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• Title Page
• Copyright Page
• Contents
• PREFACE
• 1 Introduction
• 1-1 Mathematical Representation of Signals
• 1-2 Mathematical Representation of Systems
• 1-3 Systems as Building Blocks
• 1-4 The Next Step
• 2 Sinusoids
• 2-1 Tuning-Fork Experiment
• 2-2 Review of Sine and Cosine Functions
• 2-3 Sinusoidal Signals
• 2-3.1 Relation of Frequency to Period
• 2-3.2 Phase and Time Shift
• 2-4 Sampling and Plotting Sinusoids
• 2-5 Complex Exponentials and Phasors
• 2-5.1 Review of Complex Numbers
• 2-5.2 Complex Exponential Signals
• 2-5.3 The Rotating Phasor Interpretation
• 2-5.4 Inverse Euler Formulas
• 2-6 Phasor Addition
• 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
• 2-10 Problems
• 3 Spectrum Representation
• 3-1 The Spectrum of a Sum of Sinusoids
• 3-1.1 Notation Change
• 3-1.2 Graphical Plot of the Spectrum
• 3-1.3 Analysis versus Synthesis
• 3-2 Sinusoidal Amplitude Modulation
• 3-2.1 Multiplication of Sinusoids
• 3-2.2 Beat Note Waveform
• 3-2.3 Amplitude Modulation
• 3-2.4 AM Spectrum
• 3-2.5 The Concept of Bandwidth
• 3-3 Operations on the Spectrum
• 3-3.1 Scaling or Adding a Constant
• 3-3.2 Adding Signals
• 3-3.3 Time-Shifting x(t) Multiplies a by a[sub(k)] Complex Exponential
• 3-3.4 Differentiating x(t) Multiplies a[sub(k)] by (j2&#960f[sub(k)])
• 3-3.5 Frequency Shifting
• 3-4 Periodic Waveforms
• 3-4.1 Synthetic Vowel
• 3-4.2 Sine-Cubed Signal
• 3-4.3 Example of a Non-Periodic Signal
• 3-5 Fourier Series
• 3-5.1 Fourier Series: Analysis
• 3-5.2 Analysis of a Full-Wave Rectified Sine Wave
• 3-5.3 Spectrum of the FWRS Fourier Series
• 3-5.3.1 DC Value of Fourier Series
• 3-5.3.2 Finite Synthesis of a Full-Wave Rectified Sine
• 3-6 Time–Frequency Spectrum
• 3-6.1 Stepped Frequency
• 3-6.2 Spectrogram Analysis
• 3-7 Frequency Modulation: Chirp Signals
• 3-7.1 Chirp or Linearly Swept Frequency
• 3-7.2 A Closer Look at Instantaneous Frequency
• 3-8 Summary and Links
• 3-9 Problems
• 4 Sampling and Aliasing
• 4-1 Sampling
• 4-1.1 Sampling Sinusoidal Signals
• 4-1.2 The Concept of Aliases
• 4-1.3 Sampling and Aliasing
• 4-1.4 Spectrum of a Discrete-Time Signal
• 4-1.5 The Sampling Theorem
• 4-1.6 Ideal Reconstruction
• 4-2 Spectrum View of Sampling and Reconstruction
• 4-2.1 Spectrum of a Discrete-Time Signal Obtained by Sampling
• 4-2.2 Over-Sampling
• 4-2.3 Aliasing Due to Under-Sampling
• 4-2.4 Folding Due to Under-Sampling
• 4-2.5 Maximum Reconstructed Frequency
• 4-2.6 Sampling and Reconstruction GUI
• 4-3 Discrete-to-Continuous Conversion
• 4-3.1 Interpolation with Pulses
• 4-3.2 Zero-Order Hold Interpolation
• 4-3.3 Linear Interpolation
• 4-3.4 Cubic-Spline Interpolation
• 4-3.5 Over-Sampling Aids Interpolation
• 4-3.6 Ideal Bandlimited Interpolation
• 4-4 The Sampling Theorem
• 4-5 Strobe Demonstration
• 4-5.1 Spectrum Interpretation
• 4-6 Summary and Links
• 4-7 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
• 5-4 The Unit Impulse Response and Convolution
• 5-4.1 Unit Impulse Sequence
