Could someone please explain what is happening at the "f*g" row and below? The image is located here as linked from the Wikipedia page. I want to teach myself about Fourier Transforms / Series, and I've come across "convolutions." I just want to make sure I understand this diagram correctly as I try to get a feel for what the function returns are.
Thanks.
As you saw in the Wikipedia entry, the definition of convolution is $(f \ast g) (t) = \int_{-\infty}^{\infty} f(\tau) g(t - \tau) d\tau.$
Now, notice that, in the example, $f$ is simply (assuming the constant value is $1$)$f(t) = \begin{cases}1, & a < t < b \\ 0, & \text{otherwise} \end{cases}$
Hence, the above integral becomes $$(f \ast g) (t) = \int_{a}^{b} g(t - \tau) d\tau,$$ which is just the area below the graph of $g(t - \tau)$ for $\tau \in (a,b)$. Now the drawings below the $f \ast g$ row show the overlapping of the functions $g(t- \tau)$ and $f(\tau)$ for increasing values of $t$. By analyzing the behavior (first increasing, then decreasing) of the highlighted area below the graph of $g$, we can understand the behavior of the convolution.
