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fftsubs.f
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c
c MINEOS version 1.0 by Guy Masters, John Woodhouse, and Freeman Gilbert
c
c This program is free software; you can redistribute it and/or modify
c it under the terms of the GNU General Public License as published by
c the Free Software Foundation; either version 2 of the License, or
c (at your option) any later version.
c
c This program is distributed in the hope that it will be useful,
c but WITHOUT ANY WARRANTY; without even the implied warranty of
c MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
c GNU General Public License for more details.
c
c You should have received a copy of the GNU General Public License
c along with this program; if not, write to the Free Software
c Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
c
c*************************************************************************
c
c FFT routines
c
c*************************************************************************
SUBROUTINE FACTOR(N)
C FACTOR DETERMINES THE SMALLEST EVEN NUMBER .GE. N THAT CAN BE FACTORED INTO
C PRODUCTS OF PRIMES IN THE ARRAY NP. THE RESULT IS RETURNED AS N.
C ONLY THE FIRST FOUR PRIMES ARE USED. THIS ROUTINE IS CALLED BEFORE USING FFTL.
DIMENSION NP(4)
DATA NP/2,3,5,7/
NCHK=N/2
IF(N.NE.(NCHK*2)) N=N+1
5 NF=N
DO 15 I=1,4
M=NP(I)
10 L=MOD(NF,M)
IF(L.NE.0) GO TO 15
NF=NF/M
GO TO 10
15 CONTINUE
IF(NF.EQ.1) RETURN
N=N+2
GO TO 5
END
SUBROUTINE FACDWN(N)
C FACDWN DETERMINES THE LARGEST EVEN NUMBER .LE. N THAT CAN BE FACTORED INTO
C PRODUCTS OF PRIMES IN THE ARRAY NP. THE RESULT IS RETURNED AS N.
C ONLY THE FIRST FOUR PRIMES ARE USED. THIS ROUTINE IS CALLED BEFORE USING FFTL.
DIMENSION NP(4)
DATA NP/2,3,5,7/
NCHK=N/2
IF(N.NE.(NCHK*2)) N=N-1
5 NF=N
DO 15 I=1,4
M=NP(I)
10 L=MOD(NF,M)
IF(L.NE.0) GO TO 15
NF=NF/M
GO TO 10
15 CONTINUE
IF(NF.EQ.1) RETURN
N=N-2
GO TO 5
END
SUBROUTINE FFTL(X,N,NDIR,IERR)
C IF IABS(NDIR)=1 FFTL FOURIER TRANSFORMS THE N POINTS REAL TIME SERIES IN ARRAY
C X. THE RESULT OVERWRITES X STORED AS (N+2)/2 COMPLEX NUMBERS (NON-NEGATIVE
C FREQUENCIES ONLY).
C IF IABS(NDIR)=2 FFTL FOURIER TRANSFORMS THE (N+2)/2 COMPLEX FOURIER COEFFICIENTS
C (NON-NEGATIVE FREQUENCIES ONLY) IN ARRAY X (ASSUMING THE SERIES IS HERMITIAN).
C THE RESULTING N POINT REAL TIME SERIES OVERWRITES X.
C NDIR>0 THE FORWARD TRANSFORM USES THE SIGN CONVENTION EXP(I*W*T).
C NDIR<0 THE FORWARD TRANSFORM USES THE SIGN CONVENTION EXP(-I*W*T).
C THE FORWARD TRANSFORM IS NORMALIZED SUCH THAT A SINE WAVE OF UNIT AMPLITUDE
C IS TRANSFORMED INTO DELTA FUNCTIONS OF UNIT AMPLITUDE. THE BACKWARD TRANSFORM
C IS NORMALIZED SUCH THAT TRANSFORMING FORWARD AND THEN BACK RECOVERS THE
C ORIGINAL SERIES. IERR IS NORMALLY ZERO. IF IERR.EQ.1 THEN FFT HAS NOT BEEN
C ABLE TO FACTOR THE SERIES. HOWEVER, X HAS BEEN SCRAMBLED BY REALTR. NOTE
C THAT IF N IS ODD THE LAST POINT WILL NOT BE USED IN THE TRANSFORM.
C -RPB
DIMENSION X(*)
N2=N/2
IDIR=IABS(NDIR)
GO TO (1,2),IDIR
C DO FORWARD TRANSFORM (IE. TIME TO FREQUENCY).
1 CALL FFT(X,X(2),N2,N2,N2,2,IERR)
CALL REALTR(X,X(2),N2,2)
N1=2*N2+2
SCALE=1./N
IF(NDIR.GT.0) GO TO 3
DO 5 I=4,N,2
5 X(I)=-X(I)
GO TO 3
C DO BACKWARD TRANSFORM (IE. FREQUENCY TO TIME).
2 IF(NDIR.GT.0) GO TO 6
DO 7 I=4,N,2
7 X(I)=-X(I)
6 X(2)=0.
X(2*N2+2)=0.
