A random variable X takes on values of a sequence of 4QAM sy

A random variable X takes on values of a sequence of 4QAM symbols, namely 0, 1, 2, and 3. These symbols are transmitted over a communication channel with equal probability. The channel introduces additive noise, N, equivalent to adding 0 or 1 to each of the 4QAM symbols being transmitted. The addition is performed in modulo 4. It is assumed that the channel adds a 0 with probability 0.8 and adds a 1 with probability 0.2, independently of the 4QAM symbols transmitted. Determine the probability mass function (PMF) of the random variable Y, where Y = X + N mod 4.

Solution

P(1+0) means the proabability that 1 is transmitted and the channel adds 0

0mod4 = 0

1mod4 = 1

Therefore, Nmod4 = N

pmf:

Y=1: P(1+0) = 0.25*0.8 = 0.2

Y=2:P(1+1)+P(2+0) = 0.25*0.2+0.25*0.8 = 0.25

Y=3:P(2+1)+P(3+0) = 0.25*0.2+0.25*0.8 = 0.25

Y=4:P(3+1)+P(4+0) = 0.25*0.2+0.25*0.8 = 0.25

Y=5:P(4+1) = 0.25*0.2 = 0.05