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

570 HP MLIB User’s Guide
CRC2FT/ZRC2FT Real-to-complex two-dimensional FFT
Name CRC2FT/ZRC2FT
Real-to-complex two-dimensional FFT
Purpose These subprograms compute either the forward real-to-complex or the inverse
complex-to-real, two-dimensional, discrete Fourier transform. A pair of
companion subprograms, SRC2FT and DRC2FT, performs a similar operation,
but in a space-conserving manner.
The two-dimensional, complex, forward discrete Fourier transform of a real
data set, z(n
1
,n
2
), for n
1
= 1, 2, ..., l
1
and n
2
= 1, 2, ..., l
2
, is defined by
for m
1
= 1, 2, ..., l
1
, m
2
= 1, 2, ..., l
2
, and .
The Z(m
1
,m
2
) satisfy the two-dimensional, conjugate-symmetry conditions:
and
where is the complex conjugate of Z.
Alternatively, if Z(m
1
,m
2
), for m
1
= 1, 2, ..., l
1
and m
2
= 1, 2, ..., l
2
, is a
conjugate-symmetric complex data set, the two-dimensional, real, scaled,
inverse discrete Fourier transform of Z(m
1
,m
2
) is defined by
for n
1
= 1, 2, ..., l
1
and n
2
= 1, 2, ..., l
2
.
For best performance, these subprograms require that l
1
and l
2
be products of
powers of 2, 3, and 5, that is, of the form
l
k
=2
p
k
3
q
k
5
r
k
,
where p
k
, q
k
, r
k
≥ 0, k = 1, 2, and where either l
1
= 1 or l
1
is even.
Zm
1
m
2
,() zn
1
n
2
,()e
2– πim
1
1–()n
1
1–()l
1
⁄
e
2– πim
2
1–()n
2
1–()l
2
⁄
n
2
1=
l
2
∑
n
1
1=
l
1
∑
=
i 1–=
Im Z 11,()()0,=
Zm
1
1,()Zl
1
2 m
1
–+1,()m
1
23… l
1
,, , ,=,=
Z 1 m
2
,()Z 1 l
2
, 2 m
2
–+()m
2
23… l
2
,,, ,=,=
Zm
1
m
2
,()Zl
1
2 m
1
–+ l
2
2 m
2
–+,()= m
k
23… l
k
k 12,,=,, , ,=,
Z
zn
1
n
2
,()
l
l
1
l
2
---------
Zm
1
m
2
,()e
+2πim
1
1–()n
1
1–()l
1
⁄
e
+2πim
2
1–()n
2
1–()l
2
⁄
m
2
1=
l
2
∑
m
1
1=
l
1
∑
=