A random variable X takes on values of a sequence of 4QAM symbols, namely 0, 1,

Question

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.

Explanation / Answer

Given channel adding 0 or 1 independently of symbol transmitted, P(X = a, N = b) = P(X = a) x P(N = b). Various values Y can take and their compositions are given in the following table.

X

0

1

2

3

0

1

2

3

N

0

0

0

0

1

1

1

1

Y

0

1

2

3

1

2

3

0

P(Y)

.25x.8

.25x.8

.25x.8

.25x.8

.25x.2

.25x.2

.25x.2

.25x.2

So, pmf of Y is:

y

0

1

2

3

P(y)

(.25x.8) + (.25x.2)=.25

(.25x.8) + (.25x.2)=.25

(.25x.8) + (.25x.2)=.25

(.25x.8) + (.25x.2)=.25

Thus, pmf of Y can be summarized as ‘Y takes four values 0, 1, 2, 3 with equal probability.

X

0

1

2

3

0

1

2

3

N

0

0

0

0

1

1

1

1

Y

0

1

2

3

1

2

3

0

P(Y)

.25x.8

.25x.8

.25x.8

.25x.8

.25x.2

.25x.2

.25x.2

.25x.2

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