Merge pull request #48 from dmpelt/cpu_fft_fix

Change CPU FFT implementation

3 files changed, 3382 insertions, 309 deletions

diff --git a/include/astra/Fourier.h b/include/astra/Fourier.h
index b515dc6..ff26f82 100644
--- a/include/astra/Fourier.h
+++ b/include/astra/Fourier.h
@@ -33,94 +33,50 @@ $Id$
 
 namespace astra {
 
-
-/**
- * Perform a 1D DFT or inverse DFT.
- *
- * @param iLength number of elements
- * @param pfRealIn real part of input
- * @param pfImaginaryIn imaginary part of input
- * @param pfRealOut real part of output
- * @param pfImaginaryOut imaginary part of output
- * @param iStrideIn distance between elements in pf*In
- * @param iStrideOut distance between elements in pf*Out
- * @param bInverse if true, perform an inverse DFT
- */
-
-void _AstraExport discreteFourierTransform1D(unsigned int iLength,
-                                const float32* pfRealIn,
-                                const float32* pfImaginaryIn,
-                                float32* pfRealOut,
-                                float32* pfImaginaryOut,
-                                unsigned int iStrideIn,
-                                unsigned int iStrideOut,
-                                bool bInverse);
-
-/**
- * Perform a 2D DFT or inverse DFT.
- *
- * @param iHeight number of rows
- * @param iWidth number of columns
- * @param pfRealIn real part of input
- * @param pfImaginaryIn imaginary part of input
- * @param pfRealOut real part of output
- * @param pfImaginaryOut imaginary part of output
- * @param bInverse if true, perform an inverse DFT
- */
-
-void _AstraExport discreteFourierTransform2D(unsigned int iHeight, unsigned int iWidth,
-                                const float32* pfRealIn,
-                                const float32* pfImaginaryIn,
-                                float32* pfRealOut,
-                                float32* pfImaginaryOut,
-                                bool bInverse);
-
-/**
- * Perform a 1D FFT or inverse FFT. The size must be a power of two.
- * This transform can be done in-place, so the input and output pointers
- * may point to the same data.
- *
- * @param iLength number of elements, must be a power of two
- * @param pfRealIn real part of input
- * @param pfImaginaryIn imaginary part of input
- * @param pfRealOut real part of output
- * @param pfImaginaryOut imaginary part of output
- * @param iStrideIn distance between elements in pf*In
- * @param iStrideOut distance between elements in pf*Out
- * @param bInverse if true, perform an inverse DFT
- */
-
-void _AstraExport fastTwoPowerFourierTransform1D(unsigned int iLength,
-                                    const float32* pfRealIn,
-                                    const float32* pfImaginaryIn,
-                                    float32* pfRealOut,
-                                    float32* pfImaginaryOut,
-                                    unsigned int iStrideIn,
-                                    unsigned int iStrideOut,
-                                    bool bInverse);
-
-/**
- * Perform a 2D FFT or inverse FFT. The size must be a power of two.
- * This transform can be done in-place, so the input and output pointers
- * may point to the same data.
- *
- * @param iHeight number of rows, must be a power of two
- * @param iWidth number of columns, must be a power of two
- * @param pfRealIn real part of input
- * @param pfImaginaryIn imaginary part of input
- * @param pfRealOut real part of output
- * @param pfImaginaryOut imaginary part of output
- * @param bInverse if true, perform an inverse DFT
- */
-
-void _AstraExport fastTwoPowerFourierTransform2D(unsigned int iHeight,
-                                    unsigned int iWidth,
-                                    const float32* pfRealIn,
-                                    const float32* pfImaginaryIn,
-                                    float32* pfRealOut,
-                                    float32* pfImaginaryOut,
-                                    bool bInverse);
-
+/*
+-------- Complex DFT (Discrete Fourier Transform) --------
+    [definition]
+        <case1>
+            X[k] = sum_j=0^n-1 x[j]*exp(2*pi*i*j*k/n), 0<=k<n
+        <case2>
+            X[k] = sum_j=0^n-1 x[j]*exp(-2*pi*i*j*k/n), 0<=k<n
+        (notes: sum_j=0^n-1 is a summation from j=0 to n-1)
+    [usage]
+        <case1>
+            ip[0] = 0; // first time only
+            cdft(2*n, 1, a, ip, w);
+        <case2>
+            ip[0] = 0; // first time only
+            cdft(2*n, -1, a, ip, w);
+    [parameters]
+        2*n            :data length (int)
+                        n >= 1, n = power of 2
+        a[0...2*n-1]   :input/output data (float32 *)
+                        input data
+                            a[2*j] = Re(x[j]), 
+                            a[2*j+1] = Im(x[j]), 0<=j<n
+                        output data
+                            a[2*k] = Re(X[k]), 
+                            a[2*k+1] = Im(X[k]), 0<=k<n
+        ip[0...*]      :work area for bit reversal (int *)
+                        length of ip >= 2+sqrt(n)
+                        strictly, 
+                        length of ip >= 
+                            2+(1<<(int)(log(n+0.5)/log(2))/2).
+                        ip[0],ip[1] are pointers of the cos/sin table.
+        w[0...n/2-1]   :cos/sin table (float32 *)
+                        w[],ip[] are initialized if ip[0] == 0.
+    [remark]
+        Inverse of 
+            cdft(2*n, -1, a, ip, w);
+        is 
+            cdft(2*n, 1, a, ip, w);
+            for (j = 0; j <= 2 * n - 1; j++) {
+                a[j] *= 1.0 / n;
+            }
+        .
+*/
+void cdft(int n, int isgn, float32 *a, int *ip, float32 *w);
 
 }
 
diff --git a/src/FilteredBackProjectionAlgorithm.cpp b/src/FilteredBackProjectionAlgorithm.cpp
index ccbfec6..90efd52 100644
--- a/src/FilteredBackProjectionAlgorithm.cpp
+++ b/src/FilteredBackProjectionAlgorithm.cpp
@@ -274,60 +274,57 @@ void CFilteredBackProjectionAlgorithm::performFiltering(CFloat32ProjectionData2D
 		filter[iDetector] = (2.0f * (zpDetector - iDetector)) / zpDetector;
 
 
-	float32* pfRe = new float32[iAngleCount * zpDetector];
-	float32* pfIm = new float32[iAngleCount * zpDetector];
+	float32* pf = new float32[2 * iAngleCount * zpDetector];
+	int *ip = new int[int(2+sqrt(zpDetector)+1)];
+	ip[0]=0;
+	float32 *w = new float32[zpDetector/2];
 
 	// Copy and zero-pad data
 	for (int iAngle = 0; iAngle < iAngleCount; ++iAngle) {
-		float32* pfReRow = pfRe + iAngle * zpDetector;
-		float32* pfImRow = pfIm + iAngle * zpDetector;
+		float32* pfRow = pf + iAngle * 2 * zpDetector;
 		float32* pfDataRow = _pFilteredSinogram->getData() + iAngle * iDetectorCount;
 		for (int iDetector = 0; iDetector < iDetectorCount; ++iDetector) {
-			pfReRow[iDetector] = pfDataRow[iDetector];
-			pfImRow[iDetector] = 0.0f;
+			pfRow[2*iDetector] = pfDataRow[iDetector];
+			pfRow[2*iDetector+1] = 0.0f;
 		}
 		for (int iDetector = iDetectorCount; iDetector < zpDetector; ++iDetector) {
-			pfReRow[iDetector] = 0.0f;
-			pfImRow[iDetector] = 0.0f;
+			pfRow[2*iDetector] = 0.0f;
+			pfRow[2*iDetector+1] = 0.0f;
 		}
 	}
 
 	// in-place FFT
 	for (int iAngle = 0; iAngle < iAngleCount; ++iAngle) {
-		float32* pfReRow = pfRe + iAngle * zpDetector;
-		float32* pfImRow = pfIm + iAngle * zpDetector;
-
-		fastTwoPowerFourierTransform1D(zpDetector, pfReRow, pfImRow, pfReRow, pfImRow, 1, 1, false);
+		float32* pfRow = pf + iAngle * 2 * zpDetector;
+		cdft(2*zpDetector, -1, pfRow, ip, w);
 	}
 
 	// Filter
 	for (int iAngle = 0; iAngle < iAngleCount; ++iAngle) {
-		float32* pfReRow = pfRe + iAngle * zpDetector;
-		float32* pfImRow = pfIm + iAngle * zpDetector;
+		float32* pfRow = pf + iAngle * 2 * zpDetector;
 		for (int iDetector = 0; iDetector < zpDetector; ++iDetector) {
-			pfReRow[iDetector] *= filter[iDetector];
-			pfImRow[iDetector] *= filter[iDetector];
+			pfRow[2*iDetector] *= filter[iDetector];
+			pfRow[2*iDetector+1] *= filter[iDetector];
 		}
 	}
 
 	// in-place inverse FFT
 	for (int iAngle = 0; iAngle < iAngleCount; ++iAngle) {
-		float32* pfReRow = pfRe + iAngle * zpDetector;
-		float32* pfImRow = pfIm + iAngle * zpDetector;
-
-		fastTwoPowerFourierTransform1D(zpDetector, pfReRow, pfImRow, pfReRow, pfImRow, 1, 1, true);
+		float32* pfRow = pf + iAngle * 2 * zpDetector;
+		cdft(2*zpDetector, 1, pfRow, ip, w);
 	}
 
 	// Copy data back
 	for (int iAngle = 0; iAngle < iAngleCount; ++iAngle) {
-		float32* pfReRow = pfRe + iAngle * zpDetector;
+		float32* pfRow = pf + iAngle * 2 * zpDetector;
 		float32* pfDataRow = _pFilteredSinogram->getData() + iAngle * iDetectorCount;
 		for (int iDetector = 0; iDetector < iDetectorCount; ++iDetector)
-			pfDataRow[iDetector] = pfReRow[iDetector];
+			pfDataRow[iDetector] = pfRow[2*iDetector] / zpDetector;
 	}
 
-	delete[] pfRe;
-	delete[] pfIm;
+	delete[] pf;
+	delete[] w;
+	delete[] ip;
 	delete[] filter;
 }
 
diff --git a/src/Fourier.cpp b/src/Fourier.cpp
index 584b633..5ca22e6 100644
--- a/src/Fourier.cpp
+++ b/src/Fourier.cpp
@@ -30,204 +30,3324 @@ $Id$
 
