# Continuous Time Channels with Interference

I. Ivan, M. Mitzenmacher, J. Thaler, and H. Yuen

Abstract:

Khanna and Sudan studied a natural model of continuous time channels where signals are corrupted by the effects of both noise and delay, and showed that, surprisingly, in some cases both are not enough to prevent such channels from achieving unbounded capacity. Inspired by their work, we consider channels that model continuous time communication with adversarial delay errors. The sender is allowed to subdivide time into an arbitrarily large number $M$ of micro-units in which binary symbols may be sent, but the symbols are subject to unpredictable delays and may interfere with each other. We model interference by having symbols that land in the same micro-unit of time be summed, and we study $k$-interference channels, which allow receivers to distinguish sums up to the value $k$. We consider both a channel adversary that has a limit on the maximum number of steps it can delay each symbol, and a more powerful adversary that only has a bound on the average delay.

We give precise characterizations of the threshold between finite and infinite capacity depending on the interference behavior and on the type of channel adversary: for max-bounded delay, the threshold is at $D_{\text{max}}=\Theta(M \log\min{k, M}))$, and for average bounded delay the threshold is at $D_{\text{avg}} = \Theta(\sqrt{M \cdot \min\{k, M\}})$.

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