fft8g.c

#include <ooura/fftxg.h>

/*
Fast Fourier/Cosine/Sine Transform
    dimension   :one
    data length :power of 2
    decimation  :frequency
    radix       :8, 4, 2
    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, OouraReal *, int *, OouraReal *);
    void rdft(int, int, OouraReal *, int *, OouraReal *);
    void ddct(int, int, OouraReal *, int *, OouraReal *);
    void ddst(int, int, OouraReal *, int *, OouraReal *);
    void dfct(int, OouraReal *, OouraReal *, int *, OouraReal *);
    void dfst(int, OouraReal *, OouraReal *, int *, OouraReal *);


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


const char * ooura_prec()
{
#ifdef OOURA_SINGLE_PREC
  return "float";
#else
  return "double";
#endif
}


#ifdef OOURA_SINGLE_PREC
  #define ATANr atanf
  #define COSr  cosf
  #define SINr  sinf
#else
  #define ATANr atan
  #define COSr  cos
  #define SINr  sin
#endif


/* forward declare internal functions */
static void bitrv2conj(int n, int *ip, OouraReal *a);
static void bitrv2(int n, int *ip, OouraReal *a);
static void cft1st(int n, OouraReal *a, OouraReal *w);
static void cftbsub(int n, OouraReal *a, OouraReal *w);
static void cftfsub(int n, OouraReal *a, OouraReal *w);
static void cftmdl(int n, int l, OouraReal *a, OouraReal *w);
static void dctsub(int n, OouraReal *a, int nc, OouraReal *c);
static void dstsub(int n, OouraReal *a, int nc, OouraReal *c);
static void makect(int nc, int *ip, OouraReal *c);
static void makewt(int nw, int *ip, OouraReal *w);
static void rftbsub(int n, OouraReal *a, int nc, OouraReal *c);
static void rftfsub(int n, OouraReal *a, int nc, OouraReal *c);



void cdft(int n, int isgn, OouraReal *a, int *ip, OouraReal *w)
{
    if (n > (ip[0] << 2)) {
        makewt(n >> 2, ip, w);
    }
    if (n > 4) {
        if (isgn >= 0) {
            bitrv2(n, ip + 2, a);
            cftfsub(n, a, w);
        } else {
            bitrv2conj(n, ip + 2, a);
            cftbsub(n, a, w);
        }
    } else if (n == 4) {
        cftfsub(n, a, w);
    }
}


void rdft(int n, int isgn, OouraReal *a, int *ip, OouraReal *w)
{
    int nw, nc;
    OouraReal 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) {
            bitrv2(n, ip + 2, a);
            cftfsub(n, a, w);
            rftfsub(n, a, nc, w + nw);
        } else if (n == 4) {
            cftfsub(n, a, 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);
            bitrv2(n, ip + 2, a);
            cftbsub(n, a, w);
        } else if (n == 4) {
            cftfsub(n, a, w);
        }
    }
}


void ddct(int n, int isgn, OouraReal *a, int *ip, OouraReal *w)
{
    int j, nw, nc;
    OouraReal 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);
            bitrv2(n, ip + 2, a);
            cftbsub(n, a, w);
        } else if (n == 4) {
            cftfsub(n, a, w);
        }
    }
    dctsub(n, a, nc, w + nw);
    if (isgn >= 0) {
        if (n > 4) {
            bitrv2(n, ip + 2, a);
            cftfsub(n, a, w);
            rftfsub(n, a, nc, w + nw);
        } else if (n == 4) {
            cftfsub(n, a, 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, OouraReal *a, int *ip, OouraReal *w)
{
    int j, nw, nc;
    OouraReal 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);
            bitrv2(n, ip + 2, a);
            cftbsub(n, a, w);
        } else if (n == 4) {
            cftfsub(n, a, w);
        }
    }
    dstsub(n, a, nc, w + nw);
    if (isgn >= 0) {
        if (n > 4) {
            bitrv2(n, ip + 2, a);
            cftfsub(n, a, w);
            rftfsub(n, a, nc, w + nw);
        } else if (n == 4) {
            cftfsub(n, a, 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, OouraReal *a, OouraReal *t, int *ip, OouraReal *w)
{
    int j, k, l, m, mh, nw, nc;
    OouraReal 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) {
            bitrv2(m, ip + 2, a);
            cftfsub(m, a, w);
            rftfsub(m, a, nc, w + nw);
        } else if (m == 4) {
            cftfsub(m, a, 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) {
                bitrv2(m, ip + 2, t);
                cftfsub(m, t, w);
                rftfsub(m, t, nc, w + nw);
            } else if (m == 4) {
                cftfsub(m, t, 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, OouraReal *a, OouraReal *t, int *ip, OouraReal *w)
{
    int j, k, l, m, mh, nw, nc;
    OouraReal 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) {
            bitrv2(m, ip + 2, a);
            cftfsub(m, a, w);
            rftfsub(m, a, nc, w + nw);
        } else if (m == 4) {
            cftfsub(m, a, 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) {
                bitrv2(m, ip + 2, t);
                cftfsub(m, t, w);
                rftfsub(m, t, nc, w + nw);
            } else if (m == 4) {
                cftfsub(m, t, 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, OouraReal *w)
{
    int j, nwh;
    OouraReal delta, x, y;

