1 //Title: 1-d mixed radix FFT.
3 //Copyright: Copyright (c) 1998
5 //Company: University of Wisconsin-Milwaukee.
7 // The number of DFT is factorized.
9 // Some short FFTs, such as length 2, 3, 4, 5, 8, 10, are used
10 // to improve the speed.
12 // Prime factors are processed using DFT. In the future, we can
14 // Note: there is no limit how large the prime factor can be,
15 // because for a set of data of an image, the length can be
16 // random, ie. an image can have size 263 x 300, where 263 is
17 // a large prime factor.
19 // A permute() function is used to make sure FFT can be calculated
22 // A triddle() function is used to perform the FFT.
24 // This program is for FFT of complex data, if the input is real,
25 // the program can be further improved. Because I want to use the
26 // same program to do IFFT, whose input is often complex, so I
27 // still use this program.
29 // To save the memory and improve the speed, double data are used
30 // instead of double, but I do have a double version transforms.fft.
32 // Factorize() is done in constructor, transforms.fft() is needed to be
33 // called to do FFT, this is good for use in fft2d, then
34 // factorize() is not needed for each row/column of data, since
35 // each row/column of a matrix has the same length.
40 // Maximum numbers of factors allowed.
41 //private static final int MaxFactorsNumber = 30;
42 private static final int MaxFactorsNumber = 37;
44 // cos2to3PI = cos(2*pi/3), using for 3 point FFT.
45 // cos(2*PI/3) is not -1.5
46 private static final double cos2to3PI = -1.5000f;
47 // sin2to3PI = sin(2*pi/3), using for 3 point FFT.
48 private static final double sin2to3PI = 8.6602540378444E-01f;
50 // TwotoFivePI = 2*pi/5.
51 // c51, c52, c53, c54, c55 are used in fft5().
52 // c51 =(cos(TwotoFivePI)+cos(2*TwotoFivePI))/2-1.
53 private static final double c51 = -1.25f;
54 // c52 =(cos(TwotoFivePI)-cos(2*TwotoFivePI))/2.
55 private static final double c52 = 5.5901699437495E-01f;
56 // c53 = -sin(TwotoFivePI).
57 private static final double c53 = -9.5105651629515E-01f;
58 // c54 =-(sin(TwotoFivePI)+sin(2*TwotoFivePI)).
59 private static final double c54 = -1.5388417685876E+00f;
60 // c55 =(sin(TwotoFivePI)-sin(2*TwotoFivePI)).
61 private static final double c55 = 3.6327126400268E-01f;
63 // OnetoSqrt2 = 1/sqrt(2), used in fft8().
64 private static final double OnetoSqrt2 = 7.0710678118655E-01f;
66 private static int lastRadix = 0;
68 int N; // length of N point FFT.
69 int NumofFactors; // Number of factors of N.
70 static final int maxFactor = 20; // Maximum factor of N.
72 int factors[]; // Factors of N processed in the current stage.
73 int sofar[]; // Finished factors before the current stage.
74 int remain[]; // Finished factors after the current stage.
76 double inputRe[], inputIm[]; // Input of FFT.
77 double temRe[], temIm[]; // Intermediate result of FFT.
78 double outputRe[], outputIm[]; // Output of FFT.
79 //static boolean factorsWerePrinted = false;
80 boolean factorsWerePrinted = false;
82 // Constructor: FFT of Complex data.
85 outputRe = new double[N];
86 outputIm = new double[N];
91 // Allocate memory for intermediate result of FFT.
92 temRe = new double[maxFactor];
93 temIm = new double[maxFactor];
96 public void fft(double inputRe[], double inputIm[]) {
97 // First make sure inputRe & inputIm are of the same length.
98 if (inputRe.length != N || inputIm.length != N) {
99 System.out.println("Error: the length of real part & imaginary part " +
100 "of the input to 1-d FFT are different");
103 this.inputRe = inputRe;
104 this.inputIm = inputIm;
107 //System.out.println("ready to twiddle");
109 for (int factorIndex = 0; factorIndex < NumofFactors; factorIndex++)
110 twiddle(factorIndex);
111 //System.out.println("ready to copy");
113 // Copy the output[] data to input[], so the output can be
114 // returned in the input array.
