Subspace-based exponential data fitting using linear and multilinear algebra

The exponentially damped sinusoidal (EDS) model arises in numerous signal processing applications. It is therefore of great interest to have methods able to estimate the parameters of such a model in the single-channel as well as in the multi-channel case. Because such a model naturally lends itself to subspace representation, powerful matrix approaches like HTLS in the single-channel case, HTLSstack in the multi-channel case and HTLSDstack in the decimative case have been developed to estimate the parameters of the underlying EDS model. They basically consist in stacking the signal in Hankel (single-channel) or block Hankel (multi- channel) data matrices. Then, the signal subspace is estimated by means of the singular value decomposition (SVD). The parameters of the model, namely the amplitudes, the phases, the damping factors, and the frequencies, are estimated from this subspace. Note that the sample covariance matrix counterpart is called TLS-ESPRIT, multi-channel TLS-ESPRIT and decimative TLS-ESPRIT. In these methods, the order of the model (i.e. the number of damped sinusoids) is assumed to be known. A variety of methods for estimating the model order exists. The recently developed method ESTER has been shown to outperform the existing Information Theoretic Criteria (ITC) based techniques. ESTER relies on the shift invariance property of the signal subspace. We propose an easy-to-implement SVD-based method which also exploits the same shift invariance property and outperforms the method ESTER. As far as multi-channel signals are concerned, it may be of great interest to extract only the common sinusoids. This may be for instance the case in Electroencephalogram (EEG) monitoring or material health monitoring. So far, only techniques which extract the common damped sinusoids in the twochannel case have been described. We propose a flexible and accurate method that can be applied to an arbitrary number of channels. The last part of the thesis deals with multilinear algebra, which is the algebra of higher-order tensors. Higher-order tensors can be seen as higher dimensional tables than can be addressed with more than two indices. First, we show that the matrix approaches do not exploit all the structure which is present in the theoretical decomposition. This is especially true in the multi-channel and the decimative case. In a second step we demonstrate that a higher-order representation of the problem may help to take this structure into account. We derive the higher-order counterparts of the HTLS, HTLSstack and HTLSDstack methods for estimating the parameters of an EDS model, and show by means of a higher-order dimensionality reduction algorithm that the estimation of the signal subspace, and hence the parameters of the EDS model, may be more accurate than the one obtained via the matrix approaches.