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|
c
c-----------------------------------------------------------------------
c subroutine: ffa
c fast fourier analysis subroutine
c-----------------------------------------------------------------------
c
subroutine ffa (b, nfft, ier)
c
c this subroutine replaces the real vector b(k), (k=1,2,...,n),
c with its finite discrete fourier transform. the dc term is
c returned in location b(1) with b(2) set to 0. thereafter, the
c jth harmonic is returned as a complex number stored as
c b(2*j+1) + i b(2*j+2). note that the n/2 harmonic is returned
c in b(n+1) with b(n+2) set to 0. hence, b must be dimensioned
c to size n+2.
c subroutine is called as ffa (b,n) where n=2**m and b is an
c n term real array. a real-valued, radix 8 algorithm is used
c with in-place reordering and the trig functions are computed as
c needed.
c
dimension b(2)
common /con/ pii, p7, p7two, c22, s22, pi2
c
c iw is a machine dependent write device number
c
c+noao
c iw = i1mach(2)
ier = 0
c-noao
c
pii = 4.*atan(1.)
pi8 = pii/8.
p7 = 1./sqrt(2.)
p7two = 2.*p7
c22 = cos(pi8)
s22 = sin(pi8)
pi2 = 2.*pii
n = 1
do 10 i=1,31
m = i
n = n*2
if (n.eq.nfft) go to 20
10 continue
c+noao
c write (iw,9999)
c9999 format (30h nfft not a power of 2 for ffa)
c stop
ier = 1
return
c-noao
20 continue
n8pow = m/3
c
c do a radix 2 or radix 4 iteration first if one is required
c
if (m-n8pow*3-1) 50, 40, 30
30 nn = 4
int = n/nn
call r4tr(int, b(1), b(int+1), b(2*int+1), b(3*int+1))
go to 60
40 nn = 2
int = n/nn
call r2tr(int, b(1), b(int+1))
go to 60
50 nn = 1
c
c perform radix 8 iterations
c
60 if (n8pow) 90, 90, 70
70 do 80 it=1,n8pow
nn = nn*8
int = n/nn
call r8tr(int, nn, b(1), b(int+1), b(2*int+1), b(3*int+1),
* b(4*int+1), b(5*int+1), b(6*int+1), b(7*int+1), b(1),
* b(int+1), b(2*int+1), b(3*int+1), b(4*int+1), b(5*int+1),
* b(6*int+1), b(7*int+1))
80 continue
c
c perform in-place reordering
c
90 call ord1(m, b)
call ord2(m, b)
t = b(2)
b(2) = 0.
b(nfft+1) = t
b(nfft+2) = 0.
do 100 i=4,nfft,2
b(i) = -b(i)
100 continue
return
end
c
c-----------------------------------------------------------------------
c subroutine: ffs
c fast fourier synthesis subroutine
c radix 8-4-2
c-----------------------------------------------------------------------
c
subroutine ffs (b, nfft, ier)
c
c this subroutine synthesizes the real vector b(k), where
c k=1,2,...,n. the initial fourier coefficients are placed in
c the b array of size n+2. the dc term is in b(1) with
c b(2) equal to 0.
c the jth harmonic is stored as b(2*j+1) + i b(2*j+2).
c the n/2 harmonic is in b(n+1) with b(n+2) equal to 0.
c the subroutine is called as ffs(b,n) where n=2**m and
c b is the n term real array discussed above.
c
dimension b(2)
common /con1/ pii, p7, p7two, c22, s22, pi2
c
c iw is a machine dependent write device number
c
c+noao
c iw = i1mach(2)
ier = 0
c-noao
c
pii = 4.*atan(1.)
pi8 = pii/8.
p7 = 1./sqrt(2.)
