Sampling zeros and the Euler-Frobenius polynomials

Abstract

In this paper, we show that the zeros of sampled-data systems resulting from rapid sampling of continuous-time systems preceded by a zero-order hold (ZOH) are the roots of the Euler-Frobenius polynomials, the properties of which have been studied in the context of cardinal spline interpolation and, more recently, wavelets. Using known properties of the Euler-Frobenius polynomials, we prove two conjectures of Hagiwara et al. (1993), the first of which concerns the simplicity, negative realness and interlacing properties of the sampling zeros of ZOH- and first-order hold (FOH)- sampled systems. To prove the second conjecture, we show that in the fast sampling limit, and as the continuous-time relative degree increases, the largest sampling zero for FOH-sampled systems approaches 1/e, where e is the base of the natural logarithm.