arXiv subjects

Brian Rider

Publications and source records attributed to Brian Rider.

30 records · Page 2Linked to original sources

arXiv2007

How Many Users should be Turned On in a Multi-Antenna Broadcast Channel?

This paper considers broadcast channels with L antennas at the base station and m single-antenna users, where each user has perfect channel knowledge and the base station obtains channel information through a finite rate feedback. The key observation of this paper is that the optimal number of on-users (users turned on), say s, is a function of signal-to-noise ratio (SNR) and other system parameters. Towards this observation, we use asymptotic analysis to guide the design of feedback and transmission strategies. As L, m and the feedback rates approach infinity linearly, we derive the asymptotic optimal feedback strategy and a realistic criterion to decide which users should be turned on. Define the corresponding asymptotic throughput per antenna as the spatial efficiency. It is a function of the number of on-users s, and therefore, s should be appropriately chosen. Based on the above asymptotic results, we also develop a scheme for a system with finite many antennas and users. Compared with other works where s is presumed constant, our scheme achieves a significant gain by choosing the appropriate s. Furthermore, our analysis and scheme is valid for heterogeneous systems where different users may have different path loss coefficients and feedback rates.

cs.IT↗

arXiv2007

Unequal dimensional small balls and quantization on Grassmann Manifolds

The Grassmann manifold G_{n,p}(L) is the set of all p-dimensional planes (through the origin) in the n-dimensional Euclidean space L^{n}, where L is either R or C. This paper considers an unequal dimensional quantization in which a source in G_{n,p}(L) is quantized through a code in G_{n,q}(L), where p and q are not necessarily the same. It is different from most works in literature where p\equiv q. The analysis for unequal dimensional quantization is based on the volume of a metric ball in G_{n,p}(L) whose center is in G_{n,q}(L). Our chief result is a closed-form formula for the volume of a metric ball when the radius is sufficiently small. This volume formula holds for Grassmann manifolds with arbitrary n, p, q and L, while previous results pertained only to some special cases. Based on this volume formula, several bounds are derived for the rate distortion tradeoff assuming the quantization rate is sufficiently high. The lower and upper bounds on the distortion rate function are asymptotically identical, and so precisely quantify the asymptotic rate distortion tradeoff. We also show that random codes are asymptotically optimal in the sense that they achieve the minimum achievable distortion with probability one as n and the code rate approach infinity linearly. Finally, we discuss some applications of the derived results to communication theory. A geometric interpretation in the Grassmann manifold is developed for capacity calculation of additive white Gaussian noise channel. Further, the derived distortion rate function is beneficial to characterizing the effect of beamforming matrix selection in multi-antenna communications.

cs.IT↗

arXiv2007

Effect of Finite Rate Feedback on CDMA Signature Optimization and MIMO Beamforming Vector Selection

We analyze the effect of finite rate feedback on CDMA (code-division multiple access) signature optimization and MIMO (multi-input-multi-output) beamforming vector selection. In CDMA signature optimization, for a particular user, the receiver selects a signature vector from a codebook to best avoid interference from other users, and then feeds the corresponding index back to the specified user. For MIMO beamforming vector selection, the receiver chooses a beamforming vector from a given codebook to maximize throughput, and feeds back the corresponding index to the transmitter. These two problems are dual: both can be modeled as selecting a unit norm vector from a finite size codebook to "match" a randomly generated Gaussian matrix. In signature optimization, the least match is required while the maximum match is preferred for beamforming selection. Assuming that the feedback link is rate limited, our main result is an exact asymptotic performance formulae where the length of the signature/beamforming vector, the dimensions of interference/channel matrix, and the feedback rate approach infinity with constant ratios. The proof rests on a large deviation principle over a random matrix ensemble. Further, we show that random codebooks generated from the isotropic distritution are asymptotically optimal not only on average, but also with probability one.

cs.IT↗

arXiv2006

Shape of the ground state energy density of Hill's equation with nice Gaussian potential

Consider Hill's operator Q = -D^2 + q(x) in which the potential q(x) is an almost surely continuous and rotation invariant Gaussian process on the circle of perimeter one. Viewing the classical Riccati map as a change of measure, we establish functional integral formulas for the probability density function of the ground state energy and also determine the density's shape.

math.PR↗

arXiv2006

On the shape of the ground state eigenvalue density of a random Hill's equation

Consider the Hill's operator $Q = - d^2/dx^2 + q(x)$ in which $q(x)$, $0 \le x \le 1$, is a White Noise. Denote by $f(μ)$ the probability density function of $-λ_0(q)$, the negative of the ground state eigenvalue, at $μ$. We describe the detailed asymptotics of this density as $μ\to +\infty$. This result is based on a precise Laplace analysis of a functional integral representation for $f(μ)$ established by S. Cambronero and H.P. McKean.

math.PR↗

arXiv2006

Complex determinantal processes and H1 noise

For the plane, sphere, and hyperbolic plane we consider the canonical invariant determinantal point processes with intensity rho dnu, where nu is the corresponding invariant measure. We show that as rho converges to infinity, after centering, these processes converge to invariant H1 noise. More precisely, for all functions f in the interesection of H1(nu) and L1(nu) the distribution of sum f(z) - rho/pi integral f dnu converges to Gaussian with mean 0 and variance given by ||f||_H1^2 / (4 pi).

