Polar Codes

1.2 Easy Channels There are two classes of channels that are “easy” to communicate over. ˆ Firstly, completely noisy channels with zero Shannon capacity are “easy” to communicate at capacity. Such channels are characterized by the property that observed probabilities are independent of their inputs, i.e., W (y|0) = W (y|1), ∀ y ∈ Y. (2) By Shannon’s channel coding theorem, the capacity C is 0 bits per channel use. ˆ Secondly, completely clean channels are also “easy” to communicate over. Such channels are characterized by W (y|0) ·W (y|1) = 0, ∀ y ∈ Y. (3) So, observing the output determines the input. (This is, in fact, not strictly true. If Y = R and the admissible outputs y for x = 0 is the set of rationals Q while the admissible outputs y for x = 1 is the set R \Q, deciding whether a number is rational may be difficult.)