Roundoff noise bounds derived from coefficient sensitivities for digital filters

Abstract

The sensitivities of the transfer function of a digital filter with respect to its coefficients are utilized to derive lower bounds on the roundoff noise output in the cases of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L_{\infty}</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L_{\infty}</tex> scaling for fixed-point arithmetic. General bounds are produced which apply to any filter structure if rounding is performed after multiplication and the filter has already been scaled. For the parallel and cascade forms, alternate bounds are derived which apply to rounding after multiplication or summation and which do not require prior scaling. The alternate bounds arethus independent (or nearly so) of pairing, ordering, and transposition. Examples are presented which show that the bounds are reasonably tight.