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

Chapter 5 Fast Fourier Transforms 567
Real-to-complex one-dimensional FFT SRC1FT/DRC1FT
Name SRC1FT/DRC1FT
Real-to-complex one-dimensional FFT
Purpose Given a set of real data in a real array, these subprograms compute the
nonredundant part of the complex one-dimensional forward DFT. Alternatively,
given the nonredundant part of a conjugate-symmetric complex
one-dimensional data set, these subprograms compute the real inverse discrete
Fourier transform. A pair of companion subprograms, CRC1FT and ZRC1FT,
performs a similar operation, but with the real and complex data presented in a
complex array.
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.
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
∑
=
z(n)
m 1=
l
∑
Z(m) e
+2πi(m−1)(n−1)/l
=