Synthesis of Multiple-Feedback Active Filters

In the last two decades, the question of how to realize a prescribed rational transfer function N(s) T(s) = -- = D(S) 7out Fin (1) by RC active structures was the subject of more than a thousand learned papers. The consensus at the present time seems to be as follows: (i) The active element to be used is an operational amplifier. (ii) Subnetworks (building blocks) realizing biquadratic transfer functions _ nj + »j 5 + d'Q + d[s + djs 2 _ NM Di(s) are constructed as intermediate steps. (in) Finally, the overall transfer function is realized as a cascade connection of these leading to: m T(s) = n Ti(s). i=i 527 528 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1973 Our objective in this paper is to challenge step (Hi) above and introduce an alternative synthesis method valid for a large class of transfer functions. This is done by adding one or more feedback loops to the cascade configuration. The effect of this will be that, while the transmission zeros of the overall system will remain the concatenation of those of the individual biquadratic blocks, this will not be true for the poles any more. We will assume (i) and (ii) to be correct without, however, worrying about the details of the specific configuration to be used in step (ii) above; that is to say, our most elementary building blocks will be "black boxes" realizing transfer functions of the form given by (2). Next we select a particular structure for our investigation and motivate this selection on the basis of prior work. This is followed by the development of a synthesis method for the selected structure that simultaneously proves the generality of it.