Analyzing the space of functions analog-computable via wireless multiple-access channels

Abstract

To efficiently compute linear functions of the measurements in sensor networks, it was recently shown that the superposition property of the wireless multiple-access channel can profitably be exploited. Using suitable pre - and postprocessing functions operating on real sensor readings and on the superimposed signal received by a fusion center, respectively, the natural computation property of the wireless channel can be adapted such that a much larger class of functions is efficiently computable as well. In this paper, we analyze the corresponding space of functions which are in principle computable, or at least approximable, in an analog fashion via a wireless multiple-access channel and show to what extend this impacts the communication pattern and the complexity of nodes. Finally, we change the transmission scenario to a sequence of successively received multiple-access channel output-signals and observe that the resulting questions on the computability of functions are related to the famous 13th Hilbert problem

Analog computation via wireless multiple-access channels: Universality and robustness

“…The identity map on any set A is described by idA : A → A. Definition 1 (Nomographic Functions [5]). Let A k , k ≥ 2, be a metric space.…”

“…In [5] we have shown that every f ∈ F[E n ] is essentially computable by exploiting the superposition property of a wireless MAC (i.e., mathematically N [E n ] ≡ F[E n ]). The surprising fact was that the pre-processing functions are universal and thus independent of the desired function.…”

“…In this paper we extend the work of [5] by analyzing the robustness of the universal computation scheme against variations in network topology. More precisely, after introducing the system model in Section 2, we analyze in Section 3 if the universality property is preserved when nodes drop out of the network (due to for instance failures or battery depletion) or when new sensor nodes join the network.…”

“…A natural question that arose in this context was which functions are computable 1 via an ideal MAC and we found in [5] that the corresponding function space is equal to the space of nomographic functions (see Definition 1). If no additional restrictions M. Goldenbaum and S. Stańczak were supported in part by the German Research Foundation (DFG) under grant STA 864/3-1 and H. Boche by a start-up fund of Technische Universität München.…”

“…are imposed, then in fact every real function of n variables is nomographic and thus in principle computable via the channel, where n denotes the number of active nodes in the network. A surprising observation in [5] was that the corresponding pre-processing functions are completely independent of the function the fusion center aims to evaluate at sensor readings, which results in universal computation schemes consisting of nodes with simple architecture that do not need to be updated if the function of interest changes.…”