Signal to Noise Ratio (SNR) at the output of the
demodulator is a measure of the quality of the demodulator.

SNR=signal energynoise energy
SNR
signal energy
noise energy

(1)
In the correlator described earlier,

E
s
=|
s
m
|2
E
s
s
m
2
and

σ
η
n
2=
N
0
2
σ
η
n
2
N
0
2
.
Is it possible to design a demodulator based on linear time-invariant
filters with maximum signal-to-noise ratio?

If
s
m
t
s
m
t
is the transmitted signal, then the output of the
kth
kth
filter is given as

y
k
t=∫−∞∞
r
τ
h
k
t−τd
τ
=∫−∞∞(
s
m
τ+
N
τ
)
h
k
t−τd
τ
=∫−∞∞
s
m
τ
h
k
t−τd
τ
+∫−∞∞
N
τ
h
k
t−τd
τ
y
k
t
τ
r
τ
h
k
t
τ
τ
s
m
τ
N
τ
h
k
t
τ
τ
s
m
τ
h
k
t
τ
τ
N
τ
h
k
t
τ

(2)
Sampling the output at time

TT yields

y
k
T=∫−∞∞
s
m
τ
h
k
T−τd
τ
+∫−∞∞
N
τ
h
k
T−τd
τ
y
k
T
τ
s
m
τ
h
k
T
τ
τ
N
τ
h
k
T
τ

(3)
The noise contribution:

ν
k
=∫−∞∞
N
τ
h
k
T−τd
τ
ν
k
τ
N
τ
h
k
T
τ

(4)
The expected value of the noise component is

E
ν
k
=E∫−∞∞
N
τ
h
k
T−τd
τ
=0
ν
k
τ
N
τ
h
k
T
τ
0

(5)
The variance of the noise component is the second moment since the mean
is zero and is given as

σ(
ν
k
)2=E
ν
k
2=E∫−∞∞
N
τ
h
k
T−τd
τ
∫−∞∞
N
τ
'
*
h
k
T−
τ
'
*d
τ
'
ν
k
ν
k
2
τ
N
τ
h
k
T
τ
τ
'
N
τ
'
h
k
T
τ
'

(6)
E
ν
k
2=∫−∞∞∫−∞∞
N
0
2δτ−
τ
'
h
k
T−τ
h
k
T−
τ
'
*d
τ
d
τ
'
=
N
0
2∫−∞∞|
h
k
T−τ|2d
τ
ν
k
2
τ
'
τ
N
0
2
δ
τ
τ
'
h
k
T
τ
h
k
T
τ
'
N
0
2
τ
h
k
T
τ
2

(7)
Signal Energy can be written as

∫−∞∞
s
m
τ
h
k
T−τd
τ
2
τ
s
m
τ
h
k
T
τ
2

(8)
and the signal-to-noise ratio (SNR) as

SNR=∫−∞∞
s
m
τ
h
k
T−τd
τ
2
N
0
2∫−∞∞|
h
k
T−τ|2d
τ
SNR
τ
s
m
τ
h
k
T
τ
2
N
0
2
τ
h
k
T
τ
2

(9)
The signal-to-noise ratio, can be maximized considering the well-known Cauchy-Schwarz Inequality

∫−∞∞
g
1
x
g
2
x*d
x
2≤∫−∞∞|
g
1
x|2d
x
∫−∞∞|
g
2
x|2d
x
x
g
1
x
g
2
x
2
x
g
1
x
2
x
g
2
x
2

(10)
with equality when

g
1
x=α
g
2
x
g
1
x
α
g
2
x
. Applying the inequality directly yields an upper bound on SNR

∫−∞∞
s
m
τ
h
k
T−τd
τ
2
N
0
2∫−∞∞|
h
k
T−τ|2d
τ
≤2
N
0
∫−∞∞|
s
m
τ|2d
τ
τ
s
m
τ
h
k
T
τ
2
N
0
2
τ
h
k
T
τ
2
2
N
0
τ
s
m
τ
2

(11)
with equality

h
k
opt
T−τ=α
s
m
τ*
τ
h
k
opt
T
τ
α
s
m
τ
.
Therefore, the filter to examine signal

m
m should be

τ
h
m
opt
τ=
s
m
T−τ*
τ
τ
h
m
opt
τ
s
m
T
τ

(12)
The constant factor is not relevant when one considers the
signal to noise ratio. The maximum SNR is unchanged when both
the numerator and denominator are scaled.

2
N
0
∫−∞∞|
s
m
τ|2d
τ
=2
E
s
N
0
2
N
0
τ
s
m
τ
2
2
E
s
N
0

(13)
Examples involving matched filter receivers can be found

here. An analysis in the frequency
domain is contained in

Matched Filters
in the Frequency Domain.

Another type of receiver system is the correlation receiver. A performance
analysis of both matched filters and correlator-type receivers
can be found in Performance
Analysis.