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fft4f2d.c
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/*
Fast Fourier/Cosine/Sine Transform
dimension :two
data length :power of 2
decimation :frequency
radix :4, 2, row-column
data :inplace
table :use
functions
cdft2d: Complex Discrete Fourier Transform
rdft2d: Real Discrete Fourier Transform
ddct2d: Discrete Cosine Transform
ddst2d: Discrete Sine Transform
function prototypes
void cdft2d(int, int, int, double **, int *, double *);
void rdft2d(int, int, int, double **, int *, double *);
void ddct2d(int, int, int, double **, double **, int *, double *);
void ddst2d(int, int, int, double **, double **, int *, double *);
-------- Complex DFT (Discrete Fourier Transform) --------
[definition]
<case1>
X[k1][k2] = sum_j1=0^n1-1 sum_j2=0^n2-1 x[j1][j2] *
exp(2*pi*i*j1*k1/n1) *
exp(2*pi*i*j2*k2/n2), 0<=k1<n1, 0<=k2<n2
<case2>
X[k1][k2] = sum_j1=0^n1-1 sum_j2=0^n2-1 x[j1][j2] *
exp(-2*pi*i*j1*k1/n1) *
exp(-2*pi*i*j2*k2/n2), 0<=k1<n1, 0<=k2<n2
(notes: sum_j=0^n-1 is a summation from j=0 to n-1)
[usage]
<case1>
ip[0] = 0; // first time only
cdft2d(n1, 2*n2, 1, a, ip, w);
<case2>
ip[0] = 0; // first time only
cdft2d(n1, 2*n2, -1, a, ip, w);
[parameters]
n1 :data length (int)
n1 >= 1, n1 = power of 2
2*n2 :data length (int)
n2 >= 1, n2 = power of 2
a[0...n1-1][0...2*n2-1]
:input/output data (double **)
input data
a[j1][2*j2] = Re(x[j1][j2]),
a[j1][2*j2+1] = Im(x[j1][j2]),
0<=j1<n1, 0<=j2<n2
output data
a[k1][2*k2] = Re(X[k1][k2]),
a[k1][2*k2+1] = Im(X[k1][k2]),
0<=k1<n1, 0<=k2<n2
ip[0...*]
:work area for bit reversal (int *)
length of ip >= 2+sqrt(n)
(n = max(n1, n2))
ip[0],ip[1] are pointers of the cos/sin table.
w[0...*]
:cos/sin table (double *)
length of w >= max(n1/2, n2/2)
w[],ip[] are initialized if ip[0] == 0.
[remark]
Inverse of
cdft2d(n1, 2*n2, -1, a, ip, w);
is
cdft2d(n1, 2*n2, 1, a, ip, w);
for (j1 = 0; j1 <= n1 - 1; j1++) {
for (j2 = 0; j2 <= 2 * n2 - 1; j2++) {
a[j1][j2] *= 1.0 / (n1 * n2);
}
}
.
-------- Real DFT / Inverse of Real DFT --------
[definition]
<case1> RDFT
R[k1][k2] = sum_j1=0^n1-1 sum_j2=0^n2-1 a[j1][j2] *
cos(2*pi*j1*k1/n1 + 2*pi*j2*k2/n2),
0<=k1<n1, 0<=k2<n2
I[k1][k2] = sum_j1=0^n1-1 sum_j2=0^n2-1 a[j1][j2] *
sin(2*pi*j1*k1/n1 + 2*pi*j2*k2/n2),
0<=k1<n1, 0<=k2<n2
<case2> IRDFT (excluding scale)
a[k1][k2] = (1/2) * sum_j1=0^n1-1 sum_j2=0^n2-1
(R[j1][j2] *
cos(2*pi*j1*k1/n1 + 2*pi*j2*k2/n2) +
I[j1][j2] *
sin(2*pi*j1*k1/n1 + 2*pi*j2*k2/n2)),
0<=k1<n1, 0<=k2<n2
(notes: R[n1-k1][n2-k2] = R[k1][k2],
I[n1-k1][n2-k2] = -I[k1][k2],
R[n1-k1][0] = R[k1][0],
I[n1-k1][0] = -I[k1][0],
R[0][n2-k2] = R[0][k2],
I[0][n2-k2] = -I[0][k2],
0<k1<n1, 0<k2<n2)
[usage]
<case1>
ip[0] = 0; // first time only
