Having Problem about DownConversion I hope, you are doing well. I want to ask question about baseband conversion.
Background:

I generated two AM signals in MATLAB, which one of them is shifted in time.
I multiply the signals with $\cos(2\pi f_c t)$ in order to obtain IF signal.
I filtered the IF signals with $f_c$ cut-off frequency.

In this way, I aim to downconverted to signal to baseband.
My problem is, when I try to cross-correlated the baseband signals, my result peaks at zero.
I expect to see the delay, but I could not. Am I missing something?
Notes:
I am exact sure, my cross-correlation function works perfectly. I tried this function before baseband process.
Also, in order to obtain both I and Q components, I try to multiply my signals with sin(2pifc*t). Nothing changes.
MessageSignal: Audio Signal.
If you mix up to $f_c$, but then use a filter with cut-off frequency of $f_c$, you're not downconverting your signal to baseband at all. You're just cutting off half the signal.
I think you want to re-visit the block diagram of the receiver architecture you're trying to implement. You're most definitely missing the important step, but since I'm not sure what kind of receiver you're trying to build, I can't tell you which.