Deterministic Cramér-Rao bound for a mixture of circular and strictly non-circular signals

The problem of estimating the signal parameters of a mixture of circular and strictly second-order (SO) non-circular (NC) signals impinging on an antenna array has recently attracted considerable attention. Several high-resolution algorithms have been proposed for this scenario that improve the estimation accuracy of the traditional schemes and simultaneously increase the number of resolvable signals. In this paper, we derive a closed-form expression of the deterministic Cramér-Rao bound (CRB), termed deterministic C-NC CRB, as a benchmark for this new class of algorithms. The obtained result allows to assess the maximum achievable performance gain in this scenario. The derivation is based on the Slepian-Bangs formula, which is still applicable due to the deterministic data assumption. Simulation results show that the C-NC CRB decreases when the number of strictly non-circular signals increases within a fixed number of sources. In this case, also the individual bounds of the circular signals decrease, which suggests that the presence of strictly non-circular sources reduces the estimation error of the circular signals.