Florian Roemer

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

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.

Tensor-Based Channel Estimation and Iterative Refinements for Two-Way Relaying With Multiple Antennas and Spatial Reuse

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.

Algebraic Norm-Maximizing (ANOMAX) Transmit Strategy for Two-Way Relaying With MIMO Amplify and Forward Relays

Two-way relaying is a promising scheme to achieve the ubiquitous mobile access to a reliable high data rate service, which is targeted for future mobile communication systems. In this contribution, we investigate two-way relaying with an amplify and forward relay, where the relay as well as the terminals are equipped with multiple antennas. Assuming that the terminals possess channel knowledge, the bidirectional two-way relaying channel is decoupled into two parallel effective single-user MIMO channels by subtracting the self-interference at the terminals. Thereby, any single-user MIMO technique can be applied to transmit the data. We derive an algebraic norm-maximizing (ANOMAX) transmit strategy by finding the relay amplification matrix which maximizes the weighted sum of the Frobenius norms of the effective channels and discuss the implications of this solution on the resulting signal to noise ratios. Finally, we compare ANOMAX to other existing transmission strategies via numerical computer simulations.

Sum-Rate Maximization in Two-Way AF MIMO Relaying: Polynomial Time Solutions to a Class of DC Programming Problems

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.

Tensor-based channel estimation (TENCE) for two-way relaying with multiple antennas and spatial reuse

In this paper we study two-way relaying with amplify-and-forward (AF) relays. In two-way relaying, two terminals exchange data with the help of an intermediate relay station. In order to enable mass deployment of these relays, we focus on very simple AF relays that do not have any channel state information. Hence, to separate the data streams in two-way relaying, both user terminals need reliable knowledge of all relevant channel parameters. We therefore propose the novel tensor-based channel estimation algorithm TENCE that provides both terminals with full knowledge of all channel parameters involved in the transmission. The solution is algebraic, i.e., it does not require any iterative procedures. Moreover, TENCE is applicable to arbitrary antenna configurations. We also derive criteria for the design of the pilot symbols and the corresponding relay amplification matrices. Computer simulations demonstrate the achievable channel estimation accuracy.

Multi-dimensional model order selection

Multi-dimensional model order selection (MOS) techniques achieve an improved accuracy, reliability, and robustness, since they consider all dimensions jointly during the estimation of parameters. Additionally, from fundamental identifiability results of multi-dimensional decompositions, it is known that the number of main components can be larger when compared to matrix-based decompositions. In this article, we show how to use tensor calculus to extend matrix-based MOS schemes and we also present our proposed multi-dimensional model order selection scheme based on the closed-form PARAFAC algorithm, which is only applicable to multi-dimensional data. In general, as shown by means of simulations, the Probability of correct Detection (PoD) of our proposed multi-dimensional MOS schemes is much better than the PoD of matrix-based schemes.

Analytical Performance Assessment of Multi-Dimensional Matrix- and Tensor-Based ESPRIT-Type Algorithms

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.

Sum rate maximization for multi-pair two-way relaying with single-antenna amplify and forward relays

We consider a multi-pair two-way relay network with multiple single antenna amplify-and-forward relays. The sum rate maximization problem subject to a total transmit power constraint is studied for such network. The optimization problem is non-convex. First, we show that the problem is a monotonic optimization problem and propose a polyblock approximation algorithm for obtaining the global optimum. However, this algorithm is only suitable for benchmarking because of its high computational complexity. After observing that the necessary optimality condition for our problem is similar to that of the generalized eigenvalue problem, we propose to use the generalized power iterative algorithm which can approach the global optimum recursively. Finally, we propose the total signal-to-interference-plus-noise ratio (SINR) eigen-beamformer which is a closed-form suboptimal solution that reduces the computational complexity significantly. Simulation results show that the proposed algorithms outperform the existing scheme. Moreover, the total SINR eigen-beamformer almost achieves the performance of the optimal solution.

Efficient 1-D and 2-D DOA Estimation for Non-Circular Sourceswith Hexagonal Shaped Espar Arrays

This contribution is focused on direction of arrival (DoA) estimation with a regular-hexagonal shaped ESPAR (electronically steerable parasitic antenna radiator) array that has received increased attention recently. It is shown how the estimation accuracy is improved by employing non-circular (NC) signal constellations that facilitate the application of the NC Unitary ESPRIT algorithm. It is demonstrated how this method allows the joint estimation of the azimuth and the elevation angles of up to eight uncorrelated sources with a 7-element ESPAR array. Moreover, the achievable benefits of using non-circular sources are assessed by studying deterministic Cramer-Rao bounds. It is shown that for special phase constellations between the impinging wavefronts the estimation accuracy is independent of the angular separation of the corresponding DoAs

A semi-algebraic framework for approximate CP decompositions via simultaneous matrix diagonalizations (SECSI)

In this paper, we propose a framework to compute approximate CANDECOMP / PARAFAC (CP) decompositions. Such tensor decompositions are viable tools in a broad range of applications, creating the need for versatile tools to compute such decompositions with an adjustable complexity-accuracy trade-off. To this end, we propose a novel SEmi-algebraic framework that allows the computation of approximate C P decompositions via SImultaneous Matrix Diagonalizations (SECSI). In contrast to previous Simultaneous Matrix Diagonalization (SMD)-based approaches, we use the tensor structure to construct not only one but the full set of possible SMDs. Solving all SMDs, we obtain multiple estimates of the factor matrices and present strategies to choose the best estimate in a subsequent step. This SECSI framework retains the option to choose the number of SMDs to solve and to adopt various strategies for the selection of the final solution out of the multiple estimates. A best matching scheme based on an exhaustive search as well as heuristic selection schemes are devised to flexibly adapt to specific applications. Four example algorithms with different accuracy-complexity trade-off points are compared to state-of-the-art algorithms. We obtain more reliable estimates and a reduced computational complexity.