 namespace astra {
 
+    /*
+Copyright Takuya OOURA, 1996-2001
 
-void discreteFourierTransform1D(unsigned int iLength,
-                                const float32* pfRealIn,
-                                const float32* pfImaginaryIn,
-                                float32* pfRealOut,
-                                float32* pfImaginaryOut,
-                                unsigned int iStrideIn,
-                                unsigned int iStrideOut,
-                                bool inverse)
-{
-	for (unsigned int w = 0; w < iLength; w++)
-	{
-		pfRealOut[iStrideOut*w] = pfImaginaryOut[iStrideOut*w] = 0;
-		for (unsigned int y = 0; y < iLength; y++)
-		{
-			float32 a = 2 * PI * w * y / float32(iLength);
-			if (!inverse)
-				a = -a;
-			float32 ca = cos(a);
-			float32 sa = sin(a);
-			pfRealOut[iStrideOut*w] += pfRealIn[iStrideIn*y] * ca - pfImaginaryIn[iStrideIn*y] * sa;
-			pfImaginaryOut[iStrideOut*w] += pfRealIn[iStrideIn*y] * sa + pfImaginaryIn[iStrideIn*y] * ca;   
-		}
-	}
-
-	if (inverse) {
-		for (unsigned int x = 0; x < iLength; ++x) {
-			pfRealOut[iStrideOut*x] /= iLength;
-			pfImaginaryOut[iStrideOut*x] /= iLength;
-		}
-	}
-}
-
-void discreteFourierTransform2D(unsigned int iHeight, unsigned int iWidth,
-                                const float32* pfRealIn,
-                                const float32* pfImaginaryIn,
-                                float32* pfRealOut,
-                                float32* pfImaginaryOut,
-                                bool inverse)
-{
-	float32* reTemp = new float32[iWidth * iHeight];
-	float32* imTemp = new float32[iWidth * iHeight];
-
-	//calculate the fourier transform of the columns
-	for (unsigned int x = 0; x < iWidth; x++)
-	{
-		discreteFourierTransform1D(iHeight, pfRealIn+x, pfImaginaryIn+x,
-		                           reTemp+x, imTemp+x,
-		                           iWidth, iWidth, inverse);
-	}
-
-	//calculate the fourier transform of the rows
-	for(unsigned int y = 0; y < iHeight; y++)
-	{
-		discreteFourierTransform1D(iWidth,
-		                           reTemp+y*iWidth,
-		                           imTemp+y*iWidth,
-		                           pfRealOut+y*iWidth,
-		                           pfImaginaryOut+y*iWidth,
-		                           1, 1, inverse);
-	}
-
-	delete[] reTemp;
-	delete[] imTemp;
-}
-
-/** permute the entries from pfDataIn into pfDataOut to prepare for an
- *  in-place FFT. pfDataIn may be equal to pfDataOut.
- */
-static void bitReverse(unsigned int iLength,
-                       const float32* pfDataIn, float32* pfDataOut,
-                       unsigned int iStrideShiftIn,
-                       unsigned int iStrideShiftOut)
-{
-	if (pfDataIn == pfDataOut) {
-		assert(iStrideShiftIn == iStrideShiftOut);
-		float32 t;
-		unsigned int j = 0;
-		for(unsigned int i = 0; i < iLength - 1; i++) {
-			if (i < j) {
-				t = pfDataOut[i<<iStrideShiftOut];
-				pfDataOut[i<<iStrideShiftOut] = pfDataOut[j<<iStrideShiftOut];
-				pfDataOut[j<<iStrideShiftOut] = t;
-			}
-			unsigned int k = iLength / 2;
-			while (k <= j) {
-				j -= k;
-				k /= 2;
-			}
-			j += k;
-		}
-	} else {
-		unsigned int j = 0;
-		for(unsigned int i = 0; i < iLength - 1; i++) {
-			pfDataOut[i<<iStrideShiftOut] = pfDataIn[j<<iStrideShiftIn];
-			unsigned int k = iLength / 2;
-			while (k <= j) {
-				j -= k;
-				k /= 2;
-			}
-			j += k;
-		}
-		pfDataOut[(iLength-1)<<iStrideShiftOut] = pfDataIn[(iLength-1)<<iStrideShiftOut];
-	}
-}
-
-static unsigned int log2(unsigned int n)
-{
-	unsigned int l = 0;
-	while (n > 1) {
-		n /= 2;
-		++l;
-	}
-	return l;
-}
-
-/** perform 1D FFT. iLength, iStrideIn, iStrideOut must be powers of two. */
-void fastTwoPowerFourierTransform1D(unsigned int iLength,
-                                    const float32* pfRealIn,
-                                    const float32* pfImaginaryIn,
-                                    float32* pfRealOut,
-                                    float32* pfImaginaryOut,
-                                    unsigned int iStrideIn,
-                                    unsigned int iStrideOut,
-                                    bool inverse)
-{
-	unsigned int iStrideShiftIn = log2(iStrideIn);
-	unsigned int iStrideShiftOut = log2(iStrideOut);
-	unsigned int iLogLength = log2(iLength);
-
-	bitReverse(iLength, pfRealIn, pfRealOut, iStrideShiftIn, iStrideShiftOut);
-	bitReverse(iLength, pfImaginaryIn, pfImaginaryOut, iStrideShiftIn, iStrideShiftOut);
-
-	float32 ca = -1.0;
-	float32 sa = 0.0;
-	unsigned int l1 = 1, l2 = 1;
-	for(unsigned int l=0; l < iLogLength; ++l)
-	{
-		l1 = l2;
-		l2 *= 2;
-		float32 u1 = 1.0;
-		float32 u2 = 0.0;
-		for(unsigned int j = 0; j < l1; j++)
-		{
-			for(unsigned int i = j; i < iLength; i += l2)
-			{
-				unsigned int i1 = i + l1;
-				float32 t1 = u1 * pfRealOut[i1<<iStrideShiftOut] - u2 * pfImaginaryOut[i1<<iStrideShiftOut];
-				float32 t2 = u1 * pfImaginaryOut[i1<<iStrideShiftOut] + u2 * pfRealOut[i1<<iStrideShiftOut];
-				pfRealOut[i1<<iStrideShiftOut] = pfRealOut[i<<iStrideShiftOut] - t1;
-				pfImaginaryOut[i1<<iStrideShiftOut] = pfImaginaryOut[i<<iStrideShiftOut] - t2;
-				pfRealOut[i<<iStrideShiftOut] += t1;
-				pfImaginaryOut[i<<iStrideShiftOut] += t2;
-			}
-			float32 z =  u1 * ca - u2 * sa;
-			u2 = u1 * sa + u2 * ca;
-			u1 = z;
-		}
-		sa = sqrt((1.0 - ca) / 2.0);
-		if (!inverse) 
-			sa = -sa;
-		ca = sqrt((1.0 + ca) / 2.0);
-	}
-
-	if (inverse) {
-		for (unsigned int i = 0; i < iLength; ++i) {
-			pfRealOut[i<<iStrideShiftOut] /= iLength;
-			pfImaginaryOut[i<<iStrideShiftOut] /= iLength;
-		}
-	}
-}
-
-void fastTwoPowerFourierTransform2D(unsigned int iHeight,
-                                    unsigned int iWidth,
-                                    const float32* pfRealIn,
-                                    const float32* pfImaginaryIn,
-                                    float32* pfRealOut,
-                                    float32* pfImaginaryOut,
-                                    bool inverse)
-{
-	//calculate the fourier transform of the columns
-	for (unsigned int x = 0; x < iWidth; x++)
-	{
-		fastTwoPowerFourierTransform1D(iHeight, pfRealIn+x, pfImaginaryIn+x,
-		                               pfRealOut+x, pfImaginaryOut+x,
-		                               iWidth, iWidth, inverse);
-	}
-
-	//calculate the fourier transform of the rows
-	for (unsigned int y = 0; y < iHeight; y++)
-	{
-		fastTwoPowerFourierTransform1D(iWidth,
-		                               pfRealOut+y*iWidth,
-		                               pfImaginaryOut+y*iWidth,
-		                               pfRealOut+y*iWidth,
-		                               pfImaginaryOut+y*iWidth,
-		                               1, 1, inverse);
-	}
+You may use, copy, modify and distribute this code for any purpose (include commercial use) and without fee.
+
+Source: http://www.kurims.kyoto-u.ac.jp/~ooura/fft.html
+
+Fast Fourier/Cosine/Sine Transform
+    dimension   :one
+    data length :power of 2
+    decimation  :frequency
+    radix       :split-radix
+    data        :inplace
+    table       :use
+functions
+    cdft: Complex Discrete Fourier Transform
+    rdft: Real Discrete Fourier Transform
+    ddct: Discrete Cosine Transform
+    ddst: Discrete Sine Transform
+    dfct: Cosine Transform of RDFT (Real Symmetric DFT)
+    dfst: Sine Transform of RDFT (Real Anti-symmetric DFT)
+function prototypes
+    void cdft(int, int, float32 *, int *, float32 *);
+    void rdft(int, int, float32 *, int *, float32 *);
+    void ddct(int, int, float32 *, int *, float32 *);
+    void ddst(int, int, float32 *, int *, float32 *);
+    void dfct(int, float32 *, float32 *, int *, float32 *);
+    void dfst(int, float32 *, float32 *, int *, float32 *);
+macro definitions
+    USE_CDFT_PTHREADS : default=not defined
+        CDFT_THREADS_BEGIN_N  : must be >= 512, default=8192
+        CDFT_4THREADS_BEGIN_N : must be >= 512, default=65536
+    USE_CDFT_WINTHREADS : default=not defined
+        CDFT_THREADS_BEGIN_N  : must be >= 512, default=32768
+        CDFT_4THREADS_BEGIN_N : must be >= 512, default=524288
+
+
+-------- Complex DFT (Discrete Fourier Transform) --------
+    [definition]
+        <case1>
+            X[k] = sum_j=0^n-1 x[j]*exp(2*pi*i*j*k/n), 0<=k<n
+        <case2>
+            X[k] = sum_j=0^n-1 x[j]*exp(-2*pi*i*j*k/n), 0<=k<n
+        (notes: sum_j=0^n-1 is a summation from j=0 to n-1)
+    [usage]
+        <case1>
+            ip[0] = 0; // first time only
+            cdft(2*n, 1, a, ip, w);
+        <case2>
+            ip[0] = 0; // first time only
+            cdft(2*n, -1, a, ip, w);
+    [parameters]
+        2*n            :data length (int)
+                        n >= 1, n = power of 2
+        a[0...2*n-1]   :input/output data (float32 *)
+                        input data
+                            a[2*j] = Re(x[j]), 
+                            a[2*j+1] = Im(x[j]), 0<=j<n
+                        output data
+                            a[2*k] = Re(X[k]), 
+                            a[2*k+1] = Im(X[k]), 0<=k<n
+        ip[0...*]      :work area for bit reversal (int *)
+                        length of ip >= 2+sqrt(n)
+                        strictly, 
+                        length of ip >= 
+                            2+(1<<(int)(log(n+0.5)/log(2))/2).
+                        ip[0],ip[1] are pointers of the cos/sin table.
+        w[0...n/2-1]   :cos/sin table (float32 *)
+                        w[],ip[] are initialized if ip[0] == 0.
+    [remark]
+        Inverse of 
+            cdft(2*n, -1, a, ip, w);
+        is 
+            cdft(2*n, 1, a, ip, w);
+            for (j = 0; j <= 2 * n - 1; j++) {
+                a[j] *= 1.0 / n;
+            }
+        .
+
+
+-------- Real DFT / Inverse of Real DFT --------
+    [definition]
+        <case1> RDFT
+            R[k] = sum_j=0^n-1 a[j]*cos(2*pi*j*k/n), 0<=k<=n/2
+            I[k] = sum_j=0^n-1 a[j]*sin(2*pi*j*k/n), 0<k<n/2
+        <case2> IRDFT (excluding scale)
+            a[k] = (R[0] + R[n/2]*cos(pi*k))/2 + 
+                   sum_j=1^n/2-1 R[j]*cos(2*pi*j*k/n) + 
+                   sum_j=1^n/2-1 I[j]*sin(2*pi*j*k/n), 0<=k<n
+    [usage]
+        <case1>
+            ip[0] = 0; // first time only
+            rdft(n, 1, a, ip, w);
+        <case2>
+            ip[0] = 0; // first time only
+            rdft(n, -1, a, ip, w);
+    [parameters]
+        n              :data length (int)
+                        n >= 2, n = power of 2
+        a[0...n-1]     :input/output data (float32 *)
+                        <case1>
+                            output data
+                                a[2*k] = R[k], 0<=k<n/2
+                                a[2*k+1] = I[k], 0<k<n/2
+                                a[1] = R[n/2]
+                        <case2>
+                            input data
+                                a[2*j] = R[j], 0<=j<n/2
+                                a[2*j+1] = I[j], 0<j<n/2
+                                a[1] = R[n/2]
+        ip[0...*]      :work area for bit reversal (int *)
+                        length of ip >= 2+sqrt(n/2)
+                        strictly, 
+                        length of ip >= 
+                            2+(1<<(int)(log(n/2+0.5)/log(2))/2).
+                        ip[0],ip[1] are pointers of the cos/sin table.
+        w[0...n/2-1]   :cos/sin table (float32 *)
+                        w[],ip[] are initialized if ip[0] == 0.
+    [remark]
+        Inverse of 
+            rdft(n, 1, a, ip, w);
+        is 
+            rdft(n, -1, a, ip, w);
+            for (j = 0; j <= n - 1; j++) {
+                a[j] *= 2.0 / n;
+            }
+        .
+
+
+-------- DCT (Discrete Cosine Transform) / Inverse of DCT --------
+    [definition]
+        <case1> IDCT (excluding scale)
+            C[k] = sum_j=0^n-1 a[j]*cos(pi*j*(k+1/2)/n), 0<=k<n
+        <case2> DCT
+            C[k] = sum_j=0^n-1 a[j]*cos(pi*(j+1/2)*k/n), 0<=k<n
+    [usage]
+        <case1>
+            ip[0] = 0; // first time only
+            ddct(n, 1, a, ip, w);
+        <case2>
+            ip[0] = 0; // first time only
+            ddct(n, -1, a, ip, w);
+    [parameters]
+        n              :data length (int)
+                        n >= 2, n = power of 2
+        a[0...n-1]     :input/output data (float32 *)
+                        output data
+                            a[k] = C[k], 0<=k<n
+        ip[0...*]      :work area for bit reversal (int *)
+                        length of ip >= 2+sqrt(n/2)
+                        strictly, 
+                        length of ip >= 
+                            2+(1<<(int)(log(n/2+0.5)/log(2))/2).
+                        ip[0],ip[1] are pointers of the cos/sin table.
+        w[0...n*5/4-1] :cos/sin table (float32 *)
+                        w[],ip[] are initialized if ip[0] == 0.
+    [remark]
+        Inverse of 
+            ddct(n, -1, a, ip, w);
+        is 
+            a[0] *= 0.5;
+            ddct(n, 1, a, ip, w);
+            for (j = 0; j <= n - 1; j++) {
+                a[j] *= 2.0 / n;
+            }
+        .
+
+
+-------- DST (Discrete Sine Transform) / Inverse of DST --------
+    [definition]
+        <case1> IDST (excluding scale)
+            S[k] = sum_j=1^n A[j]*sin(pi*j*(k+1/2)/n), 0<=k<n
+        <case2> DST
+            S[k] = sum_j=0^n-1 a[j]*sin(pi*(j+1/2)*k/n), 0<k<=n
+    [usage]
+        <case1>
+            ip[0] = 0; // first time only
+            ddst(n, 1, a, ip, w);
+        <case2>
+            ip[0] = 0; // first time only
