Multichannel FIR exact deconvolution in multiple variables

We present a general framework for multichannel exact deconvolution with multivariate finite impulse response (FIR) convolution and deconvolution filters using algebraic geometry. Previous work formulates the problem of multichannel FIR deconvolution into that of the left inverse of a convolution matrix which is solved by linear algebra. However, this approach requires the prior information of the support of deconvolution filters. Using algebraic geometry, we find a necessary and sufficient existence condition for FIR deconvolution filters and propose a simple algorithm based on the Gröbner basis to compute deconvolution filters. This computation algorithm obtains deconvolution filters with either minimal order or minimum number of nonzero coefficients, and no prior information of the support is required. Simulation results show that, due to the smaller size of deconvolution filters, our approach achieves better results than the liner algebra approach under an impulsive noise environment.