2 //Title: 1-d mixed radix FFT.
4 //Copyright: Copyright (c) 1998
6 //Company: University of Wisconsin-Milwaukee.
8 // The number of DFT is factorized.
10 // Some short FFTs, such as length 2, 3, 4, 5, 8, 10, are used
11 // to improve the speed.
13 // Prime factors are processed using DFT. In the future, we can
15 // Note: there is no limit how large the prime factor can be,
16 // because for a set of data of an image, the length can be
17 // random, ie. an image can have size 263 x 300, where 263 is
18 // a large prime factor.
20 // A permute() function is used to make sure FFT can be calculated
23 // A triddle() function is used to perform the FFT.
25 // This program is for FFT of complex data, if the input is real,
26 // the program can be further improved. Because I want to use the
27 // same program to do IFFT, whose input is often complex, so I
28 // still use this program.
30 // To save the memory and improve the speed, float data are used
31 // instead of double, but I do have a double version transforms.fft.
33 // Factorize() is done in constructor, transforms.fft() is needed to be
34 // called to do FFT, this is good for use in fft2d, then
35 // factorize() is not needed for each row/column of data, since
36 // each row/column of a matrix has the same length.
41 // Maximum numbers of factors allowed.
42 //private int MaxFactorsNumber = 30;
43 public int MaxFactorsNumber;
45 // cos2to3PI = cos(2*pi/3), using for 3 point FFT.
46 // cos(2*PI/3) is not -1.5
47 public double cos2to3PI;
48 // sin2to3PI = sin(2*pi/3), using for 3 point FFT.
49 public double sin2to3PI;
51 // TwotoFivePI = 2*pi/5.
52 // c51, c52, c53, c54, c55 are used in fft5().
53 // c51 =(cos(TwotoFivePI)+cos(2*TwotoFivePI))/2-1.
55 // c52 =(cos(TwotoFivePI)-cos(2*TwotoFivePI))/2.
57 // c53 = -sin(TwotoFivePI).
59 // c54 =-(sin(TwotoFivePI)+sin(2*TwotoFivePI)).
61 // c55 =(sin(TwotoFivePI)-sin(2*TwotoFivePI)).
64 // OnetoSqrt2 = 1/sqrt(2), used in fft8().
65 public double OnetoSqrt2;
69 int N; // length of N point FFT.
70 int NumofFactors; // Number of factors of N.
71 int maxFactor; // Maximum factor of N.
73 int factors[]; // Factors of N processed in the current stage.
74 int sofar[]; // Finished factors before the current stage.
75 int remain[]; // Finished factors after the current stage.
77 double inputRe[], inputIm[]; // Input of FFT.
78 double temRe[], temIm[]; // Intermediate result of FFT.
79 double outputRe[], outputIm[]; // Output of FFT.
80 boolean factorsWerePrinted;
82 // Constructor: FFT of Complex data.
85 MaxFactorsNumber = 37;
87 sin2to3PI = 8.6602540378444E-01f;
89 c52 = 5.5901699437495E-01f;
90 c53 = -9.5105651629515E-01f;
91 c54 = -1.5388417685876E+00f;
92 c55 = 3.6327126400268E-01f;
93 OnetoSqrt2 = 7.0710678118655E-01f;
96 factorsWerePrinted = false;
97 outputRe = global new double[N];
98 outputIm = global new double[N];
103 // Allocate memory for intermediate result of FFT.
104 temRe = global new double[maxFactor]; //Check usage of this
105 temIm = global new double[maxFactor];
109 public void fft(double inputRe[], double inputIm[]) {
110 // First make sure inputRe & inputIm are of the same length.
111 if (inputRe.length != N || inputIm.length != N) {
112 System.printString("Error: the length of real part & imaginary part " +
113 "of the input to 1-d FFT are different");
116 this.inputRe = inputRe;
117 this.inputIm = inputIm;
120 //System.printString("ready to twiddle");
122 for (int factorIndex = 0; factorIndex < NumofFactors; factorIndex++)
123 twiddle(factorIndex);
124 //System.printString("ready to copy");
126 // Copy the output[] data to input[], so the output can be
127 // returned in the input array.
