1 Introduction

1.1 Mathematical Representation of Signals

1.2 Mathematical Representation of Systems

1.3 Thinking About Systems

1.4 The Next Step

2 Sinusoids

2.1 An Experiment with a Tuning Fork

2.2 Review of Sine and Cosine Functions

2.3 Sinusoidal Signals

2.3.1 Relation of Frequency to Period

2.3.2 Relation of Phase Shift to Time Shift

2.4 Sampling and Plotting Sinusoids

2.5.1 Review of Complex Numbers

2.5 Complex Exponentials and Phasors

2.5.2 Complex ExponentialSignals

2.5.3 The Rotating Phasor Interpretation

2.5.4 Inverse Euler Formulas

2.6 PhasorAddition

2.6.1 Addition of Complex Numbers

2.6.2 Phasor Addition Rule

2.6.3 Phasor Addition Rule:Example

2.6.4 MATLAB Demo of Phasors

2.6.5 Summary of the Phasor Addition Rule

2.7 Physics of the Tuning Fork

2.7.1 Equations from Laws of Physics

2.7.2 General Solution to the Differential Equation

2.7.3 Listening to Tones

2.8 Time Signals:More Than Formulas

2.9 Summary and Links

Problems

3 Spectrum Representation

3.1 The Spectrum of a Sum of Sinusoids

3.1.1 Graphical Plot of the Spectrum

3.2 Beat Notes

3.2.1 Multiplication of Sinusoids

3.2.2 Beat Note Waveform

3.2.3 Amplitude Modulation

3.3 Periodic Waveforms

3.3.1 Synthetic Vowel

3.4 More Periodic Signals

3.4.1 Fourier Series:Analysis

3.4.2 The Square Wave

3.4.3 Triangle Wave

3.4.4 Example of a Non-periodic Signal

3.5 Time-Frequency Spectrum

3.5.1 Stepped Frequency

3.5.2 Spectrogram Analysis

3.6 Frequency Modulation:Chirp Signals

3.6.1 Chirp,or Linearly Swept Frequency

3.6.2 A Closer Look at Instantaneous Frequency

3.7 Summary and Links

Problems

4.1 Sampling

4 Sampling and Aliasing

4.1.1 Sampling Sinusoidal Signals

4.1.2 The SamplingTheorem

4.1.3 Aliasing

4.1.4 Folding

4.2 Spectrum View of Sampling

4.2.1 Over-Sampling

4.2.2 Aliasing Due to Under-Sampling

4.2.3 Folding Due to Under-Sampling

4.2.4 Maximum Reconstructed Frequency

4.3 Strobe Demonstration

4.3.1 Spectrum Interpretation

4.4 Discrete-to-Continuous Conversion

4.4.1 Alias Frequencies Due to Sampling

4.4.2 Interpolation with Pulses

4.4.3 Zero-Order Hold Interpolation

4.4.4 Linear Interpolation

4.4.5 Parabolic Interpolation

4.4.6 Over-Sampling Aids Interpolation

4.4.7 Ideal Bandlimited Interpolation

4.5 The Sampling Theorem

4.6 Summary and Links

Problems

5 FIR Filters

5.1 Discrete-Time Systems

5.2 The Running Average Filter

5.3 The General FIR Filter

5.3.1 An Illustration of FIR Filtering

5.3.2 The Unit Impulse Response

5.3.2.1 Unit Impulse Sequence

5.3.2.2 Unit Impulse Response Sequence

5.3.2.3 The Unit-Delay System

5.3.3 Convolution and FIR Filters

5.3.3.1 Computing the Output of a Convolution

5.3.3.2 Convolution in MATLAB

5.4 Implementation of FIR Filters

5.4.1 Building Blocks

5.4.1.1 Multiplier

