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

564 HP MLIB User’s Guide
CRC1FT/ZRC1FT Real-to-complex one-dimensional FFT
Name CRC1FT/ZRC1FT
Real-to-complex one-dimensional FFT
Purpose These subprograms compute either the forward real-to-complex or the inverse
complex-to-real, one-dimensional, DFT. Two companion subprograms, SRC1FT
and DRC1FT, perform a similar operation but in a space-conserving manner.
The one-dimensional 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 discrete Fourier transform of
Z(m) is defined by
for n = 1, 2, ..., l.
Finally, the one-dimensional, real, unscaled, inverse DFT of the
conjugate-symmetric complex data set Z(m), for m = 1, 2, ..., l, is defined by
for n = 1, 2, ..., l.
For best performance, these subprograms require that l 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
∑
=
zn() Zm()e
+2πim 1–()n 1–()l⁄
m 1=
l
∑
=
l 2
p
3
q
5
r
,=