fftsg_ld.c

#ifndef HIGHPREC
#define REAL_IS_FLOAT
#else
#define REAL_IS_DOUBLE
#endif

#if 0
#undef REAL_IS_FLOAT
#undef REAL_IS_DOUBLE
#undef REAL_IS_LONGDOUBLE
#endif

/*
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, REAL *, int *, REAL *);
    void rdft(int, int, REAL *, int *, REAL *);
    void ddct(int, int, REAL *, int *, REAL *);
    void ddst(int, int, REAL *, int *, REAL *);
    void dfct(int, REAL *, REAL *, int *, REAL *);
    void dfst(int, REAL *, REAL *, int *, REAL *);


-------- 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 (REAL *)
                        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 (REAL *)
                        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 (REAL *)
                        <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 (REAL *)
                        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 (REAL *)
                        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 (REAL *)
                        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 (REAL *)
                        <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 (REAL *)
                        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 (REAL *)
                        output data
                            a[k] = C[k], 0<=k<=n
        t[0...n/2]     :work area (REAL *)
        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 (REAL *)
                        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 (REAL *)
                        output data
                            a[k] = S[k], 0<k<n
                        (a[0] is used for work area)
        t[0...n/2-1]   :work area (REAL *)
        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 (REAL *)
                        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.
*/

#include <stdlib.h>
#include <math.h>

#ifdef REAL_IS_FLOAT
typedef float REAL;

#define SIN(x) sinf(x)
#define COS(x) cosf(x)

#define CDFT_RECURSIVE_N 1024
#endif

#ifdef REAL_IS_DOUBLE
typedef double REAL;

#define SIN(x) sin(x)
#define COS(x) cos(x)

#define CDFT_RECURSIVE_N 512
#endif

#ifdef REAL_IS_LONGDOUBLE
typedef long double REAL;

#define SIN(x) sinl(x)
#define COS(x) cosl(x)

#define CDFT_RECURSIVE_N 512
#endif

#define PI 3.1415926535897932384626433832795029L

void cdft(int n, int isgn, REAL *a, int *ip, REAL *w)
{
    void makewt(int nw, int *ip, REAL *w);
    void cftfsub(int n, REAL *a, int *ip, int nw, REAL *w);
    void cftbsub(int n, REAL *a, int *ip, int nw, REAL *w);
    int nw;
    
    nw = ip[0];
    if (n > (nw << 2)) {
        nw = n >> 2;
        makewt(nw, ip, w);
    }
    if (isgn >= 0) {
        cftfsub(n, a, ip + 2, nw, w);
    } else {
        cftbsub(n, a, ip + 2, nw, w);
    }
}


void rdft(int n, int isgn, REAL *a, int *ip, REAL *w)
{
    void makewt(int nw, int *ip, REAL *w);
    void makect(int nc, int *ip, REAL *c);
    void cftfsub(int n, REAL *a, int *ip, int nw, REAL *w);
    void cftbsub(int n, REAL *a, int *ip, int nw, REAL *w);
    void rftfsub(int n, REAL *a, int nc, REAL *c);
    void rftbsub(int n, REAL *a, int nc, REAL *c);
    int nw, nc;
    REAL 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 + 2, nw, w);
            rftfsub(n, a, nc, w + nw);
        } else if (n == 4) {
            cftfsub(n, a, ip + 2, 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 + 2, nw, w);
        } else if (n == 4) {
            cftbsub(n, a, ip + 2, nw, w);
        }
    }
}


