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fft.cpp
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fft.cpp
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/*
Radix-2 Out-of-Place DIT FFT Algorithm for 1D Real Input
TODO:
- implement IDFT
- replace complex number operations with macros
- add SIMD build support
*/
#include <complex>
#include <math.h>
using std::conj;
using std::cos;
using std::sin;
typedef double real;
typedef std::complex<real> complex;
const complex i = complex(0.0, 1.0);
const double pi=3.141592653589793238462643383279502884197169399375105820974944;
/*
Calculates twiddle factors (complex roots of unity) for an N/2-point DFT
and the associated post-processing for a real-valued 1D input
*/
complex* calculate_fft_twiddles(int N) {
complex* twiddles = new complex[N];
double phase;
for (int k=0; k<N; k++) {
phase = -2.0 * pi * k / N;
twiddles[k] = complex(cos(phase), sin(phase));
}
return twiddles;
}
/*
Combine the outputs of two DFTs
*/
void r2_butterfly(
complex* twiddles,
complex* output,
int stride,
int m
) {
complex* output2 = output + m;
complex t;
do {
t = *output2 * *twiddles;
*output2 = *output - t;
*output += t;
twiddles += 2 * stride;
output++;
output2++;
} while (--m);
}
/*
Radix-2 Cooley-Tukey FFT
*/
void fft_recursive(
complex* twiddles,
const complex* input,
int n,
complex* output,
int stride
) {
int m = n/2;
if (m == 1) {
output[0] = input[0];
output[1] = input[stride];
} else {
fft_recursive(twiddles, input, m, output, 2*stride);
fft_recursive(twiddles, input+stride, m, output+m, 2*stride);
}
r2_butterfly(twiddles, output, stride, m);
}
/*
Collapses real input of size N into complex sequence of size N/2,
calculates the N/2-point DFT, then extracts the N-point DFT of the
original N-point real input sequence
*/
void fft(
complex* twiddles,
const real* input,
int N,
complex* output
) {
// collapse real input into N/2-point complex sequence
complex* input_complex = new complex[N];
for (int k=0; k<N/2; k++) {
input_complex[k] = complex(input[2 * k], input[2 * k + 1]);
}
// perform N/2-point FFT on complex sequence
fft_recursive(twiddles, input_complex, N/2, output, 1);
// derive N-point DFT of input data from N/2-point DFT
output[N/2] = output[0].real() - output[0].imag();
output[0] = output[0].real() + output[0].imag();
output[3*N/4] = output[N/4];
output[N/4] = conj(output[N/4]);
complex T, Tc, c3, c4, c5;
for (int k=1; k<N/4; k++) {
T = output[k];
Tc = conj(output[N/2 - k]);
c3 = T + Tc;
c4 = T - Tc;
c5 = i * twiddles[k] * c4;
output[k] = 0.5 * (c3 - c5);
output[N/2 - k] = conj(0.5 * (c3 + c5));
output[N - k] = conj(output[k]);
output[N/2 + k] = conj(output[N/2 - k]);
}
}