# Denoising short, non-uniformly spaced, bandlmited sequences

Consider a bandlimited signal $x(t)$ with bandwidth $BW$. Samples of this signal are observed at non-uniformly spaced sample points in the presence of noise:

$y(t_i)=x(t_i) + n_i \qquad i\in\{0,1,\cdots,N-1\}$

where $N$ is on the order of 10-30 samples.

The samples are "densely-spaced" in the sense that:

$\frac{1}{t_i - t_{i-1}} > 2\times BW$

The noise samples can be modeled as a realization of independent, identically distributed random variables of zero mean and some variance $\sigma^2_n$. A variant of the noise model would be the inclusion of occasional outliers.

My goal is to recover estimates $\hat{x}(t_i)$ of the desired sequence which explicitly exploit the bandlimited nature of $x(t)$.

Savitzky-Golay filters can be generalized for non-uniformly spaced samples, but it's not clear how the bandwidth can be taken into account. (For uniformly spaced samples, such an interpretation is available, see "What Is a Savitzky-Golay Filter?" by R. Schafer). I guess I could use the frequency response interpretation of the SG filter for a worst case effective, constant sample frequency, but I'm wondering if other approaches may be better.

An extension would be interpolation/resampling of the data on a uniform grid.

Any suggestions?

• well, if the noise is white, it doesn't matter that it's non-uniformly sampled. if the density of non-uniform sampling of $x(t)$ is much greater than $2 \, BW$ you might want to interpolate (using some spline that is friendly to non-equally spaced points), uniformly resample, and then low-pass filter with the cutoff at approximately the $BW$. other than that, i dunno how you can differentiate the contribution of $n(t)$ that is below $BW$ from what belongs to $x(t)$ that is below $BW$. – robert bristow-johnson Jul 13 '17 at 18:02
• Yes, there is no obvious way to separate signal from noise for frequencies beneath $BW$. Do you know of spline methods which are especially well-suited to nonuniformly spaced samples? Also, since the data record is short, that would greatly limit the length of any FIR low pass filter. – rhz Jul 13 '17 at 18:31
• Also, I'd say the sample density is greater than Nyquist, but not by a huge amount, perhaps a factor of perhaps 3. – rhz Jul 13 '17 at 18:47
• low-pass filtering won't gain you much. do you know something about the spectral nature of the signal that might allow you to apply matched filtering? the simplest decent spline would be a cubic Hermite polynomial spline which preserves continuity at the 0th and 1st derivative. higher odd-orders will preserve continuity at the 2nd and higher derivatives. – robert bristow-johnson Jul 13 '17 at 20:34
• there are also wavelet-based techniques for denoising that work by separating the signal into little wavelets and ditching the wavelets with the smallest coefficients. – robert bristow-johnson Jul 13 '17 at 20:36