A simple (low pass) anti-aliasing filter has a linear drop-off, meaning the greater the difference between the frequency of the signal and the frequency of the noise, the better the signal to noise ratio. On the other hand, sampling random gaussian distributed noise will be random gaussian distributed noise no matter what frequency you sample it at.

https://en.wikipedia.org/wiki/Low-pass_filter#/media/File:Bu...

Remember that the lowpass will change the phase or time response of the original signal. If there is anything in the frequencies and phases affected by the lowpass it will get distorted.

The trick works only because the signal is spike like. I reckon it will miss some of the spikes in real life on real signals or produce spurious spikes. The statistics are not accurate at any given point in time just in the limit.

I was going to point this out (phase response distortion) but you beat me to it.

During my university period, I had to digitally sample an ECG (electrocardiogram) signal. Since the shape of the signal is important in diagnosis and must be preserved, I used a Bessel filter [1].

[1] https://en.wikipedia.org/wiki/Bessel_filter

There's no point in saying that a signal is "noise like".

The signal maybe "noise like", but it should be a noise that has a distinct bandwitch from the "real noise"

The antialias filter just reduces the amplitude of the data collected (signal plus noise) in the exact frequencies that pertains to the bandwitch of the "real noise".

i.e you have to garantee that your desired signal has a different bandwitch than the noise. In that case, thermal noise must have this proprierty in comparison to neuron signals.

PS: English is not my first language. Sorry about any grammar or orthographical errors.

If you can guarantee that this means you are actually oversampling your input signal. Sampling theorem still holds.