General synthesis of complex analogue filters C. Cuypers, N.Y. Voo, M. Teplechuk and J.I. Sewell Abstract: General methods for the synthesis of complex band-pass and band-stop analogue filters are developed for implementation by active-RC, gm-C, switched-capacitor (SC) and switched- current (SI) techniques. These principles are extended to synthesis procedures for log-domain complex realisations. Circuits for filters and group delay equalisers are presented for both ladder and cascade-biquad realisations. These have been incorporated into the filter design software XFILTER. Various designs are compared, noise and sensitivity responses being shown. 1 Introduction Re H (s) + Complex (polyphase) filters are finding many applications Re X (s) Re today [1–7], especially in the communications area. They are _ Y (s) essentially integrated circuit filters and, in order to cover the Im different frequency ranges of application, it is necessary to H (s) encompass different implementation techniques, such as input output active-RC, gm-C, switched-capacitor (SC), switched-current Im (SI) and log-domain. It is also important to develop a H (s) general synthesis method without undue restrictions on Im Im order and filter characteristic. X (s) Y (s) A complex band-pass transfer function HC(s)isderived Re from a low-pass transfer function H(s) by the frequency H (s) + shift method using the substitution (s-sÀjosh) giving HC(s) ¼ H(sÀjosh), where osh is the shift frequency. In Fig. 1 Schematic representation of a complex analogue filter general the coefficients of HC(s)arecomplex[5], therefore Re Im HC(s) can be represented as HC(s) ¼ H (s)+jH (s), where both real HRe(s) and imaginary HIm(s) parts are real techniques have been extended to include this with both transfer functions. ladder and biquad implementations. General implementa- The input and the output signals are complex tion of all these procedures in the XFILTER software is Re Im available and provides a useful design tool. Comparison of X ðsÞ¼X ðsÞþjX ðsÞ designs is shown in numerous examples. Y ðsÞ¼Y ReðsÞþjY ImðsÞ 2 Ladder-derived continuous-time complex band- The relationships between the input and output signals are pass filters Y ReðsÞ¼H ReðsÞX ReðsÞH ImðsÞX ImðsÞ The nodal equation of a passive ladder is described by Y ImðsÞ¼H ReðsÞX ImðsÞþH ImðsÞX ReðsÞ À1 and represented in Fig. 1. ðsC þ s C þ GÞV ¼ J ð1Þ The frequency shift method, which leads to two identical where C, C and G are admittance matrices of capacitors, low-pass filters connected by a cross-coupling network, has inverse inductors and resistors, respectively. V and J are been used successfully [6, 7]. This is now formalised for vectors representing the nodal voltages and input current ladder-derived filters in all the above realisation techniques. sources. The system is decomposed into two related first- Extension to the design of band-stop filters follows easily order systems, which can be implemented, directly by and essentially comprises two identical high-pass networks active-RC or gm-C circuits. The decomposition [8] is with cross-coupling elements. Cascade biquad designs are performed on the left matrix C or the right matrix C. also produced and have some value as practical realisations, For a left matrix decomposition C ¼ ClCr then mainly because of their simplicity. Equalisation of group 1 delay in these complex realisations is often required and the C W ¼ GV À GV ðÀJÞ l s r IEE, 2005 1 CrV ¼ W IEE Proceedings online no. 20040816 s doi:10.1049/ip-cds:20040816 For a right matrix decomposition C ¼ ClCr then Paper first received 14th July 2003 and in revised form 11th May 2004 1 C. Cuypers, M. Teplechuk and J.I. Sewell are with the Department of C V ¼ ðC W þ GV À JÞ Electronics and Electrical Engineering, University of Glasgow, Glasgow G12 l s l 8LT, UK 1 N.Y. Voo is with the Department of Electronics and Computer Science, I DW ¼ CrV University of Southampton, Southampton SO17 1BJ, UK s IEE Proc.-Circuits Devices Syst., Vol. 152, No. 1, February 2005 7 where W is an auxiliary vector of intermediate variables and Applying the shift frequency transformation gives ID is the identity matrix. 