Computing the Fast Fourier Transform

Problem

You want to compute the Discrete Fourier Transform (DFT) efficiently using the Fast Fourier Transform (FFT) algorithm.

Solution

The code in Example 11-33 provides a basic implementation of the FFT.

Example 11-33. FFT implementation

#include #include #include #include using namespace std; unsigned int bitReverse(unsigned int x, int log2n) { int n = 0; int mask = 0x1; for (int i=0; i < log2n; i++) { n <<= 1; n |= (x & 1); x >>= 1; } return n; } const double PI = 3.1415926536; template void fft(Iter_T a, Iter_T b, int log2n) { typedef typename iterator_traits::value_type complex; const complex J(0, 1); int n = 1 << log2n; for (unsigned int i=0; i < n; ++i) { b[bitReverse(i, log2n)] = a[i]; } for (int s = 1; s <= log2n; ++s) { int m = 1 << s; int m2 = m >> 1; complex w(1, 0); complex wm = exp(-J * (PI / m2)); for (int j=0; j < m2; ++j) { for (int k=j; k < n; k += m) { complex t = w * b[k + m2]; complex u = b[k]; b[k] = u + t; b[k + m2] = u - t; } w *= wm; } } } int main( ) { typedef complex cx; cx a[] = { cx(0,0), cx(1,1), cx(3,3), cx(4,4), cx(4, 4), cx(3, 3), cx(1,1), cx(0,0) }; cx b[8]; fft(a, b, 3); for (int i=0; i<8; ++i) cout << b[i] << " "; }

The program in Example 11-33 produces the following output:

(16,16) (-4.82843,-11.6569) (0,0) (-0.343146,0.828427) (0,0) (0.828427,-0.343146) (0,0) (-11.6569,-4.82843)

Discussion

The Fourier transform is an important equation for spectral analysis, and is required frequently in engineering and scientific applications. The FFT is an algorithm for computing a DFT that operates in N log2(N) complexity versus the expected N2 complexity of a naive implementation of a DFT. The FFT achieves such an impressive speed-up by removing redundant computations.

Finding a good FFT implementation written in idiomatic C++ (i.e., C++ that isn't mechanically ported from old Fortran or C algorithms) and that isn't severely restricted by a license is very hard. The code in Example 11-33 is based on public domain code that can be found on the digital signal processing newswgoup on usenet (comp.dsp). A big advantage of an idiomatic C++ solution over the more common C-style FFT implementations is that the standard library provides the complex template that significantly reduces the amount of code needed. The fft( ) function in Example 11-33, was written to be as simple as possible rather than focusing on efficiency.