Page 357 - Fiber Optic Communications Fund

338 Fiber Optic Communications

To find the minimum BER, the threshold r and the filter transfer function H() should be optimized. Let us
T
first consider the optimization of threshold r . P is minimum or maximum when
b
T
P b
= 0. (8.18)
r T
Differentiating Eq. (8.16) with respect to r and setting it to zero, we find
T
p (r )= p (r ), (8.19)
1 T 0 T
2
{ [ ] } { [ ] }
2
r − u (T ) r − u (T )
T
b
1
T
b
0
exp − = exp − . (8.20)
2 2 2 2
Thus, the optimum threshold r corresponds to the inter section of curves p (r) and p (r) in Fig. 8.2. From
T 0 1
Eq. (8.20), we see that
r − u (T )=±[r − u (T )]. (8.21)
T 1 b T 0 b
Taking the negative sign in Eq. (8.21), we find
r =[u (T )+ u (T )]∕2. (8.22)
T 0 b 1 b
If we choose the positive sign, it would lead to u (T )= u (T ), which is not true in our case. The optimum
b
b
1
0
threshold condition given by Eq. (8.19) is valid for arbitrary pdfs, while that given by Eq. (8.22) holds true
for Gaussian distributions. From Eq. (8.22), we see that the optimum threshold r is at the middle of u (T )
0
b
T
and u (T ). Since the conditional pdfs p (r) and p (r) are symmetrically located with respect to the optimum
b
0
1
1
threshold r (Fig. 8.2), P(0|1) and P(1|0) should be equal. Therefore, Eq. (8.17) can be rewritten as
T
2
{ [ ] }
∞ r − u (T )
1 0 b
b
P = √ ∫ exp − 2 dr. (8.23)
2 r T 2
Let
r − u (T )
0
b
z = √ , (8.24)
2
∞
1 2
b ∫ √
P = √ exp (−z ) dz
[r T −u 0 (T b )]∕ 2
( )
1 r − u (T )
T
b
0
= erfc √ , (8.25)
2
2
where erfc(⋅) is the complementary error function defined as
∞
2 2
erfc(z)= √ ∫ exp (−y ) dy. (8.26)
z
Using Eq. (8.22), Eq. (8.25) becomes
(√ )
1
P = erfc , (8.27)
b
2 8