Free Deterministic Equivalents for the Analysis of MIMO Multiple Access Channel
In this paper, a free deterministic equivalent is proposed for the capacity analysis of the multi-input multi-output (MIMO) multiple access channel (MAC) with a more general channel model compared to previous works. Specifically, a MIMO MAC with one base station (BS) equipped with several distributed antenna sets is considered. Each link between a user and a BS antenna set forms a jointly correlated Rician fading channel. The analysis is based on operator-valued free probability theory, which broadens the range of applicability of free probability techniques tremendously. By replacing independent Gaussian random matrices with operator-valued random variables satisfying certain operator-valued freeness relations, the free deterministic equivalent of the considered channel Gram matrix is obtained. The Shannon transform of the free deterministic equivalent is derived, which provides an approximate expression for the ergodic input-output mutual information of the channel. The sum-rate capacity achieving input covariance matrices are also derived based on the approximate ergodic input-output mutual information. The free deterministic equivalent results are easy to compute, and simulation results show that these approximations are numerically accurate and computationally efficient.
For the development of next generation communication systems, massive multiple-input multiple-output (MIMO) technology has been widely investigated during the last few years [1, 2, 3, 4, 5, 6]. Massive MIMO systems provide huge capacity enhancement by employing hundreds of antennas at a base station (BS). The co-location of so many antennas on a single BS is a major challenge in realizing massive MIMO, whereas dividing the BS antennas into distributed antenna sets (ASs) provides an alternative solution . In most massive MIMO literature, it is assumed that each user equipment (UE) is equipped with a single-antenna. Since multiple antenna UEs are already used in practical systems, it would be of both theoretical and practical interest to investigate the capacity of massive MIMO with distributed ASs and multiple antenna users.
In , Zhang et al. investigated the capacity of a MIMO multiple access channel (MAC) with distributed sets of correlated antennas. The results of  can be applied to a massive MIMO uplink with distributed ASs and multiple antenna UEs directly. The channel between a user and an AS in  is assumed to be a Kronecker correlated MIMO channel  with line-of-sight (LOS) components. In , Oestges concluded that the validity of the Kronecker model decreases as the array size increases. Thus, we consider in this paper a MIMO MAC with a more general channel model than that in . More precisely, we consider also distributed ASs and multiple antenna UEs, but assume that each link between a user and an AS forms a jointly correlated Rician fading channel [11, 12]. If the BS antennas become co-located, then the considered channel model reduces to that in . To the best of our knowledge, a capacity analysis for such MIMO MACs has not been addressed to date.
For the MIMO MAC under consideration, an exact capacity analysis is difficult and might be unsolvable when the number of antennas grows large. In this paper, we aim at deriving an approximate capacity expression. Deterministic equivalents , which have been addressed extensively, are successful methods to derive the approximate capacity for various MIMO channels. These deterministic equivalent approaches fall into four main categories: the Bai and Silverstein method[15, 16, 17], the Gaussian method[18, 19, 8], the replica method[20, 13] and free probability theory[21, 22].
The Bai and Silverstein method has been applied to various MIMO MACs. Couillet et al.  used it to investigate the capacity of a MIMO MAC with separately correlated channels. Combining it with the generalized Lindeberg principle , Wen et al.  derived the ergodic input-output mutual information of a MIMO MAC where the channel matrix consists of correlated non-Gaussian entries. In the Bai and Silverstein method, one needs to “guess” the deterministic equivalent of the Stieltjes transform. This limits its applicability since the deterministic equivalents of some involved models might be hard to “guess” . By using an integration by parts formula and the Nash-Poincare inequality, the Gaussian method is able to derive directly the deterministic equivalents and can be applied to random matrices with involved correlations. It is particularly suited to random matrices with Gaussian entries. Combined with the Lindeberg principle, the Gaussian method can be used to treat random matrices with non-Gaussian entries as in .
The replica method developed in statistical physics  is a widely used approach in wireless communications. It has also been applied to the MIMO MAC. Wen et al.  used it to investigate the sum-rate of multiuser MIMO uplink channels with jointly correlated Rician fading. Free probability theory  provides a better way to understand the asymptotic behavior of large dimensional random matrices. It was first applied to wireless communications by Evans and Tse to investigate the multiuser wireless communication systems .
The Bai and Silverstein method and the Gaussian method are very flexible. Both of them have been used to handle deterministic equivalents for advanced Haar models [16, 27]. Although its validity has not yet been proved , the replica method is also a powerful tool. Meanwhile, the applicability of free probability theory is commonly considered very limited as it can be only applied to large random matrices with unitarily invariant properties, such as standard Gaussian matrices and Haar unitary matrices.
The domain of applicability of free probability techniques can be broadened tremendously by operator-valued free probability theory [28, 29], which is a more general version of free probability theory and allows one to deal with random matrices with correlated entries . In , Far et al. first used operator-valued free probability theory in wireless communications to study slow-fading MIMO systems with nonseparable correlation. The results of  were then used by Pan et al. to study the approximate capacity of uplink network MIMO systems  and the asymptotic spectral efficiency of uplink MIMO-CDMA systems over arbitrarily spatially correlated Rayleigh fading channels . Quaternionic free probability used in  by Müller and Cakmak can be seen as a particular kind of operator-valued free probability.
