Digital Signal Processing Reference
In-Depth Information
1.4.2.3 The Matched Filter is a Correlator
From (
1.42
) it is possible to write that:
Z
1
y
ð
t
Þ¼
k
r
ð
k
Þ
s
ðð
t
t
o
Þþ
k
Þ
dk
¼
kR
rs
ð
t
t
o
Þ;
ð
1
:
44
Þ
1
where R
rs
(s) is the cross-correlation between the symbol and the received version
of it. Hence the matched filter is essentially a correlator. If r = s (noise-free
condition), kR(s) will have a symmetric shape with a maximum at s = 0, i.e., t -
t
o
= 0ort = t
o
. This value is y(t
o
)asin(
1.43
).
1.4.2.4 The Optimal Receiver
If there is a finite set of M symbols to be transmitted, {s
i
(t)| i = 1, 2, ..., M} and
one wants optimal reception for these symbols, one should use a bank of M filters,
each matched to one symbol only (see Fig.
1.29
). If the ith symbol s
i
(t) was
transmitted, then the ith matched filter will have a maximum at the time of optimal
reception, while the other filters will have small values. Hence the receiver will
decide that s
i
(t) was transmitted. In binary communication systems (such as
computer networks), one needs only two filters, matched to the two symbols that
represent logic ''0'' and logic ''1''.
1.5 Analog Filters
Filters play a significant role in signal processing and communication engineering.
This section will consider them in some detail.
Comparator at
t
=
T
s
1
(
t
)
y
1
(
t
)
T
∫
0
Received
Signal
s
2
(
t
)
Decision
at
t
=
T
:
s
k
was sent
where:
k
= arg[max{
Y
i
}]
OR:
Y
k
= max{
Y
i
}
y
2
(
t
)
max { Y
i
} ;
where :
Y
i
=
y
i
(
t
) |
t
=
T
i
= 1, 2,
T
∫
0
r
(
t
)
s
M
(
t
)
⋅⋅⋅
,
M
y
M
(
t
)
T
∫
0
Fig. 1.29
The optimal receiver for a finite set of symbols
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