rdft2d(n1, n2, 1, a, ip, w);
<case2>
ip[0] = 0; // first time only
rdft2d(n1, n2, -1, a, ip, w);
[parameters]
n1 :data length (int)
n1 >= 2, n1 = power of 2
n2 :data length (int)
n2 >= 2, n2 = power of 2
a[0...n1-1][0...n2-1]
:input/output data (double **)
<case1>
output data
a[k1][2*k2] = R[k1][k2] = R[n1-k1][n2-k2],
a[k1][2*k2+1] = I[k1][k2] = -I[n1-k1][n2-k2],
0<k1<n1, 0<k2<n2/2,
a[0][2*k2] = R[0][k2] = R[0][n2-k2],
a[0][2*k2+1] = I[0][k2] = -I[0][n2-k2],
0<k2<n2/2,
a[k1][0] = R[k1][0] = R[n1-k1][0],
a[k1][1] = I[k1][0] = -I[n1-k1][0],
a[n1-k1][1] = R[k1][n2/2] = R[n1-k1][n2/2],
a[n1-k1][0] = -I[k1][n2/2] = I[n1-k1][n2/2],
0<k1<n1/2,
a[0][0] = R[0][0],
a[0][1] = R[0][n2/2],
a[n1/2][0] = R[n1/2][0],
a[n1/2][1] = R[n1/2][n2/2]
<case2>
input data
a[j1][2*j2] = R[j1][j2] = R[n1-j1][n2-j2],
a[j1][2*j2+1] = I[j1][j2] = -I[n1-j1][n2-j2],
0<j1<n1, 0<j2<n2/2,
a[0][2*j2] = R[0][j2] = R[0][n2-j2],
a[0][2*j2+1] = I[0][j2] = -I[0][n2-j2],
0<j2<n2/2,
a[j1][0] = R[j1][0] = R[n1-j1][0],
a[j1][1] = I[j1][0] = -I[n1-j1][0],
a[n1-j1][1] = R[j1][n2/2] = R[n1-j1][n2/2],
a[n1-j1][0] = -I[j1][n2/2] = I[n1-j1][n2/2],
0<j1<n1/2,
a[0][0] = R[0][0],
a[0][1] = R[0][n2/2],
a[n1/2][0] = R[n1/2][0],
a[n1/2][1] = R[n1/2][n2/2]
ip[0...*]
:work area for bit reversal (int *)
length of ip >= 2+sqrt(n)
(n = max(n1, n2/2))
ip[0],ip[1] are pointers of the cos/sin table.
w[0...*]
:cos/sin table (double *)
length of w >= max(n1/2, n2/4) + n2/4
w[],ip[] are initialized if ip[0] == 0.
[remark]
Inverse of
rdft2d(n1, n2, 1, a, ip, w);
is
rdft2d(n1, n2, -1, a, ip, w);
for (j1 = 0; j1 <= n1 - 1; j1++) {
for (j2 = 0; j2 <= n2 - 1; j2++) {
a[j1][j2] *= 2.0 / (n1 * n2);
}
}
.
-------- DCT (Discrete Cosine Transform) / Inverse of DCT --------
[definition]
<case1> IDCT (excluding scale)
C[k1][k2] = sum_j1=0^n1-1 sum_j2=0^n2-1 a[j1][j2] *
cos(pi*j1*(k1+1/2)/n1) *
cos(pi*j2*(k2+1/2)/n2),
0<=k1<n1, 0<=k2<n2
<case2> DCT
C[k1][k2] = sum_j1=0^n1-1 sum_j2=0^n2-1 a[j1][j2] *
cos(pi*(j1+1/2)*k1/n1) *
cos(pi*(j2+1/2)*k2/n2),
0<=k1<n1, 0<=k2<n2
[usage]
<case1>
ip[0] = 0; // first time only
ddct2d(n1, n2, 1, a, t, ip, w);
<case2>
ip[0] = 0; // first time only
ddct2d(n1, n2, -1, a, t, ip, w);
[parameters]
n1 :data length (int)
n1 >= 2, n1 = power of 2
n2 :data length (int)
n2 >= 2, n2 = power of 2
a[0...n1-1][0...n2-1]
:input/output data (double **)
output data
a[k1][k2] = C[k1][k2], 0<=k1<n1, 0<=k2<n2
t[0...n1-1][0...n2-1]
:work area (double **)
ip[0...*]
:work area for bit reversal (int *)
length of ip >= 2+sqrt(n)
(n = max(n1, n2/2))
ip[0],ip[1] are pointers of the cos/sin table.
w[0...*]
:cos/sin table (double *)
length of w >= max(n1/2, n2/4) + max(n1, n2)
w[],ip[] are initialized if ip[0] == 0.
