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Practical Signals Theory with MATLAB Applications

Book Description

Practical Signals Theory with MATLAB Applications is organized around applications, first introducing the actual behavior of specific signals and then using them to motivate the presentation of mathematical concepts. Tervo sequences the presentation of the major transforms by their complexity: first Fourier, then Laplace, and finally the z-transform.

The goal is to help students who can't visualize phenomena from an equation to develop their intuition and learn to analyze signals by inspection.

Finally, most examples and problems are designed to use MATLAB, making the presentation more in line with modern engineering practice.

Table of Contents

  1. Coverpage
  2. Titlepage
  3. Copyright
  4. Dedication
  5. Brief Contents
  6. Contents
  7. Preface
  8. Acknowledgments
  9. 1 Introduction to Signals and Systems
    1. 1.1 Introduction
      1. 1.1.1 What Is a Signal?
      2. 1.1.2 What Is a System?
    2. 1.2 Introduction to Signal Manipulation
      1. 1.2.1 Linear Combination
      2. 1.2.2 Addition and Multiplication of Signals
      3. 1.2.3 Visualizing Signals—An Important Skill
      4. 1.2.4 Introduction to Signal Manipulation Using MATLAB
        1. Defining Signals
        2. Basic Plotting Commands
        3. Multiple Plots on One Figure
    3. 1.3 A Few Useful Signals
      1. 1.3.1 The Unit Rectangle rect(t)
      2. 1.3.2 The Unit Step u(t)
      3. 1.3.3 Reflection about t = 0
      4. 1.3.4 The Exponential ext
      5. 1.3.5 The Unit Impulse δ(t)
        1. Sketching the Unit Impulse
        2. The Sifting Property of δ(t)
        3. Sampling Function
    4. 1.4 The Sinusoidal Signal
      1. 1.4.1 The One-Sided Cosine Graph
      2. 1.4.2 Phase Change—Φ
    5. 1.5 Phase Change vs. Time Shift
      1. 1.5.1 Sine vs. Cosine
      2. 1.5.2 Combining Signals: The Gated Sine Wave
      3. 1.5.3 Combining Signals: A Dial Tone Generator
    6. 1.6 Useful Hints and Help with MATLAB
      1. 1.6.1 Annotating Graphs
    7. 1.7 Conclusions
  10. 2 Classification of Signals
    1. 2.1 Introduction
    2. 2.2 Periodic Signals
      1. 2.2.1 Sinusoid
      2. 2.2.2 Half-Wave Rectified Sinusoid
      3. 2.2.3 Full-Wave Rectified Sinusoid
      4. 2.2.4 Square Wave
      5. 2.2.5 Sawtooth Wave
      6. 2.2.6 Pulse Train
      7. 2.2.7 Rectangular Wave
      8. 2.2.8 Triangle Wave
      9. 2.2.9 Impulse Train
        1. DC Component in Periodic Signals
    3. 2.3 Odd and Even Signals
      1. 2.3.1 Combining Odd and Even Signals
      2. 2.3.2 The Constant Value s(t) = A
      3. 2.3.3 Trigonometric Identities
      4. 2.3.4 The Modulation Property
        1. A Television Tuner Box
        2. Squaring the Sinusoid
    4. 2.4 Energy and Power Signals
      1. 2.4.1 Periodic Signals = Power Signals
        1. Vrms Does not equal A/2 for All Periodic Signals
        2. MATLAB Exercise 1: Computation of Vrms
      2. 2.4.2 Comparing Signal Power: The Decibel (dB)
    5. 2.5 Complex Signals
      1. MATLAB Exercise 2: Complex Signals
    6. 2.6 Discrete Time Signals
    7. 2.7 Digital Signals
    8. 2.8 Random Signals
    9. 2.9 Useful Hints and Help with MATLAB
    10. 2.10 Conclusions
  11. 3 Linear Systems
    1. 3.1 Introduction
    2. 3.2 Definition of a Linear System
      1. 3.2.1 Superposition
      2. 3.2.2 Linear System Exercise 1: Zero State Response
        1. Zero Input → Zero Output
      3. 3.2.3 Linear System Exercise 2: Operating in a Linear Region
        1. Nonlinear Components
      4. 3.2.4 Linear System Exercise 3: Mixer
        1. A System Is Defined by Its Response Function
      5. 3.2.5 Linear Time-Invariant (LTI) Systems
      6. 3.2.6 Bounded Input, Bounded Output
      7. 3.2.7 System Behavior as a Black Box
    3. 3.3 Linear System Response Function h(t)
