Digital Signal Processing Reference
In-Depth Information
20
20
15
15
10
10
5
5
0
0
1
−
5
1
−
5
−
−
−
15
−
15
−
20
−
20
50
100
150
200
250
50
100
150
200
250
250
250
200
200
150
150
100
100
50
50
0
0
−
50
−
50
−
100
−
100
150
−
150
−
−
200
−
200
250
−
250
−
50
100
150
200
250
50
100
150
200
250
α
= 0.3
α
= 0.1
FIGURE 5.1
(
Continued
)
the myriad is also defined by the equivalent expression
N
log[
K
2
2
]
Y
K
(
n
)
=
arg min
β
+
(
X
i
(
n
)
−
β)
.
(5.6)
i
=
1
In general, for a fixed value of
K
, the minima of the cost functions in Equa-
tions 5.5 and 5.6 lead to a unique value.
To illustrate the calculation of the sample myriad and the effect of the
linearity parameter, consider the sample myriad of the set
{−
3
,
10
,
1
,
−
1
,
6
}
:
ˆ
β
=
MYRIAD
(
K
;
−
3
,
10
,
1
,
−
1
,
6
)
(5.7)
K
for
K
2. The myriad cost functions in Equation 5.5, for these three
values of
K
,are plotted in Figure 5.2. The corresponding minima are attained
at
ˆ
=
20
,
2
,
0
.
8,
ˆ
1, and
ˆ
1, respectively. The different values taken on
by the myriad as the parameter
K
is varied are best understood by the results
provided in the following properties.
β
20
=
1
.
β
2
=
0
.
β
0
.
2
=
PROPERTY 1 (Linear Property)
Given a set of samples, X
1
,X
2
,
,X
N
, the sample myriad
ˆ
...
β
K
converges to the
sample average as K
→∞
. This is,
N
1
N
ˆ
lim
K
β
=
lim
K
MYRIAD
(
K; X
1
,
...
,X
N
)
=
X
i
.
(5.8)
K
→∞
→∞
i
=
1
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