CALL REALTR(X,X(2),N2,-2)
CALL FFT(X,X(2),N2,N2,N2,-2,IERR)
N1=2*N2
SCALE=.5
C NORMALIZE THE TRANSFORM.
3 DO 4 I=1,N1
4 X(I)=SCALE*X(I)
RETURN
END
SUBROUTINE REALTR(A,B,N,ISN)
C IF ISN=1, THIS SUBROUTINE COMPLETES THE FOURIER TRANSFORM OF 2*N REAL
C DATA VALUES, WHERE THE ORIGINAL DATA VALUES ARE STORED ALTERNATELY IN
C ARRAYS A AND B, AND ARE FIRST TRANSFORMED BY A COMPLEX FOURIER TRANSFORM
c OF DIMENSION N. THE COSINE COEFFICIENTS ARE IN A(1),A(2),...A(N+1) AND
C THE SINE COEFFICIENTS ARE IN B(1),B(2),...B(N+1). A TYPICAL CALLING
C SEQUENCE IS
C CALL FFT (A,B,N,1)
C CALL REALTR (A,B,N,1)
C THE RESULTS SHOULD BE MULTIPLIED BY 0.5/N TO GIVE THE USUAL SCALING
C IF ISN=1, THE INVERSE TRANSFORMATION IS DONE, THE FIRST STEP IN EVALUATING
C A REAL FOURIER SERIES. A TYPICAL CALLING SEQUENCE IS
C CALL REALTR (A,B,N,-1)
C CALL FFT (A,B,N,-1)
C THE RESULTS SHOULD BE MULTIPLIED BY 0.5 TO GIVE THE USUAL SCALING, AND
C THE TIME DOMAIN RESULTS ALTERNATE IN ARRAYS A AND B, I.E. A(1),B(1), A(2),B(2),
C ...A(N),B(N). THE DATA MAY ALTERNATIVELY BE STORED IN A SINGLE COMPLEX
C ARRAY A, THEN THE MAGNITUDE OF ISN CHANGED TO TWO TO GIVE THE CORRECT
C INDEXING INCREMENT AND A(2) USED TO PASS THE INITIAL ADDRESS FOR THE
C SEQUENCE OF IMAGINARY VALUES,E.G.
C CALL FFT(A,A(2),N,2)
C CALL REALTR(A,A(2),N,2)
C IN THIS CASE, THE COSINE AND SINE COEFFICIENTS ALTERNATE IN A.
DIMENSION A(*),B(*)
IF(N .LE. 1) RETURN
INC=ISN
IF(INC.LT.0) INC=-INC
NK = N * INC + 2
NH = NK / 2
SD = 3.1415926535898E0/(2.E0*N)
CD = 2.E0 * SIN(SD)**2
SD = SIN(SD+SD)
SN = 0.E0
IF(ISN .GT. 0) GO TO 10
CN = -1.E0
SD = -SD
GO TO 20
10 CN = 1.E0
A(NK-1) = A(1)
B(NK-1) = B(1)
20 DO 30 J=1,NH,INC
K = NK - J
AA = A(J) + A(K)
AB = A(J) - A(K)
BA = B(J) + B(K)
BB = B(J) - B(K)
XX = CN * BA + SN * AB
YY = SN * BA - CN * AB
B(K) = YY - BB
B(J) = YY + BB
A(K) = AA - XX
A(J) = AA + XX
AA = CN - (CD * CN + SD * SN)
SN = (SD * CN - CD * SN) + SN
30 CN = AA
RETURN
END
SUBROUTINE FFT(A,B,NTOT,N,NSPAN,ISN,IERR)
C FFT IS A SINGLE PRECISION VERSION OF SINGLETON'S FFT PROCEDURE.
C MULTIVARIATE COMPLEX FOURIER TRANSFORM, COMPUTED IN PLACE USING
C MIXED-RADIX FAST FOURIER TRANSFORM ALGORITHM. BY R. C. SINGLETON,
C STANFORD RESEARCH INSTITUTE, SEPT. 1968
C ARRAYS A AND B ORIGINALLY HOLD THE REAL AND IMAGINARY COMPONENTS OF THE
C DATA, AND RETURN THE REAL AND IMAGINARY COMPONENTS OF THE RESULTING FOURIER
C COEFFICIENTS. MULTIVARIATE DATA IS INDEXED ACCORDING TO THE FORTRAN ARRAY
C ELEMENT SUCCESSOR FUNCTION, WITHOUT LIMIT ON THE NUMBER OF IMPLIED MULTIPLE
C SUBSCRIPTS. THE SUBROUTINE IS CALLED ONCE FOR EACH VARIATE. THE CALLS FOR A
C MULTIVARIATE TRANSFORM MAY BE IN ANY ORDER.
C NTOT IS THE TOTAL NUMBER OF COMPLEX DATA VALUES.
C N IS THE DIMENSION OF THE CURRENT VARIABLE.
C NSPAN/N IS THE SPACING OF CONSECUTIVE DATA VALUES FOR THE CURRENT VARIABLE.
