Table 2.

Digital equivalences to the analog integrator.

s-Domain: Continuous-Time Domain Expression	z-Domain	Discrete-Time Domain Expression	Rule	Terminology
$\frac{1}{s}\hspace{1em}\leftrightarrow \hspace{1em}y\left(t\right)=\int x\left(t\right)dt$	$\frac{1}{s}=\frac{T}{z-1}$	$y_{\mathrm{S}\mathrm{I}}\left(k\right)=y\left(k-1\right)+T·x\left(k-1\right)$	Step-invariant	Forward rule, zero-order holder
	$\frac{1}{s}=\frac{Tz}{z-1}$	$y_{\mathrm{I}\mathrm{I}}\left(k\right)=y\left(k-1\right)+T·x\left(k\right)$	Impulse-invariant	Backward rule, box car
	$\frac{1}{s}=\frac{T}{2}\frac{z+1}{z-1}$	$y_{\mathrm{B}\mathrm{L}}\left(k\right)=y\left(k-1\right)+\frac{T}{2}\left[x\left(k\right)+x\left(k-1\right)\right]$	Bilinear	Trapezoid, Tustin