Signal Sampling and Testing Under Noise

Abstract

We consider, the joint nonparametric signal sampling and detection problem when noisy samples of a signal are observed. Two distinct detection schemes are examined. In the first one, the complete data set is given in advance and we wish to test the null hypothesis that a signal takes a certain parametric form. This situation we call the off-line testing problem. In the second case, the data are collected in the sequential fashion and one would like to detect a possible departure from a reference signal as quickly as possible. In such a scheme, called the online detection, we make a decision at the every new observed data point and stop the procedure when a detector finds that the null hypothesis is false. In both cases, we examine the nonparametric situation as for a well-defined null hypothesis signal model we admit broad alternative classes that cannot be parameterized. For such a setup, we introduce signal detection methods relying on nonparametric kernel-type sampling reconstruction algorithms properly adjusted for noisy data. For the off-line testing problem we examine the L ₂ - distance detection statistics measuring the discrepancy between a parametric and a nonparametric estimate of the target signals. The asymptotic behavior of the test is given yielding a consistent detection method. The mathematical theory is based on the limit laws of quadratic forms of stationary random processes. In the on-line detection case our detector is represented as a normalized partial sum of continuous time stochastic process, for which we obtain a functional central limit theorem. The established limit theorems allow us to design a monitoring online algorithm with the desirable level of the probability of false alarm and able to detect a change with probability approaching one. The presented results allow for dependent noise processes such as a linear process.