Cramér-Rao Bound Analysis of Underdetermined Wideband DOA Estimation under the Subband Model via Frequency Decomposition

A class of Cramér-Rao bounds (CRBs) for wideband direction-of-arrival (DOA) estimation under the subband (or frequency bin) model is studied for the underdetermined case, where the number of sources is no less than that of physical sensors. A unified framework is proposed to encompass the closed-form CRB expressions for DOAs in four cases where the sources are known a priori to 1) have flat spectra/cross spectra, 2) be spatially uncorrelated, 3) be spatially uncorrelated and have proportional spectra up to unknown factors, 4) be spatially uncorrelated and have flat spectra. The relationship between the wideband CRBs and the subband ones is investigated, and the order relationship among the derived CRBs are provided. The asymptotic behavior of the CRBs with respect to the number of snapshots and the signal-to-noise ratio (SNR) is discussed. Two asymptotic expressions for sufficiently large SNR are derived in both overdetermined and underdetermined cases. Existence of the derived CRBs is examined through rank conditions of the introduced matrices, which yields upper bounds on the resolution capacities. Different from the narrowband scenario, underdetermined wideband DOA estimation is feasible even if a sparse array is not used given different a priori knowledge about the source spectra. It is possible to resolve more wideband Gaussian sources than the number of DOFs offered by the difference co-array. Finally, further interpretations of the subband model are provided, revealing the underlying connections with the multi-frequency co-array augmentation concept and the non-coherent subarray system.