void ddct(int n, int isgn, REAL *a, int *ip, REAL *w)
{
    void makewt(int nw, int *ip, REAL *w);
    void makect(int nc, int *ip, REAL *c);
    void cftfsub(int n, REAL *a, int *ip, int nw, REAL *w);
    void cftbsub(int n, REAL *a, int *ip, int nw, REAL *w);
    void rftfsub(int n, REAL *a, int nc, REAL *c);
    void rftbsub(int n, REAL *a, int nc, REAL *c);
    void dctsub(int n, REAL *a, int nc, REAL *c);
    int j, nw, nc;
    REAL 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 + 2, nw, w);
        } else if (n == 4) {
            cftbsub(n, a, ip + 2, nw, w);
        }
    }
    dctsub(n, a, nc, w + nw);
    if (isgn >= 0) {
        if (n > 4) {
            cftfsub(n, a, ip + 2, nw, w);
            rftfsub(n, a, nc, w + nw);
        } else if (n == 4) {
            cftfsub(n, a, ip + 2, 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, REAL *a, int *ip, REAL *w)
{
    void makewt(int nw, int *ip, REAL *w);
    void makect(int nc, int *ip, REAL *c);
    void cftfsub(int n, REAL *a, int *ip, int nw, REAL *w);
    void cftbsub(int n, REAL *a, int *ip, int nw, REAL *w);
    void rftfsub(int n, REAL *a, int nc, REAL *c);
    void rftbsub(int n, REAL *a, int nc, REAL *c);
    void dstsub(int n, REAL *a, int nc, REAL *c);
    int j, nw, nc;
    REAL 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 + 2, nw, w);
        } else if (n == 4) {
            cftbsub(n, a, ip + 2, nw, w);
        }
    }
    dstsub(n, a, nc, w + nw);
    if (isgn >= 0) {
        if (n > 4) {
            cftfsub(n, a, ip + 2, nw, w);
            rftfsub(n, a, nc, w + nw);
        } else if (n == 4) {
            cftfsub(n, a, ip + 2, 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, REAL *a, REAL *t, int *ip, REAL *w)
{
    void makewt(int nw, int *ip, REAL *w);
    void makect(int nc, int *ip, REAL *c);
    void cftfsub(int n, REAL *a, int *ip, int nw, REAL *w);
    void rftfsub(int n, REAL *a, int nc, REAL *c);
    void dctsub(int n, REAL *a, int nc, REAL *c);
    int j, k, l, m, mh, nw, nc;
    REAL 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 + 2, nw, w);
            rftfsub(m, a, nc, w + nw);
        } else if (m == 4) {
            cftfsub(m, a, ip + 2, 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 + 2, nw, w);
                rftfsub(m, t, nc, w + nw);
            } else if (m == 4) {
                cftfsub(m, t, ip + 2, 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, REAL *a, REAL *t, int *ip, REAL *w)
{
    void makewt(int nw, int *ip, REAL *w);
    void makect(int nc, int *ip, REAL *c);
    void cftfsub(int n, REAL *a, int *ip, int nw, REAL *w);
    void rftfsub(int n, REAL *a, int nc, REAL *c);
    void dstsub(int n, REAL *a, int nc, REAL *c);
    int j, k, l, m, mh, nw, nc;
    REAL 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 + 2, nw, w);
            rftfsub(m, a, nc, w + nw);
        } else if (m == 4) {
            cftfsub(m, a, ip + 2, 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 + 2, nw, w);
                rftfsub(m, t, nc, w + nw);
            } else if (m == 4) {
                cftfsub(m, t, ip + 2, 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 -------- */


void makewt(int nw, int *ip, REAL *w)
{
    int j, nwh, nw0, nw1;
    REAL delta, wn4r, wk1r, wk1i, wk3r, wk3i;
    
    ip[0] = nw;
    ip[1] = 1;
    if (nw > 2) {
        nwh = nw >> 1;
        //delta = atan(1.0) / nwh;
        delta = PI/4 / nwh;
        wn4r = COS(delta * nwh);
        w[0] = 1;
        w[1] = wn4r;
        if (nwh >= 4) {
            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];
                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 makect(int nc, int *ip, REAL *c)
{
    int j, nch;
    REAL delta;
    
    ip[1] = nc;
    if (nc > 1) {
        nch = nc >> 1;
        //delta = atan(1.0) / nch;
        delta = PI/4 / 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 -------- */


#ifndef CDFT_RECURSIVE_N  /* length of the recursive FFT mode */
#define CDFT_RECURSIVE_N 512  /* <= (L1 cache size) / 16 */
#endif

void cftfsub(int n, REAL *a, int *ip, int nw, REAL *w)
{
    void bitrv2(int n, int *ip, REAL *a);
    void bitrv216(REAL *a);
    void bitrv208(REAL *a);
    void cftf1st(int n, REAL *a, REAL *w);
    void cftrec1(int n, REAL *a, int nw, REAL *w);
    void cftrec2(int n, REAL *a, int nw, REAL *w);
    void cftexp1(int n, REAL *a, int nw, REAL *w);
    void cftfx41(int n, REAL *a, int nw, REAL *w);
    void cftf161(REAL *a, REAL *w);
    void cftf081(REAL *a, REAL *w);
    void cftf040(REAL *a);
    void cftx020(REAL *a);
    int m;
    
