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C++: Fast Fourier Transform

Bjarne-stroustrup
 

The purpose of this task is to calculate the FFT (Fast Fourier Transform) of an input sequence.
The most general case allows for complex numbers at the input and results in a sequence of equal length, again of complex numbers. If you need to restrict yourself to real numbers, the output should be the magnitude (i.e. sqrt(re²+im²)) of the complex result.

The classic version is the recursive Cooley–Tukey FFT. Wikipedia has pseudocode for that. Further optimizations are possible but not required.

#include <complex>
#include <iostream>
#include <valarray>

const double PI = 3.141592653589793238460;

typedef std::complex<double> Complex;
typedef std::valarray<Complex> CArray;

// Cooley–Tukey FFT (in-place)
void fft(CArray& x)
{
	const size_t N = x.size();
	if (N <= 1) return;

	// divide
	CArray even = x[std::slice(0, N/2, 2)];
	CArray  odd = x[std::slice(1, N/2, 2)];

	// conquer
	fft(even);
	fft(odd);

	// combine
	for (size_t k = 0; k < N/2; ++k)
	{
		Complex t = std::polar(1.0, -2 * PI * k / N) * odd[k];
		x[k    ] = even[k] + t;
		x[k+N/2] = even[k] - t;
	}
}

// inverse fft (in-place)
void ifft(CArray& x)
{
	// conjugate the complex numbers
	x = x.apply(std::conj);

	// forward fft
	fft( x );

	// conjugate the complex numbers again
	x = x.apply(std::conj);

	// scale the numbers
	x /= x.size();
}

int main()
{
	const Complex test[] = { 1.0, 1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0 };
	CArray data(test, 8);

	// forward fft
	fft(data);

	std::cout << "fft" << std::endl;
	for (int i = 0; i < 8; ++i)
	{
		std::cout << data[i] << std::endl;
	}

	// inverse fft
	ifft(data);

	std::cout << std::endl << "ifft" << std::endl;
	for (int i = 0; i < 8; ++i)
	{
		std::cout << data[i] << std::endl;
	}
	return 0;
}
Output:
fft
(4,0)
(1,-2.41421)
(0,0)
(1,-0.414214)
(0,0)
(1,0.414214)
(0,0)
(1,2.41421)

ifft
(1,-0)
(1,-5.55112e-017)
(1,2.4895e-017)
(1,-5.55112e-017)
(5.55112e-017,0)
(5.55112e-017,5.55112e-017)
(0,-2.4895e-017)
(5.55112e-017,5.55112e-017)

SOURCE

Content is available under GNU Free Documentation License 1.2.