C IERR IS AN ERROR RETURN INDICATOR. IT IS NORMALLY ZERO, BUT IS SET TO 1 IF
C THE NUMBER OF TERMS CANNOT BE FACTORED IN THE SPACE AVAILABLE.
C THE SIGN OF ISN DETERMINES THE SIGN OF THE COMPLEX EXPONENTIAL, AND THE
C MAGNITUDE OF ISN IS NORMALLY ONE.
C A TRI-VARIATE TRANSFORM WITH A(N1,N2,N3), B(N1,N2,N3) IS COMPUTED BY
C CALL FFT(A,B,N1*N2*N3,N1,N1,1)
C CALL FFT(A,B,N1*N2*N3,N2,N1*N2,1)
C CALL FFT(A,B,N1*N2*N3,N3,N1*N2*N3,1)
C FOR A SINGLE-VARIATE TRANSFORM,
C NTOT = N = NSPAN = (NUMBER OF COMPLEX DATA VALUES), E.G.
C CALL FFT(A,B,N,N,N,1)
C WITH MOST FORTRAN COMPILERS THE DATA CAN ALTERNATIVELY BE STORED IN A SINGLE
C COMPLEX ARRAY A, THEN THE MAGNITUDE OF ISN CHANGED TO TWO TO GIVE THE CORRECT
C INDEXING INCREMENT AND A(2) USED TO PASS THE INITIAL ADDRESS FOR THE SEQUENCE
C OF IMAGINARY VALUES, E.G.
C CALL FFT(A,A(2),NTOT,N,NSPAN,2)
C ARRAYS AT(MAXF), CK(MAXF), BT(MAXF), SK(MAXF), AND NP(MAXP) ARE USED FOR
C TEMPORARY STORAGE. IF THE AVAILABLE STORAGE IS INSUFFICIENT, THE PROGRAM IS
C TERMINATED BY THE ERROR RETURN OPTION
C MAXF MUST BE .GE. THE MAXIMUM PRIME FACTOR OF N.
C MAXP MUST BE .GT. THE NUMBER OF PRIME FACTORS OF N.
C IN ADDITION, IF THE SQUARE-FREE PORTION K OF N HAS TWO OR MORE PRIME FACTORS,
C THEN MAXP MUST BE .GE. K-1. ARRAY STORAGE IN NFAC FOR A MAXIMUM OF 11 PRIME
C FACTORS OF N. IF N HAS MORE THAN ONE SQUARE-FREE FACTOR, THE PRODUCT OF THE
C SQUARE-FREE FACTORS MUST BE .LE. 210. (2**5*7=210)
DIMENSION A(*),B(*)
DIMENSION NFAC(11),NP(209)
C ARRAY STORAGE FOR MAXIMUM PRIME FACTOR OF 23
DIMENSION AT(23),CK(23),BT(23),SK(23)
EQUIVALENCE (I,II)
C THE FOLLOWING TWO CONSTANTS SHOULD AGREE WITH THE ARRAY DIMENSIONS.
MAXF=23
MAXP=209
IERR=0
IF(N .LT. 2) RETURN
INC=ISN
C72=0.30901699437494E0
S72=0.95105651629515E0
S120=0.86602540378443E0
RAD=6.2831853071796E0
IF(ISN .GE. 0) GO TO 10
S72=-S72
S120=-S120
RAD=-RAD
INC=-INC
10 NT=INC*NTOT
KS=INC*NSPAN
KSPAN=KS
NN=NT-INC
JC=KS/N
RADF=RAD*(JC*0.5E0)
I=0
JF=0
C DETERMINE THE FACTORS OF N
M=0
K=N
GO TO 20
15 M=M+1
NFAC(M)=4
K=K/16
20 IF(K-(K/16)*16 .EQ. 0) GO TO 15
J=3
JJ=9
GO TO 30
25 M=M+1
NFAC(M)=J
K=K/JJ
30 IF(MOD(K,JJ) .EQ. 0) GO TO 25
J=J+2
JJ=J**2
IF(JJ .LE. K) GO TO 30
IF(K .GT. 4) GO TO 40
KT=M
NFAC(M+1)=K
IF(K .NE. 1) M=M+1
GO TO 80
40 IF(K-(K/4)*4 .NE. 0) GO TO 50
M=M+1
NFAC(M)=2
K=K/4
50 KT=M
J=2
60 IF(MOD(K,J) .NE. 0) GO TO 70
