# Non Equispaced / Non Uniform DFT Bandwidth

I need to construct Fourier transform of non-equispaced data.

That is, I have signal $s(t)$, $t\in[0,T]$ sampled at non-equispaced points $t_k$, $k=0...N-1$ with sample values $s_k = s(t_k)$. For Fourier transform I use approximation of integral: $$\hat S(\omega) = \int\limits_{-\infty}^\infty s(t)e^{-i\omega t}dt \approx \hat S_d(\omega) = \sum_{k=0}^{N-1}s_ke^{-i\omega t_k}\Delta t_k \tag{1}$$ where $\Delta t_k = (t_{k+1}-t_{k-1})/2$. As sampling points in frequency domain I choose $\omega_n = \frac{2\pi}{T}$.

My question is: as I can evaluate (1) for any $\omega$, what is the maximum $\omega$ such $\hat S_d(\omega)$ "adequately" represents $\hat S(\omega)$? What is the maximum of $n$ for $\omega_n$ I can use? For DFT we got Nyquist frequency. Do we have something similar for NDFT? Any references would be appreciated.

Note: I'm aware of such things as NDFT and NFFT. Though formula for NDFT, as presented in most papers, is $$\hat S_d(\omega_n) = \sum_{k=0}^{N-1}s_ke^{-i\omega_n t_k}$$ I strongly believe that I need to use formula (1) as I'm trying to build a periodogram.

And I'm not interested in fast ways of computing NDFT yet, so I'm not considering NFFT.

• i dunno what the "N" means in NDFT or NFFT. if you are representing your non-equispaced data as $$s(t) = \sum\limits_{k=0}^{N-1} s_k \delta(t-t_k)$$ then your $\hat{S}_d(\omega)$ formula is correct in the continuous-frequency domain. still not a DFT. if you want to DFT, then you have to interpolate your non-equispaced data and uniformly (re)sample it. as @rrogers had implied. dunno if i agree with the $sinc$ formula in his/her answer. i don't think i do. Jan 8, 2015 at 23:44
• Given some conditions on the non uniform sampling you may extract the exact same information as with uniform sampling.
– Royi
Feb 13, 2021 at 12:29

$$S(\omega)=\sum_{n=0}^{N-1}s_{k}sinc(\frac{\omega\triangle\left(t_{k}\right)}{2})e^{-wt_{k}}$$ Having said that, what is "adequate"? This is, more or less, subjective. The original Nyquest criterion is based upon the result that any (freq < sam/2) (I think there is some small quibbling on that) can be perfectly reconstructed; if it and the sampling lasts long enough. It says nothing about out of band signals; but the actual calculations will show aliasing.