4.5. RECEIVER SENSITIVITY 163
Figure 4.18: (a) Fluctuating signal generated at the receiver. (b) Gaussian probability densities
of 1 and 0 bits. The dashed region shows the probability of incorrect identification.
statistics of shot-noise contributionisin Eq. (4.4.6) is also approximately Gaussian for
p–i–nreceivers although that is not the case for APDs [86]–[88]. A common approx-
imation treatsisas a Gaussian random variable for bothp–i–nand APD receivers but
with different varianceσs^2 given by Eqs. (4.4.5) and (4.4.17), respectively. Since the
sum of two Gaussian random variables is also a Gaussian random variable, the sam-
pled valueIhas a Gaussian probability density function with varianceσ^2 =σs^2 +σT^2.
However, both the average and the variance are different for 1 and 0 bits sinceIpin Eq.
(4.4.6) equalsI 1 orI 0 , depending on the bit received. Ifσ 12 andσ 02 are the correspond-
ing variances, the conditional probabilities are given by
P( 0 / 1 )=
1
σ 1√
2 π∫ID−∞exp(
−
(I−I 1 )^2
2 σ 12)
dI=1
2
erfc(
I 1 −ID
σ 1√
2
)
, (4.5.3)
P( 1 / 0 )=
1
σ 0√
2 π∫∞IDexp(
−
(I−I 0 )^2
2 σ 02)
dI=1
2
erfc(
ID−I 0
σ 0√
2
)
, (4.5.4)
where erfc stands for the complementary error function, defined as [89]
erfc(x)=2
√
π∫∞xexp(−y^2 )dy. (4.5.5)By substituting Eqs. (4.5.3) and (4.5.4) in Eq. (4.5.2), the BER is given by
BER=
1
4
[
erfc(
I 1 −ID
σ 1√
2
)
+erfc(
ID−I 0
σ 0