Problem 4 A base-band digital communication system using binary signals shown in the Figure for transmission of two equiprob- able messages. The transmitted signal is s,(t), i = {1,2} and the recieved signal is r(t) = s,(t) +n(t), where n(t) is the AWGN with power-spectral density No/2. 1. In a block diagram, give the precise specifications of the optimal receiver. What are the characteristics of the matched filter and the sampler and decision device? 2. Find the error probability of the optimal receiver. 3. Now assume the channel is not ideal, but has an impulse response of e(t)=8(t) +18(t-T/2). Using the same matched filter you used in the previous part, design an optimal receiver. 4. Assuming that the channel impulse response is c(t)=6(t)+ad(t-T/2), where a is a random variable uniformly distributed on [0, 1], and using the same matched filter, design the optimal receiver. h AWGN hin Input Modulator (b) Matched filter Sampler and decision Output