M=M+1
NFAC(M)=J
K=K/J
70 J=((J+1)/2)*2+1
IF(J .LE. K) GO TO 60
80 IF(KT .EQ. 0) GO TO 100
J=KT
90 M=M+1
NFAC(M)=NFAC(J)
J=J-1
IF(J .NE. 0) GO TO 90
C COMPUTE FOURIER TRANSFORM
100 SD=RADF/(KSPAN)
CD=2.E0*SIN(SD)**2
SD=SIN(SD+SD)
KK=1
I=I+1
IF(NFAC(I) .NE. 2) GO TO 400
C TRANSFORM FOR FACTOR OF 2 (INCLUDING ROTATION FACTOR)
KSPAN=KSPAN/2
K1=KSPAN+2
210 K2=KK+KSPAN
AK=A(K2)
BK=B(K2)
A(K2)=A(KK)-AK
B(K2)=B(KK)-BK
A(KK)=A(KK)+AK
B(KK)=B(KK)+BK
KK=K2+KSPAN
IF(KK .LE. NN) GO TO 210
KK=KK-NN
IF(KK .LE. JC) GO TO 210
IF(KK .GT. KSPAN) GO TO 800
220 C1=1.E0-CD
S1=SD
230 K2=KK+KSPAN
AK=A(KK)-A(K2)
BK=B(KK)-B(K2)
A(KK)=A(KK)+A(K2)
B(KK)=B(KK)+B(K2)
A(K2)=C1*AK-S1*BK
B(K2)=S1*AK+C1*BK
KK=K2+KSPAN
IF(KK .LT. NT) GO TO 230
K2=KK-NT
C1=-C1
KK=K1-K2
IF(KK .GT. K2) GO TO 230
AK=CD*C1+SD*S1
S1=(SD*C1-CD*S1)+S1
C1=C1-AK
KK=KK+JC
IF(KK .LT. K2) GO TO 230
K1=K1+INC+INC
KK=(K1-KSPAN)/2+JC
IF(KK .LE. JC+JC) GO TO 220
GO TO 100
C TRANSFORM FOR FACTOR OF 3 (OPTIONAL CODE)
320 K1=KK+KSPAN
K2=K1+KSPAN
AK=A(KK)
BK=B(KK)
AJ=A(K1)+A(K2)
BJ=B(K1)+B(K2)
A(KK)=AK+AJ
B(KK)=BK+BJ
AK=-0.5*AJ+AK
BK=-0.5*BJ+BK
AJ=(A(K1)-A(K2))*S120
BJ=(B(K1)-B(K2))*S120
A(K1)=AK-BJ
B(K1)=BK+AJ
A(K2)=AK+BJ
B(K2)=BK-AJ
KK=K2+KSPAN
IF(KK .LT. NN) GO TO 320
KK=KK-NN
IF(KK .LE. KSPAN) GO TO 320
GO TO 700
C TRANSFORM FOR FACTOR OF 4
400 IF(NFAC(I) .NE. 4) GO TO 600
KSPNN=KSPAN
KSPAN=KSPAN/4
410 C1=1.E0
S1=0
420 K1=KK+KSPAN
K2=K1+KSPAN
K3=K2+KSPAN
AKP=A(KK)+A(K2)
AKM=A(KK)-A(K2)
AJP=A(K1)+A(K3)
AJM=A(K1)-A(K3)
A(KK)=AKP+AJP
AJP=AKP-AJP
BKP=B(KK)+B(K2)
BKM=B(KK)-B(K2)
BJP=B(K1)+B(K3)
BJM=B(K1)-B(K3)
B(KK)=BKP+BJP
BJP=BKP-BJP
IF(ISN .LT. 0) GO TO 450
AKP=AKM-BJM
AKM=AKM+BJM
BKP=BKM+AJM
BKM=BKM-AJM
IF(S1 .EQ. 0) GO TO 460
430 A(K1)=AKP*C1-BKP*S1
B(K1)=AKP*S1+BKP*C1
A(K2)=AJP*C2-BJP*S2
B(K2)=AJP*S2+BJP*C2
A(K3)=AKM*C3-BKM*S3
B(K3)=AKM*S3+BKM*C3
KK=K3+KSPAN
IF(KK .LE. NT) GO TO 420
440 C2=CD*C1+SD*S1
S1=(SD*C1-CD*S1)+S1
C1=C1-C2
C2=C1**2-S1**2
S2=2.E0*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
450 AKP=AKM+BJM
AKM=AKM-BJM
BKP=BKM-AJM
BKM=BKM+AJM
IF(S1 .NE. 0) GO TO 430
460 A(K1)=AKP
B(K1)=BKP
A(K2)=AJP
B(K2)=BJP
A(K3)=AKM
B(K3)=BKM
KK=K3+KSPAN
IF(KK .LE. NT) GO TO 420
GO TO 440
C TRANSFORM FOR FACTOR OF 5 (OPTIONAL CODE)
510 C2=C72**2-S72**2
S2=2.E0*C72*S72
520 K1=KK+KSPAN
K2=K1+KSPAN
K3=K2+KSPAN
K4=K3+KSPAN
AKP=A(K1)+A(K4)
AKM=A(K1)-A(K4)
BKP=B(K1)+B(K4)