    4. 3.4 Convolution
      1. 3.4.1 The Convolution Integral
      2. 3.4.2 Convolution Is Commutative
      3. 3.4.3 Convolution Is Associative
      4. 3.4.4 Convolution Is Distributive over Addition
      5. 3.4.5 Evaluation of the Convolution Integral
        1. Graphical Exercise 1: Convolution of a Rectangle with Itself
      6. 3.4.6 Convolution Properties
        1. Graphical Exercise 2: Convolution of Two Rectangles
        2. Graphical Exercise 3: Convolution of a Rectangle and an Exponential Decay.
        3. A Pulse Input Signal
      7. 3.4.7 Convolution with MATLAB
        1. MATLAB Exercise 1: Convolution of a Rectangle with Itself
        2. MATLAB Exercise 2: Convolution of Two Rectangles
        3. MATLAB Exercise 3: Convolution of a Rectangle with an Exponential Decay
    5. 3.5 Determining h(t) in an Unknown System
      1. 3.5.1 The Unit Impulse δ(t)Test Signal
      2. 3.5.2 Convolution and Signal Decomposition
        1. Convolution and Periodic Signals
      3. 3.5.3 An Ideal Distortionless System
        1. Deconvolution
    6. 3.6 Causality
      1. 3.6.1 Causality and Zero Input Response
    7. 3.7 Combined Systems
      1. MATLAB Exercise 4: Systems in Series
    8. 3.8 Convolution and Random Numbers
    9. 3.9 Useful Hints and Help with MATLAB
    10. 3.10 Chapter Summary
    11. 3.11 Conclusions
  12. 4 The Fourier Series
    1. Chapter Overview
    2. 4.1 Introduction
    3. 4.2 Expressing Signals by Components
      1. The Spectrum Analyzer
      2. 4.2.1 Approximating a Signal s(t) by Another: The Signal Inner Product
      3. 4.2.2 Estimating One Signal by Another
    4. 4.3 Part One—Orthogonal Signals
    5. 4.4 Orthogonality
      1. 4.4.1 An Orthogonal Signal Space
        1. Interpreting the Inner Product
      2. 4.4.2 The Signal Inner Product Formulation
      3. 4.4.3 Complete Set of Orthogonal Signals
      4. 4.4.4 What If a Complete Set Is Not Present?
      5. 4.4.5 An Orthogonal Set of Signals
        1. Defining Orthogonal Basis Signals
        2. Confirming Orthogonal Basis Signals
        3. Finding Orthogonal Components
      6. 4.4.6 Orthogonal Signals and Linearly Independent Equations
        1. MATLAB Exercise 1: Evaluating an Inner Product
    6. 4.5 Part Two—The Fourier Series
      1. 4.5.1 A Special Set of Orthogonal Functions
      2. 4.5.2 The Fourier Series—An Orthogonal Set?
    7. 4.6 Computing Fourier Series Components
      1. 4.6.1 Fourier Series Approximation to an Odd Square Wave
      2. 4.6.2 Zero-Frequency (DC) Component
    8. 4.7 Fundamental Frequency Component
      1. 4.7.1 Higher-Order Components
      2. 4.7.2 Frequency Spectrum of the Square Wave s(t)
    9. 4.8 Practical Harmonics
      1. 4.8.1 The 60 Hz Power Line
      2. 4.8.2 Audio Amplifier Specs—Total Harmonic Distortion
      3. 4.8.3 The CB Radio Booster
    10. 4.9 Odd and Even Square Waves
      1. 4.9.1 The Fourier Series Components of an Even Square Wave
    11. 4.10 Gibb’s Phenomenon
    12. 4.11 Setting Up the Fourier Series Calculation
      1. 4.11.1 Appearance of Pulse Train Frequency Components
        1. Pulse Train with 10 Percent Duty Cycle
        2. Pulse Train with 20 Percent Duty Cycle
        3. Pulse Train with 50 Percent Duty Cycle (Square Wave)
    13. 4.12 Some Common Fourier Series
    14. 4.13 Part Three—The Complex Fourier Series
      1. 4.13.1 Not All Signals Are Even or Odd
    15. 4.14 The Complex Fourier Series
      1. 4.14.1 Complex Fourier Series—The Frequency Domain
      2. 4.14.2 Comparing the Real and Complex Fourier Series
      3. 4.14.3 Magnitude and Phase
    16. 4.15 Complex Fourier Series Components
      1. 4.15.1 Real Signals and the Complex Fourier Series
