Minimum unit noise gain in non-minimal state-space realizations of digital filters

The minimum attainable unit-noise-gain in a non-minimal state-space realization of a digital filter is shown to be given by the ratio of the dimension of a minimal realization to the dimension of the non-minimal realization multiplied by the minimum unit-noise-gain of a minimal realization. It is shown that the minimum noise gain can be attained if and only if the redundant modes of the non-minimal realization are both uncontrollable and unobservable.<
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