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

Chapter 5 Fast Fourier Transforms 587
Simultaneous real-to-complex one-dimensional FFT SRCFTS/DRCFTS
Name SRCFTS/DRCFTS
Simultaneous real-to-complex one-dimensional FFT
Purpose Given a number of one-dimensional real data sets, these subprograms compute
nonredundant portions of all of their one-dimensional forward real-to-complex
discrete Fourier transforms. Alternatively, given the nonredundant parts of a
number of conjugate-symmetric one-dimensional complex data sets, these
subprograms compute the inverse complex-to-real discrete Fourier transform.
A pair of companion subprograms, CRCFTS and ZRCFTS, 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. 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 forward discrete Fourier transform 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 discrete Fourier transform of
Z(m) is defined by
for n = 1, 2, ..., l. Only the nonredundant part of Z is used.
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
∑
=