The time domain analysis and implementation of matched filters
can be found in Matched
Filters.
A frequency domain interpretation of matched filters is very
useful
SNR=∫−∞∞
s
m
τ
h
m
T−τd
τ
2
N
0
2∫−∞∞|
h
m
T−τ|2d
τ
SNR
τ
s
m
τ
h
m
T
τ
2
N
0
2
τ
h
m
T
τ
2
(1)
For the
mm-th filter,
h
m
h
m
can be expressed as
s
m
˜
T=∫−∞∞
s
m
τ
h
m
T−τd
τ
=ℱ-1
H
m
f
S
m
f=∫−∞∞
H
m
f
S
m
fei2πfTd
f
s
m
˜
T
τ
s
m
τ
h
m
T
τ
ℱ
H
m
f
S
m
f
f
H
m
f
S
m
f
2
f
T
(2)
where the second equality is because
s
~
m
s
~
m
is the filter output with input
S
m
S
m
and filter
H
m
H
m
and we can now define
H
m
^
f=
H
m
f¯e−(i2πfT)
H
m
^
f
H
m
f
2
f
T
, then
s
m
˜
T=〈
S
m
f,
H
^
m
f〉
s
m
˜
T
S
m
f
H
^
m
f
(3)
The denominator
∫−∞∞|
h
m
T−τ|2d
τ
=∫−∞∞|
h
m
τ|2d
τ
τ
h
m
T
τ
2
τ
h
m
τ
2
(4)
h
m
*
h
m
0=∫−∞∞|
H
m
f|2d
f
=〈
H
m
f,
H
m
f〉
h
m
h
m
0
f
H
m
f
2
H
m
f
H
m
f
(5)
h
m
*
h
m
0=∫−∞∞
H
m
fei2πfT
H
m
f¯e−(i2πfT)d
f
=〈
H
m
^
f,
H
m
^
f〉
h
m
h
m
0
f
H
m
f
2
f
T
H
m
f
2
f
T
H
m
^
f
H
m
^
f
(6)
Therefore,
SNR=〈
S
m
f,
H
m
^
f〉2
N
0
2〈(
H
m
^
f,
H
m
^
f)〉≤2
N
0
〈(
S
m
f,
S
m
f)〉
SNR
S
m
f
H
m
^
f
2
N
0
2
H
m
^
f
H
m
^
f
2
N
0
S
m
f
S
m
f
(7)
with equality when
H
m
^
f=α
S
m
f
H
m
^
f
α
S
m
f
(8)
or
H
m
f=
S
m
f¯e−(i2πfT)
H
m
f
S
m
f
2
f
T
(9)
s
m
˜
t=ℱ-1
s
m
f
s
m
f¯=∫−∞∞|
s
m
f|2ei2πftd
f
=∫−∞∞|
s
m
f|2cos2πftd
f
s
m
˜
t
ℱ
s
m
f
s
m
f
f
s
m
f
2
2
f
t
f
s
m
f
2
2
f
t
(10)
where
ℱ-1
ℱ
is the inverse Fourier Transform operator.
