public static void FullForwardFFT(this ComplexNumber[] data, RealNumber[] temporaryArray = null)
{
int N = data.Length;
int precisionDigits = 1;
for (int i = data.Length; --i >= 0;)
{
precisionDigits = Math.Max(precisionDigits, data[i].GetPrecisionDigits());
}
FFTRealConstants precision = new FFTRealConstants(precisionDigits);
List <int> factors = FourierTransform235.GetSlowFactorization(N).OrderBy(x => x).ToList();
ComplexNumber[] roots = null;
ComplexNumber[] temp = null;
ComplexNumber[] simpleRoots = null;
List <int> factorDigits = new List <int>();
List <int> decimalRepresentation = new List <int>();
List <ComplexNumber> incrementalProduct = new List <ComplexNumber>();
int totalProduct = 1;
int N1 = N;
int count = 1;
foreach (int factor in factors)
{
int N2 = N1;
N1 /= factor;
totalProduct *= factor;
factorDigits.Add(factor);
incrementalProduct.Clear();
int partialProduct = totalProduct;
for (int i = 0; i < factorDigits.Count; i++)
{
incrementalProduct.Add(ComplexNumber.FromPolarAngle((precision.PI << 1) / partialProduct));
partialProduct /= factorDigits[i];
}
ComplexNumber[] exactProduct = new ComplexNumber[factorDigits.Count - 1];
if (factorDigits.Count > 1)
{
exactProduct[factorDigits.Count - 2] = incrementalProduct[factorDigits.Count - 2];
for (int i = factorDigits.Count - 2; --i >= 0;)
{
exactProduct[i] = exactProduct[i + 1] * (incrementalProduct[i] * incrementalProduct[i + 2].Conjugate);
}
}
decimalRepresentation.Clear();
decimalRepresentation.AddRange(Enumerable.Repeat(0, factorDigits.Count - 1));
ComplexNumber exactPower = ComplexNumber.One;
for (int i = 0; i < count; i++)
{
DirectFFTWithStartTwiddle(data, i * N2, N1, N1, factor, precision, ref roots, ref temp, ref simpleRoots, exactPower);
getIncrement(ref exactPower, decimalRepresentation, factorDigits, exactProduct);
}
count = totalProduct;
}
if (temporaryArray == null || temporaryArray.Length < N * 2)
{
Array.Resize(ref temporaryArray, N * 2);
}
for (int i = N; --i >= 0;)
{
var number = data[i];
temporaryArray[i * 2] = number.Real;
temporaryArray[i * 2 + 1] = number.Imaginary;
}
FourierTransform235.ReversedBaseIterate(factors, (i, reversed) => data[reversed] = new ComplexNumber(temporaryArray[i * 2], temporaryArray[i * 2 + 1]));
}
private static void DirectFFTWithStartTwiddle(ComplexNumber[] data, int index, int multiplier, int length, int prime,
FFTRealConstants precision, ref ComplexNumber[] roots, ref ComplexNumber[] temp, ref ComplexNumber[] simpleRoots, ComplexNumber requiredPower)
{
if (roots == null || roots.Length < prime)
{
Array.Resize(ref roots, prime);
}
roots[1] = requiredPower;
for (int i = 2; i < prime; i++)
{
roots[i] = roots[i >> 1] * roots[(i + 1) >> 1];
}
for (; --length >= 0; index++)
{
switch (prime)
{
case 2:
{
var v0 = data[index];
var v1 = data[index + multiplier] * roots[1];
data[index] = v0 + v1;
data[index + multiplier] = v0 - v1;
}
break;
case 3:
//w^2 = -1-w
{
ComplexNumber
z0 = data[index],
z1 = data[index + multiplier] * roots[1],
z2 = data[index + 2 * multiplier] * roots[2];
ComplexNumber
t1 = z1 + z2,
t2 = z0 - (t1 >> 1),
t3 = (z1 - z2).MulI() * precision.M_SQRT_3_4;
data[index] = z0 + t1;
data[index + multiplier] = t2 + t3;
data[index + 2 * multiplier] = t2 - t3;
}
break;
case 5:
//http://www2.itap.physik.uni-stuttgart.de/lehre/vorlesungen/SS08/simmeth/fftalgorithms.pdf
{
ComplexNumber
z0 = data[index],
z1 = data[index + multiplier] * roots[1],
z2 = data[index + 2 * multiplier] * roots[2],
z3 = data[index + 3 * multiplier] * roots[3],
z4 = data[index + 4 * multiplier] * roots[4];
ComplexNumber
t1 = z1 + z4,
t2 = z2 + z3,
t3 = z1 - z4,
t4 = z2 - z3,
t5 = t1 + t2,
t6 = (t1 - t2) * precision.M_ROOT5_C2,
t7 = z0 - (t5 >> 2),
t8 = t7 + t6,
t9 = t7 - t6,
t10 = (t3 * precision.M_ROOT5_C4 + t4 * precision.M_ROOT5_C3).MulI(),
t11 = (t3 * precision.M_ROOT5_C3 - t4 * precision.M_ROOT5_C4).MulI();
data[index] = z0 + t5;
data[index + multiplier] = t8 + t10;
data[index + 2 * multiplier] = t9 + t11;
data[index + 3 * multiplier] = t9 - t11;
data[index + 4 * multiplier] = t8 - t10;
}
break;
default:
if (simpleRoots == null || simpleRoots.Length != prime)
{
Array.Resize(ref simpleRoots, prime);
simpleRoots[1] = ComplexNumber.FromPolarAngle((precision.PI << 1) / prime);
for (int i = 2; i < prime; i++)
{
simpleRoots[i] = simpleRoots[i >> 1] * simpleRoots[(i + 1) >> 1];
}
}
if (temp == null || temp.Length < prime)
{
Array.Resize(ref temp, prime);
}
var sum = temp[0] = data[index];
for (int i = prime; --i > 0;)
{
temp[i] = data[index + i * multiplier] * roots[i];
sum += temp[i];
}
data[index] = sum;
for (int i = prime; --i > 0;)
{
var result = temp[0];
for (int j = prime, k = 0; --j > 0;)
{
k -= i;
k = k < 0 ? k + prime : k; //k = i * j mod prime.
result += temp[j] * simpleRoots[k];
}
data[index + i * multiplier] = result;
}
break;
}
}
}
