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Many problems in statistical signal processing and communications can be solved using basic detection theory. For example, determining whether an airplane is located at a specific range and direction with radar and whether a received bit is a 00 or a 11 are both solved with matched filter detectors. However, many elaborations of deciding which of two models best describes a given dataset abound. For instance, suppose we have more than two models for the data. Previous modules hint at how to expand beyond two models (see (Reference) and (Reference)). However, no extensions for Neyman-Pearson detectors for more than two models exists.

To learn more about the basics of detection theory and beyond, see the books by Van Trees, Kay and McDonough and Whalen, and several modules on Connexions (search for 'likelihood ratio', 'detection theory' and 'matched filter'). The Wikipedia article on Statistical Hypothesis Testing describes what is called detection theory here more abstractly from a statistician's viewpoint.

Beyond Simple Problems

More interesting (and challenging) are situations where the data models are imprecise to some degree. The simplest case is when some model parameter is not known. For example, suppose the exact time of the radar return is not known (i.e., the airplane's range is uncertain). Here, the unknown parameter is the signal's time-of-origin. We must somehow determine that parameter and determine if the signal is actually present.

As you might expect, the likelihood ratio remains the focus of attention, now in the guise of what is known as the generalized likelihood ratio test (GLRT) (see this Connexions module).1 This technique and others opens the door to what are known as simultaneous estimation and detection algorithms.

Some unknowns may not be parametric and prevent a precise description of a model by a probability function. What do we do when the amplitude distribution function of the additive noise is not well characterized? So-called robust detection represents one attempt to address these problems. See [3] for more.

Beyond variations of model uncertainties are new approaches to solving detection problems. For example, a subtlety in the basic formulation is that all the data are available. Can we do just as well by attempting to make a decision prematurely as the data arrive? Note that always taking a fixed fewer number of samples always leads to worse performance. Instead, we acquire only the amount of data needed to make a decision. This approach is known as sequential detection or the sequential probability ratio test (SPRT). See modules in Connexions and the classic book by Wald.