• 5-4.2 Unit Impulse Response Sequence
• 5-4.2.1 The Unit-Delay System
• 5-4.3 FIR Filters and Convolution
• 5-4.3.1 Computing the Output of a Convolution
• 5-4.3.2 The Length of a Convolution
• 5-4.3.3 Convolution in MATLAB
• 5-4.3.4 Polynomial Multiplication in MATLAB
• 5-4.3.5 Filtering the Unit-Step Signal
• 5-4.3.6 Commutative Property of Convolution
• 5-4.3.7 MATLAB GUI for Convolution
• 5-5 Implementation of FIR Filters
• 5-5.1 Building Blocks
• 5-5.1.1 Multiplier
• 5-5.1.2 Adder
• 5-5.1.3 Unit Delay
• 5-5.2 Block Diagrams
• 5-5.2.1 Other Block Diagrams
• 5-5.2.2 Internal Hardware Details
• 5-6 Linear Time-Invariant (LTI) Systems
• 5-6.1 Time Invariance
• 5-6.2 Linearity
• 5-6.3 The FIR Case
• 5-7 Convolution and LTI Systems
• 5-7.1 Derivation of the Convolution Sum
• 5-7.2 Some Properties of LTI Systems
• 5-8 Cascaded LTI Systems
• 5-9 Example of FIR Filtering
• 5-10 Summary and Links
• 5-11 Problems
• 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 H(e[sup(j&#969)])
• 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-Sum Filtering
• 6-7.1 Plotting the Frequency Response
• 6-7.2 Cascade of Magnitude and Phase
• 6-7.3 Frequency Response of Running Averager
• 6-7.4 Experiment: Smoothing an Image
• 6-8 Filtering Sampled Continuous-Time Signals
• 6-8.1 Example: A Lowpass Averager for Continuous-Time Signals
• 6-8.2 Interpretation of Delay
• 6-9 Summary and Links
• 6-10 Problems
• 7 Discrete-Time Fourier Transform
• 7-1 DTFT: Fourier Transform for Discrete-Time Signals
• 7-1.1 Forward DTFT
• 7-1.2 DTFT of a Shifted Impulse Sequence
• 7-1.3 Linearity of the DTFT
• 7-1.4 Uniqueness of the DTFT
• 7-1.5 DTFT of a Pulse
• 7-1.6 DTFT of a Right-Sided Exponential Sequence
• 7-1.7 Existence of the DTFT
• 7-1.8 The Inverse DTFT
• 7-1.9 Bandlimited DTFT
• 7-1.10 Inverse DTFT for the Right-Sided Exponential
• 7-1.11 The DTFT Spectrum
• 7-2 Properties of the DTFT
• 7-2.1 Linearity Property
• 7-2.2 Time-Delay Property
• 7-2.3 Frequency-Shift Property
• 7-2.3.1 DTFT of a Finite-Length Complex Exponential
• 7-2.3.2 DTFT of a Finite-Length Real Cosine Signal
• 7-2.4 Convolution and the DTFT
• 7-2.4.1 Filtering is Convolution
• 7-2.5 Energy Spectrum and the Autocorrelation Function
• 7-2.5.1 Autocorrelation Function
• 7-3 Ideal Filters
• 7-3.1 Ideal Lowpass Filter
• 7-3.2 Ideal Highpass Filter
• 7-3.3 Ideal Bandpass Filter
• 7-4 Practical FIR Filters
• 7-4.1 Windowing
• 7-4.2 Filter Design
• 7-4.2.1 Window the Ideal Impulse Response
• 7-4.2.2 Frequency Response of Practical Filters
• 7-4.2.3 Passband Defined for the Frequency Response
• 7-4.2.4 Stopband Defined for the Frequency Response
• 7-4.2.5 Transition Zone of the LPF
• 7-4.2.6 Summary of Filter Specifications
• 7-4.3 GUI for Filter Design
• 7-5 Table of Fourier Transform Properties and Pairs
• 7-6 Summary and Links
• 7-7 Problems
• 8 Discrete Fourier Transform
• 8-1 Discrete Fourier Transform (DFT)
• 8-1.1 The Inverse DFT
• 8-1.2 DFT Pairs from the DTFT
• 8-1.2.1 DFT of Shifted Impulse
• 8-1.2.2 DFT of Complex Exponential