+            ddst(n, -1, a, ip, w);
+    [parameters]
+        n              :data length (int)
+                        n >= 2, n = power of 2
+        a[0...n-1]     :input/output data (float32 *)
+                        <case1>
+                            input data
+                                a[j] = A[j], 0<j<n
+                                a[0] = A[n]
+                            output data
+                                a[k] = S[k], 0<=k<n
+                        <case2>
+                            output data
+                                a[k] = S[k], 0<k<n
+                                a[0] = S[n]
+        ip[0...*]      :work area for bit reversal (int *)
+                        length of ip >= 2+sqrt(n/2)
+                        strictly, 
+                        length of ip >= 
+                            2+(1<<(int)(log(n/2+0.5)/log(2))/2).
+                        ip[0],ip[1] are pointers of the cos/sin table.
+        w[0...n*5/4-1] :cos/sin table (float32 *)
+                        w[],ip[] are initialized if ip[0] == 0.
+    [remark]
+        Inverse of 
+            ddst(n, -1, a, ip, w);
+        is 
+            a[0] *= 0.5;
+            ddst(n, 1, a, ip, w);
+            for (j = 0; j <= n - 1; j++) {
+                a[j] *= 2.0 / n;
+            }
+        .
+
+
+-------- Cosine Transform of RDFT (Real Symmetric DFT) --------
+    [definition]
+        C[k] = sum_j=0^n a[j]*cos(pi*j*k/n), 0<=k<=n
+    [usage]
+        ip[0] = 0; // first time only
+        dfct(n, a, t, ip, w);
+    [parameters]
+        n              :data length - 1 (int)
+                        n >= 2, n = power of 2
+        a[0...n]       :input/output data (float32 *)
+                        output data
+                            a[k] = C[k], 0<=k<=n
+        t[0...n/2]     :work area (float32 *)
+        ip[0...*]      :work area for bit reversal (int *)
+                        length of ip >= 2+sqrt(n/4)
+                        strictly, 
+                        length of ip >= 
+                            2+(1<<(int)(log(n/4+0.5)/log(2))/2).
+                        ip[0],ip[1] are pointers of the cos/sin table.
+        w[0...n*5/8-1] :cos/sin table (float32 *)
+                        w[],ip[] are initialized if ip[0] == 0.
+    [remark]
+        Inverse of 
+            a[0] *= 0.5;
+            a[n] *= 0.5;
+            dfct(n, a, t, ip, w);
+        is 
+            a[0] *= 0.5;
+            a[n] *= 0.5;
+            dfct(n, a, t, ip, w);
+            for (j = 0; j <= n; j++) {
+                a[j] *= 2.0 / n;
+            }
+        .
+
+
+-------- Sine Transform of RDFT (Real Anti-symmetric DFT) --------
+    [definition]
+        S[k] = sum_j=1^n-1 a[j]*sin(pi*j*k/n), 0<k<n
+    [usage]
+        ip[0] = 0; // first time only
+        dfst(n, a, t, ip, w);
+    [parameters]
+        n              :data length + 1 (int)
+                        n >= 2, n = power of 2
+        a[0...n-1]     :input/output data (float32 *)
+                        output data
+                            a[k] = S[k], 0<k<n
+                        (a[0] is used for work area)
+        t[0...n/2-1]   :work area (float32 *)
+        ip[0...*]      :work area for bit reversal (int *)
+                        length of ip >= 2+sqrt(n/4)
+                        strictly, 
+                        length of ip >= 
+                            2+(1<<(int)(log(n/4+0.5)/log(2))/2).
+                        ip[0],ip[1] are pointers of the cos/sin table.
+        w[0...n*5/8-1] :cos/sin table (float32 *)
+                        w[],ip[] are initialized if ip[0] == 0.
+    [remark]
+        Inverse of 
+            dfst(n, a, t, ip, w);
+        is 
+            dfst(n, a, t, ip, w);
+            for (j = 1; j <= n - 1; j++) {
+                a[j] *= 2.0 / n;
+            }
+        .
+
+
+Appendix :
+    The cos/sin table is recalculated when the larger table required.
+    w[] and ip[] are compatible with all routines.
+*/
+
+
+void cdft(int n, int isgn, float32 *a, int *ip, float32 *w)
+{
+    void makewt(int nw, int *ip, float32 *w);
+    void cftfsub(int n, float32 *a, int *ip, int nw, float32 *w);
+    void cftbsub(int n, float32 *a, int *ip, int nw, float32 *w);
+    int nw;
+    
+    nw = ip[0];
+    if (n > (nw << 2)) {
+        nw = n >> 2;
+        makewt(nw, ip, w);
+    }
+    if (isgn >= 0) {
+        cftfsub(n, a, ip, nw, w);
+    } else {
+        cftbsub(n, a, ip, nw, w);
+    }
+}
+
+
+void rdft(int n, int isgn, float32 *a, int *ip, float32 *w)
+{
+    void makewt(int nw, int *ip, float32 *w);
+    void makect(int nc, int *ip, float32 *c);
+    void cftfsub(int n, float32 *a, int *ip, int nw, float32 *w);
+    void cftbsub(int n, float32 *a, int *ip, int nw, float32 *w);
+    void rftfsub(int n, float32 *a, int nc, float32 *c);
+    void rftbsub(int n, float32 *a, int nc, float32 *c);
+    int nw, nc;
+    float32 xi;
+    
+    nw = ip[0];
+    if (n > (nw << 2)) {
+        nw = n >> 2;
+        makewt(nw, ip, w);
+    }
+    nc = ip[1];
+    if (n > (nc << 2)) {
+        nc = n >> 2;
+        makect(nc, ip, w + nw);
+    }
+    if (isgn >= 0) {
+        if (n > 4) {
+            cftfsub(n, a, ip, nw, w);
+            rftfsub(n, a, nc, w + nw);
+        } else if (n == 4) {
+            cftfsub(n, a, ip, nw, w);
+        }
+        xi = a[0] - a[1];
+        a[0] += a[1];
+        a[1] = xi;
+    } else {
+        a[1] = 0.5 * (a[0] - a[1]);
+        a[0] -= a[1];
+        if (n > 4) {
+            rftbsub(n, a, nc, w + nw);
+            cftbsub(n, a, ip, nw, w);
+        } else if (n == 4) {
+            cftbsub(n, a, ip, nw, w);
+        }
+    }
+}
+
+
+void ddct(int n, int isgn, float32 *a, int *ip, float32 *w)
+{
+    void makewt(int nw, int *ip, float32 *w);
+    void makect(int nc, int *ip, float32 *c);
+    void cftfsub(int n, float32 *a, int *ip, int nw, float32 *w);
+    void cftbsub(int n, float32 *a, int *ip, int nw, float32 *w);
+    void rftfsub(int n, float32 *a, int nc, float32 *c);
+    void rftbsub(int n, float32 *a, int nc, float32 *c);
+    void dctsub(int n, float32 *a, int nc, float32 *c);
+    int j, nw, nc;
+    float32 xr;
+    
+    nw = ip[0];
+    if (n > (nw << 2)) {
+        nw = n >> 2;
+        makewt(nw, ip, w);
+    }
+    nc = ip[1];
+    if (n > nc) {
+        nc = n;
+        makect(nc, ip, w + nw);
+    }
+    if (isgn < 0) {
+        xr = a[n - 1];
+        for (j = n - 2; j >= 2; j -= 2) {
+            a[j + 1] = a[j] - a[j - 1];
+            a[j] += a[j - 1];
+        }
+        a[1] = a[0] - xr;
+        a[0] += xr;
+        if (n > 4) {
+            rftbsub(n, a, nc, w + nw);
+            cftbsub(n, a, ip, nw, w);
+        } else if (n == 4) {
+            cftbsub(n, a, ip, nw, w);
+        }
+    }
+    dctsub(n, a, nc, w + nw);
+    if (isgn >= 0) {
+        if (n > 4) {
+            cftfsub(n, a, ip, nw, w);
+            rftfsub(n, a, nc, w + nw);
+        } else if (n == 4) {
+            cftfsub(n, a, ip, nw, w);
+        }
+        xr = a[0] - a[1];
+        a[0] += a[1];
+        for (j = 2; j < n; j += 2) {
+            a[j - 1] = a[j] - a[j + 1];
+            a[j] += a[j + 1];
+        }
+        a[n - 1] = xr;
+    }
+}
+
+
+void ddst(int n, int isgn, float32 *a, int *ip, float32 *w)
+{
+    void makewt(int nw, int *ip, float32 *w);
+    void makect(int nc, int *ip, float32 *c);
+    void cftfsub(int n, float32 *a, int *ip, int nw, float32 *w);
+    void cftbsub(int n, float32 *a, int *ip, int nw, float32 *w);
+    void rftfsub(int n, float32 *a, int nc, float32 *c);
+    void rftbsub(int n, float32 *a, int nc, float32 *c);
+    void dstsub(int n, float32 *a, int nc, float32 *c);
+    int j, nw, nc;
+    float32 xr;
+    
+    nw = ip[0];
+    if (n > (nw << 2)) {
+        nw = n >> 2;
+        makewt(nw, ip, w);
+    }
+    nc = ip[1];
+    if (n > nc) {
+        nc = n;
+        makect(nc, ip, w + nw);
+    }
+    if (isgn < 0) {
+        xr = a[n - 1];
+        for (j = n - 2; j >= 2; j -= 2) {
+            a[j + 1] = -a[j] - a[j - 1];
+            a[j] -= a[j - 1];
+        }
+        a[1] = a[0] + xr;
+        a[0] -= xr;
+        if (n > 4) {
+            rftbsub(n, a, nc, w + nw);
+            cftbsub(n, a, ip, nw, w);
+        } else if (n == 4) {
+            cftbsub(n, a, ip, nw, w);
+        }
+    }
+    dstsub(n, a, nc, w + nw);
+    if (isgn >= 0) {
+        if (n > 4) {
+            cftfsub(n, a, ip, nw, w);
+            rftfsub(n, a, nc, w + nw);
+        } else if (n == 4) {
+            cftfsub(n, a, ip, nw, w);
+        }
+        xr = a[0] - a[1];
+        a[0] += a[1];
+        for (j = 2; j < n; j += 2) {
+            a[j - 1] = -a[j] - a[j + 1];
+            a[j] -= a[j + 1];
+        }
+        a[n - 1] = -xr;
+    }
+}
+
+
+void dfct(int n, float32 *a, float32 *t, int *ip, float32 *w)
+{
+    void makewt(int nw, int *ip, float32 *w);
+    void makect(int nc, int *ip, float32 *c);
+    void cftfsub(int n, float32 *a, int *ip, int nw, float32 *w);
+    void rftfsub(int n, float32 *a, int nc, float32 *c);
+    void dctsub(int n, float32 *a, int nc, float32 *c);
+    int j, k, l, m, mh, nw, nc;
+    float32 xr, xi, yr, yi;
+    
+    nw = ip[0];
+    if (n > (nw << 3)) {
+        nw = n >> 3;
+        makewt(nw, ip, w);
+    }
+    nc = ip[1];
+    if (n > (nc << 1)) {
+        nc = n >> 1;
+        makect(nc, ip, w + nw);
+    }
+    m = n >> 1;
+    yi = a[m];
+    xi = a[0] + a[n];
+    a[0] -= a[n];
+    t[0] = xi - yi;
+    t[m] = xi + yi;
+    if (n > 2) {
+        mh = m >> 1;
+        for (j = 1; j < mh; j++) {
+            k = m - j;
+            xr = a[j] - a[n - j];
+            xi = a[j] + a[n - j];
+            yr = a[k] - a[n - k];
+            yi = a[k] + a[n - k];
+            a[j] = xr;
+            a[k] = yr;
+            t[j] = xi - yi;
+            t[k] = xi + yi;
+        }
+        t[mh] = a[mh] + a[n - mh];
+        a[mh] -= a[n - mh];
+        dctsub(m, a, nc, w + nw);
+        if (m > 4) {
+            cftfsub(m, a, ip, nw, w);
+            rftfsub(m, a, nc, w + nw);
+        } else if (m == 4) {
+            cftfsub(m, a, ip, nw, w);
+        }
+        a[n - 1] = a[0] - a[1];
+        a[1] = a[0] + a[1];
+        for (j = m - 2; j >= 2; j -= 2) {
+            a[2 * j + 1] = a[j] + a[j + 1];
+            a[2 * j - 1] = a[j] - a[j + 1];
+        }
+        l = 2;
+        m = mh;
+        while (m >= 2) {
+            dctsub(m, t, nc, w + nw);
+            if (m > 4) {
+                cftfsub(m, t, ip, nw, w);
+                rftfsub(m, t, nc, w + nw);
+            } else if (m == 4) {
+                cftfsub(m, t, ip, nw, w);
+            }
+            a[n - l] = t[0] - t[1];
+            a[l] = t[0] + t[1];
+            k = 0;
+            for (j = 2; j < m; j += 2) {
+                k += l << 2;
+                a[k - l] = t[j] - t[j + 1];
+                a[k + l] = t[j] + t[j + 1];
+            }
+            l <<= 1;
+            mh = m >> 1;
+            for (j = 0; j < mh; j++) {
+                k = m - j;
+                t[j] = t[m + k] - t[m + j];
+                t[k] = t[m + k] + t[m + j];
+            }
+            t[mh] = t[m + mh];
+            m = mh;
+        }
+        a[l] = t[0];
+        a[n] = t[2] - t[1];
+        a[0] = t[2] + t[1];
+    } else {
+        a[1] = a[0];
+        a[2] = t[0];
+        a[0] = t[1];
+    }
+}
+
+
+void dfst(int n, float32 *a, float32 *t, int *ip, float32 *w)
+{
+    void makewt(int nw, int *ip, float32 *w);
+    void makect(int nc, int *ip, float32 *c);
+    void cftfsub(int n, float32 *a, int *ip, int nw, float32 *w);
+    void rftfsub(int n, float32 *a, int nc, float32 *c);
+    void dstsub(int n, float32 *a, int nc, float32 *c);
+    int j, k, l, m, mh, nw, nc;
+    float32 xr, xi, yr, yi;
+    
+    nw = ip[0];
+    if (n > (nw << 3)) {
+        nw = n >> 3;
+        makewt(nw, ip, w);
+    }
+    nc = ip[1];
+    if (n > (nc << 1)) {
+        nc = n >> 1;
+        makect(nc, ip, w + nw);
+    }
+    if (n > 2) {
+        m = n >> 1;
+        mh = m >> 1;
+        for (j = 1; j < mh; j++) {
+            k = m - j;
+            xr = a[j] + a[n - j];
+            xi = a[j] - a[n - j];
+            yr = a[k] + a[n - k];
+            yi = a[k] - a[n - k];
+            a[j] = xr;
+            a[k] = yr;
+            t[j] = xi + yi;
+            t[k] = xi - yi;
+        }
+        t[0] = a[mh] - a[n - mh];
+        a[mh] += a[n - mh];
+        a[0] = a[m];
+        dstsub(m, a, nc, w + nw);
+        if (m > 4) {
+            cftfsub(m, a, ip, nw, w);
+            rftfsub(m, a, nc, w + nw);
+        } else if (m == 4) {
+            cftfsub(m, a, ip, nw, w);
+        }
+        a[n - 1] = a[1] - a[0];
+        a[1] = a[0] + a[1];
+        for (j = m - 2; j >= 2; j -= 2) {
+            a[2 * j + 1] = a[j] - a[j + 1];
+            a[2 * j - 1] = -a[j] - a[j + 1];
+        }
+        l = 2;
+        m = mh;
+        while (m >= 2) {
+            dstsub(m, t, nc, w + nw);
+            if (m > 4) {
+                cftfsub(m, t, ip, nw, w);
+                rftfsub(m, t, nc, w + nw);
+            } else if (m == 4) {
+                cftfsub(m, t, ip, nw, w);
+            }
+            a[n - l] = t[1] - t[0];
+            a[l] = t[0] + t[1];
+            k = 0;
+            for (j = 2; j < m; j += 2) {
+                k += l << 2;
+                a[k - l] = -t[j] - t[j + 1];
+                a[k + l] = t[j] - t[j + 1];
+            }
+            l <<= 1;
+            mh = m >> 1;
+            for (j = 1; j < mh; j++) {
+                k = m - j;
