Eigen-Inference for Multi-Source Power Estimation Romain Couillet ST-Ericsson, Sup´ elec 505 Route des Lucioles 06560 Sophia Antipolis, France Email: romain.couillet@supelec.fr Jack W. Silverstein Department of Mathematics North Carolina State University Raleigh, North Carolina 27695-8205 Email: jack@math.ncsu.edu M´ erouane Debbah Alcatel-Lucent Chair, Sup´ elec Plateau de Moulon, 3 rue Joliot-Curie 91192 Gif sur Yvette, France Email: merouane.debbah@supelec.fr Abstract—This paper introduces a new method to estimate the power transmitted by multiple signal sources, when the number of sensing devices and the available samples are sufficiently large compared to the number of sources. This work makes use of recent advances in the field of random matrix theory that prove more efficient than previous “moment-based” approaches to the problem of multi-source power detection. Simulations are performed which corroborate the theoretical claims. I. I NTRODUCTION At a time when radio resources become scarce, the alterna- tive offered by flexible radios [1] is gaining more and more interest. A flexible wireless network is a set of opportunistic entities, referred to as the secondary network, that benefit from unused spectrum resource to establish communication, with little interference to the established primary network. This is performed by letting the secondary devices sense the environ- ment for the presence of active transmissions and exchange the collected information among the secondary network. If the secondary devices can detect the number of primary sources and evaluate the power used by every individual source, their own maximum transmission power (i.e. the maximum transmit power that brings little interference to the primary network) can then be reliably estimated. The detection of the number of neighbors and the estimation of the individual transmit powers is the subject of this work. The difficulty of estimating transmit powers lies in the little information known a priori by the secondary network: the transmitted data and the transmission channels are usually inaccessible. This has motivated much work in the direction of blind detection methods [2], [3]. To solve the harder problem of power inference, it is necessary to assume that the sensed samples are of large dimension compared to the number of active sources. 1 The latter condition allows one to model the channel from the sources to the secondary users, as well as the transmit data and noise, as large random matrices; call them H, X and W, respectively. Denoting P a diagonal matrix of the source powers, the detection problem boils down to estimating the entries of P from the sole knowledge of Y = HP 1 2 X+W. Up to this day though, no computationally-cheap consistent estimator 2 for the entries of P has been proposed. Among the 1 e.g. individual secondary users are equipped with many antennas, or a large number of secondary users, each equipped with few antennas, collect their received data in a central entity. 2 an estimator ˆ P i of the i th entry P i of P is said to be consistent if ˆ P i - P i → 0 almost surely when the relevant system dimensions grow large. existing techniques are convex optimization strategies [4] or moment-based approaches [5], [6]. The latter provide consis- tent estimators of the moments of the eigenvalue distribution of P as a function of the moments of the eigenvalue distribution of YY H ; from those estimates, the entries of P themselves can be inferred. The moment-based techniques are however expected to perform worse than methods that would fully exploit the eigenvalue distribution of YY H , and not only the first moments. This problem is addressed in [7] for the sample covariance matrix model Y ′ = P 1 2 X, i.e. the entries of P are inferred from the full eigenvalue distribution of X H PX. This work generalizes this result to infer the entries of P from the observed matrix Y = HP 1 2 X + W. The novel estimator proposed here will be shown to have a very compact form, to be computationally inexpensive and to perform better than moment-based approaches. The remainder of this paper is structured as follows: Section II introduces the system model. In Section III, the novel power estimator is derived, part of the technical proofs being left to [8]. Section IV provides simulation results. Section V concludes this work. Notations: In the following, boldface lower case symbols represent vectors, capital boldface characters denote matrices (I N is the size-N identity matrix). The transpose and Hermi- tian transpose operators are denoted (·) T and (·) H , respectively. We denote by C + the set {z ∈ C, ℑ[z] > 0}. The symbol ‘ a.s. −→’ denotes almost sure convergence. II. SYSTEM MODEL Consider a wireless (primary) network in which K enti- ties are transmitting data. Transmitter k ∈{1,...,K} has transmission power P k and is equipped with n k antennas. We denote n = ∑ K k=1 n k the total number of transmit antennas in the primary network. Consider also a secondary network composed of a total of N sensing devices, e.g. a single user embedded with N antennas or N single antenna users; we shall refer to the N sensors collectively as the receiver. Denote H k ∈ C N×n k the multiple antenna channel matrix between transmitter k and the receiver. We assume that the entries of H k are independent and identically distributed (i.i.d.) with zero mean and variance 1/N . At time instant m, transmitter k emits signal x (m) k ∈ C n k , with entries assumed to be i.i.d. of zero mean and variance 1. Assume further that at time instant m the receiver is corrupted by additive white noise of variance σ 2 on every sensor; we denote σw (m) ∈ C N the