Deterministic Identification for MC ISI-Poisson Channel
Mohammad J. Salariseddigh, Vahid Jamali, Uzi Pereg, Holger Boche, Christian Deppe, Robert Schober
Abstract
Several applications of molecular communications (MC) feature an alarm-prompt behavior for which the prevalent Shannon capacity may not be the appropriate performance metric. The identification capacity as an alternative measure for such systems has been motivated and established in the literature. In this paper, we study deterministic identification (DI) for the discrete-time Poisson channel (DTPC) with intersymbol interference (ISI) where the transmitter is restricted to an average and a peak molecule release rate constraint. Such a channel serves as a model for diffusive MC systems featuring long channel impulse responses and employing molecule counting receivers. We derive lower and upper bounds on the DI capacity of the DTPC with ISI when the number of ISI channel taps <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K$</tex> may grow with the codeword length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> (e.g., due to increasing symbol rate). As a key finding, we establish that for deterministic encoding, the codebook size scales as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$2^{(n\log n)R}$</tex> assuming that the number of ISI channel taps scales as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K=2^{\kappa\log n}$</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$R$</tex> is the coding rate and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\kappa$</tex> is the ISI rate. Moreover, we show that optimizing <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\kappa$</tex> leads to an effective identification rate [bits/s] that scales linearly with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> , which is in contrast to the typical transmission rate [bits/s] that is independent of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> .