You are here

Discrete and Continuous Fourier Transforms: Analysis, Applications and Fast Algorithms

Eleanor Chu
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
, on

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, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.

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
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