Higher-Order SVD-Based Subspace Estimation to Improve the Parameter Estimation Accuracy in Multidimensional Harmonic Retrieval Problems

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Multidimensional harmonic retrieval problems are encountered in a variety of signal processing applications including radar, sonar, communications, medical imaging, and the estimation of the parameters of the dominant multipath components from MIMO channel measurements. <formula formulatype="inline"> <tex>$R$</tex></formula>-dimensional subspace-based methods, such as <formula formulatype="inline"><tex>$R$</tex></formula>-D Unitary ESPRIT, <formula formulatype="inline"> <tex>$R$</tex></formula>-D RARE, or <formula formulatype="inline"><tex>$R$</tex> </formula>-D MUSIC, are frequently used for this task. Since the measurement data is multidimensional, current approaches require stacking the dimensions into one highly structured matrix. However, in the conventional subspace estimation step, e.g., via an SVD of the latter matrix, this structure is not exploited. In this paper, we define a measurement tensor and estimate the signal subspace through a higher-order SVD. This allows us to exploit the structure inherent in the measurement data already in the first step of the algorithm which leads to better estimates of the signal subspace. We show how the concepts of forward-backward averaging and the mapping of centro-Hermitian matrices to real-valued matrices of the same size can be extended to tensors. As examples, we develop the <formula formulatype="inline"><tex>$R$</tex></formula>-D standard Tensor-ESPRIT and the <formula formulatype="inline"><tex>$R$</tex></formula>-D Unitary Tensor-ESPRIT algorithms. However, these new concepts can be applied to any multidimensional subspace-based parameter estimation scheme. Significant improvements of the resulting parameter estimation accuracy are achieved if there is at least one of the <formula formulatype="inline"><tex>$R$</tex></formula> dimensions, which possesses a number of sensors that is larger than the number of sources. This can already be observed in the two-dimensional case. </para>