Digital Signal Processing Reference
In-Depth Information
1.
Change the sign of the imaginary parts (find the complex conjugate) of
X[k].
X
*
[k].
2.
Use the DFT algorithm to find the DFT of
3.
Find the complex conjugate of the results of Step 2 for each value of
n
.
4.
Divide the results of Step 3 by
N
.
An alternative method of calculating the inverse DFT
EXAMPLE 12.11
We use the data in Table 12.3 to show the alternative procedure for computation of the
IDFT. From Table 12.3, we have
X[k] = [10, -2 + j2, -2, -2 - j2].
From the first step of the inversion procedure, we have
X
*
[k] = [10, -2 - j2, -2, -2 + j2].
X
*
[k]:
The second step is to compute the DFT of
3
X
*
[k]e
-j2pkn/4
;
4 x
*
[n] =
a
k= 0
4 x
*
[0] = 10 + (-2 - j2) + (-2) + (-2 + j2) = 4;
4 x
*
[1] = 10 + (-2 - j2)e
-jp/2
+ (-2)e
-jp
+ (-2 + j2)e
-j3p/2
= 10 + (-j)(-2 - j2) + (-1) (-2) + j(-2 + j2) = 8;
4 x
*
[2] = 10 + (-2 - j2)e
-jp
+ (-2)e
-j2p
+ (-2 + j2)e
-j3p
= 10 + (-1) (-2 - j2) + (1) (-2) + (-1) (-2 + j2) = 12;
4 x
*
[3] = 10 + (-2 - j2)e
-j3p/2
+ (-2)e
-j3p
+ (-2 + j2)e
-j9p/2
= 10 + (j)(-2 - j2) + (-1) (-2) + (-j)(-2 + j2) = 16.
Because the results of Step 2 are all real valued, finding the complex conjugate as specified in
Step 3 is not necessary. We divide the results of Step 2 by 4 to complete the evaluation:
x[n] = [1, 2, 3, 4].
The results are given in Table 12.3.
■
In the DFT, we have derived a discrete-frequency approximation of the discrete-
time Fourier transform. This transform has wide application for digital signal pro-
cessing. We have looked at the mathematical roots of the DFT as well as the
practical meaning of the development. Methods of computing the DFT and the
IDFT have been presented and used in example problems.
