Karl Werner

On Estimation of Covariance Matrices With Kronecker Product Structure

The estimation of signal covariance matrices is a crucial part of many signal processing algorithms. In some applications, the structure of the problem suggests that the underlying, true covariance matrix is the Kronecker product of two valid covariance matrices. Examples of such problems are channel modeling for multiple-input multiple-output (MIMO) communications and signal modeling of EEG data. In applications, it may also be that the Kronecker factors in turn can be assumed to possess additional, linear structure. The maximum-likelihood (ML) method for the associated estimation problem has been proposed previously. It is asymptotically efficient but has the drawback of requiring an iterative search for the maximum of the likelihood function. Two methods that are fast and noniterative are proposed in this paper. Both methods are shown to be asymptotically efficient. The first method is a noniterative variant of a well-known alternating maximization technique for the likelihood function. It performs on par with ML in simulations but has the drawback of not allowing for extra structure in addition to the Kronecker structure. The second method is based on covariance matching principles and does not suffer from this drawback. However, while the large sample performance is the same, it performs somewhat worse than the first estimator in small samples. In addition, the Cramer-Rao lower bound for the problem is derived in a compact form. The problem of estimating the Kronecker factors and the problem of detecting if the Kronecker structure is a good model for the covariance matrix of a set of samples are related. Therefore, the problem of detecting the dimensions of the Kronecker factors based on the minimum values of the criterion functions corresponding to the two proposed estimation methods is also treated in this work.

Estimating MIMO channel covariances from training data under the Kronecker model

Many algorithms for transmission in multiple input multiple output (MIMO) communication systems rely on second order statistics of the channel realizations. The problem of estimating such second order statistics of MIMO channels, based on limited amounts of training data, is treated in this article. It is assumed that the Kronecker model holds. This implies that the channel covariance is the Kronecker product of one covariance matrix that is associated with the array and the scattering at the transmitter and one that is associated with the receive array and the scattering at the receiver. The proposed estimator uses training data from a number of signal blocks (received during independent fades of the MIMO channel) to compute the estimate. This is in contrast to methods that assume that the channel realizations are directly available, or possible to estimate almost without error. It is also demonstrated how methods that make use of the training data indirectly via channel estimates can be biased. An estimator is derived that can, in an asymptotically optimal way, use, not only the structure implied by the Kronecker assumption, but also linear structure on the transmit- and receive covariance matrices. The performance of the proposed estimator is analyzed and numerical simulations illustrate the results and also provide insight into the small sample behaviour of the proposed method.

Antenna Configurations for 4x4 MIMO in LTE - Field Measurements

The 3GPP LTE standard for mobile broadband includes multi-antenna transmission modes that improve performance, both in terms of coverage, spectral efficiency and peak throughput. The antenna system design, both at the eNB and at the UE is critical to a well performing system; it should be designed with the intended performance profile in mind. Field trials were performed in order to investigate the relative performance of several four and two transmit antenna setups in an LTE system. In general, multi-antenna technology gave substantial performance gains over single antenna transmission. A closely spaced co-polarized configuration gave the best performance for users with poor channel quality while dual-polarized and well-spaced antenna configurations gave better performance for users with good channel quality. The trial also shows that UE antenna polarization is an important parameter that must be kept in mind when designing the eNB antenna system.

Reduced rank linear regression and weighted low rank approximations

This paper addresses parameter estimation in reduced rank linear regressions. This estimation problem has applications in several subject areas including system identification, sensor array processing, econometrics and statistics. A new estimation procedure, based on instrumental variable principles, is derived and analyzed. The proposed method is designed to handle noise that is both spatially and temporally autocorrelated. An asymptotical analysis shows that the proposed method outperforms previous methods when the noise is temporally correlated and that it is asymptotically efficient otherwise. A numerical study indicates that the performance is significantly improved also for finite sample set sizes. In addition, the Cramer-Rao lower bound (CRB) on unbiased estimator covariance for the data model is derived. A statistical test for rank determination is also developed. An important step in the new algorithm is the weighted low rank approximation (WLRA). As the WLRA lacks a closed form solution in its general form, two new, noniterative and approximate solutions are derived, both of them asymptotically optimal when part of the estimation procedure proposed here. These methods are also interesting in their own right since the WLRA has several applications.

DOA Estimation and Detection in Colored Noise Using Additional Noise-Only Data

In a typical array processing scenario, noise acting on the array can not be assumed spatially white. It is in many cases necessary to use quiet periods, when only noise is received, to estimate the noise covariance. If estimation of the signal parameters and noise covariance is performed jointly, performance can be improved. This is especially true when stationarity considerations limit the amount of available valid noise-only data. An asymptotically valid approximative maximum likelihood method (AML) for the estimation problem is derived in this work. The resulting criterion is, when concentrated with respect to the signal parameters, relatively simple. In numerical experiments, AML shows promising small-sample performance compared to earlier methods. The criterion function is also well suited for numerical optimization. The new criterion function allows for the development of a novel, MODE-like, non-iterative estimation procedure if the array belongs to the important class of uniform linear arrays. The resulting procedure retains the asymptotic properties of maximum likelihood, and numerical simulations indicate superior threshold performance when compared to an optimally weighted subspace fitting (WSF) formulation of MODE. For the detection problem, no method has been presented that takes the unknown noise covariance into account. Here, a well known detection scheme for WSF is extended to work in this scenario as well. The derivations of this scheme further stress the importance of using the correct weighting in WSF when the noise covariance is unknown. It is also shown that the minimum value of the criterion function associated with AML can be used for the detection purpose. Numerical experiments indicate very promising performance for the AML-detection scheme.