    ip[0] = nw;
    ip[1] = 1;
    if (nw > 2) {
        nwh = nw >> 1;
        delta = ATANr(1.0) / nwh;
        w[0] = 1;
        w[1] = 0;
        w[nwh] = COSr(delta * nwh);
        w[nwh + 1] = w[nwh];
        if (nwh > 2) {
            for (j = 2; j < nwh; j += 2) {
                x = COSr(delta * j);
                y = SINr(delta * j);
                w[j] = x;
                w[j + 1] = y;
                w[nw - j] = y;
                w[nw - j + 1] = x;
            }
            for (j = nwh - 2; j >= 2; j -= 2) {
                x = w[2 * j];
                y = w[2 * j + 1];
                w[nwh + j] = x;
                w[nwh + j + 1] = y;
            }
            bitrv2(nw, ip + 2, w);
        }
    }
}


void makect(int nc, int *ip, OouraReal *c)
{
    int j, nch;
    OouraReal delta;

    ip[1] = nc;
    if (nc > 1) {
        nch = nc >> 1;
        delta = ATANr(1.0) / nch;
        c[0] = COSr(delta * nch);
        c[nch] = 0.5 * c[0];
        for (j = 1; j < nch; j++) {
            c[j] = 0.5 * COSr(delta * j);
            c[nc - j] = 0.5 * SINr(delta * j);
        }
    }
}


/* -------- child routines -------- */


void bitrv2(int n, int *ip, OouraReal *a)
{
    int j, j1, k, k1, l, m, m2;
    OouraReal 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, OouraReal *a)
{
    int j, j1, k, k1, l, m, m2;
    OouraReal 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 cftfsub(int n, OouraReal *a, OouraReal *w)
{
    int j, j1, j2, j3, l;
    OouraReal x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i;

    l = 2;
    if (n >= 16) {
        cft1st(n, a, w);
        l = 16;
        while ((l << 3) <= n) {
            cftmdl(n, l, a, w);
            l <<= 3;
        }
    }
    if ((l << 1) < n) {
        for (j = 0; j < l; j += 2) {
            j1 = j + l;
            j2 = j1 + l;
            j3 = j2 + l;
            x0r = a[j] + a[j1];
            x0i = a[j + 1] + a[j1 + 1];
            x1r = a[j] - a[j1];
            x1i = a[j + 1] - a[j1 + 1];
            x2r = a[j2] + a[j3];
            x2i = a[j2 + 1] + a[j3 + 1];
            x3r = a[j2] - a[j3];
            x3i = a[j2 + 1] - a[j3 + 1];
            a[j] = x0r + x2r;
            a[j + 1] = x0i + x2i;
            a[j2] = x0r - x2r;
            a[j2 + 1] = x0i - x2i;
            a[j1] = x1r - x3i;
            a[j1 + 1] = x1i + x3r;
            a[j3] = x1r + x3i;
            a[j3 + 1] = x1i - x3r;
        }
    } else if ((l << 1) == n) {
        for (j = 0; j < l; j += 2) {
            j1 = j + l;
            x0r = a[j] - a[j1];
            x0i = a[j + 1] - a[j1 + 1];
            a[j] += a[j1];
            a[j + 1] += a[j1 + 1];
            a[j1] = x0r;
            a[j1 + 1] = x0i;
        }
    }
}


