1

Suppose to sample a signal $s(t)$ with bandwidth $B$ with a sampling frequency $f_c$. Suppose also that the number of sample collected is $N$ (the duration of the signal acquisition is then $T = \frac{N}{f_c}$).

I would like to have a bigger $N$ (need more data for model identification) but I can change neither $f_c$ nor $T$ due to technological and experimental issues.

I know that $f_c > 2B$. The Nyquist-Shannon theorem guarantees that the frequency information of the signal are preserved after sampling process.

My guess is that if I interpolate the sampled series with a new frequency, say $f_c' = mf_c$ for $m>0$, everything should work fine since $f'_c > f_c > 2B$. In this way, I will get a new number of samples $N' = mN$.

I don't know if this reasoning is good, I feel like performing interpolation is like "cheating", which is something I want to avoid to write in a scientific paper.

Any ideas?

the_candyman
  • 14,234
  • 4
  • 37
  • 65

1 Answers1

2

Well, yes and no.

A signal that is bandlimited with (double-sided) bandwidth $B$ can be completely reconstructed by samples taken at frequency $f_c$, as long as $f_c > 2B$.

BUT, there is no such thing as a bandlimited signal with compact support, and you only have samples that cover a limited time, so your signal isn't really bandlimited.

Taking a finite chunk of a bandlimited signal is equivalent to multiplying it by a rectangular window function. In the frequency domain, the Fourier transform of that chunk of signal is the Fourier transform of the original convolved with the Fourier transform of the window. You need to look at that to see how much of your signal energy goes into frequencies above $\frac{f_c}{2}$. This stuff will turn into errors near the beginning and end of your new samples.

Also, in order to resample you will have to apply an anti-aliasing filter. These are typically imperfect, so you'll have to look at the frequency response of the filter to ensure again that it's flat enough from $0$ to $B$, but doesn't let through too much above $\frac{f_c}{2}$. Any deviation from the ideal also means errors in the resampling process.

All that said -- If you have enough room between $B$ and $\frac{f_c}{2}$, and enough extra samples at the beginning and end that you can throw away the bad ones, then yes, resampling is very accurate.

Matt Timmermans
  • 448
  • Thanks for this answer. In another context using a construction described in the provided link, I asked in the comments if when talking about sampling a 2D signal, as a complex number, the sampling theorem should be stated as $f_s>f$ as you have 2 correlated samples per sampling period . In the context of the construction using a 3D+1 representation, the sampling frequency lower bound could be proved to be in an extended sense $f_s=\varphi f$ https://math.stackexchange.com/questions/4911385/irrational-numbers-as-periods-in-discrete-sequences-based-on-complex-exponenti – fbatrouni Jun 11 '24 at 04:31