| |

| |

| |

Useful Information (inside cover / endpaper) | |

| |

| |

| |

Identities | |

| |

| |

| |

Definite Integrals | |

| |

| |

| |

Infinite Series | |

| |

| |

| |

Orthogonality | |

| |

| |

| |

Signal Inner Product | |

| |

| |

| |

Convolution | |

| |

| |

| |

Fourier Series | |

| |

| |

| |

Complex Fourier Series | |

| |

| |

| |

Fourier Transform | |

| |

| |

| |

Laplace Transform | |

| |

| |

| |

z-Transform | |

| |

| |

| |

List of Acronyms | |

| |

| |

| |

Introduction to Signals and Systems | |

| |

| |

| |

Introduction | |

| |

| |

| |

What is a Signal? | |

| |

| |

| |

What is a System? | |

| |

| |

| |

Introduction to Signal Manipulation | |

| |

| |

| |

Linear Combination | |

| |

| |

| |

Addition and Multiplication of Signals | |

| |

| |

| |

Visualizing Signals - An Important Skill | |

| |

| |

| |

Introduction to Signal Manipulation Using MATLAB | |

| |

| |

Defining Signals | |

| |

| |

Basic Plotting Commands | |

| |

| |

Multiple Plots on One Figure | |

| |

| |

| |

A Few Useful Signals | |

| |

| |

| |

The Unit Rectangle rect(t) | |

| |

| |

| |

The Unit Step u(t) | |

| |

| |

| |

Reection about t = 0 | |

| |

| |

| |

The Exponential ext | |

| |

| |

| |

The Unit Impulse _(t) | |

| |

| |

The Sifting Property of _(t) | |

| |

| |

Sampling Function | |

| |

| |

| |

The Sinusoidal Signal | |

| |

| |

| |

The One-Sided Cosine Graph | |

| |

| |

| |

Phase Change - _ | |

| |

| |

| |

Sine vs Cosine | |

| |

| |

Combining Signals: The Gated Sine Wave | |

| |

| |

Combining Signals: A Dial Tone Generator | |

| |

| |

| |

Useful Hints and Help with MATLAB | |

| |

| |

| |

Annotating Graphs | |

| |

| |

| |

Conclusions | |

| |

| |

| |

End of Chapter Exercises | |

| |

| |

| |

Classification of Signals | |

| |

| |

| |

Introduction | |

| |

| |

| |

Periodic Signals | |

| |

| |

| |

Sinusoid | |

| |

| |

| |

Half-wave Rectified Sinusoid | |

| |

| |

| |

Full-wave Rectified Sinusoid | |

| |

| |

| |

Square Wave | |

| |

| |

| |

Sawtooth Wave | |

| |

| |

| |

Pulse Train | |

| |

| |

| |

Rectangular Wave | |

| |

| |

| |

Triangle wave | |

| |

| |

| |

Impulse Train | |

| |

| |

| |

Odd and Even Signals | |

| |

| |

| |

Combining Odd and Even signals | |

| |

| |

| |

The constant value s(t) = A | |

| |

| |

| |

Trigonometric Identities | |

| |

| |

| |

The Modulation Property | |

| |

| |

| |

Energy and Power Signals | |

| |

| |

| |

Periodic Signals = Power Signals | |

| |

| |

| |

Comparing Signal Power: The Decibel (dB) | |

| |

| |

| |

Complex Signals | |

| |

| |

| |

Discrete Time Signals | |

| |

| |

| |

Digital Signals | |

| |

| |

| |

} Random Signals | |

| |

| |

| |

Useful Hints and Help with MATLAB | |

| |

| |

| |

Conclusions | |

| |

| |

| |

End of Chapter Exercises | |

| |

| |

| |

Linear Systems | |

| |

| |

| |

Introduction | |

| |

| |

| |

Definition of a Linear System | |

| |

| |

| |

Superposition | |

| |

| |

| |

Linear System Exercise 1: Zero State Response | |

| |

| |

Zero Input ! Zero Output | |

| |

| |

| |

Linear System Exercise 2: Operating in a linear region | |

| |

| |

Non-Linear Components | |

| |

| |

| |

Linear System Exercise 3: Mixer | |

| |

| |

A System is defined by its Response Function | |

| |

| |

| |

Linear Time-Invariant (LTI) Systems | |

| |

| |

| |

Bounded Input, Bounded Output | |

| |

| |

| |

System Behavior as a Black Box | |

| |

| |

| |

Linear System Response Function h(t) | |

| |

| |

| |

Convolution | |

| |

| |

| |

The Convolution Integral | |

| |

| |

| |

Convolution is Commutative | |

| |

| |

| |

Convolution is Associative | |

| |

| |

| |

Convolution is Distributive over Addition | |

| |

| |

| |

Evaluation of the Convolution Integral | |

| |

| |

| |

Convolution Properties | |

| |

| |

A Pulse Input Signal | |

| |

| |

| |

Convolution with MATLAB | |

| |

| |