void cftbsub(int n, OouraReal *a, OouraReal *w)
{
    int j, j1, j2, j3, j4, j5, j6, j7, l;
    OouraReal 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;

    l = 2;
    if (n > 16) {
        cft1st(n, a, w);
        l = 16;
        while ((l << 3) < n) {
            cftmdl(n, l, a, w);
            l <<= 3;
        }
    }
    if ((l << 2) < n) {
        wn4r = w[2];
        for (j = 0; j < l; j += 2) {
            j1 = j + l;
            j2 = j1 + l;
            j3 = j2 + l;
            j4 = j3 + l;
            j5 = j4 + l;
            j6 = j5 + l;
            j7 = j6 + l;
            x0r = a[j] + a[j1];
            x0i = -a[j + 1] - a[j1 + 1];
            x1r = a[j] - a[j1];
            x1i = -a[j + 1] + a[j1 + 1];
            x2r = a[j2] + a[j3];
            x2i = a[j2 + 1] + a[j3 + 1];
            x3r = a[j2] - a[j3];
            x3i = a[j2 + 1] - a[j3 + 1];
            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[j4] + a[j5];
            x0i = a[j4 + 1] + a[j5 + 1];
            x1r = a[j4] - a[j5];
            x1i = a[j4 + 1] - a[j5 + 1];
            x2r = a[j6] + a[j7];
            x2i = a[j6 + 1] + a[j7 + 1];
            x3r = a[j6] - a[j7];
            x3i = a[j6 + 1] - a[j7 + 1];
            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[j1] = y1r + y5r;
            a[j1 + 1] = y1i - y5i;
            a[j5] = y1r - y5r;
            a[j5 + 1] = y1i + y5i;
            a[j3] = y3r - y7i;
            a[j3 + 1] = y3i - y7r;
            a[j7] = y3r + y7i;
            a[j7 + 1] = y3i + y7r;
            a[j] = y0r + y4r;
            a[j + 1] = y0i - y4i;
            a[j4] = y0r - y4r;
            a[j4 + 1] = y0i + y4i;
            a[j2] = y2r - y6i;
            a[j2 + 1] = y2i - y6r;
            a[j6] = y2r + y6i;
            a[j6 + 1] = y2i + y6r;
        }
    } else if ((l << 2) == n) {
        for (j = 0; j < l; j += 2) {
            j1 = j + l;
            j2 = j1 + l;
            j3 = j2 + l;
            x0r = a[j] + a[j1];
            x0i = -a[j + 1] - a[j1 + 1];
            x1r = a[j] - a[j1];
            x1i = -a[j + 1] + a[j1 + 1];
            x2r = a[j2] + a[j3];
            x2i = a[j2 + 1] + a[j3 + 1];
            x3r = a[j2] - a[j3];
            x3i = a[j2 + 1] - a[j3 + 1];
            a[j] = x0r + x2r;
            a[j + 1] = x0i - x2i;
            a[j2] = x0r - x2r;
            a[j2 + 1] = x0i + x2i;
            a[j1] = x1r - x3i;
            a[j1 + 1] = x1i - x3r;
            a[j3] = x1r + x3i;
            a[j3 + 1] = x1i + x3r;
        }
    } else {
        for (j = 0; j < l; j += 2) {
            j1 = j + l;
            x0r = a[j] - a[j1];
            x0i = -a[j + 1] + a[j1 + 1];
            a[j] += a[j1];
            a[j + 1] = -a[j + 1] - a[j1 + 1];
            a[j1] = x0r;
            a[j1 + 1] = x0i;
        }
    }
}


