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 <iostream>
#include <complex>
#include <cmath>
#include <iterator>

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<class Iter_T>
void fft(Iter_T a, Iter_T b, int log2n)
{
  typedef typename iterator_traits<Iter_T>::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<double> 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] << "\n";
}

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.