A Fast Fourier Transform (FFT) is an efficient algorithm to compute the Discrete Fourier Transform (DFT) and its inverse. There are many distinct FFT algorithms involving a wide range of mathematics, from simple complex-number arithmetic to group theory and number theory. The fast Fourier Transform is a highly efficient procedure for computing the DFT of a finite series and requires less number of computations than that of direct evaluation of DFT. It reduces the computations by taking advantage of the fact that the calculation of the coefficients of the DFT can be carried out iteratively. Due to this, FFT computation technique is used in digital spectral analysis, filter simulation, autocorrelation and pattern recognition.

The FFT is based on decomposition and breaking the transform into smaller transforms and combining them to get the total transform. FFT reduces the computation time required to compute a discrete Fourier transform and improves the performance by a factor of 100 or more over direct evaluation of the DFT.

A DFT decomposes a sequence of values into components of different frequencies. This operation is useful in many fields but computing it directly from the definition is often too slow to be practical. An FFT is a way to compute the same result more quickly: computing a DFT of N points in the obvious way, using the definition, takes O( N2 ) arithmetical operations, while an FFT can compute the same result in only O(N log N) operations.

The difference in speed can be substantial, especially for long data sets where N may be in the thousands or millions—in practice, the computation time can be reduced by several orders of magnitude in such cases, and the improvement is roughly proportional to N /log (N). This huge improvement made many DFT-based algorithms practical. FFT’s are of great importance to a wide variety of applications, from digital signal processing and solving partial...

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