Free Deconvolution for Signal Processing Applications

2011

REV-JEC

The general framework of Mobile Flexible Networks (MFN) is to design dense self-organizing, self-healing and self-energy harvesting secure networks where terminals and base stations interact and self-adapt in an intelligent manner without the need of a central controller or with the right amount of regulation to let the agents in the network exploit fully the degrees of freedom. The design depends mainly on the mobility pattern as in highly mobile environments, intelligence at the terminal reduces the cost of signalling whereas for fixed (non-mobile) networks, all the intelligence can be put on the network side. One of the big challenges is to find how to optimally split the intelligence between cognitive terminals and cognitive networks. In this paper, we discuss the challenges ahead and provide some research directions to develop the theoretical foundations of these networks 1 .

Section: Iv-c the Statistical Inference Foundationsmentioning

confidence: 99%

Theoretical Foundations of Mobile Flexible Networks

2011

REV-JEC

“…As recently shown in [6], [8], Free deconvolution relates the eigenvalue distribution of the covariance matrix (μ YY H ) with the eigenvalue distribution of the Vandermonde matrix (μ VV H ) as…”

Section: Moments Approachmentioning

confidence: 99%

“…In this paper, we provide a way to estimate the distribution of the sensors in noisy environments without any communication between the sensors. The results are based on the recent framework of free deconvolution [10], [6], [7] and asymptotic Vandermonde random matrix theory [8]. Interestingly, although the result are valid in the asymptotic regime, simulations using realistic random distribution deployment of sensors show the adequacy of the approach.…”

Section: Introductionmentioning

confidence: 99%

Estimation of the distribution of randomly deployed wireless sensors

Khan

2009 IEEE International Symposium on Information Theory

Ryan

et al. 2009

Self Cite

Abstract-The distribution of randomly deployed wireless sensors plays an important role in the quality of the methods used for data acquisition and signal reconstruction. Mathematically speaking, the estimation of the distribution of randomly deployed sensors can be related to computing the spectrum of Vandermonde matrices with non-uniform entries. In this paper, we use the recent free deconvolution framework to recover, in noisy environments, the asymptotic moments of the structured random Vandermonde matrices and relate these moments to the distribution of the randomly deployed sensors. Remarkably, the results are valid in the finite case using only a limited number of sensors and samples.

“…Definition 1: A family of unital -subalgebras will be called a free family if (6) A family of random variables are said to be free if the algebras they generate form a free family.…”

Section: Framework For Free Convolutionmentioning

confidence: 99%

“…In [4], Dozier and Silverstein explain how one can use the eigenvalue distribution of to estimate the eigenvalue distribution of by solving a given equation. In [5] and [6], we provided an algorithm for passing between the two, using the concept of multiplicative free convolution, which admits a convenient implementation. The implementation performs free convolution exactly based solely on moments.…”

mentioning

confidence: 99%

Channel Capacity Estimation Using Free-Probability Theory

Ryan

IEEE Trans. Signal Process.

2008

In many channel measurement applications, one needs to estimate some characteristics of the channels based on a limited set of measurements. This is mainly due to the highly time varying characteristics of the channel. In this contribution, it will be shown how free probability can be used for channel capacity estimation in MIMO systems. Free probability has already been applied in various application fields such as digital communications, nuclear physics and mathematical finance, and has been shown to be an invaluable tool for describing the asymptotic behaviour of many large-dimensional systems. In particular, using the concept of free deconvolution, we provide an asymptotically (w.r.t. the number of observations) unbiased capacity estimator for MIMO channels impaired with noise called the free probability based estimator. Another estimator, called the Gaussian matrix mean based estimator, is also introduced by slightly modifying the free probability based estimator. This estimator is shown to give unbiased estimation of the moments of the channel matrix for any number of observations. Also, the estimator has this property when we extend to MIMO channels with phase off-set and frequency drift, for which no estimator has been provided so far in the literature. It is also shown that both the free probability based and the Gaussian matrix mean based estimator are asymptotically unbiased capacity estimators as the number of transmit antennas go to infinity, regardless of whether phase off-set and frequency drift are present. The limitations in the two estimators are also explained. Simulations are run to assess the performance of the estimators for a low number of antennas and samples to confirm the usefulness of the asymptotic results.