5.4.1.2 Adder

5.4.1.3 Unit Delay

5.4.2 Block Diagrams

5.4.2.1 Other Block Diagrams

5.4.2.2 Internal Hardware Details

5.5 Linear Time-Invariant(LTI)Systems

5.5.1 Time Invariance

5.5.2 Linearity

5.5.3 The FIR Case

5.6 Convolution and LTI Systems

5.6.1 Derivation of the Convolution Sum

5.6.2 Some Properties of LTI Systems

5.6.2.1 Convolution as an Operator

5.6.2.2 Commutative Property of Convolution

5.6.2.3 Associative Property of Convolution

5.7 Cascaded LTI Systems

5.8 Example of FIR Filtering

Problems

5.9 Summary and Links

6 Frequency Response of FIR Filters

6.1 Sinusoidal Response of FIR Systems

6.2 Superposition and the Frequency Response

6.3 Steady State and Transient Response

6.4 Properties of the Frequency Response

6.4.1 Relation to Impulse Response and Difference Equation

6.4.2 Periodicity of ？(？)

6.4.3 Conjugate Symmetry

6.5 Graphical Representation of the Frequency Response

6.5.1 Delay System

6.5.2 First Difference System

6.5.3 A Simple Lowpass Filter

6.6 Cascaded LTI Systems

6.7 Running-Average Filtering

6.7.1 Plotting the Frequency Response

6.7.2 Cascade of Magnitude and Phase

6.7.3 Experiment:Smoothing an Image

6.8 Filtering Sampled Continuous-Time Signals

6.8.1 Example:Low-Pass Averager

6.8.2 Interpretation of Delay

6.9 Summary and Links

Problems

7 z-Transforms

7.1 Definition of the z-Transform

7.2 The z-Transform and Linear Systems

7.2.1 The z-Transform of an FIR Filter

7.3.1 The Superposition Property of the z-Transform

7.3 Properties of the z-Transform

7.3.2 The Time-Delay Property of the z-Transform

7.3.3 A General z-Transform Formula

7.4 The z-Transform as an Operator

7.4.1 Unit-Delay Operator

7.4.2 Operator Notation

7.4.3 Operator Notation in Block Diagrams

7.5 Convolution ankd the z-Transform

7.5.1 Cascading Systems

7.5.2 Factoring z-Polynomials

7.5.3 Deconvolution

7.6 Relationship Between the z-Domain and the ？-Domain

7.6.1 The z-Plane and the Unit Circle

7.6.2 The Zeros and Poles of H(z)

7.6.3 Significance of the Zeros of H(z)

7.6.4 NullingFilters

7.6.5 Graphical Relation Between z and ？

7.7 Useful Filters

7.7.1 The L-Point Running Sum Filter

7.7.2 A Complex Bandpass Filter

7.7.3 A Bandpass Filter with Real Coefficients

7.8 Practical Bandpass Filter Design

7.9 Properties of Linear Phase Filters

7.9.1 The Linear Phase Condition

7.9.2 Locations of the Zeros of FIR Linear Phase Systems

7.10 Summary and Links

Problems

8 ⅡR Filters

8.1 The General ⅡR Difference Equation

8.2 Time-Domain Response

8.2.1 Linearity and Time Invariance of ⅡR Filters

8.2.2 Impulse Response of a First-Order ⅡR System

8.2.3 Response to Finite-Length Inputs

8.2.4 Step Response of a First-Order Recursive System

8.3 System Function of an ⅡR Filter

8.3.1 The General First-Order Case

