Digital Signal Processing Reference
In-Depth Information
Now, from the expression of optimizing parameter of optimum filter (Eq.
3.30
),
∞
2
η
max
2
dt
γ
=
{
S
1
(
t
)
−
}
S
0
(
t
)
−∞
∞
S
1
(
t
)
2
S
1
(
t
)
S
0
(
t
)
dt
2
η
(3.34)
S
0
(
t
)
=
+
−
−∞
E
S
1
+
2
E
S
1
S
0
2
η
=
E
S
0
−
where,
E
S
1
,
E
S
0
are the energies of
S
1
(t) and
S
0
(t) respectively, and
E
S
1
S
0
is the joint
energy of the product waveform
√
S
1
(
t
)
S
0
(
t
).
When
S
1
(
t
)
=−
S
0
(
t
),
E
S
1
=
E
S
0
=−
E
S
1
S
0
=
E
S
.FromEq.
3.34
,
8
E
S
η
2
max
γ
=
(3.35)
With the value of optimizing parameter in Eq. (
3.35
), probability of error from Eqs.
(
3.16
) and (
3.19
), is given by
2
erfc
p
o
(
T
)
1
P
e
=
2
√
2
σ
0
2
erfc
p
o
(
T
)
1
/
2
1
=
(3.36)
2
0
8
σ
2
erfc
γ
1
/
2
2
1
=
8
2
2
max
.
From the basic concept of optimal filter,
P
e
is minimum when
γ
=
γ
Therefore,
2
erfc
γ
1
/
2
2
Max
8
1
P
e
|
Min
=
(3.37)
2
erfc
E
S
1
/
2
1
P
e
|
Min
=
η
Equation
3.37
signifies that
the error probability depends only on the signal energy
and not on the signal wave shape
.
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