Editing Minimum phase (section)

== Inverse system ==

A system <math>\mathbb{H}</math> is invertible if we can uniquely determine its input from its output. I.e., we can find a system <math>\mathbb{H}_\text{inv}</math> such that if we apply <math>\mathbb{H}</math> followed by <math>\mathbb{H}_\text{inv}</math>, we obtain the identity system <math>\mathbb{I}</math>. (See [[Inverse matrix]] for a finite-dimensional analog). That is,
<math display="block">
 \mathbb{H}_\text{inv} \mathbb{H} = \mathbb{I}.
</math>

Suppose that <math>\tilde{x}</math> is input to system <math>\mathbb{H}</math> and gives output <math>\tilde{y}</math>:
<math display="block">
 \mathbb{H} \tilde{x} = \tilde{y}.
</math>

Applying the inverse system <math>\mathbb{H}_\text{inv}</math> to <math>\tilde{y}</math> gives
<math display="block">
 \mathbb{H}_\text{inv} \tilde{y} = \mathbb{H}_\text{inv} \mathbb{H} \tilde{x} = \mathbb{I} \tilde{x} = \tilde{x}.
</math>

So we see that the inverse system <math>\mathbb{H}_{inv}</math> allows us to determine uniquely the input <math>\tilde{x}</math> from the output <math>\tilde{y}</math>.

=== Discrete-time example ===

Suppose that the system <math>\mathbb{H}</math> is a discrete-time, [[LTI system theory|linear, time-invariant]] (LTI) system described by the [[impulse response]] <math>h(n)</math> for {{mvar|n}} in {{math|'''Z'''}}. Additionally, suppose <math>\mathbb{H}_\text{inv}</math> has impulse response <math>h_\text{inv}(n)</math>. The cascade of two LTI systems is a [[convolution]]. In this case, the above relation is the following:
<math display="block">
 (h_\text{inv} * h)(n) = (h * h_\text{inv})(n) = \sum_{k=-\infty}^\infty h(k) h_\text{inv}(n - k) = \delta(n),
</math>
where <math>\delta(n)</math> is the [[Kronecker delta]], or the [[identity matrix|identity]] system in the discrete-time case. (Changing the order of <math>h_\text{inv}</math> and <math>h</math> is allowed because of commutativity of the convolution operation.) Note that this inverse system <math>\mathbb{H}_\text{inv}</math> need not be unique.