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minfft.c
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minfft.c
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/*
A minimalistic FFT library
Copyright (c) 2016-2024 Alexander Mukhin
SPDX-License-Identifier: MIT
*/
#include "minfft.h"
#include <stdlib.h>
#include <math.h>
/* constants */
static const minfft_real pi=3.141592653589793238462643383279502884L;
static const minfft_real sqrt2=1.414213562373095048801688724209698079L;
static const minfft_real invsqrt2=0.707106781186547524400844362104849039L;
/* aux structure */
struct minfft_aux {
int N; /* number of elements to transform */
void *t; /* temporary buffer */
void *e; /* exponent vector */
struct minfft_aux *sub1,*sub2; /* subtransform structures */
};
/* higher-order functions */
/* a pointer to a strided 1d complex transform routine */
typedef
void (*s_cx_1d_t)
(minfft_cmpl*,minfft_cmpl*,int,const minfft_aux*);
/* make a strided any-dimensional complex transform */
/* by repeated application of its strided one-dimensional routine */
static void
mkcx (minfft_cmpl *x, minfft_cmpl *y, int sy, const minfft_aux *a, s_cx_1d_t s_1d) {
if (a->sub2==NULL)
(*s_1d)(x,y,sy,a);
else {
int N1=a->sub1->N,N2=a->sub2->N; /* transform lengths */
int n; /* counter */
minfft_cmpl *t=a->t; /* temporary buffer */
/* strided transform of contiguous hyperplanes */
for (n=0;n<N2;++n)
mkcx(x+n*N1,t+n,N2,a->sub1,s_1d);
/* strided transform of contiguous rows */
for (n=0;n<N1;++n)
(*s_1d)(t+n*N2,y+sy*n,sy*N1,a->sub2);
}
}
/* a pointer to a strided 1d real transform routine */
typedef
void (*s_rx_1d_t)
(minfft_real*,minfft_real*,int,const minfft_aux*);
/* make a strided any-dimensional real transform */
/* by repeated application of its strided one-dimensional routine */
static void
mkrx (minfft_real *x, minfft_real *y, int sy, const minfft_aux *a, s_rx_1d_t s_1d) {
if (a->sub2==NULL)
(*s_1d)(x,y,sy,a);
else {
int N1=a->sub1->N,N2=a->sub2->N; /* transform lengths */
int n; /* counter */
minfft_real *t=a->t; /* temporary buffer */
/* strided transform of contiguous hyperplanes */
for (n=0;n<N2;++n)
mkrx(x+n*N1,t+n,N2,a->sub1,s_1d);
/* strided transform of contiguous rows */
for (n=0;n<N1;++n)
(*s_1d)(t+n*N2,y+sy*n,sy*N1,a->sub2);
}
}
/* complex transforms */
/* recursive strided one-dimensional DFT */
static void
rs_dft_1d (int N, minfft_cmpl *x, minfft_cmpl *t, minfft_cmpl *y, int sy, const minfft_cmpl *e) {
int n; /* counter */
minfft_real *xr,*tr,*er,*xi,*ti,*ei;
/* split-radix DIF */
if (N==1) {
/* terminal case */
minfft_real *xr=(minfft_real*)x,*yr=(minfft_real*)y;
minfft_real *xi=xr+1,*yi=yr+1;
/* y[0]=x[0]; */
yr[0]=xr[0];
yi[0]=xi[0];
return;
}
if (N==2) {
/* terminal case */
minfft_real *xr=(minfft_real*)x,*yr=(minfft_real*)y;
minfft_real *xi=xr+1,*yi=yr+1;
register minfft_real t0r,t1r;
register minfft_real t0i,t1i;
/* t0=x[0]+x[1]; */
t0r=xr[0]+xr[2];
t0i=xi[0]+xi[2];
/* t1=x[0]-x[1]; */
t1r=xr[0]-xr[2];
t1i=xi[0]-xi[2];
