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

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 13^th Hilbert problem.