Graphics Programs Reference
In-Depth Information
spreads over several lines (normally the spread may extend up to three lines).
This is known as spectral leakage. Since is normally unknown, this discon-
tinuity caused by an arbitrary choice of cannot be avoided. Windowing tech-
niques can be used to mitigate the effect of this discontinuity by applying
smaller weights to samples close to the edges.
f
0
T
A truncated sequence
x
()
can be viewed as one period of some periodic
ô
()
sequence
with period
N
. The discrete Fourier series expansion of
x
()
is
N
∑
1
j
2π
nk
N
---------------
x
()
=
X
k
e
(13.122)
k
=
0
It can be shown that the coefficients
X
k
are given by
N
∑
1
j
2π
nk
------------------
1
----
1
----
X
()
N
X
k
=
x
()
e
=
(13.123)
n
=
0
where
X
()
is the DFT of
x
()
. Therefore, the Discrete Power Spectrum
2
(DPS) for the band limited sequence
x
()
is the plot of
X
k
versus
k
, where
the lines are
∆
f
apart,
1
N
2
------
X
()
2
P
0
=
1
N
2
12…
N
X
()
2
)
2
------
P
k
=
{
+
X
N k
(
}
;
k
=
,, ,
----1
(13.124)
1
N
2
)
2
------
X
N
2
P
N
=
(
⁄
⁄
2
Before proceeding to the next section, we will show how to select the FFT
parameters. For this purpose, consider a band limited signal with band-
width . If the signal is not band limited, a LPF can be used to eliminate
frequencies greater than . In order to satisfy the sampling theorem, one must
choose a sampling frequency
x
()
B
B
=
1
⁄
T
s
, such that
f
s
f
s
≥
2
B
(13.125)
The truncated sequence duration
T
and the total number of samples
N
are
related by
Search WWH ::
Custom Search