/* y[0]=t0; */
yr[0]=t0r;
yi[0]=t0i;
/* y[sy]=t1; */
yr[2*sy]=t1r;
yi[2*sy]=t1i;
return;
}
if (N==4) {
/* terminal case */
minfft_real *xr=(minfft_real*)x,*yr=(minfft_real*)y;
minfft_real *xi=xr+1,*yi=yr+1;
register minfft_real t0r,t1r,t2r,t3r;
register minfft_real t0i,t1i,t2i,t3i;
/* t0=x[0]+x[2]; */
t0r=xr[0]+xr[4];
t0i=xi[0]+xi[4];
/* t1=x[1]+x[3]; */
t1r=xr[2]+xr[6];
t1i=xi[2]+xi[6];
/* t2=x[0]-x[2]; */
t2r=xr[0]-xr[4];
t2i=xi[0]-xi[4];
/* t3=I*(x[1]-x[3]); */
t3r=-xi[2]+xi[6];
t3i=xr[2]-xr[6];
/* y[0]=t0+t1; */
yr[0]=t0r+t1r;
yi[0]=t0i+t1i;
/* y[sy]=t2-t3; */
yr[2*sy]=t2r-t3r;
yi[2*sy]=t2i-t3i;
/* y[2*sy]=t0-t1; */
yr[4*sy]=t0r-t1r;
yi[4*sy]=t0i-t1i;
/* y[3*sy]=t2+t3; */
yr[6*sy]=t2r+t3r;
yi[6*sy]=t2i+t3i;
return;
}
if (N==8) {
/* terminal case */
minfft_real *xr=(minfft_real*)x,*yr=(minfft_real*)y;
minfft_real *xi=xr+1,*yi=yr+1;
register minfft_real t0r,t1r,t2r,t3r;
register minfft_real t0i,t1i,t2i,t3i;
register minfft_real t00r,t01r,t02r,t03r;
register minfft_real t00i,t01i,t02i,t03i;
register minfft_real t10r,t11r,t12r,t13r;
register minfft_real t10i,t11i,t12i,t13i;
register minfft_real ttr,tti;
/* t0=x[0]+x[4]; */
t0r=xr[0]+xr[8];
t0i=xi[0]+xi[8];
/* t1=x[2]+x[6]; */
t1r=xr[4]+xr[12];
t1i=xi[4]+xi[12];
/* t2=x[0]-x[4]; */
t2r=xr[0]-xr[8];
t2i=xi[0]-xi[8];
/* t3=I*(x[2]-x[6]); */
t3r=-xi[4]+xi[12];
t3i=xr[4]-xr[12];
/* t00=t0+t1; */
t00r=t0r+t1r;
t00i=t0i+t1i;
/* t01=t2-t3; */
t01r=t2r-t3r;
t01i=t2i-t3i;
/* t02=t0-t1; */
t02r=t0r-t1r;
t02i=t0i-t1i;
/* t03=t2+t3; */
t03r=t2r+t3r;
t03i=t2i+t3i;
/* t0=x[1]+x[5]; */
t0r=xr[2]+xr[10];
t0i=xi[2]+xi[10];
/* t1=x[3]+x[7]; */
t1r=xr[6]+xr[14];
t1i=xi[6]+xi[14];
/* t2=x[1]-x[5]; */
t2r=xr[2]-xr[10];
t2i=xi[2]-xi[10];
/* t3=I*(x[3]-x[7]); */
t3r=-xi[6]+xi[14];
t3i=xr[6]-xr[14];
/* t10=t0+t1; */
t10r=t0r+t1r;
t10i=t0i+t1i;
/* t11=(t2-t3)*invsqrt2*(1-I); */
ttr=t2r-t3r;
tti=t2i-t3i;
t11r=invsqrt2*(ttr+tti);
t11i=invsqrt2*(tti-ttr);
/* t12=(t0-t1)*(-I); */
t12r=t0i-t1i;
t12i=-t0r+t1r;
/* t13=(t2+t3)*invsqrt2*(-1-I); */
ttr=t2r+t3r;
tti=t2i+t3i;
t13r=invsqrt2*(tti-ttr);
t13i=-invsqrt2*(tti+ttr);
/* y[0]=t00+t10; */
yr[0]=t00r+t10r;
yi[0]=t00i+t10i;
/* y[sy]=t01+t11; */
yr[2*sy]=t01r+t11r;
yi[2*sy]=t01i+t11i;
/* y[2*sy]=t02+t12; */
yr[4*sy]=t02r+t12r;
yi[4*sy]=t02i+t12i;
/* y[3*sy]=t03+t13; */
yr[6*sy]=t03r+t13r;
yi[6*sy]=t03i+t13i;
/* y[4*sy]=t00-t10; */
yr[8*sy]=t00r-t10r;
yi[8*sy]=t00i-t10i;
/* y[5*sy]=t01-t11; */
yr[10*sy]=t01r-t11r;
yi[10*sy]=t01i-t11i;
/* y[6*sy]=t02-t12; */
yr[12*sy]=t02r-t12r;
yi[12*sy]=t02i-t12i;
/* y[7*sy]=t03-t13; */
yr[14*sy]=t03r-t13r;
yi[14*sy]=t03i-t13i;
return;
}
/* recursion */
xr=(minfft_real*)x,tr=(minfft_real*)t,er=(minfft_real*)e;
xi=xr+1,ti=tr+1,ei=er+1;
/* prepare sub-transform inputs */
for (n=0;n<N/4;++n) {
register minfft_real t0r,t1r,t2r,t3r;
register minfft_real t0i,t1i,t2i,t3i;
/* t0=x[n]+x[n+N/2]; */
t0r=xr[2*n]+xr[2*n+N];
t0i=xi[2*n]+xi[2*n+N];
/* t1=x[n+N/4]+x[n+3*N/4]; */
t1r=xr[2*n+N/2]+xr[2*n+3*N/2];
