Fast Fourier Transform And It's Inverse Transform

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FFT

The Formulas are too complex and the author is too lazy. Please read the references below instead:

2D-FFT Code

Function FFT is fast fourier transform.

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// Author: 围城
typedef struct complex
{  
	ElementType real;  
	ElementType imagin;
} COMPLEX;
void AddComplex(COMPLEX *src1, COMPLEX *src2, COMPLEX *dst)
{
	dst->imagin = src1->imagin+src2->imagin;
	dst->real   = src1->real+src2->real;
}
void SubComplex(COMPLEX *src1, COMPLEX *src2, COMPLEX *dst)
{
	dst->imagin = src1->imagin-src2->imagin;
	dst->real   = src1->real-src2->real;
}
void MultyComplex(COMPLEX *src1, COMPLEX *src2, COMPLEX *dst)
{
	ElementType r1,r2;
	ElementType i1,i2;
	r1=src1->real;
	r2=src2->real;
	i1=src1->imagin;
	i2=src2->imagin;
	// dst->imagin = src1->real * src2->imagin + src2->real   * src1->imagin;
	// dst->real   = src1->real * src2->real   - src1->imagin * src2->imagin;
	dst->imagin=r1*i2+r2*i1;
	dst->real=r1*r2-i1*i2;
}
void CopyComplex(COMPLEX *src, COMPLEX *dst)
{
	dst->real   = src->real;
	dst->imagin = src->imagin;
}
void getWN(Int n, Int fftSize, COMPLEX *dst)
{
	ElementType x = 2.0 * Pi * (ElementType)n / (ElementType)fftSize;
	dst->imagin = -sin(x);
	dst->real   = cos(x);
}
int isBase2(int size)
{
	int k = size;
	int z = 0;
	while (k /= 2) {
		z++;
	}
	if (size != (1<<z)) {
		return  -1;
	}
	else {
		return z;
	}
}
void fftRemap(COMPLEX *src,int size)
{
	if (1 == size) {
		return;
	}
	COMPLEX *temp = (COMPLEX*)malloc(sizeof(COMPLEX)*size);
	for (int i = 0;i < size;i++)
	{
		if(!(i%2)) {
			CopyComplex(src+i, temp+i/2);
		}
		else {
			CopyComplex(src+i, temp+(size+i)/2);
		}
	}
	for (int i = 0;i < size;i++) {
		CopyComplex(temp+i, src+i);
	}
	free(temp);
	fftRemap(src, size/2);
	fftRemap(src+size/2, size/2);
	return;
}
void RealFFT(ElementType *src, COMPLEX *dst, int size)
{
	int k = size;
	int z = 0;
	while (k /= 2) {
		z++;
	}
	k = z;
	if (size != (1<<k)) {
		exit(0);
	}
	COMPLEX *srcCpy = (COMPLEX*)malloc(sizeof(COMPLEX)*size);
	if (NULL == srcCpy) {
		exit(0);
	}
	for (int i = 0;i < size;i++)
	{
		(srcCpy+i)->real   = *(src+i);
		(srcCpy+i)->imagin = 0;
	}
	fftRemap(srcCpy, size);
	for (int i = 0;i < k;i++)
	{
		z = 0;
		for (int j = 0;j < size;j++)
		{
			if (1 == (j/(1<<i))%2)
			{
				COMPLEX wn;
				getWN(z, size, &wn);
				z += 1 << (k-i-1);
				int neighbour = j - (1<<i);