    if (n > 32) {
        m = n >> 2;
        cftf1st(n, a, &w[nw - m]);
        if (n > CDFT_RECURSIVE_N) {
            cftrec1(m, a, nw, w);
            cftrec2(m, &a[m], nw, w);
            cftrec1(m, &a[2 * m], nw, w);
            cftrec1(m, &a[3 * m], nw, w);
        } else if (m > 32) {
            cftexp1(n, a, nw, w);
        } else {
            cftfx41(n, a, nw, w);
        }
        bitrv2(n, ip, a);
    } else if (n > 8) {
        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, REAL *a, int *ip, int nw, REAL *w)
{
    void bitrv2conj(int n, int *ip, REAL *a);
    void bitrv216neg(REAL *a);
    void bitrv208neg(REAL *a);
    void cftb1st(int n, REAL *a, REAL *w);
    void cftrec1(int n, REAL *a, int nw, REAL *w);
    void cftrec2(int n, REAL *a, int nw, REAL *w);
    void cftexp1(int n, REAL *a, int nw, REAL *w);
    void cftfx41(int n, REAL *a, int nw, REAL *w);
    void cftf161(REAL *a, REAL *w);
    void cftf081(REAL *a, REAL *w);
    void cftb040(REAL *a);
    void cftx020(REAL *a);
    int m;
    
    if (n > 32) {
        m = n >> 2;
        cftb1st(n, a, &w[nw - m]);
        if (n > CDFT_RECURSIVE_N) {
            cftrec1(m, a, nw, w);
            cftrec2(m, &a[m], nw, w);
            cftrec1(m, &a[2 * m], nw, w);
            cftrec1(m, &a[3 * m], nw, w);
        } else if (m > 32) {
            cftexp1(n, a, nw, w);
        } else {
            cftfx41(n, a, nw, w);
        }
        bitrv2conj(n, ip, a);
    } else if (n > 8) {
        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, REAL *a)
{
    int j, j1, k, k1, l, m, m2;
    REAL xr, xi, yr, yi;
    
    ip[0] = 0;
    l = n;
    m = 1;
    while ((m << 3) < l) {
        l >>= 1;
        for (j = 0; j < m; j++) {
            ip[m + j] = ip[j] + l;
        }
        m <<= 1;
    }
    m2 = 2 * m;
    if ((m << 3) == l) {
        for (k = 0; k < m; k++) {
            for (j = 0; j < k; j++) {
                j1 = 2 * j + ip[k];
                k1 = 2 * k + ip[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 += m2;
                k1 += 2 * m2;
                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 += m2;
                k1 -= m2;
                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 += m2;
                k1 += 2 * m2;
                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 * k + m2 + ip[k];
            k1 = j1 + m2;
            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 = 1; k < m; k++) {
            for (j = 0; j < k; j++) {
                j1 = 2 * j + ip[k];
                k1 = 2 * k + ip[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 += m2;
                k1 += m2;
                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, REAL *a)
{
    int j, j1, k, k1, l, m, m2;
    REAL xr, xi, yr, yi;
    
    ip[0] = 0;
    l = n;
    m = 1;
    while ((m << 3) < l) {
        l >>= 1;
        for (j = 0; j < m; j++) {
            ip[m + j] = ip[j] + l;
        }
        m <<= 1;
    }
    m2 = 2 * m;
    if ((m << 3) == l) {
        for (k = 0; k < m; k++) {
            for (j = 0; j < k; j++) {
                j1 = 2 * j + ip[k];
                k1 = 2 * k + ip[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 += m2;
                k1 += 2 * m2;
                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 += m2;
                k1 -= m2;
                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 += m2;
                k1 += 2 * m2;
                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 = 2 * k + ip[k];
            a[k1 + 1] = -a[k1 + 1];
            j1 = k1 + m2;
            k1 = j1 + m2;
            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 += m2;
            a[k1 + 1] = -a[k1 + 1];
        }
    } else {
        a[1] = -a[1];
        a[m2 + 1] = -a[m2 + 1];
        for (k = 1; k < m; k++) {
            for (j = 0; j < k; j++) {
                j1 = 2 * j + ip[k];
                k1 = 2 * k + ip[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 += m2;
                k1 += m2;
                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 = 2 * k + ip[k];
            a[k1 + 1] = -a[k1 + 1];
            a[k1 + m2 + 1] = -a[k1 + m2 + 1];
        }
    }
}


