Analytical ESPRIT-based performance study: What can we gain from non-circular sources?

It is well known that parameter estimation algorithms designed to exploit the prior knowledge of the strict second-order (SO) non-circularity (NC) of incident signals can estimate the parameters of twice as many sources and achieve significant gains in reducing the estimation error. So far, the magnitude of the NC gain could only be quantified through simulations and finding a generic analytical expression for arbitrary scenarios is an intricate task. In this paper, we adopt a first-order performance analysis framework to analytically compute the asymptotic NC gain of NC Standard ESPRIT for the case of two uncorrelated strictly SO non-circular (rectilinear) sources captured by a uniform linear array (ULA). We assume a maximum phase separation of the sources, which yields the largest NC gain. For this scenario, we derive simplified asymptotic mean squared error (MSE) expressions of NC Standard ESPRIT and Standard ESPRIT, which are subsequently used to compute the NC gain. While the simplified MSE expression of Standard ESPRIT depends on the source separation, we show that if NC Standard ESPRIT is applied in this case, the two non-circular sources entirely decouple. Thus, the NC gain can theoretically approach infinity if the separation of the two sources tends to zero. Our derived expressions are verified by simulation results.