A note on constrained least squares design of M-D FIR filter based on convex projection techniques

Recently, a great deal of effort has been devoted to the design problem of "constrained least squares M-D FIR filter" because a significant improvement of the squared error is expected by a slight relaxation of the minimax error condition. Unfortunately, no design method has been reported, which has some theoretical guarantee of the convergence to the optimal solution. In this paper, we propose a class of novel design methods of "constrained least squares M-D FIR filter." The most remarkable feature is that all of the proposed methods have theoretical guarantees of convergences to the unique optimal solution under any consistent set of prescribed maximal error conditions. The proposed methods are based on "convex projection techniques" that computes the metric projection onto the intersection of multiple closed convex sets in real Hilbert space. Moreover, some of the proposed methods can still be applied even for the problem with any inconsistent set of maximal error conditions. These lead to the unique optimal solution over the set of all filters that attain the least sum of squared distances to all constraint sets.