Generalized Pascal Matrices, Inverses, Computations and Properties Using One-to-One Rational Polynomial s-z Transformations

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <?Pub Dtl=""?>This paper proposes a one-to-one mapping between the coefficients of continuous-time (<formula formulatype="inline"><tex>$s$</tex></formula>-domain) and discrete-time (<formula formulatype="inline"><?Pub Dtl=""?><tex>$z$</tex></formula>-domain) IIR transfer functions such that the <formula formulatype="inline"><tex>$s$</tex> </formula>-domain numerator/denominator coefficients can be uniquely mapped to the <formula formulatype="inline"><tex>$z$</tex></formula>-domain numerator/denominator coefficients. The one-to-one mapping provides a firm basis for proving the inverses of the so-called generalized Pascal matrices from various first-order <formula formulatype="inline"><tex>$s$</tex></formula>-<formula formulatype="inline"> <tex>$z$</tex></formula> transformations. We also derive recurrence formulas for recursively determining the inner elements of the generalized Pascal matrices from their boundary ones. Consequently, all the elements of the whole generalized Pascal matrix can be easily generated through utilizing their neighbourhood, which can be exploited for further simplifying the Pascal matrix generations. Finally, we reveal and prove some interesting properties of the generalized Pascal matrices. </para>