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The Fourier Transform and it.pdf
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傅里叶变换及其应用, 从连续时间的傅里叶变换过渡到离散时间的傅里叶变换，并详细讲解了FFT算法， 采样原理等在工程技术中广泛用到的技术。
目录：
1 Introduction
2 Groundwork
The Fourier Transform and Fourier's Integral Theorem
Conditions for the Existence of Fourier Transforms
Transforms in the Limit
Oddness and Evenness
Significance of Oddness and Evenness
Complex Conjugates
Cosine and Sine Transforms
Interpretation of the Formulas
3 Convolution
Examples of Convolution
Serial Products
Inversion of serial multiplication / The serial product in matrix notation /
Sequences as vectors
Convolution by Computer
The Autocorrelation Function and Pentagram Notation
The Triple Correlation
The Cross Correlation
The Energy Spectrum
4 Notation for Some Useful Functions
Rectangle Function of Unit Height and Base, II(x)
Triangle Function of Unit Height and Area, A(x)
Various Exponentials and Gaussian and Rayleigh Curves
Heaviside's Unit Step Function, H(x)
The Sign Function, sgn x
The Filtering or Interpolating Function, sinc x
Pictorial Representation
Summary of Special Symbols
5 The Impulse Symbol 74
The Sifting Property 78
The Sampling or Replicating Symbol III(x) 81
The Even and Odd Impulse Pairs (x) and Ii(x) 84
Derivatives of the Impulse Symbol 85
Null Functions 87
Some Functions in Two or More Dimensions 89
The Concept of Generalized Function 92
6 The Basic Theorems
A Few Transforms for Illustration
Similarity Theorem
Addition Theorem
Shift Theorem
Modulation Theorem
Convolution Theorem
Rayleigh's Theorem
Power Theorem
Autocorrelation Theorem
Derivative Theorem
Derivative of a Convolution Integral
The Transform of a Generalized Function
Proofs of Theorems
Similarity and shift theorems / Derivative theorem / Power theorem
Summary of Theorems
7 Obtaining Transforms
Integration in Closed Form
Numerical Fourier Transformation
The Slow Fourier Transform Program
Generation of Transforms by Theorems
Application of the Derivative Theorem to Segmented Functions
Measurement of Spectra
Radiofrequency spectral analysis / Optical Fourier transform spectroscopy
8 The Two Domains
Definite Integral
The First Moment
Centroid
Moment of Inertia (Second Moment)
Moments
Mean-Square Abscissa
Radius of Gyration
Contents xi
Variance
Smoothness and Compactness
Smoothness under Convolution
Asymptotic Behavior
Equivalent Width
Autocorrelation Width
Mean Square Widths
Sampling and Replication Commute
Some Inequalities
Upper limits to ordinate and slope / Schwarz's inequality
The Uncertainty Relation
Proof of uncertainty relation / Example of uncertainty relation
The Finite Difference
Running Means
Central Limit Theorem
Summary of Correspondences in the Two Domains
9 Waveforms, Spectra, Filters, and Linearity
Electrical Waveforms and Spectra
Filters
Generality of Linear Filter Theory
Digital Filtering
Interpretation of Theorems
Similarity theorem / Addition theorem / Shift theorem / Modulation theorem /
Converse of modulation theorem
Linearity and Time Invariance
Periodicity
10 Sampling and Series
Sampling Theorem
Interpolation
Rectangular Filtering in Frequency Domain
Smoothing by Running Means
Undersampling
Ordinate and glope Sampling
Interlaced Sampling
Sampling in the Presence of Noise
Fourier Series
Gibbs phenomenon / Finite Fourier transforms / Fourier coefficients
Impulse Trains That Are Periodic
The Shah Symbol Is Its Own Fourier Transform
11 The Discrete Fourier Transform and the FFT
The Discrete Transform Formula
Cyclic Convolution
Examples of Discrete Fourier Transforms
Reciprocal Property
Oddness and Evenness
Examples with Special Symmetry
Complex Conjugates
Reversal Property
Addition Theorem
Shift Theorem
Convolution Theorem
Product Theorem
Cross-Correlation
Autocorrelation
Sum of Sequence
First Value
Generalized Parseval-Rayleigh Theorem
Packing Theorem
Similarity Theorem
Examples Using MATLAB
The Fast Fourier Transform
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