Binary Detection in Gaussian Noise

[weetabixharry]

I have a vector signal, \underline{x}(t), which is afflicted with Gaussian noise \underline{n}(t). I take a finite number, L, of discrete observations and (based on those observations) want to determine whether:

(1) Only Gaussian noise is present, \left[\text{i.e. } \underline{x}(t) = \underline{n}(t)\right]
(2) Gaussian noise plus a "non-noise" term, \underline{a}m(t), are both present. \left[\text{i.e. } \underline{x}(t) = \underline{a}m(t) + \underline{n}(t)\right]

The scalar, m(t), is zero-mean and has unknown power (variance). The elements of \underline{x}(t) are independent of each other and also independent of m(t).

Given my observations, how can I estimate the probability that the signal is present?

Many thanks for any help!

 

[chiro]

Hey weetabixharry.

I'm not an expert, but I do recall the subject of Kalman filters:

http://en.wikipedia.org/wiki/Kalman_filter

Basically you should check out these kind of things where you prefix a structure for your information that assumes some noise model (like White Gaussian) and then uses a filter to not only detect noise, but also the actual non-noisy information.

There are also non-linear variants of the filter.