8.3.2 The System Function and Block-Diagram Structures

8.3.2.1 Direct Form I Structure

8.3.2.2 Direct Form Ⅱ Structure

8.3.2.3 The Transposed Form Structure

8.3.3 Relation to the Impulse Response

8.3.4 Summary of the Method

8.4 Poles and Zeros

8.4.1 Poles or Zeros at the Origin or Infinity

8.4.2 Pole Locations and Stability

8.5 Frequency Response of an ⅡR Filter

8.5.1 Frequency Response using MATLAB

8.5.2 Three-Dimensional Plot of a System Function

8.6 Three Domains

8.7 The Inverse z-Transform and Some Applications

8.7.1 Revisiting the Step Response of a First-Order System

8.7.2 A General Procedure for Inverse z-Transformation

8.8 Steady-State Response and Stability

8.9 Second-Order Filters

8.9.1 z-transform of Second-Order Filters

8.9.2 Structures for Second-Order ⅡR Systems

8.9.3 Poles and Zeros

8.9.4 Impulse Response of a Second-Order ⅡR System

8.9.4.1 Real Poles

8.9.5 Complex Poles

8.10 Frequency Response of Second-Order ⅡR Filter

8.10.1 Frequency Response via MATLAB

8.10.2 3-dB Bandwidth

8.10.3 Three-Dimensional Plot of System Functions

8.11 Example of an ⅡR Lowpass Filter

8.12 Summary and Links

Problems

9 Spectrum Analysis

9.1.1 Review of the Frequency Spectrum

9.1 Introduction and Review

9.1.2 A Spectrum Analyzer

9.2 Spectrum Analysis by Filtering

9.2.1 Frequency Shifting

9.2.2 Measuring the Average Value

9.2.3 Channel Filters

9.3 Spectrum Analysis of Periodic Signals

9.3.1 Periodic Signals

9.3.2 Spectrum of a Periodic Signal

9.3.3 Filtering with a Running Sum

9.3.4 Spectrum Analysis Using Running-Sum Filtering

9.3.5 The DFT:Discrete Fourier Transform

9.3.6 DFTExamples

9.3.7 The Fast Fourier Transform(FFT)

9.4 Spectrum Analysis of Sampled Periodic Signals

9.5 Spectrum Analysis of Nonperiodic Signals

9.5.1 Spectrum Analysis of Finite-Length Signals

9.5.2 Frequency Sampling

9.5.3 Samples of the Frequency Response

9.5.4 Spectrum Analysis of Continuing Nonperiodic Signals

9.6 The Spectrogram

9.6.1 Spectrograms in MATLAB

9.6.2 Spectrogram of a Sampled Periodic Signal

9.6.3 Resolution of the Spectrogram

9.6.3.1 Resolution Experiment

9.6.4 Spectrogram of a Musical Scale

9.6.5 Spectrogram of a Speech Signal

9.7 Filtered Speech

9.8.1 Derivation of the FFT

9.8 The Fast Fourier Transform(FFT)

9.8.1.1 FFTOperation Count

9.9 Summary and Links

Problems

Appendix A Complex Numbers

A.1 Introduction

A.2 Notation for Complex Numbers

A.2.1 Rectangular Form

A.2.2 Polar Form

A.2.3 Conversion:Rectangular and Polar

A.2.4 Difficulty in Second or Third Quadrant

A.3 Euler'sFormula

A.4 Algebraic Rules for Complex Numbers

A.3.1 Inverse Euler Formulas

A.4.1 Exercises

A.5 Geometric Views of Complex Operations

A.5.1 Geometric View of Addition

A.5.2 Geometric View of Subtraction

