Simultaneous Water Filling in Mutually Interfering Systems

Abstract

In this paper we investigate properties of simultaneous water filling for a wireless system with two mutually interfering transmitters and receivers with non-cooperative coding strategies. This is slightly different from the traditional interference channel problem which assumes that transmitters cooperate in their respective coding strategies, and that interference cancellation can be performed at the receivers. In this non-cooperative setup, greedy capacity optimization by individual transmitters through various algorithms leads to simultaneous water filling fixed points where the spectrum of the transmit covariance matrix of one user water fills over the spectrum of its corresponding interference-plus-noise covariance matrix, and in our paper we study the properties of these fixed points. We show that at a simultaneous water filling point the eigenvectors of transmit covariance matrices at each receiver are aligned, and identify three regimes which correspond to simultaneous water filling that depend on the interference gains: a) complete spectral overlap, b) partial spectral overlap, and c) spectral segregation. These imply that the transmit covariance matrices will be white in regions of both overlap and segregation, but not necessarily white overall. We also consider performance as a function of interference gain and show that complete spectral overlap is a strongly suboptimal solution over a wide range of gains. Overall, our results suggest that for strong mutual interference, an effort should be made to do joint decoding over receivers since such collaboration can provide large capacity increases. For moderate interference, distributed and/or centralized conflict resolution algorithms would be most effective since more complex collaborative methods do not afford much improvement and strictly greedy methods such as water filling perform poorly, while for weak interference a laissez faire approach seems reasonable.

Spectrum sharing and cognitive radio

“…This is an interesting generalization of the similar relations between direct and cross channel gains as reported in [11], and [12], which show that if the mutual interfering effect of two links on each other is low, it is best to share the channel. Equation (5) shows that the same concept is true in the case that a specific QoS level for the primary and secondary system is guaranteed.…”

GADIA: A Greedy Asynchronous Distributed Interference Avoidance Algorithm

“…Secondly, as mentioned in the introduction, the outcome of the Iterative Water-filling algorithm might not always be near-optimal. In fact, as pointed out by Etkin et al [15] and O. Popescu et al [42], the outcome of the Iterative Waterfilling algorithm might be the only possible outcome and far from optimal. For example, let us consider the scenario in Fig.…”

“…There are also various proposed methods for decentralized (distributed) frequency allocation in different contexts (See, for example, [13], [15], [22], [28], [34], [37], [38], [40], [41], [42], [43], [45], [51], [56], [58], and [59]). These include methods based on graph coloring for cognitive networks, greedy interference avoidance techniques, Iterative Water-filling for Digital Subscriber Lines (DSL), game theoretic approaches to dynamics spectrum allocation and methods based on auction theory.…”

“…In another strand of work, Rose et al [45], D. C. Popescu et al [40], [41], and O. Popescu et al [42] extensively study the problem of greedy interference avoidance in wireless networks. Rose et al [45] propose and study iterative algorithms (namely, the eigen-algorithm and the MMSE algorithm) for interference avoidance through waveform shaping in a synchronous network of multiple users connected to a common receiver (or co-located set of receivers).…”

“…This is clearly a Nash equilibrium point, but is far away from the optimal answer. Etkin et al [15] and O. Popescu et al [42] show that this non-optimal Nash equilibrium point might be the only Nash equilibrium, and therefore Iterative Water-filling fails for various scenarios.…”