Notes#15, ECE594I, Fall 2009, E.R. Brown 218 Statistical Detection Theory We have frequently used the signal-to-noise ratio and its derivatives (e.g., NEP) to characterize the performance of RF and THz systems. Although a very important figure-of-merit in any system, it does not tell us what the system operator ultimately needs to know, which is “what is the likelihood that the system will successfully detect a given target in a given setting, and what is the likelihood that it will miss the given target, or perhaps show the telltale signs of detection when the target is not there at all ?” These questions support the notion that detection in the presence of noise and other (false) targets is always a statistical process. So to properly predict system performance, first a statistical analysis must be made of all the possible detection outcomes, and the various conditions of the system that could have created those outcomes.1,2,3 It is inherently a reverse statistical process. That is, we know the outcome, but do not know with certainty the conditions that created it. As such, it is guided by a specific sub-field of probability theory that started with Bayes. It requires an accurate model for the system, including the signal-to-noise ratio as the key metric, and a thorough parameterization of the known target and other possible targets or environmental conditions that can create a detect for the outcome. This leads to modeling detection as a binary decision in the presence of a “target” return signal buried in additive white Gaussian noise (AWGN) and occasionally accompanied by signal from false targets or “clutter” in the local environment. Receivers are usually linear up to the detection device, and so linear super-position is assumed to hold. The output signal from the receiver can then be written: ()()()sig noiseXtXtX t=+ (1) where Xsig is the expected signal (deterministic) and Xnoise is due to electronic and thermal noise (random process) as we have discussed in detail earlier. The noise is assumed to simply add to the signal as opposed to multiply against it, non-linearly scale it, etc. Further the above formula exemplifies that now the output is a random process. 1 R. N. Mcdonough, A. D. Whalen, Detection of Signals in Noise, Academic Press, New York, 1995 2 J. V. DiFranco and W.L. Rubin, Radar Detection, SciTech Publishing, Raleigh, NC, 2004 3 W.B. Davenport and W.L. Root, An Introduction to the Theory of Random Signals and Noise, Wiley-IEEE Press, 1987.Notes#15, ECE594I, Fall 2009, E.R. Brown 219 And so the SNR as defined is not the final metric, although as will be shown a good indicator and a pivotal measurement to assess the performance of a receiver. Receiver Analysis: the Bayesian criteria Since the output signal is a random process, we must apply statistical detection theory, which is relatively simple for RF sensors if they are fully binary. That is, there are assumed to be two possible input conditions: (a) target absent (“0” condition), or (b) target present (“1” condition). And there are two possible outputs, now represented by possible ranges of the electrical variable X. If a target is absent, we have the range X0, and if the target is present we have the range X1. This mapping between target presence and output range is shown graphically in Fig. 1. And it makes sense for active RF sensors of all types, active (e.g., radar) and passive (e.g., radiometers). The primary function of the receiver is to determine which of the two input conditions exists based only on measurement of the output signal Xout over its entire range. The complication of the receiver function, and the need for a statistical detection theory, is that the ranges represented by X0 and X1 generally overlap, creating an ambiguity in the mapping. In other words, there are four possible outcomes of the fully binary target situation, as displayed graphically in Fig. 2: (a) target present and X in unambiguous region of X1, which is called a true positive; (b) target absent and X in unambiguous region of X0, which is called a true negative; (c) target absent, X in ambiguous region of X0, and mistakenly interpreted as X1, which is called a false positive (or in radar colloquial, a “false alarm”), and (d) target present, X in ambiguous region of X1, and mistakenly interpreted as X0, which is called a false negative (or in radar Fig. 1.Notes#15, ECE594I, Fall 2009, E.R. Brown 220 colloquial, a “miss”). The goal of the statistical detection theory must be to optimize the accuracy of the receiver by designing it to maximize the occurrence of the first two “correct” outcomes, and minimize the occurrence of the second two “incorrect” ones. To proceed further, we assume that probability density functions are known to describe the X0 and X1 regions of Xout space, p0(X) and p1(X), respectively. In general, these regions are not bounded as conveniently shown in Fig. 1, but are smeared out to create a continuous overlap over all the Xout space. This forces us to seek further information to optimize the receiver accuracy, and led the mathematician Bayes to realize that information must be provided about the input condition and the output result. Ideally, one should know the a priori (i.e., beforehand) probabilities π0 and π1 for the target being absent or present. And one should also know the cost, or loss factor, of each decision made. Specifically, L11 is the loss function for a true positive, L00 is the loss function for a true negative, L10 is the loss function for a false negative, and L01 is the loss function for a false positive.4 Bayes then showed that an optimum receiver function could be made in terms of the “liklihood ratio” L of the a posteriori (afterwards) densities p0 and p1, 4 In statistical terminology, L01 corresponds to type-I error and L10 a type-II error Fig. 2.Notes#15, ECE594I, Fall 2009, E.R. Brown 221 0 1 10 11100100() ( )()() ( )pXLLLXpXLLππ−≡=− (2) This led to the optimization relations, called the Bayes criteria (a) If 110 11001 00()()()LLLXLLππ−>−, then condition 0 exists (3) (b) If 110 11001 00()()()LLLXLLππ−<−, then condition 1 exists (4) It is very important for the reader to realize that these criteria are really a recipe. In other words, if the receiver produces an output X at a given decision time, the Bayes criteria tell us how to make the most