A Decomposition of a Transfer Function Minimizing Distortion and Inband Losses

A given rational transfer-function T(s) of a passive network, which is real for real s, is to be realized by an inductorless two-port. This is usually done by breaking down T(s) into functions T{(s) of the first or second degree in s. All functions of the first and those of the second degree with poles on the negative real axis are realized by passive RCnetworks with buffer amplifiers between the different stages. Those of second degree but with poles not on the negative real axis are realized by RC-active networks containing amplifiers. We deal at first with the second group of functions. The extension to the general case follows easily. The voltage swing at the input of the different stages with functions 7 (s) is often tightly limited by the threshold above which overdriving of the amplifiers (that is, distortion) occurs. A further result in many cases is high inband loss of the overall filter which cannot be overcome by amplification because of both distortion and a too low signal/noise ratio. Our task is to find a method of factoring T(s) into the different functions T,(s) such that the allowable voltage swing at the input is as high as possible without creating distortion and the inband losses as low as possible. We confine ourselves first to transfer functions T(s) = zrr = K m ' i ,,,-, ' , ' ° 7, s + om-jS + · · · o,s + b0 455