Bode gain-phase relation beyond the minimum phase condition

The Bode gain-phase relation links the phase shift introduced by a causal, linear and time-invariant system with its frequency-dependent magnitude gain. The relation, which is widely used in system theory and electronics, was first derived by H. W. Bode for a transfer function described, in the Fourier (Laplace) domain, by rational functions of frequency, where the numerator fulfills the so-called minimum-phase condition, according to which the polynomial numerator of the frequency-domain transfer function has no zeroes in the upper-half (right-half) of the complex ω-plane (s-plane). Here we discuss a general derivation that widens the range of applicability of the relation beyond the minimum phase condition, allowing the presence of zeroes and encompassing cases in which delays occur. Both these features are widely present in real physical systems, thus endowing the new proof with a broader range of validity.