2. Generalized chi-square test () theory

This test is based on the evaluation of the quadratic error between sample mean and ensemble averaged values of Q non-linear functions of the observations:

Let be the discrete version of the measured noise s(t). From a set of non-linear L_k-variate functions F_k and from appropriate choice of L_k time-lags , a set of Q sample mean generalized moments, w_k, are computed on N successive samples. The general form of such a sample mean value can be written as:

 
where .

Let denote the Q-dimensional vector formed by the juxtaposition of the sample mean w_k:

 
where symbolizes the transpose operator.

When the noise s(t) is free of RFI (termed the hypothesis throughout this paper), this vector converges to a multivariate Gaussian variable with ensemble average mean vector and ensemble average covariance matrix (Moulines et al. 1993). and depend on the statistical properties of the noise under the hypothesis. In the present case, under the hypothesis, the noise is assumed to be Gaussian. Thus, only the second order statistical properties are involved.

The test function consists in computing the quadratic error between the measured and the expected . To normalize and take the statistical links between the w_k into account, this quadratic error is weighted by the inverse of the covariance matrix . Namely,

 

Under the hypothesis, the test function is distributed asymptotically as a central chi-square variable whose degree of freedom is related to vector size Q. When no parameter of the s(t) statistics needs to be estimated, the degree of freedom is exactly Q (Moulines et al. 1993). Knowing the test distribution under the hypothesis, it is easy to establish a detection level as a function of the desired false alarm probability.