math.PR↗

arXiv2006

The noise in the circular law and the Gaussian free field

Fill an n x n matrix with independent complex Gaussians of variance 1/n. As n approaches infinity, the eigenvalues {z_k} converge to a sum of an H^1-noise on the unit disk and an independent H^{1/2}-noise on the unit circle. More precisely, for C^1 functions of suitable growth, the distribution of sum_{k=1}^n (f(z_k)-E f(z_k)) converges to that of a mean-zero Gaussian with variance given by the sum of the squares of the disk H^1 and the circle H^{1/2} norms of f. Moreover, with p_n the characteristic polynomial, log|p_n|- E log|p_n| tends to the planar Gaussian free field conditioned to be harmonic outside the unit disk. Finally, for polynomial test functions f, we prove that the limiting covariance structure is universal for a class of models including Haar distributed unitary matrices.

math.PR↗

arXiv2006

Quantization Bounds on Grassmann Manifolds and Applications to MIMO Communications

This paper considers the quantization problem on the Grassmann manifold \mathcal{G}_{n,p}, the set of all p-dimensional planes (through the origin) in the n-dimensional Euclidean space. The chief result is a closed-form formula for the volume of a metric ball in the Grassmann manifold when the radius is sufficiently small. This volume formula holds for Grassmann manifolds with arbitrary dimension n and p, while previous results pertained only to p=1, or a fixed p with asymptotically large n. Based on this result, several quantization bounds are derived for sphere packing and rate distortion tradeoff. We establish asymptotically equivalent lower and upper bounds for the rate distortion tradeoff. Since the upper bound is derived by constructing random codes, this result implies that the random codes are asymptotically optimal. The above results are also extended to the more general case, in which \mathcal{G}_{n,q} is quantized through a code in \mathcal{G}_{n,p}, where p and q are not necessarily the same. Finally, we discuss some applications of the derived results to multi-antenna communication systems.

cs.IT↗

arXiv2006

Performance Analysis of CDMA Signature Optimization with Finite Rate Feedback

We analyze the performance of CDMA signature optimization with finite rate feedback. For a particular user, the receiver selects a signature vector from a signature codebook to avoid the interference from other users, and feeds the corresponding index back to this user through a finite rate and error-free feedback link. We assume the codebook is randomly constructed where the entries are independent and isotropically distributed. It has been shown that the randomly constructed codebook is asymptotically optimal. In this paper, we consider two types of signature selection criteria. One is to select the signature vector that minimizes the interference from other users. The other one is to select the signature vector to match the weakest interference directions. By letting the processing gain, number of users and feedback bits approach infinity with fixed ratios, we derive the exact asymptotic formulas to calculate the average interference for both criteria. Our simulations demonstrate the theoretical formulas. The analysis can be extended to evaluate the signal-to-interference plus noise ratio performance for both match filter and linear minimum mean-square error receivers.

cs.IT↗

arXiv2006

On the Information Rate of MIMO Systems with Finite Rate Channel State Feedback Using Beamforming and Power On/Off Strategy

It is well known that Multiple-Input Multiple-Output (MIMO) systems have high spectral efficiency, especially when channel state information at the transmitter (CSIT) is available. When CSIT is obtained by feedback, it is practical to assume that the channel state feedback rate is finite and the CSIT is not perfect. For such a system, we consider beamforming and power on/off strategy for its simplicity and near optimality, where power on/off means that a beamforming vector (beam) is either turned on with a constant power or turned off. The main contribution of this paper is to accurately evaluate the information rate as a function of the channel state feedback rate. Name a beam turned on as an on-beam and the minimum number of the transmit and receive antennas as the dimension of a MIMO system. We prove that the ratio of the optimal number of on-beams and the system dimension converges to a constant for a given signal-to-noise ratio (SNR) when the numbers of transmit and receive antennas approach infinity simultaneously and when beamforming is perfect. Asymptotic formulas are derived to evaluate this ratio and the corresponding information rate per dimension. The asymptotic results can be accurately applied to finite dimensional systems and suggest a power on/off strategy with a constant number of on-beams. For this suboptimal strategy, we take a novel approach to introduce power efficiency factor, which is a function of the feedback rate, to quantify the effect of imperfect beamforming. By combining power efficiency factor and the asymptotic formulas for perfect beamforming case, the information rate of the power on/off strategy with a constant number of on-beams is accurately characterized.

cs.IT↗

arXiv2004

Concentration of permanent estimators for certain large matrices

Let A_n=(a_{ij})_{i,j=1}^n be an n\times n positive matrix with entries in [a,b], 0<a\le b. Let X_n=(\sqrta_{ij}x_{ij})_{i,j=1}^n be a random matrix, where {x_{ij}} are i.i.d. N(0,1) random variables. We show that for large n, \det (X_n^TX_n) concentrates sharply at the permanent of A_n, in the sense that n^{-1}\log (\det(X_n^TX_n)/perA_n)\to_{n\to\infty}0 in probability.

math.PR↗

arXiv2003

Deviations from the Circular Law

Consider Ginibre's ensemble of $N \times N$ non-Hermitian random matrices in which all entries are independent complex Gaussians of mean zero and variance $\frac{1}{N}$. As $N \uparrow \infty$ the normalized counting measure of the eigenvalues converges to the uniform measure on the unit disk in the complex plane. In this note we describe fluctuations about this {\em Circular Law}. First we obtain finite $N$ formulas for the covariance of certain linear statistics of the eigenvalues. Asymptotics of these objects coupled with a theorem of Costin and Lebowitz then result in central limit theorems for a variety of these statistics.

math.PR↗