Another binary channel has A = B = {0,1}, and no memory; the probability of a correct transmission is p, for each digit transmitted. Find the probabilities in problem 2, above, for this channel ? 1. For a particular memoryless channel we have A = {0,1}, B = {0,1,∗}, and the channel treats the input digits symmetrically; each digit has probability p of being transmitted correctly, probability q of being switched to the other digit, and probability r of being fuzzed, so that the output is ∗. Note that p +q +r = 1.
(a) Give the matrix of transition probabilities, in terms of p,q, and r.
(b) In terms of n, p, and k, what is the probability of exactly k errors (where an error is either a fuzzed digit or a switched digit) in the transmission of a binary word of length n, over this channel?
(c) Suppose that ∗ is eliminated from the output alphabet by means of coin flip, with a fair coin. Whenever ∗ is received, the coin is flipped; if heads comes up, the ∗ is read is 0, and if tails comes up, it is read as 1. What is the new matrix of transition probabilities? Is the channel now binary symmetric?
(d) Suppose that ∗ is eliminated from the output alphabet by merging it with 1. That is, whenever ∗ is received, it is read as 1 (this amounts to a coin flip with a very unfair coin). What is the new matrix of transition probabilities? Is the channel now binary symmetric?