Proof of a modified form of Shank´s conjecture on the stability of 2-D planar least square inverse polynomials and its implications

In this paper, we prove a modified form of Shanks\´ conjecture on the stability of planar least square inverse (PLSI) polynomials in 2-D by imposing restriction on the original 2-D polynomial. This restriction on the original polynomial is in fact necessary to stabilize an unstable polynomial eventually by taking double PLSI so as to maintain the magnitude spectrum same. A discussion on the optimization technique of designing 2-D recursive filters is then included and it is indicated that in practice, if the optimization in the frequency domain is done over a very close frequency grid the resulting filter transfer function will have a denominator polynomial satisfying the restriction imposed while proving the modified form of Shanks\´ conjecture. Thus we conclude that the testing the 2-D transfer functions of recursive filters for stability can be done once for all at the end of optimization. Even this testing is only an equivalent of one dimensional stability test and can be easily performed. If the filter is found unstable we can stabilize it by taking the double PLSI of the denominator polynomial of the transfer function.