How to filter very low amplitude staircase signal from a background noise I have a staircase signal of a specific pattern that is drowned in a background noise. The ratio of signal to noise amplitude is nearly $10^{-4}$. I have attached the image of both with this post. I am an absolute noob in this field so please suggest me a direction on which path I should start to hunt for the solution. 




I have tried to do fft on both of the signal and looked for any distinct signature to filter the background noise. Also I have tried with matched filtering and basic convolution techniques. But none of these seem to work. Your advice will be much appreciated. Thanks.
Linear filtering won't work here. If anything works at all then it might be something like total variation denoising, which is a non-linear technique for removing noise while preserving edges. This article is a very good read, and it also shows a Matlab implementation of the algorithm.