1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
|
c
c-----------------------------------------------------------------------
c subroutine: radix4
c computes forward or inverse complex dft via radix-4 fft.
c uses autogen technique to yield time efficient program.
c-----------------------------------------------------------------------
c
subroutine radix4(mm,iflag,jflag)
c
c input:
c mm = power of 4 (i.e., n = 4**mm complex point transform)
c (mm.ge.2 and mm.le.5)
c
c iflag = 1 on first pass for given n
c = 0 on subsequent passes for given n
c
c jflag = -1 for forward transform
c = +1 for inverse transform
c
c input/output:
c a = array of dimensions 2*n with real and imaginary parts
c of dft input/output in odd, even array components.
c
c for optimal time efficiency, common is used to pass arrays.
c this means that dimensions of arrays a, ix, and t can be
c modified to reflect maximum value of n = 4**mm to be used. note
c that array "ix" is also dimensioned in subroutine "rad4sb".
c
c i.e., a( ) ix( ) t( )
c
c m =2 32 38 27
c m<=3 128 144 135
c m<=4 512 658 567
c m<=5 2048 2996 2295
c
dimension a(2048),ix(2996),t(2295)
dimension nfac(11),np(209)
common ntypl,kkp,index,ixc
common /aa/a
common /xx/ix
c
c check for mm<2 or mm>5
c
if(mm.lt.2.or.mm.gt.5)stop
c
c initialize on first pass """"""""""""""""""""""""""""""""""""""""
c
if(iflag.eq.1) go to 9999
c
c fast fourier transform start ####################################
c
8885 kspan=2*4**mm
if(jflag.eq.1) go to 8887
c
c conjugate data for forward transform
c
do 8886 j=2,n2,2
8886 a(j)=-a(j)
go to 8889
c
c multiply data by n**(-1) if inverse transform
c
8887 do 8888 j=1,n2,2
a(j)=a(j)*xp
8888 a(j+1)=a(j+1)*xp
8889 i=3
it=ix(i-1)
go to (1,2,3,4,5,6,7,8),it
c***********************************************************************
c
c 8 multiply butterfly
c
1 kk=ix(i)
c
11 k1=kk+kspan
k2=k1+kspan
k3=k2+kspan
c
akp=a(kk)+a(k2)
akm=a(kk)-a(k2)
ajp=a(k1)+a(k3)
ajm=a(k1)-a(k3)
a(kk)=akp+ajp
c
bkp=a(kk+1)+a(k2+1)
bkm=a(kk+1)-a(k2+1)
bjp=a(k1+1)+a(k3+1)
bjm=a(k1+1)-a(k3+1)
a(kk+1)=bkp+bjp
c
bjp=bkp-bjp
c
a(k2+1)=(akp+bjp-ajp)*c707
a(k2)=a(k2+1)+bjp*cm141
c
bkp=bkm+ajm
akp=akm-bjm
c
ac0=(akp+bkp)*c924
a(k1+1)=ac0+akp*cm541
a(k1) =ac0+bkp*cm131
c
bkm=bkm-ajm
akm=akm+bjm
c
ac0=(akm+bkm)*c383
a(k3+1)=ac0+akm*c541
a(k3) =ac0+bkm*cm131
c
i=i+1
kk=ix(i)
if (kk) 111,111,11
111 i=i+2
it=ix(i-1)
go to (1,2,3,4,5,6,7,8), it
c***********************************************************************
c
c 4 multiply butterfly
c
2 kk=ix(i)
c
22 k1=kk+kspan
k2=k1+kspan
k3=k2+kspan
c
akp=a(kk)+a(k2)
akm=a(kk)-a(k2)
ajp=a(k1)+a(k3)
ajm=a(k1)-a(k3)
a(kk)=akp+ajp
c
bkp=a(kk+1)+a(k2+1)
bkm=a(kk+1)-a(k2+1)
bjp=a(k1+1)+a(k3+1)
bjm=a(k1+1)-a(k3+1)
a(kk+1)=bkp+bjp
a(k2)=-bkp+bjp
a(k2+1)=akp-ajp
c
bkp=bkm+ajm
c
a(k1+1)=(bkp+akm-bjm)*c707
a(k1)=a(k1+1)+bkp*cm141
c
akm=akm+bjm
c
a(k3+1)=(akm+ajm-bkm)*c707
a(k3)=a(k3+1)+akm*cm141
c
