7,690 research outputs found
The MIMO Iterative Waterfilling Algorithm
This paper considers the non-cooperative maximization of mutual information
in the vector Gaussian interference channel in a fully distributed fashion via
game theory. This problem has been widely studied in a number of works during
the past decade for frequency-selective channels, and recently for the more
general MIMO case, for which the state-of-the art results are valid only for
nonsingular square channel matrices. Surprisingly, these results do not hold
true when the channel matrices are rectangular and/or rank deficient matrices.
The goal of this paper is to provide a complete characterization of the MIMO
game for arbitrary channel matrices, in terms of conditions guaranteeing both
the uniqueness of the Nash equilibrium and the convergence of asynchronous
distributed iterative waterfilling algorithms. Our analysis hinges on new
technical intermediate results, such as a new expression for the MIMO
waterfilling projection valid (also) for singular matrices, a mean-value
theorem for complex matrix-valued functions, and a general contraction theorem
for the multiuser MIMO watefilling mapping valid for arbitrary channel
matrices. The quite surprising result is that uniqueness/convergence conditions
in the case of tall (possibly singular) channel matrices are more restrictive
than those required in the case of (full rank) fat channel matrices. We also
propose a modified game and algorithm with milder conditions for the uniqueness
of the equilibrium and convergence, and virtually the same performance (in
terms of Nash equilibria) of the original game.Comment: IEEE Transactions on Signal Processing (accepted
Worst-Case Robust Distributed Power Allocation in Shared Unlicensed Spectrum
This paper considers non-cooperative and fully-distributed power-allocation
for selfish transmitter-receiver pairs in shared unlicensed spectrum when
normalized-interference to each receiver is uncertain. We model each uncertain
parameter by the sum of its nominal (estimated) value and a bounded additive
error in a convex set, and show that the allocated power always converges to
its equilibrium, called robust Nash equilibrium (RNE). In the case of a bounded
and symmetric uncertainty region, we show that the power allocation problem for
each user is simplified, and can be solved in a distributed manner. We derive
the conditions for RNE's uniqueness and for convergence of the distributed
algorithm; and show that the total throughput (social utility) is less than
that at NE when RNE is unique. We also show that for multiple RNEs, the social
utility may be higher at a RNE as compared to that at the corresponding NE, and
demonstrate that this is caused by users' orthogonal utilization of bandwidth
at RNE. Simulations confirm our analysis
Optimal Linear Precoding Strategies for Wideband Non-Cooperative Systems based on Game Theory-Part II: Algorithms
In this two-part paper, we address the problem of finding the optimal
precoding/multiplexing scheme for a set of non-cooperative links sharing the
same physical resources, e.g., time and bandwidth. We consider two alternative
optimization problems: P.1) the maximization of mutual information on each
link, given constraints on the transmit power and spectral mask; and P.2) the
maximization of the transmission rate on each link, using finite order
constellations, under the same constraints as in P.1, plus a constraint on the
maximum average error probability on each link. Aiming at finding decentralized
strategies, we adopted as optimality criterion the achievement of a Nash
equilibrium and thus we formulated both problems P.1 and P.2 as strategic
noncooperative (matrix-valued) games. In Part I of this two-part paper, after
deriving the optimal structure of the linear transceivers for both games, we
provided a unified set of sufficient conditions that guarantee the uniqueness
of the Nash equilibrium. In this Part II, we focus on the achievement of the
equilibrium and propose alternative distributed iterative algorithms that solve
both games. Specifically, the new proposed algorithms are the following: 1) the
sequential and simultaneous iterative waterfilling based algorithms,
incorporating spectral mask constraints; 2) the sequential and simultaneous
gradient projection based algorithms, establishing an interesting link with
variational inequality problems. Our main contribution is to provide sufficient
conditions for the global convergence of all the proposed algorithms which,
although derived under stronger constraints, incorporating for example spectral
mask constraints, have a broader validity than the convergence conditions known
in the current literature for the sequential iterative waterfilling algorithm.Comment: Paper submitted to IEEE Transactions on Signal Processing, February
22, 2006. Revised March 26, 2007. Accepted June 5, 2007. To appear on IEEE
Transactions on Signal Processing, 200
Asynchronous Channel Training in Multi-Cell Massive MIMO
Pilot contamination has been regarded as the main bottleneck in time division
duplexing (TDD) multi-cell massive multiple-input multiple-output (MIMO)
systems. The pilot contamination problem cannot be addressed with large-scale
antenna arrays. We provide a novel asynchronous channel training scheme to
obtain precise channel matrices without the cooperation of base stations. The
scheme takes advantage of sampling diversity by inducing intentional timing
mismatch. Then, the linear minimum mean square error (LMMSE) estimator and the
zero-forcing (ZF) estimator are designed. Moreover, we derive the minimum
square error (MSE) upper bound of the ZF estimator. In addition, we propose the
equally-divided delay scheme which under certain conditions is the optimal
solution to minimize the MSE of the ZF estimator employing the identity matrix
as pilot matrix. We calculate the uplink achievable rate using maximum ratio
combining (MRC) to compare asynchronous and synchronous channel training
schemes. Finally, simulation results demonstrate that the asynchronous channel
estimation scheme can greatly reduce the harmful effect of pilot contamination
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