Apply low-pass filter to a signal

Question

I am trying to understand how I can apply a low-pass filter to a signal.

Let's say that I have the following:

$f(t) = 500\sin(500t)\cos(2kπt)$

I want to apply an ideal low-pass filter having a cutoff frequency of $500$Hz.

What steps should I follow in this particular case?

Thanks a lot!

Answer 1

You need to calculate the Fourier Transform of $f$. You'll see that it's a rect convolved with the sum of two shifted delta functions. If you understand convolution, you'll see that the result is itself the sum of two shifted rect functions. Then multiply that by a rect of appropriate width and inverse Fourier Transform.