115 for (int i = 0; i < N; i++) {
116 inputRe[i] = outputRe[i];
117 inputIm[i] = outputIm[i];
123 public void printFactors() {
124 if (factorsWerePrinted) return;
125 factorsWerePrinted = true;
126 //System.out.println("factors.length = " + factors.length + "\n");
127 for (int i = 0; i < factors.length; i++)
128 System.out.println("factors[i] = " + factors[i]);
131 private void factorize() {
132 int radices[] = {2, 3, 4, 5, 8, 10};
133 int temFactors[] = new int[MaxFactorsNumber];
135 // 1 - point FFT, no need to factorize N.
141 // N - point FFT, N is needed to be factorized.
143 int index = 0; // index of temFactors.
144 int i = radices.length - 1;
146 while ((n > 1) && (i >= 0)) {
147 if ((n % radices[i]) == 0) {
149 temFactors[index++] = radices[i];
154 // Substitute 2x8 with 4x4.
155 // index>0, in the case only one prime factor, such as N=263.
156 if ((index > 0) && (temFactors[index - 1] == 2))
157 for (i = index - 2; i >= 0; i--)
158 if (temFactors[i] == 8) {
159 temFactors[index - 1] = temFactors[i] = 4;
160 // break out of for loop, because only one '2' will exist in
161 // temFactors, so only one substitutation is needed.
166 for (int k = 2; k < Math.sqrt(n) + 1; k++)
167 while ((n % k) == 0) {
169 temFactors[index++] = k;
172 temFactors[index++] = n;
175 NumofFactors = index;
177 if(temFactors[NumofFactors-1] > 10)
183 // Inverse temFactors and store factors into factors[].
184 factors = new int[NumofFactors];
185 for (i = 0; i < NumofFactors; i++) {
186 factors[i] = temFactors[NumofFactors - i - 1];
189 // Calculate sofar[], remain[].
190 // sofar[] : finished factors before the current stage.
191 // factors[]: factors of N processed in the current stage.
192 // remain[] : finished factors after the current stage.
193 sofar = new int[NumofFactors];
194 remain = new int[NumofFactors];
196 remain[0] = N / factors[0];
198 for (i = 1; i < NumofFactors; i++) {
199 sofar[i] = sofar[i - 1] * factors[i - 1];
200 remain[i] = remain[i - 1] / factors[i];
202 } // End of function factorize().
204 private void permute() {
205 int count[] = new int[MaxFactorsNumber];
209 for (int i = 0; i < N - 1; i++) {
210 outputRe[i] = inputRe[k];
211 outputIm[i] = inputIm[k];
214 count[0] = count[0] + 1;
215 while (count[j] >= factors[j]) {
217 k = k - (j == 0?N:remain[j - 1]) + remain[j + 1];
219 count[j] = count[j] + 1;
222 outputRe[N - 1] = inputRe[N - 1];
223 outputIm[N - 1] = inputIm[N - 1];
224 } // End of function permute().
226 private void twiddle(int factorIndex) {
228 int sofarRadix = sofar[factorIndex];
229 int radix = factors[factorIndex];
230 int remainRadix = remain[factorIndex];
232 double tem; // Temporary variable to do data exchange.
234 double W = 2 * (double) Math.PI / (sofarRadix * radix);
235 double cosW = (double) Math.cos(W);
236 double sinW = -(double) Math.sin(W);
238 double twiddleRe[] = new double[radix];
239 double twiddleIm[] = new double[radix];
240 double twRe = 1.0f, twIm = 0f;
242 //Initialize twiddle addBk.address variables.
