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

To compute the output *y(t)* at a specified *t*, first the integrand *h(v)x(t - v)* is computed as a function of *v*.Then integration with respect to *v* is performed, resulting in *y(t)*.

These mathematical operations have simple graphical interpretations.First, plot *h(v)* and the "flipped and shifted" *x(t - v)* on the *v* axis, where *t* is fixed. Second, multiply the two signals and compute the signed area of the resulting function of *v* to obtain *y(t)*. These operations can be repeated for every value of *t* of interest.

To explore graphical convolution, select signals *x(t)* and *h(t)* from the provided examples below,or use the mouse to draw your own signal or to modify a selected signal. Then click at a desired value of *t* on the first *v* axis. After a moment, *h(v)* and *x(t - v)* will appear. Drag the *t* symbol along the *v* axis to change the value of *t*. For each *t*, the corresponding integrand *h(v)x(t - v)* and the output value *y(t)* will be displayed in their respective windows.

Applet by Steve Crutchfield |