t1i=xi[2*n+N/2]+xi[2*n+3*N/2];
/* t2=x[n]-x[n+N/2]; */
t2r=xr[2*n]-xr[2*n+N];
t2i=xi[2*n]-xi[2*n+N];
/* t3=I*(x[n+N/4]-x[n+3*N/4]); */
t3r=-xi[2*n+N/2]+xi[2*n+3*N/2];
t3i=xr[2*n+N/2]-xr[2*n+3*N/2];
/* t[n]=t0; */
tr[2*n]=t0r;
ti[2*n]=t0i;
/* t[n+N/4]=t1; */
tr[2*n+N/2]=t1r;
ti[2*n+N/2]=t1i;
/* t[n+N/2]=(t2-t3)*e[2*n]; */
t0r=t2r-t3r;
t0i=t2i-t3i;
t1r=t2r+t3r;
t1i=t2i+t3i;
tr[2*n+N]=t0r*er[4*n]-t0i*ei[4*n];
ti[2*n+N]=t0r*ei[4*n]+t0i*er[4*n];
/* t[n+3*N/4]=(t2+t3)*e[2*n+1]; */
tr[2*n+3*N/2]=t1r*er[4*n+2]-t1i*ei[4*n+2];
ti[2*n+3*N/2]=t1r*ei[4*n+2]+t1i*er[4*n+2];
}
/* call sub-transforms */
rs_dft_1d(N/2,t,t,y,2*sy,e+N/2);
rs_dft_1d(N/4,t+N/2,t+N/2,y+sy,4*sy,e+3*N/4);
rs_dft_1d(N/4,t+3*N/4,t+3*N/4,y+3*sy,4*sy,e+3*N/4);
}
/* strided one-dimensional DFT */
static void
s_dft_1d (minfft_cmpl *x, minfft_cmpl *y, int sy, const minfft_aux *a) {
rs_dft_1d(a->N,x,a->t,y,sy,a->e);
}
/* strided DFT of arbitrary dimension */
static void
s_dft (minfft_cmpl *x, minfft_cmpl *y, int sy, const minfft_aux *a) {
mkcx(x,y,sy,a,s_dft_1d);
}
/* user interface */
void
minfft_dft (minfft_cmpl *x, minfft_cmpl *y, const minfft_aux *a) {
s_dft(x,y,1,a);
}
/* recursive strided one-dimensional inverse DFT */
static void
rs_invdft_1d (int N, minfft_cmpl *x, minfft_cmpl *t, minfft_cmpl *y, int sy, const minfft_cmpl *e) {
int n; /* counter */
minfft_real *xr,*tr,*er,*xi,*ti,*ei;
/* split-radix DIF */
if (N==1) {
/* terminal case */
minfft_real *xr=(minfft_real*)x,*yr=(minfft_real*)y;
minfft_real *xi=xr+1,*yi=yr+1;
/* y[0]=x[0]; */
yr[0]=xr[0];
yi[0]=xi[0];
return;
}
if (N==2) {
/* terminal case */
minfft_real *xr=(minfft_real*)x,*yr=(minfft_real*)y;
minfft_real *xi=xr+1,*yi=yr+1;
register minfft_real t0r,t1r;
register minfft_real t0i,t1i;
/* t0=x[0]+x[1]; */
t0r=xr[0]+xr[2];
t0i=xi[0]+xi[2];
/* t1=x[0]-x[1]; */
t1r=xr[0]-xr[2];
t1i=xi[0]-xi[2];
/* y[0]=t0; */
yr[0]=t0r;
yi[0]=t0i;
/* y[sy]=t1; */
yr[2*sy]=t1r;
yi[2*sy]=t1i;
return;
}
if (N==4) {
/* terminal case */
minfft_real *xr=(minfft_real*)x,*yr=(minfft_real*)y;
minfft_real *xi=xr+1,*yi=yr+1;
register minfft_real t0r,t1r,t2r,t3r;
register minfft_real t0i,t1i,t2i,t3i;
/* t0=x[0]+x[2]; */
t0r=xr[0]+xr[4];
t0i=xi[0]+xi[4];
/* t1=x[1]+x[3]; */
t1r=xr[2]+xr[6];
t1i=xi[2]+xi[6];
/* t2=x[0]-x[2]; */
t2r=xr[0]-xr[4];
t2i=xi[0]-xi[4];
/* t3=I*(x[1]-x[3]); */
t3r=-xi[2]+xi[6];
t3i=xr[2]-xr[6];
/* y[0]=t0+t1; */
yr[0]=t0r+t1r;
yi[0]=t0i+t1i;
/* y[sy]=t2+t3; */
yr[2*sy]=t2r+t3r;
yi[2*sy]=t2i+t3i;
/* y[2*sy]=t0-t1; */
yr[4*sy]=t0r-t1r;
yi[4*sy]=t0i-t1i;
/* y[3*sy]=t2-t3; */
yr[6*sy]=t2r-t3r;
yi[6*sy]=t2i-t3i;
return;
}
if (N==8) {
/* terminal case */
minfft_real *xr=(minfft_real*)x,*yr=(minfft_real*)y;
minfft_real *xi=xr+1,*yi=yr+1;
register minfft_real t0r,t1r,t2r,t3r;
register minfft_real t0i,t1i,t2i,t3i;
register minfft_real t00r,t01r,t02r,t03r;
register minfft_real t00i,t01i,t02i,t03i;
register minfft_real t10r,t11r,t12r,t13r;
register minfft_real t10i,t11i,t12i,t13i;
register minfft_real ttr,tti;
/* t0=x[0]+x[4]; */
