FFT and Computer Aided Design of Filters

A fast Fourier transform (FFT) is an algorithm that samples a signal over a period of time (or space) and divides it into its frequency components.These components are single sinusoidal oscillations at distinct frequencies each with their own amplitude and phase. This transformation is illustrated in Diagram 1. Over the time period measured in the diagram, the signal contains 3 distinct dominant frequencies.

An FFT algorithm computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IFFT). Fourier analysis converts a signal from its original domain to a representation in the frequency domain and vice versa. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse (mostly zero) factors. As a result, it manages to reduce the complexity of computing the DFT from

${\displaystyle O(n^{2})}$

, which arises if one simply applies the definition of DFT, to

${\displaystyle O(n\log n)}$

, where

${\displaystyle n}$

is the data size.

Fast Fourier transforms are widely used for many applications in engineering, science, and mathematics. The basic ideas were popularized in 1965, but some algorithms had been derived as early as 1805.In 1994, Gilbert Strang described the FFT as "the most important numerical algorithm of our lifetime" and it was included in Top 10 Algorithms of 20th Century by the IEEE journal Computing in Science & Engineering.

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