High-resolution parameter estimation algorithms designed to exploit the prior knowledge about incident signals from strictly second-order (SO) non-circular (NC) sources allow for a lower estimation error and can resolve twice as many sources. In this paper, we derive the R-D NC Standard ESPRIT and the R-D NC Unitary ESPRIT algorithms that provide a significantly better performance compared to their original versions for arbitrary source signals. They are applicable to shift-invariant R-D antenna arrays and do not require a centrosymmetric array structure. Moreover, we present a first-order asymptotic performance analysis of the proposed algorithms, which is based on the error in the signal subspace estimate arising from the noise perturbation. The derived expressions for the resulting parameter estimation error are explicit in the noise realizations and asymptotic in the effective signal-to-noise ratio (SNR), i.e., the results become exact for either high SNRs or a large sample size. We also provide mean squared error (MSE) expressions, where only the assumptions of a zero mean and finite SO moments of the noise are required, but no assumptions about its statistics are necessary. As a main result, we analytically prove that the asymptotic performance of both R-D NC ESPRIT-type algorithms is identical in the high effective SNR regime. Finally, a case study shows that no improvement from strictly non-circular sources can be achieved in the special case of a single source.

Abstract-Recently, several high-resolution parameter estimation algorithms have been developed to exploit the structure of strictly second-order (SO) non-circular (NC) signals. They achieve a higher estimation accuracy and can resolve up to twice as many signal sources compared to the traditional methods for arbitrary signals. In this paper, as a benchmark for these NC methods, we derive the closed-form deterministic R-D NC Cramér-Rao bound (NC CRB) for the multi-dimensional parameter estimation of strictly non-circular (rectilinear) signal sources. Assuming a separable centro-symmetric R-D array, we show that in some special cases, the deterministic R-D NC CRB reduces to the existing deterministic R-D CRB for arbitrary signals. This suggests that no gain from strictly non-circular sources (NC gain) can be achieved in these cases. For more general scenarios, finding an analytical expression of the NC gain for an arbitrary number of sources is very challenging. Thus, in this paper, we simplify the derived NC CRB and the existing CRB for the special case of two closely-spaced strictly non-circular sources captured by a uniform linear array (ULA). Subsequently, we use these simplified CRB expressions to analytically compute the maximum achievable asymptotic NC gain for the considered two source case. The resulting expression only depends on the various physical parameters and we find the conditions that provide the largest NC gain for two sources. Our analysis is supported by extensive simulation results.

Abstract-Spatial smoothing is a widely used preprocessing scheme to improve the performance of high-resolution parameter estimation algorithms in case of coherent signals or if only a small number of snapshots is available. In this paper, we present a first-order performance analysis of the spatially smoothed versions of R-D Standard ESPRIT and R-D Unitary ESPRIT for sources with arbitrary signal constellations as well as R-D NC Standard ESPRIT and R-D NC Unitary ESPRIT for strictly second-order (SO) non-circular (NC) sources. The derived expressions are asymptotic in the effective signal-to-noise ratio (SNR), i.e., the approximations become exact for either high SNRs or a large sample size. Moreover, no assumptions on the noise statistics are required apart from a zero-mean and finite SO moments. We show that both R-D NC ESPRIT-type algorithms with spatial smoothing perform asymptotically identical in the high effective SNR regime. Generally, the performance of spatial smoothing based algorithms depends on the number of subarrays, which is a design parameter and needs to be chosen beforehand. In order to gain more insights into the optimal choice of the number of subarrays, we simplify the derived analytical R-D mean square error (MSE) expressions for the special case of a single source. The obtained MSE expression explicitly depends on the number of subarrays in each dimension, which allows us to analytically find the optimal number of subarrays for spatial smoothing. Based on this result, we additionally derive the maximum asymptotic gain from spatial smoothing and explicitly compute the asymptotic efficiency for this special case. All the analytical results are verified by simulations.

In certain applications involving direction finding, a priori knowledge of a subset of the directions to be estimated is sometimes available. Existing knowledge-aided (KA) methods apply projection and polynomial rooting techniques to exploit this information in order to improve the estimation accuracy of the unknown signal directions. In this paper, a new strategy for incorporating prior knowledge is developed for situations with a low signal-to-noise ratio (SNR) and a limited data record based on the Unitary ESPRIT algorithm. The proposed KAUnitary ESPRIT algorithm processes an enhanced covariance matrix estimate obtained by applying a shrinkage covariance estimator, which linearly combines the sample covariance matrix and an a priori known covariance matrix in an automatic fashion. Simulations show that the derived algorithm achieves significant performance gains in estimating the unknown sources and additionally provides a high robustness in the case of inaccurate prior knowledge.

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