In , Speicher and Vargas provided the free deterministic equivalent method to derive the deterministic equivalents under the operator-valued free probability framework. A free deterministic equivalent of a random matrix is a non-commutative random variable or an operator-valued random variable, and the difference between the distribution of the latter and the expected distribution of the random matrix goes to zero in the large dimension limit. They viewed the considered random matrix as a polynomial in several matrices, and obtained its free deterministic equivalent by replacing the matrices with operator-valued random variables satisfying certain freeness relations. They observed that the Cauchy transform of the free deterministic equivalent is actually the solution to the iterative deterministic equivalent equation derived by the Bai and Silverstein method or the Gaussian method. Using the free deterministic equivalent approach, they recovered the deterministic equivalent results for the advanced Haar model from .
Motivated by the results from , we propose a free deterministic equivalent for the capacity analysis of the general channel model considered in this paper. The method of free deterministic equivalents provides a relatively formalized methodology to obtain the deterministic equivalent of the Cauchy transform. By replacing independent Gaussian matrices with random matrices that are composed of non-commutative random variables and satisfying certain operator-valued freeness relations, we obtain the free deterministic equivalent of the channel Gram matrix. The Cauchy transform of the free deterministic equivalent is easy to derive by using operator-valued free probability techniques, and is asymptotically the same as that of the channel Gram matrix. Then, we compute the approximate Shannon transform of the channel Gram matrix and the approximate ergodic input-output mutual information of the channel. Furthermore, we derive the sum-rate capacity achieving input covariance matrices based on the approximate ergodic input-output mutual information.
Our considered channel model reduces to that in  when the channel between a user and an AS is a Kronecker correlated MIMO channel, and to the channel model in  when there is one AS at the BS. In this paper, we will show that the results of  and  can be recovered by using the free deterministic equivalent method. Since many existing channel models are special cases of the channel models in  and , we will also be able to provide a new approach to derive the deterministic equivalent results for them.
The rest of this article is organized as follows. The preliminaries and problem formulation are presented in Section II. The main results are provided in Section III. Simulations are contained in Section IV. The conclusion is drawn in Section V. A tutorial on free probability theory and operator-valued free probability theory is presented in Appendix A, where the free deterministic equivalents used in this paper are also introduced and a rigorous mathematical justification of the free deterministic equivalents is provided. Proofs of Lemmas and Theorems are provided in Appendices B to G.
Notations: Throughout this paper, uppercase boldface letters and lowercase boldface letters are used for matrices and vectors, respectively. The superscripts , and denote the conjugate, transpose and conjugate transpose operations, respectively. The notation denotes the mathematical expectation operator. In some cases, where it is not clear from the context, we will employ subscripts to emphasize the definition. The notation represents the composite function . We use to denote the Hadamard product of two matrices and of the same dimensions. The identity matrix is denoted by . The and zero matrices are denoted by and . We use to denote the -th entry of the matrix . The operators and represent the matrix trace and determinant, respectively. denotes a diagonal matrix with along its main diagonal. and denote and , respectively. denotes the algebra of diagonal matrices with elements in the complex field . Finally, we denote by the algebra of complex matrices and by the algebra of complex matrices.
Ii Preliminaries and Problem Formulation
In this section, we first present the definitions of the Shannon transform and the Cauchy transform, and introduce the free deterministic equivalent method with a simple channel model, while our rigorous mathematical justification of the free deterministic equivalents is provided in Appendix A. Then, we present the general model of the MIMO MAC considered in this work, followed by the problem formulation.
Ii-a Shannon Transform and Cauchy Transform
Let be an random matrix and denote the Gram matrix . Let denote the expected cumulative distribution of the eigenvalues of . The Shannon transform is defined as 
Let be a probability measure on and denote the set
The Cauchy transform for is defined by 
Let denote the Cauchy transform for . Then, we have The relation between the Cauchy transform and the Shannon transform can be expressed as 
Differentiating both sides of (3) with respect to , we obtain
Thus, if we are able to find a function whose derivative with respect to is , then we can obtain . In conclusion, if the Cauchy transform is known, then the Shannon transform can be immediately obtained by applying (4).
Ii-B Free Deterministic Equivalent Method
In this subsection, we introduce the free deterministic equivalent method, which can be used to derive the approximation of . The associated definitions, such as that of free independence, circular elements, R-cyclic matrices and semicircular elements over , are provided in Appendix A-A.
The term free deterministic equivalent was coined by Speicher and Vargas in . The considered random matrix in  was viewed as a polynomial in several deterministic matrices and several independent random matrices. The free deterministic equivalent of the considered random matrix was then obtained by replacing the matrices with operator-valued random variables satisfying certain freeness relations. Moreover, the difference between the Cauchy transform of the free deterministic equivalent and that of the considered random matrix goes to zero in the large dimension limit.