t0r=xr[0]+xr[8];
t0i=xi[0]+xi[8];
/* t1=x[2]+x[6]; */
t1r=xr[4]+xr[12];
t1i=xi[4]+xi[12];
/* t2=x[0]-x[4]; */
t2r=xr[0]-xr[8];
t2i=xi[0]-xi[8];
/* t3=I*(x[2]-x[6]); */
t3r=-xi[4]+xi[12];
t3i=xr[4]-xr[12];
/* t00=t0+t1; */
t00r=t0r+t1r;
t00i=t0i+t1i;
/* t01=t2+t3; */
t01r=t2r+t3r;
t01i=t2i+t3i;
/* t02=t0-t1; */
t02r=t0r-t1r;
t02i=t0i-t1i;
/* t03=t2-t3; */
t03r=t2r-t3r;
t03i=t2i-t3i;
/* t0=x[1]+x[5]; */
t0r=xr[2]+xr[10];
t0i=xi[2]+xi[10];
/* t1=x[3]+x[7]; */
t1r=xr[6]+xr[14];
t1i=xi[6]+xi[14];
/* t2=x[1]-x[5]; */
t2r=xr[2]-xr[10];
t2i=xi[2]-xi[10];
/* t3=I*(x[3]-x[7]); */
t3r=-xi[6]+xi[14];
t3i=xr[6]-xr[14];
/* t10=t0+t1; */
t10r=t0r+t1r;
t10i=t0i+t1i;
/* t11=(t2+t3)*invsqrt2*(1+I); */
ttr=t2r+t3r;
tti=t2i+t3i;
t11r=invsqrt2*(ttr-tti);
t11i=invsqrt2*(ttr+tti);
/* t12=(t0-t1)*I; */
t12r=-t0i+t1i;
t12i=t0r-t1r;
/* t13=(t2-t3)*invsqrt2*(-1+I); */
ttr=t2r-t3r;
tti=t2i-t3i;
t13r=-invsqrt2*(ttr+tti);
t13i=invsqrt2*(ttr-tti);
/* y[0]=t00+t10; */
yr[0]=t00r+t10r;
yi[0]=t00i+t10i;
/* y[sy]=t01+t11; */
yr[2*sy]=t01r+t11r;
yi[2*sy]=t01i+t11i;
/* y[2*sy]=t02+t12; */
yr[4*sy]=t02r+t12r;
yi[4*sy]=t02i+t12i;
/* y[3*sy]=t03+t13; */
yr[6*sy]=t03r+t13r;
yi[6*sy]=t03i+t13i;
/* y[4*sy]=t00-t10; */
yr[8*sy]=t00r-t10r;
yi[8*sy]=t00i-t10i;
/* y[5*sy]=t01-t11; */
yr[10*sy]=t01r-t11r;
yi[10*sy]=t01i-t11i;
/* y[6*sy]=t02-t12; */
yr[12*sy]=t02r-t12r;
yi[12*sy]=t02i-t12i;
/* y[7*sy]=t03-t13; */
yr[14*sy]=t03r-t13r;
yi[14*sy]=t03i-t13i;
return;
}
/* recursion */
xr=(minfft_real*)x,tr=(minfft_real*)t,er=(minfft_real*)e;
xi=xr+1,ti=tr+1,ei=er+1;
/* prepare sub-transform inputs */
for (n=0;n<N/4;++n) {
register minfft_real t0r,t1r,t2r,t3r;
register minfft_real t0i,t1i,t2i,t3i;
/* t0=x[n]+x[n+N/2]; */
t0r=xr[2*n]+xr[2*n+N];
t0i=xi[2*n]+xi[2*n+N];
/* t1=x[n+N/4]+x[n+3*N/4]; */
t1r=xr[2*n+N/2]+xr[2*n+3*N/2];
t1i=xi[2*n+N/2]+xi[2*n+3*N/2];
/* t2=x[n]-x[n+N/2]; */
t2r=xr[2*n]-xr[2*n+N];
t2i=xi[2*n]-xi[2*n+N];
/* t3=I*(x[n+N/4]-x[n+3*N/4]); */
t3r=-xi[2*n+N/2]+xi[2*n+3*N/2];
t3i=xr[2*n+N/2]-xr[2*n+3*N/2];
/* t[n]=t0; */
tr[2*n]=t0r;
ti[2*n]=t0i;
/* t[n+N/4]=t1; */
tr[2*n+N/2]=t1r;
ti[2*n+N/2]=t1i;
/* t[n+N/2]=(t2+t3)*conj(e[2*n]); */
t0r=t2r+t3r;
t0i=t2i+t3i;
t1r=t2r-t3r;
t1i=t2i-t3i;
tr[2*n+N]=t0r*er[4*n]+t0i*ei[4*n];
ti[2*n+N]=-t0r*ei[4*n]+t0i*er[4*n];
/* t[n+3*N/4]=(t2-t3)*conj(e[2*n+1]); */
tr[2*n+3*N/2]=t1r*er[4*n+2]+t1i*ei[4*n+2];
ti[2*n+3*N/2]=-t1r*ei[4*n+2]+t1i*er[4*n+2];
}
/* call sub-transforms */
rs_invdft_1d(N/2,t,t,y,2*sy,e+N/2);
rs_invdft_1d(N/4,t+N/2,t+N/2,y+sy,4*sy,e+3*N/4);
rs_invdft_1d(N/4,t+3*N/4,t+3*N/4,y+3*sy,4*sy,e+3*N/4);
}
/* strided one-dimensional inverse DFT */
static void
s_invdft_1d (minfft_cmpl *x, minfft_cmpl *y, int sy, const minfft_aux *a) {
rs_invdft_1d(a->N,x,a->t,y,sy,a->e);
}
/* strided inverse DFT of arbitrary dimension */
static void
s_invdft (minfft_cmpl *x, minfft_cmpl *y, int sy, const minfft_aux *a) {
mkcx(x,y,sy,a,s_invdft_1d);
}
/* user interface */
void
minfft_invdft (minfft_cmpl *x, minfft_cmpl *y, const minfft_aux *a) {
