Unified analysis for DOA estimation algorithms in array signal processing

doi:10.1016/0165-1684(91)90060-V

Signal Processing

Volume 25, Issue 2, November 1991, Pages 147-169

https://doi.org/10.1016/0165-1684(91)90060-V Get rights and content

Abstract

In this paper, a unified statistical performance analysis using perturbation expansions is applied to subspace-based algorithms for direction-of-arrival (DOA) estimation in array signal processing. The analysis assumes that only a finite amount of array data is available at high signal-to-noise ratio. The MUSIC, Min-Norm, State-Space Realization (TAM) and ESPRIT algorithms are analyzed in a common framework. A significant feature of this analysis is that it includes different types of error sources, such as the finite sample effect induced by additive observation noise, the sensor error effect induced by the inaccurate knowledge of sensor response and location, and the effect of a coherent noise field with unknown structure. All of the algorithms considered in this paper are based on a singular value decomposition of a data matrix. A general expression for the perturbation of singular vectors as a function of data matrix perturbations is derived and used to obtain an analytical expression for the mean-squared DOA estimation error in a simple and self-contained fashion. The tractable formulas provide insight into the algorithms. Simulation results verify the analytically predicted performance.

Zusammenfassung

In diesem Beitrag werden Unterraumalgorithmen für die Schätzung der Einfallsrichtung (direction-of-arrival, DOA) in der Array-Signalverarbeitung einer vereinheitlichten statischen Leistungsanalyse mit Hilfe einer Störungsrechnung unterzogen. Die Analyse setzt voraus, daβ nur eine endliche Menge von Arraydaten bei hohem Signal-zu-Rauschleistungsverhältnis verfügbar ist. Folgende Algorithmen werden in einem gemeinsamen Rahmen analysiert: MUSIC, Min-Norm, Zustandsraumrealisierung (TAM) und ESPRIT. Ein bezeichnendes Merkmal dieser Analyse ist daβ sie verschiedene Arten von Fehlerquellen einschlieβt, und zwar den Einfluβ der endlichen Stichprobengröβe, der durch additives Beobachtungsrauschen hervorgerufen wird, den Einfluβ des Sensorfehlers, der durch die ungenaue Kenntnis der Sensorantwort und -lage hervorgerufen wird, und den Einfluβ eines kohärenten Geräuschfeldes mit unbekannter Struktur. Alle in diesem Beitrag betrachteten Algorithmen basieren auf einer Singularwertzerlegung einer Datenmatrix. Ein allgemeiner Ausdruck für die Störung von Singularvektoren als Funktion der Störungen der Datenmatrix wird hergeleitet und benutzt, um einen analytischen Ausdruck für den mittleren quadratischen DOA-Schätzfehler in einer einfachen und in sich geschlossenen Weise zu gewinnen. Die überschaubaren Formeln bieten Einsicht in die Algorithmen. Simulationsergebnisse bestätigen die analytisch vorhergesagte Leistungsfähigkeit.

Résumé

Nous appliquons dans cet article une analyse unifiée des performances statistiques utilisant des expansions de perturbation aux algorithmes basés sur la décomposition en sous-espaces permettant l'estimation de la direction d'arrivée (DOA) en traitement de réseaux. Dans cette analyse nous supposons qu'un nombre fini de données du réseau est accessible avec un rapport signal sur bruit élevé. Nous analysons dans un cadre commun les algorithmes MUSIC, Min-Norm, réalisation dans l'espace d'état (TAM) et ESPRIT. Un aspect significatif de cette analyse est qu'elle inclut différents types de sources d'erreur, telles que l'effet d'échantillonnage fini induit par un bruit d'observation additif, l'effet d'erreur sur le capteur induit par une connaissance imparfaite de la position et de la réponse du capteur, et l'effet d'un champ de bruit cohérent de structure inconnue. Tous les algorithmes considérés dans cet article sont basés sur la décomposition en valeurs singulières de la matrice des données. Nous dérivons une expression générale de la perturbation des vecteurs singuliers en fonction des perturbations de la matrice des données et nous l'utilisons pour obtenir d'une manière simple et complète une expression analytique de l'erreur quadratique moyenne sur l'estimation de la direction d'arrivée. Les formules utilisables donnent un meilleur aperçu des algorithmes. Des résultats de simulation permettent de vérifier les performances prédites analytiquement.

