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. R-dimensional subspace-based methods, such as R-D Unitary ESPRIT, R-D RARE, or R-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 R-D standard Tensor-ESPRIT and the R-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 R 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.

Relaying is one of the key technologies to satisfy the demands of future mobile communication systems. In particular, two-way relaying is known to exploit the radio resources in a very efficient manner. In this contribution, we consider two-way relaying with amplify-and-forward (AF) MIMO relays. Since AF relays do not decode the signals, the separation of the data streams has to be performed by the terminals themselves. For this task both nodes require reliable channel knowledge of all relevant channel parameters. Therefore, we examine channel estimation schemes for two-way relaying with AF MIMO relays. We investigate a simple Least Squares (LS) based scheme for the estimation of the compound channels as well as a tensor-based channel estimation (TENCE) scheme which takes advantage of the special structure in the compound channel matrices to further improve the estimation accuracy. Note that TENCE is purely algebraic (i.e., it does not require any iterative procedures) and applicable to arbitrary antenna configurations. Then we demonstrate that the solution obtained by TENCE can be improved by an iterative refinement which is based on the structured least squares (SLS) technique. In this application, between one and four iterations are sufficient and consequently the increase in computational complexity is moderate. The iterative refinement is optional and targeted for cases where the channel estimation accuracy is critical. Moreover, we propose design rules for the training symbols as well as the relay amplification matrices during the training phase to facilitate the estimation procedures. Finally, we evaluate the achievable channel estimation accuracy of the LS-based compound channel estimation scheme as well as the tensor-based approach and its iterative refinement via numerical computer simulations.

In this paper we present a generic framework for the asymptotic performance analysis of subspace-based parameter estimation schemes. It is based on earlier results on an explicit first-order expansion of the estimation error in the signal subspace obtained via an SVD of the noisy observation matrix. We extend these results in a number of aspects. Firstly, we demonstrate that an explicit first-order expansion of the Higher-Order SVD (HOSVD)-based subspace estimate can be derived. Secondly, we show how to obtain explicit first-order expansions of the estimation error of arbitrary ESPRIT-type algorithms and provide the expressions for R-D Standard ESPRIT, R-D Unitary ESPRIT, R-D Standard Tensor-ESPRIT, as well as R-D Unitary Tensor-ESPRIT. Thirdly, we derive closed-form expressions for the mean square error (MSE) and show that they only depend on the second-order moments of the noise. Hence, to apply this framework we only need the noise to be zero mean and possess finite second order moments. Additional assumptions such as Gaussianity or circular symmetry are not needed.

High-resolution parameter estimation algorithms designed to exploit the prior knowledge about incident signals from strictly second-order (SO) noncircular (NC) sources allow for a lower estimation error and can resolve twice as many sources. In this paper, we derive the R \mathchar"702DD NC Standard ESPRIT and the R \mathchar"702DD NC Unitary ESPRIT algorithms that provide a significantly better performance compared to their original versions for arbitrary source signals. They are applicable to shift-invariant R \mathchar"702DD antenna arrays and do not require a centro-symmetric array structure. Moreover, we present a first-order asymptotic performance analysis of the proposed algorithms, which is based on the error in the signal subspace estimate arising from the noise perturbation. The derived expressions for the resulting parameter estimation error are explicit in the noise realizations and asymptotic in the effective signal-to-noise ratio (SNR), i.e., the results become exact for either high SNRs or a large sample size. We also provide mean squared error (MSE) expressions, where only the assumptions of a zero mean and finite SO moments of the noise are required, but no assumptions about its statistics are necessary. As a main result, we analytically prove that the asymptotic performance of both R \mathchar"702DD NC ESPRIT-type algorithms is identical in the high-effective SNR regime. Finally, a case study shows that no improvement from strictly non-circular sources can be achieved in the special case of a single source.

Estimating the directions of arrival of several wavefronts impinging on an array of sensors is a requirement in a variety of applications including radar, mobile communications, sonar, and seismology. Subspace based high-resolution parameter estimation schemes like ESPRIT and MUSIC have become very popular. Such parameter estimation algorithms often use forward-backward averaging to enhance their resolution, especially in the case of correlated sources. Further enhancements can be achieved if the source signals are non-circular. We derive an efficient subspace estimation scheme that exploits the non-circularity of the sources and already includes forward-backward averaging. Moreover, appropriate spatial smoothing techniques are introduced. Completely real-valued implementations of 1D and 2D unitary ESPRIT for non-circular sources are presented as examples. In these cases, NC unitary ESPRIT improves the resolution capability and the noise robustness of standard ESPRIT as well as unitary ESPRIT and can handle more sources than sensors.

Sum-rate maximization in two-way amplify-and-forward (AF) multiple-input multiple-output (MIMO) relaying belongs to the class of difference-of-convex functions (DC) programming problems. DC programming problems occur also in other signal processing applications and are typically solved using different modifications of the branch-and-bound method which, however, does not have any polynomial time complexity guarantees. In this paper, we develop two efficient polynomial time algorithms for the sum-rate maximization in two-way AF MIMO relaying. The first algorithm guarantees to find at least a Karush-Kuhn-Tucker (KKT) solution. There is a strong evidence, however, that such a solution is actually globally optimal. The second algorithm that is based on the generalized eigenvectors shows the same performance as the first one with reduced computational complexity. The objective function of the problem is represented as a product of quadratic fractional ratios and parameterized so that its convex part (versus the concave part) contains only one (or two) optimization variables. One of the algorithms is called POlynomial Time DC (POTDC) and is based on semi-definite programming (SDP) relaxation, linearization, and an iterative Newton-type search over a single parameter. The other algorithm is called RAte-maximization via Generalized EigenvectorS (RAGES) and is based on the generalized eigenvectors method and an iterative search over two (or one, in its approximate version) optimization variables. We derive an upper-bound for the optimal value of the corresponding optimization problem and show by simulations that this upper-bound is achieved by both algorithms. It provides an evidence that the algorithms find a global optimum. The proposed methods are also superior to other state-of-the-art algorithms.