Finite Difference

The finite difference approximation (FDA) amounts to replacing derivatives by finite differences, or

$$\frac{d}{dt} x(t) \buildrel$$$$\mbox{\tiny$\Delta$}$$$$\over=\lim_{\delta\to 0} \frac{x(t) - x(t-\delta)}{\delta} \approx \frac{x(n T)-x[(n-1)T]}{T}.$$

The z transform of the first-order difference operator is $(1-z^{-1})/T$. Thus, in the frequency domain, the finite-difference approximation may be performed by making the substitution

$$s \rightarrow \frac{1-z^{-1}}{T}$$

The first-order difference is first-order error accurate in $T$. Better performance can be obtained using the bilinear transform, defined by the substitution

$$s \longrightarrow c\left(\frac{1-z^{-1}}{1+z^{-1}}\right)$$   where$$\quad c=\frac{2}{T}.$$