# Question: 40 pointsthe correlation receiver in figure 413 of text book...

###### Question details

40 points)The correlation
receiver in Figure 4-13 of text book is used to demodulate and
decode a binary PAM signal. Let s_{1}(t) and s_{2}(
t) represent the two transmitted symbols, let
a_{1}(iT_{b}) represent the output of the
integrator for the ith bit if s_{1}(t) was transmitted in a
noiseless environment, and let a_{2}(iT_{b})
represent the output of the integrator for the ith bit if
s_{2}( t) was transmitted in a noiseless environment.
(hint: see text book section 4.4.4.2)

Assume that the source outputs bits that are independent but not necessarily equiprobable (let z represent the probability that a "1" was transmitted). Derive the expression for the optimal threshold setting in the presence of additive, white Gaussian noise.

Let the optimum threshold determined in Part a be represented as
τ_{opt}· Let represent the average normalized noise power
at the output of the integrator. Derive the expression for
probability of bit error in terms of the Q function.