Removing Gaussian Noise from a Signal to Get Minimum Value

I have a signal that has a minimum value that I'm trying to read.

The issue I'm having is that the signal is spread out by gaussian noise.

I have the signal at a lot of timesteps (and expect the function to be roughly exponentially distributed and being truncated at the minimum value), but can't work out how to remove that gaussian noise.

Any help would be appreciated!

(Also this may need transplanting to another stackexchange, but I feel like this was something which should have been solved multiple times previously by this community)

If you have no prior data on the signal of interest there is nothing to do actually.
The more prior you have the better you can do.

For instance, if the only information you have is the Bandwidth of your signal, the best you can do is apply an Band Pass / Low Pass Filter.

If you know a sparse representation of your signal it will be great, as you can remove all the other coefficients (Generalization of the LPF / BPF).

In case you know something about how smooth it is, you can apply Total Variation like smoothing.

Remark
When you write Gaussian Noise I assume you mean Additive White Gaussian Noise (AWGN).

• I have found this post useful and looked up Total Variation. However, I am interested in your generalisation of LPF/BPF. I don't understand what you mean by a sparse representation of the signal and removing all the other coefficients. Please could you expand?
– Hugh
Commented Oct 16, 2020 at 17:18
• Could you give exa
– Hugh
Commented Oct 17, 2020 at 7:50
• Please could you give examples? Are there standard representations? Are you thinking of model fitting? Is this written up somewhere?
– Hugh
Commented Oct 17, 2020 at 7:54
• A trivial example would be an Harmonic Signal which has sparse representation using Fourier Series as a basis. The step function can be represented in a sparse manner using Hadamard basis.
– Royi
Commented Oct 17, 2020 at 8:17