Channel Covariance Research Articles

Modeling of Wide-Band MIMO Radio Channels Based on NLoS Indoor Measurements

In this paper, we first verify a previously proposed Kronecker-structure-based narrow-band model for nonline-of-sight (NLoS) indoor multiple-input-multiple-output (MIMO) radio channels based on 5.2-GHz indoor MIMO channel measurements. It is observed that, for the narrow-band case, the measured channel coefficients are complex Gaussian distributed and, consequently, we focus on a statistical description using the first- and second-order moments of MIMO radio channels. It is shown that the MIMO channel covariance matrix can be well approximated by the Kronecker product of the covariance matrices, seen from the transmitter and receiver, respectively. A narrow-band model for NLoS indoor MIMO channels is thus verified by these results. As for the wide-band case, it is observed that the average power-delay profile of each element of the channel impulse response matrix fits the exponential decay curve and that the Kronecker structure of the second-order moments can be extended to each channel tap. A wide-band MIMO channel model is then proposed, combining a simple COST 259 single-input-single-output channel model and the Kronecker structure. Monte Carlo simulations are used to generate indoor MIMO channel realizations according to the models discussed. The results are compared with the measured data using the channel capacity and good agreement is found.

Remarks on Additivity of the Holevo Channel Capacity and of the Entanglement of Formation

The purpose of these notes is to discuss the relation between the additivity questions regarding the quantities (Holevo) capacity of a quantum channel T and entanglement of formation of a given bipartite state. In particular, using the Stinespring dilation theorem, we give a formula for the channel capacity involving entanglement of formation. This can be used to show that additivity of the latter for some states can be inferred from the additivity of capacity for certain channels. We demonstrate this connection for a family of group--covariant channels, allowing us to calculate the entanglement cost for many states, including some where a strictly smaller upper bound on the distillable entanglement is known. Group symmetry is used for more sophisticated analysis, giving formulas valid for a class of channels. This is presented in a general framework, extending recent findings of Vidal, Dur and Cirac (e-print quant-ph/0112131). We speculate on a general relation of superadditivity of the entanglement of formation, which would imply both the general additivity of this function under tensor products and of the Holevo capacity (with or without linear cost constraints).

MIMO Capacity Through Correlated Channels in the Presence of Correlated Interferers and Noise: A (Not So) Large N Analysis

The use of multiple-antenna arrays in both transmission and reception promises huge increases in the throughput of wireless communication systems. It is therefore important to analyze the capacities of such systems in realistic situations, which may include spatially correlated channels and correlated noise, as well as correlated interferers with known channel at the receiver. Here, we present an approach that provides analytic expressions for the statistics, i.e., the moments of the distribution, of the mutual information of multiple-antenna systems with arbitrary correlations, interferers, and noise. We assume that the channels of the signal and the interference are Gaussian with arbitrary covariance. Although this method is valid formally for large antenna numbers, it produces extremely accurate results even for arrays with as few as two or three antennas. We also develop a method to analytically optimize over the input signal covariance, which enables us to calculate analytic capacities when the transmitter has knowledge of the statistics of the channel (i.e., the channel covariance). In many cases of interest, this capacity is very close to the full closed-loop capacity, in which the transmitter has instantaneous channel knowledge. We apply this analytic approach to a number of examples and we compare our results with simulations to establish the validity of this approach. This method provides a simple tool to analyze the statistics of throughput for arrays of any size. The emphasis of this paper is on elucidating the novel mathematical methods used.

Optimizing MIMO antenna systems with channel covariance feedback

We consider a narrowband point-to-point communication system with n/sub T/ transmitters and n/sub R/ receivers. We assume the receiver has perfect knowledge of the channel, while the transmitter has no channel knowledge. We consider the case where the receiving antenna array has uncorrelated elements, while the elements of the transmitting array are arbitrarily correlated. Focusing on the case where n/sub T/=2, we derive simple analytic expressions for the ergodic average and the cumulative distribution function of the mutual information for arbitrary input (transmission) signal covariance. We then determine the ergodic and outage capacities and the associated optimal input signal covariances. We thus show how a transmitter with covariance knowledge should correlate its transmissions to maximize throughput. These results allow us to derive an exact condition (both necessary and sufficient) that determines when beamforming is optimal for systems with arbitrary number of transmitters and receivers.

