Partially Known Noise Amplitude Distribution

Module by: Don Johnson

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 pnn=1−ε p o nn+ε p d nn p n n 1 ε p o n n ε p d n n where p o n· p o n · is the nominal noise density, taken to be Gaussian, and p d n· 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 εε, 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=0L−1 f l rlsl−s2l2σ2 ≷ ℳ 0 ℳ 1 γ l L 1 0 f l r l s l s l 2 2 σ 2 ≷ ℳ 0 ℳ 1 γ where f l · 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 Figure 1.

The clipping function threshold z l ′ z l ′ is related to the assumed variance σ2 σ 2 of the nominal Gaussian density, the deviation parameter εε, and the signal value at the l th l th sample by the positive-valued solution of Q z l ′ +s2l2σ2slσ−Q z l ′ −s2l2σ2slσ=ε1−ε 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 εε 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 ′ 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. 1 The threshold can be established for the signal values used by the detector through the Central Limit Theorem as described in our previous discussion.

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This work is licensed by Don Johnson under a Creative Commons Attribution License (CC-BY 1.0), and is an Open Educational Resource.

Last edited by Charlet Reedstrom on Sep 15, 2003 2:55 pm GMT-5.