Timing Channel: Achievable Rate in the Finite Block-Length Regime

• Symbols denoting random variables are capitalized but symbols denoting their realization are not. • Bold face is used for vectors. • Blackboard bold or non-italic face are used for sets, calligraphic face for system of sets. • We use x̄ as a shorthand for the difference 1 − x for x ∈ [0, 1]. • Z is the set of all integers and Z+ = {z ∈ Z : z ≥ 0}. • 2 is the set {0, 1}. • [n] denotes the set {z ∈ Z : 1 ≤ z ≤ n}. • Given the input and output sets X and Y a channel is a function PY n|Xn(·|·) : A × X → [0, 1] such that for each x ∈ X PY n|Xn(·|x) is a probability measure on A. A is some σ-algebra over Y. • We define the finite sets X and Y to be Z+ × 2n−1 and 2, repectively, and associate the σ-algebras P(Xn) and P(Yn) with them. P indicates the power set. • Given a distribution PXn on X we define the distribution PY n = ∫ X PY n|Xn(y |x)PXn(dX) and the function i(x,y) = log PY n|Xn (y |x) PY n (y) for x ∈ X and y ∈ Y. This function is called the information density. • The expected value of a random variable X is denoted by E[X ]. • The mutual information between two random vectors X and Y n is E[i(x,y)]. • The capacity C(λ) of the channel is supPXn E[i(x ,y)] where PXn is such that the rate E[ ∑n−1 i=1 Xi]/(n−1) does not exceed λ. • The binary entropy function H(p) = −p log p− p̄ log p̄. • We drop subscripts whenever they are clear from the context. For example PY n|Xn(y|x) = P (yn|xn).