      2. 4.15.2 Stretching and Squeezing: Time vs. Frequency
      3. 4.15.3 Shift in Time
      4. 4.15.4 Change in Amplitude
      5. 4.15.5 Power in Periodic Signals
        1. Find the Total Power in s(t) = A cos(t) + B sin(t)
      6. 4.15.6 Parseval’s Theorem for Periodic Signals
    17. 4.16 Properties of the Complex Fourier Series
    18. 4.17 Analysis of a DC Power Supply
      1. 4.17.1 The DC Component
      2. 4.17.2 An AC-DC Converter
      3. 4.17.3 Vrms Is Always Greater Than or Equal to Vdc
      4. 4.17.4 Fourier Series: The Full-Wave Rectifier
      5. 4.17.5 Complex Fourier Series Components Cn
        1. MATLAB Exercise 2: Plotting Fourier Series Components
    19. 4.18 The Fourier Series with MATLAB
      1. 4.18.1 Essential Features of the fft() in MATLAB
        1. 1. Periodic Signals Are Defined on a Period of 2N Points
        2. 2. The Fourier Series Is Defined on 2N−1 −1 Frequency Components
      2. 4.18.2 Full-Wave Rectified Cosine (60 Hz)
      3. 4.18.3 Useful Hints and Help with MATLAB
    20. 4.19 Conclusions
  13. 5 The Fourier Transform
    1. 5.1 Introduction
      1. 5.1.1 A Fresh Look at the Fourier Series
        1. Periodic and Nonperiodic Signals
      2. 5.1.2 Approximating a Nonperiodic Signal over All Time
      3. 5.1.3 Definition of the Fourier Transform
      4. 5.1.4 Existence of the Fourier Transform
      5. 5.1.5 The Inverse Fourier Transform
    2. 5.2 Properties of the Fourier Transform
      1. 5.2.1 Linearity of the Fourier Transform
      2. 5.2.2 Value of the Fourier Transform at the Origin
      3. 5.2.3 Odd and Even Functions and the Fourier Transform
    3. 5.3 The Rectangle Signal
      1. Alternate Solution
    4. 5.4 The Sinc Function
      1. 5.4.1 Expressing a Function in Terms of sinc(t)
      2. 5.4.2 The Fourier Transform of a General Rectangle
      3. 5.4.3 Magnitude of the Fourier Transform
    5. 5.5 Signal Manipulations: Time and Frequency
      1. 5.5.1 Amplitude Variations
      2. 5.5.2 Stretch and Squeeze: The Sinc Function
      3. 5.5.3 The Scaling Theorem
      4. 5.5.4 Testing the Limits
      5. 5.5.5 A Shift in Time
      6. 5.5.6 The Shifting Theorem
      7. 5.5.7 The Fourier Transform of a Shifted Rectangle
        1. Magnitude of G(f)
        2. Phase of G(f)
      8. 5.5.8 Impulse Series—The Line Spectrum
      9. 5.5.9 Shifted Impulse δ(f − f0)
      10. 5.5.10 Fourier Transform of a Periodic Signal
    6. 5.6 Fourier Transform Pairs
      1. 5.6.1 The Illustrated Fourier Transform
    7. 5.7 Rapid Changes vs. High Frequencies
      1. 5.7.1 Derivative Theorem
      2. 5.7.2 Integration Theorem
    8. 5.8 Conclusions
  14. 6 Practical Fourier Transforms
    1. 6.1 Introduction
    2. 6.2 Convolution: Time and Frequency
      1. The Logarithm Domain
      2. 6.2.1 Simplifying the Convolution Integral
    3. 6.3 Transfer Function of a Linear System
      1. 6.3.1 Impulse Response: The Frequency Domain
      2. 6.3.2 Frequency Response Curve
    4. 6.4 Energy in Signals: Parseval’s Theorem for the Fourier Transform
      1. 6.4.1 Energy Spectral Density
    5. 6.5 Data Smoothing and the Frequency Domain
    6. 6.6 Ideal Filters
      1. 6.6.1 The Ideal Lowpass Filter Is Not Causal
    7. 6.7 A Real Lowpass Filter
      1. MATLAB Example 1: First-Order Filter
    8. 6.8 The Modulation Theorem
      1. 6.8.1 A Voice Privacy System
        1. Spectral Inversion
    9. 6.9 Periodic Signals and the Fourier Transform
      1. 6.9.1 The Impulse Train
      2. 6.9.2 General Appearance of Periodic Signals
      3. 6.9.3 The Fourier Transform of a Square Wave
        1. Changing the Pulse Train Appearance
      4. 6.9.4 Other Periodic Waveforms
    10. 6.10 The Analog Spectrum Analyzer
