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# Unknown Noise Parameters

Module by: Don Johnson. E-mail the author

When aspects of the noise, such as the variance or power spectrum, are in doubt, the detection problem becomes more difficult to solve. Although a decision rule can be derived for such problems using the techniques we have been discussing, establishing a rational threshold value is impossible in many cases. The reason for this inability is simple: all models depend on the noise, thereby disallowing a computation of a threshold based on a performance probability. The solution is innovative: derive decision rules and accompanying thresholds that do not depend on false-alarm probabilities!

Consider the case in which the variance of the noise is not known and the noise covariance matrix is written as σ2 K σ 2 K where the trace of K K is normalized to LL. The conditional density of the observations under a signal-related model is p r | i σ 2 r=1σ2L2det(2π K )e(12σ2(r s i )T K -1(r s i )) p r i σ 2 r 1 σ 2 L 2 2 K 1 2 σ 2 r s i K r s i Using the generalized-likelihood-ratio approach, the maximum value of this density with respect to σ2 σ 2 occurs when σ^ML2=(r s i )T K -1(r s i )L σ ML 2 r s i K r s i L This seemingly complicated answer is easily interpreted. The presence of K -1 K in the dot product can be considered a whitening filter. Under the i th i th model, the expected value of the observation vector is the signal. This computation amounts to subtracting the expected value from the observations, whitening the result, then averaging the squared values - the usual form for the estimate of a variance. Using this estimate for each model, the logarithm of the generalized likelihood ratio becomes L2ln(r s 0 )T K -1(r s 0 )(r s 1 )T K -1(r s 1 ) 0 1 lnη L 2 r s 0 K r s 0 r s 1 K r s 1 0 1 η Computation of the threshold remains. Both models depend on the unknown variance. However, a false-alarm probability, for example, can be computed if the probability density of the sufficient statistic does not depend on the variance of the noise. In this case, we would have what is known as a constant false-alarm rate or CFAR detector (Carlyle & Thomas; Helstrom: p.317ff). If a detector has this property, the value of the statistic will not change if the observations are scaled about their presumed mean. Unfortunately, the statistic just derived does not have this property. Let there be no signal under 0 0 . The scaling property can be checked in this zero-mean case by replacing rr by cr c r . With this substitution, the statistic becomes c2rT K -1r(cr s 1 )T K -1(cr s 1 ) c 2 r K r c r s 1 K c r s 1 . The constant cc cannot be eliminated and the detector does not have the CFAR property. If, however, the amplitude of the signal is also assumed to be in doubt, a CFAR detector emerges. Express the signal component of model ii as A s i A s i , where AA is an unknown constant. The maximum likelihood estimate of this amplitude under model ii is A^ML=rT K -1 s i s i T K -1 s i A ML r K s i s i K s i Using this estimate in the likelihood ratio, we find the decision rule for the CFAR detector. 1

L2lnrT K -1rrT K -1 s 0 2 s 0 T K -1 s 0 rT K -1rrT K -1 s 1 2 s 1 T K -1 s 1 0 1 lnη L 2 r K r r K s 0 2 s 0 K s 0 r K r r K s 1 2 s 1 K s 1 0 1 η
(1)
Now we find that when rr is replaced by cr c r , the statistic is unchanged. Thus, the probability distribution of this statistic does not depend on the unknown variance σ2 σ 2 . In most array processing applications, no signal is assumed present in model 0 0 ; in this case, 0 0 does not depend on the unknown amplitude AA and a threshold can be found to ensure a specified false-alarm rate for any value of the unknown variance. For this specific problem, the likelihood ratio can be manipulated to yield the CFAR decision rule rT K -1 s 1 2(rT K -1r)( s 1 T K -1 s 1 ) 0 1 γ r K s 1 2 r K r s 1 K s 1 0 1 γ

## Example 1

Let's extend the previous example to the CFAR statistic just discussed to the white noise case. The sufficient statistic is ϒr=rlsl2r2ls2l ϒ r r l s l 2 r l 2 s l 2 We first need to find the false-alarm probability as a function of the threshold γγ. Using the techniques described in Wishner, the probability density of ϒr ϒ r under 0 0 is given by a Beta density (see Probability Distributions), the parameters of which do not depend on either the noise variance (expectedly) or the signal values (unexpectedly). p ϒ | 0 ϒ=βϒ12L12 p ϒ 0 ϒ β ϒ 1 2 L 1 2 We express the false-alarm probability derived from this density as an incomplete Beta function (Abramowitz & Stegun), resulting in the curves shown in Figure 1.

The statistic's density under model 1 1 is related to the non-central FF distribution, expressible by the fairly simple, quickly converging, infinite sum of Beta densities p ϒ | 1 ϒ= k =0ed2d2kk!βϒk+12L12 p ϒ 1 ϒ k 0 d 2 d 2 k k β ϒ k 1 2 L 1 2 where d2 d 2 equals a signal-to-noise ratio: d2=ls2l2σ2 d 2 l s l 2 2 σ 2 . The results of using this CFAR detector are shown in Figure 2.

## Footnotes

1. K K is the normalized noise covariance matrix.

## References

1. J. W. Carlyle & J. B. Thomas. (1964, April). On nonparametric signal detectors. IEEE Trans. Info. Th., IT-10, 146-152.
2. C. W. Helstrom. (1968). Statistical Theory of Signal Detection. (second edition). Oxford: Pergamon Press.
3. R. P. Wishner. (1962, September). Distribution of the normalized periodogram detector. IRE Trans. Info. Th., IT-8, 342-349.
4. M. Abramowitz & I. A. Stegun (Eds.). (1968). Handbook of Mathematical Functions. U.S. Government Printing Office.

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