p7two = 2.*p7
c22 = cos(pi8)
s22 = sin(pi8)
pi2 = 2.*pii
n = 1
do 10 i=1,31
m = i
n = n*2
if (n.eq.nfft) go to 20
10 continue
c+noao
c write (iw,9999)
c9999 format (30h nfft not a power of 2 for ffs)
c stop
ier = 1
return
c-noao
20 continue
b(2) = b(nfft+1)
do 30 i=1,nfft
b(i) = b(i)/float(nfft)
30 continue
do 40 i=4,nfft,2
b(i) = -b(i)
40 continue
n8pow = m/3
c
c reorder the input fourier coefficients
c
call ord2(m, b)
call ord1(m, b)
c
if (n8pow.eq.0) go to 60
c
c perform the radix 8 iterations
c
nn = n
do 50 it=1,n8pow
int = n/nn
call r8syn(int, nn, b, b(int+1), b(2*int+1), b(3*int+1),
* b(4*int+1), b(5*int+1), b(6*int+1), b(7*int+1), b(1),
* b(int+1), b(2*int+1), b(3*int+1), b(4*int+1), b(5*int+1),
* b(6*int+1), b(7*int+1))
nn = nn/8
50 continue
c
c do a radix 2 or radix 4 iteration if one is required
c
60 if (m-n8pow*3-1) 90, 80, 70
70 int = n/4
call r4syn(int, b(1), b(int+1), b(2*int+1), b(3*int+1))
go to 90
80 int = n/2
call r2tr(int, b(1), b(int+1))
90 return
end
c
c-----------------------------------------------------------------------
c subroutine: r2tr
c radix 2 iteration subroutine
c-----------------------------------------------------------------------
c
c
subroutine r2tr(int, b0, b1)
dimension b0(2), b1(2)
do 10 k=1,int
t = b0(k) + b1(k)
b1(k) = b0(k) - b1(k)
b0(k) = t
10 continue
return
end
c
c-----------------------------------------------------------------------
c subroutine: r4tr
c radix 4 iteration subroutine
c-----------------------------------------------------------------------
c
subroutine r4tr(int, b0, b1, b2, b3)
dimension b0(2), b1(2), b2(2), b3(2)
do 10 k=1,int
r0 = b0(k) + b2(k)
r1 = b1(k) + b3(k)
b2(k) = b0(k) - b2(k)
b3(k) = b1(k) - b3(k)
b0(k) = r0 + r1
b1(k) = r0 - r1
10 continue
return
end
c
c-----------------------------------------------------------------------
c subroutine: r8tr
c radix 8 iteration subroutine
c-----------------------------------------------------------------------
c
subroutine r8tr(int, nn, br0, br1, br2, br3, br4, br5, br6, br7,
* bi0, bi1, bi2, bi3, bi4, bi5, bi6, bi7)
dimension l(15), br0(2), br1(2), br2(2), br3(2), br4(2), br5(2),
* br6(2), br7(2), bi0(2), bi1(2), bi2(2), bi3(2), bi4(2),
* bi5(2), bi6(2), bi7(2)
common /con/ pii, p7, p7two, c22, s22, pi2
equivalence (l15,l(1)), (l14,l(2)), (l13,l(3)), (l12,l(4)),
* (l11,l(5)), (l10,l(6)), (l9,l(7)), (l8,l(8)), (l7,l(9)),
* (l6,l(10)), (l5,l(11)), (l4,l(12)), (l3,l(13)), (l2,l(14)),
* (l1,l(15))
c
c set up counters such that jthet steps through the arguments
c of w, jr steps through starting locations for the real part of the
c intermediate results and ji steps through starting locations
c of the imaginary part of the intermediate results.
