Wireless sensor network applications often involve the computation of pre-defined functions of the measurements such as for example the arithmetic mean or maximum value. Standard approaches to this problem separate communication from computation: digitized sensor readings are transmitted interference-free to a fusion center that reconstructs each sensor reading and subsequently computes the sought function value. Such separation-based computation schemes are generally highly inefficient as a complete reconstruction of individual sensor readings at the fusion center is not necessary to compute a function of them. In particular, if the mathematical structure of the channel is suitably matched (in some sense) to the function of interest, then channel collisions induced by concurrent transmissions of different nodes can be beneficially exploited for computation purposes. This paper proposes an analog computation scheme that allows for an efficient estimate of linear and nonlinear functions over the wireless multiple-access channel. A match between the channel and the function being evaluated is thereby achieved via some pre-processing on the sensor readings and postprocessing on the superimposed signals observed by the fusion center. After analyzing the estimation error for two function examples, simulations are presented to show the potential for huge performance gains over time-and code-division multipleaccess based computation schemes.

It is known that if the objective of a wireless sensor network is not to reconstruct individual sensor readings at a fusion center but rather to compute a linear function of them, then the interference property of the wireless channel can be beneficially harnessed by letting nodes transmit simultaneously. Recently, an analog computation scheme was proposed to show that it is possible to take the advantage of the interference property even if nonlinear functions are to be computed. The scheme involves some pre-processing on the sensor readings and post-processing on the superimposed signals observed by the fusion center. Correspondingly, this paper provides a thorough base for a theory of analog-computing functions over wireless channels by specifying what is the maximum achievable. This means it is determined for networks of arbitrary topology which functions are generally analog-computable over the channel and how many wireless resources are needed. It turns out that the considerations are closely related to the famous 13th Hilbert problem and that analog-computations can be universally performed in the sense that the pre-processing at sensor nodes is independent of the function to be computed. Universality reduces the complexity of transmitters and the signaling overhead, and it is shown that this property is preserved if nodes leave or join the network. Analog-computability is therefore of high practical relevance as it allows for an efficient computation of functions in sensor networks. Index Terms-Computation over multiple-access channels, wireless sensor networks, pre-and post-processing, 13th Hilbert problem.

In this paper, a clustered wireless sensor network is considered that is modeled as a set of coupled Gaussian multiple-access channels. The objective of the network is not to reconstruct individual sensor readings at designated fusion centers but rather to reliably compute some functions thereof. Our particular attention is on real-valued functions that can be represented as a post-processed sum of pre-processed sensor readings. Such functions are called nomographic functions and their special structure permits the utilization of the interference property of the Gaussian multiple-access channel to reliably compute many linear and nonlinear functions at significantly higher rates than those achievable with standard schemes that combat interference. Motivated by this observation, a computation scheme is proposed that combines a suitable data preand post-processing strategy with a nested lattice code designed to protect the sum of pre-processed sensor readings against the channel noise. After analyzing its computation rate performance, it is shown that at the cost of a reduced rate, the scheme can be extended to compute every continuous function of the sensor readings in a finite succession of steps, where in each step a different nomographic function is computed. This demonstrates the fundamental role of nomographic representations.

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This letter studies a multiple-access transmission scheme that exploits interference for an efficient function computation in sensor networks. The central question is how much channel knowledge is generally needed and how the channel estimation effort can significantly be reduced. It is first shown that the channel magnitude at the transmitters is sufficient to achieve the same performance as with full channel state information. It is further shown that for a wide range of fading distributions, no channel state information is needed at the transmitters, provided that the receiver has access to some statistical channel knowledge and is equipped with multiple antennas.Index Terms-Computation over multiple-access channels, function estimation, channel knowledge, wireless sensor networks.