I have a vector signal, [itex]\underline{x}(t)[/itex], which is afflicted with Gaussian noise [itex]\underline{n}(t)[/itex]. I take a finite number, [itex]L[/itex], of discrete observations and (based on those observations) want to determine whether:(adsbygoogle = window.adsbygoogle || []).push({});

(1) Only Gaussian noise is present, [itex]\left[\text{i.e. } \underline{x}(t) = \underline{n}(t)\right][/itex]

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

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

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

Many thanks for any help!

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# Binary Detection in Gaussian Noise

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