Theoretical Analysis And Designs For Massive MIMO Systems Using Operator-valued Free Probability

To improve the user experience,and face the challenge of meeting the exponential growth of wireless data services and introducing new services,the next generation wireless networks will need to support high quality,high data rate,high user density,high mobility and low laten-cy communications.As one of the key technologies of the future wireless networks,massive multiple-input multiple-output(MIMO)technology has been widely investigated during the last few years.By employing hundreds of antennas at a base station(BS),massive MIMO systems provide huge potential performance enhancement,but it also poses difficult problems for system designs.In this dissertation,the theoretical research for massive MIMO is investigated based on operator-valued free probability theory.Firstly,the free deterministic equivalents of the random matrix polynomials are established based on operator-valued free probability,the distributions of the random matrix polynomials and their free deterministic equivalents are rigorously proved to be asymptotically the same,and the capacity analysis based on the free deterministic equivalents is proposed.The consid-ered random matrix polynomial consists of several square deterministic and random matrices with the same size.Starting from important definitions and results of free probability theory and operator-valued free probability theory,the free deterministic equivalent of a random ma-trix polynomial is investigated.First,by replacing the independent Gaussian entries with freely independent semicircular and circular elements,the free deterministic equivalent of the con-sidered random matrix polynomial is established.Then,when the deterministic matrices are diagonal matrices,the distribution of the considered random matrix polynomial is reproved to be asymptotically the same as that of its free deterministic equivalent,and the considered in-dependent random matrices are reproved to be asymptotically operator-valued free.Compared with the proof based on operator algebra in the literature,the proof in this paper is easier to understand.Furthermore,when the deterministic matrices are general matrices,the distribution of the considered random matrix polynomial is proved to be asymptotically the same as that of its free deterministic equivalent,and the considered independent random matrices and the de-terministic matrices are proved to be asymptotically operator-valued free.In MIMO systems,the channel Gram matrix can be seen as a random matrix polynomial,and its free deterministic equivalent can be used in deriving the deterministic equivalent of the ergodic channel mutual information.Based on an existing model from the literature,the capacity analysis based on the free deterministic equivalent method is then presented in details.Secondly,the free deterministic equivalent of the channel Gram matrix for a general mas-sive MIMO is established,the Cauchy form and the Shannon transform of the established free deterministic equivalent are derived,and the deterministic equivalent of the ergodic channel mutual information of the massive MIMO uplink and the optimal sum-rate capacity achieving input covariance matrices are obtained.The considered channel model is more general com-pared to previous works.Specifically,a massive MIMO uplink with one BS equipped with several distributed antenna arrays is considered.Each link between a user and a BS antenna ar-ray forms a jointly correlated Rician fading channel.First,the channel Gram matrix is expressed as a random matrix polynomial in independent Gaussian matrices and deterministic matrices.Then,by replacing independent Gaussian random matrices with operator-valued random vari-ables satisfying certain operator-valued freeness relations,the free deterministic equivalent of the considered channel Gram matrix is obtained.The distribution of the channel Gram matrix is proved to be asymptotically the same as that of its free deterministic equivalent.Further,the closed form expression of the Cauchy transform of the free deterministic equivalent is derived based on the operator-valued freeness conditions and the properties of operator-valued semicir-cular elements.Furthermore,by using its relation with the Cauchy transform,the closed form expression of the Shannon transform of the free deterministic equivalent is derived.Then,the deterministic equivalent of the ergodic input-output mutual information is obtained,and the re-sults are proved to be the same as those in the literature when the considered channel model reduces to those in the literature.The sum-rate capacity achieving input covariance matrices are also derived based on the approximate ergodic input-output mutual information.Simulation results show that these deterministic equivalent results are not only numerically accurate but also computationally efficient.Thirdly,the empirical moments of the channel Gram matrix for massive MIMO are rigor-ously proved to be asymptotically the same as the moments of the free deterministic equivalent of the channel Gram matrix,the closed form expressions of operator-valued moments and s-calar moments for the channel Gram matrix are derived under Rician and Rayleigh fading,and the low complexity polynomial expansion(PE)detector for massive MIMO uplink is proposed.The considered channel model employs multiple antenna user equipments(UEs).Moreover,the channel between a BS and a UE is a jointly correlated Rician fading channel.First,the moments and the empirical moments of the channel Gram matrix are rigorously proved to be asymptotically the same.Then,the free deterministic equivalent of the channel Gram matrix is established.The obtained free deterministic equivalent can be seen as the sum of several operator-valued free random variables,and its moment is the deterministic equivalent of that of the channel Gram matrix.Further,by using the properties of the sum of the operator-valued free random variables and the operator-valued moment-cumulant formula,the closed form expres-sions for the operator-valued moments and scalar moments of the constructed free deterministic equivalent are derived for general channel matrices.Furthermore,when the channel matrices re-duce to Rayleigh fading channels,the closed form expressions for the operator-valued moments and scalar moments of the constructed free deterministic equivalent are also derived.Based on the obtained moments,the low complexity PE detector is then proposed,and its mean square error(MSE)performances are also derived.Simulation results show that the proposed detector can achieve performance close to that of the MMSE detector.Lastly,the robust transmission for massive MIMO downlink that can be used in repre-sentative scenarios is proposed,the considered precoder design problem is the maximization of weighted ergodic sum-rate under imperfect channel state information(C SI)which includes channel mean and variance information,it is converted into iteratively solving quadratic opti-mization problems,the linear precoder designs based on the deterministic equivalents are pro-posed,and the optimality of the beam domain transmission when the channel means are zero matrices is proved.The considered channel model also employs multiple antenna UEs.First,the massive MIMO downlink system model for representative scenarios is established.The CSI of the users available at the BS are imperfect CSI.Specifically,it can be viewed as a jointly cor-related channel model with known channel mean and variance.The criterion of the considered precoder design is maximizing the ergodic weighted sum-rate.Then,the original complicated nonconvex linear precoder design problem is converted into iteratively solving quadratic opti-mization problems by using the minorize-maximize(MM)algorithm.In the literature,it has been proved that the MM algorithm achieves a local optimum of the original problem.The obtained surrogate optimization problem has closed form optimal solution,but it requires the expectations of several random matrices.To solve the computation problem,the free determin-istic equivalent method is used,and the robust linear precoder design based on the deterministic equivalents is then derived.Furthermore,to reduce the computational complexity of the pro-posed algorithm,two low complexity algorithms are derived.One for the general case,and the other for the case when the statistical CSI has zero means.The optimality of the beam domain transmissions when the channel means are zero matrices is also proved.Simulation results show that the proposed algorithm and its low complexity versions can achieve high spectral efficien-cy.