Linear phase and symmetries for multidimensional FIR filters over lattices

We present in this paper multidimensional FIR filters over arbitrary lattices having the linear phase property. The extended definitions of phase, group delay, and linear phase are first presented. This leads us to consider four types of filters as in the one-dimensional (1-D) case (symmetric or antisymmetric with odd or even support). The zeros in frequencies inherent to each type are presented. Hyper-octantal symmetry, in which all coefficients of a filter depend on those in one hyper-octant, is then introduced. We show conditions on lattices and filters for this kind of symmetry to be possible. We finally give equations relating dependent coefficients to independent ones.