Relationships between positive realness of continuous transfer functions and their digital counterparts

The importance of positive real transfer functions derives from the fact that they describe positive linear systems. Such systems possess the property that the time-integral of their input-output product, which is a measure of the total energy, is non-negative. This property is necessary and sufficient for guaranteeing the realisability of a transfer function as the driving point impedance or admittance of a linear passive network. The interest in carrying out formulations when the positive realness of a continuous transfer function is maintained even after discretisation through a sampler and zero-order hold device is confirmed. It is proven that positive realness may always be maintained after discretisation if all the poles of the continuous transfer function are strictly stable. This is necessary for the direct input-output transmission gain of the continuous transfer function to exceed a minimum positive threshold in order for the discrete positive realness to hold