Frequency Domain Analysis

FDA, force-drive mass $m$:

$$v_n=v_{n-1} + \frac{T}{m}f_n, \quad n=0,1,2,\ldots$$

$z$ transform ($v_{-1}=0$):

$$V(z) = z^{-1}V(z) + \frac{T}{m} F(z)$$

Driving Point Impedance (digital):

$$\zbox{R(z) \isdef \frac{F(z)}{V(z)} = m \frac{1-z^{-1}}{T}}$$

Continuous-time driving point impedance:

$$f(t)=m\frac{dv}{dt} \quad\longleftrightarrow\quad F(s)=msV(s)$$

$$\Rightarrow\quad \zbox{R(s) \isdef \frac{F(s)}{V(s)} = ms}$$

Thus, the FDA maps $s$ plane to the $z$ plane as follows:

$$\zbox{s \leftarrow \frac{1-z^{-1}}{T}}$$