Detection-estimation of more uncorrelated sources than sensors in noninteger sparse linear antenna arrays

We introduce a new approach for the detection-estimation problem for sparse linear antenna arrays comprising M identical sensors whose positions may be noninteger values (expressed in half-wavelength units). This approach considers the (noninteger) M_α-element co-array as the most appropriate virtual array to be used in connection with the augmented covariance matrix. Since the covariance matrix derived from such virtual arrays is usually very underspecified, we discuss a maximum-likelihood (ML) completion philosophy to fill in the missing elements of the partially specified Hermitian covariance matrix. Next, a transformation of the resulting unstructured ML matrix results in a sequence of properly structured positive-definite Hermitian matrices, each with their (M_α-μ) smallest eigenvalues being equal, appropriate for the candidate number of sources μ. For each candidate model (μ=1 to M_α-1), we then find the set of directions-of-arrival (DOAs) and powers that yield the minimum fitting error for the specified covariance lags in the neighbourhood of the MUSIC-initialised DOAs. Finally, these models describe a hypothesis with respect to the actual number of sources, and allow us to select the “best” hypothesis using traditional information criteria (AIC, MDL, MAP, etc.) that are based on the likelihood ratio