+                t[j] = t[m + k] + t[m + j];
+                t[k] = t[m + k] - t[m + j];
+            }
+            t[0] = t[m + mh];
+            m = mh;
+        }
+        a[l] = t[0];
+    }
+    a[0] = 0;
+}
+
+
+/* -------- initializing routines -------- */
+
+
+#include <math.h>
+
+void makewt(int nw, int *ip, float32 *w)
+{
+    void makeipt(int nw, int *ip);
+    int j, nwh, nw0, nw1;
+    float32 delta, wn4r, wk1r, wk1i, wk3r, wk3i;
+    
+    ip[0] = nw;
+    ip[1] = 1;
+    if (nw > 2) {
+        nwh = nw >> 1;
+        delta = atan(1.0) / nwh;
+        wn4r = cos(delta * nwh);
+        w[0] = 1;
+        w[1] = wn4r;
+        if (nwh == 4) {
+            w[2] = cos(delta * 2);
+            w[3] = sin(delta * 2);
+        } else if (nwh > 4) {
+            makeipt(nw, ip);
+            w[2] = 0.5 / cos(delta * 2);
+            w[3] = 0.5 / cos(delta * 6);
+            for (j = 4; j < nwh; j += 4) {
+                w[j] = cos(delta * j);
+                w[j + 1] = sin(delta * j);
+                w[j + 2] = cos(3 * delta * j);
+                w[j + 3] = -sin(3 * delta * j);
+            }
+        }
+        nw0 = 0;
+        while (nwh > 2) {
+            nw1 = nw0 + nwh;
+            nwh >>= 1;
+            w[nw1] = 1;
+            w[nw1 + 1] = wn4r;
+            if (nwh == 4) {
+                wk1r = w[nw0 + 4];
+                wk1i = w[nw0 + 5];
+                w[nw1 + 2] = wk1r;
+                w[nw1 + 3] = wk1i;
+            } else if (nwh > 4) {
+                wk1r = w[nw0 + 4];
+                wk3r = w[nw0 + 6];
+                w[nw1 + 2] = 0.5 / wk1r;
+                w[nw1 + 3] = 0.5 / wk3r;
+                for (j = 4; j < nwh; j += 4) {
+                    wk1r = w[nw0 + 2 * j];
+                    wk1i = w[nw0 + 2 * j + 1];
+                    wk3r = w[nw0 + 2 * j + 2];
+                    wk3i = w[nw0 + 2 * j + 3];
+                    w[nw1 + j] = wk1r;
+                    w[nw1 + j + 1] = wk1i;
+                    w[nw1 + j + 2] = wk3r;
+                    w[nw1 + j + 3] = wk3i;
+                }
+            }
+            nw0 = nw1;
+        }
+    }
+}
+
+
+void makeipt(int nw, int *ip)
+{
+    int j, l, m, m2, p, q;
+    
+    ip[2] = 0;
+    ip[3] = 16;
+    m = 2;
+    for (l = nw; l > 32; l >>= 2) {
+        m2 = m << 1;
+        q = m2 << 3;
+        for (j = m; j < m2; j++) {
+            p = ip[j] << 2;
+            ip[m + j] = p;
+            ip[m2 + j] = p + q;
+        }
+        m = m2;
+    }
+}
+
+
+void makect(int nc, int *ip, float32 *c)
+{
+    int j, nch;
+    float32 delta;
+    
+    ip[1] = nc;
+    if (nc > 1) {
+        nch = nc >> 1;
+        delta = atan(1.0) / nch;
+        c[0] = cos(delta * nch);
+        c[nch] = 0.5 * c[0];
+        for (j = 1; j < nch; j++) {
+            c[j] = 0.5 * cos(delta * j);
+            c[nc - j] = 0.5 * sin(delta * j);
+        }
+    }
+}
+
+
+/* -------- child routines -------- */
+
+
+#ifdef USE_CDFT_PTHREADS
+#define USE_CDFT_THREADS
+#ifndef CDFT_THREADS_BEGIN_N
+#define CDFT_THREADS_BEGIN_N 8192
+#endif
+#ifndef CDFT_4THREADS_BEGIN_N
+#define CDFT_4THREADS_BEGIN_N 65536
+#endif
+#include <pthread.h>
+#include <stdio.h>
+#include <stdlib.h>
+#define cdft_thread_t pthread_t
+#define cdft_thread_create(thp,func,argp) { \
+    if (pthread_create(thp, NULL, func, (void *) argp) != 0) { \
+        fprintf(stderr, "cdft thread error\n"); \
+        exit(1); \
+    } \
+}
+#define cdft_thread_wait(th) { \
+    if (pthread_join(th, NULL) != 0) { \
+        fprintf(stderr, "cdft thread error\n"); \
+        exit(1); \
+    } \
+}
+#endif /* USE_CDFT_PTHREADS */
+
+
+#ifdef USE_CDFT_WINTHREADS
+#define USE_CDFT_THREADS
+#ifndef CDFT_THREADS_BEGIN_N
+#define CDFT_THREADS_BEGIN_N 32768
+#endif
+#ifndef CDFT_4THREADS_BEGIN_N
+#define CDFT_4THREADS_BEGIN_N 524288
+#endif
+#include <windows.h>
+#include <stdio.h>
+#include <stdlib.h>
+#define cdft_thread_t HANDLE
+#define cdft_thread_create(thp,func,argp) { \
+    DWORD thid; \
+    *(thp) = CreateThread(NULL, 0, (LPTHREAD_START_ROUTINE) func, (LPVOID) argp, 0, &thid); \
+    if (*(thp) == 0) { \
+        fprintf(stderr, "cdft thread error\n"); \
+        exit(1); \
+    } \
+}
+#define cdft_thread_wait(th) { \
+    WaitForSingleObject(th, INFINITE); \
+    CloseHandle(th); \
+}
+#endif /* USE_CDFT_WINTHREADS */
+
+
+void cftfsub(int n, float32 *a, int *ip, int nw, float32 *w)
+{
+    void bitrv2(int n, int *ip, float32 *a);
+    void bitrv216(float32 *a);
+    void bitrv208(float32 *a);
+    void cftf1st(int n, float32 *a, float32 *w);
+    void cftrec4(int n, float32 *a, int nw, float32 *w);
+    void cftleaf(int n, int isplt, float32 *a, int nw, float32 *w);
+    void cftfx41(int n, float32 *a, int nw, float32 *w);
+    void cftf161(float32 *a, float32 *w);
+    void cftf081(float32 *a, float32 *w);
+    void cftf040(float32 *a);
+    void cftx020(float32 *a);
+#ifdef USE_CDFT_THREADS
+    void cftrec4_th(int n, float32 *a, int nw, float32 *w);
+#endif /* USE_CDFT_THREADS */
+    
+    if (n > 8) {
+        if (n > 32) {
+            cftf1st(n, a, &w[nw - (n >> 2)]);
+#ifdef USE_CDFT_THREADS
+            if (n > CDFT_THREADS_BEGIN_N) {
+                cftrec4_th(n, a, nw, w);
+            } else 
+#endif /* USE_CDFT_THREADS */
+            if (n > 512) {
+                cftrec4(n, a, nw, w);
+            } else if (n > 128) {
+                cftleaf(n, 1, a, nw, w);
+            } else {
+                cftfx41(n, a, nw, w);
+            }
+            bitrv2(n, ip, a);
+        } else if (n == 32) {
+            cftf161(a, &w[nw - 8]);
+            bitrv216(a);
+        } else {
+            cftf081(a, w);
+            bitrv208(a);
+        }
+    } else if (n == 8) {
+        cftf040(a);
+    } else if (n == 4) {
+        cftx020(a);
+    }
+}
+
+
+void cftbsub(int n, float32 *a, int *ip, int nw, float32 *w)
+{
+    void bitrv2conj(int n, int *ip, float32 *a);
+    void bitrv216neg(float32 *a);
+    void bitrv208neg(float32 *a);
+    void cftb1st(int n, float32 *a, float32 *w);
+    void cftrec4(int n, float32 *a, int nw, float32 *w);
+    void cftleaf(int n, int isplt, float32 *a, int nw, float32 *w);
+    void cftfx41(int n, float32 *a, int nw, float32 *w);
+    void cftf161(float32 *a, float32 *w);
+    void cftf081(float32 *a, float32 *w);
+    void cftb040(float32 *a);
+    void cftx020(float32 *a);
+#ifdef USE_CDFT_THREADS
+    void cftrec4_th(int n, float32 *a, int nw, float32 *w);
+#endif /* USE_CDFT_THREADS */
+    
+    if (n > 8) {
+        if (n > 32) {
+            cftb1st(n, a, &w[nw - (n >> 2)]);
+#ifdef USE_CDFT_THREADS
+            if (n > CDFT_THREADS_BEGIN_N) {
+                cftrec4_th(n, a, nw, w);
+            } else 
+#endif /* USE_CDFT_THREADS */
+            if (n > 512) {
+                cftrec4(n, a, nw, w);
+            } else if (n > 128) {
+                cftleaf(n, 1, a, nw, w);
+            } else {
+                cftfx41(n, a, nw, w);
+            }
+            bitrv2conj(n, ip, a);
+        } else if (n == 32) {
+            cftf161(a, &w[nw - 8]);
+            bitrv216neg(a);
+        } else {
+            cftf081(a, w);
+            bitrv208neg(a);
+        }
+    } else if (n == 8) {
+        cftb040(a);
+    } else if (n == 4) {
+        cftx020(a);
+    }
+}
+
+
+void bitrv2(int n, int *ip, float32 *a)
+{
+    int j, j1, k, k1, l, m, nh, nm;
+    float32 xr, xi, yr, yi;
+    
+    m = 1;
+    for (l = n >> 2; l > 8; l >>= 2) {
+        m <<= 1;
+    }
+    nh = n >> 1;
+    nm = 4 * m;
+    if (l == 8) {
+        for (k = 0; k < m; k++) {
+            for (j = 0; j < k; j++) {
+                j1 = 4 * j + 2 * ip[m + k];
+                k1 = 4 * k + 2 * ip[m + j];
+                xr = a[j1];
+                xi = a[j1 + 1];
+                yr = a[k1];
+                yi = a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 += nm;
+                k1 += 2 * nm;
+                xr = a[j1];
+                xi = a[j1 + 1];
+                yr = a[k1];
+                yi = a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 += nm;
+                k1 -= nm;
+                xr = a[j1];
+                xi = a[j1 + 1];
+                yr = a[k1];
+                yi = a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 += nm;
+                k1 += 2 * nm;
+                xr = a[j1];
+                xi = a[j1 + 1];
+                yr = a[k1];
+                yi = a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 += nh;
+                k1 += 2;
+                xr = a[j1];
+                xi = a[j1 + 1];
+                yr = a[k1];
+                yi = a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 -= nm;
+                k1 -= 2 * nm;
+                xr = a[j1];
+                xi = a[j1 + 1];
+                yr = a[k1];
+                yi = a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 -= nm;
+                k1 += nm;
+                xr = a[j1];
+                xi = a[j1 + 1];
+                yr = a[k1];
+                yi = a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 -= nm;
+                k1 -= 2 * nm;
+                xr = a[j1];
+                xi = a[j1 + 1];
+                yr = a[k1];
+                yi = a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 += 2;
+                k1 += nh;
+                xr = a[j1];
+                xi = a[j1 + 1];
+                yr = a[k1];
+                yi = a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 += nm;
+                k1 += 2 * nm;
+                xr = a[j1];
+                xi = a[j1 + 1];
+                yr = a[k1];
+                yi = a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 += nm;
+                k1 -= nm;
+                xr = a[j1];
+                xi = a[j1 + 1];
+                yr = a[k1];
+                yi = a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 += nm;
+                k1 += 2 * nm;
+                xr = a[j1];
+                xi = a[j1 + 1];
+                yr = a[k1];
+                yi = a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 -= nh;
+                k1 -= 2;
+                xr = a[j1];
+                xi = a[j1 + 1];
+                yr = a[k1];
+                yi = a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 -= nm;
+                k1 -= 2 * nm;
+                xr = a[j1];
+                xi = a[j1 + 1];
+                yr = a[k1];
+                yi = a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 -= nm;
+                k1 += nm;
+                xr = a[j1];
+                xi = a[j1 + 1];
+                yr = a[k1];
+                yi = a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 -= nm;
+                k1 -= 2 * nm;
+                xr = a[j1];
+                xi = a[j1 + 1];
+                yr = a[k1];
+                yi = a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+            }
+            k1 = 4 * k + 2 * ip[m + k];
+            j1 = k1 + 2;
+            k1 += nh;
+            xr = a[j1];
+            xi = a[j1 + 1];
+            yr = a[k1];
+            yi = a[k1 + 1];
+            a[j1] = yr;
+            a[j1 + 1] = yi;
+            a[k1] = xr;
+            a[k1 + 1] = xi;
+            j1 += nm;
+            k1 += 2 * nm;
+            xr = a[j1];
+            xi = a[j1 + 1];
+            yr = a[k1];
+            yi = a[k1 + 1];
+            a[j1] = yr;
+            a[j1 + 1] = yi;
+            a[k1] = xr;
+            a[k1 + 1] = xi;
+            j1 += nm;
+            k1 -= nm;
+            xr = a[j1];
+            xi = a[j1 + 1];
+            yr = a[k1];
+            yi = a[k1 + 1];
+            a[j1] = yr;
+            a[j1 + 1] = yi;
+            a[k1] = xr;
+            a[k1 + 1] = xi;
+            j1 -= 2;
+            k1 -= nh;
+            xr = a[j1];
+            xi = a[j1 + 1];
+            yr = a[k1];
+            yi = a[k1 + 1];
+            a[j1] = yr;
+            a[j1 + 1] = yi;
+            a[k1] = xr;
+            a[k1 + 1] = xi;
+            j1 += nh + 2;
+            k1 += nh + 2;
+            xr = a[j1];
+            xi = a[j1 + 1];
+            yr = a[k1];
+            yi = a[k1 + 1];
+            a[j1] = yr;
+            a[j1 + 1] = yi;
+            a[k1] = xr;
+            a[k1 + 1] = xi;
+            j1 -= nh - nm;
+            k1 += 2 * nm - 2;
+            xr = a[j1];
+            xi = a[j1 + 1];
+            yr = a[k1];
+            yi = a[k1 + 1];
+            a[j1] = yr;
+            a[j1 + 1] = yi;
+            a[k1] = xr;
+            a[k1 + 1] = xi;
+        }
+    } else {
+        for (k = 0; k < m; k++) {
+            for (j = 0; j < k; j++) {
+                j1 = 4 * j + ip[m + k];
+                k1 = 4 * k + ip[m + j];
+                xr = a[j1];
+                xi = a[j1 + 1];
+                yr = a[k1];
+                yi = a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 += nm;
+                k1 += nm;
+                xr = a[j1];
+                xi = a[j1 + 1];
+                yr = a[k1];
+                yi = a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 += nh;
+                k1 += 2;
+                xr = a[j1];
+                xi = a[j1 + 1];
+                yr = a[k1];
+                yi = a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 -= nm;
+                k1 -= nm;
+                xr = a[j1];
+                xi = a[j1 + 1];
+                yr = a[k1];
+                yi = a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 += 2;
+                k1 += nh;
+                xr = a[j1];
