Convolution Examples

Understanding convolution is not complete without some examples to visual the results. These examples are meant to aid in one’s ability to visualize the result of a convolution. It’s most important to flip one function horizontally and slide it through the other. As this function slides through we multiply and sum to obtain the result of the convolution.

Convolution is used mainly in two ways:

To mathematically determine how correlated two functions are
To shape a function by convolving it with another function. Examples would be filtering, smoothing, etc …

Note: There are some typos in the frequency labels for some of the sinusoids I need to fix. You can trust the graphs.

How the examples are organized

Each example will be organized in the following way. Let’s first take the equation for convolution.

$$ y[t] = \sum_{i=-n_1}^{n_2} x[i]*h[t-i] $$

For each example $ h[t] $ will be the first function we show followed by $ x[t] $ and $ y[t] $.

An Introduction to the example and why it’s important
The functions $ h[t] $ ,$ x[t] $ and $ y[t] $.

For reference the x-axis is meant to be as simple as possible and not distract from the example. Attention should be paid to the shape of the result $ y[t] $. We have 100 samples for each time unit. Thus, from 1 to 2 we have 101 samples.