# Inverse Problem / Deconvolution with Pink Noise

Hi I dived somewhat into deconvolution of systems which can be described as:

$s(t) = o(t) * h(t) + n(t)$

where $s$ is my measured 1D time resolved signal, $o$ is the original signal $h$ is the kernel or PSF pf my system, which is convolved with the original signal (with $*$ beeing the convolution operator), $n$ as sample-function for $1/f^\alpha$ noise ($\alpha > 0$).

For all I know there are several methods to deconvolve for an estimate of o. When choosing the correct method, the consideration of the noise is important. E.g. I should use the wiener deconvolution for systems with white noise, lucy-richardson for possion noise.

Which method should I use for pink noise (given the fact that there are several other (read: deeper mathematical) aspect which should be taken into account)?

On a side note: any recomendations for Papers/Books on this topic (Deconvolution of systems as described)? Thanks in advance!

Edit: For the system I know an estimate of the kernel function $h$ and I can make blank meassurements to estimate the nature of the nois sample function (it ssemst to be white noise for small freqencies and pink noise for higher freqencies).

• Just for clarification: Do you mean "O is the original signal, H is some kernel which is convolved with the original signal"? So, your question essentially is: Given $H(v)$, $S(v)$ and the statistics of $n$, how can I estimate $O(v)$ in some optimal way, right? Jan 11, 2017 at 19:17
• Yes, you got me right. I edited the question, my mistake. On a further note n is supposed to be 1/f noise. Jan 12, 2017 at 9:28
• @Royi Sure! Ended up with Wiener Deconvolution btw. The signal was indeed stationary, while the noise is not independent from the signal. But it worked well enough. On a further note: only now after far more experience in the field I can appreciate your answer fully ;) Aug 25, 2021 at 9:46

Some Remarks:

• 2nd Order Knowledge
It seems you only have a knowledge about the Auto Correlation of your data (2nd Order) and not on its distribution. Hence methods you can apply are ones which minimizes only 2nd order functions of the noise (Such as MMSE).
• Wiener Filter Minimizes the MMSE
As can be seen in the derivation of the Wiener Deconvolution Filter the only assumption made is independence from the signal. If that holds in your data this is a good solution. Pay attention that Wiener Filter model is that the input data is also stochastic and all data is wide sense stationary.
• Richardson Lucy Deconvolution
Pay attention this is a Maximum Likelihood method. Namely it assumes the data is a parameter, not a random data. It also assumes the distribution of the noise and not only its second moment,
• Decide the Cost Function
First thing you should decide what would you like to minimize according to your model. Does the MMSE makes sense? Maybe a Least Squares is good enough? May you come with other objective. This is the first step.

I think you are looking for Wiener Deconvolution: It does not require the noise to be white, but you can assume any correlation function/PSD of the noise. It gives you the optimal estimate of $O(v)$ based on the MMSE criterion.

However, the filter requires knowledge about the spectral density of your original signal. HEnce, you need some a-priori information about your signal, e.g. if it looks like white noise or if it has less components in the higher frequencies.

• Hi, thanks for the answere. I tried to flesh out the question besed on you answere Jan 12, 2017 at 12:14