Time Efficient Radix-4 Fast Fourier Transform
Return to Main Software Page
C
C-----------------------------------------------------------------------
C MAIN PROGRAM: TIME-EFFICIENT RADIX-4 FAST FOURIER TRANSFORM
C AUTHOR: L. ROBERT MORRIS
C DEPARTMENT OF SYSTEMS ENGINEERING AND COMPUTING SCIENCE
C CARLETON UNIVERSITY, OTTAWA, CANADA K1S 5B6
C
C INPUT: THE ARRAY "A" CONTAINS THE DATA TO BE TRANSFORMED
C-----------------------------------------------------------------------
C
C TEST PROGRAM FOR AUTOGEN RADIX-4 FFT
C
DIMENSION A(2048),B(2048)
COMMON /AA/A
C
IOUTD=I1MACH(2)
C
C COMPUTE DFT AND IDFT FOR N = 64, 256, AND 1024 COMPLEX POINTS
C
DO 1 MM=3,5
N=4**MM
DO 2 J=1,N
A(2*J-1)=UNI(0)
A(2*J )=UNI(0)
B(2*J-1)=A(2*J-1)
2 B(2*J )=A(2*J)
C
C FORWARD DFT
C
CALL RADIX4(MM,1,-1)
C
IF(MM.NE.3) GO TO 5
C
C LIST DFT INPUT, OUTPUT FOR N = 64 ONLY
C
WRITE(IOUTD,98)
WRITE(IOUTD,100)
DO 3 J=1,N
WRITE(IOUTD,96) B(2*J-1),B(2*J),A(2*J-1),A(2*J)
3 CONTINUE
C
C INVERSE DFT
C
5 CALL RADIX4(MM,0, 1)
C
C LIST DFT INPUT, IDFT OUTPUT FOR N = 64 ONLY
C
IF(MM.NE.3) GO TO 7
C
WRITE(IOUTD,99)
WRITE(IOUTD,100)
DO 6 J=1,N
WRITE(IOUTD,96) B(2*J-1),B(2*J),A(2*J-1),A(2*J)
6 CONTINUE
C
C CALCULATE RMS ERROR
C
7 ERR=0.0
DO 8 J=1,N
8 ERR=ERR+(A(2*J-1)-B(2*J-1))**2+(A(2*J)-B(2*J))**2
ERR=SQRT(ERR/FLOAT(N))
WRITE(IOUTD,97) MM,ERR
1 CONTINUE
C
96 FORMAT(1X,4(F10.6,2X))
97 FORMAT(1X,20H RMS ERROR FOR M =,I2,4H IS ,E14.6/)
98 FORMAT(1X,43H DFT INPUT DFT OUTPUT/)
99 FORMAT(1X,43H DFT INPUT IDFT OUTPUT/)
100 FORMAT(1X,44H REAL IMAG REAL IMAG/)
STOP
END
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
C
C-----------------------------------------------------------------------
C SUBROUTINE: RAD4SB
C USED BY SUBROUTINE RADIX4. NEVER DIRECTLY ACCESSED BY USER.
C-----------------------------------------------------------------------
C
SUBROUTINE RAD4SB(NTYPE)
C
C INPUT: NTYPE = TYPE OF BUTTERFLY INVOKED
C OUTPUT: PARAMETERS USED BY SUBROUTINE RADIX4
C
DIMENSION IX(2996)
COMMON /XX/IX
COMMON NTYPL,KKP,INDEX,IXC
IF(NTYPE.EQ.NTYPL) GO TO 7
IX(IXC)=0
IX(IXC+1)=NTYPE
IXC=IXC+2
IF(NTYPE.NE.4) GO TO 4
INDEXP=(INDEX-1)*9
IX(IXC)=KKP+1
IX(IXC+1)=INDEXP+1
IXC=IXC+2
GO TO 6
4 IX(IXC)=KKP+1
IXC=IXC+1
6 NTYPL=NTYPE
RETURN
7 IF(NTYPE.NE.4) GO TO 8
INDEXP=(INDEX-1)*9
IX(IXC)=KKP+1
IX(IXC+1)=INDEXP+1
IXC=IXC+2
RETURN
8 IX(IXC)=KKP+1
IXC=IXC+1
RETURN
END
Return to Main Software Page