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

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r (t)s1(t)dt 1 if z (i Tb) 20 Threshold 0 if z (iTb) 0 r(t)s (t) ()dt (i-1)Tb (i-1)Tb comparator Integrator Receiver Figure 4-13 An equivalent design the correlation receiver.40 points)The correlation receiver in Figure 4-13 of text book is used to demodulate and decode a binary PAM signal. Let s1(t) and s2( t) represent the two transmitted symbols, let a1(iTb) represent the output of the integrator for the ith bit if s1(t) was transmitted in a noiseless environment, and let a2(iTb) represent the output of the integrator for the ith bit if s2( 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 τopt· 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.

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