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Applying the techniques described in the previous section may be difficult to justify when the signal and/or noise modelsare uncertain. For example, we must "know" a signal down to the precise value of every sample. In other cases, we mayknow the signal's waveform, but not the waveform's amplitude as measured by a sensor. A ubiquitous example of this uncertainty is propagation loss: the range of afar-field signal can only be lower-bounded, which leads to the known waveform, unknown amplitude detection problem. Anotheruncertainty is the signal's time origin : Without this information, we do not know when to start the matchedfiltering operation! In other circumstances, the noise may have a white power spectrum but its variance is unknown. Muchworse situations (from the point of view of detection theory) can be imagined: the signal may not be known at all and onemay want to detect the presence of any disturbance in the observations other than that of well-specified noise. These problems are very realistic, butthe detection schemes as presented are inadequate to attack them. The detection results we have derived to date need tobe extended to incorporate the presence of unknowns just as we did in hypothesis testing ( Detection in the Presence of Unknowns ).
Assume that a signal's waveform is known exactly, but the amplitude is not. We need an algorithm to detect thepresence or absence of this signal observed in additive noise at an array's output. The models can be formallystated as $${}_{0}:r(l)=n(l),l\in \{0, , L-1\}$$ $${}_{1}:r(l)=As(l)+n(l),l\in \{0, , L-1\},A=?$$ As usual, $L$ observations are available and the noise is Gaussian. This problem isequivalent to an unknown parameter problem described in Detection in the Presence of Unknowns . We learned there that the first step is to ascertain the existence of a uniformly most powerful test.For each value of the unknown parameter $A$ , the logarithm of the likelihood ratio is written $$Ar^TK^{(-1)}s-A^{2}s^TK^{(-1)}s\underset{{}_{0}}{\overset{{}_{1}}{}}\ln $$ Assuming that $A> 0$ , a typical assumption in array processing problems, we write this comparison as $$r^TK^{(-1)}s\underset{{}_{0}}{\overset{{}_{1}}{}}\frac{1}{A}\ln +As^TK^{(-1)}s=$$ As the sufficient statistic does not depend on the unknown parameter and one of the models ( ${}_{0}$ ) does not depend on this parameter, a uniformly most powerful test exists: the threshold term, despite itsexplicit dependence on a variety of factors, can be determined by specifying a false-alarm probability. If thenoise is not white, the whitening filter or a spectral transformation may be used to simplify the computation ofthe sufficient statistic.
Assume that the waveform, but not the amplitude, of a signal is known. The Gaussian noise is white with a variance of $^{2}$ . The decision rule expressed in terms of a sufficient statistic becomes $$r^Ts\underset{{}_{0}}{\overset{{}_{1}}{}}$$ The false-alarm probability is given by $${P}_{F}=Q(\frac{}{\sqrt{E^{2}}})$$ where $E$ is the assumed signal energy which equals $(s)^{2}$ . The threshold $$ is thus found to be $$=\sqrt{E^{2}}Q({P}_{F})^{(-1)}$$ The probability of detection for the matched filter detector is given by $${P}_{D}=Q(\frac{-AE}{\sqrt{E^{2}}})=Q(Q({P}_{F})^{(-1)}-\sqrt{\frac{A^{2}E}{^{2}}})$$ where $A$ is the signal's actual amplitude relative to the assumed signal having energy $E$ . Thus, the observed signal, when it is present, has energe $A^{2}E$ . The probability of detection is shown in as a function of the observed signal-to-noise ratio. For any false-alarm probability, thesignal must be sufficiently energetic for its presence to be reliably determined.
All too many interesting problems exist where a uniformly most powerful decision rule cannot be found. Suppose in theproblem just described that the amplitude is known ( $A=1$ , for example), but the variance of the noise is not. Writing the covariance matrix as $^{2}\stackrel{}{K}$ , where we normalize the covariance matrix to have unit variance entries by requiring $\mathrm{tr}(\stackrel{}{K})=L$ , unknown values of $^{2}$ express the known correlation structure, unknownnoise power problem. From the results just given, the decision rule can be written so that sufficient statisticdoes not depend on the unknown variance. $$r^T\stackrel{}{K}^{(-1)}s\underset{{}_{0}}{\overset{{}_{1}}{}}^{2}\ln +s^T\stackrel{}{K}^{(-1)}s=$$ However, as both models depend on the unknown parameter, performance probabilities cannot becomputed and we cannot design a detection threshold.
Hypothesis testing ideas show the way out; estimate the
unknown parameter(s) under each model separately and thenuse these estimates in the likelihood ratio (
Non-Random
Parameters ). Using the maximum likelihood estimates
for the parameters results in the generalized likelihoodratio test for the detection problem (
Kelly,
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