void cft1st(int n, OouraReal *a, OouraReal *w)
{
    int j, k1;
    OouraReal wn4r, wtmp, wk1r, wk1i, wk2r, wk2i, wk3r, wk3i,
        wk4r, wk4i, wk5r, wk5i, wk6r, wk6i, wk7r, wk7i;
    OouraReal 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[2];
    x0r = a[0] + a[2];
    x0i = a[1] + a[3];
    x1r = a[0] - a[2];
    x1i = a[1] - a[3];
    x2r = a[4] + a[6];
    x2i = a[5] + a[7];
    x3r = a[4] - a[6];
    x3i = a[5] - a[7];
    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[8] + a[10];
    x0i = a[9] + a[11];
    x1r = a[8] - a[10];
    x1i = a[9] - a[11];
    x2r = a[12] + a[14];
    x2i = a[13] + a[15];
    x3r = a[12] - a[14];
    x3i = a[13] - 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[2] = y1r + y5r;
    a[3] = y1i + y5i;
    a[10] = y1r - y5r;
    a[11] = y1i - y5i;
    a[6] = y3r - y7i;
    a[7] = y3i + y7r;
    a[14] = y3r + y7i;
    a[15] = y3i - y7r;
    a[0] = y0r + y4r;
    a[1] = y0i + y4i;
    a[8] = y0r - y4r;
    a[9] = y0i - y4i;
    a[4] = y2r - y6i;
    a[5] = y2i + y6r;
    a[12] = y2r + y6i;
    a[13] = y2i - y6r;
    if (n > 16) {
        wk1r = w[4];
        wk1i = w[5];
        x0r = a[16] + a[18];
        x0i = a[17] + a[19];
        x1r = a[16] - a[18];
        x1i = a[17] - a[19];
        x2r = a[20] + a[22];
        x2i = a[21] + a[23];
        x3r = a[20] - a[22];
        x3i = a[21] - a[23];
        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[24] + a[26];
        x0i = a[25] + a[27];
        x1r = a[24] - a[26];
        x1i = a[25] - a[27];
        x2r = a[28] + a[30];
        x2i = a[29] + a[31];
        x3r = a[28] - a[30];
        x3i = a[29] - a[31];
        y4r = x0r + x2r;
        y4i = x0i + x2i;
        y6r = x0r - x2r;
        y6i = x0i - x2i;
        x0r = x1r - x3i;
        x0i = x1i + x3r;
        x2r = x1r + x3i;
        x2i = x3r - x1i;
        y5r = wk1i * x0r - wk1r * x0i;
        y5i = wk1i * x0i + wk1r * x0r;
        y7r = wk1r * x2r + wk1i * x2i;
        y7i = wk1r * x2i - wk1i * x2r;
        x0r = wk1r * y1r - wk1i * y1i;
        x0i = wk1r * y1i + wk1i * y1r;
        a[18] = x0r + y5r;
        a[19] = x0i + y5i;
        a[26] = y5i - x0i;
        a[27] = x0r - y5r;
        x0r = wk1i * y3r - wk1r * y3i;
        x0i = wk1i * y3i + wk1r * y3r;
        a[22] = x0r - y7r;
        a[23] = x0i + y7i;
        a[30] = y7i - x0i;
        a[31] = x0r + y7r;
        a[16] = y0r + y4r;
        a[17] = y0i + y4i;
        a[24] = y4i - y0i;
        a[25] = y0r - y4r;
        x0r = y2r - y6i;
        x0i = y2i + y6r;
        a[20] = wn4r * (x0r - x0i);
        a[21] = wn4r * (x0i + x0r);
        x0r = y6r - y2i;