BKM=B(K1)-B(K4)
AJP=A(K2)+A(K3)
AJM=A(K2)-A(K3)
BJP=B(K2)+B(K3)
BJM=B(K2)-B(K3)
AA=A(KK)
BB=B(KK)
A(KK)=AA+AKP+AJP
B(KK)=BB+BKP+BJP
AK=AKP*C72+AJP*C2+AA
BK=BKP*C72+BJP*C2+BB
AJ=AKM*S72+AJM*S2
BJ=BKM*S72+BJM*S2
A(K1)=AK-BJ
A(K4)=AK+BJ
B(K1)=BK+AJ
B(K4)=BK-AJ
AK=AKP*C2+AJP*C72+AA
BK=BKP*C2+BJP*C72+BB
AJ=AKM*S2-AJM*S72
BJ=BKM*S2-BJM*S72
A(K2)=AK-BJ
A(K3)=AK+BJ
B(K2)=BK+AJ
B(K3)=BK-AJ
KK=K4+KSPAN
IF(KK .LT. NN) GO TO 520
KK=KK-NN
IF(KK .LE. KSPAN) GO TO 520
GO TO 700
C TRANSFORM FOR ODD FACTORS
600 K=NFAC(I)
KSPNN=KSPAN
KSPAN=KSPAN/K
IF(K .EQ. 3) GO TO 320
IF(K .EQ. 5) GO TO 510
IF(K .EQ. JF) GO TO 640
JF=K
S1=RAD/(K)
C1=COS(S1)
S1=SIN(S1)
IF(JF .GT. MAXF) GO TO 998
CK(JF)=1.E0
SK(JF)=0.E0
J=1
630 CK(J)=CK(K)*C1+SK(K)*S1
SK(J)=CK(K)*S1-SK(K)*C1
K=K-1
CK(K)=CK(J)
SK(K)=-SK(J)
J=J+1
IF(J .LT. K) GO TO 630
640 K1=KK
K2=KK+KSPNN
AA=A(KK)
BB=B(KK)
AK=AA
BK=BB
J=1
K1=K1+KSPAN
650 K2=K2-KSPAN
J=J+1
AT(J)=A(K1)+A(K2)
AK=AT(J)+AK
BT(J)=B(K1)+B(K2)
BK=BT(J)+BK
J=J+1
AT(J)=A(K1)-A(K2)
BT(J)=B(K1)-B(K2)
K1=K1+KSPAN
IF(K1 .LT. K2) GO TO 650
A(KK)=AK
B(KK)=BK
K1=KK
K2=KK+KSPNN
J=1
660 K1=K1+KSPAN
K2=K2-KSPAN
JJ=J
AK=AA
BK=BB
AJ=0.E0
BJ=0.E0
K=1
670 K=K+1
AK=AT(K)*CK(JJ)+AK
BK=BT(K)*CK(JJ)+BK
K=K+1
AJ=AT(K)*SK(JJ)+AJ
BJ=BT(K)*SK(JJ)+BJ
JJ=JJ+J
IF(JJ .GT. JF) JJ=JJ-JF
IF(K .LT. JF) GO TO 670
K=JF-J
A(K1)=AK-BJ
B(K1)=BK+AJ
A(K2)=AK+BJ
B(K2)=BK-AJ
J=J+1
IF(J .LT. K) GO TO 660
KK=KK+KSPNN
IF(KK .LE. NN) GO TO 640
KK=KK-NN
IF(KK .LE. KSPAN) GO TO 640
C MULTIPLY BY ROTATION FACTOR (EXCEPT FOR FACTORS OF 2 AND 4)
700 IF(I .EQ. M) GO TO 800
KK=JC+1
710 C2=1.E0-CD
S1=SD
720 C1=C2
S2=S1
KK=KK+KSPAN
730 AK=A(KK)
A(KK)=C2*AK-S2*B(KK)
B(KK)=S2*AK+C2*B(KK)
KK=KK+KSPNN
IF(KK .LE. NT) GO TO 730
AK=S1*S2
S2=S1*C2+C1*S2
C2=C1*C2-AK
KK=KK-NT+KSPAN
IF(KK .LE. KSPNN) GO TO 730
C2=C1-(CD*C1+SD*S1)
S1=S1+(SD*C1-CD*S1)
KK=KK-KSPNN+JC
IF(KK .LE. KSPAN) GO TO 720
KK=KK-KSPAN+JC+INC
IF(KK .LE. JC+JC) GO TO 710
GO TO 100
C PERMUTE THE RESULTS TO NORMAL ORDER---DONE IN TWO STAGES
C PERMUTATION FOR SQUARE FACTORS OF N
800 NP(1)=KS
IF(KT .EQ. 0) GO TO 890
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
IF(N .NE. NTOT) GO TO 850
C PERMUTATION FOR SINGLE-VARIATE TRANSFORM (OPTIONAL CODE)
820 AK=A(KK)
A(KK)=A(K2)
A(K2)=AK
BK=B(KK)
B(KK)=B(K2)
B(K2)=BK
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
GO TO 890
C PERMUTATION FOR MULTIVARIATE TRANSFORM
