Digital Signal Processing Reference
In-Depth Information
From Schwartz inequality discussed in the previous sections, maximum value of
SNR can be obtained as,
T
S
2
η
2
E
S
η
s
2
SNR
max
=
(
t
)
dt
=
(3.42)
0
Property 3.2
The impulse response of the matched filter is the mirror image of the
signal.
From Schwartz inequality it is easily understandable that, the impulse response of
the matched filter is
k
×
s
(
T
S
−
t
)
;0
≤
t
≤
T
S
h
(
t
)
=
(3.43)
0;
elsewhere
This implies that the impulse response of the filter that is responsible for maxi-
mum signal to noise ratio, is the mirror image of the message signal, delayed by the
symbol duration.
Property 3.3
Matched filter does not contribute inter symbol interference (ISI).
For the delay T
S
and the convolution operation, the match filer output is always non-
zero for t > 2T
S
. Hence, match filter, though introduces a delay, does not introduce
ISI as it makes the output to go to zero at the next sampling instant.
3.5 The Correlator
The correlator is a
coherent
baseband receiver system. The term 'correlation' means
similarity. The present system takes the noise accumulated signal from the commu-
nication channel and generates a known difference waveform with respect to raw
signal component, locally. The similarity of all the components on the received
signal is then measured with the local signal. Obviously the similarity with the
signal component is much higher than the similarity (correlation) with the noise
component. As the system maximizes the correlation, the system is called as
correlator.
The in-system arrangement is shown in the Fig.
3.12
. The received signal along-
with noise is multiplied by a locally generated signal
S
1
(t)
S
0
(t)
. The output
is then passed through an integrator (LPF) whose response is samples at t
−
T
S
.
Immediately after each sampling, the energy storing element of the integrator is
discharged.
The output signal and noise are
=
T
S
1
τ
S
o
(
T
S
)
=
S
i
(
t
)
[
S
1
(
t
)
−
S
0
(
t
)
]
dt
(3.44)
0
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