We begin with the simpler case where the network only consists of PLs 1,2 and VL 1 whose outputs are y1, y2 and y(v)1 . The probability that two PLs and the VL all give
identical decision can be decomposed as
The binary symmetric nature of both PLs gives
Pr distributed, and w is a zero mean Gaussian random variable with variance var(w) = N0/2 = 1/2SNR1, we obtain
The first integrand of (B.23) can be expressed as a standard bivariate Gaussian
distribution function Q(x, y; ρ) which, in turn, yields the Craig form as [37, (4.17)]
Using the method described in [37, ch.5] and the identity [37, (5.A.11)]
∫ (
Invoking the relation [37, (6.42)]
Q(−x, −y; ρ) = 1 − Q(x) − Q(y) + Q(x, y; ρ) x, y ≥ 0
and (B.25), we express the conditional correct (pairwise) SMP as
Pr
(by1 =by1(v) = 1|x = 1)
= Pr
(by1 = by(v)1 =−1|x = −1)
=
∫
h
Pr (w >−h, m > −h|x = −1, h) f(h)dh
=1− e1− e(v)1 + pem(e1, a(v)1 )def= pcm(e1, a(v)1 ) (B.26)
Summarizing (B.21)—(B.26), we then obtain
p12(v1)= e2pem(e1, a(v)1 ) + (1− e2)pcm(e1, a(v)1 ) (B.27)
which is (4.27) in the main text. The other probabilities, (4.30)-(4.33), can be similarly derived with the aid of the following two identities [37, (6.42)]:
Q(x, y, ρ) = Q(x)− Q(x, −y, −ρ), x ≥ 0, y < 0 (B.28) Q(x, y, ρ) = Q(y)− Q(−x, y, −ρ), x < 0, y ≥ 0 (B.29)
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