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In this paper a complex-valued formulation of the modal superposition equation is provided and shown to be equivalent to the original, real-valued Blind Modal IDentification (BMID) problem. The complex-valued variant involves the analytic form of the physical and modal responses. The formulation is shown to be more concise and straightforward than the original. It is noted that complex-valued mode shapes can be obtained using a complex version of the two-step Joint Approximate Diagonalization (JAD) algorithm. Using this approach the modal response pairing step of the original BMID method is eliminated. Since the development of the original BMID method, several new, one-step JAD algorithms have been devised. Many of the algorithms can be extended to identify complex mixing matrices. A complex version of the one-step JAD method known as the Weighted Exhaustive Diagonalization with Gauss itErations algorithm is utilized to solve for the complex mode shapes and modal responses. By using this simplified formulation, the whitening step is eliminated, as well as the modal response pairing step, which is necessary in the original BMID algorithm. Performance of the new Complex BMID (CBMID) algorithm is evaluated by application to synthesized data from a threedegrees-of-freedom system with complex modes, application to measured laboratory data on a structural frame and application to measured output-only data from the Heritage Court Tower building. It is seen that the CBMID method results in essentially the same estimates of modal responses, complex mode shapes, natural frequencies and modal damping compared to results from BMID. Furthermore, it is shown that modal parameters from BMID and CBMID are very consistent with those obtained from state-of-the-art methods, such as the Eigensystem Realization Algorithm and the covariance-driven Stochastic Subspace Identification method.

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A modal identification algorithm is developed, combining techniques from Second Order Blind Source Separation (SOBSS) and State Space Realization (SSR) theory. In this hybrid algorithm, a set of correlation matrices is generated using time-shifted, analytic data and assembled into several Hankel matrices. Dissimilar left and right matrices are found, which diagonalize the set of nonhermetian Hankel matrices. The complex-valued modal matrix is obtained from this decomposition. The modal responses, modal auto-correlation functions and discrete-time plant matrix (in state space modal form) are subsequently identified. System eigenvalues are computed from the plant matrix to obtain the natural frequencies and modal fractions of critical damping. Joint Approximate Diagonalization (JAD) of the Hankel matrices enables the under determined (more modes than sensors) problem to be effectively treated without restrictions on the number of sensors required. Because the analytic signal is used, the redundant complex conjugate pairs are eliminated, reducing the system order (number of modes) to be identified half. This enables smaller Hankel matrix sizes and reduced computational effort. The modal auto-correlation functions provide an expedient means of screening out spurious computational modes or modes corresponding to noise sources, eliminating the need for a consistency diagram. In addition, the reduction in the number of modes enables the modal responses to be identified when there are at least as many sensors as independent (not including conjugate pairs) modes. A further benefit of the algorithm is that identification of dissimilar left and right diagonalizers preclude the need for windowing of the analytic data. The effectiveness of the new modal identification method is demonstrated using vibration data from a 6 DOF simulation, 4-story building simulation and the Heritage court tower building.

In recent literature, attempts have been made to apply Independent Component Analysis (ICA) techniques to the modal identification problem. Paramount to this task is establishing a relationship between the source components realized using the suggested ICA technique and modal responses. In this paper, the relationship between independent components and free-vibration modal responses is discussed. Theoretical arguments are presented for responses of undamped systems and arguments are extended to damped responses.