• 8-1.3 Computing the DFT
• 8-1.4 Matrix Form of the DFT and IDFT
• 8-2 Properties of the DFT
• 8-2.1 DFT Periodicity for X[k]
• 8-2.2 Negative Frequencies and the DFT
• 8-2.3 Conjugate Symmetry of the DFT
• 8-2.3.1 Ambiguity at X[N/2]
• 8-2.4 Frequency-Domain Sampling and Interpolation
• 8-2.5 DFT of a Real Cosine Signal
• 8-3 Inherent Time-Domain Periodicity of x[n] in the DFT
• 8-3.1 DFT Periodicity for x[n]
• 8-3.2 The Time Delay Property for the DFT
• 8-3.2.1 Zero Padding
• 8-3.3 The Convolution Property for the DFT
• 8-4 Table of Discrete Fourier Transform Properties and Pairs
• 8-5 Spectrum Analysis of Discrete Periodic Signals
• 8-5.1 Periodic Discrete-Time Signal: Discrete Fourier Series
• 8-5.2 Sampling Bandlimited Periodic Signals
• 8-5.3 Spectrum Analysis of Periodic Signals
• 8-6 Windows
• 8-6.1 DTFT of Windows
• 8-7 The Spectrogram
• 8-7.1 An Illustrative Example
• 8-7.2 Time-Dependent DFT
• 8-7.3 The Spectrogram Display
• 8-7.4 Interpretation of the Spectrogram
• 8-7.4.1 Frequency Resolution
• 8-7.5 Spectrograms in MATLAB
• 8-8 The Fast Fourier Transform (FFT)
• 8-8.1 Derivation of the FFT
• 8-8.1.1 FFT Operation Count
• 8-9 Summary and Links
• 8-10 Problems
• 9 z-Transforms
• 9-1 Definition of the z-Transform
• 9-2 Basic z-Transform Properties
• 9-2.1 Linearity Property of the z-Transform
• 9-2.2 Time-Delay Property of the z-Transform
• 9-2.3 A General z-Transform Formula
• 9-3 The z-Transform and Linear Systems
• 9-3.1 Unit-Delay System
• 9-3.2 The z[sup(-1)] Notation in Block Diagrams
• 9-3.3 The z-Transform of an FIR Filter
• 9-3.4 z-Transform of the Impulse Response
• 9-3.5 Roots of a z-Transform Polynomial
• 9-4 Convolution and the z-Transform
• 9-4.1 Cascading Systems
• 9-4.2 Factoring z-Polynomials
• 9-4.3 Deconvolution
• 9-5 Relationship Between the z-Domain and the Domain
• 9-5.1 The z-Plane and the Unit Circle
• 9-5.2 The z-Transform and the DFT
• 9-6 The Zeros and Poles of H(z)
• 9-6.1 Pole-Zero Plot
• 9-6.2 Significance of the Zeros of H(z)
• 9-6.3 Nulling Filters
• 9-6.4 Graphical Relation Between z and &#969
• 9-6.5 Three-Domain Movies
• 9-7 Simple Filters
• 9-7.1 Generalize the L-Point Running-Sum Filter
• 9-7.2 A Complex Bandpass Filter
• 9-7.3 A Bandpass Filter with Real Coefficients
• 9-8 Practical Bandpass Filter Design
• 9-9 Properties of Linear-Phase Filters
• 9-9.1 The Linear-Phase Condition
• 9-9.2 Locations of the Zeros of FIR Linear- Phase Systems
• 9-10 Summary and Links
• 9-11 Problems
• 10 IIR Filters
• 10-1 The General IIR Difference Equation
• 10-2 Time-Domain Response
• 10-2.1 Linearity and Time Invariance of IIR Filters
• 10-2.2 Impulse Response of a First-Order IIR System
• 10-2.3 Response to Finite-Length Inputs
• 10-2.4 Step Response of a First-Order Recursive System
• 10-3 System Function of an IIR Filter
• 10-3.1 The General First-Order Case
• 10-3.2 H(z) from the Impulse Response
• 10-4 The System Function and Block Diagram Structures
• 10-4.1 Direct Form I Structure
• 10-4.2 Direct Form II Structure
• 10-4.3 The Transposed Form Structure
