Mario Goldenbaum

Robust Analog Function Computation via Wireless Multiple-Access Channels

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 post-processing 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 multiple-access based computation schemes.

On Function Computation via Wireless Sensor Multiple-Access Channels

In wireless sensor networks, the identity of a particular sensor node and a complete reconstruction of sensed data at a designated sink node may be not needed. Indeed, the objective is often to compute a certain function of the sensed data. Such desired functions can be for example the arithmetic mean, the geometric mean, polynomials and other functions that adequately match the mathematical characteristic of the underlying multiple-access channel. In this paper, we propose a simple practical approach to compute desired functions of sensor network data, exploiting explicitly the mathematical characteristic of the wireless sensor multiple-access channel (WSMAC). In contrast to traditional schemes that are designed to combat interference caused by other connections, we exploit this interference with the goal of computing the desired functions, which is in a sense a paradigm shift. This leads directly to a higher data rate in terms of function computation or a higher SNR in comparison to other schemes like time division multiple access (TDMA). Our approach needs no extensive symbol or phase synchronization, since the measured values are converted into the tranmit power of a specific random transmit sequence with unit norm. Only a coarse block synchronization is necessary so that the proposed scheme is easy to implement.

Computing functions via simo multiple-access channels: Howmuch channel knowledge is needed?

We view a wireless sensor network as a collection of sensor nodes that observe sources of information, process the picked up data and send it to a sink node, with the goal of computing a desired function of the measurements. To this end, we consider a previously proposed coding scheme that exploits the underlying fading multiple-access channel (MAC) to efficiently estimate the function values. The main problem addressed in this paper is how much channel state information (CSI) is needed at the sensor nodes to obtain sufficiently good estimates? First we show that there is no performance loss, independent of fading distributions, if, instead of perfect CSI, each sensor node has only access to the modulus of its channel coefficient. In the case of multiple antenna elements at the sink node and specific independent distributed fading environments, it is shown that CSI at sensor nodes is not necessary and a very simple correction of fading effects can be performed at the sink based on some statistical channel knowledge. In many cases, fading improves the estimation accuracy due to the multiple-access nature of the channel.

Harnessing Interference for Analog Function Computation in Wireless Sensor Networks

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.

On the effect of feedback delay in the downlink of multiuser OFDM systems

Allocating wireless resources often relies on the accurate feedback of channel state information (CSI). In this paper, we study how strongly a multiuser OFDM downlink with single-cell scheduling and channel estimation suffers from feedback delay. Unlike previous work, we study this degradation for optimal joint power and rate allocation under fairness constraints. Comparing the performance of the ideal case to delayed CSI shows that (i) adjusting the schedulers fairness cannot mitigate the strong performance loss due to feedback delay but that (ii) simple linear channel prediction is a powerful tool to do so.

Fast average consensus in clustered wireless sensor networks by superposition gossiping

In this paper we propose a gossip algorithm for average consensus in clustered wireless sensor networks called superposition gossiping, where the nodes in each cluster exploit the natural superposition property of wireless multiple-access channels to significantly decrease local averaging times. More precisely, the considered network is organized into single-hop clusters and in each cluster average values are computed at a designated cluster head via the wireless channel and subsequently broadcasted to update the entire cluster. Since the clusters are activated randomly in a time division multiple-access fashion, we can apply well-established techniques for analyzing gossip algorithms to prove the convergence of the algorithm to the average consensus in the second moment and almost surely, provided that some connectivity condition between clusters is fulfilled. Finally, we follow a semidefinite programming approach to optimize wake up probabilities of cluster heads that further accelerates convergence.

Computing the geometric mean over multiple-access channels: Error analysis and comparisons

The paper deals with the problem of data transmission and function computation of the sensed data in wireless sensor networks, in which multiple sensor nodes transmit their data to one sink node over a wireless multiple-access channel. We focus on the problem of computing the geometric mean at the sink node by merging the data transmission and function computation into one step via an explicit utilization of channel collisions caused by simultaneous transmissions of sensor nodes. The paper provides the analysis of the estimation error and compares the scheme with traditional time division multiple-access based schemes to indicate potential for significant performance gains.

On the Channel Estimation Effort for Analog Computation over Wireless Multiple-Access Channels

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.

Nomographic Functions: Efficient Computation in Clustered Gaussian Sensor Networks

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 pre- and 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.

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

Recently, it has been shown that the superposition property of wireless multiple-access channels can be exploited to compute functions in sensor networks much more efficiently. By using appropriate pre- and post-processing functions operating on real sensor readings and the superimposed signal received by a fusion center, every function of the measurements is in principle computable by means of the wireless channel in which the pre-processing functions, and therefore the transmitting nodes, do not depend on the function of interest. In this paper we extend these general considerations by examining how robust this kind of universality is against variations in network topology due to nodes that drop out of the network or due to new nodes that connect to the network.