| FFT1D |
Performs the Fast Fourier Transform (FFT) or Inverse-FFT of a 1-Dimensional discrete complex signal This is a slow CPU version purely designed to test the Fourier transform and for debugging purpose The idea behind the "Fast" Discrete Fourier Transform is that we can recursively subdivide the analyze of a domain into sub-domains that are easier to compute, and re-use the results each stage. It turns a O(N²) computation into log2(N) stages of O(N) computations. Basically, a DFT is: N-1 X_k = Sum[ x_n * e^(-2PI * n/N * k) ] n=0 For k€[0,N[ Assuming a power of two length N, we can separate the even and odd powers and write: N/2-1 X_k = Sum[ x_(2n) * e^(-i * 2PI * (2n)/N * k) ] n=0 N/2-1 + Sum[ x_(2n+1) * e^(-i * 2PI * (2n+1)/N * k) ] n=0 That we rewrite: X_k = E_k + e^(-i * 2PI * k / N) * O_k With: N/2-1 E_k = Sum[ x_(2n) * e^(-i * 2PI * (2n)/N * k) ] n=0 N/2-1 O_k = Sum[ x_(2n+1) * e^(-i * 2PI * (2n)/N * k) ] n=0 Noticing that: e^(-i * 2PI * (2n)/N * (k+N/2)) = e^(-i * 2PI * (2n)/N * k) * e^(-i * 2PI * (2n/N)*(N/2) ) = e^(-i * 2PI * (2n)/N * k) * e^(-i * 2PI * n) = e^(-i * 2PI * (2n)/N * k) E_(k+N/2) = E_k and O_(k+N/2) = O_k Therefore we can rewrite: X_k = E_k + e^(-i * 2PI * k / N) * O_k if 0 <= k < N/2 X_k = E_(k-N/2) + e^(-i * 2PI * k / N) * O_(k-N/2) if N/2 <= k < N (that is the part where we reuse existing results) Noticing the "twiddle factor" also simplifies: e^(-i * 2PI * (k+N/2) / N) = e^(-i * 2PI * k / N) * e^(-i * PI) = -e^(-i * 2PI * k / N) Thus we get the final result for 0 <= k < N/2 : X_k = E_k + e^(-i * 2PI * k / N) * O_k X_(k+N/2) = E_k - e^(-i * 2PI * k / N) * O_k Expressing the DFT of length N recursively in terms of two DFTs of length N/2 allows to reach N*log(N) speeds instead of the traditional N² form. |
