Channel
$$ \mathcal{X} \to \boxed{P[Y|X]} \to \mathcal{Y} \\ $$Definition
A system $ (\mathcal{X}, P[Y|X], \mathcal{Y}) $.
- $ \mathcal{X} $ : An Input Alphabet. $ \mathcal{Y} $ : An Output Alphabet.
- $ P[Y|X] $ : A Probability Transition Matrix.
Memoryless Property
Output only depends on the current input: $$ \begin{aligned} p_{Y^n|X^n}(y^n | x^n) = \prod_{i=1}^{n} p_{Y|X}(y_i | x_i) \end{aligned} $$