Editing Linear time-invariant system (section)

=== Examples ===
{{bulleted list
| A simple example of an LTI operator is the delay operator <math>D\{x[n]\} \mathrel{\stackrel{\text{def}}{=}} x[n-1]</math>.
* <math> D \left( c_1 \cdot x_1[n] + c_2 \cdot x_2[n] \right) = c_1 \cdot x_1[n - 1] + c_2\cdot x_2[n - 1] = c_1\cdot Dx_1[n] + c_2\cdot Dx_2[n]</math> &nbsp; (i.e., it is linear)
* <math> D\{x[n - m]\} = x[n - m - 1] = x[(n - 1) - m] = D\{x\}[n - m]</math> &nbsp; (i.e., it is time invariant)
The Z transform of the delay operator is a simple multiplication by ''z''<sup>−1</sup>. That is,
<math display="block"> \mathcal{Z}\left\{Dx[n]\right\} = z^{-1} X(z). </math>
| Another simple LTI operator is the averaging operator
<math display="block"> \mathcal{A}\left\{x[n]\right\} \mathrel{\stackrel{\text{def}}{=}} \sum_{k=n-a}^{n+a} x[k].</math>

Because of the linearity of sums,
<math display="block">\begin{align}
  \mathcal{A}\left\{c_1 x_1[n] + c_2 x_2[n] \right\}
    &= \sum_{k=n-a}^{n+a} \left( c_1 x_1[k] + c_2 x_2[k] \right)\\
    &= c_1 \sum_{k=n-a}^{n+a} x_1[k] + c_2 \sum_{k=n-a}^{n+a} x_2[k]\\
    &= c_1 \mathcal{A}\left\{x_1[n] \right\} + c_2 \mathcal{A}\left\{x_2[n] \right\},
\end{align}</math>

and so it is linear. Because,
<math display="block">\begin{align}
  \mathcal{A}\left\{x[n-m]\right\}
    &= \sum_{k=n-a}^{n+a} x[k-m]\\
    &= \sum_{k'=(n-m)-a}^{(n-m)+a} x[k']\\
    &= \mathcal{A}\left\{x\right\}[n-m],
\end{align}</math>

it is also time invariant.
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