This post details about Why is FFT fast?, What makes FFT faster than DFT?, How much faster is FFT?

## Why is FFT fast?

FFT, or fast Fourier transform, is quickly primarily due to its algorithmic structure and use of mathematical properties that reduce the number of operations required to calculate the discrete Fourier transform (DFT) of a sequence.

Unlike traditional DFT, which calculates each frequency component individually with complexity O(N2)O(N^2)O(N2), FFT exploits the divide-and-conquer strategy as well as symmetries and periodicities in the equations of Fourier transformation. This approach significantly reduces the number of calculations required, making FFT much faster than directly calculating DFT for large sequences of data points.

### What makes FFT faster than DFT?

FFT is faster than direct calculation of DFT mainly because of its computational efficiency in terms of time complexity.

Direct calculation of DFT involves O(N2)O(N^2)O(N2) operations, where Nnn is the number of data points in the sequence. In contrast, FFT reduces this complexity to o(nlogn)o(n log n)o(nlogn), which is a significant improvement for large nnn. This efficiency comes from the recursive division of the DFT calculation into smaller subproblems, combined with the use of complex roots of unity and symmetric properties of the Fourier transform.

### How much faster is FFT?

The speedup obtained by FFT compared to direct DFT calculation depends on the size of the NNN input sequence.

For a large NNN, FFT can be orders of magnitude faster than the direct method. Reducing the computational complexity from o(n2)o(n^2)o(n2) to o(nlogn)o(n log n)o(nlogn) means that the FFT becomes faster as Nnn increase. For example, for n = 1024n = 1024n = 1024, FFT can be about 100 times faster than direct DFT calculation, and the speed advantage becomes even more pronounced as NNN gets larger.

FFT is more computationally efficient mainly because of its reduced number of operations required to calculate the Fourier transform of a sequence.

This efficiency is achieved by decomposing the DFT calculation into smaller, simpler subproblems and exploiting mathematical properties such as symmetry and periodicity in the frequency domain. By decomposing the calculation recursively and using efficient algorithms like Cooley-Tukey FFT or other variants, FFT achieves optimal performance suitable for real-time processing and high-speed applications in various areas of signal processing, communications, audio processing and scientific computing.

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