[remark]
Inverse of
ddct2d(n1, n2, -1, a, t, ip, w);
is
for (j1 = 0; j1 <= n1 - 1; j1++) {
a[j1][0] *= 0.5;
}
for (j2 = 0; j2 <= n2 - 1; j2++) {
a[0][j2] *= 0.5;
}
ddct2d(n1, n2, 1, a, t, ip, w);
for (j1 = 0; j1 <= n1 - 1; j1++) {
for (j2 = 0; j2 <= n2 - 1; j2++) {
a[j1][j2] *= 4.0 / (n1 * n2);
}
}
.
-------- DST (Discrete Sine Transform) / Inverse of DST --------
[definition]
<case1> IDST (excluding scale)
S[k1][k2] = sum_j1=1^n1 sum_j2=1^n2 A[j1][j2] *
sin(pi*j1*(k1+1/2)/n1) *
sin(pi*j2*(k2+1/2)/n2),
0<=k1<n1, 0<=k2<n2
<case2> DST
S[k1][k2] = sum_j1=0^n1-1 sum_j2=0^n2-1 a[j1][j2] *
sin(pi*(j1+1/2)*k1/n1) *
sin(pi*(j2+1/2)*k2/n2),
0<k1<=n1, 0<k2<=n2
[usage]
<case1>
ip[0] = 0; // first time only
ddst2d(n1, n2, 1, a, t, ip, w);
<case2>
ip[0] = 0; // first time only
ddst2d(n1, n2, -1, a, t, ip, w);
[parameters]
n1 :data length (int)
n1 >= 2, n1 = power of 2
n2 :data length (int)
n2 >= 2, n2 = power of 2
a[0...n1-1][0...n2-1]
:input/output data (double **)
<case1>
input data
a[j1][j2] = A[j1][j2], 0<j1<n1, 0<j2<n2,
a[j1][0] = A[j1][n2], 0<j1<n1,
a[0][j2] = A[n1][j2], 0<j2<n2,
a[0][0] = A[n1][n2]
(i.e. A[j1][j2] = a[j1 % n1][j2 % n2])
output data
a[k1][k2] = S[k1][k2], 0<=k1<n1, 0<=k2<n2
<case2>
output data
a[k1][k2] = S[k1][k2], 0<k1<n1, 0<k2<n2,
a[k1][0] = S[k1][n2], 0<k1<n1,
a[0][k2] = S[n1][k2], 0<k2<n2,
a[0][0] = S[n1][n2]
(i.e. S[k1][k2] = a[k1 % n1][k2 % n2])
t[0...n1-1][0...n2-1]
:work area (double **)
ip[0...*]
:work area for bit reversal (int *)
length of ip >= 2+sqrt(n)
(n = max(n1, n2/2))
ip[0],ip[1] are pointers of the cos/sin table.
w[0...*]
:cos/sin table (double *)
length of w >= max(n1/2, n2/4) + max(n1, n2)
w[],ip[] are initialized if ip[0] == 0.
[remark]
Inverse of
ddst2d(n1, n2, -1, a, t, ip, w);
is
for (j1 = 0; j1 <= n1 - 1; j1++) {
a[j1][0] *= 0.5;
}
for (j2 = 0; j2 <= n2 - 1; j2++) {
a[0][j2] *= 0.5;
}
ddst2d(n1, n2, 1, a, t, ip, w);
for (j1 = 0; j1 <= n1 - 1; j1++) {
for (j2 = 0; j2 <= n2 - 1; j2++) {
a[j1][j2] *= 4.0 / (n1 * n2);
}
}
.
