Problem 3. Suppose we receive a BPSK signal of the form r(t) = A₁ cos(2π fet) +n(t), 0≤t≤T = where n(t) is a noise process, T = 1 sec, fc = 1 KHz and A₁ ±2 with equal probability. The signal is demodulated in an optimal manner and then an optimal threshold is used in making a decision. For demodulation assume the received signal r(t) is mixed with a cosine wave of amplitude 1 with the same frequency and phase of the transmitted signal of interest and then low pass filtered by integrating from 0 to T. With this approach suppose at the output of the demodulator (just prior to the threshold comparator) the noise (call it n) is characterized by the density 1 1 2 -2.0 ≤ n < 0 p(n) = 1 1 = 2 4 -n, 0≤ n ≤ 2.0 0 elsewhere. a. Using the optimal detector (with the corresponding optimal threshold) as described above compute the probability of a correct decision, Pc, od for the value of A₁. You may assume that fe is known to the receiver. b. Now suppose an interfering signal is also present at the same frequency so that the received signal is r(t) = A₁ cos(2π fet) + A2 cos(2π fet +3π/4)+n(t) 0≤t≤T. where A2 = 0.707. Find the overall probability of a correct decision at the receiver (you may assume that no adjustments to the receiver design are made due to the jammer). c. Same setup as part (b) but now assume that during a BPSK symbol time the interferer is present with probability 1/2 (if not present then the interferer is completely absent during that symbol time). Find the overall probability of a correct decision at the receiver (again using the same receiver design as part (a) without any adjustments in the receiver design). d. Same setup as part (c) but now the receiver designer is aware that the interferer is present with probability 1/2 and knows the amplitude, frequency and phase of the interferer. The receiver designer adjusts the threshold used for decisions to optimize the probability of correct decision that takes into account the possible presence of the interferer. So for any particular BPSK symbol the designer knows the interferer is present with probability 1/2 but does not know if the interferer is actually present for any particular BPSK symbol time (the designer just knows the interferer is present with probability 1/2). Find the new threshold to use for decision making and find the overall probability of a correct decision at the receiver using the new threshold.