• 10-5 Poles and Zeros
• 10-5.1 Roots in MATLAB
• 10-5.2 Poles or Zeros at z = 0 or &#8734
• 10-5.3 Output Response from Pole Location
• 10-6 Stability of IIR Systems
• 10-6.1 The Region of Convergence and Stability
• 10-7 Frequency Response of an IIR Filter
• 10-7.1 Frequency Response Using MATLAB
• 10-7.2 Three-Dimensional Plot of a System Function
• 10-8 Three Domains
• 10-9 The Inverse z-Transform and Applications
• 10-9.1 Revisiting the Step Response of a First-Order System
• 10-9.2 A General Procedure for Inverse z-Transformation
• 10-10 Steady-State Response and Stability
• 10-11 Second-Order Filters
• 10-11.1 z-Transform of Second-Order Filters
• 10-11.2 Structures for Second-Order IIR Systems
• 10-11.3 Poles and Zeros
• 10-11.4 Impulse Response of a Second-Order IIR System
• 10-11.4.1 Distinct Real Poles
• 10-11.5 Complex Poles
• 10-12 Frequency Response of Second-Order IIR Filter
• 10-12.1 Frequency Response via MATLAB
• 10-12.2 3-dB Bandwidth
• 10-12.3 Three-Dimensional Plot of a System Function
• 10-12.4 Pole-Zero Placing with the PeZ GUI
• 10-13 Example of an IIR Lowpass Filter
• 10-14 Summary and Links
• 10-15 Problems
• 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’s Formula
• A-3.1 Inverse Euler Formulas
• A-4 Algebraic Rules for Complex Numbers
• A-4.1 Complex Number 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 the Inverse, z[sup(-1)]
• A-5.6 Geometric View of the Conjugate, z[sup(*)]
• 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
• A-8 Problems
• B: Programming in MATLAB
• B-1 MATLAB Help
• B-2 Matrix Operations and Variables
• B-2.1 The Colon Operator
• B-2.2 Matrix and Array 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-4.1 MATLAB Built-In Functions
• B-4.2 Program Flow
• B-5 MATLAB Scripts
• 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
• C: Fourier Series
• C-1 Fourier Series Derivation
• C-1.1 Fourier Integral Derivation
• C-2 Examples of Fourier Analysis
• C-2.1 The Pulse Wave
• C-2.1.1 Spectrum of a Pulse Wave
• C-2.1.2 Finite Synthesis of a Pulse Wave
• C-2.2 Triangular Wave
• C-2.2.1 Spectrum of a Triangular Wave
• C-2.2.2 Finite Synthesis of a Triangular Wave
• C-2.3 Half-Wave Rectified Sine
• C-2.3.1 Finite Synthesis of a Half-Wave Rectified Sine
• C-3 Operations on Fourier Series
• C-3.1 Scaling or Adding a Constant
• C-3.2 Adding Signals
• C-3.3 Time-Scaling Property
• C-3.4 Time-Shifting Property
• C-3.5 Differentiation Property
• C-3.6 Frequency-Shifting Property and Multiplying by a Sinusoid
• C-4 Average Power, Convergence, and Optimality
• C-4.1 Derivation of Parseval’s Theorem
• C-4.2 Convergence of Fourier Synthesis
• C-4.3 Minimum Mean-Square Approximation
• C-5 The Spectrum in Pulsed-Doppler Radar Waveform Design
• C-5.1 Measuring Range
• C-5.2 Measuring Velocity from Doppler Shift
• C-5.3 Pulsed-Doppler Radar Waveform
• C-5.4 Measuring the Doppler Shift
• C-6 Problems
• D: Laboratory Projects
• Index
• A
• B
• C
• D
• E
• F
• G
• H
• I
• L
• M
• N
• O
• P
• R
• S
• T
• U
• V
• W
• Z