Digital Signal Processing Reference
In-Depth Information
with respect to the DFT coefficient index
r
, by the inverse DFT matrix
G
.Itis
straightforward to show that
G
F
=
F
G
=
I
N
, where
I
N
is the identity
matrix of order
N
.
Example 12.4 repeats Example 12.1 using the matrix-vector representation
for the DFT.
Example 12.4
Calculate the four-point DFT of the aperiodic signal
x
[
k
] considered in
Example 12.1.
Solution
Arranging the values of the DT sequence in the signal vector
x
, we obtain
T
,
x
=
[2
3
−
11
where superscript T represents the transpose operation for a vector. Using
Eq. (12.19), we obtain
X
[0]
11
x
[0]
1
1
=
−
j(2
π/
N
)e
−
j(4
π/
N
)
−
j(6
π/
N
)
X
[1]
X
[2]
X
[3]
1e
e
x
[1]
x
[2]
x
[3]
−
j(4
π/
N
)
−
j(8
π/
N
)
−
j(12
π/
N
)
1e
e
e
−
j(6
π/
N
)
−
j(12
π/
N
)
−
j(18
π/
N
)
1e
e
e
DFT matrix:
F
11
=
1
1
2
−
1
1
5
3
−
j2
−
3
3
+
j2
−
j(2
π/
4)
−
j(4
π/
4)
−
j(6
π/
4)
1e
e
e
=
.
−
j(4
π/
4)
−
j(8
π/
4)
−
j(12
π/
4)
1e
e
e
−
j(6
π/
4)
−
j(12
π/
4)
−
j(18
π/
4)
1e
e
e
DFT matrix:
F
The above values for the DFT coefficients are the same as the ones obtained in
Example 12.1.
Example 12.5
Calculate the inverse DFT of
X
[
r
] considered in Example 12.2.
Solution
Arranging the values of the DFT coefficients in the DFT vector
X
, we obtain
−
33
+
j2]
T
.
X
=
[5
3
−
j2
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