164 CHAPTER 4. OPTICAL RECEIVERS
Equation (4.5.6) shows that the BER depends on thedecision threshold ID.In
practice,IDis optimized to minimize the BER. The minimum occurs whenIDis chosen
such that
(ID−I 0 )^2
2 σ 02
=
(I 1 −ID)^2
2 σ 12+ln(
σ 1
σ 0)
. (4.5.7)
The last term in this equation is negligible in most cases of practical interest, andIDis
approximately obtained from
(ID−I 0 )/σ 0 =(I 1 −ID)/σ 1 ≡Q. (4.5.8)An explicit expression forIDis
ID=
σ 0 I 1 +σ 1 I 0
σ 0 +σ 1. (4.5.9)
Whenσ 1 =σ 0 ,ID=(I 1 +I 0 )/2, which corresponds to setting the decision threshold
in the middle. This is the situation for mostp–i–nreceivers whose noise is dominated
by thermal noise (σTσs) and is independent of the average current. By contrast,
shot noise is larger for bit 1 than for bit 0, sinceσs^2 varies linearly with the average
current. In the case of APD receivers, the BER can be minimized by setting the decision
threshold in accordance with Eq. (4.5.9).
The BER with the optimum setting of the decision threshold is obtained by using
Eqs. (4.5.6) and (4.5.8) and depends only on theQparameter as
BER=
1
2
erfc(
Q
√
2
)
≈
exp(−Q^2 / 2 )
Q√
2 π, (4.5.10)
where the parameterQis obtained from Eqs. (4.5.8) and (4.5.9) and is given by
Q=
I 1 −I 0
σ 1 +σ 0. (4.5.11)
The approximate form of BER is obtained by using the asymptotic expansion [89]
of erfc(Q/
√
2 )and is reasonably accurate forQ>3. Figure 4.19 shows how the BER
varies with theQparameter. The BER improves asQincreases and becomes lower than
10 −^12 forQ>7. The receiver sensitivity corresponds to the average optical power for
whichQ≈6, since BER≈ 10 −^9 whenQ=6. The next subsection provides an explicit
expression for the receiver sensitivity.
4.5.2 Minimum Received Power
Equation (4.5.10) can be used to calculate the minimum optical power that a receiver
needs to operate reliably with a BER below a specified value. For this purpose theQ
parameter should be related to the incident optical power. For simplicity, consider the
case in which 0 bits carry no optical power so thatP 0 =0, and henceI 0 =0. The power
P 1 in 1 bits is related toI 1 as
I 1 =MRP 1 = 2 MRP ̄rec, (4.5.12)