Page 491 - Fiber Optic Communications Fund

472 Fiber Optic Communications

by J complex coefficients (or 2J real coefficients) of the expansion in a set of orthonormal basis functions.
Let us represent the signal and noise fields using an orthonormal set of basis functions as
J−1
∑
s (T)= s F (T), (10.352)
j j
in
j=0
J−1
∑
n(T)= n F (T), (10.353)
j j
j=0
where {F (T)} is a set of orthonormal functions,
j
∞
∗
F (T)F (T)dT = 1if j = k
∫ j k
−∞
= 0 otherwise. (10.354)
Because of the orthogonality of the basis functions, it follows that
∞
∗
n = ∫ n(T)F (T)dT. (10.355)
j
j
−∞
Using Eqs. (10.355) and (10.348)–(10.350), we obtain
< n >= 0, (10.356)
j
∗
< n n > = if j = k
j k
= 0 otherwise, (10.357)

< n n ) >= 0. (10.358)
j k
Using Eqs. (10.352) and (10.353) in Eq. (10.347), we find
J−1
∑
s (T)= (s + n )F (T). (10.359)
out j j j
j=0
Suppose ‘1’ is transmitted (a = 1). We choose F (T)= p(T) so that
0
0
√
s = E if j = 0
j
= 0 otherwise. (10.360)
Eq. (10.359) can be written as

J−1
√ ∑
s (T)=( E + n )p(T)+ n F (T). (10.361)
out 0 j j
j=1
Let us assume that the signal power is much larger than the noise power and s (T) is real. Let
in
n(T)= n (T)+ in (T), (10.362)
r i