Anne Ferreol

On the virtual array concept for higher order array processing

For about two decades, many fourth order (FO) array processing methods have been developed for both direction finding and blind identification of non-Gaussian signals. One of the main interests in using FO cumulants only instead of second-order (SO) ones in array processing applications relies on the increase of both the effective aperture and the number of sensors of the considered array, which eventually introduces the FO Virtual Array concept presented elsewhere and allows, in particular, a better resolution and the processing of more sources than sensors. To still increase the resolution and the number of sources to be processed from a given array of sensors, new families of blind identification, source separation, and direction finding methods, at an order m=2q (q/spl ges/2) only, have been developed recently. In this context, the purpose of this paper is to provide some important insights into the mechanisms and, more particularly, to both the resolution and the maximal processing capacity, of numerous 2qth order array processing methods, whose previous methods are part of, by extending the Virtual Array concept to an arbitrary even order for several arrangements of the data statistics and for arrays with space, angular and/or polarization diversity.

High-Resolution Direction Finding From Higher Order Statistics: The

From the beginning of the 1980s, many second-order (SO) high-resolution direction-finding methods, such as the MUSIC method (or 2-MUSIC), have been developed mainly to process efficiently the multisource environments. Despite of their great interests, these methods suffer from serious drawbacks such as a weak robustness to both modeling errors and the presence of a strong colored background noise whose spatial coherence is unknown, poor performance in the presence of several poorly angularly separated sources from a limited duration observation and a maximum of N-1 sources to be processed from an array of N sensors. Mainly to overcome these limitations and in particular to increase both the resolution and the number of sources to be processed from an array of N sensors, fourth-order (FO) high-resolution direction-finding methods have been developed, from the end of the 1980s, to process non-Gaussian sources, omnipresent in radio communications, among which the 4-MUSIC method is the most popular. To increase even more the resolution, the robustness to modeling errors, and the number of sources to be processed from a given array of sensors, and thus to minimize the number of sensors in operational contexts, we propose in this paper an extension of the MUSIC method to an arbitrary even order 2q (qges1), giving rise to the 2q-MUSIC methods. The performance analysis of these new methods show off new important results for direction-finding applications and in particular the best performances, with respect to 2-MUSIC and 4-MUSIC, of 2q-MUSIC methods with q>2, despite their higher variance, when some resolution is required

2rm q

For more than a decade, fourth-order (FO) direction finding (DF) methods have been developed for non-Gaussian signals. Recently, it has been shown, through the introduction of the virtual cross-correlation (VCC) concept, that the use of FO cumulants for the DF problem increases the effective aperture of an arbitrary antenna array, which eventually introduces the virtual array concept. The purpose of this correspondence is first to present this virtual array (VA) concept through an alternative way that is easier and more direct to handle than the VCC tool and, second, to present further results associated with this concept, not only for arrays with space diversity but also for arrays with angular and/or polarization diversity.

-MUSIC Algorithm

Most of the second-order (SO) and higher order (HO) blind source separation methods developed in the 1990s aim at blindly separating statistically independent sources that are assumed zero-mean, stationary, and ergodic. Nevertheless, in many situations of practical interest, such as in radiocommunication contexts, the sources are nonstationary and very often (quasi)-cyclostationary (digital modulations). In these conditions, it becomes important to wonder whether the performance of these current SO and HO blind source separation methods, which have been developed for stationary sources, may be affected by the potential nonstationarity of the latter. Limiting the analysis to the SO and fourth-order (FO) cumulant-based blind source separation methods, the purpose of this paper is to bring some answers to this important question through the behavior analysis of the empirical SO and FO cumulants estimator in the presence of zero-mean (quasi)-cyclostationary sources.

On the virtual array concept for the fourth-order direction finding problem

For about two decades, numerous methods have been developed to blindly identify overdetermined (P/spl les/N) mixtures of P statistically independent narrowband (NB) sources received by an array of N sensors. These methods exploit the information contained in the second-order (SO), the fourth-order (FO) or both the SO and FO statistics of the data. However, in practical situations, the probability of receiving more sources than sensors increases with the reception bandwidth and the use of blind identification (BI) methods able to process underdetermined mixtures of sources, for which P>N may be required. Although such methods have been developed over the past few years, they all present serious limitations in practical situations related to the radiocommunications context. For this reason, the purpose of this paper is to propose a new attractive BI method, exploiting the information contained in the FO data statistics only, that is able to process underdetermined mixtures of sources without the main limitations of the existing methods, provided that the sources have different trispectrum and nonzero kurtosis with the same sign. A new performance criterion that is able to quantify the identification quality of a given source and allowing the quantitative comparison of two BI methods for each source, is also proposed in the paper. Finally, an application of the proposed method is presented through the introduction of a powerful direction-finding method built from the blindly identified mixture matrix.

