Digital Signal Processing Reference
In-Depth Information
(
)
C
S
n
=
cos
sin
2
p
2
n N
(
)
n
=
p
n N
The output sequence
X
(
k
) after the final stage 3 is also shown in Figure F.1. For
example,
()
=+ + =+
()
+
()
=
()
=+ + =+
X
X
022020221204
1
CS
CS
2
2
1
2 1
2
1 414
.
+=
0
3 41
.
M
()
=+
()
+ -
X
7
0
0
CS
7
2 7
1 414
.
(F.16)
This resulting output sequence can be verified from the
X
(
k
) obtained with the FFT,
using
(
{
=
{
[
()
]
}
-
{
[
()
]
}
DHT
xn
Re
DFT
xn
Im DFT
xn
(F.17)
For example, from the eight-point FFT in Exercise 6.1,
X
(1)
=
1
-
j
2.41, and
X
(
{
=
Re
X
1
1
(
{
=-
Im
1
2 41
.
Using (F.17),
(
{}
=
()
=--
(
)
=
DHT
x
1
X
1
1
2 41
.
3 41
.
as in (F.16). Conversely, the FFT can be obtained from the FHT using
{
[
()
]
}
=
{
[
(
)
]
+
[
()
]
}
1
2
1
2
Re DFT
xn
DHT
xN
-
n
DHT
xn
{
[
()
]
}
=
{
[
(
)
]
-
[
()
]
}
Im DFT
xn
DHT
xN
-
n
DHT
xn
(F.18)
For example, using (F.18) to obtain
X
(1)
=
1
-
j
2.41 from the FHT,
(
{
=
{
}
=- +
{
}
=
()
+
()
Re
Im
X
1
1
2
X
7
X
1
1
2
1 41
.
3 41
.
1
(
{
=
[
()
-
()
]
=- -
{
}
=-
X
1
1
2
X
7
X
1
1
2
1 41
.
3 41
.
2 41
.
(F.19)
where the left-hand side of (F.18) is associated with the FFT and the right-hand side
with the FHT.
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