Robust broadband beamforming and digital filter design methods and applications

Abstract: This doctoral thesis consists of a summary and six parts corresponding to six different papers. There are two published and three submitted journal papers, and one research report. The summary will highlight the main results and emphasize the interrelationships between different parts. The thesis comprises two main themes: robust broadband beamforming and digital filter design. In most of the papers in this thesis these concepts are closely related. Some of the results on digital filter design actually concern robust filter design, and one of the main contributions of this thesis concerns the possibility of anomalous designs when using conventional methods in the FIR filter design of broadband beamformers. Part I deals with robustness of broadband adaptive beamformers in the sense that there is an uncertainty in the frequency content and spatial location of the desired target signal. The adaptive beamformer should perform well over a region in space and frequency. This is achieved by defining the Spatial Filter designed Generalized Sidelobe Canceller (SFGSC), and by using an appropriate digital filter design. The novelty of the SFGSC method is the implementation of target signal constraints by using a filter design approach. With this strategy the constraints are approximated over a given design domain, rather than exactly implemented as with the conventional Generalized Sidelobe Canceller (GSC). With this new method (in contrast to the GSC) the SFGSC is able to handle a continuum of constraints. Part II introduces a quadratic programming formulation of the weighted Chebyshev FIR filter design problem for a broadband beamformer in the near-field. This technique may be used to design the digital filters of the broadband SFGSC described in Part I. Part III reveals that the digital filters of a broadband beamformer are incompletely specified whenever the beamformer is specified only in space and frequency. Using conventional filter design criteria with such incomplete specifications may lead to excessively large filter coefficients. Robust weighted least squares and weighted Chebyshev design criteria are introduced in order to avoid this anomaly. "Robustness" in this context means insensitivity to model imperfections such as sensor element placement errors, amplifier mismatch, etc. Again, the filters designed are typically components of the SFGSC described in Part I. Part IV addresses the problem of the non-uniqueness of the Chebyshev approximation for two-dimensional linear phase digital FIR filters. It is shown that the unique Chebyshev approximation having minimum Euclidean filter weight norm can be obtained by using a wellconditioned quadratic programming formulation. This is the same quadratic program that was used to define one of the robust beamformer designs in Part III. Part V deals with robust design for one-dimensional non-linear phase FIR filters which are incompletely specified. The conventional weighted Chebyshev solution can be obtained by using a quadratic programming formulation similar to that given in Part II. A robust weighted Chebyshev design criterion is defined by a modified quadratic program, similar to one of the robust design methods given in Part III. Part VI emphasizes the robustness of adaptive beamformers with respect to channel mismatch, sensor positioning, etc. In particular, the paper addresses the difficulty of mathematically modeling a beamformer for a small enclosure such as a car compartment. A calibrating scheme is proposed which is independent of array geometry and channel matching, and which calibrates the adaptive array to the given acoustic environment and to the given electronic equipment. Results from real measurements in a car compartment are included. An international patent is applied for based on this paper.