*/
void cdft2d(int n1, int n2, int isgn, double **a, int *ip, double *w)
{
void makewt(int nw, int *ip, double *w);
void bitrv2col(int n1, int n, int *ip, double **a);
void bitrv2row(int n, int n2, int *ip, double **a);
void cftbcol(int n1, int n, double **a, double *w);
void cftbrow(int n, int n2, double **a, double *w);
void cftfcol(int n1, int n, double **a, double *w);
void cftfrow(int n, int n2, double **a, double *w);
int n;
n = n1 << 1;
if (n < n2) {
n = n2;
}
if (n > (ip[0] << 2)) {
makewt(n >> 2, ip, w);
}
if (n2 > 4) {
bitrv2col(n1, n2, ip + 2, a);
}
if (n1 > 2) {
bitrv2row(n1, n2, ip + 2, a);
}
if (isgn < 0) {
cftfcol(n1, n2, a, w);
cftfrow(n1, n2, a, w);
} else {
cftbcol(n1, n2, a, w);
cftbrow(n1, n2, a, w);
}
}
void rdft2d(int n1, int n2, 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 bitrv2col(int n1, int n, int *ip, double **a);
void bitrv2row(int n, int n2, int *ip, double **a);
void cftbcol(int n1, int n, double **a, double *w);
void cftbrow(int n, int n2, double **a, double *w);
void cftfcol(int n1, int n, double **a, double *w);
void cftfrow(int n, int n2, double **a, double *w);
void rftbcol(int n1, int n, double **a, int nc, double *c);
void rftfcol(int n1, int n, double **a, int nc, double *c);
int n, nw, nc, n1h, i, j;
double xi;
n = n1 << 1;
if (n < n2) {
n = n2;
}
nw = ip[0];
if (n > (nw << 2)) {
nw = n >> 2;
makewt(nw, ip, w);
}
nc = ip[1];
if (n2 > (nc << 2)) {
nc = n2 >> 2;
makect(nc, ip, w + nw);
}
n1h = n1 >> 1;
if (isgn < 0) {
for (i = 1; i <= n1h - 1; i++) {
j = n1 - i;
xi = a[i][0] - a[j][0];
a[i][0] += a[j][0];
a[j][0] = xi;
xi = a[j][1] - a[i][1];
a[i][1] += a[j][1];
a[j][1] = xi;
}
if (n1 > 2) {
bitrv2row(n1, n2, ip + 2, a);
}
cftfrow(n1, n2, a, w);
for (i = 0; i <= n1 - 1; i++) {
a[i][1] = 0.5 * (a[i][0] - a[i][1]);
a[i][0] -= a[i][1];
}
if (n2 > 4) {
rftfcol(n1, n2, a, nc, w + nw);
bitrv2col(n1, n2, ip + 2, a);
}
cftfcol(n1, n2, a, w);
} else {
if (n2 > 4) {
bitrv2col(n1, n2, ip + 2, a);
}
cftbcol(n1, n2, a, w);
if (n2 > 4) {
rftbcol(n1, n2, a, nc, w + nw);
}
for (i = 0; i <= n1 - 1; i++) {
xi = a[i][0] - a[i][1];
a[i][0] += a[i][1];
a[i][1] = xi;
}
if (n1 > 2) {
bitrv2row(n1, n2, ip + 2, a);
}
cftbrow(n1, n2, a, w);
for (i = 1; i <= n1h - 1; i++) {
j = n1 - i;
a[j][0] = 0.5 * (a[i][0] - a[j][0]);
a[i][0] -= a[j][0];
a[j][1] = 0.5 * (a[i][1] + a[j][1]);
a[i][1] -= a[j][1];
}
}
}
void ddct2d(int n1, int n2, int isgn, 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 bitrv2col(int n1, int n, int *ip, double **a);
void bitrv2row(int n, int n2, int *ip, double **a);
void cftbcol(int n1, int n, double **a, double *w);