+                xi = a[j1 + 1];
+                yr = a[k1];
+                yi = a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 += nm;
+                k1 += nm;
+                xr = a[j1];
+                xi = a[j1 + 1];
+                yr = a[k1];
+                yi = a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 -= nh;
+                k1 -= 2;
+                xr = a[j1];
+                xi = a[j1 + 1];
+                yr = a[k1];
+                yi = a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 -= nm;
+                k1 -= nm;
+                xr = a[j1];
+                xi = a[j1 + 1];
+                yr = a[k1];
+                yi = a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+            }
+            k1 = 4 * k + ip[m + k];
+            j1 = k1 + 2;
+            k1 += nh;
+            xr = a[j1];
+            xi = a[j1 + 1];
+            yr = a[k1];
+            yi = a[k1 + 1];
+            a[j1] = yr;
+            a[j1 + 1] = yi;
+            a[k1] = xr;
+            a[k1 + 1] = xi;
+            j1 += nm;
+            k1 += nm;
+            xr = a[j1];
+            xi = a[j1 + 1];
+            yr = a[k1];
+            yi = a[k1 + 1];
+            a[j1] = yr;
+            a[j1 + 1] = yi;
+            a[k1] = xr;
+            a[k1 + 1] = xi;
+        }
+    }
+}
+
+
+void bitrv2conj(int n, int *ip, float32 *a)
+{
+    int j, j1, k, k1, l, m, nh, nm;
+    float32 xr, xi, yr, yi;
+    
+    m = 1;
+    for (l = n >> 2; l > 8; l >>= 2) {
+        m <<= 1;
+    }
+    nh = n >> 1;
+    nm = 4 * m;
+    if (l == 8) {
+        for (k = 0; k < m; k++) {
+            for (j = 0; j < k; j++) {
+                j1 = 4 * j + 2 * ip[m + k];
+                k1 = 4 * k + 2 * ip[m + j];
+                xr = a[j1];
+                xi = -a[j1 + 1];
+                yr = a[k1];
+                yi = -a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 += nm;
+                k1 += 2 * nm;
+                xr = a[j1];
+                xi = -a[j1 + 1];
+                yr = a[k1];
+                yi = -a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 += nm;
+                k1 -= nm;
+                xr = a[j1];
+                xi = -a[j1 + 1];
+                yr = a[k1];
+                yi = -a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 += nm;
+                k1 += 2 * nm;
+                xr = a[j1];
+                xi = -a[j1 + 1];
+                yr = a[k1];
+                yi = -a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 += nh;
+                k1 += 2;
+                xr = a[j1];
+                xi = -a[j1 + 1];
+                yr = a[k1];
+                yi = -a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 -= nm;
+                k1 -= 2 * nm;
+                xr = a[j1];
+                xi = -a[j1 + 1];
+                yr = a[k1];
+                yi = -a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 -= nm;
+                k1 += nm;
+                xr = a[j1];
+                xi = -a[j1 + 1];
+                yr = a[k1];
+                yi = -a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 -= nm;
+                k1 -= 2 * nm;
+                xr = a[j1];
+                xi = -a[j1 + 1];
+                yr = a[k1];
+                yi = -a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 += 2;
+                k1 += nh;
+                xr = a[j1];
+                xi = -a[j1 + 1];
+                yr = a[k1];
+                yi = -a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 += nm;
+                k1 += 2 * nm;
+                xr = a[j1];
+                xi = -a[j1 + 1];
+                yr = a[k1];
+                yi = -a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 += nm;
+                k1 -= nm;
+                xr = a[j1];
+                xi = -a[j1 + 1];
+                yr = a[k1];
+                yi = -a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 += nm;
+                k1 += 2 * nm;
+                xr = a[j1];
+                xi = -a[j1 + 1];
+                yr = a[k1];
+                yi = -a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 -= nh;
+                k1 -= 2;
+                xr = a[j1];
+                xi = -a[j1 + 1];
+                yr = a[k1];
+                yi = -a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 -= nm;
+                k1 -= 2 * nm;
+                xr = a[j1];
+                xi = -a[j1 + 1];
+                yr = a[k1];
+                yi = -a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 -= nm;
+                k1 += nm;
+                xr = a[j1];
+                xi = -a[j1 + 1];
+                yr = a[k1];
+                yi = -a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 -= nm;
+                k1 -= 2 * nm;
+                xr = a[j1];
+                xi = -a[j1 + 1];
+                yr = a[k1];
+                yi = -a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+            }
+            k1 = 4 * k + 2 * ip[m + k];
+            j1 = k1 + 2;
+            k1 += nh;
+            a[j1 - 1] = -a[j1 - 1];
+            xr = a[j1];
+            xi = -a[j1 + 1];
+            yr = a[k1];
+            yi = -a[k1 + 1];
+            a[j1] = yr;
+            a[j1 + 1] = yi;
+            a[k1] = xr;
+            a[k1 + 1] = xi;
+            a[k1 + 3] = -a[k1 + 3];
+            j1 += nm;
+            k1 += 2 * nm;
+            xr = a[j1];
+            xi = -a[j1 + 1];
+            yr = a[k1];
+            yi = -a[k1 + 1];
+            a[j1] = yr;
+            a[j1 + 1] = yi;
+            a[k1] = xr;
+            a[k1 + 1] = xi;
+            j1 += nm;
+            k1 -= nm;
+            xr = a[j1];
+            xi = -a[j1 + 1];
+            yr = a[k1];
+            yi = -a[k1 + 1];
+            a[j1] = yr;
+            a[j1 + 1] = yi;
+            a[k1] = xr;
+            a[k1 + 1] = xi;
+            j1 -= 2;
+            k1 -= nh;
+            xr = a[j1];
+            xi = -a[j1 + 1];
+            yr = a[k1];
+            yi = -a[k1 + 1];
+            a[j1] = yr;
+            a[j1 + 1] = yi;
+            a[k1] = xr;
+            a[k1 + 1] = xi;
+            j1 += nh + 2;
+            k1 += nh + 2;
+            xr = a[j1];
+            xi = -a[j1 + 1];
+            yr = a[k1];
+            yi = -a[k1 + 1];
+            a[j1] = yr;
+            a[j1 + 1] = yi;
+            a[k1] = xr;
+            a[k1 + 1] = xi;
+            j1 -= nh - nm;
+            k1 += 2 * nm - 2;
+            a[j1 - 1] = -a[j1 - 1];
+            xr = a[j1];
+            xi = -a[j1 + 1];
+            yr = a[k1];
+            yi = -a[k1 + 1];
+            a[j1] = yr;
+            a[j1 + 1] = yi;
+            a[k1] = xr;
+            a[k1 + 1] = xi;
+            a[k1 + 3] = -a[k1 + 3];
+        }
+    } else {
+        for (k = 0; k < m; k++) {
+            for (j = 0; j < k; j++) {
+                j1 = 4 * j + ip[m + k];
+                k1 = 4 * k + ip[m + j];
+                xr = a[j1];
+                xi = -a[j1 + 1];
+                yr = a[k1];
+                yi = -a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 += nm;
+                k1 += nm;
+                xr = a[j1];
+                xi = -a[j1 + 1];
+                yr = a[k1];
+                yi = -a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 += nh;
+                k1 += 2;
+                xr = a[j1];
+                xi = -a[j1 + 1];
+                yr = a[k1];
+                yi = -a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 -= nm;
+                k1 -= nm;
+                xr = a[j1];
+                xi = -a[j1 + 1];
+                yr = a[k1];
+                yi = -a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 += 2;
+                k1 += nh;
+                xr = a[j1];
+                xi = -a[j1 + 1];
+                yr = a[k1];
+                yi = -a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 += nm;
+                k1 += nm;
+                xr = a[j1];
+                xi = -a[j1 + 1];
+                yr = a[k1];
+                yi = -a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 -= nh;
+                k1 -= 2;
+                xr = a[j1];
+                xi = -a[j1 + 1];
+                yr = a[k1];
+                yi = -a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+                j1 -= nm;
+                k1 -= nm;
+                xr = a[j1];
+                xi = -a[j1 + 1];
+                yr = a[k1];
+                yi = -a[k1 + 1];
+                a[j1] = yr;
+                a[j1 + 1] = yi;
+                a[k1] = xr;
+                a[k1 + 1] = xi;
+            }
+            k1 = 4 * k + ip[m + k];
+            j1 = k1 + 2;
+            k1 += nh;
+            a[j1 - 1] = -a[j1 - 1];
+            xr = a[j1];
+            xi = -a[j1 + 1];
+            yr = a[k1];
+            yi = -a[k1 + 1];
+            a[j1] = yr;
+            a[j1 + 1] = yi;
+            a[k1] = xr;
+            a[k1 + 1] = xi;
+            a[k1 + 3] = -a[k1 + 3];
+            j1 += nm;
+            k1 += nm;
+            a[j1 - 1] = -a[j1 - 1];
+            xr = a[j1];
+            xi = -a[j1 + 1];
+            yr = a[k1];
+            yi = -a[k1 + 1];
+            a[j1] = yr;
+            a[j1 + 1] = yi;
+            a[k1] = xr;
+            a[k1 + 1] = xi;
+            a[k1 + 3] = -a[k1 + 3];
+        }
+    }
+}
+
+
+void bitrv216(float32 *a)
+{
+    float32 x1r, x1i, x2r, x2i, x3r, x3i, x4r, x4i, 
+        x5r, x5i, x7r, x7i, x8r, x8i, x10r, x10i, 
+        x11r, x11i, x12r, x12i, x13r, x13i, x14r, x14i;
+    
+    x1r = a[2];
+    x1i = a[3];
+    x2r = a[4];
+    x2i = a[5];
+    x3r = a[6];
+    x3i = a[7];
+    x4r = a[8];
+    x4i = a[9];
+    x5r = a[10];
+    x5i = a[11];
+    x7r = a[14];
+    x7i = a[15];
+    x8r = a[16];
+    x8i = a[17];
+    x10r = a[20];
+    x10i = a[21];
+    x11r = a[22];
+    x11i = a[23];
+    x12r = a[24];
+    x12i = a[25];
+    x13r = a[26];
+    x13i = a[27];
+    x14r = a[28];
+    x14i = a[29];
+    a[2] = x8r;
+    a[3] = x8i;
+    a[4] = x4r;
+    a[5] = x4i;
+    a[6] = x12r;
+    a[7] = x12i;
+    a[8] = x2r;
+    a[9] = x2i;
+    a[10] = x10r;
+    a[11] = x10i;
+    a[14] = x14r;
+    a[15] = x14i;
+    a[16] = x1r;
+    a[17] = x1i;
+    a[20] = x5r;
+    a[21] = x5i;
+    a[22] = x13r;
+    a[23] = x13i;
+    a[24] = x3r;
+    a[25] = x3i;
+    a[26] = x11r;
+    a[27] = x11i;
+    a[28] = x7r;
+    a[29] = x7i;
+}
+
+
+void bitrv216neg(float32 *a)
+{
+    float32 x1r, x1i, x2r, x2i, x3r, x3i, x4r, x4i, 
+        x5r, x5i, x6r, x6i, x7r, x7i, x8r, x8i, 
+        x9r, x9i, x10r, x10i, x11r, x11i, x12r, x12i, 
+        x13r, x13i, x14r, x14i, x15r, x15i;
+    
+    x1r = a[2];
+    x1i = a[3];
+    x2r = a[4];
+    x2i = a[5];
+    x3r = a[6];
+    x3i = a[7];
+    x4r = a[8];
+    x4i = a[9];
+    x5r = a[10];
+    x5i = a[11];
+    x6r = a[12];
+    x6i = a[13];
+    x7r = a[14];
+    x7i = a[15];
+    x8r = a[16];
+    x8i = a[17];
+    x9r = a[18];
+    x9i = a[19];
+    x10r = a[20];
+    x10i = a[21];
+    x11r = a[22];
+    x11i = a[23];
+    x12r = a[24];
+    x12i = a[25];
+    x13r = a[26];
+    x13i = a[27];
+    x14r = a[28];
+    x14i = a[29];
+    x15r = a[30];
+    x15i = a[31];
+    a[2] = x15r;
+    a[3] = x15i;
+    a[4] = x7r;
+    a[5] = x7i;
+    a[6] = x11r;
+    a[7] = x11i;
+    a[8] = x3r;
+    a[9] = x3i;
+    a[10] = x13r;
+    a[11] = x13i;
+    a[12] = x5r;
+    a[13] = x5i;
+    a[14] = x9r;
+    a[15] = x9i;
+    a[16] = x1r;
+    a[17] = x1i;
+    a[18] = x14r;
+    a[19] = x14i;
+    a[20] = x6r;
+    a[21] = x6i;
+    a[22] = x10r;
+    a[23] = x10i;
+    a[24] = x2r;
+    a[25] = x2i;
+    a[26] = x12r;
+    a[27] = x12i;
+    a[28] = x4r;
+    a[29] = x4i;
+    a[30] = x8r;
+    a[31] = x8i;
+}
+
+
+void bitrv208(float32 *a)
+{
+    float32 x1r, x1i, x3r, x3i, x4r, x4i, x6r, x6i;
+    
+    x1r = a[2];
+    x1i = a[3];
+    x3r = a[6];
+    x3i = a[7];
+    x4r = a[8];
+    x4i = a[9];
+    x6r = a[12];
+    x6i = a[13];
+    a[2] = x4r;
+    a[3] = x4i;
+    a[6] = x6r;
+    a[7] = x6i;
+    a[8] = x1r;
+    a[9] = x1i;
+    a[12] = x3r;
+    a[13] = x3i;
+}
+
+
+void bitrv208neg(float32 *a)
+{
+    float32 x1r, x1i, x2r, x2i, x3r, x3i, x4r, x4i, 
+        x5r, x5i, x6r, x6i, x7r, x7i;
+    
+    x1r = a[2];
+    x1i = a[3];
+    x2r = a[4];
+    x2i = a[5];
+    x3r = a[6];
+    x3i = a[7];
+    x4r = a[8];
+    x4i = a[9];
+    x5r = a[10];
+    x5i = a[11];
+    x6r = a[12];
+    x6i = a[13];
+    x7r = a[14];
+    x7i = a[15];
+    a[2] = x7r;
+    a[3] = x7i;
+    a[4] = x3r;
+    a[5] = x3i;
+    a[6] = x5r;
+    a[7] = x5i;
+    a[8] = x1r;
+    a[9] = x1i;
+    a[10] = x6r;
+    a[11] = x6i;
+    a[12] = x2r;
+    a[13] = x2i;
+    a[14] = x4r;
+    a[15] = x4i;
+}
+
+
+void cftf1st(int n, float32 *a, float32 *w)
+{
+    int j, j0, j1, j2, j3, k, m, mh;
+    float32 wn4r, csc1, csc3, wk1r, wk1i, wk3r, wk3i, 
+        wd1r, wd1i, wd3r, wd3i;
+    float32 x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i, 
+        y0r, y0i, y1r, y1i, y2r, y2i, y3r, y3i;
+    
+    mh = n >> 3;
+    m = 2 * mh;
+    j1 = m;
+    j2 = j1 + m;
+    j3 = j2 + m;
+    x0r = a[0] + a[j2];
+    x0i = a[1] + a[j2 + 1];
+    x1r = a[0] - a[j2];
+    x1i = a[1] - a[j2 + 1];
+    x2r = a[j1] + a[j3];
+    x2i = a[j1 + 1] + a[j3 + 1];
+    x3r = a[j1] - a[j3];
+    x3i = a[j1 + 1] - a[j3 + 1];
+    a[0] = x0r + x2r;
+    a[1] = x0i + x2i;
+    a[j1] = x0r - x2r;
+    a[j1 + 1] = x0i - x2i;