1 Using the frequency shift transformation (s-sÀjosh), the CV ¼ ðJ À GV À gAW þ joshCVÞ left matrix decomposition becomes s 1 C W ¼ ðgAT V þ jo C WÞ 1 L s sh L C lW ¼ ðCV À joshC lW s Which leads to gm-C complex band-pass filter implementa- À joshGV þ joshJÞGV þ J tions utilising two cross-coupled low-pass ladder realisa- 1 tions, though the single value of transconductance is C V ¼ ðW þ jo C VÞ r s sh r sacrificed in the cross-coupling terms. and the right matrix decomposition 3 Ladder-derived sampled-data complex band- À1 pass filters CV ¼ ðC W þ GV À J À jo CVÞ s l sh For switched networks (SC and SI), the frequency shifting 1 Àjy I W ¼ ðC V þ jo WÞ mechanism [4] is expressed as HC(z) ¼ Hlp(ze ), where D s r sh y ¼ osh T,withosh the shift frequency and T the sampling The complex transfer functions are available from two sets time. Equation (1), after bilinear transformation and some manipulation, gives of equations. Left matrix decompositions 1 For real transfer functions: A þ FB þ D V ¼ J00 c 1 C W Re ¼ ðCVRe þ o C W Im where l s sh l Im Im Re Re 2 T þ oshGV À oshJ ÞGV þ J A ¼ C þ C þ G; B ¼ 2T G; D ¼ 2G and T 2 1 C VRe ¼ ðW Re À o C VImÞ J00 ¼ð1 þ zÞJ: r s sh r zÀ1 1 For imaginary transfer functions C ¼ and F ¼ 1 À zÀ1 1 À zÀ1 Im 1 Im Re represent forward and backward Euler integrators, respec- C lW ¼ ðCV À oshClW s tively [8]. Re Re Im Im À oshGV þ oshJ ÞGV þ J A left matrix decomposition is Im 1 Im Re AlW ¼ðFB þ DÞV À 2ðJÞ C rV ¼ ðW À oshC rV Þ s À1 ArV ¼WW À Al ðJÞ Right matrix decompositions In general A ¼ LU, A ¼ UL, A ¼ IA, A ¼ AI are the various For real transfer functions: decompositions. Substituting LDI integrator operators into the above Re 1 Re Re Re Im equations CV ¼ ðClW þ GV À J þ oshCV Þ s 1 A W BV DV 2 J Re 1 Re Im l ¼ À1 À À ð Þ I DW ¼ ðCrV À oshW Þ 1 À z s zÀ1 A V ¼ W À AÀ1ðJÞ For imaginary transfer functions: r 1 À zÀ1 l 1 Applying a frequency shift (z-zeÀjy) to obtain the complex CV Im ¼ ðC W Im þ GV Im À JIm À o CV ReÞ s l sh transfer functions gives 1 À1 jy I W Im ¼ ðC VIm À o W ReÞ AlW ¼AlWz e À BV À DV D s r sh þ DVzÀ1ejy À 2ðJÞþ2ðJÞzÀ1ejy The complex band-pass filter realisation therefore consists À1 jy À1 jy À1 of two identical real low-pass filters cross-coupled via ArV ¼ArVz e þ Wz e À Al ðJÞ resistances (terms in o ). A range of active-RC implemen- À1 À1 jy sh þ Al ðJÞz e tations is available from a number of decompositions [8]. jy A number of gm-C ladder decompositions are aimed at Substituting e ¼ cosy+jsiny will lead to two low-pass producing equal transconductance values, one of these, the filters with cross-coupling and modified internal topologies, TC topological decomposition [9], is used to produce a which increase the complexity of circuit design. However, T D complex band-pass filter. With C ¼ ADA ,whereD is a for a sufficiently high clock rate (oclk/osh440), cosy 1 diagonal matrix of inverse inductance values and A is an then incidence matrix, the following pair of equations is AlW ¼ðFB À DÞV À 2ðJÞþj 2ðJÞC sin y equivalent to (1), (g is a scaling conductance and 2 À1 þ jAlWC sin y þ jDVC sin y C L ¼ g D ): A V WC AÀ1 J jAÀ1 J C sin y 1 r ¼ À l ð Þþ l ð Þ CV ¼ ðJ À GV À gAWÞ s þ jArVC sin y þ jWC sin y 1 C W ¼ ðgAT VÞ Complex transfer functions are described by two sets of L s equations: 8 IEE Proc.-Circuits Devices Syst., Vol. 152, No. 1, February 2005 For real transfer functions Applying the frequency shift operator and approximating Re À1 Re cos yD1 leads to the system equations AlW ¼ðFB À Dð1 À z ÞÞV À 2ðJÞ 1 þ zÀ1 Im Im Im AI ¼f½BlW þ DIþ J À 2ðJ ÞF sin y À AlW F sin y À DV F sin y 1 À zÀ1 Re Re À1 Re À1 Im þ jJc sin y þ jAIc sin y ArV ¼ W C À Al ðJ ÞAl ðJ ÞF sin y Im Im À ArV F sin y À W F sin y I DW ¼ cBrI þ jI DWc sin y þ jBrIc sin y For imaginary transfer functions: Which again yield two identical cross-coupled low-pass Im À1 Im networks. AlW ¼ðFB À Dð1 À z ÞÞV À 2ðJÞ The same theory applies equally to the design of complex Re Re Re analogue band-stop filters, with the result that two identical þ 2ðJ ÞC sin y þ AlW C sin y þ DV C sin y high-pass networks with cross-connecting elements are Im Im À1 Im À1 Re required.