Digital Signal Processing Reference
In-Depth Information
Factorizing the odd terms we have
{
}
0
k
2
k
1
k
2
k
Xk
()
=
x
()
0
W
+
x
()
2
W
+
W
x
()
1
+
x
()
3
W
,
0
≤≤
k
3
4
4
4
4
(3.11)
1
k
=
Xk
()
+
W
Xk
(),
0
≤
k
≤
3
1
4
2
Step 2: Computation of
X
1
(
k
)
0
k
2
k
Xk xW
()
=
()
0
+
xW
()
2
, 0
≤
k
≤
3
(3.12)
1
4
4
Examination of Equation 3.12 yields the following interesting relations:
X
1
(2)
= X
1
(0)
= x
(0)
+ x
(2)
X
1
(3)
= X
1
(1)
= x
(0)
- x
(2)
Hence, we have to calculate, only the following terms:
00
20
Xx
()
0
=
()
0
Wx
+
()
2
W
(3.13a)
1
4
4
and
01
21
Xx
()
1
=
()
0
Wx
+
()
2
W
(3.13b)
1
4
4
Step 3: Computation of
X
2
(
k
)
x W
k
2
Xk x
()
=
()
1
+
()
3
, 0
≤
k
≤
3
(3.14)
2
4
Examination of Equation 3.14 yields similar interesting relations:
X
2
(2) =
X
2
(0) =
x
(1)
+ x
(3)
X
2
(3) =
X
2
(1) =
x
(1)
- x
(3)
Hence, we have to calculate, only the following terms:
20
Xx
()
0
=
()
1
+
x
()
3
W
(3.15a)
2
4
and
21
Xx
()
1
=
()
1
+
x
()
3
W
(3.15b)
2
4