128 for (int i = 0; i < N; i++) {
129 inputRe[i] = outputRe[i];
130 inputIm[i] = outputIm[i];
136 public void printFactors() {
137 if (factorsWerePrinted) return;
138 factorsWerePrinted = true;
139 for (int i = 0; i < factors.length; i++)
140 System.printString("factors[i] = " + factors[i]);
143 public void factorize() {
144 int radices[] = new int[6];
151 int temFactors[] = new int[MaxFactorsNumber];
153 // 1 - point FFT, no need to factorize N.
159 // N - point FFT, N is needed to be factorized.
161 int index = 0; // index of temFactors.
162 int i = radices.length - 1;
164 while ((n > 1) && (i >= 0)) {
165 if ((n % radices[i]) == 0) {
167 temFactors[index++] = radices[i];
172 // Substitute 2x8 with 4x4.
173 // index>0, in the case only one prime factor, such as N=263.
174 if ((index > 0) && (temFactors[index - 1] == 2)) {
176 for (i = index - 2; (i >= 0) && (test == 0); i--) {
177 if (temFactors[i] == 8) {
178 temFactors[index - 1] = temFactors[i] = 4;
179 // break out of for loop, because only one '2' will exist in
180 // temFactors, so only one substitutation is needed.
188 for (int k = 2; k < Math.sqrt(n) + 1; k++)
189 while ((n % k) == 0) {
191 temFactors[index++] = k;
194 temFactors[index++] = n;
197 NumofFactors = index;
198 //if(temFactors[NumofFactors-1] > 10)
203 // Inverse temFactors and store factors into factors[].
204 factors = new int[NumofFactors];
205 for (i = 0; i < NumofFactors; i++) {
206 factors[i] = temFactors[NumofFactors - i - 1];
209 // Calculate sofar[], remain[].
210 // sofar[] : finished factors before the current stage.
211 // factors[]: factors of N processed in the current stage.
212 // remain[] : finished factors after the current stage.
213 sofar = new int[NumofFactors];
214 remain = new int[NumofFactors];
216 remain[0] = N / factors[0];
218 for (i = 1; i < NumofFactors; i++) {
219 sofar[i] = sofar[i - 1] * factors[i - 1];
220 remain[i] = remain[i - 1] / factors[i];
222 } // End of function factorize().
224 private void permute() {
225 int count[] = new int[MaxFactorsNumber];
229 for (int i = 0; i < N - 1; i++) {
230 outputRe[i] = inputRe[k];
231 outputIm[i] = inputIm[k];
234 count[0] = count[0] + 1;
235 while (count[j] >= factors[j]) {
237 k = k - (j == 0?N:remain[j - 1]) + remain[j + 1];
239 count[j] = count[j] + 1;
242 outputRe[N - 1] = inputRe[N - 1];
243 outputIm[N - 1] = inputIm[N - 1];
244 } // End of function permute().
247 private void twiddle(int factorIndex) {
249 int sofarRadix = sofar[factorIndex];
250 int radix = factors[factorIndex];
251 int remainRadix = remain[factorIndex];
253 double tem; // Temporary variable to do data exchange.
255 double W = 2 * (double) Math.PI / (sofarRadix * radix);
256 double cosW = (double) Math.cos(W);
257 double sinW = -(double) Math.sin(W);
259 double twiddleRe[] = new double[radix];
260 double twiddleIm[] = new double[radix];
261 double twRe = 1.0f, twIm = 0f;
263 //Initialize twiddle addBk.address variables.
