Modulo 8 Channel and a Binary Source with Erasure Distortion
(35 points)
Consider the memoryless channel described by
Y = (X + Z) mod 8,
where the channel input X, output Y , and the noise Z are real valued. For example,
if x = 6.4 and z = 1.8 then y = 0.2. The noise is uniformly distributed on [−B, B],
i. e. Z ∼ Uniform[−B, B] and is independent of the input X. Assume that 0 < B ≤ 4
is a known channel parameter.
(a) Find the channel capacity as a function of B.
(b) A communication system is suggested where the permitted channel input values
are restricted to the set {1, 3, 5, 7}. Show that, when B = 1, it is still possible to
achieve the channel capacity. In fact, to achieve this capacity with zero probability
of error and with a very simple scheme.