On the behavior of current second and higher order blind source separation methods for cyclostationary sources

This paper provides a new analytic expression of the bias and RMS error (root mean square) error of the estimated direction of arrival (DOA) in the presence of modeling errors. In , first-order approximations of the RMS error are derived, which are accurate for small enough perturbations. However, the previously available expressions are not able to capture the behavior of the estimation algorithm into the threshold region. In order to fill this gap, we provide a second-order performance analysis, which is valid in a larger interval of modeling errors. To this end, it is shown that the DOA estimation error for each signal source can be expressed as a ratio of Hermitian forms, with a stochastic vector containing the modeling error. Then, an analytic expression for the moments of such a Hermitian forms ratio is provided. Finally, a closed-form expression for the performance (bias and RMS error) is derived. Simulation results indicate that the new result is accurate into the region where the algorithm breaks down.

Fourth-order blind identification of underdetermined mixtures of sources (FOBIUM)

This paper considers the statistical performance of the MUSIC method under the condition that two closely spaced sources impinging on an array of sensors are effectively resolved, i.e., the spectrum exhibits two peaks in the neighborhood of the true directions-of-arrival (DOA). The MUSIC algorithm is known to have an infinite resolution power in theory. However, in the presence of modeling errors, sources can not be resolved with certainty, even if the array correlation matrix is perfectly known. The focus of this paper is to predict the bias and variance of the DOA estimates taking into account the possible resolution failure of MUSIC. This performance prediction, based on our recent mathematical investigation, is new to the best of our knowledge. A general mathematical framework to derive closed form expressions of the bias and variance versus the model mismatch, conditioned on a general statistical resolution test is proposed. In order to illustrate our mathematical approach, statistical tests with one and two conditions, respectively, are investigated. The accuracy of the performance prediction is illustrated in a simulation study. It is found that the proposed approach outperforms the “classical” technique, which ignores the possible resolution failure of the MUSIC algorithm. Therefore, our results provide better tools for determining the necessary antenna calibration accuracy to achieve some targeted specifications on the estimator performance.

On the asymptotic performance analysis of subspace DOA estimation in the presence of modeling errors: case of MUSIC

The problem of blind separation of overdetermined mixtures of sources, that is, with fewer sources than (or as many sources as) sensors, is addressed in this paper. A new method, called Independent Component Analysis using Redundancies in the quadricovariance (ICAR), is proposed in order to process complex data. This method, without any whitening operation, only exploits some redundancies of a particular quadricovariance matrix of the data. Computer simulations demonstrate that ICAR offers in general good results and even outperforms classical methods in several situations: ICAR i) succeeds in separating sources with low signal-to-noise ratios, ii) does not require sources with different second-order or/and first-order spectral densities, iii) is asymptotically not affected by the presence of a Gaussian noise with unknown spatial correlation, iv) is not sensitive to an over estimation of the number of sources.

Statistical Analysis of the MUSIC Algorithm in the Presence of Modeling Errors, Taking Into Account the Resolution Probability

Most of the second-order (SO) and higher order (HO) blind source separation (BSS) methods developed this last decade aim at blindly separating statistically independent sources that are assumed zero-mean, stationary, and ergodic. Nevertheless, in many situations of practical interest, such as in radiocommunications contexts, the sources are nonstationary and very often cyclostationary (digital modulations). The behavior of the current SO and fourth-order (FO) cumulant-based BSS methods in the presence of cyclostationary sources has been analyzed, recently, in a previous paper by Ferre/spl acute/ol and Chevalier, assuming zero-mean sources. However, some cyclostationary sources used in practical situations are not zero-mean but have a first-order (FIO) cyclostationarity property, which is, in particular, the case for some amplitude modulated (AM) signals and for some nonlinearly modulated digital sources such as frequency shift keying (FSK) or some continuous phase frequency shift keying (CPFSK) sources. For such sources, the results presented in the previous paper by Ferre/spl acute/ol and Chevalier no longer hold, and the purpose of this paper is to analyze the behavior and to propose adaptations of the current SO BSS methods for sources that are both FIO and SO cyclostationary and cyclo-ergodic. An extension for deterministic sources is also proposed in the paper.

ICAR: a tool for blind source separation using fourth-order statistics only

The problem of resolving closely spaced signal sources using an antenna array remains a difficult one, although several estimation methods are available in the literature. When the array correlation matrix is known, the resolution capability of subspace algorithms is infinitely high. However, in the presence of modeling errors the resolution deteriorates, even for a known correlation matrix. In this paper, we analyze the MUSIC method, by way of three different definitions of the resolution. Assuming Gaussian circular random modeling errors, we determine the corresponding expressions of the probability of source resolution versus the model mismatch. A first series of simulations validates the mathematical expression of the three resolution probabilities. A second series of simulations is used to select among them the tightest one to the empirical one. The results are useful, e.g., for determining the necessary antenna calibration accuracy to achieve a target performance.