Proceedings VII European Signal Processing Conference (EUSIPCO’94), vol. 3, pp. 1582-1585, Edinburgh, 13-16 Sept. 1994 ———————————————————————————————————————————————————————— High Order Transformations for Flexible IIR Filter Design Gerald D. CAIN, Artur KRUKOWSKI and Izzet KALE School of Electronic and Manufacturing Systems Engineering, University of Westminster 115 New Cavendish St., London W1M 8JS, UK, Fax: +44 [ 71] 580 4319 E-Mail: kale@cmsa.westminster.ac.uk Abstract. Extensions of popular transformations for IIR filters are given which employ high order mapping filters. Easy control of prototype transfer function features in multiband renditions is demonstrated. A wider interpretation of transformation is also suggested which permits “less-than-N-band replication” (at a cost in dimensionality , phase fidelity and attention to stability enforcement) that is believed to be of considerable benefit in practical design situations. 1. Introduction Conversion of an existing FIR or IIR filter design to a modified IIR form is often done by means of allpass transformations. Although the resulting designs are considerably more expensive, in terms of dimensionality, than the original prototype, the ease of use (in fixed or variable application) is a big advantage. Up to now the definitive mapping equations are those put forward by Constantinides [1] and since adopted as "industry standard". These well-known equations are geared up to map lowpass to bandpass and several other highly stylised combinations. They were the culmination of preceding work [2]- [4] which pioneered departure from the earliest transformation work by Broome, where a simple modulation approach (suffering from severe aliasing) was used [5]. Recent work [6], [7] has further strengthened the general utility of both of these methods. Here we pursue only an extension of the Constantinides approach. The basic form of mapping in common use is:     H (z) H H z r p 2  (1) where H (z) r is the resulting filter when a prototype filter H (z) p is acted upon by a second-order mapping filter: H (z) z z z z 2 2 2          1 2 2 1 1 (2) The two degrees of freedom providable by   and   choices are under-used by the usual restrictive set of “flat-top” classical mappings like lowpass to bandpass. Instead, any two transfer function features can be migrated to (almost) any two other frequency locations if   and   are chosen so as to keep the poles of H 2 (z) strictly outside the unit circle (since H 2 (z) is substituted for z in the original prototype transfer function). Moreover, as first pointed out by Constantinides, the selection of outside sign influences whether the original feature at zero can be moved (the minus sign, a condition we refer to as "DC mobility") or whether the Nyquist frequency can be migrated (the "Nyquist mobility" case - as we call it - arising when the leading sign is positive). Unfortunately there is not total freedom in re-deployment of any pair of frequency constraint points; in Section 2 below we outline both the generalised second-order  1 ,   selection relation and the forbidden combinations, through a pair of simple inequalities. In this paper we treat also the case of transformations of higher order than two. Though the enhanced design flexibility this offers is readily evident, there has been little work reported in this area. Although Mullis and colleagues have given one very useful multiband solution to the general mapping problem [8]-[11], it seems that scant application experience of that (or any other) method has been related in IIR design literature. Here we present an N th order generalization of our second-order linear equation solution method as an alternative to the Mullis approach and demonstrate its utility in the context of several examples. Our method permits migration of selected constraint points, as well as embracing the multiband possibilities of Mullis. 2. Extensions to Second-Order Mappings We like to think of (2) as relating “old” and “new” z-domain images: z z z z z old new new new 2 new      2 1 2 2 1 1     (3) where the upper (+) sign is applicable for “Nyquist mobility”, while the lower one is for the “DC mobility”. We can independently specify two distinct migrations z z old1 new1  , z z old2 new2  and use these to solve the two simultaneous equations which arise from eq.(3):   1 2   EC - FB AE - BD and AF - CD AE - BD (4) where we shorten “old” to “o” and “new” to “n” in the subscripts utilized in these terms: