Partially Known Noise Amplitude Distribution 1.5 2003/06/16 2003/09/15 14:47:34.004 GMT-5 Don Johnson dhj@rice.edu Don Johnson dhj@rice.edu Eileen Krause erkrause@rice.edu Kyle Clarkson kclarks@rice.edu Elizabeth Gregory lizzardg@rice.edu Mariyah Poonawala mariyah@rice.edu Matthew Jeanes mjeanes@rice.edu Jeffrey Silverman jsilv@rice.edu The previous sections assumed that the probability distribution of the noise was known precisely, and furthermore, that this distribution was Gaussian. Deviations from this assumption occur frequently in applications (Machell and Penrod, Middleton, Milne and Ganton). We expect Gaussian noise in situations where noise sources are many and of roughly equal strength: the Central Limit Theorem suggests that if these noise sources are independent (or even mildly dependent), their superposition will be Gaussian. As shown in this discussion, the Central Limit Theorem converges very slowly and deviations from this model, particularly in the tails of the distribution, are a fact of life. Furthermore, unexpected, deviant noise sources also occur and these distort the noise amplitude distribution. Examples of the phenomenon are lightning (causing momentary, large, skewed changes in the nominal amplitude distribution in electromagnetic sensors) and ice fractures in polar seas (evoking similar distributional changes in accoustic noise). These changes are momentary and their effects on the amplitude distribution are, by and large, unpredictable. For these situations, we invoke ideas from robust model evaluation to design robust detectors, ones insensitive to deviations from the Gaussian model (El-Sawy and Vandelinde, Kassam and Poor, Poor pp.175-187). We assume that the noise component in the observations consists of statistically independent elements, each having a probability amplitude density of the form p n n 1 ε p o n n ε p d n n where p o n · is the nominal noise density, taken to be Gaussian, and p d n · is the deviation of the actual density from the nominal, also a density. This ε-contamination model (Huber; 1981) is parameterized by ε, the uncertainty variable, a positive number less than one that defines how large the deviations from the nominal can be. As shown in Robust Hypothesis Testing, the decision rule for the robust detector is l L 1 0 f l r l s l s l 2 2 σ 2 ≷ ℳ 0 ℳ 1 γ where f l · is a memoryless nonlinearity having the form of a clipper (see Robust Hypothesis Testing). The block diagram of this receiver is shown diagrammatically in .

The robust detector consists of a linear scaling and shifting operation followed by a unit-slope clipper, whose clipping thresholds depends on the value of the signal. The key element is the clipper, which serves to censor large excursions of the observed from the value of the signal.

The clipping function threshold z l ′ is related to the assumed variance σ 2 of the nominal Gaussian density, the deviation parameter ε, and the signal value at the l th sample by the positive-valued solution of Q z l ′ s l 2 2 σ 2 s l σ Q z l ′ s l 2 2 σ 2 s l σ ε 1 ε An example of solving for ε is shown in this figure. The characteristics of the clipper vary with each signal value: the clipper is exquisitely tuned to the proper signal value, ignoring values that deviate from the signal. Furthermore, note that this detector relies on large signal values relative to the noise. If the signal values are small, the above equation for z l ′ has no solution. Essentially, the robust detector ignores those values since the signal is too weak to prevent the noise density deviations from entirely confusing the models. In the next section, we present a structure applicable in small signal-to-noise ratio cases. The threshold can be established for the signal values used by the detector through the Central Limit Theorem as described in our previous discussion. A.H. El-Sawy and V.D. Vandelinde Robust detection of known signals. IEEE Trans. Info. Th. 1977 IT-23 722-727 S.A. Kassam and H.V. Poor Robust techniques for signal processing: A survey Proc. IEEE 1985 73 433-481 F.W. Machell and C.X. Penrod Probability density functions of ocean acoustic noise processes Statistical Signal Processing Marcel Dekker 1984 E.J. Wegman and J.G. Smith 211-221 New York D. Middleton Statistical-physical models of electromagnetic interference IEEE Trans. Electromag. Compat. 1977 EMC-17 106-127 A.R. Milne and J.H. Ganton Ambient noise under Arctic sea ice J. Acoust. Soc. Am. 1964 36 855-863 H.V. Poor An Introduction to Signal Detection and Estimation Springer-Verlag 1988 New York P.J. Huber Robust Statistics John Wiley and Sons 1981 New York