LOGDOMAIN COMPLEX FILTER DESIGN WITH XFILTER M.A. Teplechuk and J. I. Sewell Department of Electronics and Electrical Engineering, University of Glasgow, Glasgow, GI2 8LT, UK M.TeDlechuk@elec.ela.ac.uk, i .sewell@elec.ela.ac.uk ABSTRACT To obtain a complex network, a linear frequency shift Synthesis procedures for log-domain complex filters and should be carried out in the frequency group delay equalisers are presented for both ladder and cascade-biauad realisations. These synthesis procedures domains 3 s - jw,, SYNTHESIS Direct log-domain ladder filter synthesis is well developed [6] and the complex log-domain filter synthesis method applies linear. transformation to the existing method. A prototype passive ladder filter nodal admittance equationin time domain can be expressed as: c;+ r IV +GV = uv, (I) have been implemented in the design software XFILTER. I INTRODUCTION For a number of years there has been considerable interest in log-domain filter design [I]. Similarly, cornpledpolyphase filters have found no less attention in recent years [2]. The combination of these two techniques as the design of complex log-domain filters offers rather exciting extension of boundaries in analogue signal processing. The low-voltage, wide-dynamic range, high frequency nature of the log-domain approach embedded in the complex filter common to many quadrature receiver architectures should provide an efficient topology for front-end RF receivers. Numerous log-domain filter design techniques have been proposed [1,3,4] and potentially all of these could be extended into complex realisations. This paper utilises the unified method for high-order log-domain filter design [6] in coniunction with the general principles on the synthesis y. - V. - x, - -eb <=Ujek -zTL,ieh -Gj,eh 'et, 5 * k v. - ~ v, +Jw.,h$ek +J~.Jr,,rn.ek (8) , - Cr,,,eT x, . ", -ex ID, - '!- k cV+r~v+~~-;~.,cv-;~~,rv=u~,~ (2) where termjw,,C' and jw,T are coefficient matrices of frequency dependent negative elements. In order to simulate the system represented by equation (2) using integrators, a set of equations which are linear with respect to the first order derivative operator is required. Matrix After introducing an intermediate variable can be factorised as [XI: r = rLrn (3) x=r,jv (4) equation (2) can be rewritten: CV+r,X+GV-jw,,,CV-jw,hr,T,V = U ! ' & (5) Equations (4) and (5) describe the linear system. which can be rewritten as: of ladder derived complex analogue filters [5]: The resultant realisations comprise two identical lowpass 1.. = uv, -r,x-cv -[-jo,,clv - [-;w~~r~v 16) \-, ladder derived filters with : cross-coupling elements to filters with cross-coupling to form a bandstop where 1,. is the identit" matrix. produce a bandpass characteristic or two identical highpass \I, x = r,v " The externally linear vectors of V and X can be characteristic. Cascade biquad designs are also produced and have Some value as practical realisations mainly because of their simplicity. Equalisation of group delay in these comdex realisations is often reauired and the exponent,al~y mapped to non-linear internal [I] using * 1 techniques bave been extended to include' this with both ladder and biquad implementations. Comparison of k designs is shown in a typical example. - =Wt, and Wy =-eii Y, (7) where k=VT for bipolar and k=nVT for CMOS weak- where C, rand G are coefficient matrices representing capacitances, inverse inductances and conductances Setting vn =e and re-writing in log-domain W respectively. U is a column vector of input connections. variables (8) becomes ~7803-7761-3103iE17.00 02003 IEEE 1-545