    11. 6.11 Conclusions
  15. 7 The Laplace Transform
    1. 7.1 Introduction
    2. 7.2 The Laplace Transform
      1. 7.2.1 The Frequency Term ejwt
      2. 7.2.2 The Exponential Term eσt
      3. 7.2.3 The s-Domain
    3. 7.3 Exploring the s-Domain
      1. 7.3.1 A Pole at the Origin
        1. Graphing the Function H(s) = 1/s
      2. 7.3.2 Decaying Exponential
      3. 7.3.3 A Sinusoid
        1. The Generalized Cosine: A = cos(wt + Φ)
      4. 7.3.4 A Decaying Sinusoid
      5. 7.3.5 An Unstable System
    4. 7.4 Visualizing the Laplace Transform
      1. 7.4.1 First-Order Lowpass Filter
      2. 7.4.2 Pole Position Determines Frequency Response
      3. 7.4.3 Second-Order Lowpass Filter
        1. Resonance Frequency
        2. Multiple Poles and Zeros
      4. 7.4.4 Two-Sided Laplace Transform
      5. 7.4.5 The Bode Plot
        1. Bode Plot—Multiple Poles and Zeros
        2. Laplace Transform Exercise 1: Calculating the Laplace Transform
      6. 7.4.6 System Analysis in MATLAB
    5. 7.5 Properties of the Laplace Transform
    6. 7.6 Differential Equations
      1. 7.6.1 Solving a Differential Equation
        1. Compound Interest
      2. 7.6.2 Transfer Function as Differential Equations
    7. 7.7 Laplace Transform Pairs
      1. 7.7.1 The Illustrated Laplace Transform
    8. 7.8 Circuit Analysis with the Laplace Transform
      1. 7.8.1 Voltage Divider
      2. 7.8.2 A First-Order Lowpass Filter
      3. 7.8.3 A First-Order Highpass Filter
      4. 7.8.4 A Second-Order Filter
        1. Lowpass Filter
        2. Bandpass Filter
        3. Highpass Filter
        4. Analysis of a Second-Order System
        5. Series RLC Circuit Analysis
    9. 7.9 State Variable Analysis
      1. 7.9.1 State Variable Analysis—First-Order System
      2. 7.9.2 First-Order State Space Analysis with MATLAB
      3. 7.9.3 State Variable Analysis —Second-Order System
      4. 7.9.4 Matrix Form of the State Space Equations
      5. 7.9.5 Second-Order State Space Analysis with MATLAB
      6. 7.9.6 Differential Equation
      7. 7.9.7 State Space and Transfer Functions with MATLAB
    10. 7.10 Conclusions
  16. 8 Discrete Signals
    1. 8.1 Introduction
    2. 8.2 Discrete Time vs. Continuous Time Signals
      1. 8.2.1 Digital Signal Processing
    3. 8.3 A Discrete Time Signal
      1. 8.3.1 A Periodic Discrete Time Signal
    4. 8.4 Data Collection and Sampling Rate
      1. 8.4.1 The Selection of a Sampling Rate
      2. 8.4.2 Bandlimited Signal
      3. 8.4.3 Theory of Sampling
      4. 8.4.4 The Sampling Function
      5. 8.4.5 Recovering a Waveform from Samples
      6. 8.4.6 A Practical Sampling Signal
      7. 8.4.7 Minimum Sampling Rate
      8. 8.4.8 Nyquist Sampling Rate
      9. 8.4.9 The Nyquist Sampling Rate Is a Theoretical Minimum
      10. 8.4.10 Sampling Rate and Alias Frequency
      11. 8.4.11 Practical Aliasing
      12. 8.4.12 Analysis of Aliasing
      13. 8.4.13 Anti-Alias Filter
    5. 8.5 Introduction to Digital Filtering
      1. 8.5.1 Impulse Response Function
      2. 8.5.2 A Simple Discrete Response Function
      3. 8.5.3 Delay Blocks Are a Natural Consequence of Sampling
      4. 8.5.4 General Digital Filtering
      5. 8.5.5 The Fourier Transform of Sampled Signals
      6. 8.5.6 The Discrete Fourier Transform (DFT)
      7. 8.5.7 A Discrete Fourier Series
      8. 8.5.8 Computing the Discrete Fourier Transform (DFT)
      9. 8.5.9 The Fast Fourier Transform (FFT)
    6. 8.6 Illustrative Examples
      1. MATLAB Exercise 1: The FFT and the Inverse FFT
      2. 8.6.1 FFT and Sample Rate
      3. 8.6.2 Practical DFT Issues
        1. Constructing the Ideal Discrete Signal
    7. 8.7 Discrete Time Filtering with MATLAB