Optimal Utilization of Signal-Free Samples for Array Processing in Unknown Colored Noise Fields

Many algorithms for direction-of-arrival (DOA) estimation require the noise covariance matrix to be known or to possess a known structure. In many cases, the noise covariance is, in fact, estimated from separate measurements. This paper addresses the combined effects of finite sample sizes, both in the estimated noise covariance matrix and in the data with signals present. It is assumed that a batch of signal-free samples is available in addition to the signal-containing samples. No assumption is made on the structure of the noise covariance. In this paper, the asymptotic covariance of the weighted subspace fitting (WSF) algorithm is derived for the case in which the data are whitened using an estimated noise covariance. The expression obtained suggests an optimal weighting that improves performance compared to the standard choice. In addition, a new method based on covariance matching is proposed. Both methods are asymptotically statistically efficient. The Cramer-Rao lower bound (CRB) on the covariance of the estimate for the data model is also derived. Monte Carlo simulations show promising small sample performance for the two new methods and confirm the asymptotic results

LTE-A Field Measurements: 8x8 MIMO and Carrier Aggregation

The introduction of 8×8 MIMO and carrier aggregation in the 3GPP LTE Rel. 10 opens up for increased user throughput. The potential gains using these techniques have been evaluated in a field measurement campaign with a testbed implementation. A downlink throughput exceeding 1 Gbps has been achieved combining 8×8 MIMO in an outdoor macro scenario with carrier aggregation using three component carriers (3×20 MHz). The relation between the achievable throughput and the channel richness arising from the physical environment and antenna spacing was demonstrated. The performance of MIMO setups ranging from 1×2 up to 8×8 was evaluated in indoor-to-indoor, outdoor-to-indoor, and outdoor-to-outdoor deployments. It was observed that each added transmit or receive antenna increased the throughput. These gains were achieved with a compact UE antenna that is reasonable in size for implementation in a consumer device.

ML Estimation of Covariance Matrices with Kronecker and Persymmetric Structure

Estimation of covariance matrices is often an integral part in many signal processing algorithms. In some applications, the covariance matrices can be assumed to have certain structure. Imposing this structure in the estimation typically leads to improved accuracy and robustness (e.g., to small sample effects). In MIMO communications or in signal modelling of EEG data the full covariance matrix can sometimes be modelled as the Kronecker product of two smaller covariance matrices. These smaller matrices may also be structured, e.g., being Toeplitz or at least persymmetric. In this paper we discuss a recently proposed closed form maximum likelihood (ML) based method for the estimation of the Kronecker factor matrices. We also extend the previously presented method to be able to impose the persymmetric constraint into the estimator. Numerical examples show that the mean square errors of the new estimator attains the Cramér-Rao bound even for very small sample sizes.

MIMO Performance at 700MHz: Field Trials of LTE with Handheld UE

MIMO is one of the techniques used in LTE Release 8 to achieve very high data rates. A field trial was performed in a pre-commercial LTE network. The objective is to investigate how well MIMO works with realistically designed handhelds in band 13 (746-756 MHz in downlink). In total, three different handheld designs were tested using antenna mockups. In addition to the mockups, a reference antenna design with less stringent restrictions on physical size and excellent properties for MIMO was used. The trial comprised test drives in areas with different characteristics and with different network load levels. The effects of hands holding the devices and the effect of using the device inside a test vehicle were also investigated. In general, it is very clear from the trial that MIMO works very well and gives a substantial performance improvement at the tested carrier frequency if the antenna design of the hand-held is well made with respect to MIMO. In fact, the best of the handhelds performed similar to the reference antenna.

Kronecker Structured Covariance Matrix Estimation

The estimation of signal covariance matrices is a crucial part of many signal processing algorithms. In some applications, the structure of the problem suggests that the underlying, true, covariance matrix is the Kronecker product of two matrices. Examples of such problems are channel modelling for MIMO communications and signal modelling of EEG data. In applications it may also be that the Kronecker factors in turn can be assumed to possess additional, linear, structure. The maximum likelihood (ML) estimator for the problem has been proposed previously. It is asymptotically efficient but has the drawback of requiring an iterative search. Two methods that are both non-iterative and asymptotically efficient are proposed in this paper. The first method is derived from a well-known iterative maximization technique for the likelihood function. It performs on par with ML in simulations, but has the drawback of not allowing for extra structure in addition to the Kronecker structure. The second method is based on covariance matching principles, and does not suffer from this drawback. However, while the large sample performance is shown to be identical to ML, it performs somewhat worse in small samples than the first estimator. In addition, the Cramer-Rao lower bound (CRB) for the problem is derived in a compact form.