The Illustrated z-TransformE MATLAB Reference GuideE.1 Defining SignalsE.1.1 MATLAB VariablesE.1.2 The Time AxisE.1.3 Common SignalsE.2 Complex NumbersE.3 Plot CommandsE.4 Signal OperationsE.5 Defining SystemsE.5.1 System DefinitionE.5.2 System Analysis