c
l(1) = nn/8
do 40 k=2,15
if (l(k-1)-2) 10, 20, 30
10 l(k-1) = 2
20 l(k) = 2
go to 40
30 l(k) = l(k-1)/2
40 continue
piovn = pii/float(nn)
ji = 3
jl = 2
jr = 2
do 120 j1=2,l1,2
do 120 j2=j1,l2,l1
do 120 j3=j2,l3,l2
do 120 j4=j3,l4,l3
do 120 j5=j4,l5,l4
do 120 j6=j5,l6,l5
do 120 j7=j6,l7,l6
do 120 j8=j7,l8,l7
do 120 j9=j8,l9,l8
do 120 j10=j9,l10,l9
do 120 j11=j10,l11,l10
do 120 j12=j11,l12,l11
do 120 j13=j12,l13,l12
do 120 j14=j13,l14,l13
do 120 jthet=j14,l15,l14
th2 = jthet - 2
if (th2) 50, 50, 90
50 do 60 k=1,int
t0 = br0(k) + br4(k)
t1 = br1(k) + br5(k)
t2 = br2(k) + br6(k)
t3 = br3(k) + br7(k)
t4 = br0(k) - br4(k)
t5 = br1(k) - br5(k)
t6 = br2(k) - br6(k)
t7 = br3(k) - br7(k)
br2(k) = t0 - t2
br3(k) = t1 - t3
t0 = t0 + t2
t1 = t1 + t3
br0(k) = t0 + t1
br1(k) = t0 - t1
pr = p7*(t5-t7)
pi = p7*(t5+t7)
br4(k) = t4 + pr
br7(k) = t6 + pi
br6(k) = t4 - pr
br5(k) = pi - t6
60 continue
if (nn-8) 120, 120, 70
70 k0 = int*8 + 1
kl = k0 + int - 1
do 80 k=k0,kl
pr = p7*(bi2(k)-bi6(k))
pi = p7*(bi2(k)+bi6(k))
tr0 = bi0(k) + pr
ti0 = bi4(k) + pi
tr2 = bi0(k) - pr
ti2 = bi4(k) - pi
pr = p7*(bi3(k)-bi7(k))
pi = p7*(bi3(k)+bi7(k))
tr1 = bi1(k) + pr
ti1 = bi5(k) + pi
tr3 = bi1(k) - pr
ti3 = bi5(k) - pi
pr = tr1*c22 - ti1*s22
pi = ti1*c22 + tr1*s22
bi0(k) = tr0 + pr
bi6(k) = tr0 - pr
bi7(k) = ti0 + pi
bi1(k) = pi - ti0
pr = -tr3*s22 - ti3*c22
pi = tr3*c22 - ti3*s22
bi2(k) = tr2 + pr
bi4(k) = tr2 - pr
bi5(k) = ti2 + pi
bi3(k) = pi - ti2
80 continue
go to 120
90 arg = th2*piovn
c1 = cos(arg)
s1 = sin(arg)
c2 = c1**2 - s1**2
s2 = c1*s1 + c1*s1
c3 = c1*c2 - s1*s2
s3 = c2*s1 + s2*c1
c4 = c2**2 - s2**2
s4 = c2*s2 + c2*s2
c5 = c2*c3 - s2*s3
s5 = c3*s2 + s3*c2
c6 = c3**2 - s3**2
s6 = c3*s3 + c3*s3
c7 = c3*c4 - s3*s4
s7 = c4*s3 + s4*c3
int8 = int*8
j0 = jr*int8 + 1
k0 = ji*int8 + 1
jlast = j0 + int - 1
do 100 j=j0,jlast
k = k0 + j - j0
tr1 = br1(j)*c1 - bi1(k)*s1
ti1 = br1(j)*s1 + bi1(k)*c1
tr2 = br2(j)*c2 - bi2(k)*s2
ti2 = br2(j)*s2 + bi2(k)*c2
tr3 = br3(j)*c3 - bi3(k)*s3
ti3 = br3(j)*s3 + bi3(k)*c3
tr4 = br4(j)*c4 - bi4(k)*s4
ti4 = br4(j)*s4 + bi4(k)*c4
tr5 = br5(j)*c5 - bi5(k)*s5
ti5 = br5(j)*s5 + bi5(k)*c5
tr6 = br6(j)*c6 - bi6(k)*s6
ti6 = br6(j)*s6 + bi6(k)*c6
tr7 = br7(j)*c7 - bi7(k)*s7
ti7 = br7(j)*s7 + bi7(k)*c7
c
t0 = br0(j) + tr4