s_invdft(x,y,1,a);
}
/* real transforms */
/* strided one-dimensional real DFT */
static void
s_realdft_1d (minfft_real *x, minfft_cmpl *z, int sz, const minfft_aux *a) {
int n; /* counter */
int N=a->N; /* transform length */
minfft_cmpl *e=a->e; /* exponent vector */
minfft_cmpl *w=(minfft_cmpl*)x; /* alias */
minfft_cmpl *t=a->t; /* temporary buffer */
minfft_real *zr=(minfft_real*)z;
minfft_real *zi=zr+1;
register minfft_real ur,vr,ui,vi;
minfft_real *tr,*er,*ti,*ei;
if (N==1) {
/* trivial case */
zr[0]=x[0];
zi[0]=0;
return;
}
if (N==2) {
/* trivial case */
register minfft_real t0,t1;
t0=x[0];
t1=x[1];
/* z[0]=t0+t1; */
zr[0]=t0+t1;
zi[0]=0;
/* z[sz]=t0-t1; */
zr[2*sz]=t0-t1;
zi[2*sz]=0;
return;
}
/* reduce to complex DFT of length N/2 */
/* do complex DFT */
s_dft_1d(w,t,1,a->sub1);
/* recover results */
tr=(minfft_real*)t,er=(minfft_real*)e;
ti=tr+1,ei=er+1;
/* u=t[0]; */
ur=tr[0];
ui=ti[0];
/* z[0]=creal(u)+cimag(u); */
zr[0]=ur+ui;
zi[0]=0;
/* z[sz*N/2]=creal(u)-cimag(u); */
zr[sz*N]=ur-ui;
zi[sz*N]=0;
for (n=1;n<N/4;++n) {
register minfft_real ttr,ter;
register minfft_real tti,tei;
/* u=(t[n]+conj(t[N/2-n]))/2; */
ur=(tr[2*n]+tr[N-2*n])/2;
ui=(ti[2*n]-ti[N-2*n])/2;
/* v=(t[n]-conj(t[N/2-n]))*e[n]/(2*I); */
ttr=tr[2*n]-tr[N-2*n];
tti=ti[2*n]+ti[N-2*n];
ter=ei[2*n]; /* te=e[n]/I */
tei=-er[2*n];
vr=(ttr*ter-tti*tei)/2;
vi=(ttr*tei+tti*ter)/2;
/* z[sz*n]=u+v; */
zr[2*sz*n]=ur+vr;
zi[2*sz*n]=ui+vi;
/* z[sz*(N/2-n)]=conj(u-v); */
zr[sz*(N-2*n)]=ur-vr;
zi[sz*(N-2*n)]=-ui+vi;
}
/* z[sz*N/4]=conj(t[N/4]); */
zr[sz*N/2]=tr[N/2];
zi[sz*N/2]=-ti[N/2];
}
/* real DFT of arbitrary dimension */
void
minfft_realdft (minfft_real *x, minfft_cmpl *z, const minfft_aux *a) {
if (a->sub2==NULL)
s_realdft_1d(x,z,1,a);
else {
int N1=a->sub1->N,N2=a->sub2->N; /* transform lengths */
int n; /* counter */
minfft_cmpl *t=a->t; /* temporary buffer */
/* strided real DFT of contiguous rows */
for (n=0;n<N2;++n)
s_realdft_1d(x+n*N1,t+n,N2,a->sub1);
/* strided complex DFT of contiguous hyperplanes */
for (n=0;n<N1/2+1;++n)
s_dft(t+n*N2,z+n,N1/2+1,a->sub2);
}
}
/* one-dimensional inverse real DFT */
static void
invrealdft_1d (minfft_cmpl *z, minfft_real *y, const minfft_aux *a) {
int n; /* counter */
int N=a->N; /* transform length */
minfft_cmpl *e=a->e; /* exponent vector */
minfft_cmpl *w=(minfft_cmpl*)y; /* alias */
minfft_cmpl *t=a->t; /* temporary buffer */
minfft_real *zr=(minfft_real*)z;
minfft_real *zi=zr+1;
minfft_real *tr,*er,*ti,*ei;
if (N==1) {
/* trivial case */
y[0]=zr[0];
return;
}
if (N==2) {
/* trivial case */
register minfft_real t0,t1; /* temporary values */
t0=zr[0];
t1=zr[2];
y[0]=t0+t1;
y[1]=t0-t1;
return;
}
/* reduce to inverse complex DFT of length N/2 */
/* prepare complex DFT inputs */
tr=(minfft_real*)t,er=(minfft_real*)e;
ti=tr+1,ei=er+1;
/* t[0]=(z[0]+z[N/2])+I*(z[0]-z[N/2]); */
tr[0]=zr[0]+zr[N];
ti[0]=zr[0]-zr[N];
for (n=1;n<N/4;++n) {
register minfft_real ur,vr;
register minfft_real ui,vi;