| |

Determining h(t) in an Unknown System | |

| |

| |

| |

The Unit Impulse _(t) Test Signal | |

| |

| |

| |

Convolution and Signal Decomposition | |

| |

| |

| |

An Ideal Distortionless System | |

| |

| |

| |

Causality | |

| |

| |

| |

Causality and Zero Input Response | |

| |

| |

| |

Combined Systems | |

| |

| |

| |

} Convolution and Random Numbers | |

| |

| |

| |

Useful Hints and Help with MATLAB | |

| |

| |

| |

Chapter Summary | |

| |

| |

| |

Conclusions | |

| |

| |

| |

End of Chapter Exercises | |

| |

| |

| |

The Fourier Series | |

| |

| |

| |

Introduction | |

| |

| |

| |

Expressing Signals by Components | |

| |

| |

| |

Approximating a Signal s(t) by Another: The Signal Inner Product | |

| |

| |

| |

Estimating One Signal by Another | |

| |

| |

| |

Part One - Orthogonal Signals | |

| |

| |

| |

Orthogonality | |

| |

| |

| |

An Orthogonal Signal Space | |

| |

| |

| |

The Signal Inner Product Formulation | |

| |

| |

| |

Complete Set of Orthogonal Signals | |

| |

| |

| |

What if a Complete Set is not Present? | |

| |

| |

| |

An Orthogonal Set of Signals | |

| |

| |

| |

Orthogonal Signals and Linearly Independent Equations | |

| |

| |

| |

Part Two - The Fourier Series | |

| |

| |

| |

A Special set of Orthogonal Functions | |

| |

| |

| |

The Fourier Series - An Orthogonal Set? | |

| |

| |

| |

Computing Fourier Series Components | |

| |

| |

| |

Fourier Series Approximation to an Odd Square Wave | |

| |

| |

| |

Zero-Frequency (DC) Component | |

| |

| |

| |

Fundamental Frequency Component | |

| |

| |

| |

Higher Order Components | |

| |

| |

| |

Frequency Spectrum of the Square Wave s(t) | |

| |

| |

| |

Practical Harmonics | |

| |

| |

| |

The 60 Hz Power Line | |

| |

| |

| |

Audio Amplifier Specs - Total Harmonic Distortion | |

| |

| |

| |

The CB Radio Booster | |

| |

| |

| |

Odd and Even Square Waves | |

| |

| |

| |

The Fourier Series Components of an Even Square Wave | |

| |

| |

| |

Gibb's Phenomenon | |

| |

| |

| |

Setting-Up the Fourier Series Calculation | |

| |

| |

| |

Appearance of Pulse Train Frequency Components | |

| |

| |

| |

Some Common Fourier Series | |

| |

| |

| |

Part Three: The Complex Fourier Series | |

| |

| |

| |

Not all Signals are Even or Odd | |

| |

| |

| |

The Complex Fourier Series | |

| |

| |

| |

Complex Fourier Series - The Frequency Domain | |

| |

| |

| |

Comparing the Real and Complex Fourier Series | |

| |

| |

| |

Magnitude and Phase | |

| |

| |

| |

Complex Fourier Series Components | |

| |

| |

| |

Real Signals and the Complex Fourier Series | |

| |

| |

| |

Stretching and Squeezing: Time vs Frequency | |

| |

| |

| |

Shift in Time | |

| |

| |

| |

Change in Amplitude | |

| |

| |

| |

Power in Periodic Signals | |

| |

| |

| |

Parseval's Theorem for Periodic Signals | |

| |

| |

| |

Properties of the Complex Fourier Series | |

| |

| |

| |

Analysis of a DC Power Supply | |

| |

| |

| |

The DC Component | |

| |

| |

| |

An AC-DC Converter | |

| |

| |

| |

Vrms is always greater than or equal to Vdc | |

| |

| |

| |

Fourier Series: The Full-wave Rectifier | |

| |

| |

| |

Complex Fourier series components Cn | |

| |

| |

| |

The Fourier Series with MATLAB | |

| |

| |

| |

Essential features of the fft() in MATLAB | |

| |

| |

| |

Full-wave Rectified Cosine (60 Hz) | |

| |

| |

| |

Useful Hints and Help with MATLAB | |

| |

| |

| |

Conclusions | |

| |

| |

| |

End of Chapter Exercises | |

| |

| |

| |

Orthogonal Signals | |

| |

| |

| |

The Fourier Series | |

| |

| |

| |

The Fourier Transform | |

| |

| |

| |

Introduction | |

| |

| |

| |

A Fresh Look at the Fourier Series | |

| |

| |

| |

Approximating a Non-Periodic Signal Over All Time | |

| |

| |

| |

Definition of the Fourier Transform | |

| |

| |

| |

Existence of the Fourier Transform | |

| |

| |

| |

The Inverse Fourier Transform | |

| |