t1 = bi0(k) + ti4
tr4 = br0(j) - tr4
ti4 = bi0(k) - ti4
t2 = tr1 + tr5
t3 = ti1 + ti5
tr5 = tr1 - tr5
ti5 = ti1 - ti5
t4 = tr2 + tr6
t5 = ti2 + ti6
tr6 = tr2 - tr6
ti6 = ti2 - ti6
t6 = tr3 + tr7
t7 = ti3 + ti7
tr7 = tr3 - tr7
ti7 = ti3 - ti7
c
tr0 = t0 + t4
ti0 = t1 + t5
tr2 = t0 - t4
ti2 = t1 - t5
tr1 = t2 + t6
ti1 = t3 + t7
tr3 = t2 - t6
ti3 = t3 - t7
t0 = tr4 - ti6
t1 = ti4 + tr6
t4 = tr4 + ti6
t5 = ti4 - tr6
t2 = tr5 - ti7
t3 = ti5 + tr7
t6 = tr5 + ti7
t7 = ti5 - tr7
br0(j) = tr0 + tr1
bi7(k) = ti0 + ti1
bi6(k) = tr0 - tr1
br1(j) = ti1 - ti0
br2(j) = tr2 - ti3
bi5(k) = ti2 + tr3
bi4(k) = tr2 + ti3
br3(j) = tr3 - ti2
pr = p7*(t2-t3)
pi = p7*(t2+t3)
br4(j) = t0 + pr
bi3(k) = t1 + pi
bi2(k) = t0 - pr
br5(j) = pi - t1
pr = -p7*(t6+t7)
pi = p7*(t6-t7)
br6(j) = t4 + pr
bi1(k) = t5 + pi
bi0(k) = t4 - pr
br7(j) = pi - t5
100 continue
jr = jr + 2
ji = ji - 2
if (ji-jl) 110, 110, 120
110 ji = 2*jr - 1
jl = jr
120 continue
return
end
c
c-----------------------------------------------------------------------
c subroutine: r4syn
c radix 4 synthesis
c-----------------------------------------------------------------------
c
subroutine r4syn(int, b0, b1, b2, b3)
dimension b0(2), b1(2), b2(2), b3(2)
do 10 k=1,int
t0 = b0(k) + b1(k)
t1 = b0(k) - b1(k)
t2 = b2(k) + b2(k)
t3 = b3(k) + b3(k)
b0(k) = t0 + t2
b2(k) = t0 - t2
b1(k) = t1 + t3
b3(k) = t1 - t3
10 continue
return
end
c
c-----------------------------------------------------------------------
c subroutine: r8syn
c radix 8 synthesis subroutine
c-----------------------------------------------------------------------
c
subroutine r8syn(int, nn, br0, br1, br2, br3, br4, br5, br6, br7,
* bi0, bi1, bi2, bi3, bi4, bi5, bi6, bi7)
dimension l(15), br0(2), br1(2), br2(2), br3(2), br4(2), br5(2),
* br6(2), br7(2), bi0(2), bi1(2), bi2(2), bi3(2), bi4(2),
* bi5(2), bi6(2), bi7(2)
common /con1/ pii, p7, p7two, c22, s22, pi2
equivalence (l15,l(1)), (l14,l(2)), (l13,l(3)), (l12,l(4)),
* (l11,l(5)), (l10,l(6)), (l9,l(7)), (l8,l(8)), (l7,l(9)),
* (l6,l(10)), (l5,l(11)), (l4,l(12)), (l3,l(13)), (l2,l(14)),
* (l1,l(15))
l(1) = nn/8
do 40 k=2,15
if (l(k-1)-2) 10, 20, 30
10 l(k-1) = 2
20 l(k) = 2
go to 40
30 l(k) = l(k-1)/2
40 continue
piovn = pii/float(nn)
ji = 3
jl = 2
jr = 2
c
do 120 j1=2,l1,2
