This thesis deals with three estimation problems motivated by spatial signal processing using arrays of sensors. All three problems are approached using tools from estimation theory, including asymptotical analysis of performance and Cramér-Rao lower bound; Monte Carlo methods are used to evaluate small sample performance.
The first part of this thesis treats direction of arrival estimation for narrowband signals. Most algorithms require the noise covariance matrix to be known or to possess a known structure. In many cases, the noise covariance is estimated from a separate batch of signal-free samples; in a non-stationary environment this sample set can be small. By deriving the Cramér-Rao bound in a form that can be compared to well-known results, we investigate the combined effects of finite sample sizes, both in the estimated noise covariance matrix and in the data with signals present. Under the same data model, we derive the asymptotical covariance of weighted subspace fitting, where the signal-free samples are used for whitening. The obtained expression suggests optimal weights that improve performance compared to the standard choice and that result in an asymptotically efficient estimate. In addition, we propose a new, asymptotically efficient, method based on the likelihood function. If the array is uniform and linear, then an iterative search can be avoided. We propose two such algorithms, based on the two general, iterative, algorithms discussed. We also treat the detection problem, and provide results that are useful in a joint detection and estimation algorithm based on the proposed estimators.
Parameter estimation for the reduced rank linear regression is the second estimation problem treated in the thesis. It appears in, for example, system identification and signal processing for communications. We propose a new method based on instrumental variable principles and we analyze its asymptotical performance. The new method is asymptotically efficient if the noise is temporally white, and outperforms previously suggested algorithms when the noise is temporally correlated. As part of the estimation algorithm, the closest low rank approximation of a matrix, as measured under a weighted norm, has to be calculated. This problem lacks solution in the general case. We propose two new methods that can be computed in fixed time; both methods are approximate but asymptotically optimal as part of the estimation procedure in question. We also propose a new algorithm for the related rank detection problem.
The third problem is that of estimating the covariance matrix of a multivariate stochastic process. In some applications, the structure of the problem suggests that the underlying, true, covariance matrix is the Kronecker product of two matrix factors. The covariance matrix of the channel realizations in multiple input multiple output (MIMO) communications systems can, under certain assumptions, have such Kronecker product structure. Moreover, the factor matrices can sometimes, in turn, be assumed to possess additional structure. We propose two asymptotically efficient estimators for the case where the channel realizations can be assumed known. Both estimators can be computed in fixed time; they differ in their small sample performance and in their ability to incorporate extra structure in the Kronecker factors. In a practical MIMO system, the channel realizations have to be estimated from training data. If the amount of training data is limited, then it is better to treat the training data, rather than the channel estimates, as inputs to the channel covariance estimator. We derive and analyze an estimator based on this new data model. This estimate can be computed in fixed time and the estimator is also able to optimally use extra structure in the factor matrices
Stockholm: KTH , 2007. , viii, 201 p.