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]; +} + +