+    a[j2] = x1r - x3i;
+    a[j2 + 1] = x1i + x3r;
+    a[j3] = x1r + x3i;
+    a[j3 + 1] = x1i - x3r;
+    wn4r = w[1];
+    csc1 = w[2];
+    csc3 = w[3];
+    wd1r = 1;
+    wd1i = 0;
+    wd3r = 1;
+    wd3i = 0;
+    k = 0;
+    for (j = 2; j < mh - 2; j += 4) {
+        k += 4;
+        wk1r = csc1 * (wd1r + w[k]);
+        wk1i = csc1 * (wd1i + w[k + 1]);
+        wk3r = csc3 * (wd3r + w[k + 2]);
+        wk3i = csc3 * (wd3i + w[k + 3]);
+        wd1r = w[k];
+        wd1i = w[k + 1];
+        wd3r = w[k + 2];
+        wd3i = w[k + 3];
+        j1 = j + m;
+        j2 = j1 + m;
+        j3 = j2 + m;
+        x0r = a[j] + a[j2];
+        x0i = a[j + 1] + a[j2 + 1];
+        x1r = a[j] - a[j2];
+        x1i = a[j + 1] - a[j2 + 1];
+        y0r = a[j + 2] + a[j2 + 2];
+        y0i = a[j + 3] + a[j2 + 3];
+        y1r = a[j + 2] - a[j2 + 2];
+        y1i = a[j + 3] - a[j2 + 3];
+        x2r = a[j1] + a[j3];
+        x2i = a[j1 + 1] + a[j3 + 1];
+        x3r = a[j1] - a[j3];
+        x3i = a[j1 + 1] - a[j3 + 1];
+        y2r = a[j1 + 2] + a[j3 + 2];
+        y2i = a[j1 + 3] + a[j3 + 3];
+        y3r = a[j1 + 2] - a[j3 + 2];
+        y3i = a[j1 + 3] - a[j3 + 3];
+        a[j] = x0r + x2r;
+        a[j + 1] = x0i + x2i;
+        a[j + 2] = y0r + y2r;
+        a[j + 3] = y0i + y2i;
+        a[j1] = x0r - x2r;
+        a[j1 + 1] = x0i - x2i;
+        a[j1 + 2] = y0r - y2r;
+        a[j1 + 3] = y0i - y2i;
+        x0r = x1r - x3i;
+        x0i = x1i + x3r;
+        a[j2] = wk1r * x0r - wk1i * x0i;
+        a[j2 + 1] = wk1r * x0i + wk1i * x0r;
+        x0r = y1r - y3i;
+        x0i = y1i + y3r;
+        a[j2 + 2] = wd1r * x0r - wd1i * x0i;
+        a[j2 + 3] = wd1r * x0i + wd1i * x0r;
+        x0r = x1r + x3i;
+        x0i = x1i - x3r;
+        a[j3] = wk3r * x0r + wk3i * x0i;
+        a[j3 + 1] = wk3r * x0i - wk3i * x0r;
+        x0r = y1r + y3i;
+        x0i = y1i - y3r;
+        a[j3 + 2] = wd3r * x0r + wd3i * x0i;
+        a[j3 + 3] = wd3r * x0i - wd3i * x0r;
+        j0 = m - j;
+        j1 = j0 + m;
+        j2 = j1 + m;
+        j3 = j2 + m;
+        x0r = a[j0] + a[j2];
+        x0i = a[j0 + 1] + a[j2 + 1];
+        x1r = a[j0] - a[j2];
+        x1i = a[j0 + 1] - a[j2 + 1];
+        y0r = a[j0 - 2] + a[j2 - 2];
+        y0i = a[j0 - 1] + a[j2 - 1];
+        y1r = a[j0 - 2] - a[j2 - 2];
+        y1i = a[j0 - 1] - a[j2 - 1];
+        x2r = a[j1] + a[j3];
+        x2i = a[j1 + 1] + a[j3 + 1];
+        x3r = a[j1] - a[j3];
+        x3i = a[j1 + 1] - a[j3 + 1];
+        y2r = a[j1 - 2] + a[j3 - 2];
+        y2i = a[j1 - 1] + a[j3 - 1];
+        y3r = a[j1 - 2] - a[j3 - 2];
+        y3i = a[j1 - 1] - a[j3 - 1];
+        a[j0] = x0r + x2r;
+        a[j0 + 1] = x0i + x2i;
+        a[j0 - 2] = y0r + y2r;
+        a[j0 - 1] = y0i + y2i;
+        a[j1] = x0r - x2r;
+        a[j1 + 1] = x0i - x2i;
+        a[j1 - 2] = y0r - y2r;
+        a[j1 - 1] = y0i - y2i;
+        x0r = x1r - x3i;
+        x0i = x1i + x3r;
+        a[j2] = wk1i * x0r - wk1r * x0i;
+        a[j2 + 1] = wk1i * x0i + wk1r * x0r;
+        x0r = y1r - y3i;
+        x0i = y1i + y3r;
+        a[j2 - 2] = wd1i * x0r - wd1r * x0i;
+        a[j2 - 1] = wd1i * x0i + wd1r * x0r;
+        x0r = x1r + x3i;
+        x0i = x1i - x3r;
+        a[j3] = wk3i * x0r + wk3r * x0i;
+        a[j3 + 1] = wk3i * x0i - wk3r * x0r;
+        x0r = y1r + y3i;
+        x0i = y1i - y3r;
+        a[j3 - 2] = wd3i * x0r + wd3r * x0i;
+        a[j3 - 1] = wd3i * x0i - wd3r * x0r;
+    }
+    wk1r = csc1 * (wd1r + wn4r);
+    wk1i = csc1 * (wd1i + wn4r);
+    wk3r = csc3 * (wd3r - wn4r);
+    wk3i = csc3 * (wd3i - wn4r);
+    j0 = mh;
+    j1 = j0 + m;
+    j2 = j1 + m;
+    j3 = j2 + m;
+    x0r = a[j0 - 2] + a[j2 - 2];
+    x0i = a[j0 - 1] + a[j2 - 1];
+    x1r = a[j0 - 2] - a[j2 - 2];
+    x1i = a[j0 - 1] - a[j2 - 1];
+    x2r = a[j1 - 2] + a[j3 - 2];
+    x2i = a[j1 - 1] + a[j3 - 1];
+    x3r = a[j1 - 2] - a[j3 - 2];
+    x3i = a[j1 - 1] - a[j3 - 1];
+    a[j0 - 2] = x0r + x2r;
+    a[j0 - 1] = x0i + x2i;
+    a[j1 - 2] = x0r - x2r;
+    a[j1 - 1] = x0i - x2i;
+    x0r = x1r - x3i;
+    x0i = x1i + x3r;
+    a[j2 - 2] = wk1r * x0r - wk1i * x0i;
+    a[j2 - 1] = wk1r * x0i + wk1i * x0r;
+    x0r = x1r + x3i;
+    x0i = x1i - x3r;
+    a[j3 - 2] = wk3r * x0r + wk3i * x0i;
+    a[j3 - 1] = wk3r * x0i - wk3i * x0r;
+    x0r = a[j0] + a[j2];
+    x0i = a[j0 + 1] + a[j2 + 1];
+    x1r = a[j0] - a[j2];
+    x1i = a[j0 + 1] - a[j2 + 1];
+    x2r = a[j1] + a[j3];
+    x2i = a[j1 + 1] + a[j3 + 1];
+    x3r = a[j1] - a[j3];
+    x3i = a[j1 + 1] - a[j3 + 1];
+    a[j0] = x0r + x2r;
+    a[j0 + 1] = x0i + x2i;
+    a[j1] = x0r - x2r;
+    a[j1 + 1] = x0i - x2i;
+    x0r = x1r - x3i;
+    x0i = x1i + x3r;
+    a[j2] = wn4r * (x0r - x0i);
+    a[j2 + 1] = wn4r * (x0i + x0r);
+    x0r = x1r + x3i;
+    x0i = x1i - x3r;
+    a[j3] = -wn4r * (x0r + x0i);
+    a[j3 + 1] = -wn4r * (x0i - x0r);
+    x0r = a[j0 + 2] + a[j2 + 2];
+    x0i = a[j0 + 3] + a[j2 + 3];
+    x1r = a[j0 + 2] - a[j2 + 2];
+    x1i = a[j0 + 3] - a[j2 + 3];
+    x2r = a[j1 + 2] + a[j3 + 2];
+    x2i = a[j1 + 3] + a[j3 + 3];
+    x3r = a[j1 + 2] - a[j3 + 2];
+    x3i = a[j1 + 3] - a[j3 + 3];
+    a[j0 + 2] = x0r + x2r;
+    a[j0 + 3] = x0i + x2i;
+    a[j1 + 2] = x0r - x2r;
+    a[j1 + 3] = x0i - x2i;
+    x0r = x1r - x3i;
+    x0i = x1i + x3r;
+    a[j2 + 2] = wk1i * x0r - wk1r * x0i;
+    a[j2 + 3] = wk1i * x0i + wk1r * x0r;
+    x0r = x1r + x3i;
+    x0i = x1i - x3r;
+    a[j3 + 2] = wk3i * x0r + wk3r * x0i;
+    a[j3 + 3] = wk3i * x0i - wk3r * x0r;
+}
+
+
+void cftb1st(int n, float32 *a, float32 *w)
+{
+    int j, j0, j1, j2, j3, k, m, mh;
+    float32 wn4r, csc1, csc3, wk1r, wk1i, wk3r, wk3i, 
+        wd1r, wd1i, wd3r, wd3i;
+    float32 x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i, 
+        y0r, y0i, y1r, y1i, y2r, y2i, y3r, y3i;
+    
+    mh = n >> 3;
+    m = 2 * mh;
+    j1 = m;
+    j2 = j1 + m;
+    j3 = j2 + m;
+    x0r = a[0] + a[j2];
+    x0i = -a[1] - a[j2 + 1];
+    x1r = a[0] - a[j2];
+    x1i = -a[1] + a[j2 + 1];
+    x2r = a[j1] + a[j3];
+    x2i = a[j1 + 1] + a[j3 + 1];
+    x3r = a[j1] - a[j3];
+    x3i = a[j1 + 1] - a[j3 + 1];
+    a[0] = x0r + x2r;
+    a[1] = x0i - x2i;
+    a[j1] = x0r - x2r;
+    a[j1 + 1] = x0i + x2i;
+    a[j2] = x1r + x3i;
+    a[j2 + 1] = x1i + x3r;
+    a[j3] = x1r - x3i;
+    a[j3 + 1] = x1i - x3r;
+    wn4r = w[1];
+    csc1 = w[2];
+    csc3 = w[3];
+    wd1r = 1;
+    wd1i = 0;
+    wd3r = 1;
+    wd3i = 0;
+    k = 0;
+    for (j = 2; j < mh - 2; j += 4) {
+        k += 4;
+        wk1r = csc1 * (wd1r + w[k]);
+        wk1i = csc1 * (wd1i + w[k + 1]);
+        wk3r = csc3 * (wd3r + w[k + 2]);
+        wk3i = csc3 * (wd3i + w[k + 3]);
+        wd1r = w[k];
+        wd1i = w[k + 1];
+        wd3r = w[k + 2];
+        wd3i = w[k + 3];
+        j1 = j + m;
+        j2 = j1 + m;
+        j3 = j2 + m;
+        x0r = a[j] + a[j2];
+        x0i = -a[j + 1] - a[j2 + 1];
+        x1r = a[j] - a[j2];
+        x1i = -a[j + 1] + a[j2 + 1];
+        y0r = a[j + 2] + a[j2 + 2];
+        y0i = -a[j + 3] - a[j2 + 3];
+        y1r = a[j + 2] - a[j2 + 2];
+        y1i = -a[j + 3] + a[j2 + 3];
+        x2r = a[j1] + a[j3];
+        x2i = a[j1 + 1] + a[j3 + 1];
+        x3r = a[j1] - a[j3];
+        x3i = a[j1 + 1] - a[j3 + 1];
+        y2r = a[j1 + 2] + a[j3 + 2];
+        y2i = a[j1 + 3] + a[j3 + 3];
+        y3r = a[j1 + 2] - a[j3 + 2];
+        y3i = a[j1 + 3] - a[j3 + 3];
+        a[j] = x0r + x2r;
+        a[j + 1] = x0i - x2i;
+        a[j + 2] = y0r + y2r;
+        a[j + 3] = y0i - y2i;
+        a[j1] = x0r - x2r;
+        a[j1 + 1] = x0i + x2i;
+        a[j1 + 2] = y0r - y2r;
+        a[j1 + 3] = y0i + y2i;
+        x0r = x1r + x3i;
+        x0i = x1i + x3r;
+        a[j2] = wk1r * x0r - wk1i * x0i;
+        a[j2 + 1] = wk1r * x0i + wk1i * x0r;
+        x0r = y1r + y3i;
+        x0i = y1i + y3r;
+        a[j2 + 2] = wd1r * x0r - wd1i * x0i;
+        a[j2 + 3] = wd1r * x0i + wd1i * x0r;
+        x0r = x1r - x3i;
+        x0i = x1i - x3r;
+        a[j3] = wk3r * x0r + wk3i * x0i;
+        a[j3 + 1] = wk3r * x0i - wk3i * x0r;
+        x0r = y1r - y3i;
+        x0i = y1i - y3r;
+        a[j3 + 2] = wd3r * x0r + wd3i * x0i;
+        a[j3 + 3] = wd3r * x0i - wd3i * x0r;
+        j0 = m - j;
+        j1 = j0 + m;
+        j2 = j1 + m;
+        j3 = j2 + m;
+        x0r = a[j0] + a[j2];
+        x0i = -a[j0 + 1] - a[j2 + 1];
+        x1r = a[j0] - a[j2];
+        x1i = -a[j0 + 1] + a[j2 + 1];
+        y0r = a[j0 - 2] + a[j2 - 2];
+        y0i = -a[j0 - 1] - a[j2 - 1];
+        y1r = a[j0 - 2] - a[j2 - 2];
+        y1i = -a[j0 - 1] + a[j2 - 1];
+        x2r = a[j1] + a[j3];
+        x2i = a[j1 + 1] + a[j3 + 1];
+        x3r = a[j1] - a[j3];
+        x3i = a[j1 + 1] - a[j3 + 1];
+        y2r = a[j1 - 2] + a[j3 - 2];
+        y2i = a[j1 - 1] + a[j3 - 1];
+        y3r = a[j1 - 2] - a[j3 - 2];
+        y3i = a[j1 - 1] - a[j3 - 1];
+        a[j0] = x0r + x2r;
+        a[j0 + 1] = x0i - x2i;
+        a[j0 - 2] = y0r + y2r;
+        a[j0 - 1] = y0i - y2i;
+        a[j1] = x0r - x2r;
+        a[j1 + 1] = x0i + x2i;
+        a[j1 - 2] = y0r - y2r;
+        a[j1 - 1] = y0i + y2i;
+        x0r = x1r + x3i;
+        x0i = x1i + x3r;
+        a[j2] = wk1i * x0r - wk1r * x0i;
+        a[j2 + 1] = wk1i * x0i + wk1r * x0r;
+        x0r = y1r + y3i;
+        x0i = y1i + y3r;
+        a[j2 - 2] = wd1i * x0r - wd1r * x0i;
+        a[j2 - 1] = wd1i * x0i + wd1r * x0r;
+        x0r = x1r - x3i;
+        x0i = x1i - x3r;
+        a[j3] = wk3i * x0r + wk3r * x0i;
+        a[j3 + 1] = wk3i * x0i - wk3r * x0r;
+        x0r = y1r - y3i;
+        x0i = y1i - y3r;
+        a[j3 - 2] = wd3i * x0r + wd3r * x0i;
+        a[j3 - 1] = wd3i * x0i - wd3r * x0r;
+    }
+    wk1r = csc1 * (wd1r + wn4r);
+    wk1i = csc1 * (wd1i + wn4r);
+    wk3r = csc3 * (wd3r - wn4r);
+    wk3i = csc3 * (wd3i - wn4r);
+    j0 = mh;
+    j1 = j0 + m;
+    j2 = j1 + m;
+    j3 = j2 + m;
+    x0r = a[j0 - 2] + a[j2 - 2];
+    x0i = -a[j0 - 1] - a[j2 - 1];
+    x1r = a[j0 - 2] - a[j2 - 2];
+    x1i = -a[j0 - 1] + a[j2 - 1];
+    x2r = a[j1 - 2] + a[j3 - 2];
+    x2i = a[j1 - 1] + a[j3 - 1];
+    x3r = a[j1 - 2] - a[j3 - 2];
+    x3i = a[j1 - 1] - a[j3 - 1];
+    a[j0 - 2] = x0r + x2r;
+    a[j0 - 1] = x0i - x2i;
+    a[j1 - 2] = x0r - x2r;
+    a[j1 - 1] = x0i + x2i;
+    x0r = x1r + x3i;
+    x0i = x1i + x3r;
+    a[j2 - 2] = wk1r * x0r - wk1i * x0i;
+    a[j2 - 1] = wk1r * x0i + wk1i * x0r;
+    x0r = x1r - x3i;
+    x0i = x1i - x3r;
+    a[j3 - 2] = wk3r * x0r + wk3i * x0i;
+    a[j3 - 1] = wk3r * x0i - wk3i * x0r;
+    x0r = a[j0] + a[j2];
+    x0i = -a[j0 + 1] - a[j2 + 1];
+    x1r = a[j0] - a[j2];
+    x1i = -a[j0 + 1] + a[j2 + 1];
+    x2r = a[j1] + a[j3];
+    x2i = a[j1 + 1] + a[j3 + 1];
+    x3r = a[j1] - a[j3];
+    x3i = a[j1 + 1] - a[j3 + 1];
+    a[j0] = x0r + x2r;
+    a[j0 + 1] = x0i - x2i;
+    a[j1] = x0r - x2r;
+    a[j1 + 1] = x0i + x2i;
+    x0r = x1r + x3i;
+    x0i = x1i + x3r;
+    a[j2] = wn4r * (x0r - x0i);
+    a[j2 + 1] = wn4r * (x0i + x0r);
+    x0r = x1r - x3i;
+    x0i = x1i - x3r;
+    a[j3] = -wn4r * (x0r + x0i);
+    a[j3 + 1] = -wn4r * (x0i - x0r);
+    x0r = a[j0 + 2] + a[j2 + 2];
+    x0i = -a[j0 + 3] - a[j2 + 3];
+    x1r = a[j0 + 2] - a[j2 + 2];
+    x1i = -a[j0 + 3] + a[j2 + 3];
+    x2r = a[j1 + 2] + a[j3 + 2];
+    x2i = a[j1 + 3] + a[j3 + 3];
+    x3r = a[j1 + 2] - a[j3 + 2];
+    x3i = a[j1 + 3] - a[j3 + 3];
+    a[j0 + 2] = x0r + x2r;
+    a[j0 + 3] = x0i - x2i;
+    a[j1 + 2] = x0r - x2r;
+    a[j1 + 3] = x0i + x2i;
+    x0r = x1r + x3i;
+    x0i = x1i + x3r;
+    a[j2 + 2] = wk1i * x0r - wk1r * x0i;
+    a[j2 + 3] = wk1i * x0i + wk1r * x0r;
+    x0r = x1r - x3i;
+    x0i = x1i - x3r;
+    a[j3 + 2] = wk3i * x0r + wk3r * x0i;
+    a[j3 + 3] = wk3i * x0i - wk3r * x0r;
+}
+
+
+#ifdef USE_CDFT_THREADS
+struct cdft_arg_st {
+    int n0;
+    int n;
+    float32 *a;
+    int nw;
+    float32 *w;
+};
+typedef struct cdft_arg_st cdft_arg_t;
+
+
+void cftrec4_th(int n, float32 *a, int nw, float32 *w)
+{
+    void *cftrec1_th(void *p);
+    void *cftrec2_th(void *p);
+    int i, idiv4, m, nthread;
+    cdft_thread_t th[4];
+    cdft_arg_t ag[4];
+    
+    nthread = 2;
+    idiv4 = 0;
+    m = n >> 1;
+    if (n > CDFT_4THREADS_BEGIN_N) {
+        nthread = 4;
+        idiv4 = 1;
+        m >>= 1;
+    }
+    for (i = 0; i < nthread; i++) {
+        ag[i].n0 = n;
+        ag[i].n = m;
+        ag[i].a = &a[i * m];
+        ag[i].nw = nw;
+        ag[i].w = w;
+        if (i != idiv4) {
+            cdft_thread_create(&th[i], cftrec1_th, &ag[i]);
+        } else {
+            cdft_thread_create(&th[i], cftrec2_th, &ag[i]);
+        }
+    }
+    for (i = 0; i < nthread; i++) {
+        cdft_thread_wait(th[i]);
+    }
+}
+
+
+void *cftrec1_th(void *p)
+{
+    int cfttree(int n, int j, int k, float32 *a, int nw, float32 *w);
+    void cftleaf(int n, int isplt, float32 *a, int nw, float32 *w);
+    void cftmdl1(int n, float32 *a, float32 *w);
+    int isplt, j, k, m, n, n0, nw;
+    float32 *a, *w;
+    
+    n0 = ((cdft_arg_t *) p)->n0;
+    n = ((cdft_arg_t *) p)->n;
+    a = ((cdft_arg_t *) p)->a;
+    nw = ((cdft_arg_t *) p)->nw;
+    w = ((cdft_arg_t *) p)->w;
+    m = n0;
+    while (m > 512) {
+        m >>= 2;
+        cftmdl1(m, &a[n - m], &w[nw - (m >> 1)]);
+    }
+    cftleaf(m, 1, &a[n - m], nw, w);
+    k = 0;
+    for (j = n - m; j > 0; j -= m) {
+        k++;
+        isplt = cfttree(m, j, k, a, nw, w);
+        cftleaf(m, isplt, &a[j - m], nw, w);
+    }
+    return (void *) 0;
+}
+
+
+void *cftrec2_th(void *p)
+{
