# 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

## 1 Answer

Not a lot of references that include nonuniform, band limited, and filtering

D. Bonacci and B. Lacaze, "Lowpass/bandpass signal reconstruction and digital filtering from nonuniform samples," 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), South Brisbane, QLD, 2015, pp. 3626-3630. doi: 10.1109/ICASSP.2015.7178647 Abstract: This paper considers the problem of non uniform sampling in the case of finite energy functions and random processes, not necessarily approaching to zero as time goes to infinity. The proposed method allows to perform exact signal reconstruction, spectral estimation or linear filtering directly from the non-uniform samples. The method can be applied to either lowpass, or bandpass signals. keywords: {band-pass filters;digital filters;linear phase filters;low-pass filters;random processes;signal reconstruction;signal sampling;band-pass signal reconstruction;digital filter;finite energy function;linear filter;low-pass signal reconstruction;nonuniform sampling problem;random process;spectral estimation;Baseband;Estimation;Fourier series;Interpolation;Nonuniform sampling;Random processes;Time-frequency analysis;Nonuniform filtering;Periodic nonuniform sampling;Sampling theory;Signal reconstruction;Spectral estimation}, URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7178647&isnumber=7177909

H. Johansson and P. Lowenborg, "Reconstruction of nonuniform sampled bandlimited signals using digital fractional filters," ISCAS 2001. The 2001 IEEE International Symposium on Circuits and Systems (Cat. No.01CH37196), Sydney, NSW, 2001, pp. 593-596 vol. 2. doi: 10.1109/ISCAS.2001.921140 Abstract: This paper considers the problem of reconstructing nonuniformly sampled bandlimited signals using a synthesis system composed of digital fractional delay filters. The overall system can be viewed as a generalization of time-interleaved ADC systems. By generalizing these systems, it is possible to eliminate the errors that are introduced in practice due to time-skew errors keywords: {analogue-digital conversion;bandlimited signals;digital filters;filtering theory;signal reconstruction;digital fractional filters;nonuniform sampled bandlimited signals;signal reconstruction;synthesis system;time-interleaved ADC systems;time-skew errors;Analog-digital conversion;Delay;Digital filters;Electronic mail;Filter bank;Frequency;Interleaved codes;Nonuniform sampling;Sampling methods;Signal synthesis}, URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=921140&isnumber=19923

Y. C. Eldar and A. V. Oppenheim, "Filter bank interpolation and reconstruction from generalized and recurrent nonuniform samples," 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.00CH37100), Istanbul, 2000, pp. 324-327 vol.1. doi: 10.1109/ICASSP.2000.861961 Abstract: This paper introduces a filter bank interpretation of various sampling strategies, which leads to efficient interpolation and reconstruction methods. An identity, referred to as the interpolation identity, is used to obtain particularly efficient discrete-time systems for interpolation to uniform Nyquist samples, either for further processing in that form or for conversion to continuous-time. The interpolation identity also leads to a new class of sampling theorems including an extension of Papoulis' (1997) generalized sampling expansion keywords: {channel bank filters;discrete time filters;filtering theory;interpolation;signal reconstruction;signal sampling;discrete-time systems;filter bank interpolation;generalized sampling expansion;interpolation identity;recurrent nonuniform samples;sampling theorems;signal reconstruction;uniform Nyquist samples;Data acquisition;Digital signal processing;Filter bank;Fourier transforms;Interpolation;Iterative algorithms;Laboratories;Sampling methods;Signal processing;Signal sampling}, URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=861961&isnumber=18684