I'm studing for an exam and I'm stuck on a simple exercise about convolution between two functions. It says:
A system has a triangular impulse response (LSF) centered at the origin of the plane $h(x)=Λ(x)$. We input an image with two impulses, defined as $f(x)=δ(x-x_0)+δ(x-2x_0)$. Get the output of the system.
I was thinking to use this formula: $\int_{-\infty }^{+\infty} f(x-x_0) h(x_0) dx_0$
So it would become:
$\int_{-\infty }^{+\infty} (δ(x-2x_0)+δ(x-3x_0)) Λ(x_0) dx_0$
I don't know if it makes sense, so i'll appreciate any help.
Thank you.
Best Answer
I think you've misunderstood what $x_0$ is. It is a constant, not a variable, so you should not be integrating over it. The variable you integrate over in the convolution is a dummy variable that won't appear in the final answer, so it give a name that doesn't match any names you already have for variables. In this case, the convolution formula would written as $$(f\star g)(x) = \int_{-\infty}^\infty f(x-y)g(y)dy = \int_{-\infty}^\infty f(y)g(x-y)dy$$
Can you try writing the convolution again?