# Convolution and linear time-invariant systems

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## Convolution and its numerical approximation

The output $y\left(t\right)$ of a continuous-time linear time-invariant (LTI) system is related to its input $x\left(t\right)$ and the system impulse response $h\left(t\right)$ through the convolution integral expressed as (for details on the theory of convolution and LTI systems, refer to signals and systems textbooks, for example, references [link] - [link] ):

$y\left(t\right)=\underset{-\infty }{\overset{\infty }{\int }}h\left(t-\tau \right)x\left(\tau \right)\mathrm{d\tau }$

For a computer program to perform the above continuous-time convolution integral, a numerical approximation of the integral is needed noting that computer programs operate in a discrete – not continuous – fashion. One way to approximate the continuous functions in the Equation (1) integral is to use piecewise constant functions. Define ${\delta }_{\Delta }\left(t\right)$ to be a rectangular pulse of width $\Delta$ and height 1, centered at $t=0$ :

${\delta }_{\Delta }\left(t\right)=\left\{\begin{array}{cc}1& -\Delta /2\le t\le \Delta /2\\ 0& \text{otherwise}\end{array}$

Approximate a continuous function $x\left(t\right)$ with a piecewise constant function ${x}_{\Delta }\left(t\right)$ as a sequence of pulses spaced every $\Delta$ seconds in time with heights $x\left(\mathrm{k\Delta }\right)$ :

${x}_{\Delta }\left(t\right)=\sum _{k=-\infty }^{\infty }x\left(\mathrm{k\Delta }\right){\delta }_{\Delta }\left(t-\mathrm{k\Delta }\right)$

It can be shown in the limit as $\Delta \to 0,{x}_{\Delta }\left(t\right)\to x\left(t\right)$ . As an example, [link] shows the approximation of a decaying exponential $x\left(t\right)=\text{exp}\left(-\frac{t}{2}\right)$ starting from 0 using $\Delta =1$ . Similarly, $h\left(t\right)$ can be approximated by

${h}_{\Delta }\left(t\right)=\sum _{k=-\infty }^{\infty }h\left(\mathrm{k\Delta }\right){\delta }_{\Delta }\left(t-\mathrm{k\Delta }\right)$

One can thus approximate the convolution integral by convolving the two piecewise constant signals as follows:

${y}_{\Delta }\left(t\right)=\underset{-\infty }{\overset{\infty }{\int }}{h}_{\Delta }\left(t-\tau \right){x}_{\Delta }\left(\tau \right)\mathrm{d\tau }$

Notice that ${y}_{\Delta }\left(t\right)$ is not necessarily a piecewise constant. For computer representation purposes, discrete output values are needed, which can be obtained by further approximating the convolution integral as indicated below:

${y}_{\Delta }\left(\mathrm{n\Delta }\right)=\Delta \sum _{k=-\infty }^{\infty }x\left(\mathrm{k\Delta }\right)h\left(\left(n-k\right)\Delta \right)$

If one represents the signals ${h}_{\Delta }\left(t\right)$ and ${x}_{\Delta }\left(t\right)$ in a .m file by vectors containing the values of the signals at $t=\mathrm{n\Delta }$ , then Equation (5) can be used to compute an approximation to the convolution of $x\left(t\right)$ and $h\left(t\right)$ . Compute the discrete convolution sum $\sum _{k=-\infty }^{\infty }x\left(\mathrm{k\Delta }\right)h\left(\left(n-k\right)\Delta \right)$ with the built-in LabVIEW MathScript command conv . Then, multiply this sum by $\Delta$ to get an estimate of $y\left(t\right)$ at $t=\mathrm{n\Delta }$ Note that as $\Delta$ is made smaller, one gets a closer approximation to $y\left(t\right)$ .

## Convolution properties

Convolution satisfies the following three properties (see [link] ):

• Commutative property
$x\left(t\right)\ast h\left(t\right)=h\left(t\right)\ast x\left(t\right)$
• Associative property
$x\left(t\right)\ast {h}_{1}\left(t\right)\ast {h}_{2}\left(t\right)=x\left(t\right)\ast \left\{{h}_{1}\left(t\right)\ast {h}_{2}\left(t\right)\right\}$
• Distributive property
$x\left(t\right)\ast \left\{{h}_{1}\left(t\right)+{h}_{2}\left(t\right)\right\}=x\left(t\right)\ast {h}_{1}\left(t\right)+x\left(t\right)\ast {h}_{2}\left(t\right)$

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