HP MLIB User's Guide Vol. 2 7th Ed.

Chapter 5 Fast Fourier Transforms 579
Real-to-complex three-dimensional FFT SRC3FT/DRC3FT
Name SRC3FT/DRC3FT
Real-to-complex three-dimensional FFT
Purpose Given a set of three-dimensional real data, these subprograms compute the
nonredundant portion of the complex three-dimensional forward DFT.
Alternatively, given the nonredundant part of a conjugate-symmetric,
three-dimensional, complex data set, these subprograms compute the real
inverse DFT. A pair of companion subprograms, CRC3FT and ZRC3FT,
performs similar operations, but with the real or complex data presented in a
complex array. These companion subprograms require more storage than the
ones described here.
The three-dimensional, complex, forward a DFT of a real data set z(n
1
,n
2
,n
3
),
for n
1
= 1, 2, ..., l
1
, n
2
= 1, 2, ..., l
2
, and n
3
= 1, 2, ..., l
3
, is defined by
for m
1
= 1, 2, ..., l
1
, m
2
= 1, 2, ..., l
2
, m
3
= 1, 2, ..., l
3
, and .
The Z(m
1
,m
2
,m
3
) satisfy the three-dimensional conjugate-symmetry conditions:
and
where is the complex conjugate of Z.
Zm
1
m
2
m
3
,,() zn
1
n
2
n
3
,,()
e
2– πim
1
1–()n
1
1–()l
1
⁄
e
2– πim
2
1–()n
2
1–()l
2
⁄
e
2– πim
3
1–()n
3
1–()
l
⁄
×
n
3
1=
l
3
∑
n
2
1=
l
2
∑
n
1
1=
l
1
∑
=
i 1–=
Im Z 111,,()()0,=
Zm
1
11,,()Zl
1
2 m
1
–+11,,()= m
1
23… l
1
,,, ,=,
Z 1 m
2
1,,()Z 1 l
2
, 2 m
1
1,–+()= m
2
23… l
2
,,, ,=,
Z 11m
3
,,()Z 11l
3
,, 2 m
1
–+()= m
3
23… l
3,
,, ,=,
Zm
1
m
2
, 1,()Zl
1
2 m
1
l
2
2 m
2
1,–+,–+()= m
k
23… l
k
k 12,=,,, , ,=,
Zm
1
1 m
3
,,()Zl
1
2 m
1
1 l
3
, 2 m
3
–+,–+()= m
k
23… l
k
k 13,=,,, , ,=,
Z 1 m
2
m
3
,,()Z 1 l
2
, 2 m
2
l
3
2 m
3
–+,–+()= m
k
23… l
k
k 23,=,,, , ,=,
Zm
1
m
2
, m
3
,()Zl
1
2 m
1
l
2
2 m
2
l
3
2 m
3
–+,–+,–+()= m
k
23… l
k
k 123,,,=,, , ,=,
Z