On error surfaces of sufficient order adaptive IIR filters: proofs and counterexamples to a unimodality conjecture

It is shown that although a conjecture proposed by S. Stearns (ibid., vol.ASSP-29, p.763-6, June 1981) concerning sufficient conditions to guarantee unimodality of error surfaces for adaptive pole-zero infinite-impulse-response (IIR) filters is valid for first- and second-order filters, it is not true in general without an additional constraint introduced by T. Soderstrom (Autom., vol.11, p.93-9, Jan. 1982). Proofs of the conjecture for a first- and second-order adaptive filters are given. Counterexamples are presented to illustrate the failure of this conjecture for higher order filters when the condition of Soderstrom is not satisfied for both matching order and overparameterizing cases