| |

| |

Properties of the Fourier Transform | |

| |

| |

| |

Linearity of the Fourier Transform | |

| |

| |

| |

Value of the Fourier transform at the Origin | |

| |

| |

| |

Odd and Even Functions and the Fourier Transform | |

| |

| |

| |

The Rectangle Signal | |

| |

| |

| |

The Sinc Function | |

| |

| |

| |

Expressing a Function in Terms of sinc(t) | |

| |

| |

| |

The Fourier Transform of a General Rectangle | |

| |

| |

| |

Magnitude of the Fourier Transform | |

| |

| |

| |

Signal Manipulations: Time and Frequency | |

| |

| |

| |

Amplitude Variations | |

| |

| |

| |

Stretch and Squeeze: The Sinc Function | |

| |

| |

| |

The Scaling Theorem | |

| |

| |

| |

Testing the Limits | |

| |

| |

| |

A Shift in Time | |

| |

| |

| |

The Shifting Theorem | |

| |

| |

| |

The Fourier Transform of a Shifted Rectangle | |

| |

| |

| |

Impulse Series - The Line Spectrum | |

| |

| |

| |

Shifted Impulse _(f f0) | |

| |

| |

| |

Fourier Transform of a Periodic Signal | |

| |

| |

| |

Fourier Transform Pairs | |

| |

| |

| |

The Illustrated Fourier Transform | |

| |

| |

| |

Rapid Changes vs High Frequencies | |

| |

| |

| |

Derivative Theorem | |

| |

| |

| |

Integration Theorem | |

| |

| |

| |

Conclusions | |

| |

| |

| |

End of Chapter Exercises | |

| |

| |

| |

Practical Fourier Transforms | |

| |

| |

| |

Introduction | |

| |

| |

| |

Convolution: Time and Frequency | |

| |

| |

| |

Simplifying the Convolution Integral | |

| |

| |

| |

Transfer Function of a Linear System | |

| |

| |

| |

Impulse Response: The Frequency Domain | |

| |

| |

| |

Frequency Response Curve | |

| |

| |

| |

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

| |

| |

| |

Energy Spectral Density | |

| |

| |

| |

Data Smoothing and the Frequency Domain | |

| |

| |

| |

Ideal Filters | |

| |

| |

| |

The Ideal Lowpass Filter is not Causal | |

| |

| |

| |

A Real Lowpass Filter | |

| |

| |

| |

The Modulation Theorem | |

| |

| |

| |

A Voice Privacy System | |

| |

| |

| |

Periodic Signals and the Fourier Transform | |

| |

| |

| |

The Impulse Train | |

| |

| |

| |

General Appearance of Periodic Signals | |

| |

| |

| |

The Fourier Transform of a Square wave | |

| |

| |

| |

Other Periodic Waveforms | |

| |

| |

| |

The Analog Spectrum Analyzer | |

| |

| |

| |

Conclusions | |

| |

| |

| |

End of Chapter Exercises | |

| |

| |

| |

The Laplace Transform | |

| |

| |

| |

Introduction | |

| |

| |

| |

The Laplace Transform | |

| |

| |

| |

The Frequency Term ej!t | |

| |

| |

| |

The Exponential Term e_t | |

| |

| |

| |

The s-domain | |

| |

| |

| |

Exploring the s-domain | |

| |

| |

| |

A Pole at the origin | |

| |

| |

Graphing the function H(s) = 1=s | |

| |

| |

| |

Decaying Exponential | |

| |

| |

| |

A Sinusoid | |

| |

| |

The Generalized Cosine: A = cos(!t + _) | |

| |

| |

| |

A Decaying Sinusoid | |

| |

| |

| |

An Unstable System | |

| |

| |

| |

Visualizing the Laplace Transform | |

| |

| |

| |

First Order Lowpass Filter | |

| |

| |

| |

Pole Position Determines Frequency Response | |

| |

| |

| |

Second Order Lowpass Filter | |

| |

| |

Resonance Frequency | |

| |

| |

Multiple Poles and Zeros | |

| |

| |

| |

Two-Sided Laplace Transform | |

| |

| |

| |

The Bode Plot | |

| |

| |

Bode Plot - Multiple Poles and Zeros | |

| |

| |

| |

System Analysis in MATLAB | |

| |

| |

| |

Properties of the Laplace Transform | |

| |

| |

| |

Differential Equations | |

| |

| |

| |

Solving a Differential Equation | |

| |

| |

Compound Interest | |

| |

| |

| |

Transfer Function as Differential Equations | |

| |

| |

| |

Laplace Transform Pairs | |

| |

| |

| |

The Illustrated Laplace Transform | |

| |

| |

| |

Circuit Analysis with the Laplace Transform | |

| |

| |

| |

Voltage Divider | |

| |

| |

| |

A First-Order Lowpass Filter | |

| |

| |

| |

A First-Order Highpass Filter | |

