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Abstract
Tensors are higher order generalization of vectors and matrices which can be used to describe signals indexed by more than two indices. This paper introduces a tensor framework for minimum mean square error (MMSE) estimation for multi-domain signals and data using the Einstein Product. The framework addresses both proper and improper complex tensors. The multi-domain nature of tensors has been harnessed to provide an augmented representation of improper complex tensors to account for covariance and pseudo-covariance. The classical notions of linear and widely linear MMSE estimators are extended to tensor case leading to the notion of multi-linear and widely multi-linear MMSE estimation. The Tucker product based n-mode Wiener filtering approach more commonly used in tensor estimation has been shown to be a special case of the proposed multi-linear MMSE estimation. An application of the tensor based estimation in a multiple antenna Orthogonal Frequency Division Multiplexing (MIMO OFDM) system is presented where the tensor formulation allows a convenient treatment of inter-carrier interference. A comparison between the tensor estimation and per sub-carrier estimation used for MIMO OFDM is presented which shows a significant performance advantage of using the tensor framework.
Highlights
Minimum mean square error (MMSE) estimation has been extensively used in various disciplines including seismology, radio astronomy, sonar, speech and image processing, radar, medical signal processing, and communications [1]
We present a generic tensor framework for MMSE estimation based on the Einstein product, which is concerned with estimating a signal in tensor form from a tensor based noisy observation
The tensor multi-linear (TL) and widely multi-linear (TWL) MMSE estimation techniques for multi-domain signals have been formulated while keeping the multi-way structure of the signals intact
Summary
Minimum mean square error (MMSE) estimation has been extensively used in various disciplines including seismology, radio astronomy, sonar, speech and image processing, radar, medical signal processing, and communications [1]. The Tucker product based technique is known as the n-mode Wiener filtering approach [30] This method aims to find N separate factor matrices along each mode of the order N tensor to be estimated. The mode-n product between the observed noisy tensor and the factor matrices is used to find the estimate Such an approach has been employed in various applications including image processing [30], speech processing [31], and communication systems [26]. Many situations in practical cases involve signals that naturally involve multiple indices such as MIMO OFDM, Generalized Frequency Division Multiplexing (GFDM), Filter-bank Multi-carrier (FBMC) systems [15] Representing such signals by tensors is natural. PANDEY AND LEIB: A TENSOR FRAMEWORK FOR MULTI-LINEAR COMPLEX MMSE ESTIMATION faster approaches to implement Newton method for tensor inversion
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