243 int dataOffset = 0, groupOffset = 0, address = 0;
245 for (int dataNo = 0; dataNo < sofarRadix; dataNo++) {
246 //System.out.println("datano="+dataNo);
247 if (sofarRadix > 1) {
252 for (int i = 2; i < radix; i++) {
255 twiddleRe[i] = twRe * twiddleRe[i - 1] - twIm * twiddleIm[i - 1];
256 twiddleIm[i] = twIm * twiddleRe[i - 1] + twRe * twiddleIm[i - 1];
258 tem = cosW * twRe - sinW * twIm;
259 twIm = sinW * twRe + cosW * twIm;
262 for (int groupNo = 0; groupNo < remainRadix; groupNo++) {
263 //System.out.println("groupNo="+groupNo);
264 if ((sofarRadix > 1) && (dataNo > 0)) {
265 temRe[0] = outputRe[address];
266 temIm[0] = outputIm[address];
269 address = address + sofarRadix;
270 temRe[blockIndex] = twiddleRe[blockIndex] * outputRe[address] -
271 twiddleIm[blockIndex] * outputIm[address];
272 temIm[blockIndex] = twiddleRe[blockIndex] * outputIm[address] +
273 twiddleIm[blockIndex] * outputRe[address];
275 } while (blockIndex < radix);
277 for (int i = 0; i < radix; i++) {
278 //System.out.println("temRe.length="+temRe.length);
279 //System.out.println("i = "+i);
280 temRe[i] = outputRe[address];
281 temIm[i] = outputIm[address];
282 address += sofarRadix;
284 //System.out.println("radix="+radix);
287 tem = temRe[0] + temRe[1];
288 temRe[1] = temRe[0] - temRe[1];
290 tem = temIm[0] + temIm[1];
291 temIm[1] = temIm[0] - temIm[1];
295 double t1Re = temRe[1] + temRe[2];
296 double t1Im = temIm[1] + temIm[2];
297 temRe[0] = temRe[0] + t1Re;
298 temIm[0] = temIm[0] + t1Im;
300 double m1Re = cos2to3PI * t1Re;
301 double m1Im = cos2to3PI * t1Im;
302 double m2Re = sin2to3PI * (temIm[1] - temIm[2]);
303 double m2Im = sin2to3PI * (temRe[2] - temRe[1]);
304 double s1Re = temRe[0] + m1Re;
305 double s1Im = temIm[0] + m1Im;
307 temRe[1] = s1Re + m2Re;
308 temIm[1] = s1Im + m2Im;
309 temRe[2] = s1Re - m2Re;
310 temIm[2] = s1Im - m2Im;
328 address = groupOffset;
329 for (int i = 0; i < radix; i++) {
330 outputRe[address] = temRe[i];
331 outputIm[address] = temIm[i];
332 address += sofarRadix;
334 groupOffset += sofarRadix * radix;
335 address = groupOffset;
337 groupOffset = ++dataOffset;
338 address = groupOffset;
340 } // End of function twiddle().
342 // The two arguments dataRe[], dataIm[] are mainly for using in fft8();
343 private void fft4(double dataRe[], double dataIm[]) {
344 double t1Re,t1Im, t2Re,t2Im;
345 double m2Re,m2Im, m3Re,m3Im;
347 t1Re = dataRe[0] + dataRe[2];
348 t1Im = dataIm[0] + dataIm[2];
349 t2Re = dataRe[1] + dataRe[3];
350 t2Im = dataIm[1] + dataIm[3];
352 m2Re = dataRe[0] - dataRe[2];
353 m2Im = dataIm[0] - dataIm[2];
354 m3Re = dataIm[1] - dataIm[3];
355 m3Im = dataRe[3] - dataRe[1];
357 dataRe[0] = t1Re + t2Re;
358 dataIm[0] = t1Im + t2Im;
359 dataRe[2] = t1Re - t2Re;
360 dataIm[2] = t1Im - t2Im;
361 dataRe[1] = m2Re + m3Re;
362 dataIm[1] = m2Im + m3Im;
363 dataRe[3] = m2Re - m3Re;
364 dataIm[3] = m2Im - m3Im;
365 } // End of function fft4().
