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

582 HP MLIB User’s Guide
CRCFTS/ZRCFTS Simultaneous real-to-complex one-dimensional FFT
Name CRCFTS/ZRCFTS
Simultaneous real-to-complex one-dimensional FFT
Purpose Given a number of one-dimensional data sets, these subprograms compute all
of their one-dimensional forward real-to-complex or inverse complex-to-real
discrete Fourier transforms. A pair of companion subprograms, SRCFTS and
DRCFTS, performs the same operation, but in a space-conserving manner.
Other subprograms, documented elsewhere in this chapter, are more suited for
computing just one real-to-complex or complex-to-real transform.
The one-dimensional complex forward DFT of a real data set z(n), for
n = 1, 2, ..., l, is defined by
for m = 1, 2, ..., l and .
The Z(m) satisfies the one-dimensional conjugate-symmetry conditions:
Im(Z(1)) = 0
and
If m is even, then in addition
Im(Z(l/2+1)) = 0
Alternatively, if Z(m), for m = 1, 2, ..., l, is a conjugate-symmetric complex
data set, the one-dimensional, real, scaled, inverse DFT of Z(m) is defined by
for n = 1, 2, ..., l.
These subprograms perform forward real-to-complex or inverse complex-to-real
transform operations simultaneously on a number of data sets. For best
performance, they require that the length l of the data sets be a product of
powers of 2, 3, and 5, that is, of the form
where p, q, r ≥ 0, and where either l = 1 or l is even.
Zm() zn()e
2– πim 1–()n 1–()l⁄
n 1=
l
∑
=
i 1–=
Zm() Zl 2 m–+()= m 23… l 1+()2⁄,, ,=,
zn()
1
l
---
Zm()e
+2πim 1–()n 1–()l⁄
m 1=
l
∑
=
l 2
p
3
q
5
r
,=