Digital Signal Processing Reference
In-Depth Information
20
1
16
Window
15
0.5
3
10
29
0
−0.5
5
−
1
0
0
10
20
30
40
0
10
20
30
40
Samples
Frequency bin
number
(a)
(b)
Figure 4.7 Non-rectangular windowing effect
4.4.2 Frequency Resolution
The given frequency in
0.5 to 0.5 is divided into 32 divisions, giving a resolution
of
N. The greater the observation time, the better the resolution in the
spectrum. This gives us an opportunity to look at the DFT as a bank of filters
separated at
f
¼
1
=
f . Due to conjugate symmetry, the useful range is 0 to 15 bins.
4.4.3 Decimation in Time
We could also split the input sequence into odd and even sequences then compute
the DFT. We can write (4.15) in a different way as
X
k
¼
X
16
þ
X
14
15
x
n
W
nk
x
n
W
nk
16
:
ð
4
:
16
Þ
n
¼
even
n
¼
odd
For the even case, n takes the form 2m; for the odd case, n takes the form 2m
þ
1.
With this in mind we write (4.16) as
X
k
¼
X
þ
X
7
7
x
odd
W
ð
2m
þ
1
Þ
k
x
even
W
2mk
16
16
m
¼
0
m
¼
0
!
!
:
X
þ
W
16
X
7
7
x
even
W
mk
x
odd
W
mk
¼
ð
4
:
17
Þ
8
8
m
¼
0
m
¼
0
There are two steps in obtaining a 16-point DFT using an 8-point DFT. First, we
divide the time sequence as odd and even or as two sets of alternating points. We
compute the 8-point DFT of the two sequences, shown as step A in Figure 4.8.
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