4.13 Unknown signal waveform

The most general unknown signal parameter problem occurs when the signal itself is unknown. This phrasing of the detectionproblem can be applied to two different sorts of situations. The signal's waveform may not be known precisely because ofpropagation effects (rapid multipath, for example) or because of source uncertainty. Another situation is the "Hello, is anyoneout there?" problem: you want to determine if any non-noise-like quantity is present in the observations. These problems imposesevere demands on the detector, which must function with little a priori knowledge of the signal's structure. Consequently, we cannot expect superlative performance. $${}_0:r(l)=n(l)$$ $${}_1:r(l)=s(l)+n(l),s(l)=?$$ The noise is assumed to be Gaussian with a covariance matrix $K$ . The conditional density under ${}_1$ is given by $$p(r, {}_1, s, r)=\frac{1}{\sqrt{\det (2\pi K)}}e^{-(\frac{1}{2}(r-s)^TK^{(-1)}(r-s))}$$ Using the generalized likelihood ratio test, the maximum value of this density with respect to the unknown "parameters" - thesignal values - occurs when $r=s$ . $$\max\{s , p(r, {}_1, s, r)\}=\frac{1}{\sqrt{\det (2\pi K)}}$$ The other model does not depend on the signal and the generalized likelihood ratio test for the unknown signalproblem, often termed the square-law detector , is