void cftbrow(int n, int n2, double **a, double *w);
void cftfcol(int n1, int n, double **a, double *w);
void cftfrow(int n, int n2, double **a, double *w);
void rftbcol(int n1, int n, double **a, int nc, double *c);
void rftfcol(int n1, int n, double **a, int nc, double *c);
void dctbsub(int n1, int n2, double **a, int nc, double *c);
void dctfsub(int n1, int n2, double **a, int nc, double *c);
int n, nw, nc, n1h, n2h, i, ix, ic, j, jx, jc;
double xi;
n = n1 << 1;
if (n < n2) {
n = n2;
}
nw = ip[0];
if (n > (nw << 2)) {
nw = n >> 2;
makewt(nw, ip, w);
}
nc = ip[1];
if (n1 > nc || n2 > nc) {
if (n1 > n2) {
nc = n1;
} else {
nc = n2;
}
makect(nc, ip, w + nw);
}
n1h = n1 >> 1;
n2h = n2 >> 1;
if (isgn >= 0) {
for (i = 0; i <= n1 - 1; i++) {
for (j = 1; j <= n2h - 1; j++) {
jx = j << 1;
t[i][jx] = a[i][j];
t[i][jx + 1] = a[i][n2 - j];
}
}
t[0][0] = a[0][0];
t[0][1] = a[0][n2h];
t[n1h][0] = a[n1h][0];
t[n1h][1] = a[n1h][n2h];
for (i = 1; i <= n1h - 1; i++) {
ic = n1 - i;
t[i][0] = a[i][0];
t[ic][1] = a[i][n2h];
t[i][1] = a[ic][0];
t[ic][0] = a[ic][n2h];
}
dctfsub(n1, n2, t, nc, w + nw);
if (n1 > 2) {
bitrv2row(n1, n2, ip + 2, t);
}
cftfrow(n1, n2, t, w);
for (i = 0; i <= n1 - 1; i++) {
t[i][1] = 0.5 * (t[i][0] - t[i][1]);
t[i][0] -= t[i][1];
}
if (n2 > 4) {
rftfcol(n1, n2, t, nc, w + nw);
bitrv2col(n1, n2, ip + 2, t);
}
cftfcol(n1, n2, t, w);
for (i = 0; i <= n1h - 1; i++) {
ix = i << 1;
ic = n1 - 1 - i;
for (j = 0; j <= n2h - 1; j++) {
jx = j << 1;
jc = n2 - 1 - j;
a[ix][jx] = t[i][j];
a[ix][jx + 1] = t[i][jc];
a[ix + 1][jx] = t[ic][j];
a[ix + 1][jx + 1] = t[ic][jc];
}
}
} else {
for (i = 0; i <= n1h - 1; i++) {
ix = i << 1;
ic = n1 - 1 - i;
for (j = 0; j <= n2h - 1; j++) {
jx = j << 1;
jc = n2 - 1 - j;
t[i][j] = a[ix][jx];
t[i][jc] = a[ix][jx + 1];
t[ic][j] = a[ix + 1][jx];
t[ic][jc] = a[ix + 1][jx + 1];
}
}
if (n2 > 4) {
bitrv2col(n1, n2, ip + 2, t);
}
cftbcol(n1, n2, t, w);
if (n2 > 4) {
rftbcol(n1, n2, t, nc, w + nw);
}
for (i = 0; i <= n1 - 1; i++) {
xi = t[i][0] - t[i][1];
t[i][0] += t[i][1];
t[i][1] = xi;
}
if (n1 > 2) {
bitrv2row(n1, n2, ip + 2, t);
}
cftbrow(n1, n2, t, w);
dctbsub(n1, n2, t, nc, w + nw);
for (i = 0; i <= n1 - 1; i++) {
for (j = 1; j <= n2h - 1; j++) {
jx = j << 1;
a[i][j] = t[i][jx];
a[i][n2 - j] = t[i][jx + 1];
}
}
a[0][0] = t[0][0];
a[0][n2h] = t[0][1];
a[n1h][0] = t[n1h][0];
a[n1h][n2h] = t[n1h][1];
for (i = 1; i <= n1h - 1; i++) {
ic = n1 - i;
a[i][0] = t[i][0];
a[i][n2h] = t[ic][1];
a[ic][0] = t[i][1];
a[ic][n2h] = t[ic][0];
}
}
}
void ddst2d(int n1, int n2, int isgn, 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 bitrv2col(int n1, int n, int *ip, double **a);
