Editing Linear time-invariant system (section)

===Impulse response and convolution===
The behavior of a linear, continuous-time, time-invariant system with input signal ''x''(''t'') and output signal ''y''(''t'') is described by the convolution integral:<ref>Crutchfield, p. 1. ''Welcome!''</ref>
:{|
| <math>y(t) = (x * h)(t)</math>
| <math>\mathrel{\stackrel{\mathrm{def}}{=}} \int\limits_{-\infty}^{\infty} x(t - \tau)\cdot h(\tau) \, \mathrm{d}\tau</math>
|-
|
| <math>= \int\limits_{-\infty}^\infty x(\tau)\cdot h(t - \tau) \,\mathrm{d}\tau,</math> {{spaces|5}} (using [[Convolution#Commutativity|commutativity]])
|}

where <math display="inline"> h(t)</math> is the system's response to an [[Dirac delta function|impulse]]: <math display="inline">x(\tau) = \delta(\tau)</math>. <math display="inline"> y(t) </math> is therefore proportional to a weighted average of the input function <math display="inline">x(\tau)</math>. The weighting function is <math display="inline"> h(-\tau)</math>, simply shifted by amount <math display="inline"> t</math>. As <math display="inline"> t</math> changes, the weighting function emphasizes different parts of the input function. When <math display="inline"> h(\tau)</math> is zero for all negative <math display="inline"> \tau</math>, <math display="inline"> y(t)</math> depends only on values of <math display="inline"> x</math> prior to time <math display="inline"> t</math>, and the system is said to be [[Causal system|causal]].

To understand why the convolution produces the output of an LTI system, let the notation <math display="inline"> \{x(u-\tau);\ u\}</math> represent the function <math display="inline"> x(u-\tau)</math> with variable <math display="inline"> u</math> and constant <math display="inline"> \tau</math>. And let the shorter notation <math display="inline"> \{x\}</math> represent <math display="inline"> \{x(u);\ u\}</math>. Then a continuous-time system transforms an input function, <math display="inline"> \{x\},</math> into an output function, <math display="inline">\{y\}</math>. And in general, every value of the output can depend on every value of the input. This concept is represented by:
<math display="block">y(t) \mathrel{\stackrel{\text{def}}{=}} O_t\{x\},</math>
where <math display="inline"> O_t</math> is the transformation operator for time <math display="inline"> t</math>. In a typical system, <math display="inline"> y(t)</math> depends most heavily on the values of <math display="inline"> x</math> that occurred near time <math display="inline"> t</math>. Unless the transform itself changes with <math display="inline"> t</math>, the output function is just constant, and the system is uninteresting.

For a linear system, <math display="inline"> O</math> must satisfy {{EquationNote|Eq.1}}:

{{NumBlk|:|<math>
O_t\left\{\int\limits_{-\infty}^\infty c_{\tau}\ x_{\tau}(u) \, \mathrm{d}\tau ;\ u\right\} = \int\limits_{-\infty}^\infty c_\tau\ \underbrace{y_\tau(t)}_{O_t\{x_\tau\}} \, \mathrm{d}\tau.
</math>|{{EquationRef|Eq.2}}}}

And the time-invariance requirement is:

{{NumBlk|:|<math>
\begin{align}
  O_t\{x(u - \tau);\ u\} &\mathrel{\stackrel{\quad}{=}} y(t - \tau)\\
                         &\mathrel{\stackrel{\text{def}}{=}} O_{t-\tau}\{x\}.\,
\end{align}
</math>
| {{EquationRef|Eq.3}}
}}

In this notation, we can write the '''impulse response''' as <math display="inline"> h(t) \mathrel{\stackrel{\text{def}}{=}} O_t\{\delta(u);\ u\}.</math>

Similarly:
:{|
| <math>h(t - \tau)</math>
| <math>\mathrel{\stackrel{\text{def}}{=}} O_{t-\tau}\{\delta(u);\ u\}</math>
|-
|
| <math>= O_t\{\delta(u - \tau);\ u\}.</math> {{spaces|5}} (using {{EquationNote|Eq.3}})
|}

Substituting this result into the convolution integral:
<math display="block">
\begin{align}
  (x * h)(t) &= \int_{-\infty}^\infty x(\tau)\cdot h(t - \tau) \,\mathrm{d}\tau \\[4pt]
             &= \int_{-\infty}^\infty x(\tau)\cdot O_t\{\delta(u-\tau);\ u\} \, \mathrm{d}\tau,\,
\end{align}
</math>

which has the form of the right side of {{EquationNote|Eq.2}} for the case <math display="inline"> c_\tau = x(\tau)</math> and <math display="inline"> x_\tau(u) = \delta(u-\tau).</math>

{{EquationNote|Eq.2}} then allows this continuation:
<math display="block">
\begin{align}
  (x * h)(t) &= O_t\left\{\int_{-\infty}^\infty x(\tau)\cdot \delta(u-\tau) \, \mathrm{d}\tau;\ u \right\}\\[4pt]
             &= O_t\left\{x(u);\ u \right\}\\
             &\mathrel{\stackrel{\text{def}}{=}} y(t).\,
\end{align}
</math>

In summary, the input function, <math display="inline"> \{x\}</math>, can be represented by a continuum of time-shifted impulse functions, combined "linearly", as shown at {{EquationRef|Eq.1}}.  The system's linearity property allows the system's response to be represented by the corresponding continuum of impulse <u>responses</u>, combined in the same way. And the time-invariance property allows that combination to be represented by the convolution integral.

The mathematical operations above have a simple graphical simulation.<ref>Crutchfield, p. 1. ''Exercises''</ref>