Page 362 - Fiber Optic Communications Fund

Performance Analysis 343

x (t)
1
T b
x(t) ʃ (∙)dt r (T )
1
b
∑ If r (T ) > r (T )
1
0
b
b
1 (T b
0 select x (t),
1
Comparator
otherwise select
T b
n(t) ʃ (∙)dt r (T ) x 0 (t).
0
b
0
x (t)
0
Figure 8.7 Realization of the matched filter using a correlator when E = E .
1 0
If r(T b ) > 0
T b select x 1 (t),
Received signal r(T b ) Decision
ʃ (∙)dt
device otherwise
0
select x 0 (t).
x 1 (t)
Figure 8.8 A realization of the matched filter when x (t)=−x (t).
0
1
Example 8.1
Find the matched filter and its output at t = T for the following signals.
b
(a)
{
±A 0 < t < T
x(t)= B (8.54)
0 otherwise.
(b)
[ ]
⎧ (t−T B ∕2) 2
⎪±A exp − 0 < t < T
x(t)= 2T 2 B (8.55)
⎨ 0
0 otherwise.
⎪
⎩
Assume T << T .
B
0
Solution:
(a) To transmit ‘1’ (‘0’), A (−A) is sent over the interval 0 < t < T . Suppose ‘1’ is transmitted. The Fourier
B
transform of x(t) is
2A sin (T ∕2) −iT b ∕2
T B
b
̃ x ()= A exp (−it)dt = e (8.56)
1 ∫
0
=−̃x (). (8.57)
0