void bitrv216(REAL *a)
{
    REAL 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(REAL *a)
{
    REAL 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(REAL *a)
{
    REAL 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(REAL *a)
{
    REAL 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, REAL *a, REAL *w)
{
    int j, j0, j1, j2, j3, k, m, mh;
    REAL wn4r, csc1, csc3, wk1r, wk1i, wk3r, wk3i, 
        wd1r, wd1i, wd3r, wd3i;
    REAL 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, REAL *a, REAL *w)
{
    int j, j0, j1, j2, j3, k, m, mh;
    REAL wn4r, csc1, csc3, wk1r, wk1i, wk3r, wk3i, 
        wd1r, wd1i, wd3r, wd3i;
    REAL 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 cftrec1(int n, REAL *a, int nw, REAL *w)
{
    void cftrec1(int n, REAL *a, int nw, REAL *w);
    void cftrec2(int n, REAL *a, int nw, REAL *w);
    void cftmdl1(int n, REAL *a, REAL *w);
    void cftexp1(int n, REAL *a, int nw, REAL *w);
    int m;
    
    m = n >> 2;
    cftmdl1(n, a, &w[nw - 2 * m]);
    if (n > CDFT_RECURSIVE_N) {
        cftrec1(m, a, nw, w);
        cftrec2(m, &a[m], nw, w);
        cftrec1(m, &a[2 * m], nw, w);
        cftrec1(m, &a[3 * m], nw, w);
    } else {
        cftexp1(n, a, nw, w);
    }
}


void cftrec2(int n, REAL *a, int nw, REAL *w)
{
    void cftrec1(int n, REAL *a, int nw, REAL *w);
    void cftrec2(int n, REAL *a, int nw, REAL *w);
    void cftmdl2(int n, REAL *a, REAL *w);
    void cftexp2(int n, REAL *a, int nw, REAL *w);
    int m;
    
    m = n >> 2;
    cftmdl2(n, a, &w[nw - n]);
    if (n > CDFT_RECURSIVE_N) {
        cftrec1(m, a, nw, w);
        cftrec2(m, &a[m], nw, w);
        cftrec1(m, &a[2 * m], nw, w);
        cftrec2(m, &a[3 * m], nw, w);
    } else {
        cftexp2(n, a, nw, w);
    }
}


void cftexp1(int n, REAL *a, int nw, REAL *w)
{
    void cftmdl1(int n, REAL *a, REAL *w);
    void cftmdl2(int n, REAL *a, REAL *w);
    void cftfx41(int n, REAL *a, int nw, REAL *w);
    void cftfx42(int n, REAL *a, int nw, REAL *w);
    int j, k, l;
    
    l = n >> 2;
    while (l > 128) {
        for (k = l; k < n; k <<= 2) {
            for (j = k - l; j < n; j += 4 * k) {
                cftmdl1(l, &a[j], &w[nw - (l >> 1)]);
                cftmdl2(l, &a[k + j], &w[nw - l]);
                cftmdl1(l, &a[2 * k + j], &w[nw - (l >> 1)]);
            }
        }
        cftmdl1(l, &a[n - l], &w[nw - (l >> 1)]);
        l >>= 2;
    }
    for (k = l; k < n; k <<= 2) {
        for (j = k - l; j < n; j += 4 * k) {
            cftmdl1(l, &a[j], &w[nw - (l >> 1)]);
            cftfx41(l, &a[j], nw, w);
            cftmdl2(l, &a[k + j], &w[nw - l]);
            cftfx42(l, &a[k + j], nw, w);
            cftmdl1(l, &a[2 * k + j], &w[nw - (l >> 1)]);
            cftfx41(l, &a[2 * k + j], nw, w);
        }
    }
    cftmdl1(l, &a[n - l], &w[nw - (l >> 1)]);
    cftfx41(l, &a[n - l], nw, w);
}


void cftexp2(int n, REAL *a, int nw, REAL *w)
{
    void cftmdl1(int n, REAL *a, REAL *w);
    void cftmdl2(int n, REAL *a, REAL *w);
    void cftfx41(int n, REAL *a, int nw, REAL *w);
    void cftfx42(int n, REAL *a, int nw, REAL *w);
    int j, k, l, m;
    
