There are multiple ways to visualize the convolution operation, each gives unique insights.
Flip-and-slide
Discrete convolution is often visualized as flipping one of the input signals (we call it the kernel or filter) and using it as weights to compute a moving average over another signal.
Similarly, continuous convolution can be visualized by a similar process where the result represents the overlapping area between two functions. In this visualization:
The variable can be interpreted as the amount of “sliding” between the two functions (can also think of it as the time passed)
The dummy variable represents the integration variable, allowing us to calculate the overlapping area (integrate) at each point of the sliding process.
Diagonal Sum
An alternative visualization of the convolution theorem is to view values of the results as a sum or integral of a diagonal element when we lay the input into 2 dimension. 1 This visualization leads to some interesting insides such as treating polynomial multiplication as convolution2.
In the discrete case, we have a straightforward sum:
In the continuous case, the convolution is the area under the curve traced by the diagonal line divided by . 1
Info
The reason that we have that is that the step size on the diagonal line is multiplied by compared to the x or y axis.
Think the Riemann sum approximation of the area of the slice:
This is about the Riemann approximation of the convolution, but with an extra constant :