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The output $y(t)$ of a continuous-time linear time-invariant (LTI) system is related to its input $x(t)$ and the system impulse response $h(t)$ 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] ):
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 ${\delta}_{\Delta}(t)$ to be a rectangular pulse of width $\Delta $ and height 1, centered at $t=0$ :
Approximate a continuous function $x(t)$ with a piecewise constant function ${x}_{\Delta}(t)$ as a sequence of pulses spaced every $\Delta $ seconds in time with heights $x(\mathrm{k\Delta})$ :
It can be shown in the limit as $\Delta \to \mathrm{0,}{x}_{\Delta}(t)\to x(t)$ . As an example, [link] shows the approximation of a decaying exponential $x(t)=\text{exp}(-\frac{t}{2})$ starting from 0 using $\Delta =1$ . Similarly, $h(t)$ can be approximated by
One can thus approximate the convolution integral by convolving the two piecewise constant signals as follows:
Notice that ${y}_{\Delta}(t)$ 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:
If one represents the signals
${h}_{\Delta}(t)$ and
${x}_{\Delta}(t)$ 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
$x(t)$ and
$h(t)$ . Compute the discrete convolution sum
$\sum _{k=-\infty}^{\infty}x(\mathrm{k\Delta})h((n-k)\Delta )$ with the built-in LabVIEW MathScript command
conv
. Then, multiply this sum by
$\Delta $ to get an estimate of
$y(t)$ at
$t=\mathrm{n\Delta}$ Note that as
$\Delta $ is made smaller, one gets a closer approximation to
$y(t)$ .
Convolution satisfies the following three properties (see [link] ):
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