Digital Signal Processing Reference
In-Depth Information
That was a bit tedious, but there is some insight to be gained
from the results of these simple examples.
Table 12.2
shows how the DFT is able to represent the signal
energy in each frequency bin. The first example has all the energy
at DC. The second and third examples are complex exponentials
at frequency
/ 2 radians/sample, which corresponds to DFT
output X
2
. The magnitude of the DFT outputs is the same for both
examples, since the only difference of the inputs is the phase. The
fourth example is the most interesting. In this case, the input
frequency is close to
u ΒΌ p
p
/ 4 radians/sample, which corresponds to
DFT output X
1
.SoX
1
does capture most of the energy of the
Table 12.2
DFT
Output
Magnitude
e
D j2
p
i/4
e
D j(2
p
(i D 1) / 4)
e
D j2.1
p
i/8
x
i
[
{1,1,1,1,1,1,1,1}
x
i
[
x
i
[
x
i
[
Output X
0
8
0
0
0.39
Output X
1
0
0
0
7.99
Output X
2
0
8
8
0.43
Output X
3
0
0
0
0.23
Output X
4
0
0
0
0.17
Output X
5
0
0
0
0.16
Output X
6
0
0
0
0.17
Output X
7
0
0
0
0.22
signal. But small amounts of energy spill into other frequency
bins, particularly the adjacent bins.
We can increase the frequency sorting ability of the DFT by
increasing the value of N. This will narrow each frequency bin
because the frequency spectrum is divided into N sections in the
DFT. This will result in a given frequency component being more
selectively represented by a particular frequency bin. For
example, the frequency response plots of the filters contained in
the FIR chapter are computed with a value of N equal to 1024.
This means the spectrum was divided into 1024 sections, and the
response computed for each particular frequency. When plotted
together, this gives a very good representation of the complete
frequency spectrum.