      1. 8.7.1 A Discrete Rectangle
      2. 8.7.2 A Cosine Test Signal
      3. 8.7.3 Check Calculation
    8. 8.8 Conclusions
  17. 9 The z-Transform
    1. 9.1 Introduction
    2. 9.2 The z-Transform
      1. 9.2.1 Fourier Transform, Laplace Transform, and z-transform
      2. 9.2.2 Definition of the z-Transform
      3. 9.2.3 The z-Plane and the Fourier Transform
    3. 9.3 Calculating the z-Transform
      1. 9.3.1 Unit Step u[n]
      2. 9.3.2 Exponential an u[n]
      3. 9.3.3 Sinusoid cos(nw0) u[n] and sin(nw0) u[n]
      4. 9.3.4 Differentiation
      5. 9.3.5 The Effect of Sampling Rate
    4. 9.4 A Discrete Time Laplace Transform
    5. 9.5 Properties of the z-Transform
    6. 9.6 z-Transform Pairs
    7. 9.7 Transfer Function of a Discrete Linear System
    8. 9.8 MATLAB Analysis with the z-Transform
      1. 9.8.1 First-Order Lowpass Filter
      2. 9.8.2 Pole-Zero Diagram
      3. 9.8.3 Bode Plot
      4. 9.8.4 Impulse Response
      5. 9.8.5 Calculating Frequency Response
      6. 9.8.6 Pole Position Determines Frequency Response
    9. 9.9 Digital Filtering—FIR Filter
      1. 9.9.1 A One-Pole FIR Filter
      2. 9.9.2 A Two-Pole FIR Filter
      3. 9.9.3 Higher-Order FIR Filters
        1. Frequency Response
        2. Pole-Zero Diagram
        3. Phase Response
        4. Step Response
    10. 9.10 Digital Filtering—IIR Filter
      1. 9.10.1 A One-Pole IIR Filter
      2. 9.10.2 IIR versus FIR
      3. 9.10.3 Higher-Order IIR Filters
      4. 9.10.4 Combining FIR and IIR Filters
    11. 9.11 Conclusions
  18. 10 Introduction to Communications
    1. 10.1 Introduction
      1. 10.1.1 A Baseband Signal m(t)
      2. 10.1.2 The Need for a Carrier Signal
      3. 10.1.3 A Carrier Signal c(t)
      4. 10.1.4 Modulation Techniques
      5. 10.1.5 The Radio Spectrum
    2. 10.2 Amplitude Modulation
      1. 10.2.1 Transmitted Carrier Double Sideband—(AM-TCDSB)
      2. 10.2.2 Demodulation of AM Signals
      3. 10.2.3 Graphical Analysis
      4. 10.2.4 AM Demodulation—Diode Detector
      5. 10.2.5 Examples of Diode Detection
    3. 10.3 Suppressed Carrier Transmission
      1. 10.3.1 Demodulation of Single Sideband Signals
      2. 10.3.2 Percent Modulation and Overmodulation
    4. 10.4 Superheterodyne Receiver
      1. 10.4.1 An Experiment with Intermediate Frequency
      2. 10.4.2 When Receivers Become Transmitters
      3. 10.4.3 Image Frequency
      4. 10.4.4 Beat Frequency Oscillator
    5. 10.5 Digital Communications
      1. 10.5.1 Modulation Methods
      2. 10.5.2 Morse Code
      3. 10.5.3 On Off Keying (OOK)
      4. 10.5.4 Bandwidth Considerations
      5. 10.5.5 Receiving a Morse Code Signal
    6. 10.6 Phase Shift Keying
      1. 10.6.1 Differential Coding
      2. 10.6.2 Higher-Order Modulation Schemes
    7. 10.7 Conclusions
  19. A The Illustrated Fourier Transform
  20. B The Illustrated Laplace Transform
  21. C The Illustrated z-Transform
  22. D MATLAB Reference Guide
    1. D.1 Defining Signals
      1. D.1.1 MATLAB Variables
      2. D.1.2 The Time Axis
      3. D.1.3 Common Signals
    2. D.2 Complex Numbers
    3. D.3 Plot Commands
    4. D.4 Signal Operations
    5. D.5 Defining Systems
      1. D.5.1 System Definition
        1. 1. Transfer Function
        2. 2. Zeros and Poles and Gain
        3. 3. State Space Model
        4. 4. Discrete Time Systems
      2. D.5.2 System Analysis
    6. D.6 Example System Definition and Test
  23. E Reference Tables
    1. E.1 Fourier Transform
      1. E.1.1 Fourier Transform Theorems
    2. E.2 Laplace Transform
      1. E.2.1 Laplace Transform Theorems
    3. E.3 z-Transform
      1. E.3.1 z–Transform Theorems
  24. Bibliography
  25. Index