264 int dataOffset = 0, groupOffset = 0, address = 0;
266 for (int dataNo = 0; dataNo < sofarRadix; dataNo++) {
267 //System.printString("datano="+dataNo);
268 if (sofarRadix > 1) {
273 for (int i = 2; i < radix; i++) {
276 twiddleRe[i] = twRe * twiddleRe[i - 1] - twIm * twiddleIm[i - 1];
277 twiddleIm[i] = twIm * twiddleRe[i - 1] + twRe * twiddleIm[i - 1];
279 tem = cosW * twRe - sinW * twIm;
280 twIm = sinW * twRe + cosW * twIm;
283 for (int groupNo = 0; groupNo < remainRadix; groupNo++) {
284 //System.printString("groupNo="+groupNo);
285 if ((sofarRadix > 1) && (dataNo > 0)) {
286 temRe[0] = outputRe[address];
287 temIm[0] = outputIm[address];
290 address = address + sofarRadix;
291 temRe[blockIndex] = twiddleRe[blockIndex] * outputRe[address] -
292 twiddleIm[blockIndex] * outputIm[address];
293 temIm[blockIndex] = twiddleRe[blockIndex] * outputIm[address] +
294 twiddleIm[blockIndex] * outputRe[address];
296 } while (blockIndex < radix);
298 for (int i = 0; i < radix; i++) {
299 //System.printString("temRe.length="+temRe.length);
300 //System.printString("i = "+i);
301 temRe[i] = outputRe[address];
302 temIm[i] = outputIm[address];
303 address += sofarRadix;
305 //System.printString("radix="+radix);
308 tem = temRe[0] + temRe[1];
309 temRe[1] = temRe[0] - temRe[1];
311 tem = temIm[0] + temIm[1];
312 temIm[1] = temIm[0] - temIm[1];
316 double t1Re = temRe[1] + temRe[2];
317 double t1Im = temIm[1] + temIm[2];
318 temRe[0] = temRe[0] + t1Re;
319 temIm[0] = temIm[0] + t1Im;
321 double m1Re = cos2to3PI * t1Re;
322 double m1Im = cos2to3PI * t1Im;
323 double m2Re = sin2to3PI * (temIm[1] - temIm[2]);
324 double m2Im = sin2to3PI * (temRe[2] - temRe[1]);
325 double s1Re = temRe[0] + m1Re;
326 double s1Im = temIm[0] + m1Im;
328 temRe[1] = s1Re + m2Re;
329 temIm[1] = s1Im + m2Im;
330 temRe[2] = s1Re - m2Re;
331 temIm[2] = s1Im - m2Im;
349 address = groupOffset;
350 for (int i = 0; i < radix; i++) {
351 outputRe[address] = temRe[i];
352 outputIm[address] = temIm[i];
353 address += sofarRadix;
355 groupOffset += sofarRadix * radix;
356 address = groupOffset;
358 groupOffset = ++dataOffset;
359 address = groupOffset;
361 } // End of function twiddle().
364 // The two arguments dataRe[], dataIm[] are mainly for using in fft8();
365 private void fft4(double dataRe[], double dataIm[]) {
366 double t1Re,t1Im, t2Re,t2Im;
367 double m2Re,m2Im, m3Re,m3Im;
369 t1Re = dataRe[0] + dataRe[2];
370 t1Im = dataIm[0] + dataIm[2];
371 t2Re = dataRe[1] + dataRe[3];
372 t2Im = dataIm[1] + dataIm[3];
374 m2Re = dataRe[0] - dataRe[2];
375 m2Im = dataIm[0] - dataIm[2];
376 m3Re = dataIm[1] - dataIm[3];
377 m3Im = dataRe[3] - dataRe[1];
379 dataRe[0] = t1Re + t2Re;
380 dataIm[0] = t1Im + t2Im;
381 dataRe[2] = t1Re - t2Re;
382 dataIm[2] = t1Im - t2Im;
383 dataRe[1] = m2Re + m3Re;
384 dataIm[1] = m2Im + m3Im;
385 dataRe[3] = m2Re - m3Re;
386 dataIm[3] = m2Im - m3Im;
387 } // End of function fft4().
