Subspace Methods and Exploitation of Special Array Structures

In this chapter we provide an overview of subspace-based parameter estimation schemes for uniform arrays, non-uniform arrays, and other specific array structures. The popularity of many of these special array structures is due to the availability of search-free low computational complexity direction of arrival (or spatial frequency) estimation algorithms to exploit the particularly structure of the array. More precisely, we mainly focus on the study and the comparison between several subspace-based algorithms. The latter can be classified into spectral search-based and search-free techniques. The spectral searching schemes include MUSIC, weighted subspace fitting algorithms, and rank reduction schemes divised for sensor arrays composed of multiple fully-calibrated subarrays with unknown subarray displacements. On the other hand, the search-free schemes can be partitioned into two subclasses: (i) Polynomial-rooting techniques, in which, we describe and compare MODE, the root-MUSIC algorithm and its variants for the uniform linear array (ULA) configuration, and the interpolated root-MUSIC, the manifold separation, and the Fourier domain root-MUSIC schemes in the non-uniform array context. (ii) Matrix-shifting techniques, in which, we present and compare the ESPRIT algorithm adapted for array geometries that exhibit a shift invariance structure and its variants, as the GESPRIT and Unitary ESPRIT algorithms. By numerical simulations, we show that in the one-dimensional case, the threshold of the root-MUSIC algorithm occurs at a higher SNR than for ESPRIT-based algorithms, in which the Unitary ESPRIT scheme performs best among all ESPRIT-based schemes. For the case of multidimensional parameter estimation, we introduce -D matrix-based and tensor-based algorithms. We demonstrate that multidimensional signals can be represented by tensors which provide a natural formulation of the -dimensional signals and their properties (such as the -D shift invariances needed for matrix shifting techniques). Based on this representation, an improved HOSVD-based signal subspace estimate is proposed. We show that this subspace estimate performs a more efficient denoising of the data which leads to a tensor gain in terms of an enhanced estimation accuracy. This subspace estimate can be combined with arbitrary existing multidimensional subspace-based parameter estimation schemes. Then we discuss the tensor-based schemes -D Standard Tensor-ESPRIT and -D Unitary Tensor-ESPRIT. They outperform the matrix based -D ESPRIT-type algorithms due to the enhanced subspace estimate obtained from the HOSVD. We also show that strict-sense non-circular sources can be exploited to virtually double the number of available sensors by an augmentation of the measurement matrix. Based on this idea, the -D NC Standard ESPRIT and the -D NC Unitary ESPRIT algorithm are derived. As a result, the number of resolvable wavefronts is doubled and the achievable estimation accuracy is improved. Finally, the family of NC Tensor-ESPRIT-type algorithms is introduced to combine both benefits, the strict-sense non-circular source symbols and the multidimensional structure of the signals. This is a non-trivial task, since the augmentation of the measurement matrix performed for -D NC Unitary ESPRIT destroys the structure needed for the Tensor-ESPRIT-type algorithms. This challenge can be solved by defining a mode-wise augmentation of the measurement tensor