On flat gain with frequency- dependent terminations

The problem of matching frequency-dependent generator and load impedances to each other with a lossless reciprocal equalizer so as to achieve constant transducer gain over a passband ("selective flat gain") is considered. Despite the fact that selective flat gain is achievable from a resistive source to a complex load (single matching) it is shown, using a potential analogy and properties of dipole layer potential functions, that for double matching (complex source and complex load) selective flat gain to an arbitrary tolerance is never physically realizable even if the number of equalizer elements becomes infinite. Various examples are given including a calculation of a bound on ultimate tolerance deviation from flat gain when load and generator are both parallel impedances and the number of equalizer elements becomes infinite. The case of "all-pass flat gain" matching when generator and load have only RHP zeros of transmission is also considered. Surprisingly in this case flat gain can sometimes be achieved if special restrictions based on a Nevanlinna-Pick matrix criterion are satisfied, but selective flat gain is always unrealizable.