| |

| |

| |

A Second Order Filter | |

| |

| |

Lowpass Filter | |

| |

| |

Bandpass Filter | |

| |

| |

Highpass Filter | |

| |

| |

Analysis of a Second Order System | |

| |

| |

Series RLC Circuit Analysis | |

| |

| |

| |

State Variable Analysis | |

| |

| |

| |

State Variable Analysis - First Order System | |

| |

| |

| |

First Order State Space Analysis with MATLAB | |

| |

| |

| |

State Variable Analysis - Second Order System | |

| |

| |

| |

Matrix Form of the State Space Equations | |

| |

| |

| |

Second Order State Space Analysis with MATLAB | |

| |

| |

| |

Differential Equation | |

| |

| |

| |

State Space and Transfer Functions with MATLAB | |

| |

| |

| |

Conclusions | |

| |

| |

| |

End of Chapter Exercises | |

| |

| |

| |

Discrete Signals | |

| |

| |

| |

Introduction | |

| |

| |

| |

Discrete Time vs Continuous Time Signals | |

| |

| |

| |

1 Digital Signal Processing | |

| |

| |

| |

A Discrete Time Signal | |

| |

| |

| |

1 A Periodic Discrete Time Signal | |

| |

| |

| |

Data Collection and Sampling Rate | |

| |

| |

| |

The Selection of a Sampling Rate | |

| |

| |

| |

Bandlimited Signal | |

| |

| |

| |

Theory of Sampling | |

| |

| |

| |

The Sampling Function | |

| |

| |

| |

Recovering a Waveform from Samples | |

| |

| |

| |

A Practical Sampling Signal | |

| |

| |

| |

Minimum Sampling Rate | |

| |

| |

| |

Nyquist Sampling Rate | |

| |

| |

| |

The Nyquist Sampling Rate is a Theoretical Minimum | |

| |

| |

| |

Sampling Rate and Alias Frequency | |

| |

| |

| |

Practical Aliasing | |

| |

| |

| |

Analysis of Aliasing | |

| |

| |

| |

Anti-Alias Filter | |

| |

| |

| |

Introduction to Digital Filtering | |

| |

| |

| |

Impulse Response Function | |

| |

| |

| |

A Simple Discrete Response Function | |

| |

| |

| |

Delay Blocks are a Natural Consequence of Sampling | |

| |

| |

| |

General Digital Filtering | |

| |

| |

| |

The Fourier Transform of Sampled Signals | |

| |

| |

| |

The Discrete Fourier Transform (DFT) | |

| |

| |

| |

A Discrete Fourier Series | |

| |

| |

| |

Computing the Discrete Fourier Transform (DFT) | |

| |

| |

| |

The Fast Fourier Transform (FFT) | |

| |

| |

| |

Illustrative Examples | |

| |

| |

| |

FFT and Sample Rate | |

| |

| |

| |

Practical DFT Issues | |

| |

| |

Constructing the Ideal Discrete Signal | |

| |

| |

A Typical Discrete Signal | |

| |

| |

A DFT Window | |

| |

| |

| |

Discrete Time Filtering with MATLAB | |

| |

| |

| |

A Discrete Rectangle | |

| |

| |

| |

A Cosine Test Signal | |

| |

| |

| |

Check Calculation | |

| |

| |

| |

Conclusions | |

| |

| |

| |

End of Chapter Exercises | |

| |

| |

| |

The z-Transform | |

| |

| |

| |

Introduction | |

| |

| |

| |

The z-Transform | |

| |

| |

| |

Fourier Transform, Laplace Transform, z-transform | |

| |

| |

| |

Definition of the z-Transform | |

| |

| |

| |

The z-Plane and the Fourier Transform | |

| |

| |

| |

Calculating the z-Transform | |

| |

| |

| |

Unit Step u[n] | |

| |

| |

| |

Exponential an u[n] | |

| |

| |

| |

Sinusoid cos(n!0) u[n] and sin(n!0) u[n] | |

| |

| |

| |

Differentiation | |

| |

| |

| |

The effect of Sampling Rate | |

| |

| |

| |

A Discrete Time Laplace Transform | |

| |

| |

| |

Properties of the z-Transform | |

| |

| |

| |

z-Transform Pairs | |

| |

| |

| |

Transfer Function of a Discrete Linear System | |

| |

| |

| |

MATLAB Analysis with the z-transform | |

| |

| |

| |

First Order Lowpass Filter | |

| |

| |

| |

Pole-Zero Diagram | |

| |

| |

| |

Bode Plot | |

| |

| |

| |

Impulse Response | |

| |

| |

| |

Calculating Frequency Response | |

| |

| |

| |

Pole Position Determines Frequency Response | |

| |

| |

| |

Digital Filtering - FIR Filter | |

| |

| |

| |

A One Pole FIR Filter | |

| |

| |

| |

A Two Pole FIR Filter | |

| |

| |

| |

Higher Order FIR Filters | |

| |

| |

Frequency Response | |