				COMPLEX temp;
				temp.real   = (srcCpy+neighbour)->real;
				temp.imagin = (srcCpy+neighbour)->imagin;
				MultyComplex(srcCpy+j, &wn, srcCpy+j);
				AddComplex(&temp, srcCpy+j, srcCpy+neighbour);
				SubComplex(&temp, srcCpy+j, srcCpy+j);
			}
			else {
				z = 0;
			}
		}
	}
	for (int i = 0;i < size;i++) {
		CopyComplex(srcCpy+i, dst+i);
	}
	free(srcCpy);
}
void ColumnVector(COMPLEX *src, COMPLEX *dst, int fftWidthSize, int fftHeightSize)
{
	for (int row = 0;row < fftHeightSize;row++) {
		CopyComplex(src+fftWidthSize*row, dst+row);
	}
}
void FFT(COMPLEX *src, COMPLEX *dst, int size)
{
	int k = size;
	int z = 0;
	while (k /= 2) {
		z++;
	}
	k = z;
	if (size != (1<<k)) {
		exit(0);
	}
	COMPLEX *srcCpy=(COMPLEX*)malloc(sizeof(COMPLEX)*size);
	if (NULL == srcCpy) {
		exit(0);
	}
	for (int i = 0;i < size;i++) {
		CopyComplex(src+i, srcCpy+i);
	}
	fftRemap(srcCpy, size);
	for (int i = 0;i < k;i++)
	{
		z = 0;
		for (int j = 0;j < size;j++)
		{
			if (1 == (j/(1<<i))%2)
			{
				COMPLEX wn;
				getWN(z, size, &wn);
				z += 1 << (k-i-1);
				int neighbour = j - (1<<i);
				COMPLEX temp;
				temp.real   = (srcCpy+neighbour)->real;
				temp.imagin = (srcCpy+neighbour)->imagin;
				MultyComplex(srcCpy+j, &wn,srcCpy+j);
				AddComplex(&temp, srcCpy+j, srcCpy+neighbour);
				SubComplex(&temp, srcCpy+j, srcCpy+j);
			}
			else {
				z = 0;
			}
		}
	}
	for (int i = 0;i < size;i++) {
		CopyComplex(srcCpy+i, dst+i);
	}
	free(srcCpy);
}
void IColumnVector(COMPLEX *src, COMPLEX *dst, int fftWidthSize, int fftHeightSize)
{
	for (int row = 0;row < fftHeightSize;row++) {
		CopyComplex(src+row, dst+fftWidthSize*row);
	}
}
int FFT2D(ElementType *src, COMPLEX *dst, int fftWidthSize, int fftHeightSize)
{
	if (-1 == isBase2(fftWidthSize) || -1 == isBase2(fftHeightSize)) {
		exit(0);
	}
	COMPLEX *row = (COMPLEX*)malloc(sizeof(COMPLEX)*fftHeightSize*fftWidthSize);
	if (NULL == row) {
		return -1;
	}
	for (int i = 0;i < fftHeightSize;i++) {
		RealFFT(src+fftWidthSize*i, row+fftWidthSize*i, fftWidthSize);
	}
	COMPLEX *column = (COMPLEX*)malloc(sizeof(COMPLEX)*fftHeightSize);
	if (NULL == column) {
		return -1;
	}
	for (int i = 0;i < fftWidthSize;i++)
	{
		ColumnVector(row+i, column, fftWidthSize, fftHeightSize);
		FFT(column, column, fftHeightSize);
		IColumnVector(column, row+i, fftWidthSize, fftHeightSize);
	}
	for (int i = 0;i < fftHeightSize*fftWidthSize;i++) {
		CopyComplex(row+i, dst+i);
	}
	free(row);
	free(column);
	return 0;
}