850 K=KK+JC
860 AK=A(KK)
A(KK)=A(K2)
A(K2)=AK
BK=B(KK)
B(KK)=B(K2)
B(K2)=BK
KK=KK+INC
K2=K2+INC
IF(KK .LT. K) GO TO 860
KK=KK+KS-JC
K2=K2+KS-JC
IF(KK .LT. NT) GO TO 850
K2=K2-NT+KSPAN
KK=KK-NT+JC
IF(K2 .LT. KS) GO TO 850
870 K2=K2-NP(J)
J=J+1
K2=NP(J+1)+K2
IF(K2 .GT. NP(J)) GO TO 870
J=1
880 IF(KK .LT. K2) GO TO 850
KK=KK+JC
K2=KSPAN+K2
IF(K2 .LT. KS) GO TO 880
IF(KK .LT. KS) GO TO 870
JC=K3
890 IF(2*KT+1 .GE. M) RETURN
KSPNN=NP(KT+1)
C PERMUTATION FOR SQUARE-FREE FACTORS OF N
J=M-KT
NFAC(J+1)=1
900 NFAC(J)=NFAC(J)*NFAC(J+1)
J=J-1
IF(J .NE. KT) GO TO 900
KT=KT+1
NN=NFAC(KT)-1
IF(NN .GT. MAXP) GO TO 998
JJ=0
J=0
GO TO 906
902 JJ=JJ-K2
K2=KK
K=K+1
KK=NFAC(K)
904 JJ=KK+JJ
IF(JJ .GE. K2) GO TO 902
NP(J)=JJ
906 K2=NFAC(KT)
K=KT+1
KK=NFAC(K)
J=J+1
IF(J .LE. NN) GO TO 904
C DETERMINE THE PERMUTATION CYCLES OF LENGTH GREATER THAN 1
J=0
GO TO 914
910 K=KK
KK=NP(K)
NP(K)=-KK
IF(KK .NE. J) GO TO 910
K3=KK
914 J=J+1
KK=NP(J)
IF(KK .LT. 0) GO TO 914
IF(KK .NE. J) GO TO 910
NP(J)=-J
IF(J .NE. NN) GO TO 914
MAXF=INC*MAXF
C REORDER A AND B, FOLLOWING THE PERMUTATION CYCLES
GO TO 950
924 J=J-1
IF(NP(J) .LT. 0) GO TO 924
JJ=JC
926 KSPAN=JJ
IF(JJ .GT. MAXF) KSPAN=MAXF
JJ=JJ-KSPAN
K=NP(J)
KK=JC*K+II+JJ
K1=KK+KSPAN
K2=0
928 K2=K2+1
AT(K2)=A(K1)
BT(K2)=B(K1)
K1=K1-INC
IF(K1 .NE. KK) GO TO 928
932 K1=KK+KSPAN
K2=K1-JC*(K+NP(K))
K=-NP(K)
936 A(K1)=A(K2)
B(K1)=B(K2)
K1=K1-INC
K2=K2-INC
IF(K1 .NE. KK) GO TO 936
KK=K2
IF(K .NE. J) GO TO 932
K1=KK+KSPAN
K2=0
940 K2=K2+1
A(K1)=AT(K2)
B(K1)=BT(K2)
K1=K1-INC
IF(K1 .NE. KK) GO TO 940
IF(JJ .NE. 0) GO TO 926
IF(J .NE. 1) GO TO 924
950 J=K3+1
NT=NT-KSPNN
II=NT-INC+1
IF(NT .GE. 0) GO TO 924
RETURN
C ERROR FINISH, INSUFFICIENT ARRAY STORAGE
998 IERR=1
RETURN
END
SUBROUTINE FFTLDP(X,N,NDIR,DT,IERR)
C FFTLDP IS A DOUBLE PRECISION VERSION OF FFTL. SEE NOTES TO FFTL FOR USAGE
C THE SCALING IN FFTLDP DIFFERS FROM THAT IN FFTL :
C THE FORWARD TRANSFORM IS SPECTRUM(W)=DT*SUM(0,N-1)X(J)*EXP(-I*W*J*DT)
C THE SPACING IN W IS 2*PI/(N*DT)
IMPLICIT REAL*8(A-H,O-Z)
DIMENSION X(*)
N2=N/2
IDIR=NDIR
IF(IDIR.LT.0) IDIR=-IDIR
GO TO (1,2),IDIR
C DO FORWARD TRANSFORM (IE. TIME TO FREQUENCY).
1 CALL FFTDP(X,X(2),N2,N2,N2,2,IERR)
CALL RLTRDP(X,X(2),N2,2)
N1=2*N2+2
SCALE=0.5D0*DT
IF(NDIR.GT.0) GO TO 3
DO 5 I=4,N,2
5 X(I)=-X(I)
GO TO 3
C DO BACKWARD TRANSFORM (IE. FREQUENCY TO TIME).