Optimal Transmission with Imperfect Channel State Information at theTransmit Antenna Array

We study the optimal transmission strategy of a multiple-input single-output wireless communication link. The receiver has perfect channel state information while the transmitter has only long-term channel state information in terms of the channel covariance matrix. It was recently shown that the optimal eigenvectors of the transmit covariance matrix correspond with the eigenvalues of the channel covariance matrix. However, the optimal eigenvalues are difficult to compute. We study the properties of these optimal capacity achieving eigenvalues, and present a necessary and sufficient condition for the optimal eigenvalues of the transmit covariance matrix. Furthermore, we develop a necessary and sufficient condition for achieving capacity when transmitting in all directions. We compare the capacity gain of an optimal diversity system with a system which works with beamforming, and we derive an upper bound. We answer the main questions regarding the system design using the developed results. Additionally, we show in which way the multiplexing gain can be computed in case the channel covariance matrix is given. We compute the maximum number of required parallel data streams, and we define a multiplexing function in order to obtain a measure for the available multiplexing gain. Furthermore, we show that the capacity gain is small considering the additional complexity at the receiver. We illustrate all results by numerical simulations.

On the asymptotic eigenvalue distribution of concatenated vector-valued fading channels

The linear vector-valued channel x |/spl rarr/ /spl Pi//sub n/ M/sub n/x + z with z and M/sub n/ denoting additive white Gaussian noise and independent random matrices, respectively, is analyzed in the asymptotic regime as the dimensions of the matrices and vectors involved become large. The asymptotic eigenvalue distribution of the channel's covariance matrix is given in terms of an implicit equation for its Stieltjes transform as well as an explicit expression for its moments. Additionally, almost all eigenvalues are shown to converge toward zero as the number of factors grows over all bounds. This effect cumulates the total energy in a vanishing number of dimensions. The channel model addressed generalizes the model introduced Muller (see IEEE Trans. Inform. Theory) for communication via large antenna arrays to N-fold scattering per propagation path. As a byproduct, the multiplicative free convolution is shown to extend to a certain class of asymptotically large non-Gaussian random covariance matrices.

Performance analysis of a deterministic channel estimator for block transmission systems with null guard intervals

A deterministic algorithm was proposed for channel identification in block communication systems. The method assumed that the channel has a finite impulse response (FIR) and that null guard intervals of length greater than the channel order are inserted between successive blocks to prevent interblock interference and allow block synchronization. In the absence of noise, the algorithm provides error-free channel estimates, using a finite number of received data, without requiring training sequences and without imposing a restriction neither on the channel, except for finite order and time invariance, nor on the symbol constellation. Using small perturbation analysis, we derive approximate expressions of the estimated channel covariance matrix, which are used to quantify the resilience of the estimation algorithm to additive noise and channel fluctuations. Specifically, we consider channel fluctuations induced by transmitter/receiver relative motion, asynchronism, and oscillators' phase noise. We also compare the channel estimation accuracy with the Cramer-Rao bound (CRB) and prove that our estimation method is statistically efficient at practical SNR values for any data block length. Finally, we validate our theoretical analysis with simulations and compare our transmission scheme with an alternative system using training sequences for channel estimation.

Downlink channel covariance matrix (DCCM) estimation and its applications in wireless DS-CDMA systems

The downlink channel covariance matrix (DCCM) is of vital importance in determining downlink beamforming weights for base station (BS) antenna array systems. For the frequency-division-duplex (FDD) mode, DCCM is difficult to obtain due to a lack of direct measurement of downlink channel responses. In this paper, a novel technique is proposed for estimating DCCM using uplink channel responses only, which does not need direction-of-arrival (DOA) estimation and its association. The downlink beamforming scheme is then proposed for wireless DS-CDMA systems, using the obtained DCCM information together with the so-called virtual uplink beamforming and power control technique. Computer simulations show that using the BS antenna array together with this new beamforming technique can provide larger system capacity than traditional DOA-based approaches, which just direct the main beam toward the desired user.

Spatial frequency tuned covariance channels for red–green and luminance-modulated gratings: psychophysical data from human adults

Both chromatic and luminance-modulated stimuli are served by multiple spatial-frequency-tuned channels. This experiment investigated the independence versus interdependence of spatial frequency channels that serve the detection of red–green chromatic versus yellow–black luminance-modulated stimuli at low spatial frequencies. Contrast thresholds for both chromatic and luminance-modulated gratings were measured within 12 individual subjects using a repeated-measures design. Spatial frequencies ranged from 0.27 to 2.16 c/deg. A covariance structure analysis of individual differences was applied to the data. We computed statistical sources of individual variability, used them to define covariance channels, and determined the number and frequency tuning of these channels. For luminance-modulated gratings, two covariance channels were found, including one above and one below 1 c/deg [cf. Peterzell, & Teller (1996). Individual differences in contrast sensitivity functions: the coarsest spatial pattern analyzer. Vision Research, 36, 3077–3085]. For chromatic gratings, correlations between thresholds for most spatial frequencies were uniformly high, yielding a single covariance channel covering all but the highest spatial frequency tested. A combined analysis of both data sets recovered the same three covariance channels, and showed that detection thresholds for low-frequency red–green chromatic and luminance-modulated stimuli are served by separate, statistically independent processes.

