Optimal Linear Constraints and Subspace Adaptive Filtering
Daniel Rabideau
Lincoln Laboratory
Massachusetts Institute of Technology
In subspace adaptive processing, a multidimensional signal is first partitioned into multiple subspaces. Within each subspace, adaptive filter coefficients are then computed and applied with the goal of rejecting the interference within that subspace. Finally, desired signals are recovered by recombining the adaptively nulled subspaces. Subspace adaptation tends to be less costly to implement, and requires less training data than fully adaptive processing (in fact, fully adaptive processing often performs very badly due to the unavailability of sufficient numbers of homogeneous training samples). As a result, subspace adaptation is frequently proposed for wideband sensor applications, such as high range resolution radar (both synthetically and instantaneously wideband), wireless comm., and nonstationary multipath mitigation. One problem with subspace adaptation is that the adaptive weights have a statistically random jitter, causing distortion of desired signals. While point constraints are useful for lowering the distortion of some signals, significant improvements can be made. In this paper, we derive insightful statistical expressions for the subspace-recombined signals. The expressions tell us what to expect our signal power to be, as a function of both its array response (i.e. position) and the various adaptation parameters. One use for such expressions is to quantify sidelobe rejection. For example, it is shown that low sidelobes within each subspace adaptive filter are not enough to ensure low sidelobes in the reconstructed signal space. Moreover, the expected sidelobe levels (in the reconstructed space), and the probabilities of exceeding these levels, are quantified. Another way to use these expressions is to gain insight into exactly how the many adaptation parameters should be modified to achieve some desired level of performance. In particular, we derive the statistically optimal filtering constraints that produce the minimum sidelobes in the recombined signal space. Application of these techniques to the subband adaptive filtering problem will be discussed.
*This work was sponsored by DARPA under Air Force contract F19628-95-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the author and are not necessarily endorsed by the USAF.