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Publisher:

Chapman & Hall/CRC

Publication Date:

2008

Number of Pages:

400

Format:

Hardcover

Price:

89.95

ISBN:

9781420063639

Category:

Textbook

[Reviewed by , on ]

Allen Stenger

11/8/2008

This is an engineering cookbook that covers the more elementary aspects of computing discrete Fourier transforms (DFTs). It attempts to integrate the continuous Fourier transform into the discussion but this is not very successful.

The book has a good bit of material on the pitfalls of sampling and calculation, and many numerical examples and lots of graphs. It includes two chapters on material I had not seen before: DFTs with a large prime number of samples.

The big weakness of the book is that it has no applications. For example, there is a lot of material on windowing and on finite-impulse response (FIR) filters but no mention of why these are useful.

A good book on this subject for mathematicians, that does a better job of combining the discrete and continuous (but is much more advanced) is Claude Gasquet & Patrick Witomski's *Fourier Analysis and Applications* (Springer, 1999). Good books for engineers are Oppenheim & Willsky & Hamid's *Signals and Systems* (Prentice-Hall, 2nd editiion 1996), and the more advanced book by Oppenheim & Schafer & Buck, *Discrete-Time Signal Processing* (Prentice-Hall, 2nd edition 1999); however, these are not cookbooks.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.

** Preface**

*Fundamentals, Analysis, and Applications*

**Analytical and Graphical Representation of Function Contents**

Time and Frequency Contents of a Function

The Frequency-Domain Plots as Graphical Tools

Identifying the Cosine and Sine Modes

Using Complex Exponential Modes

Using Cosine Modes with Phase or Time Shifts

Periodicity and Commensurate Frequencies

Review of Results and Techniques

Expressing Single Component Signals

General Form of a Sinusoid in Signal Application

Fourier Series: A Topic to Come

Terminology

**Sampling and Reconstruction of Functions—Part I**

DFT and Band-Limited Periodic Signal

Frequencies Aliased by Sampling

Connection: Anti-Aliasing Filter

Alternate Notations and Formulas

Sampling Period and Alternate Forms of DFT

Sample Size and Alternate Forms of DFT

**The Fourier Series**

Formal Expansions

Time-Limited Functions

Even and Odd Functions

Half-Range Expansions

Fourier Series Using Complex Exponential Modes

Complex-Valued Functions

Fourier Series in Other Variables

Truncated Fourier Series and Least Squares

Orthogonal Projections and Fourier Series

Convergence of the Fourier Series

Accounting for Aliased Frequencies in DFT

**DFT and Sampled Signals**

Deriving the DFT and IDFT Formulas

Direct Conversion between Alternate Forms

DFT of Concatenated Sample Sequences

DFT Coefficients of a Commensurate Sum

Frequency Distortion by Leakage

The Effects of Zero Padding

Computing DFT Defining Formulas Per Se

**Sampling and Reconstruction of Functions—Part II**

Sampling Nonperiodic Band-Limited Functions

Deriving the Fourier Transform Pair

The Sine and Cosine Frequency Contents

Tabulating Two Sets of Fundamental Formulas

Connections with Time/Frequency Restrictions

Fourier Transform Properties

Alternate Form of the Fourier Transform

Computing the Fourier Transform

Computing the Fourier Coefficients

**Sampling and Reconstruction of Functions—Part III**

Impulse Functions and Their Properties

Generating the Fourier Transform Pairs

Convolution and Fourier Transform

Periodic Convolution and Fourier Series

Convolution with the Impulse Function

Impulse Train as a Generalized Function

Impulse Sampling of Continuous-Time Signals

Nyquist Sampling Rate Rediscovered

Sampling Theorem for Band-Limited Signal

Sampling of Band-Pass Signals

**The Fourier Transform of a Sequence**

Deriving the Fourier Transform of a Sequence

Properties of the Fourier Transform of a Sequence

Generating the Fourier Transform Pairs

Duality in Connection with the Fourier Series

The Fourier Transform of a Periodic Sequence

The DFT Interpretation

**The Discrete Fourier Transform of a Windowed Sequence**

A Rectangular Window of Infinite Width

A Rectangular Window of Appropriate Finite Width

Frequency Distortion by Improper Truncation

Windowing a General Nonperiodic Sequence

Frequency-Domain Properties of Windows

Applications of the Windowed DFT

**Discrete Convolution and the DFT**

Linear Discrete Convolution

Periodic Discrete Convolution

The Chirp Fourier Transform

**Applications of the DFT in Digital Filtering and Filters**

The Background

Application-Oriented Terminology

Revisit Gibbs Phenomenon from the Filtering Viewpoint

Experimenting with Digital Filtering and Filter Design

*Fast Algorithms*

**Index Mapping and Mixed-Radix FFTs**

Algebraic DFT versus FFT-Computed DFT

The Role of Index Mapping

The Recursive Equation Approach

Other Forms by Alternate Index Splitting

**Kronecker Product Factorization and FFTs**

Reformulating the Two-Factor Mixed-Radix FFT

From Two-Factor to Multifactor Mixed-Radix FFT

Other Forms by Alternate Index Splitting

Factorization Results by Alternate Expansion

Unordered FFT for Scrambled Input

Utilities of the Kronecker Product Factorization

**The Family of Prime Factor FFT Algorithms**

Connecting the Relevant Ideas

Deriving the Two-Factor PFA

Matrix Formulation of the Two-Factor PFA

Matrix Formulation of the Multifactor PFA

Number Theory and Index Mapping by Permutations

The In-Place and In-Order PFA

Efficient Implementation of the PFA

**On Computing the DFT of Large Prime Length**

Performance of FFT for Prime *N*

Fast Algorithm I: Approximating the FFT

Fast Algorithm II: Using Bluestein’s FFT

**Bibliography**

**Index**

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