# 24. The Fast Fourier Transform and Applications

## Abstract

*Once the*[

*FFT*]

*method was established it became clear that it had a long and interesting prehistory going back as far as Gauss. But until the advent of computing machines it was a solution looking for a problem*.

*Fourier Analysis*(1988)

*Life as we know it would be very different without the FFT*.

*Computational Frameworks for the Fast Fourier Transform*(1992)

**24.1. The Fast Fourier Transform**

*y*=

*F*

_{n}

*x*, where

*O*(

*n*

^{2}) operations are required. The fast Fourier transform (FFT) is a way to compute

*y*in just

*O*(

*n*log

*n*) operations. This represents a dramatic reduction in complexity.

*F*

_{n}.

**Theorem 24.1**(Cooley-Tukey radix 2 factorization).

*If n*= 2

^{t}

*then the DFT matrix F*

_{n}

*may be factorized as*

*where P*

_{n}

*is a permutation matrix and*

**Proof.**See Van Loan [1182, 1992, Thm. 1.3.3].

*y*=

*F*

_{n}

*x*as

*A*

_{k}(two nonzeros per row) that yields the

*O*(

*n*log

*n*) operation count.

*P*

_{n}in (24.1). However, the way in which the weights ${\omega}_{k}^{j}$ are computed does affect the accuracy. We will assume that computed weights ${\hat{\omega}}_{k}^{j}$ are used that satisfy, for all

*j*and

*k*,

*cu*, μ =

*cu*log

*j*, and μ =

*cuj*, where

*c*is a constant that depends on the method; see Van Loan [1182, 1992, §1.4].

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## Information & Authors

### Information

#### Published In

**ISBN (Print)**: 0898715210

**ISBN (Print)**: 978-0-89871-521-7

**ISBN (Online)**: 978-0-89871-802-7

#### Copyright

#### History

**Published online**: 23 March 2012

#### Keywords

#### Keywords

#### Keywords

#### Keywords

- QA297 .H53 2002Numerical analysis—Data processing, Computer algorithms

### Authors

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