Question

In a ternary (three level) communications system, 3 is transmitted three times more frequently than a 1 while a 2 is transmitted two times more frequently than 1.

Probability of receiving a 2 when is 1 transmitted is α/2 (same for receiving a 3).

Probability of receiving a 1 when 2 is transmitted is β/2 (same for receiving a 3).

The probability of receiving a 2 when 3 is transmitted is γ/2 (same for receiving a 1).

(a) Draw the transition or the structure of the channel indicating the various probabilities

(b) At the receiver a '1' is observed. Using part (a) or directly, estimate the probability that the observed 1 was transmitted as a 1?

(c) Using parts (a), (b), or directly, what is the probability of receiving a 2' or a ‘3'?

Answer 1

(a) The transition diagram will look like this:

For example, when 1 is transmitted, the probability of receiving 1 is , the probability of receiving 2 is , and the probability of receiving 3 is also .

Also, let x be the probability of transmitting 1. Then, probability of transmitting 2 is 2x and the probability of transmitting 3 is 3x. So,

So, probability of transmitting 1, 2, and 3 are 1/6, 2/6 = 1/3, and 3/6 = 1/2 respectively.

(b) From the transition diagram, we can see that 1 can observed because of three transitions. So, P(1 was transmitted | 1 was observed) = P(1 was transmitted and 1 was observed) / P(1 was observed)

(c) Again, from the transition diagram, we can see that P(receiving 2 or 3)

= P(sending 1) * P(receiving 1 as 2 or 3) + P(sending 2) * P(receiving 2 as 2 or 3) + P(sending 3) * P(receiving 3 as 2 or 3)