Digital Signal Processing Reference
In-Depth Information
When
1
,
2, no closed expressions exist for the density functions, making
it necessary to resort to series expansions or integral transforms to describe
them.
7
Symmetric
α
=
α
-stable densities maintain many of the features of the Gaussian
density. They are smooth, unimodal, symmetric with respect to the mode,
and bell-shaped. Figure 5.1 illustrates the impulsive behavior of symmetric
α
-stable processes as the characteristic exponent
α
is varied. Each of the plots
shows an i.i.d. “zero-centered” symmetric
-stable signal with unitary geo-
metric power.* To give a better feeling of the impulsive structure of the data,
the signals are plotted twice under two different scales. As can be appreciated,
the Gaussian signal (
α
α
=
2) does not show impulsive behavior. For values of
α
9), the structure of the signal is still similar to the Gaussian,
although little impulsiveness can now be observed. As the value of
close to 2 (
α
=
1
.
α
is de-
creased, the impulsive behavior of the
-stable process increases progres-
sively, exhibiting the highest levels of impulsiveness for very small values of
α
α
.
5.3
Running Myriad Smoothers
The sample myriad emerges as the maximum likelihood estimate of location
under a set of several distributions within the family of
-stable distributions,
including the well-known Cauchy distributions. Since their introduction by
Fisher in 1922,
15
myriad-type estimators have been studied and applied under
very different contexts as an efficient alternative to cope with the presence of
impulsive noise.
16-21
The most general form of the myriad, where the potential
of tuning the so-called
linearity parameter
in order to control its behavior is fully
exploited, was first introduced by Gonzalez and Arce in 1996.
22
Depending
on the value of this free parameter, the sample myriad can present drastically
different behaviors, ranging from
highly resistant
mode-type estimators to the
familiar (Gaussian-efficient) sample average. This rich variety of operation
modes is the key concept explaining important optimality properties of the
myriad in the class of symmetric
α
α
-stable distributions.
] and a fixed
positive (tunable) value of
K
, the running myriad smoother output at time
n
is computed as
Given an observation vector
X
(
n
)
=
[
X
1
(
n
)
,X
2
(
n
)
,
...
,X
N
(
n
)
Y
K
(
n
)
=
MYRIAD[
K
;
X
1
(
n
)
,X
2
(
n
)
,
...
,X
N
(
n
)
]
N
[
K
2
2
]
=
arg min
β
+
(
X
i
(
n
)
−
β)
.
(5.5)
i
=
1
* The geometric power is an indicator of signal strength suited to the class of processes with
infinite variance.
14
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