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r@na sl ?` + @ a@ + 74s0T + 7 aT + 7: ^4}.t 44 V 4R /1R + ^d 1R + `1R + b02 0r e + ih i@ + Q<j WR SUBROUTINE FOUR2 (DATA,N,NDIM,ISIGN,IFORM)
C COOLEY-TUKEY FAST FOURIER TRANSFORM IN USASI BASIC FORTRAN.
C MULTI-DIMENSIONAL TRANSFORM, EACH DIMENSION A POWER OF TWO,
C COMPLEX OR REAL DATA.
C TRANSFORM(K1,K2,...) = SUM(DATA(J1,J2,...)*EXP(ISIGN*2*PI*SQRT(-1)
C *((J1-1)*(K1-1)/N(1)+(J2-1)*(K2-1)/N(2)+...))), SUMMED FOR ALL
C J1 AND K1 FROM 1 TO N(1), J2 AND K2 FROM 1 TO N(2),
C ETC. FOR ALL NDIM SUBSCRIPTS. NDIM MUST BE POSITIVE AND
C EACH N(IDIM) MUST BE A POWER OF TWO. ISIGN IS +1 OR -1.
C LET NTOT = N(1)*N(2)*...*N(NDIM). THEN A -1 TRANSFORM
C FOLLOWED BY A +1 ONE (OR VICE VERSA) RETURNS NTOT
C TIMES THE ORIGINAL DATA. IFORM = 1, 0 OR -1, AS DATA IS
C COMPLEX, REAL OR THE FIRST HALF OF A COMPLEX ARRAY. TRANSFORM
C VALUES ARE RETURNED TO ARRAY DATA. THEY ARE COMPLEX, REAL OR
C THE FIRST HALF OF A COMPLEX ARRAY, AS IFORM = 1, -1 OR 0.
C THE TRANSFORM OF A REAL ARRAY (IFORM = 0) DIMENSIONED N(1) BY N(2)
C BY ... WILL BE RETURNED IN THE SAME ARRAY, NOW CONSIDERED TO
C BE COMPLEX OF DIMENSIONS N(1)/2+1 BY N(2) BY .... NOTE THAT IF
C IFORM = 0 OR -1, N(1) MUST BE EVEN, AND ENOUGH ROOM MUST BE
C RESERVED. THE MISSING VALUES MAY BE OBTAINED BY COMPLEX CONJUGA-
C TION. THE REVERSE TRANSFORMATION, OF A HALF COMPLEX ARRAY DIMEN-
C SIONED N(1)/2+1 BY N(2) BY ..., IS ACCOMPLISHED BY SETTING IFORM
C TO -1. IN THE N ARRAY, N(1) MUST BE THE TRUE N(1), NOT N(1)/2+1.
C THE TRANSFORM WILL BE REAL AND RETURNED TO THE INPUT ARRAY.
C RUNNING TIME IS PROPORTIONAL TO NTOT*LOG2(NTOT), RATHER THAN
C THE NAIVE NTOT**2. FURTHERMORE, LESS ERROR IS BUILT UP.
C WRITTEN BY NORMAN BRENNER OF MIT LINCOLN LABORATORY, JANUARY 1969.
C SEE-- IEEE AUDIO TRANSACTIONS (JUNE 1967), SPECIAL ISSUE ON FFT.
