Graphics Programs Reference
In-Depth Information
T
=
s
(13.126)
or equivalently,
N
----
f
s
=
(13.127)
It follows that
N
----
f
s
=
≥
2
B
(13.128)
and the frequency resolution is
f
s
1
---
2
B
N
1
NT
s
-------
∆
f
=
---------
=
----
=
≥
(13.129)
13.13. Windowing Techniques
Truncation of the sequence
x
()
can be accomplished by computing the
product,
x
w
()
x
()
w
()
=
(13.130)
where
f
()
;
n
=
01…
,, ,
N
1
w
()
=
(13.131)
0
otherwise
where . The finite sequence is called a windowing sequence, or
simply a window. The windowing process should not impact the phase
response of the truncated sequence. Consequently, the sequence must
retain linear phase. This can be accomplished by making the window symmet-
rical with respect to its central point.
f
() 1
≤
w
()
w
()
If for all we have what is known as the rectangular window. It
leads to the Gibbs phenomenon which manifests itself as an overshoot and a
ripple before and after a discontinuity.
Fig. 13.2
shows the amplitude spectrum
of a rectangular window. Note that the first side lobe is at below the
main lobe. Windows that place smaller weights on the samples near the edges
will have lesser overshoot at the discontinuity points (lower side lobes); hence,
they are more desirable than a rectangular window. However, sidelobes reduc-
tion is offset by a widening of the main lobe. Therefore, the proper choice of a
windowing sequence is continuous trade-off between side lobe reduction and
f
() 1
=
n
13.46
dB
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