Table of Contents

Table of Contents

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Preface

Chapter 1: Mathematical Preliminaries Download

	1.A	Introduction
	1.B	Review of some useful concepts of geometry and algebra

Scalar arithmetic
Vector arithmetic
Multiplication
Vector length
Summary

1.C

Review of phasors and complex numbers

Phasor length and the magnitude of complex numbers
Statistics of complex numbers

Chapter 2: Sinusoids, Phasors, and Matrices Download

	2.A	Phasor representation of sinusoidal waveforms
	2.B	Matrix algebra

Rotation matrices
Basis vectors
Orthogonal decomposition

Chapter 3: Fourier Analysis of Discrete Functions Download

3.A
3.B
3.C
3.D
3.E
3.F
3.G
3.H
3.I
3.J

Introduction
A Function Sampled at 1 point
A Function Sampled at 2 points
Fourier Analysis is a Linear Transformation
Fourier Analysis is a Change in Basis Vectors
A Function Sampled at 3 points
A Function Sampled at D points
Tidying Up
Parseval's Theorem
A Statistical Connection

Chapter 4: The Frequency Domain Download

4.A
4.B
4.C
4.D
4.E
4.F
4.G
4.H
4.I

Spectral Analysis
Physical Units
Cartesian vs. Polar Form
Complex Form of Spectral Analysis
Complex Fourier Coefficients
Relationship between Complex and Trigonometric Fourier Coefficients
Discrete Fourier Transforms in Two or More Dimensions
Matlab's implementation of the DFT
Parseval's theorem, revisited

Chapter 5: Continuous Functions Download

5.A
5.B
5.C
5.D

Introduction
Inner products and orthogonality
Symmetry
Complex-valued functions

Chapter 6: Fourier Analysis of Continuous Functions Download

6.A
6.B
6.C
6.D

Introduction
The Fourier Model
Practicalities of Obtaining the Fourier Coefficients
Theorems

Linearity
Shift theorem
Scaling theorem
Differentiation theorem
Integration theorem

Chapter 7: Sampling Theory Download

7.A
7.B
7.C
7.D
7.E
7.F

Introduction
The Sampling Theorem
Aliasing
Parseval's Theorem
Truncation Errors
Truncated Fourier Series & Regression Theory

Chapter 8: Statistical Description of Fourier Coefficients Download

8.A
8.B
8.C
8.D
8.E
8.F

Introduction
Statistical Assumptions
Mean and Variance of Fourier Coefficients for Noisy Signals
Probability Distribution of Fourier Coefficients for Noisy Signals
Distribution of Fourier Coefficients for Random Signals
Signal Averaging

Chapter 9: Hypothesis Testing for Fourier Coefficients Download

9.A
9.B
9.C
9.D
9.E
9.F

Introduction
Regression analysis
Band-limited signals
Confidence intervals
Mulitvariate statistical analysis of Fourier coefficients
References

Chapter 10: Directional Data Analysis Download

10.A
10.B
10.C
10.D
10.E
10.F

Introduction
Determination of mean direction and concentration about the mean
Hypothesis testing
Grouped data
The Fourier connection
Higher harmonics

Chapter 11: The Fourier Transform Download

11.A
11.B
11.C
11.D
11.E
11.F
11.G

Introduction
The Inverse Cosine and Sine Transforms
The Forward Cosine and Sine Transforms
An Example
Complex Form of the Fourier Transform
Fourier's Theorem
Relationship between Complex & Trigonometric Transforms

Chapter 12: Properties of The Fourier Transform Download

12.A
12.B

Introduction
Theorems

Linearity
Scaling
Time/Space Shift
Frequency Shift
Modulation
Differentiation
Integration
Transform of a transform
Central ordinate
Equivalent width
Convolution
Derivative of a convolution
Cross-correlation
Auto-correlation
Parseval/Rayleigh

12.C
12.D
12.E
12.F
12.G

The convolution operation
Delta functions
Complex conjugate relations
Symmetry relations
Variations on the convolution theorem

Chapter 13: Signal Analysis Download

13.A
13.B
13.C
13.D
13.E
13.F
13.G

Introduction
Windowing
Sampling with an array of windows
Aliasing
Reconstruction by interpolation
Non-point sampling
The coverage factor rule

Bibliography Download

Fourier Series and Transforms
Statistics of Fourier Coefficients
Directional Data Analysis
Random Signals and Noise
Probability Theory & Stochastic Processes
Signal Detection Theory

Homework

Homework Set 01           Download
Homework Set 02           Download
Homework Set 03           Download
Homework Set 04           Download
Homework Set 05           Download
Homework Set 06           Download
Homework Set 07           Download
Homework Set 08           Download
Homework Set 09           Download
Homework Set 10           Download
Homework Set 11           Download
Homework Set 12           Download