Convolution and linear time-invariant systems

Convolution and its numerical approximation

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 properties

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

• Commutative property

• Associative property

• Distributive property