3. Ternary signaling. Let the signal S be a random variable defined as follows: S = ???? ??? ?1 with probability 1 3 0 with probability 1 3 +1 with probability 1 3. The signal is sent over a channel with additive Laplacian noise Z, i.e., Z is a Laplacian random variable with pdf fZ(z) = ? 2 e??|z|, ??

(a) Find fY |S(y|s) for s = ?1, 0, +1 . (b) Find the optimal decoding rule ?SMAP(y) for deciding whether S is ?1, 0 or +1. Give your answer in terms of ranges of values of y. (c) Find the probability of decoding error for ?SMAP(y) in terms of ?.

3. Ternary signaling. Let the signal S be a random variable dened as follows: 1 with probability :1; S = 0 with probability % +1 With probability %. The signal is sent over a channel with additive Laplacian noise Z, i.e., Z is a Laplacian random variable with pdf A fz(z) = iceAM , 00