Björn E. Ottersten

On the complexity of sphere decoding in digital communications

Sphere decoding has been suggested by a number of authors as an efficient algorithm to solve various detection problems in digital communications. In some cases, the algorithm is referred to as an algorithm of polynomial complexity without clearly specifying what assumptions are made about the problem structure. Another claim is that although worst-case complexity is exponential, the expected complexity of the algorithm is polynomial. Herein, we study the expected complexity where the problem size is defined to be the number of symbols jointly detected, and our main result is that the expected complexity is exponential for fixed signal-to-noise ratio (SNR), contrary to previous claims. The sphere radius, which is a parameter of the algorithm, must be chosen to ensure a nonvanishing probability of solving the detection problem. This causes the exponential complexity since the squared radius must grow linearly with problem size. The rate of linear increase is, however, dependent on the noise variance, and thus, the rate of the exponential function is strongly dependent on the SNR. Therefore sphere decoding can be efficient for some SNR and problems of moderate size, even though the number of operations required by the algorithm strictly speaking always grows as an exponential function of the problem size.

Sensor array processing based on subspace fitting

Algorithms for estimating unknown signal parameters from the measured output of a sensor array are considered in connection with the subspace fitting problem. The methods considered are the deterministic maximum likelihood method (ML), ESPRIT, and a recently proposed multidimensional signal subspace method. These methods are formulated in a subspace-fitting-based framework, which provides insight into their algebraic and asymptotic relations. It is shown that by introducing a specific weighting matrix, the multidimensional signal subspace method can achieve the same asymptotic properties as the ML method. The asymptotic distribution of the estimation error is derived for a general subspace weighting, and the weighting that provides minimum variance estimates is identified. The resulting optimal technique is termed the weighted subspace fitting (WSF) method. Numerical examples indicate that the asymptotic variance of the WSF estimates coincides with the Cramer-Rao bound. The performance improvement compared to the other techniques is found to be most prominent for highly correlated signals. >

Combining beamforming and orthogonal space-time block coding

Multiple transmit and receive antennas can be used in wireless systems to achieve high data rate communication. Efficient space-time codes have been developed that utilize a large portion of the available capacity. These codes are designed under the assumption that the transmitter has no knowledge about the channel. In this work, on the other hand, we consider the case when the transmitter has partial, but not perfect, knowledge about the channel and how to improve a predetermined code so that this fact is taken into account. A performance criterion is derived for a frequency-nonselective fading channel and then utilized to optimize a linear transformation of the predetermined code. The resulting optimization problem turns out to be convex and can thus be efficiently solved using standard methods. In addition, a particularly efficient solution method is developed for the special case of independently fading channel coefficients. The proposed transmission scheme combines the benefits of conventional beamforming with those given by orthogonal space-time block coding. Simulation results for a narrow-band system with multiple transmit antennas and one or more receive antennas demonstrate significant gains over conventional methods in a scenario with nonperfect channel knowledge.

Detection and estimation in sensor arrays using weighted subspace fitting

The problem of signal parameter estimation of narrowband emitter signals impinging on an array of sensors is addressed. A multidimensional estimation procedure that applies to arbitrary array structures and signal correlation is proposed. The method is based on the recently introduced weighted subspace fitting (WSF) criterion and includes schemes for both detecting the number of sources and estimating the signal parameters. A Gauss-Newton-type method is presented for solving the multidimensional WSF and maximum-likelihood optimization problems. The global and local properties of the search procedure are investigated through computer simulations. Most methods require knowledge of the number of coherent/noncoherent signals present. A scheme for consistently estimating this is proposed based on an asymptotic analysis of the WSF cost function. The performance of the detection scheme is also investigated through simulations. >

Propagation delay estimation in asynchronous direct-sequence code-division multiple access systems

In an asynchronous direct-sequence code-division multiple access (DS-CDMA) communication system, the parameter estimation problem, i.e., estimating the propagation delay, attenuation and phase shift of each users transmitted signal, may be complicated by the so-called near-far problem. The near-far problem occurs when the amplitudes of the users received signals are very dissimilar, as the case might be in many important applications. In particular, the standard method for estimating the propagation delays will fail in a near-far situation. Several new estimators, the maximum likelihood, an approximative maximum likelihood and a subspace-based estimator, are therefore proposed and are shown to be robust against the near-far problem. No knowledge of the transmitted bits is assumed, and the proposed estimators can thus be used for both acquisition and tracking. In addition, the Cramer-Rao bound is derived for the parameter estimation problem.

