For a communication system z = x + w, x is the signal to be transmitted over the noisy channel and is a binary random variable with equal probability being -1 or 1, w is a zero mean unit variance Gaussian noise, and z is the received signal. (a) Based on the received signal z, a decision needs to be made to determine which signal has been transmitted. Design such a decision making policy based on the maximum likelihood principle, what is the probability of error? (b) Use MATLAB to simulate the system, generate 100,000 random trials, use the decision making policy designed in (a) to detect the signal. How is the actual error rate comparing to the analytical result predicted by (a). (c) Repeat (a) and (b) assuming the signal x strength is doubled and is a binary random variable with equal probability being -2 or 2 and the noise is the same as before.