Why are analytic FFT and computational FFT not matching?

1. What are the limitations of FFT?
2. How does FFT improve the speed of computation?
3. What are the two basic classes of FFT algorithm?
4. How accurate is FFT?

What are the limitations of FFT?

A disadvantage associated with the FFT is the restricted range of waveform data that can be transformed and the need to apply a window weighting function (to be defined) to the waveform to compensate for spectral leakage (also to be defined). An alternative to the FFT is the discrete Fourier transform (DFT).

How does FFT improve the speed of computation?

An FFT can be seen as transforming the DFT matrix into multiple matrices which when multiplied together give you the same result, crucially some of the terms in the FFT matricies are 0 so all multiplications with these terms can be omitted meaning you end up with fewer multiplications overall.

What are the two basic classes of FFT algorithm?

There are two main families of FFT algorithms: the Cooley-Tukey algorithm and the Prime Factor algorithm.

How accurate is FFT?

Fast Fourier transform (FFT)-based computations can be far more accurate than the slow transforms suggest. Discrete Fourier transforms computed through the FFT are far more accurate than slow transforms, and convolutions computed via FFT are far more accurate than the direct results.