367 // The two arguments dataRe[], dataIm[] are mainly for using in fft10();
368 private void fft5(double dataRe[], double dataIm[]) {
369 double t1Re,t1Im, t2Re,t2Im, t3Re,t3Im, t4Re,t4Im, t5Re,t5Im;
370 double m1Re,m1Im, m2Re,m2Im, m3Re,m3Im, m4Re,m4Im, m5Re,m5Im;
371 double s1Re,s1Im, s2Re,s2Im, s3Re,s3Im, s4Re,s4Im, s5Re,s5Im;
373 t1Re = dataRe[1] + dataRe[4];
374 t1Im = dataIm[1] + dataIm[4];
375 t2Re = dataRe[2] + dataRe[3];
376 t2Im = dataIm[2] + dataIm[3];
377 t3Re = dataRe[1] - dataRe[4];
378 t3Im = dataIm[1] - dataIm[4];
379 t4Re = dataRe[3] - dataRe[2];
380 t4Im = dataIm[3] - dataIm[2];
384 dataRe[0] = dataRe[0] + t5Re;
385 dataIm[0] = dataIm[0] + t5Im;
389 m2Re = c52 * (t1Re - t2Re);
390 m2Im = c52 * (t1Im - t2Im);
391 m3Re = -c53 * (t3Im + t4Im);
392 m3Im = c53 * (t3Re + t4Re);
402 s1Re = dataRe[0] + m1Re;
403 s1Im = dataIm[0] + m1Im;
409 dataRe[1] = s2Re + s3Re;
410 dataIm[1] = s2Im + s3Im;
411 dataRe[2] = s4Re + s5Re;
412 dataIm[2] = s4Im + s5Im;
413 dataRe[3] = s4Re - s5Re;
414 dataIm[3] = s4Im - s5Im;
415 dataRe[4] = s2Re - s3Re;
416 dataIm[4] = s2Im - s3Im;
417 } // End of function fft5().
419 private void fft8() {
420 double data1Re[] = new double[4];
421 double data1Im[] = new double[4];
422 double data2Re[] = new double[4];
423 double data2Im[] = new double[4];
426 // To improve the speed, use direct assaignment instead for loop here.
427 data1Re[0] = temRe[0];
428 data2Re[0] = temRe[1];
429 data1Re[1] = temRe[2];
430 data2Re[1] = temRe[3];
431 data1Re[2] = temRe[4];
432 data2Re[2] = temRe[5];
433 data1Re[3] = temRe[6];
434 data2Re[3] = temRe[7];
436 data1Im[0] = temIm[0];
437 data2Im[0] = temIm[1];
438 data1Im[1] = temIm[2];
439 data2Im[1] = temIm[3];
440 data1Im[2] = temIm[4];
441 data2Im[2] = temIm[5];
442 data1Im[3] = temIm[6];
443 data2Im[3] = temIm[7];
445 fft4(data1Re, data1Im);
446 fft4(data2Re, data2Im);
448 tem = OnetoSqrt2 * (data2Re[1] + data2Im[1]);
449 data2Im[1] = OnetoSqrt2 * (data2Im[1] - data2Re[1]);
452 data2Im[2] = -data2Re[2];
454 tem = OnetoSqrt2 * (data2Im[3] - data2Re[3]);
455 data2Im[3] = -OnetoSqrt2 * (data2Re[3] + data2Im[3]);
458 temRe[0] = data1Re[0] + data2Re[0];
459 temRe[4] = data1Re[0] - data2Re[0];
460 temRe[1] = data1Re[1] + data2Re[1];
461 temRe[5] = data1Re[1] - data2Re[1];
462 temRe[2] = data1Re[2] + data2Re[2];
463 temRe[6] = data1Re[2] - data2Re[2];
464 temRe[3] = data1Re[3] + data2Re[3];
465 temRe[7] = data1Re[3] - data2Re[3];
467 temIm[0] = data1Im[0] + data2Im[0];
468 temIm[4] = data1Im[0] - data2Im[0];
469 temIm[1] = data1Im[1] + data2Im[1];
470 temIm[5] = data1Im[1] - data2Im[1];
471 temIm[2] = data1Im[2] + data2Im[2];
472 temIm[6] = data1Im[2] - data2Im[2];
473 temIm[3] = data1Im[3] + data2Im[3];
474 temIm[7] = data1Im[3] - data2Im[3];
475 } // End of function fft8().
477 private void fft10() {
478 double data1Re[] = new double[5];
479 double data1Im[] = new double[5];
480 double data2Re[] = new double[5];
481 double data2Im[] = new double[5];
483 // To improve the speed, use direct assaignment instead for loop here.