DIMENSION DATA(1), N(1)
NTOT=1
DO 10 IDIM=1,NDIM
10 NTOT=NTOT*N(IDIM)
IF (IFORM) 70,20,20
20 NREM=NTOT
DO 60 IDIM=1,NDIM
NREM=NREM/N(IDIM)
NPREV=NTOT/(N(IDIM)*NREM)
NCURR=N(IDIM)
IF (IDIM-1+IFORM) 30,30,40
30 NCURR=NCURR/2
40 CALL BITRV (DATA,NPREV,NCURR,NREM)
CALL COOL2 (DATA,NPREV,NCURR,NREM,ISIGN)
IF (IDIM-1+IFORM) 50,50,60
50 CALL FIXRL (DATA,N(1),NREM,ISIGN,IFORM)
NTOT=(NTOT/N(1))*(N(1)/2+1)
60 CONTINUE
RETURN
70 NTOT=(NTOT/N(1))*(N(1)/2+1)
NREM=1
DO 100 JDIM=1,NDIM
IDIM=NDIM+1-JDIM
NCURR=N(IDIM)
IF (IDIM-1) 80,80,90
80 NCURR=NCURR/2
CALL FIXRL (DATA,N(1),NREM,ISIGN,IFORM)
NTOT=NTOT/(N(1)/2+1)*N(1)
90 NPREV=NTOT/(N(IDIM)*NREM)
CALL BITRV (DATA,NPREV,NCURR,NREM)
CALL COOL2 (DATA,NPREV,NCURR,NREM,ISIGN)
100 NREM=NREM*N(IDIM)
RETURN
END
SUBROUTINE BITRV (DATA,NPREV,N,NREM)
C SHUFFLE THE DATA BY BIT REVERSAL.
C DIMENSION DATA(NPREV,N,NREM)
C COMPLEX DATA
C EXCHANGE DATA(J1,J4REV,J5) WITH DATA(J1,J4,J5) FOR ALL J1 FROM 1
C TO NPREV, ALL J4 FROM 1 TO N (WHICH MUST BE A POWER OF TWO), AND
C ALL J5 FROM 1 TO NREM. J4REV-1 IS THE BIT REVERSAL OF J4-1. E.G.
C SUPPOSE N = 32. THEN FOR J4-1 = 10011, J4REV-1 = 11001, ETC.
DIMENSION DATA(1)
IP0=2
IP1=IP0*NPREV
IP4=IP1*N
IP5=IP4*NREM
I4REV=1
C I4REV = 1+(J4REV-1)*IP1
DO 60 I4=1,IP4,IP1
C I4 = 1+(J4-1)*IP1
IF (I4-I4REV) 10,30,30
10 I1MAX=I4+IP1-IP0
DO 20 I1=I4,I1MAX,IP0
C I1 = 1+(J1-1)*IP0+(J4-1)*IP1
DO 20 I5=I1,IP5,IP4
C I5 = 1+(J1-1)*IP0+(J4-1)*IP1+(J5-1)*IP4
I5REV=I4REV+I5-I4
C I5REV = 1+(J1-1)*IP0+(J4REV-1)*IP1+(J5-1)*IP4
TEMPR=DATA(I5)
TEMPI=DATA(I5+1)
DATA(I5)=DATA(I5REV)
DATA(I5+1)=DATA(I5REV+1)
DATA(I5REV)=TEMPR
20 DATA(I5REV+1)=TEMPI
C ADD ONE WITH DOWNWARD CARRY TO THE HIGH ORDER BIT OF J4REV-1.
30 IP2=IP4/2
40 IF (I4REV-IP2) 60,60,50
50 I4REV=I4REV-IP2
IP2=IP2/2
IF (IP2-IP1) 60,40,40
60 I4REV=I4REV+IP2
RETURN
END
SUBROUTINE COOL2 (DATA,NPREV,N,NREM,ISIGN)
C DISCRETE FOURIER TRANSFORM OF LENGTH N. IN-PLACE COOLEY-TUKEY
C ALGORITHM, BIT-REVERSED TO NORMAL ORDER, SANDE-TUKEY PHASE SHIFTS.
C DIMENSION DATA(NPREV,N,NREM)
C COMPLEX DATA
C DATA(J1,K4,J5) = SUM(DATA(J1,J4,J5)*EXP(ISIGN*2*PI*I*(J4-1)*
C (K4-1)/N)), SUMMED OVER J4 = 1 TO N FOR ALL J1 FROM 1 TO NPREV,
C K4 FROM 1 TO N AND J5 FROM 1 TO NREM. N MUST BE A POWER OF TWO.
