Editing Minimum phase (section)

=== Continuous-time frequency analysis ===

Analysis for the continuous-time case proceeds in a similar manner, except that we use the [[Laplace transform]] for frequency analysis. The time-domain equation is
<math display="block">
 (h * h_\text{inv})(t) = \delta(t),
</math>
where <math>\delta(t)</math> is the [[Dirac delta function]]{{snd}} the identity operator in the continuous-time case because of the sifting property with any signal {{math|''x''(''t'')}}:
<math display="block">
 (\delta * x)(t) = \int_{-\infty}^\infty \delta(t - \tau) x(\tau) \,d\tau = x(t).
</math>

Applying the [[Laplace transform]] gives the following relation in the [[s-plane]]:
<math display="block">
 H(s) H_\text{inv}(s) = 1,
</math>
from which we realize that
<math display="block">
 H_\text{inv}(s) = \frac{1}{H(s)}.
</math>

Again, for simplicity, we consider only the case of a [[rational function|rational]] [[transfer function]] {{math|''H''(''s'')}}. Causality and stability imply that all [[pole (complex analysis)|poles]] of {{math|''H''(''s'')}} must be strictly inside the left-half [[s-plane]] (see [[BIBO stability#Continuous-time signals|stability]]). Suppose
<math display="block">
 H(s) = \frac{A(s)}{D(s)},
</math>
where {{math|''A''(''s'')}} and {{math|''D''(''s'')}} are [[polynomial]] in {{mvar|s}}. Causality and stability imply that the [[pole (complex analysis)|poles]]{{snd}} the [[Root of a function|roots]] of {{math|''D''(''s'')}}{{snd}} must be inside the left-half [[s-plane]]. We also know that
<math display="block">
 H_\text{inv}(s) = \frac{D(s)}{A(s)},
</math>
so causality and stability for <math>H_\text{inv}(s)</math> imply that its [[pole (complex analysis)|poles]]{{snd}} the roots of {{math|''A''(''s'')}}{{snd}} must be strictly inside the left-half [[s-plane]]. These two constraints imply that both the zeros and the poles of a minimum-phase system must be strictly inside the left-half [[s-plane]].