# Convolution Revisited

Consider an LTI system with impulse response *h*: *Integers* →
*Reals*. Recall that the output is defined by the convolution sum

.

where *x*(*n*)* *is the input. An individual term in the
summation is *x*(*k*)* h*(*n* −
*k*), which is a delayed version of the impulse response scaled by *x*(*k*).
Recall further that the impulse response of a moving average filter is

*h*(*n*) = 1/*L* when 0
≤
*n* < *L*

*h*(*n*) = 0 otherwise.

The following applet illustrates the convolution sum by allowing to only include selected terms from the summation.

If you include just one term, you get a response that is just a scaled and delayed version of the impulse response of the running average system. If you include two widely separated terms, then you get two distinct delayed and scaled versions of the impulse response. However, if you include two closely spaced terms, then the scaled impulses responses sum to give a more complicated response.

In this example, the filter is a four point (*L* = 4) running average filter.
The original signal is a (hypothetical) closing stock price for XYZ corporation
over 32 days. Notice that the moving average filter smooths the signal somewhat.
But it also introduces a time delay.