Abstract
We present sum-set inequalities specialized to the generalized degrees of freedom (GDoF) framework. These are information theoretic lower bounds on the entropy of bounded density linear combinations of discrete, power-limited dependent random variables in terms of the joint entropies of arbitrary linear combinations of new random variables that are obtained by power level partitioning of the original random variables. The bounds are useful instruments to obtain GDoF characterizations for wireless interference networks, especially with multiple antenna nodes, subject to arbitrary channel strength and channel uncertainty levels.
Highlights
Originating in additive combinatorics, sum-set inequalities are bounds on the cardinalities of sumsets
Let G(Z) ⊂ G denote the set of all bounded density channel coefficients that appear in Z = Lb(X2, X2), and let W be a random variable such that conditioned on any Go ⊂ (G/G(Z)) ∪ {W }, the channel coefficients G(Z) satisfy the bounded density assumption
We present a class of sum-set inequalities for bounded set linear combinations of random variables typically encountered in the generalized degrees of freedom (GDoF) framework
Summary
Originating in additive combinatorics, sum-set inequalities are bounds on the cardinalities of sumsets (given X1, X2, the sumset X1 + X2 {x1 + x2 : x1 ∈ X1, x2 ∈ X2}). Crossing over to network information theory, sum-set inequalities represent bounds on the entropies of sums of random variables, typically expressed in terms of the entropies of the constituent random variables. Prominent examples of such inequalities include Ruzsa’s sum-triangle inequality in additive combinatorics [1] and the entropy power inequality in information theory [2]. Sum-set inequalities are essential to the study of the capacity of wireless interference networks This is true for the studies of capacity approximations known as generalized degrees of freedom (GDoF) [3] through deterministic models [4] which de-emphasize the additive noise to place the focus exclusively on the interactions between signals. The intricacies of the sum-set structure are such that even a coarse metric like the degrees of freedom (DoF) for constant channel realizations turns out to be sensitive to fragile details of no conceivable practical relevance – e.g., whether the channel coefficients take rational or irrational values [7]
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