How to calculate the convolution of $\operatorname{sinc}t$ and another function?

Question

I have for example this function: $B(t)=\cos(2\pi t)+2\sin(2\pi t)$

Now I have a LPF function: $H(t)=C\operatorname{sinc}(wt)$

I want to calculate this convolution: $ y=(B*H)$

but I am finding problems doing this, I looked up how do the integral for the $\operatorname{sinc}$ function but not in convolution... and it was kinda of complicated, so I think there is something I am missing out...

Any hint? Thanks in advance.

update( is there is a way to do this with the help of fourier transform ? i think this is the trick maybe :/ )

Answer 1

Hint:

The transform of the convolution is the product of the transforms.

The transform of the cardinal sine is a square window.

The transform of the sinusoid is a pair of Dirac deltas.

Hence the product is either a pair of Dirac deltas, corresponding to the original sinusoid, or zero, depending on whether the Dirac fall inside the window or not.