# Question: In this problem we extend the results of Exercise 3 36

In this problem, we extend the results of Exercise 3.36 to the case when there are more than two possible messages sent. Suppose now that the communication system sends one of four possible messages, M = 1, M = 2, M = 3 and, each with equal probability. The corresponding conditional PDFs of the voltage measured at the receiver are of the form

Find and plot Pr (M = m| X =x) as a function of for σ2.

Determine the range of x for which Pr (M= 0| X =x) > Pr (m =m| X = x) for all m ≠ 0. This will be the range of x for which the receiver will decide in favor of the message M = 0.

Determine the range of for which Pr (M= 0| X =x) > Pr (m =m| X = x) for all m ≠ 1. This will be the range of for which the receiver will decide in favor of the message M =1.

Based on your results of parts (b) and (c) and the symmetry of the problem, can you infer the ranges of for which the receiver will decide in favor of the other two messages, M = 2 and M =3?

Find and plot Pr (M = m| X =x) as a function of for σ2.

Determine the range of x for which Pr (M= 0| X =x) > Pr (m =m| X = x) for all m ≠ 0. This will be the range of x for which the receiver will decide in favor of the message M = 0.

Determine the range of for which Pr (M= 0| X =x) > Pr (m =m| X = x) for all m ≠ 1. This will be the range of for which the receiver will decide in favor of the message M =1.

Based on your results of parts (b) and (c) and the symmetry of the problem, can you infer the ranges of for which the receiver will decide in favor of the other two messages, M = 2 and M =3?

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