# Binary Detection in Gaussian Noise

1. Feb 7, 2013

### 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!

2. Feb 7, 2013

### 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.