Using prior knowledge to recover signal

[pvm]

I have a signal corrupted with normally distributed uncorrelated white noise. The noise has zero mean and known variance sigma1. I'd like to recover the signal as far as possible. However, the only thing I know is that the signal itself is normally distributed also, with mean zero and known variance sigma2, and some significant autocorrelation.

Can I use this knowledge to help me recover/improve the signal?

 

[chiro]

Hey pvm.

You are going to have to specify more constraints.

Usually what electrical engineers do is they structure the signal to take into account the noise profile and often use redundancy (that means using the same information multiple times in many ways) to reduce the error and therefore increase the confidence that the signal is what it is.

Error correction coding formalizes a lot of this and you can read a book (something like Hamming's book should be a good introduction).

You also might want to look at the idea of a Kalman Filter for this sort of thing.

 

[pvm]

Many thanks for responding. Actually I am already using a Kalman Filter - I was trying to capture the essence of the problem without the details, perhaps I simplified it too far :)

In a bit more detail, let's say that I have an acceleration signal with additive Gaussian noise. I double integrate this in the KF keeping the velocity and position as state. Actually there are other state parameters related to calibration, but let's put these aside. So with no observations, just the predictive model, the velocity for example will take a random walk with linearly increasing variance. The position will have quadratically increasing variance. However, I have some knowledge about the position: it is mean zero with known variance (as the device being sensed is not actually moving much). At present I use a pseudo-observation on the position in the KF which works fairly well to constrain its mean and variance, and give good values for the calibration parameters. But I have to find the pseudo-observation covariance empirically... (i.e. trial and error!). I'd really like some theoretical solution. There's a fair bit of material around on including constraints in KF design, but I have something a bit different I think: knowledge of the actual pdf of the resulting state variable... I just don't know how to apply it.

Any thoughts?

 

[chiro]

The only way to really answer this question is to know the design of your signal (which is what I mean by redundancy since it involves that).

When it comes to statistics the way to reduce uncertainty is through redundancy and I'll explain this with a simpler example.

Let's say you have a distribution X and a sample S which has data points. Often what is required is that you have to find something that is constant (usually a parameter - call it p) and you need to combine the information in S to get an estimate for p - which is related to X.

So all of S has something in common with X which has something in common with p which you are trying to figure out.

In a signal you have S (as the signal) which has something in common with the information transmitted (call it X) which has something in common with the actual information (call it p).

You take your information you are sending in the channel (p), design X so that it has the right redundancy to get p within some error tolerance and it is S that is actually being measured. The same process for say getting a mean is used to setup your signal except instead of measuring a single parameter you are measuring data in a particular format - and probably a vector of values with specific constraints.

The idea is the same though - and this is the basis for electrical/computer/telecommunications engineering regarding signals processing (even if it isn't specified in the same way as I am specifying above).

I don't have experience with Electrical Engineering textbooks but I do know quite a bit of statistics and if you can specify the information relevant to what is mentioned above (information structure - p, distribution X and actual transmitted information S) then I can make more sense of it.

To be a bit more specific what is required is that you often have a channel model with a noise component and one has to construct the distribution X as a function of p so that S can be taken and test statistics are constructed to estimate p. That is often Normally distributed with a co-variance matrix along with physical constraints which are used to find the redundancy structure which is actually encoded and sent.

You construct the distribution to minimize the noise component and this is a minimization problem with respect to the test statistic being used which is a function of the information you collect.

I'd need to know extra information in order to give more advice.