Having Problem about DownConversion
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.