# How to estimate the autocorrelation from nonuniformly spaced data

Assume a continues-time random process $X(t)$ sampled nonuniformely in time to acquire discrete signal $x[n]$. The sampling times are known but the autocorrelation is not. Is there an accurate approach to estimate the autocorrelation function? This problem is challenging since at some lags there is no sample to average.

Any suggestion even in a special case, e.g. a simple autocorrelation function with an exponential decay (continuous-time AR process of order one) would be appreciated.

Edit

I have already read some papers, so I would like either an analytical answer or a peace of algorithm/code to address this problem for instance for an AR(1) process assuming non-uniformly spaced samples(e.g. in uniformly distributed random times).

• I'd declare this an interpolation problem – but interpolation requires that you make some assumptions on the signal, i.e. something that you'd need to add to your problem description Sep 3, 2016 at 15:30
• @ Marcus Müller Also is related to spectral estimation, since autocorrelation is the inverse FT of the PSD. $X(t)$ is band-limited if it helps. If you know a solution under any further assumption (such as bounded derivatives etc.), I will also appreciate it.
– msm
Sep 3, 2016 at 19:51
• Well, assumptions about your signals is something that you should make, isn't it? But band-limitation is a good one; what about finding the minimum-number-of-sinusoids signal that explains all the samples you have? Sep 3, 2016 at 19:52
• You are right. The only assumption I have is $X(t)$ being band limited. I suppose from that one could truncate the Forurier representation somewhere and get the minimum number...
– msm
Sep 3, 2016 at 19:55
• Interesting! I had similar question in my mind, how to estimate active frequency bins (not through FFT) using non-uniform (under) sampled data? Sep 16, 2016 at 7:13

As suggested by Marcus Müller, interpolation in the time domain could be a solution. I never had to perform such a task, and the outcomes may depend in the nonuniformity of your sampling. I propose a use of the Wiener-Khinchin theorem:

The Fourier transform of the autocorrelation function is the power spectrum, or equivalently, the autocorrelation is the inverse Fourier transform of the power spectrum

You may use for instance:

and get a regularly sampled power spectrum, which I hope you could inverse Fourier back to an estimate of the autocorrelation. As I have no practical experience, if you intend to try this, I'd really appreciate you could share the outcomes, good or bad, along with the specificities of your signal:

• what kind of nonuniformity (additive random, jitter, lacunary)?
• type of signal and noise?