Generalized Consistent Estimation in Arbitrarily High Dimensional Signal Processing

The theory of statistical signal processing finds a wide variety of applications in the fields of data communications, such as in channel estimation, equalization and symbol detection, and sensor array processing, as in beamforming, and radar systems. Indeed, a large number of these applications can be interpreted in terms of a parametric estimation problem, typically approached by a linear filtering operation acting upon a set of multidimensional observations. Moreover, in many cases, the underlying structure of the observable signals is linear in the parameter to be inferred. This dissertation is devoted to the design and evaluation of statistical signal processing methods under realistic implementation conditions encountered in practice. Traditional statistical signal processing techniques intrinsically provide a good performance under the availability of a particularly high number of observations of fixed dimension. Indeed, the original optimality conditions cannot be theoretically guaranteed unless the number of samples increases asymptotically to infinity. Under this assumption, a statistical characterization can be often afforded by using the large-sample theory of sample covariance matrices. In practice, though, the application of these methods to the implementation of, for instance, training schemes in communication systems and adaptive procedures for radar detection problems, must rely on an observation window of finite length. Moreover, the dimension of the received signal samples (e.g. number of array sensors in multi-antenna systems) and the observations window size are most often comparable in magnitude. Under these situations, approaches based on the classical multivariate statistical analysis significantly lose efficiency or cannot even be applied. As a consequence, the performance of practical solutions in some real situations might turn out to be unacceptable. In this dissertation, a theoretical framework for characterizing the efficiency loss incurred by classical multivariate statistical approaches in conventional signal processing applications under the practical conditions mentioned above is provided. Based on the theory of the spectral analysis of large-dimensional random matrices, or random matrix theory (RMT a family of new statistical inference methods overcoming the limitations of traditional inferential schemes under comparably large sample-size and observation dimension is derived. Specifically, the new class of consistent estimators generalize conventional implementations by proving to be consistent even for arbitrarily high-dimensional observations (i.e., for a limited number of samples per filtering degree-of-freedom). In particular, the proposed theoretical framework is shown to properly characterize the performance of multi-antenna systems with training preambles in the more meaningful asymptotic regime defined by both sample size and dimension increasing without bound at the same rate. Moreover, the problem of optimum reduced-rank linear filtering is reviewed and extended to satisfy the previous generalized consistency definition. On the other hand, a double-limit asymptotic characterization of a set of vector-valued quadratic forms involving the negative powers of the observation covariance is provided that generalizes existing results on the limiting eigenvalue moments of the inverse Wishart distribution. Using these results, a new generalized consistent eigenspectrum estimator based on the inverse-shifted power method is derived that uniquely relies on the SCM and does not require matrix eigendecomposition. The effectiveness of the previous spectral estimator is demonstrated upon its application to the construction of an improved source power estimator that is robust to inaccuracies in the knowledge of both noise level and true covariance matrix. In order to alleviate the computation complexity issue associated with practical implementations involving matrix inversions, a solution to the two previous problems is afforded in terms of the positive powers of the SCM. To that effect, a class of generalized consistent estimators of the covariance eigenspectrum and the power level are obtained on the Krylov subspace defined by the true covariance matrix and the signature vector associated with the intended parameter. In practice, filtering solutions are very often required to robustly operate not only under sample-size constraints but also under the availability of an imprecise knowledge of the signature vector. Finally, a signal-mismatch robust filtering architecture is proposed that is consistent in the doubly-asymptotic regime.