The theory and rationale behind matched filter receivers can be
found in
Matched Filters.
Example 1
s
1
t=t
s
1
t
t
for
0≤t≤T
0
t
T
s
2
t=-t
s
2
t
t
for
0≤t≤T
0
t
T
h
1
t=T-t
h
1
t
T
t
for
0≤t≤T
0
t
T
h
2
t=-T+t
h
2
t
T
t
for
0≤t≤T
0
t
T
∀t,0≤t≤2T:
s
1
˜
t=∫-∞∞
s
1
τ
h
1
t-τdτ
t
0
t
2
T
s
1
˜
t
τ
s
1
τ
h
1
t
τ
(1)
s
1
˜
t=∫0tτT-t+τdτ=12T-tτ2|0t+13τ3|0t=t22T-t3
s
1
˜
t
τ
0
t
τ
T
t
τ
0
t
1
2
T
t
τ
2
0
t
1
3
τ
3
t
2
2
T
t
3
(2)
s
1
˜
T=T33
s
1
˜
T
T
3
3
(3)
Compared to the correlator-type demodulation
ψ
1
t=
s
1
t
E
s
ψ
1
t
s
1
t
E
s
(4)
s
11
=∫0T
s
1
τ
ψ
1
τdτ
s
11
τ
0
T
s
1
τ
ψ
1
τ
(5)
∫0t
s
1
τ
ψ
1
τdτ=1
E
s
∫0tττdτ=1
E
s
13t3
τ
0
t
s
1
τ
ψ
1
τ
1
E
s
τ
0
t
τ
τ
1
E
s
1
3
t
3
(6)
Example 2
Assume binary data is transmitted at the rate of
1T
1
T
Hertz.
0⇒b=1⇒
s
1
t=st
0
b
1
s
1
t
s
t
for
0≤t≤T
0
t
T
1⇒b=-1⇒
s
2
t=-st
1
b
-1
s
2
t
s
t
for
0≤t≤T
0
t
T
X
t
=∑i=-PP
b
i
st-iT
X
t
i
P
P
b
i
s
t
i
T
(7)