do 120 j2=j1,l2,l1
do 120 j3=j2,l3,l2
do 120 j4=j3,l4,l3
do 120 j5=j4,l5,l4
do 120 j6=j5,l6,l5
do 120 j7=j6,l7,l6
do 120 j8=j7,l8,l7
do 120 j9=j8,l9,l8
do 120 j10=j9,l10,l9
do 120 j11=j10,l11,l10
do 120 j12=j11,l12,l11
do 120 j13=j12,l13,l12
do 120 j14=j13,l14,l13
do 120 jthet=j14,l15,l14
th2 = jthet - 2
if (th2) 50, 50, 90
50 do 60 k=1,int
t0 = br0(k) + br1(k)
t1 = br0(k) - br1(k)
t2 = br2(k) + br2(k)
t3 = br3(k) + br3(k)
t4 = br4(k) + br6(k)
t6 = br7(k) - br5(k)
t5 = br4(k) - br6(k)
t7 = br7(k) + br5(k)
pr = p7*(t7+t5)
pi = p7*(t7-t5)
tt0 = t0 + t2
tt1 = t1 + t3
t2 = t0 - t2
t3 = t1 - t3
t4 = t4 + t4
t5 = pr + pr
t6 = t6 + t6
t7 = pi + pi
br0(k) = tt0 + t4
br1(k) = tt1 + t5
br2(k) = t2 + t6
br3(k) = t3 + t7
br4(k) = tt0 - t4
br5(k) = tt1 - t5
br6(k) = t2 - t6
br7(k) = t3 - t7
60 continue
if (nn-8) 120, 120, 70
70 k0 = int*8 + 1
kl = k0 + int - 1
do 80 k=k0,kl
t1 = bi0(k) + bi6(k)
t2 = bi7(k) - bi1(k)
t3 = bi0(k) - bi6(k)
t4 = bi7(k) + bi1(k)
pr = t3*c22 + t4*s22
pi = t4*c22 - t3*s22
t5 = bi2(k) + bi4(k)
t6 = bi5(k) - bi3(k)
t7 = bi2(k) - bi4(k)
t8 = bi5(k) + bi3(k)
rr = t8*c22 - t7*s22
ri = -t8*s22 - t7*c22
bi0(k) = (t1+t5) + (t1+t5)
bi4(k) = (t2+t6) + (t2+t6)
bi1(k) = (pr+rr) + (pr+rr)
bi5(k) = (pi+ri) + (pi+ri)
t5 = t1 - t5
t6 = t2 - t6
bi2(k) = p7two*(t6+t5)
bi6(k) = p7two*(t6-t5)
rr = pr - rr
ri = pi - ri
bi3(k) = p7two*(ri+rr)
bi7(k) = p7two*(ri-rr)
80 continue
go to 120
90 arg = th2*piovn
c1 = cos(arg)
s1 = -sin(arg)
c2 = c1**2 - s1**2
s2 = c1*s1 + c1*s1
c3 = c1*c2 - s1*s2
s3 = c2*s1 + s2*c1
c4 = c2**2 - s2**2
s4 = c2*s2 + c2*s2
c5 = c2*c3 - s2*s3
s5 = c3*s2 + s3*c2
c6 = c3**2 - s3**2
s6 = c3*s3 + c3*s3
c7 = c3*c4 - s3*s4
s7 = c4*s3 + s4*c3
int8 = int*8
j0 = jr*int8 + 1
k0 = ji*int8 + 1
jlast = j0 + int - 1
do 100 j=j0,jlast
k = k0 + j - j0
tr0 = br0(j) + bi6(k)
ti0 = bi7(k) - br1(j)
tr1 = br0(j) - bi6(k)
ti1 = bi7(k) + br1(j)
tr2 = br2(j) + bi4(k)
ti2 = bi5(k) - br3(j)
tr3 = bi5(k) + br3(j)
ti3 = bi4(k) - br2(j)
tr4 = br4(j) + bi2(k)
ti4 = bi3(k) - br5(j)
t0 = br4(j) - bi2(k)
t1 = bi3(k) + br5(j)
tr5 = p7*(t0+t1)
ti5 = p7*(t1-t0)
tr6 = br6(j) + bi0(k)
ti6 = bi1(k) - br7(j)
t0 = br6(j) - bi0(k)
t1 = bi1(k) + br7(j)
tr7 = -p7*(t0-t1)
ti7 = -p7*(t1+t0)
t0 = tr0 + tr2
t1 = ti0 + ti2
t2 = tr1 + tr3
t3 = ti1 + ti3