| |

| |

Pole Zero Diagram | |

| |

| |

Phase Response | |

| |

| |

Step Response | |

| |

| |

| |

Digital Filtering - IIR Filter | |

| |

| |

| |

A One Pole IIR Filter | |

| |

| |

| |

IIR vs FIR | |

| |

| |

| |

Higher Order IIR Filters | |

| |

| |

| |

Combining FIR and IIR Filters | |

| |

| |

| |

Conclusions | |

| |

| |

| |

End of Chapter Exercises | |

| |

| |

| |

Introduction to Communications | |

| |

| |

| |

Introduction | |

| |

| |

| |

A Baseband Signal m(t) | |

| |

| |

| |

The need for a Carrier Signal | |

| |

| |

| |

A Carrier Signal c(t) | |

| |

| |

| |

Modulation Techniques | |

| |

| |

| |

The Radio Spectrum | |

| |

| |

| |

Amplitude Modulation | |

| |

| |

| |

Transmitted Carrier Double Sideband - (AM-TCDSB) | |

| |

| |

| |

Demodulation of AM Signals | |

| |

| |

| |

Graphical Analysis | |

| |

| |

| |

AM Demodulation - Diode Detector | |

| |

| |

| |

Examples of Diode Detection | |

| |

| |

| |

Suppressed Carrier Transmission | |

| |

| |

| |

Demodulation of Single Sideband Signals | |

| |

| |

| |

Percent Modulation and Overmodulation | |

| |

| |

| |

Superheterodyne Receiver | |

| |

| |

| |

An Experiment with Intermediate Frequency | |

| |

| |

| |

When Receivers become Transmitters | |

| |

| |

| |

Image Frequency | |

| |

| |

| |

Beat Frequency Oscillator | |

| |

| |

| |

Digital Communications | |

| |

| |

| |

Modulation Methods | |

| |

| |

| |

Morse Code | |

| |

| |

| |

On O_ Keying (OOK) | |

| |

| |

| |

Bandwidth Considerations | |

| |

| |

| |

Phase Shift Keying | |

| |

| |

| |

Differential Coding | |

| |

| |

| |

Higher-Order Modulation Schemes | |

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Conclusions | |

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End of Chapter Exercises | |

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The Illustrated Fourier Transform | |

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The Illustrated Laplace Transform | |

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The Illustrated z-Transform | |

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MATLAB Reference Guide | |

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Defining Signals | |

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MATLAB Variables | |

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The Time Axis | |

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Common Signals | |

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Non-Periodic | |

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Periodic | |

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Complex Numbers | |

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Plot Commands | |

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Signal Operations | |

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Defining Systems | |

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System Definition | |

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Transfer Function | |

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Zeros and Poles and Gain | |

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State Space Model | |

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Discrete Time Systems | |

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System Analysis | |

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Example System Definition and Test | |

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Reference Tables | |

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Fourier Transform | |

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Fourier Transform Theorems | |

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Laplace Transform | |

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Laplace Transform Theorems | |

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z-Transform | |

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z-Transform Theorems | |