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

Chapter 5 Fast Fourier Transforms 573
Real-to-complex two-dimensional FFT SRC2FT/DRC2FT
Name SRC2FT/DRC2FT
Real-to-complex two-dimensional FFT
Purpose Given a set of two-dimensional real data, these subprograms compute the
nonredundant portion of the complex, two-dimensional, forward DFT.
Alternatively, given the nonredundant part of a conjugate-symmetric
two-dimensional complex data set, these subprograms compute the real inverse
discrete Fourier transform. A pair of companion subprograms, CRC2FT and
ZRC2FT, performs similar operations, but with the real or complex data
presented in a complex array. The companion subprograms require more
storage than the ones described here.
The two-dimensional complex forward DFT 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, the two-dimensional, real, inverse DFT of Z(m
1
,m
2
), for
m
1
= 1, 2, ..., l
1
and m
2
= 1, 2, ..., l
2
, is defined by
for n
1
= 1, 2, ..., l
1
and n
2
= 1, 2, ..., l
2
.
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
1
–+()= 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
,()
1
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
∑
=