register minfft_real ttr,ter;
register minfft_real tti,tei;
/* u=z[n]+conj(z[N/2-n]); */
ur=zr[2*n]+zr[N-2*n];
ui=zi[2*n]-zi[N-2*n];
/* v=I*(z[n]-conj(z[N/2-n]))*conj(e[n]); */
ttr=zr[2*n]-zr[N-2*n];
tti=zi[2*n]+zi[N-2*n];
/* te=I*conj(e[n]) */
ter=ei[2*n];
tei=er[2*n];
vr=ttr*ter-tti*tei;
vi=ttr*tei+tti*ter;
/* t[n]=u+v; */
tr[2*n]=ur+vr;
ti[2*n]=ui+vi;
/* t[N/2-n]=conj(u-v); */
tr[N-2*n]=ur-vr;
ti[N-2*n]=-ui+vi;
}
/* t[N/4]=2*conj(z[N/4]); */
tr[N/2]=2*zr[N/2];
ti[N/2]=-2*zi[N/2];
/* do inverse complex DFT */
s_invdft_1d(t,w,1,a->sub1);
}
/* inverse real DFT of arbitrary dimension */
void
minfft_invrealdft (minfft_cmpl *z, minfft_real *y, const minfft_aux *a) {
if (a->sub2==NULL)
invrealdft_1d(z,y,a);
else {
int N1=a->sub1->N,N2=a->sub2->N; /* transform lengths */
int n; /* counter */
minfft_cmpl *t=a->t; /* temporary buffer */
minfft_real *zr=(minfft_real*)z,*tr=(minfft_real*)t;
minfft_real *zi=zr+1,*ti=tr+1;
int k;
/* transpose */
for (n=0;n<N2;++n)
for (k=0;k<N1/2+1;++k) {
/* t[n+N2*k]=z[(N1/2+1)*n+k]; */
tr[2*n+2*N2*k]=zr[2*(N1/2+1)*n+2*k];
ti[2*n+2*N2*k]=zi[2*(N1/2+1)*n+2*k];
}
/* strided complex DFT of contiguous hyperplanes */
for (n=0;n<N1/2+1;++n)
s_invdft(t+n*N2,z+n,N1/2+1,a->sub2);
/* inverse real DFT of contiguous rows */
for (n=0;n<N2;++n)
invrealdft_1d(z+n*(N1/2+1),y+n*N1,a->sub1);
}
}
/* real symmetric transforms */
/* strided one-dimensional DCT-2 */
static void
s_dct2_1d (minfft_real *x, minfft_real *y, int sy, const minfft_aux *a) {
int n; /* counter */
int N=a->N; /* transform length */
minfft_real *t=a->t; /* temporary buffer */
minfft_cmpl *z=(minfft_cmpl*)t; /* its alias */
minfft_cmpl *e=a->e; /* exponent vector */
minfft_real *er,*ei;
if (N==1) {
/* trivial case */
y[0]=2*x[0];
return;
}
/* reduce to real DFT of length N */
/* prepare sub-transform inputs */
for (n=0;n<N/2;++n) {
t[n]=x[2*n];
t[N/2+n]=x[N-1-2*n];
}
/* do real DFT in-place */
s_realdft_1d(t,z,1,a->sub1);
/* recover results */
er=(minfft_real*)e;
ei=er+1;
/* y[0]=2*creal(z[0]); */
y[0]=2*t[0];
for (n=1;n<N/2;++n) {
/* y[sy*n]=2*creal(z[n]*e[n]); */
y[sy*n]=2*(t[2*n]*er[2*n]-t[2*n+1]*ei[2*n]);
/* y[sy*(N-n)]=-2*cimag(z[n]*e[n]); */
y[sy*(N-n)]=-2*(t[2*n]*ei[2*n]+t[2*n+1]*er[2*n]);
}
/* y[sy*N/2]=sqrt2*creal(z[N/2]); */
y[sy*N/2]=sqrt2*t[N];
}
/* strided DCT-2 of arbitrary dimension */
static void
s_dct2 (minfft_real *x, minfft_real *y, int sy, const minfft_aux *a) {
mkrx(x,y,sy,a,s_dct2_1d);
}
/* user interface */
void
minfft_dct2 (minfft_real *x, minfft_real *y, const minfft_aux *a) {
s_dct2(x,y,1,a);
}
/* strided one-dimensional DST-2 */
static void
s_dst2_1d (minfft_real *x, minfft_real *y, int sy, const minfft_aux *a) {
int n; /* counter */
int N=a->N; /* transform length */
minfft_real *t=a->t; /* temporary buffer */
minfft_cmpl *z=(minfft_cmpl*)t; /* its alias */
minfft_cmpl *e=a->e; /* exponent vector */
minfft_real *er,*ei;
if (N==1) {
/* trivial case */
y[0]=2*x[0];
return;
}
/* reduce to real DFT of length N */
