Performance Analysis of Multi-Dimensional ESPRIT-Type Algorithms for Arbitrary and Strictly Non-Circular Sources With Spatial Smoothing

Abstract

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.

Multidimensional ESPRIT for Damped and Undamped Signals: Algorithm, Computations, and Perturbation Analysis

“…Statistical performances of subspace 1-D estimation methods have been extensively studied in the case of undamped sinusoids [21], [22], [23] and damped ones [24]. Analytical performances of tensor-based ESPRIT-type algorithms have been assessed for undamped signals [25] , and more recently, for the case of spatial smoothing [26]. Statistical performance of some related methods have been also studied, but only in the case of undamped sinusoids [8], [9].…”

“…For damped signals, a new study was presented for the case of 1-D damped single-tone [27], resulting in new closed-from expressions. An extension of the results of [27] to the case of 2-D ESPRIT was initiated in [28] independently of [26].…”

“…We discuss the difference between the N -D ESPRIT and the multidimensional ESPRIT of [6], and other methods, such as IMDF [9] and MEMP [4]. We also give explicitly recovery (identifiability) conditions for the N -D ESPRIT algorithm, that are, up to our knowledge, not discussed in [6], [26].…”

“…Our derivations of the first-order perturbations are self-contained and are based on rigorous proofs. In particular, we base our results on the recent full expressions for the first-order perturbations of the SVD [29,Theorem 1], [30,Proposition 9], unlike the state-of-the-art papers [25], [26] (and earlier papers [21], [31]) that neglect the term containing the change of basis of the signal subspace. In our paper, we fill this gap and provide a rigorous proof that the aforementioned term does not influence the first-order perturbation of the modes (similarly to 1-D ESPRIT as shown in [29], [30]).…”

“…Finally, we derive closedform expressions for the variances of the perturbations in the N -D damped and undamped single-tone case. For the single tone case for undamped signals, we obtain the results as in [26,Theorem 3]. However, our final formula is simpler than the one of [26].…”