Spatial frequency tuned covariance channels for red–green and luminance-modulated gratings: psychophysical data from human infants

This study concerns the spatial-frequency-tuned channels underlying infants’ contrast sensitivity functions (CSFs) for red–green chromatic stimuli, and their relationship to the channels underlying infants’ CSFs for luminance-modulated stimuli. Behavioral (forced-choice preferential-looking) techniques and stationary stimuli were used. In experiment 1, contrast thresholds were measured in 4- and 6-month-olds, using isoluminant red–green gratings with spatial frequencies ranging from 0.27 to 1.53 c/deg. In experiment 2, contrast thresholds were measured in 4-month-olds, using both red–green and luminance-modulated gratings in the same low spatial frequency range. Covariance analyses of individual differences were performed. Experiment 1 revealed one dominant covariance channel for the detection of red–green gratings, with a second channel contributing to detection of the highest spatial frequencies used. Experiment 2 revealed two to three channels serving color and luminance; but surprisingly these channels were not statistically separable for luminance versus chromatic stimuli. Thus, covariance channels for color and luminance that are independent for adults [Peterzell & Teller (2000). Spatial frequency tuned covariance channels for red–green and luminance-modulated gratings: psychophysical data from human adults. Vision Research, 40, 417–430] are apparently interdependent in infants. These data suggest that for infants, detection thresholds for chromatic and luminance-modulated stimuli may be limited by common mechanisms.

Covariance matrix properties and multiple symbol/soft decision detection in slow flat Rician fading

For a small number of symbols N and slow flat fading channels, it is shown that covariance matrices encountered in practice have two nonnegligible eigenvalues, the first much larger than the second, with a symmetric eigenvector associated with the first eigenvalue, and a skew symmetric eigenvector associated with the second eigenvalue. The first eigenvector is well approximated by a conditional mean, and the second eigenvector represents a small drift about the mean. The eigenvalues and eigenvectors of the slow flat fading channel covariance matrix are shown to be strongly related to those of a certain conditional covariance matrix. The maximum likelihood (ML) rules for block hard decision and symbol-by-symbol hard decision, and a rule for soft decision detection of M-DPSK, all using multiple symbol information, are obtained for the Rician channel as a function of N. The eigenvalue-eigenvector results lead to practical implementations of all rules. For small to moderate N, it is shown that a simple open-loop algorithm, of complexity N log N, attains the performance of the ML decision rules for an E/sub s//N/sub 0/ range of interest for several land mobile satellite systems. The ML decision rules are seen to give rapidly diminishing returns as N increases, showing that simple noncoherent techniques can have very effective performance for the Rician fading channel. Lastly, several conclusions are drawn about the asymptotic channel behavior, including the Rayleigh channel. The work is directly applicable to the Australian and North American land mobile satellite systems. >

Joint design of block source codes and modulation signal sets

The problem of designing block source codes and modulation signal sets that are both energy and bandwidth constrained is considered. For the class of linear estimator-based decoders, necessary conditions for optimality for the encoder, decoder and modulation signal set are derived. An algorithm that iteratively solves these necessary conditions to converge to a locally optimum solution has been developed. By studying the performance of the previous class of digital communication systems in the limit of infinite encoding rates, it is demonstrated that the MSE of a bandwidth and energy constrained digital system is bounded from below by that of a block pulse amplitude modulation system. This bound is readily computable in terms of the eigenvalues of the source and channel covariance matrices. The results indicate that for a correlated source, a sufficiently noisy channel and specific source block sizes and bandwidths, the digital system performance coincides with the optimum performance theoretically attainable. Further, significant performance improvements over the standard VQ-based system are demonstrated when the channel is noisy. >

Vector coding for partial response channels

A linear technique for combining equalization and coset codes on partial response channels with additive white Gaussian noise is developed. The technique, vector coding, uses a set of transmit filters or 'vectors' to partition the channel into an independent set of parallel intersymbol interference (ISI)-free channels for any given finite (or infinite) block length. The optimal transmit vectors for such channel partitioning are shown to be the eigenvectors of the channel covariance matrix for the specified block length, and the gains of the individual channels are the eigenvalues. An optimal bit allocation and energy distribution, are derived for the set of parallel channels, under an accurate extension of the continuous approximation for power in optimal multidimensional signal sets for constellations with unequal signal spacing in different dimensions. Examples are presented that demonstrate performance advantages with respect to zero-forcing decision feedback methods that use the same coset code on the same partial response channel. Only resampling the channel at an optimal rate and assuming no errors in the feedback path will bring the performance of the decision feedback methods up to the level of the vector coded system.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>