    m = n >> 1;
    l = n >> 2;
    while (l > 128) {
        for (k = l; k < m; k <<= 2) {
            for (j = k - l; j < m; j += 2 * k) {
                cftmdl1(l, &a[j], &w[nw - (l >> 1)]);
                cftmdl1(l, &a[m + j], &w[nw - (l >> 1)]);
            }
            for (j = 2 * k - l; j < m; j += 4 * k) {
                cftmdl2(l, &a[j], &w[nw - l]);
                cftmdl2(l, &a[m + j], &w[nw - l]);
            }
        }
        l >>= 2;
    }
    for (k = l; k < m; k <<= 2) {
        for (j = k - l; j < m; j += 2 * k) {
            cftmdl1(l, &a[j], &w[nw - (l >> 1)]);
            cftfx41(l, &a[j], nw, w);
            cftmdl1(l, &a[m + j], &w[nw - (l >> 1)]);
            cftfx41(l, &a[m + j], nw, w);
        }
        for (j = 2 * k - l; j < m; j += 4 * k) {
            cftmdl2(l, &a[j], &w[nw - l]);
            cftfx42(l, &a[j], nw, w);
            cftmdl2(l, &a[m + j], &w[nw - l]);
            cftfx42(l, &a[m + j], nw, w);
        }
    }
}


void cftmdl1(int n, REAL *a, REAL *w)
{
    int j, j0, j1, j2, j3, k, m, mh;
    REAL wn4r, wk1r, wk1i, wk3r, wk3i;
    REAL 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, REAL *a, REAL *w)
{
    int j, j0, j1, j2, j3, k, kr, m, mh;
    REAL wn4r, wk1r, wk1i, wk3r, wk3i, wd1r, wd1i, wd3r, wd3i;
    REAL 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, REAL *a, int nw, REAL *w)
{
    void cftf161(REAL *a, REAL *w);
    void cftf162(REAL *a, REAL *w);
    void cftf081(REAL *a, REAL *w);
    void cftf082(REAL *a, REAL *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 - 16]);
        cftf082(&a[16], &w[nw - 16]);
        cftf081(&a[32], &w[nw - 16]);
        cftf081(&a[48], &w[nw - 16]);
    }
}


void cftfx42(int n, REAL *a, int nw, REAL *w)
{
    void cftf161(REAL *a, REAL *w);
    void cftf162(REAL *a, REAL *w);
    void cftf081(REAL *a, REAL *w);
    void cftf082(REAL *a, REAL *w);
    
    if (n == 128) {
        cftf161(a, &w[nw - 8]);
        cftf162(&a[32], &w[nw - 32]);
        cftf161(&a[64], &w[nw - 8]);
        cftf162(&a[96], &w[nw - 32]);
    } else {
        cftf081(a, &w[nw - 16]);
        cftf082(&a[16], &w[nw - 16]);
        cftf081(&a[32], &w[nw - 16]);
        cftf082(&a[48], &w[nw - 16]);
    }
}


void cftf161(REAL *a, REAL *w)
{
    REAL 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];
    wk1i = wn4r * w[2];
    wk1r = wk1i + w[2];
    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(REAL *a, REAL *w)
{
    REAL 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(REAL *a, REAL *w)
{
    REAL 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(REAL *a, REAL *w)
{
    REAL 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[4];
    wk1i = w[5];
    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(REAL *a)
{
    REAL 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[4] = x0r - x2r;
    a[5] = x0i - x2i;
    a[2] = x1r - x3i;
    a[3] = x1i + x3r;
    a[6] = x1r + x3i;
    a[7] = x1i - x3r;
}


void cftb040(REAL *a)
{
    REAL 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[4] = x0r - x2r;
    a[5] = x0i - x2i;
    a[2] = x1r + x3i;
    a[3] = x1i - x3r;
    a[6] = x1r - x3i;
    a[7] = x1i + x3r;
}


void cftx020(REAL *a)
{
    REAL 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, REAL *a, int nc, REAL *c)
{
    int j, k, kk, ks, m;
    REAL 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, REAL *a, int nc, REAL *c)
{
    int j, k, kk, ks, m;
    REAL 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, REAL *a, int nc, REAL *c)
{
    int j, k, kk, ks, m;
    REAL 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, REAL *a, int nc, REAL *c)
{
    int j, k, kk, ks, m;
    REAL 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];
}