Digital Signal Processing Reference
In-Depth Information
at time
t
for better classifying the phoneme /a/ at time
t
+1.Next,amore
detailed explanation of the matched filter is given.
q
(
t
)=
/p/
/a/
tt
+1
/a/
t
t
/a/
−
10
0
1
/p/
/p/
/p/
MIMO
channel
Channel
matched
filter
Fig. 6.4.
Example of how the channel models the phonotactic in /p a/ and how the
matched filter uses this information for improving the classification of the phoneme
/a/ at time instance
t
+1.
6.1.6 Channel Matched Filter
Following with the block diagram of Fig. 6.1, the received signal
x
(
t
)isintro-
duced to the channel matched filter. In telecommunication, the matched filter
is the optimal LTI system which maximizes the signal to noise ratio (SNR) in
the presence of Additive White Gaussian Noise (AWGN) channel [Linder 05].
The impulse response of the matched filter is the transposed conjugated time-
inverted of the channel impulse response
H
∗
T
t
). However, since all signals
involved in this analysis are real signals, then
H
∗
T
(
(
−
t
)=
H
T
(
−
−
t
). The out-
put signal of the matched filter is obtained by convolution:
y
(
t
)=
H
T
(
−
t
)
∗
x
(
t
)
(6.13)
⎡
⎣
⎤
⎦
h
11
(
−
t
)
∗
x
1
(
t
)+
···
+
h
n
1
(
−
t
)
∗
x
n
(
t
)
x
n
(
t
)
..................
h
1
n
(
h
12
(
−
t
)
∗
x
1
(
t
)+
···
+
h
n
2
(
−
t
)
∗
=
(6.14)
−
t
)
∗
x
1
(
t
)+
···
+
h
nn
(
−
t
)
∗
x
n
(
t
)
If all impulse responses are centered at the origin
t
=0,andthetime-
length of the positive and negative parts is equal to
d
2
then, (6.14) can be
expressed as:
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