post detection noise filtering

Hi all,
I wonder if it is useful to filter the output of an envelope detector to get an SNR gain. In fact I have an IF pulse passed through a filter whose bandwidth is more than that of the signal. Then the pulse's envelope is detected by a logamp. I wonder if it is useful to filter the detected signal to limit the bandwidth of the noise. Do you have any idea to calculate the SNR of the output of the detector analytically? Do you know of any document on this issue?

Thanks.

The SNR is stated at the input of the detector.
But in FM communication there is SINAD (Signal-to-Noise and Distortion) which is measured at the audio output.

https://en.wikipedia.org/wiki/SINAD

First of all I want to know if the quality of the detected signal is improved through post detection filtering, no matter how this quality is quantified. If yes then we can seek a way to find how much this improvement is.

No, the quality is not improved, and at the best is not distorted, because as any other filter in this world, it will introduce some distortion (due to ripple, group delay, etc).

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!

The baseband (video) noise filtering it will improve the quality of the signal, but the distortion of the signal introduced by the filter should be counted. This is the reason that SINAD characterization takes into account both, distortion and noise.
There are a lot of analog receivers that use DSP noise filtering in audio/baseband/video, to improve the quality of the demodulated signal.