We present a method for estimating conditionally Gaussian random vectors with random covariance matrices, which uses techniques from the field of machine learning. Such models are typical in communication systems, where the covariance matrix of the channel vector depends on random parameters, e.g., angles of propagation paths. If the covariance matrices exhibit certain Toeplitz and shift-invariance structures, the complexity of the minimum mean squared error (MMSE) channel estimator can be reduced to @math floating point operations, where @math is the channel dimension. While in the absence of structure the complexity is much higher, we obtain a similarly efficient (but suboptimal) estimator by using the MMSE estimator of the structured model as a blueprint for the architecture of a neural network. This network learns the MMSE estimator for the unstructured model, but only within the given class of estimators that contains the MMSE estimator for the structured model. Numerical simulations with typical spatial channel models demonstrate the generalization properties of the chosen class of estimators to realistic channel models.
The ML-based channel estimation techniques exploit the Toeplitz structure of the covariance matrix, which is a result of regular array geometries and the model with a continuous power density function @math . In the 3GPP models, this power density function usually has a very limited angular support, i.e., @math is approximately zero except for @math in the vicinity of the cluster centers @math . The resulting covariance matrices have a very low numerical rank @cite_20 . As a consequence, under such a model, any given realization of a channel vector admits a sparse approximation in a given dictionary @math , where all but @math entries of @math are zero. The vector @math can be found by solving the sparse approximation problem where @math denotes the number of nonzero entries of @math . This combinatorial optimization problem can be solved efficiently with methods from the area of compressive sensing, e.g., the orthogonal matching pursuit (OMP) algorithm @cite_8 or iterative hard thresholding (IHT) @cite_28 .
@cite_28 @cite_20 @cite_8
1999640431 2339667469 2133105246 1632803383
Covariance matrices from stationary time series are Toeplitz. Multichannel and multidimensional processes have covariance matrices of block Toeplitz form. In these cases and many other situations, one knows that the actual covariance matrix belongs to a particular subclass of covariance matrices. This paper discusses a method for estimating a covariance matrix of specified structure from vector samples of the random process. The theoretical foundation of the method is to assume that the random process is zero-mean multivariate Gaussian, and to find the maximum-likelihood covariance matrix that has the specified structure. An existence proof is given and the solution is interpreted in terms of a minimum-entropy principle. The necessary gradient conditions that must be satisfied by the maximum-likelihood solution are derived and unique and nonunique analytic solutions for some simple problems are presented. A major contribution of this paper is an iterative algorithm that solves the necessary gradient equations for moderate-sized problems with reasonable computational ease. Theoretical convergence properties of the basic algorithm are investigated and robust modifications discussed. In doing maximum-entropy spectral analysis of a sine wave in white noise from a single vector sample, this new estimation procedure causes no splitting of the spectral line in contrast to the Burg technique. We propose an efficient open-loop channel estimator for a millimeter-wave (mm-wave) hybrid multiple-input multiple-output (MIMO) system consisting of radio-frequency (RF) beamformers with large antenna arrays followed by a baseband MIMO processor. A sparse signal recovery problem exploiting the sparse nature of mm-wave channels is formulated for channel estimation based on the parametric channel model with quantized angles of departures arrivals (AoDs AoAs), called the angle grids. The problem is solved by the orthogonal matching pursuit (OMP) algorithm employing a redundant dictionary consisting of array response vectors with finely quantized angle grids. We suggest the use of non-uniformly quantized angle grids and show that such grids reduce the coherence of the redundant dictionary. The lower and upper bounds of the sum-of-squared errors of the proposed OMP-based estimator are derived analytically: the lower bound is derived by considering the oracle estimator that assumes the knowledge of AoDs AoAs, and the upper bound is derived based on the results of the OMP performance guarantees. The design of training vectors (or sensing matrix) is particularly important in hybrid MIMO systems, because the RF beamformer prevents the use of independent and identically distributed random training vectors, which are popular in compressed sensing. We design training vectors so that the total coherence of the equivalent sensing matrix is minimized for a given RF beamforming matrix, which is assumed to be unitary. It is observed that the estimation accuracy can be improved significantly by randomly permuting the columns of the RF beamforming matrix. The simulation results demonstrate the advantage of the proposed OMP with a redundant dictionary over the existing methods such as the least squares method and the OMP based on the virtual channel model. Statistical inference and information processing of high-dimensional data often require an efficient and accurate estimation of their second-order statistics. With rapidly changing data, limited processing power and storage at the acquisition devices, it is desirable to extract the covariance structure from a single pass over the data and a small number of stored measurements. In this paper, we explore a quadratic (or rank-one) measurement model which imposes minimal memory requirements and low computational complexity during the sampling process, and is shown to be optimal in preserving various low-dimensional covariance structures. Specifically, four popular structural assumptions of covariance matrices, namely, low rank, Toeplitz low rank, sparsity, jointly rank-one and sparse structure, are investigated, while recovery is achieved via convex relaxation paradigms for the respective structure. The proposed quadratic sampling framework has a variety of potential applications, including streaming data processing, high-frequency wireless communication, phase space tomography and phase retrieval in optics, and noncoherent subspace detection. Our method admits universally accurate covariance estimation in the absence of noise, as soon as the number of measurements exceeds the information theoretic limits. We also demonstrate the robustness of this approach against noise and imperfect structural assumptions. Our analysis is established upon a novel notion called the mixed-norm restricted isometry property (RIP- @math ), as well as the conventional RIP- @math for near-isotropic and bounded measurements. In addition, our results improve upon the best-known phase retrieval (including both dense and sparse signals) guarantees using PhaseLift with a significantly simpler approach. The tremendous bandwidth available in the millimeter wave frequencies above 10 GHz have made these bands an attractive candidate for next-generation cellular systems. However, reliable communication at these frequencies depends critically on beamforming with very high-dimensional antenna arrays. Estimating the channel sufficiently accurately to perform beamforming can be challenging due to both low coherence time and a large number of antennas. Also, the measurements used for channel estimation may need to be made with analog beamforming, where the receiver can “look” in only one direction at a time. This paper presents a novel method for estimation of the receive-side spatial covariance matrix of a channel from a sequence of power measurements made in different angular directions. It is shown that maximum likelihood estimation of the covariance matrix reduces to a non-negative matrix completion problem. We show that the non-negative nature of the covariance matrix reduces the number of measurements required when the matrix is low-rank. The fast iterative methods are presented to solve the problem. Simulations are presented for both single-path and multi-path channels using models derived from real measurements in New York City at 28 GHz.