Question

Graphically compute the convolution of a square pulse with itself, where a square pulse is defined as {1 if 0<=x<=1, 0 otherwise}.

Answer 1

I know a triangle would result, but I'm just not sure where exactly in the x-axis would this triangle be on. (0,2)? (-1,1)? (-0.5, 1.5)?

Answer 2

sorry, im not good with graph stuff :/i would hate to give u the wrong answer:(

Answer 3

Haha that's okay. Thanks anyway. :D

Answer 4

convolution means, as one signal moves across the other, how does the overlapping area change

Answer 5

wikipedia should have an animation there

Answer 6

@electrokid: Yes I know what convolution means, but I just wanted to make sure which one is the right answer, because my friend and I have been arguing about this haha.

Answer 7

if f(t) is the window function
$$g(t)=f(t)\ast f(t)$$
for
$$t<0\quad g(t)=0$$


$$0\leq t \leq 1\\
g(t)=\int \limits_{-\infty}^{\infty}f(\tau)f(t-\tau)d\tau=\int\limits_0^t1\times1d\tau=t$$

$$1<t\leq2\\
g(t)=\int \limits_{-\infty}^{\infty}f(\tau)f(t-\tau)d\tau=\int\limits_{-1+t}^{1}1\times1 d\tau=1-(-1+t)=2-t$$

$$2<t\\
g(t)=\int \limits_{-\infty}^{\infty}f(\tau)f(t-\tau)d\tau=0$$
$$\therefore\begin{equation*}
g(t) =
\begin{cases}
t & \text{if $0\leq t \leq 1$}\\
2-t & \text{if $1<x\le 2$}\\
0 &\text{otherwise}
\end{cases}
\end{equation*}$$

Answer 8

to find the limits you can use the property,

If f(x) is defined on a<x<b and g(x) on c<x<d,
$$f(x)\ast g(x)$$ is defined on a+b<x<c+d

Answer 9

Thanks!

Answer 10

you're welcome :)