R-dimensional esprit-type algorithms for strictly second-order non-circular sources and their performance analysis

Abstract

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.

Direct Derivation of the Stochastic CRB of DOA Estimation for Rectilinear Sources

“…The purpose of the next section is to directly derive the SCRB of the parameter θ alone from the FIM (6). Noting that the parameters θ k and φ k are non-linearly related in the extended steering vector a k , closed-form expressions of the SCRB of the couple ω…”

“…Various DOA estimation algorithms such as MUSIC [1], [2], root-MUSIC [3], standard ESPRIT [4] and unitary ESPRIT [5], [6] have been adapted to exploit the structure of rectilinearity or strictly second-order noncircularity of signals, which include commonly used digital modulation schemes such as BPSK and ASK. These algorithms are known to achieve a higher estimation accuracy and can resolve up to twice as many sources compared to the traditional DOA algorithms.…”

Subspace leakage analysis of sample data covariance matrix

“…D Using ( 8) together with the series expansion of Pin ( 11), ( 12), and ( 13) up to the 0 2 P term, the facts that P pl.. = pl.. P = 0, (pl.. ) 2 = pl.. , (p) 2 = P, and Lemma 1, we can compute p as…”

“…In [ 7], numerical examples are used to show that even a small portion of the estimated sig nal subspace energy residing in the true noise subspace ( de noted by inter-subspace leakage) can result in performance breakdown. The notion of leakage comes originally from the performance assessment based on perturbed subspace estima tion [8][9][10][11][12]. The results in [ 8] are asymptotic in the number of samples, and the results in [9][10][11][12] are asymptotic in the ef fective SNR.…”

“…The notion of leakage comes originally from the performance assessment based on perturbed subspace estima tion [8][9][10][11][12]. The results in [ 8] are asymptotic in the number of samples, and the results in [9][10][11][12] are asymptotic in the ef fective SNR. Here, we formally define the subspace leakage notion as a Frobenius norm of the perturbation matrix.…”