/* prepare sub-transform inputs */
for (n=0;n<N/2;++n) {
t[n]=x[2*n];
t[N/2+n]=-x[N-1-2*n];
}
/* do real DFT in-place */
s_realdft_1d(t,z,1,a->sub1);
/* recover results */
er=(minfft_real*)e;
ei=er+1;
/* y[sy*(N-1)]=2*creal(z[0]); */
y[sy*(N-1)]=2*t[0];
for (n=1;n<N/2;++n) {
/* y[sy*(n-1)]=-2*cimag(z[n]*e[n]); */
y[sy*(n-1)]=-2*(t[2*n]*ei[2*n]+t[2*n+1]*er[2*n]);
/* y[sy*(N-n-1)]=2*creal(z[n]*e[n]); */
y[sy*(N-n-1)]=2*(t[2*n]*er[2*n]-t[2*n+1]*ei[2*n]);
}
/* y[sy*(N/2-1)]=sqrt2*creal(z[N/2]); */
y[sy*(N/2-1)]=sqrt2*t[N];
}
/* strided DST-2 of arbitrary dimension */
static void
s_dst2 (minfft_real *x, minfft_real *y, int sy, const minfft_aux *a) {
mkrx(x,y,sy,a,s_dst2_1d);
}
/* user interface */
void
minfft_dst2 (minfft_real *x, minfft_real *y, const minfft_aux *a) {
s_dst2(x,y,1,a);
}
/* strided one-dimensional DCT-3 */
static void
s_dct3_1d (minfft_real *x, minfft_real *y, int sy, const minfft_aux *a) {
int n; /* counter */
int N=a->N; /* transform length */
minfft_cmpl *z=a->t; /* temporary buffer */
minfft_real *t=(minfft_real*)z; /* its alias */
minfft_cmpl *e=a->e; /* exponent vector */
minfft_real *er,*zr,*ei,*zi;
if (N==1) {
/* trivial case */
y[0]=x[0];
return;
}
/* reduce to inverse real DFT of length N */
/* prepare sub-transform inputs */
er=(minfft_real*)e,zr=(minfft_real*)z;
ei=er+1,zi=zr+1;
/* z[0]=x[0]; */
zr[0]=x[0];
zi[0]=0;
for (n=1;n<N/2;++n) {
/* z[n]=conj((x[n]+I*x[N-n])*e[n]); */
zr[2*n]=x[n]*er[2*n]-x[N-n]*ei[2*n];
zi[2*n]=-x[n]*ei[2*n]-x[N-n]*er[2*n];
}
/* z[N/2]=sqrt2*x[N/2]; */
zr[N]=sqrt2*x[N/2];
zi[N]=0;
/* do inverse real DFT in-place */
invrealdft_1d(z,t,a->sub1);
/* recover results */
for (n=0;n<N/2;++n) {
y[sy*2*n]=t[n];
y[sy*(N-1-2*n)]=t[N/2+n];
}
}
/* strided DCT-3 of arbitrary dimension */
static void
s_dct3 (minfft_real *x, minfft_real *y, int sy, const minfft_aux *a) {
mkrx(x,y,sy,a,s_dct3_1d);
}
/* user interface */
void
minfft_dct3 (minfft_real *x, minfft_real *y, const minfft_aux *a) {
s_dct3(x,y,1,a);
}
/* strided one-dimensional DST-3 */
static void
s_dst3_1d (minfft_real *x, minfft_real *y, int sy, const minfft_aux *a) {
int n; /* counter */
int N=a->N; /* transform length */
minfft_cmpl *z=a->t; /* temporary buffer */
minfft_real *t=(minfft_real*)z; /* its alias */
minfft_cmpl *e=a->e; /* exponent vector */
minfft_real *er,*zr,*ei,*zi;
if (N==1) {
/* trivial case */
y[0]=x[0];
return;
}
/* reduce to inverse real DFT of length N */
/* prepare sub-transform inputs */
er=(minfft_real*)e,zr=(minfft_real*)z;
ei=er+1,zi=zr+1;
/* z[0]=x[N-1]; */
zr[0]=x[N-1];
zi[0]=0;
for (n=1;n<N/2;++n) {
/* z[n]=conj((x[N-n-1]+I*x[n-1])*e[n]); */
zr[2*n]=x[N-n-1]*er[2*n]-x[n-1]*ei[2*n];
zi[2*n]=-x[N-n-1]*ei[2*n]-x[n-1]*er[2*n];
}
/* z[N/2]=sqrt2*x[N/2-1]; */
zr[N]=sqrt2*x[N/2-1];
zi[N]=0;
/* do inverse real DFT in-place */
invrealdft_1d(z,t,a->sub1);
/* recover results */
for (n=0;n<N/2;++n) {
y[sy*2*n]=t[n];
y[sy*(N-1-2*n)]=-t[N/2+n];
}
}
/* strided DST-3 of arbitrary dimension */
static void