        x0i = y2r + y6i;
        a[28] = wn4r * (x0r - x0i);
        a[29] = wn4r * (x0i + x0r);
        k1 = 4;
        for (j = 32; j < n; j += 16) {
            k1 += 4;
            wk1r = w[k1];
            wk1i = w[k1 + 1];
            wk2r = w[k1 + 2];
            wk2i = w[k1 + 3];
            wtmp = 2 * wk2i;
            wk3r = wk1r - wtmp * wk1i;
            wk3i = wtmp * wk1r - wk1i;
            wk4r = 1 - wtmp * wk2i;
            wk4i = wtmp * wk2r;
            wtmp = 2 * wk4i;
            wk5r = wk3r - wtmp * wk1i;
            wk5i = wtmp * wk1r - wk3i;
            wk6r = wk2r - wtmp * wk2i;
            wk6i = wtmp * wk2r - wk2i;
            wk7r = wk1r - wtmp * wk3i;
            wk7i = wtmp * wk3r - wk1i;
            x0r = a[j] + a[j + 2];
            x0i = a[j + 1] + a[j + 3];
            x1r = a[j] - a[j + 2];
            x1i = a[j + 1] - a[j + 3];
            x2r = a[j + 4] + a[j + 6];
            x2i = a[j + 5] + a[j + 7];
            x3r = a[j + 4] - a[j + 6];
            x3i = a[j + 5] - a[j + 7];
            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[j + 8] + a[j + 10];
            x0i = a[j + 9] + a[j + 11];
            x1r = a[j + 8] - a[j + 10];
            x1i = a[j + 9] - a[j + 11];
            x2r = a[j + 12] + a[j + 14];
            x2i = a[j + 13] + a[j + 15];
            x3r = a[j + 12] - a[j + 14];
            x3i = a[j + 13] - a[j + 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);
            x0r = y1r + y5r;
            x0i = y1i + y5i;
            a[j + 2] = wk1r * x0r - wk1i * x0i;
            a[j + 3] = wk1r * x0i + wk1i * x0r;
            x0r = y1r - y5r;
            x0i = y1i - y5i;
            a[j + 10] = wk5r * x0r - wk5i * x0i;
            a[j + 11] = wk5r * x0i + wk5i * x0r;
            x0r = y3r - y7i;
            x0i = y3i + y7r;
            a[j + 6] = wk3r * x0r - wk3i * x0i;
            a[j + 7] = wk3r * x0i + wk3i * x0r;
            x0r = y3r + y7i;
            x0i = y3i - y7r;
            a[j + 14] = wk7r * x0r - wk7i * x0i;
            a[j + 15] = wk7r * x0i + wk7i * x0r;
            a[j] = y0r + y4r;
            a[j + 1] = y0i + y4i;
            x0r = y0r - y4r;
            x0i = y0i - y4i;
            a[j + 8] = wk4r * x0r - wk4i * x0i;
            a[j + 9] = wk4r * x0i + wk4i * x0r;
            x0r = y2r - y6i;
            x0i = y2i + y6r;
            a[j + 4] = wk2r * x0r - wk2i * x0i;
            a[j + 5] = wk2r * x0i + wk2i * x0r;
            x0r = y2r + y6i;
            x0i = y2i - y6r;
            a[j + 12] = wk6r * x0r - wk6i * x0i;
            a[j + 13] = wk6r * x0i + wk6i * x0r;
        }
    }
}