i=i+1
kk=ix(i)
if (kk) 222,222,22
222 i=i+2
it=ix(i-1)
go to (1,2,3,4,5,6,7,8), it
c***********************************************************************
c
c 8 multiply butterfly
c
3 kk=ix(i)
c
33 k1=kk+kspan
k2=k1+kspan
k3=k2+kspan
c
akp=a(kk)+a(k2)
akm=a(kk)-a(k2)
ajp=a(k1)+a(k3)
ajm=a(k1)-a(k3)
a(kk)=akp+ajp
c
bkp=a(kk+1)+a(k2+1)
bkm=a(kk+1)-a(k2+1)
bjp=a(k1+1)+a(k3+1)
bjm=a(k1+1)-a(k3+1)
a(kk+1)=bkp+bjp
c
ajp=akp-ajp
c
a(k2+1)=(ajp+bjp-bkp)*c707
a(k2)=a(k2+1)+ajp*cm141
c
bkp=bkm+ajm
akp=akm-bjm
c
ac0=(akp+bkp)*c383
a(k1+1)=ac0+akp*c541
a(k1) =ac0+bkp*cm131
c
bkm=bkm-ajm
akm=akm+bjm
c
ac0=(akm+bkm)*cm924
a(k3+1)=ac0+akm*c541
a(k3) =ac0+bkm*c131
c
i=i+1
kk=ix(i)
if (kk) 333,333,33
333 i=i+2
it=ix(i-1)
go to (1,2,3,4,5,6,7,8), it
c***********************************************************************
c
c general 9 multiply butterfly
c
4 kk=ix(i)
c
44 k1=kk+kspan
k2=k1+kspan
k3=k2+kspan
c
akp=a(kk)+a(k2)
akm=a(kk)-a(k2)
ajp=a(k1)+a(k3)
ajm=a(k1)-a(k3)
a(kk)=akp+ajp
c
bkp=a(kk+1)+a(k2+1)
bkm=a(kk+1)-a(k2+1)
bjp=a(k1+1)+a(k3+1)
bjm=a(k1+1)-a(k3+1)
a(kk+1)=bkp+bjp
c
ajp=akp-ajp
bjp=bkp-bjp
c
j=ix(i+1)
c
ac0=(ajp+bjp)*t(j+8)
a(k2+1)=ac0+ajp*t(j+6)
a(k2) =ac0+bjp*t(j+7)
c
bkp=bkm+ajm
akp=akm-bjm
c
ac0=(akp+bkp)*t(j+5)
a(k1+1)=ac0+akp*t(j+3)
a(k1) =ac0+bkp*t(j+4)
c
bkm=bkm-ajm
akm=akm+bjm
c
ac0=(akm+bkm)*t(j+2)
a(k3+1)=ac0+akm*t(j)
a(k3) =ac0+bkm*t(j+1)
c
i=i+2
kk=ix(i)
if (kk) 444,444,44
444 i=i+2
it=ix(i-1)
go to (1,2,3,4,5,6,7,8), it
c***********************************************************************
c
c 0 multiply butterfly
c
5 kk=ix(i)
c
55 k1=kk+kspan
k2=k1+kspan
k3=k2+kspan
c
akp=a(kk)+a(k2)
akm=a(kk)-a(k2)
ajp=a(k1)+a(k3)
ajm=a(k1)-a(k3)
a(kk)=akp+ajp
a(k2)=akp-ajp
c
bkp=a(kk+1)+a(k2+1)
bkm=a(kk+1)-a(k2+1)
bjp=a(k1+1)+a(k3+1)
bjm=a(k1+1)-a(k3+1)
a(kk+1)=bkp+bjp
a(k2+1)=bkp-bjp
c
a(k3+1)=bkm-ajm
a(k1+1)=bkm+ajm
a(k3)=akm+bjm
a(k1)=akm-bjm
c
i=i+1
kk=ix(i)
if (kk) 555,555,55
555 i=i+2
it=ix(i-1)
go to (1,2,3,4,5,6,7,8), it
c***********************************************************************
c
c offset reduced
c
6 kspan=kspan/4
i=i+2
it=ix(i-1)
go to (1,2,3,4,5,6,7,8), it
c***********************************************************************
c
c bit reversal (shuffling)
c
7 ip1=ix(i)
77 ip2=ix(i+1)
t1=a(ip2)
a(ip2)=a(ip1)
a(ip1)=t1
t1=a(ip2+1)
a(ip2+1)=a(ip1+1)
a(ip1+1)=t1
i=i+2
ip1=ix(i)
if (ip1) 777,777,77
777 i=i+2
it=ix(i-1)
go to (1,2,3,4,5,6,7,8), it
c***********************************************************************
8 if(jflag.eq.1) go to 888
c
c conjugate output if forward transform
c
do 88 j=2,n2,2
88 a(j)=-a(j)
888 return
c
c fast fourier transform ends #####################################
c
c initialization phase starts. done only once
c
9999 ixc=1
n=4**mm
xp=n
xp=1./xp
ntot=n
n2=n*2
nspan=n
n1test=n/16
n2test=n/8
n3test=(3*n)/16
nspan4=nspan/4
ibase=0
isn=1
inc=isn
rad=8.0*atan(1.0)
pi=4.*atan(1.0)
c707=sin(pi/4.)
