/*Fast Fourier/Cosine/Sine Transform    dimension   :one    data length :power of 2    decimation  :frequency    radix       :8, 4, 2    data        :inplace    table       :usefunctions    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, double *, int *, double *);    void rdft(int, int, double *, int *, double *);    void ddct(int, int, double *, int *, double *);    void ddst(int, int, double *, int *, double *);    void dfct(int, double *, double *, int *, double *);    void dfst(int, double *, double *, int *, double *);-------- 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 (double *)                        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 (double *)                        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 (double *)                        <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 (double *)                        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 (double *)                        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 (double *)                        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 (double *)                        <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 (double *)                        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 (double *)                        output data                            a[k] = C[k], 0<=k<=n        t[0...n/2]     :work area (double *)        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 (double *)                        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 (double *)                        output data                            a[k] = S[k], 0<k<n                        (a[0] is used for work area)        t[0...n/2-1]   :work area (double *)        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 (double *)                        w[],ip[] are initialized if ip[0] == 0.    [remark]        Inverse of             dfst(n, a, t, ip, w);        is             dfst(n, a, t, ip, w);            for (j = 1; j <= n - 1; j++) {                a[j] *= 2.0 / n;            }        .Appendix :    The cos/sin table is recalculated when the larger table required.    w[] and ip[] are compatible with all routines.*/void cdft(int n, int isgn, double *a, int *ip, double *w){    void makewt(int nw, int *ip, double *w);    void bitrv2(int n, int *ip, double *a);    void bitrv2conj(int n, int *ip, double *a);    void cftfsub(int n, double *a, double *w);    void cftbsub(int n, double *a, double *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, double *a, int *ip, double *w){    void makewt(int nw, int *ip, double *w);    void makect(int nc, int *ip, double *c);    void bitrv2(int n, int *ip, double *a);    void cftfsub(int n, double *a, double *w);    void cftbsub(int n, double *a, double *w);    void rftfsub(int n, double *a, int nc, double *c);    void rftbsub(int n, double *a, int nc, double *c);    int nw, nc;    double 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, double *a, int *ip, double *w){    void makewt(int nw, int *ip, double *w);    void makect(int nc, int *ip, double *c);    void bitrv2(int n, int *ip, double *a);    void cftfsub(int n, double *a, double *w);    void cftbsub(int n, double *a, double *w);    void rftfsub(int n, double *a, int nc, double *c);    void rftbsub(int n, double *a, int nc, double *c);    void dctsub(int n, double *a, int nc, double *c);    int j, nw, nc;    double 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, double *a, int *ip, double *w){    void makewt(int nw, int *ip, double *w);