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Truncated singular value decomposition is a reduced version of the singular value decomposition in which only a few largest singular values are retained. This paper presents a novel perturbation analysis for the truncated singular value decomposition for real matrices. First, we describe perturbation expansions for the singular value truncation of order r. We extend perturbation results for the singular subspace decomposition to derive the first-order perturbation expansion of the truncated operator about a matrix with rank greater than or equal to r. Observing that the first-order expansion can be greatly simplified when the matrix has exact rank r, we further show that the singular value truncation admits a simple second-order perturbation expansion about a rank-r matrix. Second, we introduce the first-known error bound on the linear approximation of the truncated singular value decomposition of a perturbed rank-r matrix. Our bound only depends on the least singular value of the unperturbed matrix and the norm of the perturbation matrix. Intriguingly, while the singular subspaces are known to be extremely sensitive to additive noises, the newly established error bound holds universally for perturbations with arbitrary magnitude. Finally, we demonstrate an application of our results to the analysis of the mean squared error associated with the TSVD-based matrix denoising solution.
Performance analysis of G-MUSIC based DOA estimator with random linear array: A single source case
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On the whole, we assume that each sensor in the RLA is in operation with a known probability p. For DOA estimation, subspace-based approaches are widely used because of their low complexity and robustness relative to the maximum-likelihood (ML) techniques, which are sensitive to signal and noise covariance matrix modeling errors [1–4]. Popular subspace-based algorithms for DOA estimation include MUltiple SIgnal Classification (MUSIC) [5–9], and Estimating Signal Parameter via Rotational Invariance Techniques (ESPRIT) [10].
We analyze the performance of direction-of-arrival (DOA) estimation based on the G-MUSIC algorithm for a single source using a random linear array, i.e., only a random subset of sensor array is in operation. The classical asymptotic analysis assumes that the number of snapshots tends to infinity whereas the number of sensors remains fixed. In contrast, this work studies the scenario where both quantities tend to infinity at the same rate. This general asymptotic regime provides a more accurate description of the practical situation where these two quantities are finite and comparable. We prove the consistency of the DOA estimator and derive an analytic expression for the mean square error (MSE) performance in the general asymptotic situation. The asymptotic MSE turns out to be a function of signal-to-noise ratio, probability of operating sensors and ratio of number of sensors to number of snapshots. Numerical results are presented to confirm the effectiveness of our theoretical calculations.
Performance improvement of direction finding algorithms in non-homogeneous environment through data fusion
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The latter are then fused on the basis of their accuracy to obtain a fused DOA using the federated fusion algorithm [21]. The DOA accuracy expression is obtained from the well-established performance analysis of DOA MUSIC-based algorithms [22–25]. The aim of this paper is to demonstrate that the proposed approach based on data fusion concepts and the well-known performance analysis can improve significantly subspace based algorithms.
This paper proposes a new effective approach to improve the performance of DOA (Direction Of Arrival) algorithms when bursts affect the array data. The proposed approach is based on the combining of data fusion techniques and the results of theoretical performance analysis of conventional DOA algorithms. For this purpose, the received array data is first split in M time-segments. Then, the DOAs are estimated from each data segment using a conventional DOA algorithm. The obtained estimates are fused using the federated fusion algorithm according to their statistical accuracy obtained from the well-documented performance analysis of the considered algorithm. As proof of concept of the proposed approach, numerical experiments have been conducted by considering the MUSIC algorithm. The obtained results show that the new algorithm outperforms the conventional one in terms of accuracy in a non-homogeneous environment. Therefore, it exhibits enhanced robustness capability. Moreover, it reduces the memory cost and computational complexity which makes it suitable for real time applications. To our knowledge, it is the first time that theoretical performance analysis results are exploited for the derivation of new subspace-DOA methods.
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The MUSIC [15] algorithm is a representative subspace based algorithm. There has been considerable research work in either analyzing its performance [10,16–18,3,4] or developing more advanced robust MUSIC algorithms [1,13,5]. In a multipath environment, the MUSIC algorithm can be used to estimate the AoAs of signals impinging on the receive antenna array simultaneously [8].
Using the MUSIC (MUltiple SIgnal Classification) algorithm to estimate the angle-of-arrival (AoA), we derive a new asymptotic error variance bound when the transmitted signal can be pre-processed. We next propose a precoder design to achieve this bound. However, such an optimal precoder requires channel state information at the transmitter (CSIT) exclusive of the receiver array which cannot be separately estimated practically. A more feasible precoder design, which leverages on the feedback CSIT estimated at the receiver, is next proposed. Using the performance of the optimal precoder which achieves the bound asymptotically as a benchmark, the practical precoder design performs close to the optimal precoder even in the high-resolution scenario. Both precoder schemes exhibit performance improvement compared with the case when no precoder is used.
Analytical Performance Assessment of 2-D Tensor ESPRIT in Terms of Physical Parameters
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Theme paperUnified analysis for DOA estimation algorithms in array signal processing

Abstract

Zusammenfassung

Résumé

Improving the resolution performance of eigenstructure-based direction-finding algorithm

Performance analysis of Root-MUSIC

IEEE Trans. Acoust. Speech Signal Process.

Performance analysis of ESPRIT and TAM in determining the direction of arrival of plane waves in noise

IEEE Trans. Acoust. Speech Signal Process.

Performance of high resolution frequencies estimation methods compared to the Cramer-Rao bounds

IEEE Trans. Acoust. Speech Signal Process.

A performance analysis of the adaptive algorithms in the presence of calibration errors

Theoretical performance prediction of the MUSIC algorithm

A sensitivity analysis of the MUSIC algorithm

Perturbation analysis of TK method for harmonic retrieval problems

IEEE Trans. Acoust. Speech Signal Process.

Matrix pencil method and its performance

The statistical performance of the MUSIC and the Minimum-Norm algorithms in resolving plane-waves in noise

IEEE Trans. Acoust. Speech Signal Process.

Estimating the angles of arrival of multiple plane waves

IEEE Trans. Aerospace Electron. Systems

State-space and singular-value decomposition-based approximation methods for the harmonic retrieval problem

J. Opt. Soc. Amer.

The Theory of Matrices

MUSIC performance prediction by matrix approximation at high SNR

Min-Norm Linear Prediction for arbitrary sensor array

Theme paper
Unified analysis for DOA estimation algorithms in array signal processing