2 IF(NDIR.GT.0) GO TO 6
DO 7 I=4,N,2
7 X(I)=-X(I)
6 X(2)=0.D0
N1=2*N2
X(N1+2)=0.D0
CALL RLTRDP(X,X(2),N2,-2)
CALL FFTDP(X,X(2),N2,N2,N2,-2,IERR)
SCALE=1.D0/(N*DT)
C NORMALIZE THE TRANSFORM.
3 DO 4 I=1,N1
4 X(I)=SCALE*X(I)
RETURN
END
SUBROUTINE RLTRDP(A,B,N,ISN)
C RLTRDP IS A DOUBLE PRECISION VERSION OF REALTR. SEE REALTR FOR NOTES
C ON USAGE.
IMPLICIT REAL*8(A-H,O-Z)
DIMENSION A(*),B(*)
IF(N .LE. 1) RETURN
INC=ISN
IF(INC.LT.0) INC=-INC
NK = N * INC + 2
NH = NK / 2
SD = 3.1415926535898D0/(2.D0*N)
CD = 2.D0 * DSIN(SD)**2
SD = DSIN(SD+SD)
SN = 0.D0
IF(ISN .GT. 0) GO TO 10
CN = -1.D0
SD = -SD
GO TO 20
10 CN = 1.D0
A(NK-1) = A(1)
B(NK-1) = B(1)
20 DO 30 J=1,NH,INC
K = NK - J
AA = A(J) + A(K)
AB = A(J) - A(K)
BA = B(J) + B(K)
BB = B(J) - B(K)
XX = CN * BA + SN * AB
YY = SN * BA - CN * AB
B(K) = YY - BB
B(J) = YY + BB
A(K) = AA - XX
A(J) = AA + XX
AA = CN - (CD * CN + SD * SN)
SN = (SD * CN - CD * SN) + SN
30 CN = AA
RETURN
END
SUBROUTINE FFTDP(A,B,NTOT,N,NSPAN,ISN,IERR)
C FFTDP IS A DOUBLE PRECISION VERSION OF SINGLETON'S FFT PROCEDURE. SEE NOTES
C TO FFT FOR USAGE
IMPLICIT REAL*8(A-H),REAL*8(O-Z)
DIMENSION A(*),B(*)
DIMENSION NFAC(11),NP(209)
C ARRAY STORAGE FOR MAXIMUM PRIME FACTOR OF 23
DIMENSION AT(23),CK(23),BT(23),SK(23)
EQUIVALENCE (I,II)
C THE FOLLOWING TWO CONSTANTS SHOULD AGREE WITH THE ARRAY DIMENSIONS.
MAXF=23
MAXP=209
IERR=0
IF(N .LT. 2) RETURN
INC=ISN
C72=0.30901699437494D0
S72=0.95105651629515D0
S120=0.86602540378443D0
RAD=6.2831853071796D0
IF(ISN .GE. 0) GO TO 10
S72=-S72
S120=-S120
RAD=-RAD
INC=-INC
10 NT=INC*NTOT
KS=INC*NSPAN
KSPAN=KS
NN=NT-INC
JC=KS/N
RADF=RAD*(JC*0.5D0)
I=0
JF=0
C DETERMINE THE FACTORS OF N
M=0
K=N
GO TO 20
15 M=M+1
NFAC(M)=4
K=K/16
20 IF(K-(K/16)*16 .EQ. 0) GO TO 15
J=3
JJ=9
GO TO 30
25 M=M+1
NFAC(M)=J
K=K/JJ
30 IF(MOD(K,JJ) .EQ. 0) GO TO 25
J=J+2
JJ=J**2
IF(JJ .LE. K) GO TO 30
IF(K .GT. 4) GO TO 40
KT=M
NFAC(M+1)=K
IF(K .NE. 1) M=M+1
GO TO 80
40 IF(K-(K/4)*4 .NE. 0) GO TO 50
M=M+1
NFAC(M)=2
K=K/4
50 KT=M
J=2
60 IF(MOD(K,J) .NE. 0) GO TO 70
M=M+1
NFAC(M)=J
K=K/J
70 J=((J+1)/2)*2+1
IF(J .LE. K) GO TO 60
80 IF(KT .EQ. 0) GO TO 100
J=KT
90 M=M+1
NFAC(M)=NFAC(J)
J=J-1
IF(J .NE. 0) GO TO 90
C COMPUTE FOURIER TRANSFORM
100 SD=RADF/(KSPAN)
CD=2.D0*DSIN(SD)**2
SD=DSIN(SD+SD)
KK=1
I=I+1
IF(NFAC(I) .NE. 2) GO TO 400
C TRANSFORM FOR FACTOR OF 2 (INCLUDING ROTATION FACTOR)