IFFT

references:

The source of 2D-IFFT is rare. Readers might understand 2D-IFFT by reading author’s code.

2D-IFFT Code

Function IFFT is inverse fast fourier transform.

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// Author: 围城
typedef struct complex
{  
	ElementType real;  
	ElementType imagin;
} COMPLEX;
void AddComplex(COMPLEX *src1, COMPLEX *src2, COMPLEX *dst)
{
	dst->imagin = src1->imagin+src2->imagin;
	dst->real   = src1->real+src2->real;
}
void SubComplex(COMPLEX *src1, COMPLEX *src2, COMPLEX *dst)
{
	dst->imagin = src1->imagin-src2->imagin;
	dst->real   = src1->real-src2->real;
}
void MultyComplex(COMPLEX *src1, COMPLEX *src2, COMPLEX *dst)
{
	ElementType r1,r2;
	ElementType i1,i2;
	r1=src1->real;
	r2=src2->real;
	i1=src1->imagin;
	i2=src2->imagin;
	// dst->imagin = src1->real * src2->imagin + src2->real   * src1->imagin;
	// dst->real   = src1->real * src2->real   - src1->imagin * src2->imagin;
	dst->imagin=r1*i2+r2*i1;
	dst->real=r1*r2-i1*i2;
}
void CopyComplex(COMPLEX *src, COMPLEX *dst)
{
	dst->real   = src->real;
	dst->imagin = src->imagin;
}
void getWN(Int n, Int fftSize, COMPLEX *dst)
{
	ElementType x = 2.0 * Pi * (ElementType)n / (ElementType)fftSize;
	dst->imagin = -sin(x);
	dst->real   = cos(x);
}
int isBase2(int size)
{
	int k = size;
	int z = 0;
	while (k /= 2) {
		z++;
	}
	if (size != (1<<z)) {
		return  -1;
	}
	else {
		return z;
	}
}
void fftRemap(COMPLEX *src,int size)
{
	if (1 == size) {
		return;
	}
	COMPLEX *temp = (COMPLEX*)malloc(sizeof(COMPLEX)*size);
	for (int i = 0;i < size;i++)
	{
		if(!(i%2)) {
			CopyComplex(src+i, temp+i/2);
		}
		else {
			CopyComplex(src+i, temp+(size+i)/2);
		}
	}
	for (int i = 0;i < size;i++) {
		CopyComplex(temp+i, src+i);
	}
	free(temp);
	fftRemap(src, size/2);
	fftRemap(src+size/2, size/2);
	return;
}
void ColumnVector(COMPLEX *src, COMPLEX *dst, int fftWidthSize, int fftHeightSize)
{
	for (int row = 0;row < fftHeightSize;row++) {
		CopyComplex(src+fftWidthSize*row, dst+row);
	}
}
void IFFT(COMPLEX *src, COMPLEX *dst, int size)
{
	for (int i = 0;i < size;i++) {
		(src+i)->imagin = -(src+i)->imagin;
	}
	fftRemap(src, size);
	int z, k;
	if (-1 != (z=isBase2(size))) {
		k = isBase2(size);
	}
	else {
		exit(0);
	}
	for (int i = 0;i < k;i++)
	{
		z = 0;
		for (int j = 0;j < size;j++)
		{
			if (1 == (j/(1<<i))%2)
			{
				COMPLEX wn;
				getWN(z, size, &wn);
				z += 1 << (k-i-1);
				int neighbour = j - (1<<i);
				COMPLEX temp;
				CopyComplex(src+neighbour, &temp);
				MultyComplex(src+j, &wn, src+j);
				AddComplex(&temp, src+j, src+neighbour);
				SubComplex(&temp, src+j, src+j);
			}
			else {
				z = 0;
			}
		}
	}
	for (int i = 0;i < size;i++)
	{
		(dst+i)->imagin = (1./size) * (src+i)->imagin;
		(dst+i)->real   = (1./size) * (src+i)->real;
	}
}
void IColumnVector(COMPLEX *src, COMPLEX *dst, int fftWidthSize, int fftHeightSize)
{
	for (int row = 0;row < fftHeightSize;row++) {
		CopyComplex(src+row, dst+fftWidthSize*row);
	}
}
int IFFT2D(COMPLEX *src, COMPLEX *dst, int fftWidthSize, int fftHeightSize)
{
	if (-1 == isBase2(fftWidthSize) || -1 == isBase2(fftHeightSize)) {
		exit(0);
	}
	COMPLEX *row=(COMPLEX*)malloc(sizeof(COMPLEX)*fftHeightSize*fftWidthSize);
	if (NULL == row) {
		return -1;
	}
	COMPLEX *column=(COMPLEX*)malloc(sizeof(COMPLEX)*fftHeightSize);
	if (NULL == column) {
		return -1;
	}
	for(int i = 0;i < fftWidthSize;i++)
	{
		ColumnVector(src+i, column, fftWidthSize, fftHeightSize);
		IFFT(column, column, fftHeightSize);
		IColumnVector(column, src+i, fftWidthSize, fftHeightSize);
	}
	for (int i = 0;i < fftWidthSize*fftHeightSize;i++) {
		(src+i)->imagin = -(src+i)->imagin;
	}
	for (int i = 0;i < fftHeightSize;i++) {
		IFFT(src+fftWidthSize*i, row+fftWidthSize*i, fftWidthSize);
	}
	for (int i = 0;i < fftHeightSize*fftWidthSize;i++)
	{
		CopyComplex(row+i, dst+i);
		// *(dst+i) = (row+i)->real;
	}
	free(row);
	free(column);
	return 0;
}