+    int cfttree(int n, int j, int k, float32 *a, int nw, float32 *w);
+    void cftleaf(int n, int isplt, float32 *a, int nw, float32 *w);
+    void cftmdl2(int n, float32 *a, float32 *w);
+    int isplt, j, k, m, n, n0, nw;
+    float32 *a, *w;
+    
+    n0 = ((cdft_arg_t *) p)->n0;
+    n = ((cdft_arg_t *) p)->n;
+    a = ((cdft_arg_t *) p)->a;
+    nw = ((cdft_arg_t *) p)->nw;
+    w = ((cdft_arg_t *) p)->w;
+    k = 1;
+    m = n0;
+    while (m > 512) {
+        m >>= 2;
+        k <<= 2;
+        cftmdl2(m, &a[n - m], &w[nw - m]);
+    }
+    cftleaf(m, 0, &a[n - m], nw, w);
+    k >>= 1;
+    for (j = n - m; j > 0; j -= m) {
+        k++;
+        isplt = cfttree(m, j, k, a, nw, w);
+        cftleaf(m, isplt, &a[j - m], nw, w);
+    }
+    return (void *) 0;
+}
+#endif /* USE_CDFT_THREADS */
+
+
+void cftrec4(int n, float32 *a, int nw, float32 *w)
+{
+    int cfttree(int n, int j, int k, float32 *a, int nw, float32 *w);
+    void cftleaf(int n, int isplt, float32 *a, int nw, float32 *w);
+    void cftmdl1(int n, float32 *a, float32 *w);
+    int isplt, j, k, m;
+    
+    m = n;
+    while (m > 512) {
+        m >>= 2;
+        cftmdl1(m, &a[n - m], &w[nw - (m >> 1)]);
+    }
+    cftleaf(m, 1, &a[n - m], nw, w);
+    k = 0;
+    for (j = n - m; j > 0; j -= m) {
+        k++;
+        isplt = cfttree(m, j, k, a, nw, w);
+        cftleaf(m, isplt, &a[j - m], nw, w);
+    }
+}
+
+
+int cfttree(int n, int j, int k, float32 *a, int nw, float32 *w)
+{
+    void cftmdl1(int n, float32 *a, float32 *w);
+    void cftmdl2(int n, float32 *a, float32 *w);
+    int i, isplt, m;
+    
+    if ((k & 3) != 0) {
+        isplt = k & 1;
+        if (isplt != 0) {
+            cftmdl1(n, &a[j - n], &w[nw - (n >> 1)]);
+        } else {
+            cftmdl2(n, &a[j - n], &w[nw - n]);
+        }
+    } else {
+        m = n;
+        for (i = k; (i & 3) == 0; i >>= 2) {
+            m <<= 2;
+        }
+        isplt = i & 1;
+        if (isplt != 0) {
+            while (m > 128) {
+                cftmdl1(m, &a[j - m], &w[nw - (m >> 1)]);
+                m >>= 2;
+            }
+        } else {
+            while (m > 128) {
+                cftmdl2(m, &a[j - m], &w[nw - m]);
+                m >>= 2;
+            }
+        }
+    }
+    return isplt;
+}
+
+
+void cftleaf(int n, int isplt, float32 *a, int nw, float32 *w)
+{
+    void cftmdl1(int n, float32 *a, float32 *w);
+    void cftmdl2(int n, float32 *a, float32 *w);
+    void cftf161(float32 *a, float32 *w);
+    void cftf162(float32 *a, float32 *w);
+    void cftf081(float32 *a, float32 *w);
+    void cftf082(float32 *a, float32 *w);
+    
+    if (n == 512) {
+        cftmdl1(128, a, &w[nw - 64]);
+        cftf161(a, &w[nw - 8]);
+        cftf162(&a[32], &w[nw - 32]);
+        cftf161(&a[64], &w[nw - 8]);
+        cftf161(&a[96], &w[nw - 8]);
+        cftmdl2(128, &a[128], &w[nw - 128]);
+        cftf161(&a[128], &w[nw - 8]);
+        cftf162(&a[160], &w[nw - 32]);
+        cftf161(&a[192], &w[nw - 8]);
+        cftf162(&a[224], &w[nw - 32]);
+        cftmdl1(128, &a[256], &w[nw - 64]);
+        cftf161(&a[256], &w[nw - 8]);
+        cftf162(&a[288], &w[nw - 32]);
+        cftf161(&a[320], &w[nw - 8]);
+        cftf161(&a[352], &w[nw - 8]);
+        if (isplt != 0) {
+            cftmdl1(128, &a[384], &w[nw - 64]);
+            cftf161(&a[480], &w[nw - 8]);
+        } else {
+            cftmdl2(128, &a[384], &w[nw - 128]);
+            cftf162(&a[480], &w[nw - 32]);
+        }
+        cftf161(&a[384], &w[nw - 8]);
+        cftf162(&a[416], &w[nw - 32]);
+        cftf161(&a[448], &w[nw - 8]);
+    } else {
+        cftmdl1(64, a, &w[nw - 32]);
+        cftf081(a, &w[nw - 8]);
+        cftf082(&a[16], &w[nw - 8]);
+        cftf081(&a[32], &w[nw - 8]);
+        cftf081(&a[48], &w[nw - 8]);
+        cftmdl2(64, &a[64], &w[nw - 64]);
+        cftf081(&a[64], &w[nw - 8]);
+        cftf082(&a[80], &w[nw - 8]);
+        cftf081(&a[96], &w[nw - 8]);
+        cftf082(&a[112], &w[nw - 8]);
+        cftmdl1(64, &a[128], &w[nw - 32]);
+        cftf081(&a[128], &w[nw - 8]);
+        cftf082(&a[144], &w[nw - 8]);
+        cftf081(&a[160], &w[nw - 8]);
+        cftf081(&a[176], &w[nw - 8]);
+        if (isplt != 0) {
+            cftmdl1(64, &a[192], &w[nw - 32]);
+            cftf081(&a[240], &w[nw - 8]);
+        } else {
+            cftmdl2(64, &a[192], &w[nw - 64]);
+            cftf082(&a[240], &w[nw - 8]);
+        }
+        cftf081(&a[192], &w[nw - 8]);
+        cftf082(&a[208], &w[nw - 8]);
+        cftf081(&a[224], &w[nw - 8]);
+    }
+}
+
+
+void cftmdl1(int n, float32 *a, float32 *w)
+{
+    int j, j0, j1, j2, j3, k, m, mh;
+    float32 wn4r, wk1r, wk1i, wk3r, wk3i;
+    float32 x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i;
+    
+    mh = n >> 3;
+    m = 2 * mh;
+    j1 = m;
+    j2 = j1 + m;
+    j3 = j2 + m;
+    x0r = a[0] + a[j2];
+    x0i = a[1] + a[j2 + 1];
+    x1r = a[0] - a[j2];
+    x1i = a[1] - a[j2 + 1];
+    x2r = a[j1] + a[j3];
+    x2i = a[j1 + 1] + a[j3 + 1];
+    x3r = a[j1] - a[j3];
+    x3i = a[j1 + 1] - a[j3 + 1];
+    a[0] = x0r + x2r;
+    a[1] = x0i + x2i;
+    a[j1] = x0r - x2r;
+    a[j1 + 1] = x0i - x2i;
+    a[j2] = x1r - x3i;
+    a[j2 + 1] = x1i + x3r;
+    a[j3] = x1r + x3i;
+    a[j3 + 1] = x1i - x3r;
+    wn4r = w[1];
+    k = 0;
+    for (j = 2; j < mh; j += 2) {
+        k += 4;
+        wk1r = w[k];
+        wk1i = w[k + 1];
+        wk3r = w[k + 2];
+        wk3i = w[k + 3];
+        j1 = j + m;
+        j2 = j1 + m;
+        j3 = j2 + m;
+        x0r = a[j] + a[j2];
+        x0i = a[j + 1] + a[j2 + 1];
+        x1r = a[j] - a[j2];
+        x1i = a[j + 1] - a[j2 + 1];
+        x2r = a[j1] + a[j3];
+        x2i = a[j1 + 1] + a[j3 + 1];
+        x3r = a[j1] - a[j3];
+        x3i = a[j1 + 1] - a[j3 + 1];
+        a[j] = x0r + x2r;
+        a[j + 1] = x0i + x2i;
+        a[j1] = x0r - x2r;
+        a[j1 + 1] = x0i - x2i;
+        x0r = x1r - x3i;
+        x0i = x1i + x3r;
+        a[j2] = wk1r * x0r - wk1i * x0i;
+        a[j2 + 1] = wk1r * x0i + wk1i * x0r;
+        x0r = x1r + x3i;
+        x0i = x1i - x3r;
+        a[j3] = wk3r * x0r + wk3i * x0i;
+        a[j3 + 1] = wk3r * x0i - wk3i * x0r;
+        j0 = m - j;
+        j1 = j0 + m;
+        j2 = j1 + m;
+        j3 = j2 + m;
+        x0r = a[j0] + a[j2];
+        x0i = a[j0 + 1] + a[j2 + 1];
+        x1r = a[j0] - a[j2];
+        x1i = a[j0 + 1] - a[j2 + 1];
+        x2r = a[j1] + a[j3];
+        x2i = a[j1 + 1] + a[j3 + 1];
+        x3r = a[j1] - a[j3];
+        x3i = a[j1 + 1] - a[j3 + 1];
+        a[j0] = x0r + x2r;
+        a[j0 + 1] = x0i + x2i;
+        a[j1] = x0r - x2r;
+        a[j1 + 1] = x0i - x2i;
+        x0r = x1r - x3i;
+        x0i = x1i + x3r;
+        a[j2] = wk1i * x0r - wk1r * x0i;
+        a[j2 + 1] = wk1i * x0i + wk1r * x0r;
+        x0r = x1r + x3i;
+        x0i = x1i - x3r;
+        a[j3] = wk3i * x0r + wk3r * x0i;
+        a[j3 + 1] = wk3i * x0i - wk3r * x0r;
+    }
+    j0 = mh;
+    j1 = j0 + m;
+    j2 = j1 + m;
+    j3 = j2 + m;
+    x0r = a[j0] + a[j2];
+    x0i = a[j0 + 1] + a[j2 + 1];
+    x1r = a[j0] - a[j2];
+    x1i = a[j0 + 1] - a[j2 + 1];
+    x2r = a[j1] + a[j3];
+    x2i = a[j1 + 1] + a[j3 + 1];
+    x3r = a[j1] - a[j3];
+    x3i = a[j1 + 1] - a[j3 + 1];
+    a[j0] = x0r + x2r;
+    a[j0 + 1] = x0i + x2i;
+    a[j1] = x0r - x2r;
+    a[j1 + 1] = x0i - x2i;
+    x0r = x1r - x3i;
+    x0i = x1i + x3r;
+    a[j2] = wn4r * (x0r - x0i);
+    a[j2 + 1] = wn4r * (x0i + x0r);
+    x0r = x1r + x3i;
+    x0i = x1i - x3r;
+    a[j3] = -wn4r * (x0r + x0i);
+    a[j3 + 1] = -wn4r * (x0i - x0r);
+}
+
+
+void cftmdl2(int n, float32 *a, float32 *w)
+{
+    int j, j0, j1, j2, j3, k, kr, m, mh;
+    float32 wn4r, wk1r, wk1i, wk3r, wk3i, wd1r, wd1i, wd3r, wd3i;
+    float32 x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i, y0r, y0i, y2r, y2i;
+    
+    mh = n >> 3;
+    m = 2 * mh;
+    wn4r = w[1];
+    j1 = m;
+    j2 = j1 + m;
+    j3 = j2 + m;
+    x0r = a[0] - a[j2 + 1];
+    x0i = a[1] + a[j2];
+    x1r = a[0] + a[j2 + 1];
+    x1i = a[1] - a[j2];
+    x2r = a[j1] - a[j3 + 1];
+    x2i = a[j1 + 1] + a[j3];
+    x3r = a[j1] + a[j3 + 1];
+    x3i = a[j1 + 1] - a[j3];
+    y0r = wn4r * (x2r - x2i);
+    y0i = wn4r * (x2i + x2r);
+    a[0] = x0r + y0r;
+    a[1] = x0i + y0i;
+    a[j1] = x0r - y0r;
+    a[j1 + 1] = x0i - y0i;
+    y0r = wn4r * (x3r - x3i);
+    y0i = wn4r * (x3i + x3r);
+    a[j2] = x1r - y0i;
+    a[j2 + 1] = x1i + y0r;
+    a[j3] = x1r + y0i;
+    a[j3 + 1] = x1i - y0r;
+    k = 0;
+    kr = 2 * m;
+    for (j = 2; j < mh; j += 2) {
+        k += 4;
+        wk1r = w[k];
+        wk1i = w[k + 1];
+        wk3r = w[k + 2];
+        wk3i = w[k + 3];
+        kr -= 4;
+        wd1i = w[kr];
+        wd1r = w[kr + 1];
+        wd3i = w[kr + 2];
+        wd3r = w[kr + 3];
+        j1 = j + m;
+        j2 = j1 + m;
+        j3 = j2 + m;
+        x0r = a[j] - a[j2 + 1];
+        x0i = a[j + 1] + a[j2];
+        x1r = a[j] + a[j2 + 1];
+        x1i = a[j + 1] - a[j2];
+        x2r = a[j1] - a[j3 + 1];
+        x2i = a[j1 + 1] + a[j3];
+        x3r = a[j1] + a[j3 + 1];
+        x3i = a[j1 + 1] - a[j3];
+        y0r = wk1r * x0r - wk1i * x0i;
+        y0i = wk1r * x0i + wk1i * x0r;
+        y2r = wd1r * x2r - wd1i * x2i;
+        y2i = wd1r * x2i + wd1i * x2r;
+        a[j] = y0r + y2r;
+        a[j + 1] = y0i + y2i;
+        a[j1] = y0r - y2r;
+        a[j1 + 1] = y0i - y2i;
+        y0r = wk3r * x1r + wk3i * x1i;
+        y0i = wk3r * x1i - wk3i * x1r;
+        y2r = wd3r * x3r + wd3i * x3i;
+        y2i = wd3r * x3i - wd3i * x3r;
+        a[j2] = y0r + y2r;
+        a[j2 + 1] = y0i + y2i;
+        a[j3] = y0r - y2r;
+        a[j3 + 1] = y0i - y2i;
+        j0 = m - j;
+        j1 = j0 + m;
+        j2 = j1 + m;
+        j3 = j2 + m;
+        x0r = a[j0] - a[j2 + 1];
+        x0i = a[j0 + 1] + a[j2];
+        x1r = a[j0] + a[j2 + 1];
+        x1i = a[j0 + 1] - a[j2];
+        x2r = a[j1] - a[j3 + 1];
+        x2i = a[j1 + 1] + a[j3];
+        x3r = a[j1] + a[j3 + 1];
+        x3i = a[j1 + 1] - a[j3];
+        y0r = wd1i * x0r - wd1r * x0i;
+        y0i = wd1i * x0i + wd1r * x0r;
+        y2r = wk1i * x2r - wk1r * x2i;
+        y2i = wk1i * x2i + wk1r * x2r;
+        a[j0] = y0r + y2r;
+        a[j0 + 1] = y0i + y2i;
+        a[j1] = y0r - y2r;
+        a[j1 + 1] = y0i - y2i;
+        y0r = wd3i * x1r + wd3r * x1i;
+        y0i = wd3i * x1i - wd3r * x1r;
+        y2r = wk3i * x3r + wk3r * x3i;
+        y2i = wk3i * x3i - wk3r * x3r;
+        a[j2] = y0r + y2r;
+        a[j2 + 1] = y0i + y2i;
+        a[j3] = y0r - y2r;
+        a[j3 + 1] = y0i - y2i;
+    }
+    wk1r = w[m];
+    wk1i = w[m + 1];
+    j0 = mh;
+    j1 = j0 + m;
+    j2 = j1 + m;
+    j3 = j2 + m;
+    x0r = a[j0] - a[j2 + 1];
+    x0i = a[j0 + 1] + a[j2];
+    x1r = a[j0] + a[j2 + 1];
+    x1i = a[j0 + 1] - a[j2];
+    x2r = a[j1] - a[j3 + 1];
+    x2i = a[j1 + 1] + a[j3];
+    x3r = a[j1] + a[j3 + 1];
+    x3i = a[j1 + 1] - a[j3];
+    y0r = wk1r * x0r - wk1i * x0i;
+    y0i = wk1r * x0i + wk1i * x0r;
+    y2r = wk1i * x2r - wk1r * x2i;
+    y2i = wk1i * x2i + wk1r * x2r;
+    a[j0] = y0r + y2r;
+    a[j0 + 1] = y0i + y2i;
+    a[j1] = y0r - y2r;
+    a[j1 + 1] = y0i - y2i;
+    y0r = wk1i * x1r - wk1r * x1i;
+    y0i = wk1i * x1i + wk1r * x1r;
+    y2r = wk1r * x3r - wk1i * x3i;
+    y2i = wk1r * x3i + wk1i * x3r;
+    a[j2] = y0r - y2r;
+    a[j2 + 1] = y0i - y2i;
+    a[j3] = y0r + y2r;
+    a[j3 + 1] = y0i + y2i;
+}
+
+
+void cftfx41(int n, float32 *a, int nw, float32 *w)
+{
+    void cftf161(float32 *a, float32 *w);
+    void cftf162(float32 *a, float32 *w);
+    void cftf081(float32 *a, float32 *w);
+    void cftf082(float32 *a, float32 *w);
+    
+    if (n == 128) {
+        cftf161(a, &w[nw - 8]);
+        cftf162(&a[32], &w[nw - 32]);
+        cftf161(&a[64], &w[nw - 8]);
+        cftf161(&a[96], &w[nw - 8]);
+    } else {
+        cftf081(a, &w[nw - 8]);
+        cftf082(&a[16], &w[nw - 8]);
+        cftf081(&a[32], &w[nw - 8]);
+        cftf081(&a[48], &w[nw - 8]);
+    }
+}
+
+
+void cftf161(float32 *a, float32 *w)
+{
+    float32 wn4r, wk1r, wk1i, 
+        x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i, 
+        y0r, y0i, y1r, y1i, y2r, y2i, y3r, y3i, 
+        y4r, y4i, y5r, y5i, y6r, y6i, y7r, y7i, 
+        y8r, y8i, y9r, y9i, y10r, y10i, y11r, y11i, 
+        y12r, y12i, y13r, y13i, y14r, y14i, y15r, y15i;
+    
+    wn4r = w[1];
+    wk1r = w[2];
+    wk1i = w[3];
+    x0r = a[0] + a[16];
+    x0i = a[1] + a[17];
+    x1r = a[0] - a[16];
+    x1i = a[1] - a[17];
+    x2r = a[8] + a[24];
+    x2i = a[9] + a[25];
+    x3r = a[8] - a[24];
+    x3i = a[9] - a[25];
+    y0r = x0r + x2r;
+    y0i = x0i + x2i;
+    y4r = x0r - x2r;
+    y4i = x0i - x2i;
+    y8r = x1r - x3i;
+    y8i = x1i + x3r;
+    y12r = x1r + x3i;
+    y12i = x1i - x3r;
+    x0r = a[2] + a[18];
+    x0i = a[3] + a[19];
+    x1r = a[2] - a[18];
+    x1i = a[3] - a[19];
+    x2r = a[10] + a[26];
+    x2i = a[11] + a[27];
+    x3r = a[10] - a[26];
+    x3i = a[11] - a[27];
+    y1r = x0r + x2r;
+    y1i = x0i + x2i;
+    y5r = x0r - x2r;
+    y5i = x0i - x2i;
+    x0r = x1r - x3i;
+    x0i = x1i + x3r;
+    y9r = wk1r * x0r - wk1i * x0i;
+    y9i = wk1r * x0i + wk1i * x0r;
+    x0r = x1r + x3i;
+    x0i = x1i - x3r;
+    y13r = wk1i * x0r - wk1r * x0i;
+    y13i = wk1i * x0i + wk1r * x0r;
+    x0r = a[4] + a[20];
+    x0i = a[5] + a[21];
+    x1r = a[4] - a[20];
+    x1i = a[5] - a[21];
+    x2r = a[12] + a[28];
+    x2i = a[13] + a[29];
+    x3r = a[12] - a[28];
+    x3i = a[13] - a[29];