A.5.3 Geometric View of Multiplication

A.5.4 Geometric View of Division

A.5.5 Geometric View of Inverse

A.5.6 Geometric Viewof Conjugate

A.6 Powers and Roots

A.6.1 Roots of Unity

A.6.1.1 Procedure for Finding Multiple Roots

A.7 Summary and Links

Problems

Appendix B Programming in MATLAB

B.1 MATLAB Help

B.2 Matrix Operations and Variables

B.2.1 The Colon Operator

B.2.2 Matrix andArray Operations

B.2.2.1 A Review of Matrix Multiplication

B.2.2.2 Pointwise Array Operations

B.3 Plots and Graphics

B.3.1 Figure Windows

B.3.2 Multiple Plots

B.3.3 Printing and Saving Graphics

B.4 Programming Constructs

B.5 MATLAB Scripts

B.4.2 Program Flow

B.4.1 MATLAB Built-in Functions

B.6 Writing a MATLAB Function

B.6.1 Creating A Clip Function

B.6.2 Debugging a MATLAB M-file

B.7 Programming Tips

B.7.1 Avoiding Loops

B.7.2 Repeating Rows or Columns

B.7.3 Vectorizing Logical Operations

B.7.4 Creating an Impulse

B.7.5 The Find Function

B.7.6 Seek to Vectorize

B.7.7 Programming Style

Appendix C Laboratory Projects

C.1.2 Warm-up

C.1.2.1 Basic Commands

C.1 Laboratory:Introduction to MATLAB

C.1.1 Overview and Goals

C.1.2.2 MATLAB Array Indexing

C.1.2.3 MATLAB Script Files

C.1.2.4 MATLAB Demos

C.1.2.5 MATLAB Sound

C.1.2.6 Functions

C.1.2.7 Vectorization

C.1.3 Exercises:Using MATLAB

C.1.3.1 Manipulating Sinusoids with MATLAB

C.1.4 Lab Review Questions

C.2.1.1 Complex Numbers in MATLAB

C.2.1 Overview

C.2 Laboratory:Introduction to Complex Exponentials

C.2.1.2 Sinusoid Addition Using Complex Exponentials

C.2.1.3 Harmonic Sinusoids

C.2.2 Warm-up

C.2.2.1 Complex Numbers

C.2.2.2 Sinusoidal Synthesis with an M-File

C.2.3 Exercises:Complex Exponentials

C.2.3.1 Representation of Sinusoids with Complex Exponentials

C.2.3.2 Verify Addition of Sinusoids Using Complex Exponentials

C.2.4 Periodic Waveforms

C.3 Laboratory:Synthesis of Sinusoidal Signals

C.3.1 Overview

C.3.2 Warm-up:Music Synthesis

C.3.2.2 Theory of Sampling

C.3.2.1 D-to-A Conversion

C.3.2.3 Piano Keyboard

C.3.3 Lab:Synthesis of MusicalNotes

C.3.3.1 Spectrogram of the Music

C.3.3.2 FürElise

C.3.3.3 Musical Tweaks

C.3.3.4 Programming Tips

C.3.3.5 Alternative Piece:Jesu,Joy of Man's Desiring

C.3.3.6 Alternative Piece:Minuet in G

C.3.3.7 Alternative Piece:Beethoven's Fifth Symphony

C.3.3.8 Alternative Piece:Twinkle,Twinkle,Little Star

C.3.4 Sound Evaluation Criteria

C.4.1.2 Frequency Modulated Signals

C.4.1.1 Amplitude Modulation

C.4 Laboratory:AM and FM Sinusoidal Signals

C.4.1 Overview

C.4.1.3 Chirp,or Linearly Swept Frequency

C.4.1.4 Advanced Topic:Spectrograms

C.4.2 Warm-up

C.4.2.1 MATLAB Synthesis of Chirp Signals

C.4.3 Lab A:Chirps and Beats

C.4.3.1 Synthesize a Chirp

C.4.3.2 Beat Notes

C.4.3.3 More on Spectrograms(Optional)

