Abstract. A linear system described by autoregressive equations with a given input/ output structure cannot be transformed to standard state-space form if the implied input/ output relation is non proper. Instead, a realization in descriptor form must be used. In this paper, it is shown how to obtain minimal descriptor realizations from autoregressive equations without separating finite and infinite frequencies, and without going through a reduction process. External equivalence is used, so that even situations in which there is no transfer matrix can be considered. The approach is based on the so-called pencil representation of linear systems, and it is shown that there is a natural realization of autoregressive equations in pencil form. Jn this way, the link between the realization theories of Willems and Fuhrmann can also be clarified.
Convolutional codes are considered with code sequences modelled as semi-infinite Laurent series. It is wellknown that a convolutional code C over a finite group G has a minimal trellis representation that can be derived from code sequences. It is also wellknown that, for the case that G is a finite field, any polynomial encoder of C can be algebraically manipulated to yield a minimal polynomial encoder whose controller canonical realization is a minimal trellis. In this paper we seek to extend this result to the finite ring case G = Zpr by introducing a socalled "p-encoder". We show how to manipulate a polynomial encoding of a noncatastrophic convolutional code over Zpr to produce a particular type of p-encoder ("minimal p-encoder") whose controller canonical realization is a minimal trellis with nonlinear features. The minimum number of trellis states is then expressed as p γ , where γ is the sum of the row degrees of the minimal p-encoder. In particular, we show that any convolutional code over Zpr admits a delay-free p-encoder which implies the novel result that delay-freeness is not a property of the code but of the encoder, just as in the field case. We conjecture that a similar result holds with respect to catastrophicity, i.e., any catastrophic convolutional code over Zpr admits a noncatastrophic p-encoder.
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