void cftmdl(int n, int l, OouraReal *a, OouraReal *w)
{
    int j, j1, j2, j3, j4, j5, j6, j7, k, k1, m;
    OouraReal wn4r, wtmp, wk1r, wk1i, wk2r, wk2i, wk3r, wk3i,
        wk4r, wk4i, wk5r, wk5i, wk6r, wk6i, wk7r, wk7i;
    OouraReal x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i,
        y0r, y0i, y1r, y1i, y2r, y2i, y3r, y3i,
        y4r, y4i, y5r, y5i, y6r, y6i, y7r, y7i;

    m = l << 3;
    wn4r = w[2];
    for (j = 0; j < l; j += 2) {
        j1 = j + l;
        j2 = j1 + l;
        j3 = j2 + l;
        j4 = j3 + l;
        j5 = j4 + l;
        j6 = j5 + l;
        j7 = j6 + l;
        x0r = a[j] + a[j1];
        x0i = a[j + 1] + a[j1 + 1];
        x1r = a[j] - a[j1];
        x1i = a[j + 1] - a[j1 + 1];
        x2r = a[j2] + a[j3];
        x2i = a[j2 + 1] + a[j3 + 1];
        x3r = a[j2] - a[j3];
        x3i = a[j2 + 1] - a[j3 + 1];
        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[j4] + a[j5];
        x0i = a[j4 + 1] + a[j5 + 1];
        x1r = a[j4] - a[j5];
        x1i = a[j4 + 1] - a[j5 + 1];
        x2r = a[j6] + a[j7];
        x2i = a[j6 + 1] + a[j7 + 1];
        x3r = a[j6] - a[j7];
        x3i = a[j6 + 1] - a[j7 + 1];
        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[j1] = y1r + y5r;
        a[j1 + 1] = y1i + y5i;
        a[j5] = y1r - y5r;
        a[j5 + 1] = y1i - y5i;
        a[j3] = y3r - y7i;
        a[j3 + 1] = y3i + y7r;
        a[j7] = y3r + y7i;
        a[j7 + 1] = y3i - y7r;
        a[j] = y0r + y4r;
        a[j + 1] = y0i + y4i;
        a[j4] = y0r - y4r;
        a[j4 + 1] = y0i - y4i;
        a[j2] = y2r - y6i;
        a[j2 + 1] = y2i + y6r;
        a[j6] = y2r + y6i;
        a[j6 + 1] = y2i - y6r;
    }
    if (m < n) {
        wk1r = w[4];
        wk1i = w[5];
        for (j = m; j < l + m; j += 2) {
            j1 = j + l;
            j2 = j1 + l;
            j3 = j2 + l;
            j4 = j3 + l;
            j5 = j4 + l;
            j6 = j5 + l;
            j7 = j6 + l;
            x0r = a[j] + a[j1];
            x0i = a[j + 1] + a[j1 + 1];
            x1r = a[j] - a[j1];
            x1i = a[j + 1] - a[j1 + 1];
            x2r = a[j2] + a[j3];
            x2i = a[j2 + 1] + a[j3 + 1];
            x3r = a[j2] - a[j3];
            x3i = a[j2 + 1] - a[j3 + 1];
            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[j4] + a[j5];
            x0i = a[j4 + 1] + a[j5 + 1];
            x1r = a[j4] - a[j5];
            x1i = a[j4 + 1] - a[j5 + 1];
            x2r = a[j6] + a[j7];
            x2i = a[j6 + 1] + a[j7 + 1];
            x3r = a[j6] - a[j7];
            x3i = a[j6 + 1] - a[j7 + 1];
            y4r = x0r + x2r;
            y4i = x0i + x2i;
            y6r = x0r - x2r;
            y6i = x0i - x2i;
            x0r = x1r - x3i;
            x0i = x1i + x3r;
            x2r = x1r + x3i;
            x2i = x3r - x1i;
            y5r = wk1i * x0r - wk1r * x0i;
            y5i = wk1i * x0i + wk1r * x0r;
            y7r = wk1r * x2r + wk1i * x2i;
            y7i = wk1r * x2i - wk1i * x2r;
            x0r = wk1r * y1r - wk1i * y1i;
            x0i = wk1r * y1i + wk1i * y1r;
            a[j1] = x0r + y5r;
            a[j1 + 1] = x0i + y5i;
            a[j5] = y5i - x0i;
            a[j5 + 1] = x0r - y5r;
            x0r = wk1i * y3r - wk1r * y3i;
            x0i = wk1i * y3i + wk1r * y3r;
            a[j3] = x0r - y7r;
            a[j3 + 1] = x0i + y7i;
            a[j7] = y7i - x0i;
            a[j7 + 1] = x0r + y7r;
            a[j] = y0r + y4r;
            a[j + 1] = y0i + y4i;
            a[j4] = y4i - y0i;
            a[j4 + 1] = y0r - y4r;
            x0r = y2r - y6i;
            x0i = y2i + y6r;
            a[j2] = wn4r * (x0r - x0i);
            a[j2 + 1] = wn4r * (x0i + x0r);
            x0r = y6r - y2i;