C METHOD--LET IPREV TAKE THE VALUES 1, 2 OR 4, 4 OR 8, ..., N/16,
C N/4, N. THE CHOICE BETWEEN 2 OR 4, ETC., DEPENDS ON WHETHER N IS
C A POWER OF FOUR. DEFINE IFACT = 2 OR 4, THE NEXT FACTOR THAT
C IPREV MUST TAKE, AND IREM = N/(IFACT*IPREV). THEN--
C DIMENSION DATA(NPREV,IPREV,IFACT,IREM,NREM)
C COMPLEX DATA
C DATA(J1,J2,K3,J4,J5) = SUM(DATA(J1,J2,J3,J4,J5)*EXP(ISIGN*2*PI*I*
C (K3-1)*((J3-1)/IFACT+(J2-1)/(IFACT*IPREV)))), SUMMED OVER J3 = 1
C TO IFACT FOR ALL J1 FROM 1 TO NPREV, J2 FROM 1 TO IPREV, K3 FROM
C 1 TO IFACT, J4 FROM 1 TO IREM AND J5 FROM 1 TO NREM. THIS IS
C A PHASE-SHIFTED DISCRETE FOURIER TRANSFORM OF LENGTH IFACT.
C FACTORING N BY FOURS SAVES ABOUT TWENTY FIVE PERCENT OVER FACTOR-
C ING BY TWOS. DATA MUST BE BIT-REVERSED INITIALLY.
C IT IS NOT NECESSARY TO REWRITE THIS SUBROUTINE INTO COMPLEX
C NOTATION SO LONG AS THE FORTRAN COMPILER USED STORES REAL AND
C IMAGINARY PARTS IN ADJACENT STORAGE LOCATIONS. IT MUST ALSO
C STORE ARRAYS WITH THE FIRST SUBSCRIPT INCREASING FASTEST.
DIMENSION DATA(1)
TWOPI=6.2831853072*FLOAT(ISIGN)
IP0=2
IP1=IP0*NPREV
IP4=IP1*N
IP5=IP4*NREM
IP2=IP1
C IP2=IP1*IPROD
NPART=N
10 IF (NPART-2) 60,30,20
20 NPART=NPART/4
GO TO 10
C DO A FOURIER TRANSFORM OF LENGTH TWO
30 IF (IP2-IP4) 40,160,160
40 IP3=IP2*2
C IP3=IP2*IFACT
DO 50 I1=1,IP1,IP0
C I1 = 1+(J1-1)*IP0
DO 50 I5=I1,IP5,IP3
C I5 = 1+(J1-1)*IP0+(J4-1)*IP3+(J5-1)*IP4
I3A=I5
I3B=I3A+IP2
C I3 = 1+(J1-1)*IP0+(J2-1)*IP1+(J3-1)*IP2+(J4-1)*IP3+(J5-1)*IP4
TEMPR=DATA(I3B)
TEMPI=DATA(I3B+1)
DATA(I3B)=DATA(I3A)-TEMPR
DATA(I3B+1)=DATA(I3A+1)-TEMPI
DATA(I3A)=DATA(I3A)+TEMPR
50 DATA(I3A+1)=DATA(I3A+1)+TEMPI
IP2=IP3
C DO A FOURIER TRANSFORM OF LENGTH FOUR (FROM BIT REVERSED ORDER)
60 IF (IP2-IP4) 70,160,160
70 IP3=IP2*4
C IP3=IP2*IFACT
C COMPUTE TWOPI THRU WR AND WI IN DOUBLE PRECISION, IF AVAILABLE.
THETA=TWOPI/FLOAT(IP3/IP1)
SINTH=SIN(THETA/2.)
WSTPR=-2.*SINTH*SINTH
WSTPI=SIN(THETA)
WR=1.
WI=0.
