4.24 Non-parametric detection

In situations when no nominal density can be reasonably assigned or when the possible extent of deviations from the nominalcannot be assessed, non-parametric detection theory can rise to the occasion ( Gisbon and Melsa , Kassam and Thomas ). In this framework first explored in Non-Parametric Model Evaluation , little is assumed about the form of the noise density. Assume thatmodel ${}_0$ corresponds to the noise-only situation and ${}_1$ to the presence of a signal. Moreover, assume that the noise density has zero median: any noise value is equally likelyto be positive or negative. This assumption does not necessarily demand that the density be symmetric about the origin, but suchdensities do have zero median. Given these assumptions, the formalism of non-parametric model evaluation yields the signtest as the best decision rule. As described in the simpler model evaluation context, ${}_1$ had constant, positive mean for each observation. Signal values are usually unequal and change sign; we mustextend the sign test to this more realistic situation. Noting that the statistic of the sign test did not depend on the valueof the mean but on its sign, the sign of each observation should be "matched" with the sign of each signal value. A kind ofmatched filter results, where $\mathrm{sign}(r(l))$ is match-filtered with $\mathrm{sign}(s(l))$ .

To find the threshold, the ubiquitous Central Limit Theorem can be invoked. Under ${}_0()$ , the expected value of summation is $\frac{L}{2}$ and its variance $\frac{L}{4}$ . The false alarm probability for a given value of $$ is therefore approximately given by $$P_F=Q(\frac{-\frac{L}{2}}{\sqrt{\frac{L}{4}}})$$ and the threshold easily found. We find the probability of detection with similar techniques, assuming a Gaussian densitydescribes the distribution of the sum when a signal is present. Letting $P_l$ denote the probability the observation and the signal value agree in sign at the $l^{\mathrm{th}}()$ sample, the sum's expected value is $\sum P_l()$ and its variance $\sum P_l(1-P_l)$ . Using the Central Limit Theorem approximation, the probabilityof detection is given by $$P_D=Q(\frac{-\sum P_l}{\sqrt{\sum P_l(1-P_l)}})$$ For a symmetric as well as zero-median density for the noise amplitude, this probability is given by $P_l=\int_{-\left|s(l)\right|} \,d n$∞ p n n . If Gaussian noise were present, this probability would be $1-Q(\frac{s(l)}{})$ and for Laplacian noise $1-\frac{1}{2}e^{-\left(\frac{\left|s(l)\right|}{\sqrt{\frac{^2}{2}}}\right)}$ .

The non-parametric detector expressed by has many attractive properties for array processing applications. First, the detector does not require knowledge ofthe amplitude of the signal. In addition, note that the false-alarm probability does not depend on the variance of the noise; the sign detector is thereforeCFAR. Another property of the sign detector is its robustness: we have implicitly assumed that the nosie values have the worst-case probability density - the Laplacian. A more practical property is the one bit of precisionrequired by the quantities used in the computation of the sufficient statistic: each observation is passed through aninfinite clipper (a one-bit quantizer) and matched (ANDed) with a one bit representation of the signal. A less desirableproperty is the dependence of the sign detector's performance on the signal waveform. A signal having a few dominant peak valuesmay be less frequently detected than an equal energy one having a more constant envelope. As the following example demonstrates,the loss in performance compared to a detector specially tailored to the signal and noise properties can be small.