Digital Signal Processing Reference
In-Depth Information
Tutorial 32
Q: The pulse x(t) shown in Fig. a below is sent along a communication line. At
the receiver, a matched filter h(t) = x(-t) is used for optimal detection.
1. Assume that the noise at the receiver has a flat spectrum, but is bandlimited to
10 kHz. The variance of noise is r
2
. What is the maximum signal power to
noise power ratio (SNR) at the input of the matched filter (i.e., what is SNR
i
at
the time of optimal detection)?
2. What is the maximum SNR at the output of the matched filter, SNR
o
?
3. What is the ratio
ð
SNR
o
j
max
Þ=ð
SNR
i
j
max
Þ
?
4. Repeat the above steps for s(t) in Fig. b.
(a)
(b)
x
(
t
)
s
(
t
)
1
1
t
t
0
T
=1
0
T
=1
Transmitter ( Tx )
AWGN Channel
Receiver ( Rx )
Sampler
[at
t
=
t
o
]
n
(
t
)
MF
z
(
t
) =
y
(
t
) +
n
o
(
t
)
r
(
t
) =
x
(
t
) +
n
(
t
)
x
(
t
)
H
(
f
)
SNR
i
(
t
) = |
x
(
t
)|
2
/
2
SNR
o
(
t
) = |
y
(
t
)|
2
/
2
σ
σ
o
G
n
(
f
)
x
(
t
)
h
(
t
)
η
/2
1
1
t
f
t
0
1
−
B
0
B
−1
0
Solution: Since h(t) = x(t
o
- t) will maximize SNR
o
at t = t
o
, we expect.
SNR
o
|
max
at t = 0 here, as t
o
= 0. Note that t
o
is the time of optimal reception. We
can reach this result from x(t)*h(t) as follows:
y
ð
t
Þ¼
x
ð
t
Þ
h
ð
t
Þ¼
Z
1
v
¼
k
t
¼
Z
1
x
ð
k
Þ
h
ð
t
k
Þ
|{z}
¼
x
ð
k
t
Þ
dk
!
x
ð
v
þ
t
Þ
x
ð
v
Þ
dv
¼
R
x
ð
t
Þ
1
1
Following Tutorial 23, we find y(t) as shown below.
y
(
t
)
1
t
−1
0
1
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