Hilbert Transform and Gain/Phase Error Bounds for Rational Functions

Abstract

It is well known that a function analytic in the right half plane can be constructed from its real part alone, or (modulo an additive constant) from its imaginary part alone via the Hilbert transform. It is also known that a stable minimum phase transfer function can be reconstructed from its gain alone, or (modulo a multiplicative constant) from its phase alone, via the Bode gain/phase relations. This paper considers the question of the continuity of these constructions, for example, whether small phase errors imply small errors in the calculated transfer function. This is considered in the context of rational functions, and the bound obtained depends on the McMillan degree of the function.