DO 150 I2=1,IP2,IP1
C I2 = 1+(J2-1)*IP1
IF (I2-1) 90,90,80
80 W2R=WR*WR-WI*WI
W2I=2.*WR*WI
W3R=W2R*WR-W2I*WI
W3I=W2R*WI+W2I*WR
90 I1MAX=I2+IP1-IP0
DO 140 I1=I2,I1MAX,IP0
C I1 = 1+(J1-1)*IP0+(J2-1)*IP1
DO 140 I5=I1,IP5,IP3
C I5 = 1+(J1-1)*IP0+(J2-1)*IP1+(J4-1)*IP3+(J5-1)*IP4
I3A=I5
I3B=I3A+IP2
I3C=I3B+IP2
I3D=I3C+IP2
C I3 = 1+(J1-1)*IP0+(J2-1)*IP1+(J3-1)*IP2+(J4-1)*IP3+(J5-1)*IP4
IF (I2-1) 110,110,100
C APPLY THE PHASE SHIFT FACTORS
100 TEMPR=DATA(I3B)
DATA(I3B)=W2R*DATA(I3B)-W2I*DATA(I3B+1)
DATA(I3B+1)=W2R*DATA(I3B+1)+W2I*TEMPR
TEMPR=DATA(I3C)
DATA(I3C)=WR*DATA(I3C)-WI*DATA(I3C+1)
DATA(I3C+1)=WR*DATA(I3C+1)+WI*TEMPR
TEMPR=DATA(I3D)
DATA(I3D)=W3R*DATA(I3D)-W3I*DATA(I3D+1)
DATA(I3D+1)=W3R*DATA(I3D+1)+W3I*TEMPR
110 T0R=DATA(I3A)+DATA(I3B)
T0I=DATA(I3A+1)+DATA(I3B+1)
T1R=DATA(I3A)-DATA(I3B)
T1I=DATA(I3A+1)-DATA(I3B+1)
T2R=DATA(I3C)+DATA(I3D)
T2I=DATA(I3C+1)+DATA(I3D+1)
T3R=DATA(I3C)-DATA(I3D)
T3I=DATA(I3C+1)-DATA(I3D+1)
DATA(I3A)=T0R+T2R
DATA(I3A+1)=T0I+T2I
DATA(I3C)=T0R-T2R
DATA(I3C+1)=T0I-T2I
IF (ISIGN) 120,120,130
120 T3R=-T3R
T3I=-T3I
130 DATA(I3B)=T1R-T3I
DATA(I3B+1)=T1I+T3R
DATA(I3D)=T1R+T3I
140 DATA(I3D+1)=T1I-T3R
TEMPR=WR
WR=WSTPR*TEMPR-WSTPI*WI+TEMPR
150 WI=WSTPR*WI+WSTPI*TEMPR+WI
IP2=IP3
GO TO 60
160 RETURN
END
SUBROUTINE FIXRL (DATA,N,NREM,ISIGN,IFORM)
C FOR IFORM = 0, CONVERT THE TRANSFORM OF A DOUBLED-UP REAL ARRAY,
C CONSIDERED COMPLEX, INTO ITS TRUE TRANSFORM. SUPPLY ONLY THE
C FIRST HALF OF THE COMPLEX TRANSFORM, AS THE SECOND HALF HAS
C CONJUGATE SYMMETRY. FOR IFORM = -1, CONVERT THE FIRST HALF
C OF THE TRUE TRANSFORM INTO THE TRANSFORM OF A DOUBLED-UP REAL
C ARRAY. N MUST BE EVEN.
C USING COMPLEX NOTATION AND SUBSCRIPTS STARTING AT ZERO, THE
C TRANSFORMATION IS--
C DIMENSION DATA(N,NREM)
C ZSTP = EXP(ISIGN*2*PI*I/N)
C DO 10 I2=0,NREM-1
C DATA(0,I2) = CONJ(DATA(0,I2))*(1+I)
C DO 10 I1=1,N/4
C Z = (1+(2*IFORM+1)*I*ZSTP**I1)/2
C I1CNJ = N/2-I1
C DIF = DATA(I1,I2)-CONJ(DATA(I1CNJ,I2))
C TEMP = Z*DIF
C DATA(I1,I2) = (DATA(I1,I2)-TEMP)*(1-IFORM)
C 10 DATA(I1CNJ,I2) = (DATA(I1CNJ,I2)+CONJ(TEMP))*(1-IFORM)
C IF I1=I1CNJ, THE CALCULATION FOR THAT VALUE COLLAPSES INTO
C A SIMPLE CONJUGATION OF DATA(I1,I2).