            x0i = y2r + y6i;
            a[j6] = wn4r * (x0r - x0i);
            a[j6 + 1] = wn4r * (x0i + x0r);
        }
        k1 = 4;
        for (k = 2 * m; k < n; k += m) {
            k1 += 4;
            wk1r = w[k1];
            wk1i = w[k1 + 1];
            wk2r = w[k1 + 2];
            wk2i = w[k1 + 3];
            wtmp = 2 * wk2i;
            wk3r = wk1r - wtmp * wk1i;
            wk3i = wtmp * wk1r - wk1i;
            wk4r = 1 - wtmp * wk2i;
            wk4i = wtmp * wk2r;
            wtmp = 2 * wk4i;
            wk5r = wk3r - wtmp * wk1i;
            wk5i = wtmp * wk1r - wk3i;
            wk6r = wk2r - wtmp * wk2i;
            wk6i = wtmp * wk2r - wk2i;
            wk7r = wk1r - wtmp * wk3i;
            wk7i = wtmp * wk3r - wk1i;
            for (j = k; j < l + k; j += 2) {
                j1 = j + l;
                j2 = j1 + l;
                j3 = j2 + l;
                j4 = j3 + l;
                j5 = j4 + l;
                j6 = j5 + l;
                j7 = j6 + l;
                x0r = a[j] + a[j1];
                x0i = a[j + 1] + a[j1 + 1];
                x1r = a[j] - a[j1];
                x1i = a[j + 1] - a[j1 + 1];
                x2r = a[j2] + a[j3];
                x2i = a[j2 + 1] + a[j3 + 1];
                x3r = a[j2] - a[j3];
                x3i = a[j2 + 1] - a[j3 + 1];
                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[j4] + a[j5];
                x0i = a[j4 + 1] + a[j5 + 1];
                x1r = a[j4] - a[j5];
                x1i = a[j4 + 1] - a[j5 + 1];
                x2r = a[j6] + a[j7];
                x2i = a[j6 + 1] + a[j7 + 1];
                x3r = a[j6] - a[j7];
                x3i = a[j6 + 1] - a[j7 + 1];
                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);
                x0r = y1r + y5r;
                x0i = y1i + y5i;
                a[j1] = wk1r * x0r - wk1i * x0i;
                a[j1 + 1] = wk1r * x0i + wk1i * x0r;
                x0r = y1r - y5r;
                x0i = y1i - y5i;
                a[j5] = wk5r * x0r - wk5i * x0i;
                a[j5 + 1] = wk5r * x0i + wk5i * x0r;
                x0r = y3r - y7i;
                x0i = y3i + y7r;
                a[j3] = wk3r * x0r - wk3i * x0i;
                a[j3 + 1] = wk3r * x0i + wk3i * x0r;
                x0r = y3r + y7i;
                x0i = y3i - y7r;
                a[j7] = wk7r * x0r - wk7i * x0i;
                a[j7 + 1] = wk7r * x0i + wk7i * x0r;
                a[j] = y0r + y4r;
                a[j + 1] = y0i + y4i;
                x0r = y0r - y4r;
                x0i = y0i - y4i;
                a[j4] = wk4r * x0r - wk4i * x0i;
                a[j4 + 1] = wk4r * x0i + wk4i * x0r;
                x0r = y2r - y6i;
                x0i = y2i + y6r;
                a[j2] = wk2r * x0r - wk2i * x0i;
                a[j2 + 1] = wk2r * x0i + wk2i * x0r;
                x0r = y2r + y6i;
                x0i = y2i - y6r;
                a[j6] = wk6r * x0r - wk6i * x0i;
                a[j6 + 1] = wk6r * x0i + wk6i * x0r;
            }
        }
    }
}


void rftfsub(int n, OouraReal *a, int nc, OouraReal *c)
{
    int j, k, kk, ks, m;
    OouraReal 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, OouraReal *a, int nc, OouraReal *c)
{
    int j, k, kk, ks, m;
    OouraReal wkr, wki, xr, xi, yr, yi;

    a[1] = -a[1];
    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[j + 1];
        a[k] += yr;
        a[k + 1] = yi - a[k + 1];
    }
    a[m + 1] = -a[m + 1];
}


void dctsub(int n, OouraReal *a, int nc, OouraReal *c)
{
    int j, k, kk, ks, m;
    OouraReal 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, OouraReal *a, int nc, OouraReal *c)
{
    int j, k, kk, ks, m;
    OouraReal 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];
}