# Fast Fourier Transform Concepts | bartleby

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## Fast Fourier transform

DTF and IDTF computations can be computed using fast Fourier transform algorithms. It was first introduced by Carl Friedrich Gauss in 1805. Later on, it was developed throughout the 19^{th} century. It takes the approach of Danielson-Lanczos Lemma to divide the data sets into successively smaller problems, not by factors of 2, but by whatever small prime factors happen to divide N-point. It is invertible in the form of inverse fast Fourier transform (IFFT).

### FFT Algorithm

#### Cooley–Tukey algorithm

The algorithm is based on the divide and conquer method. The DFT is broken down into smaller sizes. The division of the data sets is based on radix-2 cases where the transform is divided into two smaller pieces at the middle, after every step. The basic concept is of breaking down a DFT into sections that can be considered as micro DFTs.

The division is done to make parts of size N-point=N-1 N-2, in the case of a two-dimensional DFT.

#### Prime factor Fast Fourier Transform

With similarities to mixed radix method of Cooley-Tukey algorithm, Prime factor algorithm (PFA). The algorithm is used when N1 and N2 are relatively prime numbers. PFA required reindexing of the data samples and this property is a disadvantage to the algorithm.

#### Bruun’s Fast Fourier Transform

The algorithm uses the recursive polynomial factorization method to compute the DFT of the input data. It was developed by G. Bruun in 1978 and was modified throughout the 19^{th} century. It is also known as Bruun factorization.

#### Bluestein’s Fast Fourier Transform

It is a type of chirp Z- transform (CZT). It was invented by Leo Bluestein in 1968. It uses the convolution theorem to receive an output.

#### Hexagonal Fast Fourier Transform

With hexagonal sampling, Hexagonal FFT is used on two-dimensional signals. The transform uses the hexagonal efficient coordinate system for efficient outcomes. It was invented by Mersereau. It uses linear transforms to two-dimensional samples of data, to create subarrays.

#### Rader’s Fast Fourier Transform

The cyclic convolution is used in the algorithm to express the divided parts. The algorithm is based on the periodic nature of DFT.

### Applications

The main application of the FFT algorithm is digital signal processing. In 1965, the application of FFT was introduced in O(N log N). Applications range from nuclear tests to national security. Material analysis and processing use FFT as an indispensable part.

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