Concerning the minimum number of resonators and the minimum unloaded Q needed in a filter

Given a selective circuit or filter specification, two important questions which arise are: what is the minimum number of elements which can be used to satisfy the specification, and what is the minimum unloaded Q which these elements must have? This paper answers these questions by means of three pairs of design equations, Equations 6 and 11, Equations 7 and 12, and Equations 8 and 13 for filter networks supplying attenuation zeros only (i.e. filters equivalent to constant K type filters) and the corresponding three pairs of equations, Equations 22 and 32 Equations 23 and 33, and Equations 24 and 34, for filters supplying both attenuation zeros and attenuation poles (i.e. filters equivalent to m-derived type filters).