390 // The two arguments dataRe[], dataIm[] are mainly for using in fft10();
391 private void fft5(double dataRe[], double dataIm[]) {
392 double t1Re,t1Im, t2Re,t2Im, t3Re,t3Im, t4Re,t4Im, t5Re,t5Im;
393 double m1Re,m1Im, m2Re,m2Im, m3Re,m3Im, m4Re,m4Im, m5Re,m5Im;
394 double s1Re,s1Im, s2Re,s2Im, s3Re,s3Im, s4Re,s4Im, s5Re,s5Im;
396 t1Re = dataRe[1] + dataRe[4];
397 t1Im = dataIm[1] + dataIm[4];
398 t2Re = dataRe[2] + dataRe[3];
399 t2Im = dataIm[2] + dataIm[3];
400 t3Re = dataRe[1] - dataRe[4];
401 t3Im = dataIm[1] - dataIm[4];
402 t4Re = dataRe[3] - dataRe[2];
403 t4Im = dataIm[3] - dataIm[2];
407 dataRe[0] = dataRe[0] + t5Re;
408 dataIm[0] = dataIm[0] + t5Im;
412 m2Re = c52 * (t1Re - t2Re);
413 m2Im = c52 * (t1Im - t2Im);
414 m3Re = -c53 * (t3Im + t4Im);
415 m3Im = c53 * (t3Re + t4Re);
425 s1Re = dataRe[0] + m1Re;
426 s1Im = dataIm[0] + m1Im;
432 dataRe[1] = s2Re + s3Re;
433 dataIm[1] = s2Im + s3Im;
434 dataRe[2] = s4Re + s5Re;
435 dataIm[2] = s4Im + s5Im;
436 dataRe[3] = s4Re - s5Re;
437 dataIm[3] = s4Im - s5Im;
438 dataRe[4] = s2Re - s3Re;
439 dataIm[4] = s2Im - s3Im;
440 } // End of function fft5().
444 private void fft8() {
445 double data1Re[] = new double[4];
446 double data1Im[] = new double[4];
447 double data2Re[] = new double[4];
448 double data2Im[] = new double[4];
451 // To improve the speed, use direct assaignment instead for loop here.
452 data1Re[0] = temRe[0];
453 data2Re[0] = temRe[1];
454 data1Re[1] = temRe[2];
455 data2Re[1] = temRe[3];
456 data1Re[2] = temRe[4];
457 data2Re[2] = temRe[5];
458 data1Re[3] = temRe[6];
459 data2Re[3] = temRe[7];
461 data1Im[0] = temIm[0];
462 data2Im[0] = temIm[1];
463 data1Im[1] = temIm[2];
464 data2Im[1] = temIm[3];
465 data1Im[2] = temIm[4];
466 data2Im[2] = temIm[5];
467 data1Im[3] = temIm[6];
468 data2Im[3] = temIm[7];
470 fft4(data1Re, data1Im);
471 fft4(data2Re, data2Im);
473 tem = OnetoSqrt2 * (data2Re[1] + data2Im[1]);
474 data2Im[1] = OnetoSqrt2 * (data2Im[1] - data2Re[1]);
477 data2Im[2] = -data2Re[2];
479 tem = OnetoSqrt2 * (data2Im[3] - data2Re[3]);
480 data2Im[3] = -OnetoSqrt2 * (data2Re[3] + data2Im[3]);
483 temRe[0] = data1Re[0] + data2Re[0];
484 temRe[4] = data1Re[0] - data2Re[0];
485 temRe[1] = data1Re[1] + data2Re[1];
486 temRe[5] = data1Re[1] - data2Re[1];
487 temRe[2] = data1Re[2] + data2Re[2];
488 temRe[6] = data1Re[2] - data2Re[2];
489 temRe[3] = data1Re[3] + data2Re[3];
490 temRe[7] = data1Re[3] - data2Re[3];
492 temIm[0] = data1Im[0] + data2Im[0];
493 temIm[4] = data1Im[0] - data2Im[0];
494 temIm[1] = data1Im[1] + data2Im[1];
495 temIm[5] = data1Im[1] - data2Im[1];
496 temIm[2] = data1Im[2] + data2Im[2];
497 temIm[6] = data1Im[2] - data2Im[2];
498 temIm[3] = data1Im[3] + data2Im[3];
499 temIm[7] = data1Im[3] - data2Im[3];
500 } // End of function fft8().
504 private void fft10() {
505 double data1Re[] = new double[5];
506 double data1Im[] = new double[5];
507 double data2Re[] = new double[5];
508 double data2Im[] = new double[5];
510 // To improve the speed, use direct assaignment instead for loop here.