Convex Optimization-Based Beamforming

In this article, an overview of advanced convex optimization approaches to multisensor beamforming is presented, and connections are drawn between different types of optimization-based beamformers that apply to a broad class of receive, transmit, and network beamformer design problems. It is demonstrated that convex optimization provides an indispensable set of tools for beamforming, enabling rigorous formulation and effective solution of both long-standing and emerging design problems.

Estimation of nominal direction of arrival and angular spread using an array of sensors

The problem of estimating the nominal direction of arrival and angular spread of a source surrounded by a large number of local scatterers using an array of sensors is addressed. This type of propagation occurs frequently in, for example, mobile communications. The maximum likelihood estimator is considered and two computationally less complex estimators are also proposed. They are based on least-squares fits of the sample covariance to the theoretical covariance matrix derived from the measurement model. The performance of the least-squares-based algorithm is analyzed and based on this, an optimally weighted least-squares criterion is proposed. The weighted least-squares algorithm is shown to be asymptotically efficient. Results of numerical experiments are presented to indicate small sample behavior of the estimators. The nominal direction-of-arrival (DOA) estimates are compared with those provided by a standard subspace algorithm. Finally, the methods are applied to experimental array data to determine spread angles for non line of sight scenarios.

Exact and Large Sample ML Techniques for Parameter Estimation and Detection in Array Processing

Sensor array signal processing deals with the problem of extracting information from a collection of measurements obtained from sensors distributed in space. The number of signals present is assumed to be finite, and each signal is parameterized by a finite number of parameters. Based on measurements of the array output, the objective is to estimate the signals and their parameters. This research area has attracted considerable interest for several years. A vast number of algorithms has appeared in the literature for estimating unknown signal parameters from the measured output of a sensor array.

Cooperative Multicell Precoding: Rate Region Characterization and Distributed Strategies With Instantaneous and Statistical CSI

Base station cooperation is an attractive way of increasing the spectral efficiency in multiantenna communication. By serving each terminal through several base stations in a given area, intercell interference can be coordinated and higher performance achieved, especially for terminals at cell edges. Most previous work in the area has assumed that base stations have common knowledge of both data dedicated to all terminals and full or partial channel state information (CSI) of all links. Herein, we analyze the case of distributed cooperation where each base station has only local CSI, either instantaneous or statistical. In the case of instantaneous CSI, the beamforming vectors that can attain the outer boundary of the achievable rate region are characterized for an arbitrary number of multiantenna transmitters and single-antenna receivers. This characterization only requires local CSI and justifies distributed precoding design based on a novel virtual signal-to-interference noise ratio (SINR) framework, which can handle an arbitrary SNR and achieves the optimal multiplexing gain. The local power allocation between terminals is solved heuristically. Conceptually, analogous results for the achievable rate region characterization and precoding design are derived in the case of local statistical CSI. The benefits of distributed cooperative transmission are illustrated numerically, and it is shown that most of the performance with centralized cooperation can be obtained using only local CSI.

Robust Cognitive Beamforming With Bounded Channel Uncertainties

This paper studies the robust beamforming design for a multi-antenna cognitive radio (CR) network, which transmits to multiple secondary users (SUs) and coexists with a primary network of multiple users. We aim to maximize the minimum of the received signal-to-interference-plus-noise ratios (SINRs) of the SUs, subject to the constraints of the total SU transmit power and the received interference power at the primary users (PUs) by optimizing the beamforming vectors at the SU transmitter based on imperfect channel state information (CSI). To model the uncertainty in CSI, we consider a bounded region for both cases of channel matrices and channel covariance matrices. As such, the optimization is done while satisfying the interference constraints for all possible CSI error realizations. We shall first derive equivalent conditions for the interference constraints and then convert the problems into the form of semi-definite programming (SDP) with the aid of rank relaxation, which leads to iterative algorithms for obtaining the robust optimal beamforming solution. Results demonstrate the achieved robustness and the performance gain over conventional approaches and that the proposed algorithms can obtain the exact robust optimal solution with high probability.