On the Monotonicity of Frequency Response Gains

Linear time-invariant single-input-single-output systems with nonnegative impulse responses, commonly called externally positive systems, carry well-known monotonicity properties such as: (i) the static gain equals the $H_{\infty}$-norm (peak of the Bode magnitude diagram), (ii) monotone inputs are mapped to monotone outputs, (iii) the transfer function is totally monotone on the positive reals. In this paper, we complement these properties by proving monotonicity properties of the frequency response gain with the help of variation diminishing theory. While our results give new insights into proving monotonicity properties of the gains of positive systems, they are not limited to such systems, and extend to systems that preserve the periodic monotonicity of their inputs. In particular, our results also provide an interesting sufficient condition for positive dominance.