Convex optimization for approximations of positive real functions

Passive systems are intimately connected to positive real functions. These are analytical functions P (s) that map the right half complex plane s ∈ C to itself with the symmetry condition P (s∗)∗ = P (s), and can be identified with Herglotz functions, which map the upper half complex plane to itself. There is a representation theorem for these functions, stating that P (s) = sC + 1 π ∫ ∞ −∞ sRe{P (jξ)} ξ2 + s2 dξ On the frequency axis s = jω, this can be used to represent the imaginary part of an arbitrary positive real function in terms of the Hilbert transform of the real part (and vice versa), ImP (jω) = jωC + 1 jπ ∫∞ −∞ Re{P (jξ)} ω−ξ dξ. Allowing a pole at ω = 0, and taking the symmetry of the positive real function into account, an admittance function Y (ω) = P (jω) = G(ω) + jB(ω) can be discretized as