Page 361 - Fiber Optic Communications Fund

342 Fiber Optic Communications

8.2.1 Realization of the Matched Filter
Suppose the input to the matched filter is y(t) (= x(t)+ n(t)). The decision is based on the signal r(t), which
is given by
∞
r(t)=[ −1 {̃y()H()}] = y()h(t − ) d. (8.48)
∫
−∞
In Eq. (8.48), we have used the fact that the product in the spectral domain becomes convolution in the time
domain. From Eq. (8.42), we have
h(t)= x (T − t)− x (T − t), (8.49)
1 b 0 b
h(t − )= x [T −(t − )] − x [T −(t − )]. (8.50)
1 b 0 b
Hence,
∞
r(t)= y()[x (T + − t)− x (T + − t)] d. (8.51)
∫
0
b
b
1
−∞
The decision is made based on the sample of r(t) at t = T :
b
∞
r(T )= y()[x ()− x ()]d = r (T )− r (T ), (8.52)
b ∫ 1 0 1 b 0 b
−∞
where
T b
r (T )= y()x ()d, j = 0, 1. (8.53)
j b ∫ j
0
In Eq. (8.53), we have made use of the fact that x (t) is zero when t < 0 and t > T . Thus, the matched filter
j b
can be realized by the correlation receiver shown in Fig. 8.6. If the energies of the signal u (t) and u (t) are
1 0
equal, i.e., E = E , from Eq. (8.47) we have r = 0. In this case, the equivalent form of correlation receiver is
1 0 T
shown in Fig. 8.7. If x (t)=−x (t), a simplified form of correlation receiver as shown in Fig. 8.8 may be used.
0 1

x 1 (t)
T r (T )
x(t) b 1 b If r(T ) > r T
b
∑ ʃ (∙)dt + select x 1 (t)
0
Decision
∑
T * r(T b ) device
n(t) b Otherwise
ʃ (∙)dt select x (t)
0 r (T ) 0
b
0
x (t)
0
Figure 8.6 Realization of the matched filter using correlators.