Editing Linear time-invariant system (section)

=== Impulse response and convolution ===

Let <math>\{x[m - k];\ m\}</math> represent the sequence <math>\{x[m - k];\text{ for all integer values of } m\}.</math>

And let the shorter notation <math>\{x\}</math> represent <math>\{x[m];\ m\}.</math>

A discrete system transforms an input sequence, <math>\{x\}</math> into an output sequence, <math>\{y\}.</math> In general, every element of the output can depend on every element of the input.  Representing the transformation operator by <math>O</math>, we can write:
<math display="block">y[n] \mathrel{\stackrel{\text{def}}{=}} O_n\{x\}.</math>

Note that unless the transform itself changes with ''n'', the output sequence is just constant, and the system is uninteresting. (Thus the subscript, ''n''.) In a typical system, ''y''[''n''] depends most heavily on the elements of ''x'' whose indices are near ''n''.

For the special case of the [[Kronecker delta function]], <math>x[m] = \delta[m],</math> the output sequence is the '''impulse response''':
<math display="block">h[n] \mathrel{\stackrel{\text{def}}{=}} O_n\{\delta[m];\ m\}.</math>

For a linear system, <math>O</math> must satisfy:
{{NumBlk|:|<math>
  O_n\left\{\sum_{k=-\infty}^{\infty} c_k\cdot x_k[m];\ m\right\} =
  \sum_{k=-\infty}^{\infty} c_k\cdot O_n\{x_k\}.
</math>|{{EquationRef|Eq.4}}}}

And the time-invariance requirement is:
{{NumBlk|:|<math>
\begin{align}
  O_n\{x[m-k];\ m\} &\mathrel{\stackrel{\quad}{=}} y[n-k]\\
                    &\mathrel{\stackrel{\text{def}}{=}} O_{n-k}\{x\}.\,
\end{align}
</math>|{{EquationRef|Eq.5}}}}

In such a system, the impulse response, <math>\{h\}</math>, characterizes the system completely. That is, for any input sequence, the output sequence can be calculated in terms of the input and the impulse response. To see how that is done, consider the identity:
<math display="block">x[m] \equiv \sum_{k=-\infty}^{\infty} x[k] \cdot \delta[m - k],</math>

which expresses <math>\{x\}</math> in terms of a sum of weighted delta functions.

Therefore:
<math display="block">\begin{align}
  y[n] = O_n\{x\}
    &= O_n\left\{\sum_{k=-\infty}^\infty x[k]\cdot \delta[m-k];\ m \right\}\\
    &= \sum_{k=-\infty}^\infty x[k]\cdot O_n\{\delta[m-k];\ m\},\,
\end{align}</math>

where we have invoked {{EquationNote|Eq.4}} for the case <math>c_k = x[k]</math> and <math>x_k[m] = \delta[m-k]</math>.

And because of {{EquationNote|Eq.5}}, we may write:
<math display="block">\begin{align}
  O_n\{\delta[m-k];\ m\} &\mathrel{\stackrel{\quad}{=}} O_{n-k}\{\delta[m];\ m\} \\
                         &\mathrel{\stackrel{\text{def}}{=}} h[n-k].
\end{align}</math>

Therefore:

:{|
| <math>y[n]</math>
| <math>= \sum_{k=-\infty}^{\infty} x[k] \cdot h[n - k]</math>
|-
|
| <math>= \sum_{k=-\infty}^{\infty} x[n-k] \cdot h[k],</math> {{spaces|5}} ([[Convolution#Commutativity|commutativity]])
|}

which is the familiar discrete convolution formula.  The operator <math>O_n</math> can therefore be interpreted as proportional to a weighted average of the function ''x''[''k''].
The weighting function is ''h''[−''k''], simply shifted by amount ''n''. As ''n'' changes, the weighting function emphasizes different parts of the input function.  Equivalently, the system's response to an impulse at ''n''=0 is a "time" reversed copy of the unshifted weighting function.  When ''h''[''k''] is zero for all negative ''k'', the system is said to be [[Causal system|causal]].