So I'm currently studying ASK (amplitude shift keying) and how it is modulated and demodulated. The modulation take out data signal, $M(t)$, and multiplies it with our carrier frequency, lets say $\cos(2\pi f t)$, and that's our modulated signal.
For the demodulation, we our taking our received signal and multiplying it with our carrier frequency, which in the frequency domain shifts our received signal to the left and right by our carrier frequency. So we end up with out signal both at the origin and at 2 times our carrier frequency. After that we can do a low pass filter to recover out message signal $M(t)$.
So my question is, what happened to our correlator? I thought in digital communications you take your received signal and pass it through a match filter, which can also be implemented as a correlator. In discrete time this is a multiplication with a signal and then a summation, right? What it seems like we're doing is just multiplying just a single point from the received signal with a single point of our carrier frequency. I've simulated this in MATLAB, and I can see what happens to the signal, I'm just confused why this simple multiplication be being called a correlator.
That being said, can this multiplication and low pass filter be done in the frequency domain by any chance? I know a convolution (or correlation) in the time domain is just a multiplication in the frequency domain, but i'm not sure how to do what we're doing for the ASK demodulation in the frequency domain.
Thanks for your time and input!