# How to choose sampling rate for non-band limited signal given as an expression?

I have a signal given as an expression $x(t)$. In frequency domain most of the information is around 0 and higher frequencies progressively get smaller and smaller compared to DC component.

I was wondering how to determine the sampling rate I should use if I wanted to compute discrete samples and pass them through some filter?

I was considering that if I pick some $F_s$ sampling rate, aliasing will cause frequencies above $|F_s/2|$ to be folded back into the region $F_s/2$. Using Parseval's theorem the sum of square errors because of this appears to be $\int_{|\Omega > F_s/2|}|X(j\Omega)|^2d\Omega$ . So I could try to figure out the acceptable error and pick $F_s$ based on that. But that integral goes to infinity and what if the integral doesn't converge, or it's not something I can compute?

I've also seen discussions on anti-aliasing filters that mention the stop band attenuation should be higher than the SNR (eg anti-aliasing filter). In my case I don't really have an anti-aliasing filter because I'm just computing x(t). But it seems if I want 16 bits resolution, SNR=98dB so if I compute the frequency at which $20log(\frac{|X(j\Omega)|}{|X(0)|})<-98$ it will be as if I used anti-aliasing filter. However, I don't understand why the stop-band attenuation is chosen to be equal to SNR, and what that means in terms of signal error. Because while -98dB is really small compared to the rest of the signal, there are infinitely many aliased frequencies being folded and added up.

Thanks for any clarification.

Edit

I have frequency vs relative magnitude (relative to magnitude at 0 frequency) plot calculated by sampling at 100kHz and 1MHz and taking DFT of the samples.

What reasonable conclusion about required sampling rate can I make based on something like this? How can I justify it?

For example at sampling rate 100kHz, the smallest frequency component appears to be 158 dB smaller than the largest (or 10^(7.9) times smaller). Is that small enough to be unimportant to overall signal? How should I go about determining what's small enough?

It depends on the spectral distribution of the signal under concern. If, for example, as you said the signal has most of its energy around zero frequency and gets progressively smaller as frequency increases then it may be possible to sample it with an acceptable error, provided that the progress is rapid enough. So it depends on the progress: Provided that it decays fast enough, you may find an acceptable sample rate which would provide a bounded error. After all many practical signals will have nonzero (but very small) energy above their half the sampling rate even after aa filtering, but nevertheless they are sampled with sufficient accuracy.

And your bit resolution vs stop band attenuation concern. That is the noise floor of the N-bit representation. Any error less than noise floor, and being uncorrelated with the signal, may be considered ok.

Finaly please note that, when the aliased frequencies fold up (say at 0 hz) for every next shifted spectra to be added, the folded tail will be progressiveley smaller, as it comes from progressiveley farther away from its center. Hence the sum of such infinitely many decaying terms may converge. Provided the folded terms are getting smaller and smaller.