Editing Bilinear transform (section)

== Stability and minimum-phase property preserved ==
A continuous-time causal filter is [[BIBO stability|stable]] if the [[Pole (complex analysis)|poles]] of its transfer function fall in the left half of the [[complex number|complex]] [[s-plane]]. A discrete-time causal filter is stable if the poles of its transfer function fall inside the [[unit circle]] in the [[complex plane|complex z-plane]]. The bilinear transform maps the left half of the complex s-plane to the interior of the unit circle in the z-plane. Thus, filters designed in the continuous-time domain that are stable are converted to filters in the discrete-time domain that preserve that stability.

Likewise, a continuous-time filter is [[minimum-phase]] if the [[Zero (complex analysis)|zeros]] of its transfer function fall in the left half of the complex s-plane. A discrete-time filter is minimum-phase if the zeros of its transfer function fall inside the unit circle in the complex z-plane. Then the same mapping property assures that continuous-time filters that are minimum-phase are converted to discrete-time filters that preserve that property of being minimum-phase.