s_dst3 (minfft_real *x, minfft_real *y, int sy, const minfft_aux *a) {
mkrx(x,y,sy,a,s_dst3_1d);
}
/* user interface */
void
minfft_dst3 (minfft_real *x, minfft_real *y, const minfft_aux *a) {
s_dst3(x,y,1,a);
}
/* strided one-dimensional DCT-4 */
static void
s_dct4_1d (minfft_real *x, minfft_real *y, int sy, const minfft_aux *a) {
int n; /* counter */
int N=a->N; /* transform length */
minfft_cmpl *t=a->t; /* temporary buffer */
minfft_cmpl *e=a->e; /* exponent vector */
minfft_real *tr,*er,*ti,*ei;
if (N==1) {
/* trivial case */
y[0]=sqrt2*x[0];
return;
}
/* reduce to complex DFT of length N/2 */
/* prepare sub-transform inputs */
tr=(minfft_real*)t,er=(minfft_real*)e;
ti=tr+1,ei=er+1;
for (n=0;n<N/2;++n) {
/* t[n]=*e++*(x[2*n]+I*x[N-1-2*n]); */
tr[2*n]=er[2*n]*x[2*n]-ei[2*n]*x[N-1-2*n];
ti[2*n]=er[2*n]*x[N-1-2*n]+ei[2*n]*x[2*n];
}
/* do complex DFT in-place */
s_dft_1d(t,t,1,a->sub1);
/* recover results */
er+=N;
ei+=N;
for (n=0;n<N/2;++n) {
/* y[sy*2*n]=2*creal(*e++*t[n]); */
y[sy*2*n]=2*(er[4*n]*tr[2*n]-ei[4*n]*ti[2*n]);
/* y[sy*(2*n+1)]=2*creal(*e++*conj(t[N/2-1-n])); */
y[sy*(2*n+1)]=2*(er[4*n+2]*tr[N-2-2*n]+ei[4*n+2]*ti[N-2-2*n]);
}
}
/* strided DCT-4 of arbitrary dimension */
static void
s_dct4 (minfft_real *x, minfft_real *y, int sy, const minfft_aux *a) {
mkrx(x,y,sy,a,s_dct4_1d);
}
/* user interface */
void
minfft_dct4 (minfft_real *x, minfft_real *y, const minfft_aux *a) {
s_dct4(x,y,1,a);
}
/* strided one-dimensional DST-4 */
static void
s_dst4_1d (minfft_real *x, minfft_real *y, int sy, const minfft_aux *a) {
int n; /* counter */
int N=a->N; /* transform length */
minfft_cmpl *t=a->t; /* temporary buffer */
minfft_cmpl *e=a->e; /* exponent vector */
minfft_real *tr,*er,*ti,*ei;
if (N==1) {
/* trivial case */
y[0]=sqrt2*x[0];
return;
}
/* reduce to complex DFT of length N/2 */
/* prepare sub-transform inputs */
tr=(minfft_real*)t,er=(minfft_real*)e;
ti=tr+1,ei=er+1;
for (n=0;n<N/2;++n) {
/* t[n]=-*e++*(x[2*n]-I*x[N-1-2*n]); */
tr[2*n]=-er[2*n]*x[2*n]-ei[2*n]*x[N-1-2*n];
ti[2*n]=er[2*n]*x[N-1-2*n]-ei[2*n]*x[2*n];
}
/* do complex DFT in-place */
s_dft_1d(t,t,1,a->sub1);
/* recover results */
er+=N;
ei+=N;
for (n=0;n<N/2;++n) {
/* y[sy*2*n]=2*cimag(*e++*t[n]); */
y[sy*2*n]=2*(er[4*n]*ti[2*n]+ei[4*n]*tr[2*n]);
/* y[sy*(2*n+1)]=2*cimag(*e++*conj(t[N/2-1-n])); */
y[sy*(2*n+1)]=2*(-er[4*n+2]*ti[N-2-2*n]+ei[4*n+2]*tr[N-2-2*n]);
}
}
/* strided DST-4 of arbitrary dimension */
static void
s_dst4 (minfft_real *x, minfft_real *y, int sy, const minfft_aux *a) {
mkrx(x,y,sy,a,s_dst4_1d);
}
/* user interface */
void
minfft_dst4 (minfft_real *x, minfft_real *y, const minfft_aux *a) {
s_dst4(x,y,1,a);
}
/* routines for computing N-th roots of unity */
/* prototype */
static minfft_real nsin (int, int);
/* cos(2*pi*n/N) */
static minfft_real
ncos (int n, int N) {
/* reduce n to 0..N/8 */
if (n<0)
return ncos(-n,N);
else if (n>=N/2)
return -ncos(n-N/2,N);
else if (n>=N/4)
return -nsin(n-N/4,N);
else if (n>N/8)
return nsin(N/4-n,N);
else