+    y2r = x0r + x2r;
+    y2i = x0i + x2i;
+    y6r = x0r - x2r;
+    y6i = x0i - x2i;
+    x0r = x1r - x3i;
+    x0i = x1i + x3r;
+    y10r = wn4r * (x0r - x0i);
+    y10i = wn4r * (x0i + x0r);
+    x0r = x1r + x3i;
+    x0i = x1i - x3r;
+    y14r = wn4r * (x0r + x0i);
+    y14i = wn4r * (x0i - x0r);
+    x0r = a[6] + a[22];
+    x0i = a[7] + a[23];
+    x1r = a[6] - a[22];
+    x1i = a[7] - a[23];
+    x2r = a[14] + a[30];
+    x2i = a[15] + a[31];
+    x3r = a[14] - a[30];
+    x3i = a[15] - a[31];
+    y3r = x0r + x2r;
+    y3i = x0i + x2i;
+    y7r = x0r - x2r;
+    y7i = x0i - x2i;
+    x0r = x1r - x3i;
+    x0i = x1i + x3r;
+    y11r = wk1i * x0r - wk1r * x0i;
+    y11i = wk1i * x0i + wk1r * x0r;
+    x0r = x1r + x3i;
+    x0i = x1i - x3r;
+    y15r = wk1r * x0r - wk1i * x0i;
+    y15i = wk1r * x0i + wk1i * x0r;
+    x0r = y12r - y14r;
+    x0i = y12i - y14i;
+    x1r = y12r + y14r;
+    x1i = y12i + y14i;
+    x2r = y13r - y15r;
+    x2i = y13i - y15i;
+    x3r = y13r + y15r;
+    x3i = y13i + y15i;
+    a[24] = x0r + x2r;
+    a[25] = x0i + x2i;
+    a[26] = x0r - x2r;
+    a[27] = x0i - x2i;
+    a[28] = x1r - x3i;
+    a[29] = x1i + x3r;
+    a[30] = x1r + x3i;
+    a[31] = x1i - x3r;
+    x0r = y8r + y10r;
+    x0i = y8i + y10i;
+    x1r = y8r - y10r;
+    x1i = y8i - y10i;
+    x2r = y9r + y11r;
+    x2i = y9i + y11i;
+    x3r = y9r - y11r;
+    x3i = y9i - y11i;
+    a[16] = x0r + x2r;
+    a[17] = x0i + x2i;
+    a[18] = x0r - x2r;
+    a[19] = x0i - x2i;
+    a[20] = x1r - x3i;
+    a[21] = x1i + x3r;
+    a[22] = x1r + x3i;
+    a[23] = x1i - x3r;
+    x0r = y5r - y7i;
+    x0i = y5i + y7r;
+    x2r = wn4r * (x0r - x0i);
+    x2i = wn4r * (x0i + x0r);
+    x0r = y5r + y7i;
+    x0i = y5i - y7r;
+    x3r = wn4r * (x0r - x0i);
+    x3i = wn4r * (x0i + x0r);
+    x0r = y4r - y6i;
+    x0i = y4i + y6r;
+    x1r = y4r + y6i;
+    x1i = y4i - y6r;
+    a[8] = x0r + x2r;
+    a[9] = x0i + x2i;
+    a[10] = x0r - x2r;
+    a[11] = x0i - x2i;
+    a[12] = x1r - x3i;
+    a[13] = x1i + x3r;
+    a[14] = x1r + x3i;
+    a[15] = x1i - x3r;
+    x0r = y0r + y2r;
+    x0i = y0i + y2i;
+    x1r = y0r - y2r;
+    x1i = y0i - y2i;
+    x2r = y1r + y3r;
+    x2i = y1i + y3i;
+    x3r = y1r - y3r;
+    x3i = y1i - y3i;
+    a[0] = x0r + x2r;
+    a[1] = x0i + x2i;
+    a[2] = x0r - x2r;
+    a[3] = x0i - x2i;
+    a[4] = x1r - x3i;
+    a[5] = x1i + x3r;
+    a[6] = x1r + x3i;
+    a[7] = x1i - x3r;
+}
+
+
+void cftf162(float32 *a, float32 *w)
+{
+    float32 wn4r, wk1r, wk1i, wk2r, wk2i, wk3r, wk3i, 
+        x0r, x0i, x1r, x1i, x2r, x2i, 
+        y0r, y0i, y1r, y1i, y2r, y2i, y3r, y3i, 
+        y4r, y4i, y5r, y5i, y6r, y6i, y7r, y7i, 
+        y8r, y8i, y9r, y9i, y10r, y10i, y11r, y11i, 
+        y12r, y12i, y13r, y13i, y14r, y14i, y15r, y15i;
+    
+    wn4r = w[1];
+    wk1r = w[4];
+    wk1i = w[5];
+    wk3r = w[6];
+    wk3i = -w[7];
+    wk2r = w[8];
+    wk2i = w[9];
+    x1r = a[0] - a[17];
+    x1i = a[1] + a[16];
+    x0r = a[8] - a[25];
+    x0i = a[9] + a[24];
+    x2r = wn4r * (x0r - x0i);
+    x2i = wn4r * (x0i + x0r);
+    y0r = x1r + x2r;
+    y0i = x1i + x2i;
+    y4r = x1r - x2r;
+    y4i = x1i - x2i;
+    x1r = a[0] + a[17];
+    x1i = a[1] - a[16];
+    x0r = a[8] + a[25];
+    x0i = a[9] - a[24];
+    x2r = wn4r * (x0r - x0i);
+    x2i = wn4r * (x0i + x0r);
+    y8r = x1r - x2i;
+    y8i = x1i + x2r;
+    y12r = x1r + x2i;
+    y12i = x1i - x2r;
+    x0r = a[2] - a[19];
+    x0i = a[3] + a[18];
+    x1r = wk1r * x0r - wk1i * x0i;
+    x1i = wk1r * x0i + wk1i * x0r;
+    x0r = a[10] - a[27];
+    x0i = a[11] + a[26];
+    x2r = wk3i * x0r - wk3r * x0i;
+    x2i = wk3i * x0i + wk3r * x0r;
+    y1r = x1r + x2r;
+    y1i = x1i + x2i;
+    y5r = x1r - x2r;
+    y5i = x1i - x2i;
+    x0r = a[2] + a[19];
+    x0i = a[3] - a[18];
+    x1r = wk3r * x0r - wk3i * x0i;
+    x1i = wk3r * x0i + wk3i * x0r;
+    x0r = a[10] + a[27];
+    x0i = a[11] - a[26];
+    x2r = wk1r * x0r + wk1i * x0i;
+    x2i = wk1r * x0i - wk1i * x0r;
+    y9r = x1r - x2r;
+    y9i = x1i - x2i;
+    y13r = x1r + x2r;
+    y13i = x1i + x2i;
+    x0r = a[4] - a[21];
+    x0i = a[5] + a[20];
+    x1r = wk2r * x0r - wk2i * x0i;
+    x1i = wk2r * x0i + wk2i * x0r;
+    x0r = a[12] - a[29];
+    x0i = a[13] + a[28];
+    x2r = wk2i * x0r - wk2r * x0i;
+    x2i = wk2i * x0i + wk2r * x0r;
+    y2r = x1r + x2r;
+    y2i = x1i + x2i;
+    y6r = x1r - x2r;
+    y6i = x1i - x2i;
+    x0r = a[4] + a[21];
+    x0i = a[5] - a[20];
+    x1r = wk2i * x0r - wk2r * x0i;
+    x1i = wk2i * x0i + wk2r * x0r;
+    x0r = a[12] + a[29];
+    x0i = a[13] - a[28];
+    x2r = wk2r * x0r - wk2i * x0i;
+    x2i = wk2r * x0i + wk2i * x0r;
+    y10r = x1r - x2r;
+    y10i = x1i - x2i;
+    y14r = x1r + x2r;
+    y14i = x1i + x2i;
+    x0r = a[6] - a[23];
+    x0i = a[7] + a[22];
+    x1r = wk3r * x0r - wk3i * x0i;
+    x1i = wk3r * x0i + wk3i * x0r;
+    x0r = a[14] - a[31];
+    x0i = a[15] + a[30];
+    x2r = wk1i * x0r - wk1r * x0i;
+    x2i = wk1i * x0i + wk1r * x0r;
+    y3r = x1r + x2r;
+    y3i = x1i + x2i;
+    y7r = x1r - x2r;
+    y7i = x1i - x2i;
+    x0r = a[6] + a[23];
+    x0i = a[7] - a[22];
+    x1r = wk1i * x0r + wk1r * x0i;
+    x1i = wk1i * x0i - wk1r * x0r;
+    x0r = a[14] + a[31];
+    x0i = a[15] - a[30];
+    x2r = wk3i * x0r - wk3r * x0i;
+    x2i = wk3i * x0i + wk3r * x0r;
+    y11r = x1r + x2r;
+    y11i = x1i + x2i;
+    y15r = x1r - x2r;
+    y15i = x1i - x2i;
+    x1r = y0r + y2r;
+    x1i = y0i + y2i;
+    x2r = y1r + y3r;
+    x2i = y1i + y3i;
+    a[0] = x1r + x2r;
+    a[1] = x1i + x2i;
+    a[2] = x1r - x2r;
+    a[3] = x1i - x2i;
+    x1r = y0r - y2r;
+    x1i = y0i - y2i;
+    x2r = y1r - y3r;
+    x2i = y1i - y3i;
+    a[4] = x1r - x2i;
+    a[5] = x1i + x2r;
+    a[6] = x1r + x2i;
+    a[7] = x1i - x2r;
+    x1r = y4r - y6i;
+    x1i = y4i + y6r;
+    x0r = y5r - y7i;
+    x0i = y5i + y7r;
+    x2r = wn4r * (x0r - x0i);
+    x2i = wn4r * (x0i + x0r);
+    a[8] = x1r + x2r;
+    a[9] = x1i + x2i;
+    a[10] = x1r - x2r;
+    a[11] = x1i - x2i;
+    x1r = y4r + y6i;
+    x1i = y4i - y6r;
+    x0r = y5r + y7i;
+    x0i = y5i - y7r;
+    x2r = wn4r * (x0r - x0i);
+    x2i = wn4r * (x0i + x0r);
+    a[12] = x1r - x2i;
+    a[13] = x1i + x2r;
+    a[14] = x1r + x2i;
+    a[15] = x1i - x2r;
+    x1r = y8r + y10r;
+    x1i = y8i + y10i;
+    x2r = y9r - y11r;
+    x2i = y9i - y11i;
+    a[16] = x1r + x2r;
+    a[17] = x1i + x2i;
+    a[18] = x1r - x2r;
+    a[19] = x1i - x2i;
+    x1r = y8r - y10r;
+    x1i = y8i - y10i;
+    x2r = y9r + y11r;
+    x2i = y9i + y11i;
+    a[20] = x1r - x2i;
+    a[21] = x1i + x2r;
+    a[22] = x1r + x2i;
+    a[23] = x1i - x2r;
+    x1r = y12r - y14i;
+    x1i = y12i + y14r;
+    x0r = y13r + y15i;
+    x0i = y13i - y15r;
+    x2r = wn4r * (x0r - x0i);
+    x2i = wn4r * (x0i + x0r);
+    a[24] = x1r + x2r;
+    a[25] = x1i + x2i;
+    a[26] = x1r - x2r;
+    a[27] = x1i - x2i;
+    x1r = y12r + y14i;
+    x1i = y12i - y14r;
+    x0r = y13r - y15i;
+    x0i = y13i + y15r;
+    x2r = wn4r * (x0r - x0i);
+    x2i = wn4r * (x0i + x0r);
+    a[28] = x1r - x2i;
+    a[29] = x1i + x2r;
+    a[30] = x1r + x2i;
+    a[31] = x1i - x2r;
+}
+
+
+void cftf081(float32 *a, float32 *w)
+{
+    float32 wn4r, x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i, 
+        y0r, y0i, y1r, y1i, y2r, y2i, y3r, y3i, 
+        y4r, y4i, y5r, y5i, y6r, y6i, y7r, y7i;
+    
+    wn4r = w[1];
+    x0r = a[0] + a[8];
+    x0i = a[1] + a[9];
+    x1r = a[0] - a[8];
+    x1i = a[1] - a[9];
+    x2r = a[4] + a[12];
+    x2i = a[5] + a[13];
+    x3r = a[4] - a[12];
+    x3i = a[5] - a[13];
+    y0r = x0r + x2r;
+    y0i = x0i + x2i;
+    y2r = x0r - x2r;
+    y2i = x0i - x2i;
+    y1r = x1r - x3i;
+    y1i = x1i + x3r;
+    y3r = x1r + x3i;
+    y3i = x1i - x3r;
+    x0r = a[2] + a[10];
+    x0i = a[3] + a[11];
+    x1r = a[2] - a[10];
+    x1i = a[3] - a[11];
+    x2r = a[6] + a[14];
+    x2i = a[7] + a[15];
+    x3r = a[6] - a[14];
+    x3i = a[7] - a[15];
+    y4r = x0r + x2r;
+    y4i = x0i + x2i;
+    y6r = x0r - x2r;
+    y6i = x0i - x2i;
+    x0r = x1r - x3i;
+    x0i = x1i + x3r;
+    x2r = x1r + x3i;
+    x2i = x1i - x3r;
+    y5r = wn4r * (x0r - x0i);
+    y5i = wn4r * (x0r + x0i);
+    y7r = wn4r * (x2r - x2i);
+    y7i = wn4r * (x2r + x2i);
+    a[8] = y1r + y5r;
+    a[9] = y1i + y5i;
+    a[10] = y1r - y5r;
+    a[11] = y1i - y5i;
+    a[12] = y3r - y7i;
+    a[13] = y3i + y7r;
+    a[14] = y3r + y7i;
+    a[15] = y3i - y7r;
+    a[0] = y0r + y4r;
+    a[1] = y0i + y4i;
+    a[2] = y0r - y4r;
+    a[3] = y0i - y4i;
+    a[4] = y2r - y6i;
+    a[5] = y2i + y6r;
+    a[6] = y2r + y6i;
+    a[7] = y2i - y6r;
+}
+
+
+void cftf082(float32 *a, float32 *w)
+{
+    float32 wn4r, wk1r, wk1i, x0r, x0i, x1r, x1i, 
+        y0r, y0i, y1r, y1i, y2r, y2i, y3r, y3i, 
+        y4r, y4i, y5r, y5i, y6r, y6i, y7r, y7i;
+    
+    wn4r = w[1];
+    wk1r = w[2];
+    wk1i = w[3];
+    y0r = a[0] - a[9];
+    y0i = a[1] + a[8];
+    y1r = a[0] + a[9];
+    y1i = a[1] - a[8];
+    x0r = a[4] - a[13];
+    x0i = a[5] + a[12];
+    y2r = wn4r * (x0r - x0i);
+    y2i = wn4r * (x0i + x0r);
+    x0r = a[4] + a[13];
+    x0i = a[5] - a[12];
+    y3r = wn4r * (x0r - x0i);
+    y3i = wn4r * (x0i + x0r);
+    x0r = a[2] - a[11];
+    x0i = a[3] + a[10];
+    y4r = wk1r * x0r - wk1i * x0i;
+    y4i = wk1r * x0i + wk1i * x0r;
+    x0r = a[2] + a[11];
+    x0i = a[3] - a[10];
+    y5r = wk1i * x0r - wk1r * x0i;
+    y5i = wk1i * x0i + wk1r * x0r;
+    x0r = a[6] - a[15];
+    x0i = a[7] + a[14];
+    y6r = wk1i * x0r - wk1r * x0i;
+    y6i = wk1i * x0i + wk1r * x0r;
+    x0r = a[6] + a[15];
+    x0i = a[7] - a[14];
+    y7r = wk1r * x0r - wk1i * x0i;
+    y7i = wk1r * x0i + wk1i * x0r;
+    x0r = y0r + y2r;
+    x0i = y0i + y2i;
+    x1r = y4r + y6r;
+    x1i = y4i + y6i;
+    a[0] = x0r + x1r;
+    a[1] = x0i + x1i;
+    a[2] = x0r - x1r;
+    a[3] = x0i - x1i;
+    x0r = y0r - y2r;
+    x0i = y0i - y2i;
+    x1r = y4r - y6r;
+    x1i = y4i - y6i;
+    a[4] = x0r - x1i;
+    a[5] = x0i + x1r;
+    a[6] = x0r + x1i;
+    a[7] = x0i - x1r;
+    x0r = y1r - y3i;
+    x0i = y1i + y3r;
+    x1r = y5r - y7r;
+    x1i = y5i - y7i;
+    a[8] = x0r + x1r;
+    a[9] = x0i + x1i;
+    a[10] = x0r - x1r;
+    a[11] = x0i - x1i;
+    x0r = y1r + y3i;
+    x0i = y1i - y3r;
+    x1r = y5r + y7r;
+    x1i = y5i + y7i;
+    a[12] = x0r - x1i;
+    a[13] = x0i + x1r;
+    a[14] = x0r + x1i;
+    a[15] = x0i - x1r;
+}
+
+
+void cftf040(float32 *a)
+{
+    float32 x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i;
+    
+    x0r = a[0] + a[4];
+    x0i = a[1] + a[5];
+    x1r = a[0] - a[4];
+    x1i = a[1] - a[5];
+    x2r = a[2] + a[6];
+    x2i = a[3] + a[7];
+    x3r = a[2] - a[6];
+    x3i = a[3] - a[7];
+    a[0] = x0r + x2r;
+    a[1] = x0i + x2i;
+    a[2] = x1r - x3i;
+    a[3] = x1i + x3r;
+    a[4] = x0r - x2r;
+    a[5] = x0i - x2i;
+    a[6] = x1r + x3i;
+    a[7] = x1i - x3r;
+}
+
+
+void cftb040(float32 *a)
+{
+    float32 x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i;
+    
+    x0r = a[0] + a[4];
+    x0i = a[1] + a[5];
+    x1r = a[0] - a[4];
+    x1i = a[1] - a[5];
+    x2r = a[2] + a[6];
+    x2i = a[3] + a[7];
+    x3r = a[2] - a[6];
+    x3i = a[3] - a[7];
+    a[0] = x0r + x2r;
+    a[1] = x0i + x2i;
+    a[2] = x1r + x3i;
+    a[3] = x1i - x3r;
+    a[4] = x0r - x2r;
+    a[5] = x0i - x2i;
+    a[6] = x1r - x3i;
+    a[7] = x1i + x3r;
+}
+
+
+void cftx020(float32 *a)
+{
+    float32 x0r, x0i;
+    
+    x0r = a[0] - a[2];
+    x0i = a[1] - a[3];
+    a[0] += a[2];
+    a[1] += a[3];
+    a[2] = x0r;
+    a[3] = x0i;
+}
+
+
+void rftfsub(int n, float32 *a, int nc, float32 *c)
+{
+    int j, k, kk, ks, m;
+    float32 wkr, wki, xr, xi, yr, yi;
+    
+    m = n >> 1;
+    ks = 2 * nc / m;
+    kk = 0;
+    for (j = 2; j < m; j += 2) {
+        k = n - j;
+        kk += ks;
+        wkr = 0.5 - c[nc - kk];
+        wki = c[kk];
+        xr = a[j] - a[k];
+        xi = a[j + 1] + a[k + 1];
+        yr = wkr * xr - wki * xi;
+        yi = wkr * xi + wki * xr;
+        a[j] -= yr;
+        a[j + 1] -= yi;
+        a[k] += yr;
+        a[k + 1] -= yi;
+    }
+}
+
+
+void rftbsub(int n, float32 *a, int nc, float32 *c)
+{
+    int j, k, kk, ks, m;
+    float32 wkr, wki, xr, xi, yr, yi;
+    
+    m = n >> 1;
+    ks = 2 * nc / m;
+    kk = 0;
+    for (j = 2; j < m; j += 2) {
+        k = n - j;
+        kk += ks;
+        wkr = 0.5 - c[nc - kk];
+        wki = c[kk];
+        xr = a[j] - a[k];
+        xi = a[j + 1] + a[k + 1];
+        yr = wkr * xr + wki * xi;
+        yi = wkr * xi - wki * xr;
+        a[j] -= yr;
+        a[j + 1] -= yi;
+        a[k] += yr;
+        a[k + 1] -= yi;
+    }
+}
+
+
+void dctsub(int n, float32 *a, int nc, float32 *c)
+{
+    int j, k, kk, ks, m;
+    float32 wkr, wki, xr;
+    
+    m = n >> 1;
+    ks = nc / n;
+    kk = 0;
+    for (j = 1; j < m; j++) {
+        k = n - j;
+        kk += ks;
+        wkr = c[kk] - c[nc - kk];
+        wki = c[kk] + c[nc - kk];
+        xr = wki * a[j] - wkr * a[k];
+        a[j] = wkr * a[j] + wki * a[k];
+        a[k] = xr;
+    }
+    a[m] *= c[0];
+}
+
+
+void dstsub(int n, float32 *a, int nc, float32 *c)
+{
+    int j, k, kk, ks, m;
+    float32 wkr, wki, xr;
+    
+    m = n >> 1;
+    ks = nc / n;
+    kk = 0;
+    for (j = 1; j < m; j++) {
+        k = n - j;
+        kk += ks;
+        wkr = c[kk] - c[nc - kk];
+        wki = c[kk] + c[nc - kk];
+        xr = wki * a[k] - wkr * a[j];
+        a[k] = wkr * a[k] + wki * a[j];
+        a[j] = xr;
+    }
+    a[m] *= c[0];
 }
 
 }