void bitrv2row(int n, int n2, int *ip, double **a);
void cftbcol(int n1, int n, double **a, double *w);
void cftbrow(int n, int n2, double **a, double *w);
void cftfcol(int n1, int n, double **a, double *w);
void cftfrow(int n, int n2, double **a, double *w);
void rftbcol(int n1, int n, double **a, int nc, double *c);
void rftfcol(int n1, int n, double **a, int nc, double *c);
void dstbsub(int n1, int n2, double **a, int nc, double *c);
void dstfsub(int n1, int n2, double **a, int nc, double *c);
int n, nw, nc, n1h, n2h, i, ix, ic, j, jx, jc;
double xi;
n = n1 << 1;
if (n < n2) {
n = n2;
}
nw = ip[0];
if (n > (nw << 2)) {
nw = n >> 2;
makewt(nw, ip, w);
}
nc = ip[1];
if (n1 > nc || n2 > nc) {
if (n1 > n2) {
nc = n1;
} else {
nc = n2;
}
makect(nc, ip, w + nw);
}
n1h = n1 >> 1;
n2h = n2 >> 1;
if (isgn >= 0) {
for (i = 0; i <= n1 - 1; i++) {
for (j = 1; j <= n2h - 1; j++) {
jx = j << 1;
t[i][jx] = a[i][j];
t[i][jx + 1] = a[i][n2 - j];
}
}
t[0][0] = a[0][0];
t[0][1] = a[0][n2h];
t[n1h][0] = a[n1h][0];
t[n1h][1] = a[n1h][n2h];
for (i = 1; i <= n1h - 1; i++) {
ic = n1 - i;
t[i][0] = a[i][0];
t[ic][1] = a[i][n2h];
t[i][1] = a[ic][0];
t[ic][0] = a[ic][n2h];
}
dstfsub(n1, n2, t, nc, w + nw);
if (n1 > 2) {
bitrv2row(n1, n2, ip + 2, t);
}
cftfrow(n1, n2, t, w);
for (i = 0; i <= n1 - 1; i++) {
t[i][1] = 0.5 * (t[i][0] - t[i][1]);
t[i][0] -= t[i][1];
}
if (n2 > 4) {
rftfcol(n1, n2, t, nc, w + nw);
bitrv2col(n1, n2, ip + 2, t);
}
cftfcol(n1, n2, t, w);
for (i = 0; i <= n1h - 1; i++) {
ix = i << 1;
ic = n1 - 1 - i;
for (j = 0; j <= n2h - 1; j++) {
jx = j << 1;
jc = n2 - 1 - j;
a[ix][jx] = t[i][j];
a[ix][jx + 1] = -t[i][jc];
a[ix + 1][jx] = -t[ic][j];
a[ix + 1][jx + 1] = t[ic][jc];
}
}
} else {
for (i = 0; i <= n1h - 1; i++) {
ix = i << 1;
ic = n1 - 1 - i;
for (j = 0; j <= n2h - 1; j++) {
jx = j << 1;
jc = n2 - 1 - j;
t[i][j] = a[ix][jx];
t[i][jc] = -a[ix][jx + 1];
t[ic][j] = -a[ix + 1][jx];
t[ic][jc] = a[ix + 1][jx + 1];
}
}
if (n2 > 4) {
bitrv2col(n1, n2, ip + 2, t);
}
cftbcol(n1, n2, t, w);
if (n2 > 4) {
rftbcol(n1, n2, t, nc, w + nw);
}
for (i = 0; i <= n1 - 1; i++) {
xi = t[i][0] - t[i][1];
t[i][0] += t[i][1];
t[i][1] = xi;
}
if (n1 > 2) {
bitrv2row(n1, n2, ip + 2, t);
}
cftbrow(n1, n2, t, w);
dstbsub(n1, n2, t, nc, w + nw);
for (i = 0; i <= n1 - 1; i++) {
for (j = 1; j <= n2h - 1; j++) {
jx = j << 1;
a[i][j] = t[i][jx];
a[i][n2 - j] = t[i][jx + 1];
}
}
a[0][0] = t[0][0];
a[0][n2h] = t[0][1];
a[n1h][0] = t[n1h][0];
a[n1h][n2h] = t[n1h][1];
for (i = 1; i <= n1h - 1; i++) {
ic = n1 - i;
a[i][0] = t[i][0];
a[i][n2h] = t[ic][1];
a[ic][0] = t[i][1];
a[ic][n2h] = t[ic][0];
}
}
}
