Page 359 - Fiber Optic Communications Fund

340 Fiber Optic Communications

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curve |H()| and, therefore, decreases. If the filter is too narrow (Fig. 8.4(b)), a significant fraction of the
signal component is truncated by the filter and, therefore, the numerator of Eq. (8.29) becomes too small. The
∗
optimum filter transfer function can be obtained by setting the variation of with respect to H() and H ()
to zero:

= 0 and = 0. (8.30)
H H ∗
To find the variations given by Eq. (8.30), let us replace the integrals of Eq. (8.29) by summations:
[ ] 2
∑ [ ]
̃ x ( )− ̃x ( ) H( ) exp (−i T )Δ n
n
n
n b
n
1
0
n
= lim ∑ . (8.31)
2
Δ n →0 N 0 |H( )| Δ n
n
n
∗
can be optimized by setting its partial derivatives with respect to H( ) and H ( ) to zero. Note that
n
n
∗
H( ) and H ( ) are independent variables. Alternatively, Re[H( )] and Im[H( )] can also be chosen as
n
n
n
n
independent variables:
√
2 N[̃x ( )− ̃x ( )] exp (−i T ) ∗ N
n
1
n b
n
0
= − N H ( ) = 0, (8.32)
0
n
H( ) D D 2
n
where N and D denote the numerator and denominator of Eq. (8.29), respectively. Simplifying Eq. (8.32), we
obtain
∗ ∗
H( )= k[̃x ( )− ̃x ( )] exp (i T ),
n 1 n 0 n n b
∗ ∗
H()= k[̃x ()− ̃x ()] exp (iT ), (8.33)
1 0 b
where ( )
2D 1
k = √ (8.34)
N N 0
is an arbitrary constant, which we set to unity from now on. The same result can be obtained by setting the
∗
variation of with respect to H () to zero. The filter with the transfer function given by Eq. (8.33) is called
a matched filter. Using Eq. (8.33) in Eq. (8.29), we obtain
∞
1 2
max = N ∫ |̃x ()− ̃x ()| d
0
1
0 −∞
2 T b 2
= [x (t)− x (t)] dt, (8.35)
0
1
N ∫ 0
0
where we have made use of Parseval’s relations. Let
T b
E = ∫ x (t)x (t) dt, j = 1, 0, (8.36)
j
jk
k
0
E ≡ E . (8.37)
jj j
Using Eqs. (8.36) and (8.37) in Eqs. (8.35) and (8.27), we obtain
2
= [E + E − 2E ], (8.38)
max 1 0 10
N
0
( )
√
1 max
P b,min = erfc . (8.39)
2 8