The large-system performance of maximum-a-posterior estimation is studied considering a general distortion function when the observation vector is received through a linear system with additive white Gaussian noise. The analysis considers the system matrix to be chosen from the large class of rotationally invariant random matrices. We take a statistical mechanical approach by introducing a spin glass corresponding to the estimator, and employing the replica method for the large-system analysis. In contrast to earlier replica based studies, our analysis evaluates the general replica ansatz of the corresponding spin glass and determines the asymptotic distortion of the estimator for any structure of the replica correlation matrix. Consequently, the replica symmetric as well as the replica symmetry breaking ansatz with b steps of breaking is deduced from the given general replica ansatz. The generality of our distortion function lets us derive a more general form of the maximum-a-posterior decoupling principle. Based on the general replica ansatz, we show that for any structure of the replica correlation matrix, the vector-valued system decouples into a bank of equivalent decoupled linear systems followed by maximum-a-posterior estimators. The structure of the decoupled linear system is further studied under both the replica symmetry and the replica symmetry breaking assumptions. For b steps of symmetry breaking, the decoupled system is found to be an additive system with a noise term given as the sum of an independent Gaussian random variable with b correlated impairment terms. The general decoupling property of the maximum-a-posterior estimator leads to the idea of a replica simulator which represents the replica ansatz through the state evolution of a transition system described by its corresponding decoupled system. As an application of our study, we investigate large compressive sensing systems by considering the ℓp norm minimization recovery schemes. Our numerical investigations show that the replica symmetric ansatz for ℓ0 norm recovery fails to give an accurate approximation of the mean square error as the compression rate grows, and therefore, the replica symmetry breaking ansätze are needed in order to assess the performance precisely. Index TermsMaximum-a-posterior estimation, linear vector channel, decoupling principle, equivalent single-user system, compressive sensing, zero norm, replica method, statistical physics, replica symmetry breaking, replica simulatorThe results of this manuscript were presented in parts at 2016 IEEE Information Theory Workshop (ITW) [78] and 2017 IEEE Information Theory and Applications Workshop (ITA) [79]. ). 2 then, expresses the large-system performance regarding the distortion function d(·; ·). The performance analysis of the estimator requires (2) to be explicitly computed, and then,x = g(y|A) substituted in the distortion function.This task, however, is not trivial for many choices of the utility function and the source support , and becomes unfeasible as n grows large....
This paper proposes the nonlinear Least Square Error (LSE) precoders for multiuser MIMO broadcast channels. The output signals of LSE Precoders are limited to be chosen from a predefined set which let these precoders address several constraints such as peak power limitation, constant envelope transmission and discrete constellations. We study the large-system performance of these precoders via the replica method from statistical physics, and derive a closed-form expression for the asymptotic distortion. Our results demonstrate that an LSE precoder with the output peak-to-average power ratio of 3 dB can track the performance of the Regularized Zero Forcing (RZF) precoder closely. As the peakto-average power ratio reduces to one, the constant envelope precoder is recovered. The investigations depict that the performance of the RZF precoder is achieved by the constant envelope precoder with 20% of more transmit antennas. For M -PSK constellations, our analysis gives a lower-bound on the asymptotic distortion which is tight for moderate antenna-to-user ratios and deviates as the ratio grows.We improve this bound by deriving the replica solution under one-step of replica symmetry breaking.Our numerical investigations for this case show that the bound is tight for antenna-to-user ratios less than 5.
Abstract-The large-system decoupling property of a MAP estimator is studied when it estimates the i.i.d. vector x from the observation y = Ax + z with A being chosen from a wide range of matrix ensembles, and the noise vector z being i.i.d. and Gaussian. Using the replica method, we show that the marginal joint distribution of any two corresponding input and output symbols converges to a deterministic distribution which describes the input-output distribution of a single user system followed by a MAP estimator. Under the bRSB assumption, the single user system is a scalar channel with additive noise where the noise term is given by the sum of an independent Gaussian random variable and b correlated interference terms. As the bRSB assumption reduces to RS, the interference terms vanish which results in the formerly studied RS decoupling principle.
This paper proposes the class of Generalized Least-Square-Error (GLSE) precoders for multiuser massive MIMO systems. For a generic transmit constellation, GLSE precoders minimize the interference at user terminals assuring that given constraints on the transmit signals are satisfied. The general form of these precoders enables us to impose multiple restrictions at the transmit signal such as limited peak power and restricted number of active transmit antennas. Invoking the replica method from statistical mechanics, we study the performance of GLSE precoders in the large-system limit. We show that the output symbols of these precoders are identically distributed and their statistics are described with an equivalent scalar GLSE precoder. Using the asymptotic results, we further address some applications of the GLSE precoders; namely forming transmit signals over a restricted alphabet and transmit antenna selection. Our investigations demonstrate that a computationally efficient GLSE precoder requires 41% fewer active transmit antennas than conventional selection protocols in order to achieve a desired level of input-output distortion. Index TermsPrecoding, massive MIMO, transmit antenna selection, peak-to-average power ratio, replica method, general replica solution. Parts of this work have been presented in 21th International ITG Workshop on Smart Antennas (WSA) and 2017 IEEE International Symposium on Information Theory (ISIT) [1], [2].This is the extended version of the paper submitted to IEEE Transactions on Wireless Communications.
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