cm141=-2.*c707
c383=sin(pi/8.)
c924=cos(pi/8.)
cm924=-c924
c541=c924-c383
cm541=-c541
c131=c924+c383
cm131=-c131
10 nt=inc*ntot
ks=inc*nspan
kspan=ks
jc=ks/n
radf=rad*float(jc)*.5
i=0
c
c determine the factors of n
c all factors must be 4 for this version
c
m=0
k=n
15 m=m+1
nfac(m)=4
k=k/4
20 if(k-(k/4)*4.eq.0) go to 15
kt=1
if(n.ge.256) kt=2
kspan0=kspan
ntypl=0
c
100 ndelta=kspan0/kspan
index=0
sd=radf/float(kspan)
cd=2.0*sin(sd)**2
sd=sin(sd+sd)
kk=1
i=i+1
c
c transform for a factor of 4
c
kspan=kspan/4
ix(ixc)=0
ix(ixc+1)=6
ixc=ixc+2
c
410 c1=1.0
s1=0.0
420 k1=kk+kspan
k2=k1+kspan
k3=k2+kspan
if(s1.eq.0.0) go to 460
430 if(kspan.ne.nspan4) go to 431
t(ibase+5)=-(s1+c1)
t(ibase+6)=c1
t(ibase+4)=s1-c1
t(ibase+8)=-(s2+c2)
t(ibase+9)=c2
t(ibase+7)=s2-c2
t(ibase+2)=-(s3+c3)
t(ibase+3)=c3
t(ibase+1)=s3-c3
ibase=ibase+9
c
431 kkp=(kk-1)*2
if(index.ne.n1test) go to 150
call rad4sb(1)
go to 5035
150 if(index.ne.n2test) go to 160
call rad4sb(2)
go to 5035
160 if(index.ne.n3test) go to 170
call rad4sb(3)
go to 5035
170 call rad4sb(4)
5035 kk=k3+kspan
if(kk.le.nt) go to 420
440 index=index+ndelta
c2=c1-(cd*c1+sd*s1)
s1=(sd*c1-cd*s1)+s1
c1=c2
c2=c1*c1-s1*s1
s2=c1*s1+c1*s1
c3=c2*c1-s2*s1
s3=c2*s1+s2*c1
kk=kk-nt+jc
if(kk.le.kspan) go to 420
kk=kk-kspan+inc
if(kk.le.jc) go to 410
if(kspan.eq.jc) go to 800
go to 100
460 kkp=(kk-1)*2
call rad4sb(5)
5050 kk=k3+kspan
if(kk.le.nt) go to 420
go to 440
c
800 ix(ixc)=0
ix(ixc+1)=7
ixc=ixc+2
c
c compute parameters to permute the results to normal order
c done in two steps
c permutation for square factors of n
c
np(1)=ks
k=kt+kt+1
if(m.lt.k) k=k-1
j=1
np(k+1)=jc
810 np(j+1)=np(j)/nfac(j)
np(k)=np(k+1)*nfac(j)
j=j+1
k=k-1
if(j.lt.k) go to 810
k3=np(k+1)
kspan=np(2)
kk=jc+1
k2=kspan+1
j=1
c
c permutation for single variate transform
c
820 kkp=(kk-1)*2
k2p=(k2-1)*2
ix(ixc)=kkp+1
ix(ixc+1)=k2p+1
ixc=ixc+2
kk=kk+inc
k2=kspan+k2
if(k2.lt.ks) go to 820
830 k2=k2-np(j)
j=j+1
k2=np(j+1)+k2
if(k2.gt.np(j)) go to 830
j=1
840 if(kk.lt.k2) go to 820
kk=kk+inc
k2=kspan+k2
if(k2.lt.ks) go to 840
if(kk.lt.ks) go to 830
jc=k3
ix(ixc)=0
ix(ixc+1)=8
go to 8885
end
|