tr2 = tr0 - tr2
ti2 = ti0 - ti2
tr3 = tr1 - tr3
ti3 = ti1 - ti3
t4 = tr4 + tr6
t5 = ti4 + ti6
t6 = tr5 + tr7
t7 = ti5 + ti7
ttr6 = ti4 - ti6
ti6 = tr6 - tr4
ttr7 = ti5 - ti7
ti7 = tr7 - tr5
br0(j) = t0 + t4
bi0(k) = t1 + t5
br1(j) = c1*(t2+t6) - s1*(t3+t7)
bi1(k) = c1*(t3+t7) + s1*(t2+t6)
br2(j) = c2*(tr2+ttr6) - s2*(ti2+ti6)
bi2(k) = c2*(ti2+ti6) + s2*(tr2+ttr6)
br3(j) = c3*(tr3+ttr7) - s3*(ti3+ti7)
bi3(k) = c3*(ti3+ti7) + s3*(tr3+ttr7)
br4(j) = c4*(t0-t4) - s4*(t1-t5)
bi4(k) = c4*(t1-t5) + s4*(t0-t4)
br5(j) = c5*(t2-t6) - s5*(t3-t7)
bi5(k) = c5*(t3-t7) + s5*(t2-t6)
br6(j) = c6*(tr2-ttr6) - s6*(ti2-ti6)
bi6(k) = c6*(ti2-ti6) + s6*(tr2-ttr6)
br7(j) = c7*(tr3-ttr7) - s7*(ti3-ti7)
bi7(k) = c7*(ti3-ti7) + s7*(tr3-ttr7)
100 continue
jr = jr + 2
ji = ji - 2
if (ji-jl) 110, 110, 120
110 ji = 2*jr - 1
jl = jr
120 continue
return
end
c
c-----------------------------------------------------------------------
c subroutine: ord1
c in-place reordering subroutine
c-----------------------------------------------------------------------
c
subroutine ord1(m, b)
dimension b(2)
c
k = 4
kl = 2
n = 2**m
do 40 j=4,n,2
if (k-j) 20, 20, 10
10 t = b(j)
b(j) = b(k)
b(k) = t
20 k = k - 2
if (k-kl) 30, 30, 40
30 k = 2*j
kl = j
40 continue
return
end
c
c-----------------------------------------------------------------------
c subroutine: ord2
c in-place reordering subroutine
c-----------------------------------------------------------------------
c
subroutine ord2(m, b)
dimension l(15), b(2)
equivalence (l15,l(1)), (l14,l(2)), (l13,l(3)), (l12,l(4)),
* (l11,l(5)), (l10,l(6)), (l9,l(7)), (l8,l(8)), (l7,l(9)),
* (l6,l(10)), (l5,l(11)), (l4,l(12)), (l3,l(13)), (l2,l(14)),
* (l1,l(15))
n = 2**m
l(1) = n
do 10 k=2,m
l(k) = l(k-1)/2
10 continue
do 20 k=m,14
l(k+1) = 2
20 continue
ij = 2
do 40 j1=2,l1,2
do 40 j2=j1,l2,l1
do 40 j3=j2,l3,l2
do 40 j4=j3,l4,l3
do 40 j5=j4,l5,l4
do 40 j6=j5,l6,l5
do 40 j7=j6,l7,l6
do 40 j8=j7,l8,l7
do 40 j9=j8,l9,l8
do 40 j10=j9,l10,l9
do 40 j11=j10,l11,l10
do 40 j12=j11,l12,l11
do 40 j13=j12,l13,l12
do 40 j14=j13,l14,l13
do 40 ji=j14,l15,l14
if (ij-ji) 30, 40, 40
30 t = b(ij-1)
b(ij-1) = b(ji-1)
b(ji-1) = t
t = b(ij)
b(ij) = b(ji)
b(ji) = t
40 ij = ij + 2
return
end
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