DIMENSION DATA(1)
TWOPI=6.283185307*FLOAT(ISIGN)
IP0=2
IP1=IP0*(N/2)
IP2=IP1*NREM
IF (IFORM) 10,70,70
C PACK THE REAL INPUT VALUES (TWO PER COLUMN)
10 J1=IP1+1
DATA(2)=DATA(J1)
IF (NREM-1) 70,70,20
20 J1=J1+IP0
I2MIN=IP1+1
DO 60 I2=I2MIN,IP2,IP1
DATA(I2)=DATA(J1)
J1=J1+IP0
IF (N-2) 50,50,30
30 I1MIN=I2+IP0
I1MAX=I2+IP1-IP0
DO 40 I1=I1MIN,I1MAX,IP0
DATA(I1)=DATA(J1)
DATA(I1+1)=DATA(J1+1)
40 J1=J1+IP0
50 DATA(I2+1)=DATA(J1)
60 J1=J1+IP0
70 DO 80 I2=1,IP2,IP1
TEMPR=DATA(I2)
DATA(I2)=DATA(I2)+DATA(I2+1)
80 DATA(I2+1)=TEMPR-DATA(I2+1)
IF (N-2) 200,200,90
90 THETA=TWOPI/FLOAT(N)
SINTH=SIN(THETA/2.)
ZSTPR=-2.*SINTH*SINTH
ZSTPI=SIN(THETA)
ZR=(1.-ZSTPI)/2.
ZI=(1.+ZSTPR)/2.
IF (IFORM) 100,110,110
100 ZR=1.-ZR
ZI=-ZI
110 I1MIN=IP0+1
I1MAX=IP0*(N/4)+1
DO 190 I1=I1MIN,I1MAX,IP0
DO 180 I2=I1,IP2,IP1
I2CNJ=IP0*(N/2+1)-2*I1+I2
IF (I2-I2CNJ) 150,120,120
120 IF (ISIGN*(2*IFORM+1)) 130,140,140
130 DATA(I2+1)=-DATA(I2+1)
140 IF (IFORM) 170,180,180
150 DIFR=DATA(I2)-DATA(I2CNJ)
DIFI=DATA(I2+1)+DATA(I2CNJ+1)
TEMPR=DIFR*ZR-DIFI*ZI
TEMPI=DIFR*ZI+DIFI*ZR
DATA(I2)=DATA(I2)-TEMPR
DATA(I2+1)=DATA(I2+1)-TEMPI
DATA(I2CNJ)=DATA(I2CNJ)+TEMPR
DATA(I2CNJ+1)=DATA(I2CNJ+1)-TEMPI
IF (IFORM) 160,180,180
160 DATA(I2CNJ)=DATA(I2CNJ)+DATA(I2CNJ)
DATA(I2CNJ+1)=DATA(I2CNJ+1)+DATA(I2CNJ+1)
170 DATA(I2)=DATA(I2)+DATA(I2)
DATA(I2+1)=DATA(I2+1)+DATA(I2+1)
180 CONTINUE
TEMPR=ZR-.5
ZR=ZSTPR*TEMPR-ZSTPI*ZI+ZR
190 ZI=ZSTPR*ZI+ZSTPI*TEMPR+ZI
C RECURSION SAVES TIME, AT A SLIGHT LOSS IN ACCURACY. IF AVAILABLE,
C USE DOUBLE PRECISION TO COMPUTE ZR AND ZI.
200 IF (IFORM) 270,210,210
C UNPACK THE REAL TRANSFORM VALUES (TWO PER COLUMN)
210 I2=IP2+1
I1=I2
J1=IP0*(N/2+1)*NREM+1
GO TO 250
220 DATA(J1)=DATA(I1)
DATA(J1+1)=DATA(I1+1)
I1=I1-IP0
J1=J1-IP0
230 IF (I2-I1) 220,240,240
240 DATA(J1)=DATA(I1)
DATA(J1+1)=0.
250 I2=I2-IP1
J1=J1-IP0
DATA(J1)=DATA(I2+1)
DATA(J1+1)=0.
I1=I1-IP0
J1=J1-IP0
IF (I2-1) 260,260,230
260 DATA(2)=0.
270 RETURN
END