SUBSPACE ESTIMATION USING FACTOR ANALYSIS Ahmad Mouri Sardarabadi, and Alle-Jan van der Veen Faculty of Electrical Engineering, Mathematics and Computer Science Delft University of Technology (TU Delft) e-mail:{A.Mourisardarabadi, A.j.veen}@tudelft.nl ABSTRACT Many subspace estimation techniques assume either that the sys- tem has a calibrated array or that the noise covariance matrix is known. If the noise covariance matrix is unknown, training or other calibration techniques are used to find it. In this paper another approach to the problem of unknown noise covariance is presented. The complex factor analysis (FA) and a new ex- tended version of this model are used to model the covariance matrix. The steep algorithm for finding the MLE of the model parameters is presented. The Fisher information and an expres- sion for the Cram´ er–Rao bound are derived. The practical use of the model is illustrated using simulated and experimental data. Index Terms— Factor analysis, complex factor analysis, sub- space estimation, Cram´ er-Rao bound, maximum-likelihood. 1. INTRODUCTION Many array processing techniques rely on estimating the signal sub- space using eigenvalue decomposition (EVD) of the covariance ma- trix. This method does not have an inherit model for the noise and its usage for subspace estimation is limited to the case where the noise covariance matrix could approximately be modeled as σ 2 I . In this paper we are interested in estimating the signal subspace even if the noise covariance does not have this special form and is un- known. Also each receiving element of the array could have a dif- ferent unknown gain. These complications requires a data–model that includes unknown noise powers and is scale–invariant. The FA model has both of these required properties and is used to find the desired subspace. After its first formulation by Spearman in 1904, the FA model has been used in various fields such as psychology, social sciences, natural science, etc [1, 2, 3]. Variations of the FA model have also been explored for blind source separation and array calibration [4, 5]. Even though the work presented here could be used in various fields, the focus is on its usage for radio–astronomy. Spatial fil- tering of the strong sources or interference and removing the ex- tended emissions are two possible applications of the FA model in this field. This paper gives some new results needed for extending the FA model to complex case and also extends the model to account for a more general noise models. The MLE of the model parameters is estimated using the steep method. The Fisher information and the Fisher score are presented here could also be used for the scoring algorithm as presented in [6]. 2. DATA MODEL For a system with p receiving elements that is exposed to m sources, a commonly used narrow-band model has the form x(t)= A0s0(t)+ n(t). (1) Where x is a p × 1 vector of received signals, A0 is the p × m array- response matrix, s0 is an m × 1 vector of source signals and n is a p × 1 vector representing all the noise contributions in the system. This data model suffers from some ambiguities that need to be addressed before attempting to estimate the model parameters. Given any invertable matrix Z and any unitary matrix Q, the model could be rewritten as x(t)= A0ZQQ H Z -1 s0(t)+ n(t). (2) It is assumed that the sources and noise contributions are uncor- related and have proper complex Gaussian distributions CN (0, Rs 0 ) and CN (0, Rn ) respectively. In this situation the covariance matrix of received signal is given by Rx = A0ZQQ H Z -1 Rs 0 Z -H QQ H Z H A H 0 + Rn . (3) The matrix Z could always be chosen in such a way that the Z -1 Rs 0 Z -H = I m. By introducing A = A0ZQ the covariance matrix of the received signal becomes Rx = AA H + Rn . (4) The only ambiguity left is the choice of the unitary matrix Q. In order to choose Q, first the case of known noise covariance is considered. Let R0 = A0Rs 0 A H 0 = AA H and assume that Rn is known, then the EVD of the whitened covariance matrix R - 1 2 n Rx R - 1 2 n − I p = R - 1 2 n R0R - 1 2 n could be used to estimate A. Consider the following eigenvalue problem R - 1 2 n R0R - 1 2 n U = U Γ, (5) where Γ is a diagonal matrix. If we require U = R - 1 2 n A then (5) becomes R - 1 2 n R0R - 1 2 n R - 1 2 n A = R - 1 2 n A(A H R -1 n A) = R - 1 2 n AΓ. (6) One way to make sure that (6) holds, is by setting A H R -1 n A = Γ. (7) The matrix Q is now chosen in such a way that (7) holds. With the choice of Q there are no ambiguities left. Finally we assume that the noise contributions are uncorrelated so that Rx = AA H + D. (8) where D is a diagonal matrix. For the remainder of this paper we refer to (8) as the (classical) FA model. 2012 IEEE 7th Sensor Array and Multichannel Signal Processing Workshop (SAM) 978-1-4673-1071-0/12/$31.00 ©2012 IEEE 477