511 data1Re[0] = temRe[0];
512 data2Re[0] = temRe[5];
513 data1Re[1] = temRe[2];
514 data2Re[1] = temRe[7];
515 data1Re[2] = temRe[4];
516 data2Re[2] = temRe[9];
517 data1Re[3] = temRe[6];
518 data2Re[3] = temRe[1];
519 data1Re[4] = temRe[8];
520 data2Re[4] = temRe[3];
522 data1Im[0] = temIm[0];
523 data2Im[0] = temIm[5];
524 data1Im[1] = temIm[2];
525 data2Im[1] = temIm[7];
526 data1Im[2] = temIm[4];
527 data2Im[2] = temIm[9];
528 data1Im[3] = temIm[6];
529 data2Im[3] = temIm[1];
530 data1Im[4] = temIm[8];
531 data2Im[4] = temIm[3];
533 fft5(data1Re, data1Im);
534 fft5(data2Re, data2Im);
536 temRe[0] = data1Re[0] + data2Re[0];
537 temRe[5] = data1Re[0] - data2Re[0];
538 temRe[6] = data1Re[1] + data2Re[1];
539 temRe[1] = data1Re[1] - data2Re[1];
540 temRe[2] = data1Re[2] + data2Re[2];
541 temRe[7] = data1Re[2] - data2Re[2];
542 temRe[8] = data1Re[3] + data2Re[3];
543 temRe[3] = data1Re[3] - data2Re[3];
544 temRe[4] = data1Re[4] + data2Re[4];
545 temRe[9] = data1Re[4] - data2Re[4];
547 temIm[0] = data1Im[0] + data2Im[0];
548 temIm[5] = data1Im[0] - data2Im[0];
549 temIm[6] = data1Im[1] + data2Im[1];
550 temIm[1] = data1Im[1] - data2Im[1];
551 temIm[2] = data1Im[2] + data2Im[2];
552 temIm[7] = data1Im[2] - data2Im[2];
553 temIm[8] = data1Im[3] + data2Im[3];
554 temIm[3] = data1Im[3] - data2Im[3];
555 temIm[4] = data1Im[4] + data2Im[4];
556 temIm[9] = data1Im[4] - data2Im[4];
557 } // End of function fft10().
561 public double sqrt(double d) {
567 private void fftPrime(int radix) {
569 double W = 2 * (double) Math.PI / radix;
570 double cosW = (double) Math.cos(W);
571 double sinW = -(double) Math.sin(W);
572 double WRe[] = new double[radix];
573 double WIm[] = new double[radix];
580 for (int i = 2; i < radix; i++) {
581 WRe[i] = cosW * WRe[i - 1] - sinW * WIm[i - 1];
582 WIm[i] = sinW * WRe[i - 1] + cosW * WIm[i - 1];
585 // FFT of prime length data, using DFT, can be improved in the future.
586 double rere, reim, imre, imim;
588 int max = (radix + 1) / 2;
590 double tem1Re[] = new double[max];
591 double tem1Im[] = new double[max];
592 double tem2Re[] = new double[max];
593 double tem2Im[] = new double[max];
595 for (j = 1; j < max; j++) {
596 tem1Re[j] = temRe[j] + temRe[radix - j];
597 tem1Im[j] = temIm[j] - temIm[radix - j];
598 tem2Re[j] = temRe[j] - temRe[radix - j];
599 tem2Im[j] = temIm[j] + temIm[radix - j];
602 for (j = 1; j < max; j++) {
605 temRe[radix - j] = temRe[0];
606 temIm[radix - j] = temIm[0];
608 for (int i = 1; i < max; i++) {
609 rere = WRe[k] * tem1Re[i];
610 imim = WIm[k] * tem1Im[i];
611 reim = WRe[k] * tem2Im[i];
612 imre = WIm[k] * tem2Re[i];
614 temRe[radix - j] += rere + imim;
615 temIm[radix - j] += reim - imre;
616 temRe[j] += rere - imim;
617 temIm[j] += reim + imre;
624 for (j = 1; j < max; j++) {
625 temRe[0] = temRe[0] + tem1Re[j];
626 temIm[0] = temIm[0] + tem2Im[j];
628 } // End of function fftPrime().
631 } // End of class FFT2d