Digital Signal Processing Reference
In-Depth Information
(
)
-
Â
N
21
21
cas
p
2
nK
N
2
Ê
Ë
ˆ
¯
(
)
=
()
Hk
+
bn
(F.15)
n
=
0
A more complete development of the FHT can be found in [3]. We now illustrate
the FHT with two exercises: an 8-point FHT and a 16-point FHT. We will then
readily verify these results from the FFT exercises in Chapter 6.
Exercise F.1: Eight-Point Fast Hartley Transform
Let the rectangular sequence
x
(
n
) be represented by
x
(0)
=
x
(1)
=
x
(2)
=
x
(3)
=
1,
0. The flow graph in Figure F.1 is used to find
X
(
k
).
We will now use
X
(
k
) instead of
H
(
k
). The sequence is first permuted and the inter-
mediate results after the first two stages are as shown in Figure F.1. The coefficients
C
n
and S
n
are (with
N
and
x
(4)
=
x
(5)
=
x
(6)
=
x
(7)
=
=
8)
FIGURE F.1.
Eight-point FHT flow graph.
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