484 data1Re[0] = temRe[0];
485 data2Re[0] = temRe[5];
486 data1Re[1] = temRe[2];
487 data2Re[1] = temRe[7];
488 data1Re[2] = temRe[4];
489 data2Re[2] = temRe[9];
490 data1Re[3] = temRe[6];
491 data2Re[3] = temRe[1];
492 data1Re[4] = temRe[8];
493 data2Re[4] = temRe[3];
494 data1Im[0] = temIm[0];
495 data2Im[0] = temIm[5];
496 data1Im[1] = temIm[2];
497 data2Im[1] = temIm[7];
498 data1Im[2] = temIm[4];
499 data2Im[2] = temIm[9];
500 data1Im[3] = temIm[6];
501 data2Im[3] = temIm[1];
502 data1Im[4] = temIm[8];
503 data2Im[4] = temIm[3];
505 fft5(data1Re, data1Im);
506 fft5(data2Re, data2Im);
508 temRe[0] = data1Re[0] + data2Re[0];
509 temRe[5] = data1Re[0] - data2Re[0];
510 temRe[6] = data1Re[1] + data2Re[1];
511 temRe[1] = data1Re[1] - data2Re[1];
512 temRe[2] = data1Re[2] + data2Re[2];
513 temRe[7] = data1Re[2] - data2Re[2];
514 temRe[8] = data1Re[3] + data2Re[3];
515 temRe[3] = data1Re[3] - data2Re[3];
516 temRe[4] = data1Re[4] + data2Re[4];
517 temRe[9] = data1Re[4] - data2Re[4];
519 temIm[0] = data1Im[0] + data2Im[0];
520 temIm[5] = data1Im[0] - data2Im[0];
521 temIm[6] = data1Im[1] + data2Im[1];
522 temIm[1] = data1Im[1] - data2Im[1];
523 temIm[2] = data1Im[2] + data2Im[2];
524 temIm[7] = data1Im[2] - data2Im[2];
525 temIm[8] = data1Im[3] + data2Im[3];
526 temIm[3] = data1Im[3] - data2Im[3];
527 temIm[4] = data1Im[4] + data2Im[4];
528 temIm[9] = data1Im[4] - data2Im[4];
529 } // End of function fft10().
531 public double sqrt(double d) {
535 private void fftPrime(int radix) {
537 double W = 2 * (double) Math.PI / radix;
538 double cosW = (double) Math.cos(W);
539 double sinW = -(double) Math.sin(W);
540 double WRe[] = new double[radix];
541 double WIm[] = new double[radix];
548 for (int i = 2; i < radix; i++) {
549 WRe[i] = cosW * WRe[i - 1] - sinW * WIm[i - 1];
550 WIm[i] = sinW * WRe[i - 1] + cosW * WIm[i - 1];
553 // FFT of prime length data, using DFT, can be improved in the future.
554 double rere, reim, imre, imim;
556 int max = (radix + 1) / 2;
558 double tem1Re[] = new double[max];
559 double tem1Im[] = new double[max];
560 double tem2Re[] = new double[max];
561 double tem2Im[] = new double[max];
563 for (j = 1; j < max; j++) {
564 tem1Re[j] = temRe[j] + temRe[radix - j];
565 tem1Im[j] = temIm[j] - temIm[radix - j];
566 tem2Re[j] = temRe[j] - temRe[radix - j];
567 tem2Im[j] = temIm[j] + temIm[radix - j];
570 for (j = 1; j < max; j++) {
573 temRe[radix - j] = temRe[0];
574 temIm[radix - j] = temIm[0];
576 for (int i = 1; i < max; i++) {
577 rere = WRe[k] * tem1Re[i];
578 imim = WIm[k] * tem1Im[i];
579 reim = WRe[k] * tem2Im[i];
580 imre = WIm[k] * tem2Re[i];
582 temRe[radix - j] += rere + imim;
583 temIm[radix - j] += reim - imre;
584 temRe[j] += rere - imim;
585 temIm[j] += reim + imre;
592 for (j = 1; j < max; j++) {
593 temRe[0] = temRe[0] + tem1Re[j];
594 temIm[0] = temIm[0] + tem2Im[j];
596 } // End of function fftPrime().
598 } // End of class FFT2d