Sorry, I may not have made my point clear.
Although filtering may distort the signal but it can reject out of band noise as well and the net result can be an improved signal quality. This is the idea of filtering, isn't it? My question is if it is useful to band limit the noise after envelope detection. In my case, the signal bandwidth is 2MHz. The natural choice would be filtering the received signal with a bandpass filter of similar bandwidth and then pass it through the detector (logamp). However because of a possible maximum drift of +/-5 MHz in the carrier frequency I should keep the input filter as wide as 12Mhz to let the signal come in if the carrier is 5MHz away from the filter center frequency. After the detection however, just the envelope of the received signal is remained regardless of its carrier frequency so the bandwidth of the signal at the detector output will be 2MHz, while I have let a noise of 12MHz bandwidth enter the detector. Now I want to know if I can put a filter of 2MHz bandwidth to reject this excess noise. That's it!