Minimum Phase Systems

One further consequence of the delay is that determining the phase response of the measured system is more complicated. The delay is responsible for a linear phase term since $\delta (n - k) \longleftrightarrow e^{-j2\pi fk/f_S}$. If the delay is known (or measured), then it may be removed by multiplying the measured spectrum by $e^{j2\pi fk/f_S}$. However, if the system being measured is known to be minimum phase, then this method may be applied to find the minimum phase frequency response corresponding to the measured frequency response.

The transfer function measurement toolbox assumes that the system being measured is minimum phase. This is a valid assumption in many cases. For instance, all strictly positive real transfer functions are minimum phase. Dissipative systems are strictly positive real (and therefore minimum phase) if the appropriate quantity is measured and the sensor and motor are collocated. For example, if $s(n)$ controls a motor exerting a force on a dissipative system, and $r(n)$ is the velocity at that same point, the corresponding transfer function will be minimum phase. This holds for other dual variable pairs such as torque and angular velocity, voltage and current, and pressure and fluid flow.

For systems that are not minimum phase, such as systems involving a transmission delay between the input and output quantities, the phase plotted by the transfer function measurement toolbox is not the system phase response, but rather the minimum phase response corresponding to the measured system phase response.