Biomedical Engineering Reference
In-Depth Information
A problem that arises with this is that usually only a single realization of
this random process is available to estimate its power spectrum. Furthermore,
we do not have the true autocorrelation function
γ
xx
(
τ
) and cannot compute
the power density spectrum directly. If we use the single realization alone, it
is common to compute the time-average autocorrelation function taken over
an interval 2
T
0
. This is written mathematically as
T
0
1
2
T
0
x
∗
(
t
)
x
(
t
+
τ
)d
t
R
xx
(
τ
)=
(2.72)
−T
0
Using this, we can then proceed to estimate the power density spectrum as
follows:
T
0
1
2
T
0
R
xx
(
τ
)e
−j
2
πF τ
d
τ
P
xx
(
τ
)=
(2.73)
−T
0
Note that this is an estimate since the true power density spectrum is
obtained in the limit as
T
0
. There are two approaches to computing
P
xx
(
τ
) similar to the ones described earlier. If the signal
x
(
t
) is sampled
(digitized), an equivalent form of the time-averaged autocorrelation function
can be derived, that is,
→∞
N−m−
1
1
r
xx
(
m
)=
x
∗
(
n
)
x
(
n
+
m
)
(2.74)
N
−
m
n
=0
for
m
=0
,
1
,...,N
1 under the assumption that the
x
(
t
) is sampled at a rate
of
F
s
>
2
B
, where
B
is the highest frequency in the power density spectrum
of the random process. It can be shown (Oppenheim et al., 1999) that this
estimate of the autocorrelation function is “consistent” and “asymptotically
unbiased” in the sense that it approaches the true autocorrelation function
as
N
−
. With the estimate of the autocorrelation function, we can now
estimate the power density spectrum using
→∞
N−
1
r
xx
(
m
)e
−j
2
πfm
P
xx
(
τ
)=
(2.75)
m
=
−
(
N−
1)
which can be simplified as
x
(
n
)e
−j
2
πfn
2
N
−
1
P
xx
(
τ
)=
1
N
1
N
|
2
=
X
(
f
)
|
(2.76)
n
=0
where
X
(
f
) is the Fourier transform of the sample sequence
x
(
n
). This esti-
mated form of power density spectrum is known as a
periodogram
, but there
are several problems due to the use of this estimate, such as smoothing effects,
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