A study and extension of second-order blind source separation to operational modal analysis J. Antoni a,n,1 , S. Chauhan b a Laboratory Roberval, CNRS UMR 6253, University of Technology of Compiegne, Centre de Recherche de Royallieu, BP 20529-60205 Ce´dex, Compi  egne, France b Bruel & Kjaer Sound and Vibration Measurement A/S Skodsborgvej 307, DK 2850 Naerum, Denmark article info Article history: Received 6 October 2011 Received in revised form 8 September 2012 Accepted 12 September 2012 Handling Editor: K. Shin Available online 16 October 2012 abstract Second-order blind source separation (SOBSS) has gained recent interest in operational modal analysis (OMA), since it is able to separate a set of system responses into modal coordinates from which the system poles can be extracted by single-degree-of-freedom techniques. In addition, SOBSS returns a mixing matrix whose columns are the estimates of the system mode shapes. The objective of this paper is threefold. First, a theoretical analysis of current SOBSS methods is conducted within the OMA framework and its precise conditions of applicability are established. Second, a new separation method is proposed that fixes current limitations of SOBSS: It returns estimate of complex mode shapes, it can deal with more active modes than the number of available sensors, and it shows superior performance in the case of heavily damped and/or strongly coupled modes. Third, a theoretical connection is drawn between SOBSS and stochastic subspace identification (SSI), which stands as one of the points of reference in OMA. All approaches are finally compared by means of numerical simulations. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction During last decades, blind source separation (BSS) techniques have gained tremendous popularity amongst the signal processing community due to their ability to extract source signals and the underlying mixing system from the output response signals without requiring any training data or apriori information [1–4]. These techniques have found research interest and application in a variety of areas including medical imaging, wireless communications, speech enhancement, data mining, and lately in vibration and acoustics [5–8]. Recently BSS techniques have also attracted interest of researchers in the area of structural dynamics. The feasibility of using BSS to separate individual mode shapes from output-only measurement –from which modal parameters can subsequently be estimated using simple single-degree-of-freedom techniques–has been demonstrated in several publications, thus providing a truly valid and probably simpler alternative to other identification methods used in operational modal analysis (OMA). The correspondence between the mode shapes of a structure and the columns of the mixing matrix in BSS was first recognized in Ref. [9]. The application of independent component analysis (ICA 2 ) by means of a higher-order BSS algorithm was then proposed to estimate weakly damped modes (less than 1% damping). Elaborating on this finding, Refs. [10,11] demonstrated that SOBI (second-order blind identification), a second-order BSS (SOBSS) algorithm, could achieve Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/jsvi Journal of Sound and Vibration 0022-460X/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2012.09.016 n Corresponding author. Tel.: þ33 6357 61759. E-mail addresses: jerome.antoni@utc.fr (J. Antoni), schauhan@bksv.com (S. Chauhan). 1 Current address: Laboratory Vibrations Acoustics, University of LyonF-69621 Villeurbanne Ce ´ dex, France. 2 Independent component analysis, a particular case of BSS, refers to the decomposition of a set of signals into independent sources. The condition of independence is stronger than decorrelation which is enforced in second-order BSS. See Ref. [28] for a complete exposition of ICA and second-order BSS. Journal of Sound and Vibration 332 (2013) 1079–1106