fft4g.c

几个FFT算法

/*
Fast Fourier/Cosine/Sine Transform
    dimension   :one
    data length :power of 2
    decimation  :frequency
    radix       :4, 2
    data        :inplace
    table       :use
functions
    cdft: Complex Discrete Fourier Transform
    rdft: Real Discrete Fourier Transform
    
function prototypes
    void cdft(int, int, double *, int *, double *);
    void rdft(int, int, 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;
            }
        .



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);
        }
    }
}

/* -------- initializing routines -------- */

#include <math.h>

void makewt(int nw, int *ip, double *w)
{
    void bitrv2(int n, int *ip, double *a);
    int j, nwh;
    double delta, x, y;
    
    ip[0] = nw;
    ip[1] = 1;
    if (nw > 2) 
	{
        nwh = nw >> 1;
        delta = atan(1.0) / nwh;
        w[0] = 1;
        w[1] = 0;
        w[nwh] = cos(delta * nwh);
        w[nwh + 1] = w[nwh];
        if (nwh > 2) 
		{
            for (j = 2; j < nwh; j += 2) 
			{
                x = cos(delta * j);
                y = sin(delta * j);
                w[j] = x;
                w[j + 1] = y;
                w[nw - j] = y;
                w[nw - j + 1] = x;
            }
            bitrv2(nw, ip + 2, w);
        }
    }
}


void makect(int nc, int *ip, double *c)
{
    int j, nch;
    double delta;
    
    ip[1] = nc;
    if (nc > 1) 
	{
        nch = nc >> 1;
        delta = atan(1.0) / nch;
        c[0] = cos(delta * nch);
        c[nch] = 0.5 * c[0];
        for (j = 1; j < nch; j++) 
		{
            c[j] = 0.5 * cos(delta * j);
            c[nc - j] = 0.5 * sin(delta * j);
        }
    }
}


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


void bitrv2(int n, int *ip, double *a)
{
    int j, j1, k, k1, l, m, m2;
    double 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, double *a)
{
    int j, j1, k, k1, l, m, m2;
    double 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;