/* -------- initializing routines -------- */
#include <math.h>
void makewt(int nw, int *ip, double *w)
{
void bitrv2(int n, int *ip, double *a);
int nwh, j;
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];
for (j = 2; j <= nwh - 2; 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 nch, j;
double delta;
ip[1] = nc;
if (nc > 1) {
nch = nc >> 1;
delta = atan(1.0) / nch;
c[0] = 0.5;
c[nch] = 0.5 * cos(delta * nch);
for (j = 1; j <= nch - 1; 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;
ip[0] = 0;
l = n;
m = 1;
while ((m << 2) < l) {
l >>= 1;
for (j = 0; j <= m - 1; j++) {
ip[m + j] = ip[j] + l;
}
m <<= 1;
}
if ((m << 2) > l) {
for (k = 1; k <= m - 1; k++) {
for (j = 0; j <= k - 1; j++) {
j1 = (j << 1) + ip[k];
k1 = (k << 1) + ip[j];
xr = a[j1];
xi = a[j1 + 1];
a[j1] = a[k1];
a[j1 + 1] = a[k1 + 1];
a[k1] = xr;
a[k1 + 1] = xi;
}
}
} else {
m2 = m << 1;
for (k = 1; k <= m - 1; k++) {
for (j = 0; j <= k - 1; j++) {
j1 = (j << 1) + ip[k];
k1 = (k << 1) + ip[j];
xr = a[j1];
xi = a[j1 + 1];
a[j1] = a[k1];
a[j1 + 1] = a[k1 + 1];
a[k1] = xr;
a[k1 + 1] = xi;
j1 += m2;
k1 += m2;
xr = a[j1];
xi = a[j1 + 1];
a[j1] = a[k1];
a[j1 + 1] = a[k1 + 1];
a[k1] = xr;
a[k1 + 1] = xi;
}
}
}
}
void bitrv2col(int n1, int n, int *ip, double **a)
{
int i, j, j1, k, k1, l, m, m2;
double xr, xi;
ip[0] = 0;
l = n;
m = 1;
while ((m << 2) < l) {
l >>= 1;
for (j = 0; j <= m - 1; j++) {
ip[m + j] = ip[j] + l;
}
m <<= 1;
}
if ((m << 2) > l) {
for (i = 0; i <= n1 - 1; i++) {
for (k = 1; k <= m - 1; k++) {
for (j = 0; j <= k - 1; j++) {
j1 = (j << 1) + ip[k];
k1 = (k << 1) + ip[j];
xr = a[i][j1];
xi = a[i][j1 + 1];
a[i][j1] = a[i][k1];
a[i][j1 + 1] = a[i][k1 + 1];
a[i][k1] = xr;
a[i][k1 + 1] = xi;
}
}
}
} else {
m2 = m << 1;
for (i = 0; i <= n1 - 1; i++) {
for (k = 1; k <= m - 1; k++) {
for (j = 0; j <= k - 1; j++) {
j1 = (j << 1) + ip[k];
k1 = (k << 1) + ip[j];
xr = a[i][j1];
xi = a[i][j1 + 1];
a[i][j1] = a[i][k1];
a[i][j1 + 1] = a[i][k1 + 1];
a[i][k1] = xr;
a[i][k1 + 1] = xi;
j1 += m2;
k1 += m2;
xr = a[i][j1];
xi = a[i][j1 + 1];
a[i][j1] = a[i][k1];
a[i][j1 + 1] = a[i][k1 + 1];
a[i][k1] = xr;
a[i][k1 + 1] = xi;
}
}
}
}
}
void bitrv2row(int n, int n2, int *ip, double **a)
{
int i, j, j1, k, k1, l, m;
double xr, xi;
ip[0] = 0;
l = n;
m = 1;
while ((m << 1) < l) {
l >>= 1;
for (j = 0; j <= m - 1; j++) {
ip[m + j] = ip[j] + l;
}
m <<= 1;
}
if ((m << 1) > l) {
for (k = 1; k <= m - 1; k++) {
for (j = 0; j <= k - 1; j++) {
j1 = j + ip[k];
k1 = k + ip[j];
for (i = 0; i <= n2 - 2; i += 2) {
xr = a[j1][i];
xi = a[j1][i + 1];
a[j1][i] = a[k1][i];
a[j1][i + 1] = a[k1][i + 1];
a[k1][i] = xr;
a[k1][i + 1] = xi;
