The uniqueness in designing multidimensional causal recursive digital filters possessing magnitude hyperspherical symmetry

It is shown that magnitude hyperspherically symmetric transfer functions of multidimensional (MD) causal recursive digital filters must have numerator and denominator polynomials that are separately magnitude hyperspherically symmetric. Further, the exact reference-domain magnitude hyperspherically symmetric denominator polynomial is of infinite order, possessing only one free parameter, and the magnitude hyperspherically symmetric numerator polynomial itself has to be a radial even function. The corresponding MD design problem is shown to be essentially a one-dimensional design problem. Filter transfer functions having good symmetry and moderate degree can be designed by using the presented procedure