KSPAN=KSPAN/2
K1=KSPAN+2
210 K2=KK+KSPAN
AK=A(K2)
BK=B(K2)
A(K2)=A(KK)-AK
B(K2)=B(KK)-BK
A(KK)=A(KK)+AK
B(KK)=B(KK)+BK
KK=K2+KSPAN
IF(KK .LE. NN) GO TO 210
KK=KK-NN
IF(KK .LE. JC) GO TO 210
IF(KK .GT. KSPAN) GO TO 800
220 C1=1.D0-CD
S1=SD
230 K2=KK+KSPAN
AK=A(KK)-A(K2)
BK=B(KK)-B(K2)
A(KK)=A(KK)+A(K2)
B(KK)=B(KK)+B(K2)
A(K2)=C1*AK-S1*BK
B(K2)=S1*AK+C1*BK
KK=K2+KSPAN
IF(KK .LT. NT) GO TO 230
K2=KK-NT
C1=-C1
KK=K1-K2
IF(KK .GT. K2) GO TO 230
AK=CD*C1+SD*S1
S1=(SD*C1-CD*S1)+S1
C1=C1-AK
KK=KK+JC
IF(KK .LT. K2) GO TO 230
K1=K1+INC+INC
KK=(K1-KSPAN)/2+JC
IF(KK .LE. JC+JC) GO TO 220
GO TO 100
C TRANSFORM FOR FACTOR OF 3 (OPTIONAL CODE)
320 K1=KK+KSPAN
K2=K1+KSPAN
AK=A(KK)
BK=B(KK)
AJ=A(K1)+A(K2)
BJ=B(K1)+B(K2)
A(KK)=AK+AJ
B(KK)=BK+BJ
AK=-0.5*AJ+AK
BK=-0.5*BJ+BK
AJ=(A(K1)-A(K2))*S120
BJ=(B(K1)-B(K2))*S120
A(K1)=AK-BJ
B(K1)=BK+AJ
A(K2)=AK+BJ
B(K2)=BK-AJ
KK=K2+KSPAN
IF(KK .LT. NN) GO TO 320
KK=KK-NN
IF(KK .LE. KSPAN) GO TO 320
GO TO 700
C TRANSFORM FOR FACTOR OF 4
400 IF(NFAC(I) .NE. 4) GO TO 600
KSPNN=KSPAN
KSPAN=KSPAN/4
410 C1=1.D0
S1=0
420 K1=KK+KSPAN
K2=K1+KSPAN
K3=K2+KSPAN
AKP=A(KK)+A(K2)
AKM=A(KK)-A(K2)
AJP=A(K1)+A(K3)
AJM=A(K1)-A(K3)
A(KK)=AKP+AJP
AJP=AKP-AJP
BKP=B(KK)+B(K2)
BKM=B(KK)-B(K2)
BJP=B(K1)+B(K3)
BJM=B(K1)-B(K3)
B(KK)=BKP+BJP
BJP=BKP-BJP
IF(ISN .LT. 0) GO TO 450
AKP=AKM-BJM
AKM=AKM+BJM
BKP=BKM+AJM
BKM=BKM-AJM
IF(S1 .EQ. 0) GO TO 460
430 A(K1)=AKP*C1-BKP*S1
B(K1)=AKP*S1+BKP*C1
A(K2)=AJP*C2-BJP*S2
B(K2)=AJP*S2+BJP*C2
A(K3)=AKM*C3-BKM*S3
B(K3)=AKM*S3+BKM*C3
KK=K3+KSPAN
IF(KK .LE. NT) GO TO 420
440 C2=CD*C1+SD*S1
S1=(SD*C1-CD*S1)+S1
C1=C1-C2
C2=C1**2-S1**2
S2=2.D0*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
450 AKP=AKM+BJM
AKM=AKM-BJM
BKP=BKM-AJM
BKM=BKM+AJM
IF(S1 .NE. 0) GO TO 430
460 A(K1)=AKP
B(K1)=BKP
A(K2)=AJP
B(K2)=BJP
A(K3)=AKM
B(K3)=BKM
KK=K3+KSPAN
IF(KK .LE. NT) GO TO 420
GO TO 440
C TRANSFORM FOR FACTOR OF 5 (OPTIONAL CODE)
510 C2=C72**2-S72**2
S2=2.D0*C72*S72
520 K1=KK+KSPAN
K2=K1+KSPAN
K3=K2+KSPAN
K4=K3+KSPAN
AKP=A(K1)+A(K4)
AKM=A(K1)-A(K4)
BKP=B(K1)+B(K4)
BKM=B(K1)-B(K4)
AJP=A(K2)+A(K3)
AJM=A(K2)-A(K3)
BJP=B(K2)+B(K3)
BJM=B(K2)-B(K3)
AA=A(KK)
BB=B(KK)
A(KK)=AA+AKP+AJP
B(KK)=BB+BKP+BJP
AK=AKP*C72+AJP*C2+AA
BK=BKP*C72+BJP*C2+BB
AJ=AKM*S72+AJM*S2
BJ=BKM*S72+BJM*S2
A(K1)=AK-BJ
A(K4)=AK+BJ
B(K1)=BK+AJ
B(K4)=BK-AJ
AK=AKP*C2+AJP*C72+AA
BK=BKP*C2+BJP*C72+BB