Koichi Kurahashi
FFT power Spectrum on AQM1248 LCD. - FRDM-KL46Z - inner LCD - inner MAG3110 Magnetometer - AQM1248 micro graphical LCD - Dr.Ooura's very fast FFT library thanks.
Dependencies: MAG3110 SLCD aqm1248a_lcd mbed
FRDM-KL46Zに内蔵されているMAG3110で磁力を測定し、FFTでパワースペクトルを求めてグラフ表示しています。と言っても自分ではほとんどコードは書いておらず、すべては
- 内蔵LCD
- 内蔵MAG3110
- AQM1248
- 大浦先生のFFTライブラリ
以上のライブラリのおかげです。ありがとうございます。
プログラムとしては：
- Intervalを使ってバッファにMAG3110からのデータを詰め込む
- メインループではバッファを監視し、バッファが一杯になったらFFTかけてスペクトル表示
を繰り返しているだけです。せめてRTOSを使ってFFT〜スペクトル表示も別タスクにしないと…。
関連ブログ：http://jiwashin.blogspot.com/2015/05/fft.html
なお、AQM1248ライブラリのソースを拝見するとサポートしているのは「LPC1768とKL05」という感じです。KL46では動作確認しましたが、その他のプラットフォーム上で使用する場合には、ピンアサインなどを十分確認してください。その点に気をつければとても使い勝手の良いライブラリです。開発者の方に改めてお礼申し上げます。
なお、AQM1248とKL46との接続は以下の通りです：
| AQM1248 | KL46 |
| Vcc | 3.3v |
| CS | D10 |
| RESET | D9 |
| RS | D8 |
| SCLK | D13 |
| SDI | D11 |
Diff: fft4g.cpp
- Revision:
- 0:47be4d9de4b9
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/fft4g.cpp Fri May 01 19:26:11 2015 +0000
@@ -0,0 +1,1346 @@
+/*
+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];
+}
+
+