2. [Communication in Gaussian Noise] A wireless communication system consists of a transmitter and a receiver. The transmitter sends a signal x, and the receiver observes Y = x+Z, where Z is a noise term, modeled as a Gaussian random variable with mean z = 0 and variance = 1. (a) Suppose the transmitted signal is x = 1. What is the pdf of the received signal Y? (b) Now suppose the transmitted signal can be either x = -1 or x = 1. The receiver uses the following decoding rule: if Y > 0, it declares that x = 1; if Y < 0, it declares that x=-1. Assuming that the transmitter sends -1 or +1 with probability 1/2 each, what is the receiver's error probability? (c) Now suppose the transmitted signal x can be chosen from three possible values: x = -2, x = 0 and x = 2. The receiver now uses the following decoding rule: if Y < -1, it declares x = -2, if Y> 1, it declares x = 2, and otherwise it declares x = 0. Assuming the transmitter sends each possible symbol with probability 1/3, what is the receiver's error probability? 700 x 391