C.4.4 Lab B:FM Synthesis of Instrument Sounds

C.4.4.1 Generating the Bell Envelopes

C.4.4.2 Parameters for the Bell

C.4.4.3 The Bell Sound

C.4.4.4 Comments about the Bell

C.4.5 Woodwinds

C.4.5.1 Generating the Envelopes for Woodwinds

C.4.5.2 Scaling the Clarinet Envelopes

C.4.5.3 Clarinet Envelopes

C.4.5.4 Parameters for the Clarinet

C.4.5.5 Experiment with the Clarinet Sound

C.5 Laboratory:FIR Filtering of Sinusoidal Waveforms

C.5.1 Overview of Filtering

C.5.1.1 Frequency Response of FIR Filters

C.5.2 Warm-up

C.5.2.1 Frequency Response of the 3-Point Averager

C.5.3.1 Filtering Cosine Waves

C.5.3 Lab:FIR Filters

C.5.3.3 Linearity of the Filter

C.5.3.2 First-Difference Filter

C.5.3.4 Time Invariance of the Filter

C.5.3.5 Cascading Two Systems

C.6 Laboratory:Filtering Sampled Waveforms

C.6.1 Overview of Linear Filters

C.6.2 Warm-up

C.6.2.1 Properties of Discrete-TimeFilters

C.6.3 Laboratory:Sampling and Filters

C.6.3.1 Filtering a Stair-Step Signal

C.6.3.2 Implementation of Five-Point Averager

C.6.3.3 Implementation of First-Difference System

C.6.3.6 Comparison of Systems of Figs.C.11 and C.12

C.6.3.7 Filtering the Speech Waveform

C.6.3.4 Implementation of First Cascade(Fig.C.11)

C.6.3.5 Implementation of Second Cascade(Fig.C.12)

C.7 Laboratory:Everyday Sinusoidal Signals

C.7.1 Background

C.7.1.1 Background A:Telephone Touch Tone Dialing

C.7.1.2 DTMFDecoding

C.7.1.3 Background B:Amplitude Modulation(AM)

C.7.1.4 AM Demodulation

C.7.1.5 Envelope Detection(Peak Tracking)

C.7.1.6 LTI filter-based demodulation

C.7.1.7 Notch Filters for Demodulation

C.7.2.1 DTMF Dial Function

C.7.2 Warm-up A:DTMF Synthesis

C.7.3 Warm-up B:Tone Amplitude Modulation

C.7.4 Laboratory A:DTMFDecoding

C.7.4.1 Filter Design

C.7.4.2 A Scoring Function

C.7.4.3 DTMF Decode Function

C.7.4.4 Telephone Numbers

C.7.5 Laboratory B:AM Waveform Detection

C.7.6 Optional:Amplitude Modulation with Speech

C.8 Laboratory:Filtering and Edge Detection of Images

C.8.1 Overview

C.8.1.1 Digital Images

C.8.1.2 Displaying Images

C.8.2 Warm-up:Display of Images

C.8.1.3 Image Filtering

C.8.2.1 Display Test

C.8.3 Laboratory:Filtering Images

C.8.3.1 One-Dimensional Filtering

C.8.3.2 Blurring an Image

C.8.3.3 More Image Filters

C.8.3.4 Frequency Content of an Image

C.8.3.5 The Method of Synthetic Highs

C.8.3.6 Nonlinear Filters

C.8.3.7 Edges in an Image

C.8.3.8 The Slope-Threshold Function

C.8.3.9 What's Nonlinear about Edge Detection?

C.9.2 Warm-up:Linear Interpolation

C.9.1 Overview

C.9 Laboratory:Sampling and Zooming of Images

C.9.3 Laboratory:Sampling of Images

C.9.3.1 Reconstruction of Images

C.9.3.2 Zooming for an Image

C.10 Laboratory:The z-,n-,and ？-Domains

C.10.1 Objective

C.10.2 Warm-up

C.10.3 Laboratory:Relationships Between z-,n-,and ？-domains

C.10.4 Real Poles

C.10.5 Complex Poles

C.10.6 Filter Design

C.11.2.1 Spectrogram Computation

C.11.2 Warm-up:System Components

C.11.1 Overview

C.11 Laboratory:Extracting Frequencies of Musical Tones

C.11.2.2 Generating the Window

C.11.2.3 Display the Spectrogram

C.11.2.4 Finding Peaks

C.11.3 Design of the Music-Writing System

C.11.3.1 Block Diagram for the System

C.11.3.2 Write a Spectrogram Function

C.11.3.3 Parameters of the Spectrogram

C.11.3.4 Peak Picking and Editing

C.11.3.5 Writing the Musical Score

C.11.4 Testing the Music Extraction Program

Appendix D About the CD

Index