MATCHED FILTER THRESHOLD ADJUSTMENT FOR SKEWED NOISE USING GRAM-CHARLIER EXPANSION

A Matched Filter is commonly used as an optimal linear detector of a known signal in a noisy environment. The filter’s output is used in a Likelihood Ratio Test (LRT), which is usually set under the assumption that the filter’s output has a Gaussian distribution, both under the null hypothesis and under the alternative hypothesis. When the noise is Gaussian, this assumption is perfectly justified. When the noise is non-Gaussian, this assumption is usually a good approximation if the filter is “long enough”, thanks to the Central Limit Theorem. However, if the noise is skewed and the filter is relatively short, the distribution of the filter’s output (under both hypotheses) departs from Gaussianity (and is usually too complicated to derive in an explicit form). In this paper we show that if the skewness of the noise is known, the departure of the filter’s output distribution from Gaussianity can be well-approximated using a Gram-Charlier expansion (which depends on the filter’s coefficients), leading to a more accurate determination of the decision threshold, which we derive in closed form. We demonstrate the resulting improvement in the overall decision-error probability in simulation.