CooleyTukey

Cooley–Tukey algorithm, also known as Cooley-Tukey fast Fourier transform (FFT), is an efficient algorithm for calculating the discrete Fourier transform (DFT) of a sequence. The algorithm was independently developed by James Cooley and John Tukey in 1965.

The Cooley-Tukey algorithm is based on the divide-and-conquer approach, where the DFT of a sequence is calculated by breaking it down into smaller DFTs, which are then combined using a set of base cases and the recursive formula. The algorithm starts by dividing the input sequence into smaller subsets, called radix-2 FFTs, which are then combined to calculate the DFT of the original sequence.

The Cooley-Tukey algorithm has a time complexity of O(n log n), making it much faster than the straightforward DFT calculation, which has a time complexity of O(n^2). This is particularly important in many fields, such as signal processing, where large datasets need to be processed quickly.

The Cooley-Tukey algorithm has several advantages over other FFT algorithms, including its stability and robustness, as well as its ability to handle large datasets. However, it also has some limitations, such as requiring the input sequence to be a power of 2 in size, and not being suitable for highly irregular data.

Despite these limitations, the Cooley-Tukey algorithm has become widely used in many applications, including telecommunications, audio processing, and data compression. It is also the basis for other FFT algorithms, such as the mixed radix FFT and the split-radix FFT. Overall, the Cooley-Tukey algorithm is a fundamental component of many digital signal processing techniques.