On Reduced Rank Linear Regression and Direction Estimation in Unknown Colored Noise Fields

Abstract: Two estimation problems are treated in this thesis. Estimators are suggested and the asymptotical properties of the estimates are investigated analytically. Numerical simulations are used to assess small-sample performance. In addition, performance bounds are calculated. The first problem treated is parameter estimation for the reduced rank linear regression. A new method based on instrumental variable principles is proposed and its asymptotical performance analyzed. In addition, the Cram'{e}r-Rao bound for the problem is derived for a general Gaussian noise model. The new method is asymptotically efficient (it has the smallest possible covariance) if the noise is temporally white, and outperforms previously suggested algorithms when the noise is temporally correlated. The approximation of a matrix with one of lower rank under a weighted norm is needed as part of the estimation algorithm. Two new, computationally efficient, methods are suggested. While the general matrix approximation problem has no known closed form solution, the proposed methods are asymptotically optimal as part of the estimation procedure in question. A new algorithm is also suggested for the related rank detection problem. The second part of this thesis treats direction of arrival estimation for narrowband signals using an array of sensors. Most algorithms require the noise covariance matrix to be known (up to a scaling factor) or to possess a known structure. In many cases the noise covariance is in fact estimated from a separate batch of signal-free samples. This work addresses the combined effects of finite sample sizes both in the estimated noise covariance matrix and in the data with signals present. No assumption is made on the structure of the noise covariance. The asymptotical covariance of weighted subspace fitting (WSF) is derived for the case in which the data are whitened using the noise covariance estimate. The obtained expression suggests an optimal weighting that improves performance compared to the standard choice. In addition, a new method based on covariance matching is proposed. Both methods are asymptotically statistically efficient. The Cramér-Rao bound for the problem is derived, and the expression becomes surprisingly simple.