/*
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
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);
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 dstsub(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);
}
}
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, double *a, double *t, 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 rftfsub(int n, double *a, int nc, double *c);
void dctsub(int n, double *a, int nc, double *c);
int j, k, l, m, mh, nw, nc;
double 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, double *a, double *t, 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 rftfsub(int n, double *a, int nc, double *c);
void dstsub(int n, double *a, int nc, double *c);
int j, k, l, m, mh, nw, nc;
double 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, 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;
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, double *a, double *w)
{
void cft1st(int n, double *a, double *w);
void cftmdl(int n, int l, double *a, double *w);
int j, j1, j2, j3, l;
double x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i;
l = 2;
if (n > 8) {
cft1st(n, a, w);
l = 8;
while ((l << 2) < n) {
cftmdl(n, l, a, w);
l <<= 2;
}
}
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[j1 + 1];
a[j1] = x0r;
a[j1 + 1] = x0i;
}
}
}
void cftbsub(int n, double *a, double *w)
{
void cft1st(int n, double *a, double *w);
void cftmdl(int n, int l, double *a, double *w);
int j, j1, j2, j3, l;
double x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i;
l = 2;
if (n > 8) {
cft1st(n, a, w);
l = 8;
while ((l << 2) < n) {
cftmdl(n, l, a, w);
l <<= 2;
}
}
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, double *a, double *w)
{
int j, k1, k2;
double wk1r, wk1i, wk2r, wk2i, wk3r, wk3i;
double x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i;
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];
a[0] = x0r + x2r;
a[1] = x0i + x2i;
a[4] = x0r - x2r;
a[5] = x0i - x2i;
a[2] = x1r - x3i;
a[3] = x1i + x3r;
a[6] = x1r + x3i;
a[7] = x1i - x3r;
wk1r = w[2];
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];
a[8] = x0r + x2r;
a[9] = x0i + x2i;
a[12] = x2i - x0i;
a[13] = x0r - x2r;
x0r = x1r - x3i;
x0i = x1i + x3r;
a[10] = wk1r * (x0r - x0i);
a[11] = wk1r * (x0r + x0i);
x0r = x3i + x1r;
x0i = x3r - x1i;
a[14] = wk1r * (x0i - x0r);
a[15] = wk1r * (x0i + x0r);
k1 = 0;
for (j = 16; j < n; j += 16) {
k1 += 2;
k2 = 2 * k1;
wk2r = w[k1];
wk2i = w[k1 + 1];
wk1r = w[k2];
wk1i = w[k2 + 1];
wk3r = wk1r - 2 * wk2i * wk1i;
wk3i = 2 * wk2i * wk1r - 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];
a[j] = x0r + x2r;
a[j + 1] = x0i + x2i;
x0r -= x2r;
x0i -= x2i;
a[j + 4] = wk2r * x0r - wk2i * x0i;
a[j + 5] = wk2r * x0i + wk2i * x0r;
x0r = x1r - x3i;
x0i = x1i + x3r;
a[j + 2] = wk1r * x0r - wk1i * x0i;
a[j + 3] = wk1r * x0i + wk1i * x0r;
x0r = x1r + x3i;
x0i = x1i - x3r;
a[j + 6] = wk3r * x0r - wk3i * x0i;
a[j + 7] = wk3r * x0i + wk3i * x0r;
wk1r = w[k2 + 2];
wk1i = w[k2 + 3];
wk3r = wk1r - 2 * wk2r * wk1i;
wk3i = 2 * wk2r * wk1r - wk1i;
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];
a[j + 8] = x0r + x2r;
a[j + 9] = x0i + x2i;
x0r -= x2r;
x0i -= x2i;
a[j + 12] = -wk2i * x0r - wk2r * x0i;
a[j + 13] = -wk2i * x0i + wk2r * x0r;
x0r = x1r - x3i;
x0i = x1i + x3r;
a[j + 10] = wk1r * x0r - wk1i * x0i;
a[j + 11] = wk1r * x0i + wk1i * x0r;
x0r = x1r + x3i;
x0i = x1i - x3r;
a[j + 14] = wk3r * x0r - wk3i * x0i;
a[j + 15] = wk3r * x0i + wk3i * x0r;
}
}
void cftmdl(int n, int l, double *a, double *w)
{
int j, j1, j2, j3, k, k1, k2, m, m2;
double wk1r, wk1i, wk2r, wk2i, wk3r, wk3i;
double x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i;
m = l << 2;
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;
}
wk1r = w[2];
for (j = m; j < l + m; 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] = x2i - x0i;
a[j2 + 1] = x0r - x2r;
x0r = x1r - x3i;
x0i = x1i + x3r;
a[j1] = wk1r * (x0r - x0i);
a[j1 + 1] = wk1r * (x0r + x0i);
x0r = x3i + x1r;
x0i = x3r - x1i;
a[j3] = wk1r * (x0i - x0r);
a[j3 + 1] = wk1r * (x0i + x0r);
}
k1 = 0;
m2 = 2 * m;
for (k = m2; k < n; k += m2) {
k1 += 2;
k2 = 2 * k1;
wk2r = w[k1];
wk2i = w[k1 + 1];
wk1r = w[k2];
wk1i = w[k2 + 1];
wk3r = wk1r - 2 * wk2i * wk1i;
wk3i = 2 * wk2i * wk1r - wk1i;
for (j = k; j < l + k; 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;
x0r -= x2r;
x0i -= x2i;
a[j2] = wk2r * x0r - wk2i * x0i;
a[j2 + 1] = wk2r * x0i + wk2i * x0r;
x0r = x1r - x3i;
x0i = x1i + x3r;
a[j1] = wk1r * x0r - wk1i * x0i;
a[j1 + 1] = wk1r * x0i + wk1i * x0r;
x0r = x1r + x3i;
x0i = x1i - x3r;
a[j3] = wk3r * x0r - wk3i * x0i;
a[j3 + 1] = wk3r * x0i + wk3i * x0r;
}
wk1r = w[k2 + 2];
wk1i = w[k2 + 3];
wk3r = wk1r - 2 * wk2r * wk1i;
wk3i = 2 * wk2r * wk1r - wk1i;
for (j = k + m; j < l + (k + m); 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;
x0r -= x2r;
x0i -= x2i;
a[j2] = -wk2i * x0r - wk2r * x0i;
a[j2 + 1] = -wk2i * x0i + wk2r * x0r;
x0r = x1r - x3i;
x0i = x1i + x3r;
a[j1] = wk1r * x0r - wk1i * x0i;
a[j1 + 1] = wk1r * x0i + wk1i * x0r;
x0r = x1r + x3i;
x0i = x1i - x3r;
a[j3] = wk3r * x0r - wk3i * x0i;
a[j3 + 1] = wk3r * x0i + wk3i * x0r;
}
}
}
void rftfsub(int n, double *a, int nc, double *c)
{
int j, k, kk, ks, m;
double 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, double *a, int nc, double *c)
{
int j, k, kk, ks, m;
double 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, double *a, int nc, double *c)
{
int j, k, kk, ks, m;
double 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, double *a, int nc, double *c)
{
int j, k, kk, ks, m;
double 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];
}