}
}
}
} else {
for (k = 1; k <= m - 1; k++) {
for (j = 0; j <= k - 1; j++) {
j1 = j + ip[k];
k1 = k + ip[j];
for (i = 0; i <= n2 - 2; i += 2) {
xr = a[j1][i];
xi = a[j1][i + 1];
a[j1][i] = a[k1][i];
a[j1][i + 1] = a[k1][i + 1];
a[k1][i] = xr;
a[k1][i + 1] = xi;
}
j1 += m;
k1 += m;
for (i = 0; i <= n2 - 2; i += 2) {
xr = a[j1][i];
xi = a[j1][i + 1];
a[j1][i] = a[k1][i];
a[j1][i + 1] = a[k1][i + 1];
a[k1][i] = xr;
a[k1][i + 1] = xi;
}
}
}
}
}
void cftbcol(int n1, int n, double **a, double *w)
{
int i, j, j1, j2, j3, k, k1, ks, l, m;
double wk1r, wk1i, wk2r, wk2i, wk3r, wk3i;
double x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i;
for (i = 0; i <= n1 - 1; i++) {
l = 2;
while ((l << 1) < n) {
m = l << 2;
for (j = 0; j <= l - 2; j += 2) {
j1 = j + l;
j2 = j1 + l;
j3 = j2 + l;
x0r = a[i][j] + a[i][j1];
x0i = a[i][j + 1] + a[i][j1 + 1];
x1r = a[i][j] - a[i][j1];
x1i = a[i][j + 1] - a[i][j1 + 1];
x2r = a[i][j2] + a[i][j3];
x2i = a[i][j2 + 1] + a[i][j3 + 1];
x3r = a[i][j2] - a[i][j3];
x3i = a[i][j2 + 1] - a[i][j3 + 1];
a[i][j] = x0r + x2r;
a[i][j + 1] = x0i + x2i;
a[i][j2] = x0r - x2r;
a[i][j2 + 1] = x0i - x2i;
a[i][j1] = x1r - x3i;
a[i][j1 + 1] = x1i + x3r;
a[i][j3] = x1r + x3i;
a[i][j3 + 1] = x1i - x3r;
}
if (m < n) {
wk1r = w[2];
for (j = m; j <= l + m - 2; j += 2) {
j1 = j + l;
j2 = j1 + l;
j3 = j2 + l;
x0r = a[i][j] + a[i][j1];
x0i = a[i][j + 1] + a[i][j1 + 1];
x1r = a[i][j] - a[i][j1];
x1i = a[i][j + 1] - a[i][j1 + 1];
x2r = a[i][j2] + a[i][j3];
x2i = a[i][j2 + 1] + a[i][j3 + 1];
x3r = a[i][j2] - a[i][j3];
x3i = a[i][j2 + 1] - a[i][j3 + 1];
a[i][j] = x0r + x2r;
a[i][j + 1] = x0i + x2i;
a[i][j2] = x2i - x0i;
a[i][j2 + 1] = x0r - x2r;
x0r = x1r - x3i;
x0i = x1i + x3r;
a[i][j1] = wk1r * (x0r - x0i);
a[i][j1 + 1] = wk1r * (x0r + x0i);
x0r = x3i + x1r;
x0i = x3r - x1i;
a[i][j3] = wk1r * (x0i - x0r);
a[i][j3 + 1] = wk1r * (x0i + x0r);
}
k1 = 1;
ks = -1;
for (k = (m << 1); k <= n - m; k += m) {
k1++;
ks = -ks;
wk1r = w[k1 << 1];
wk1i = w[(k1 << 1) + 1];
wk2r = ks * w[k1];
wk2i = w[k1 + ks];
wk3r = wk1r - 2 * wk2i * wk1i;
wk3i = 2 * wk2i * wk1r - wk1i;
for (j = k; j <= l + k - 2; j += 2) {
j1 = j + l;
j2 = j1 + l;
j3 = j2 + l;
x0r = a[i][j] + a[i][j1];
x0i = a[i][j + 1] + a[i][j1 + 1];
x1r = a[i][j] - a[i][j1];
x1i = a[i][j + 1] - a[i][j1 + 1];
x2r = a[i][j2] + a[i][j3];
x2i = a[i][j2 + 1] + a[i][j3 + 1];
x3r = a[i][j2] - a[i][j3];
x3i = a[i][j2 + 1] - a[i][j3 + 1];
a[i][j] = x0r + x2r;
a[i][j + 1] = x0i + x2i;
x0r -= x2r;
x0i -= x2i;
a[i][j2] = wk2r * x0r - wk2i * x0i;
a[i][j2 + 1] = wk2r * x0i + wk2i * x0r;
x0r = x1r - x3i;
x0i = x1i + x3r;
a[i][j1] = wk1r * x0r - wk1i * x0i;