However, the method in  only showed how to obtain the free deterministic equivalents for the case where the random matrices are standard Gaussian matrices and Haar unitary matrices. A method similar to that in  was presented by Speicher in , which appeared earlier than . The method in  showed that the random matrix with independent Gaussian entries having different variances can be replaced by the random matrix with free (semi)circular elements having different variances. But, it only considered a very simple case, and the replacement process had no rigorous mathematical proof. Moreover, the free deterministic equivalents were not mentioned in .
In this paper, we introduce in Appendix A-B the free deterministic equivalents for the case where all the matrices are square and have the same size, and the random matrices are Hermitian and composed of independent Gaussian entries with different variances. Similarly to , the free deterministic equivalent of a polynomial in matrices is defined. The replacement process used is that in . Moreover, a rigorous mathematical justification of the free deterministic equivalents we introduce is also provided in Appendix A-B and Appendix A-C.
In , the deterministic equivalent results of  were rederived. But the description in  is not easy to follow. To show how the introduced free deterministic equivalents can be used to derive the approximation of the Cauchy transform , we use the channel model in  as a toy example and restate the method used in  as follows.
The channel matrix in  consists of an deterministic matrix and an random matrix , i.e., . The entries of are independent zero mean complex Gaussian random variables with variances .
Let denote , denote the algebra of complex random variables and denote the algebra of complex random matrices. We define by
where each is a complex random variable. Hereafter, we use the notations and for brevity.
Let be an matrix defined by 
The matrix is even, i.e., all the odd moments of are zeros, and
Let be a diagonal matrix with . The -valued Cauchy transform is given by
When and , we have that
for each . Furthermore, we write as
Since , we have related the calculation of with that of .
We define and by
Then, we have that .
The free deterministic equivalent of is constructed as follows. Let be a unital algebra, be a non-commutative probability space and denote an matrix with entries from . The entries are freely independent centered circular elements with variances . Let denote , denote
It follows that . The matrix is the free deterministic equivalent of .
We define by
where each is a non-commutative random variable from . Then, is a -valued probability space.
From the discussion of the free deterministic equivalents provided in Appendix A-B, we have that and are asymptotically the same. Let denote . The relation between and is the same as that between and . Thus, we also have that and are asymptotically the same and is the deterministic equivalent of . For convenience, we also call the free deterministic equivalent of . In the following, we derive the Cauchy transform by using operator-valued free probability techniques.
Since its elements on and above the diagonal are freely independent, we have that is an R-cyclic matrix. From Theorem 8.2 of , we then have that and are free over . The -valued Cauchy transform of the sum of two -valued free random variables is given by (130) in Appendix A-A. Applying (130), we have that
where is the -valued R-transform of .
Let denote , where . From Theorem 7.2 of , we obtain that is semicircular over , and thus its -valued R-transform is given by
Equations (20) and (21) are equivalent to the ones provided by Theorem 2.4 of . Finally, the Cauchy transform is obtained by .
In conclusion, the free deterministic equivalent method provides a way to derive the approximation of the Cauchy transform . The fundamental step is to construct the free deterministic equivalent of . After the construction, the Cauchy transform can be derived by using operator-valued free probability techniques. Moreover, is the deterministic equivalent of .
Ii-C General Channel Model of MIMO MAC
We consider a frequency-flat fading MIMO MAC channel with one BS and UEs. The BS antennas are divided into distributed ASs. The -th AS is equipped with antennas. The -th UE is equipped with antennas. Furthermore, we assume and . Let denote the transmitted vector of the -th UE. The covariance matrices of are given by
where is the total transmitted power of the -th UE, and is an positive semidefinite matrix with the constraint . The received signal for a single symbol interval can be written as
where is the channel matrix between the BS and the -th UE, and is a complex Gaussian noise vector distributed as . The channel matrix is normalized as
Furthermore, has the following structure
where and are defined by
where and are deterministic unitary matrices, is an deterministic matrix with nonnegative elements, and is a complex Gaussian random matrix with independent and identically distributed (i.i.d.), zero mean and unit variance entries. The jointly correlated channel model not only accounts for the correlation at both link ends, but also characterizes their mutual dependence. It provides a more adequate model for realistic massive MIMO channels since the validity of the widely used Kronecker model decreases as the number of antennas increases. Furthermore, the justification of using the jointly correlated channel model for massive MIMO channels has been provided in [41, 42, 43]. We assume that the channel matrices of different links are independent in this paper, i.e., when or , we have that
where and . Let denote . We define as . The parameterized one-sided correlation matrix is given by
where , and is an diagonal matrix valued function with the diagonal entries obtained by
Similarly, the other parameterized one-sided correlation matrix is expressed as
where , the notation denotes the diagonal block of , i.e., the submatrix of obtained by extracting the entries of the rows and columns with indices from to , and is an diagonal matrix valued function with the diagonal entries computed by
The channel model described above is suitable for describing cellular systems employing cooperative multipoint (CoMP) processing , and also conforms with the framework of cloud radio access networks (C-RANs) . Moreover, it embraces many existing channel models as special cases. When , the MIMO MAC in  is described. Let be an matrix of all s, be an diagonal matrix with positive entries and be an diagonal matrix with positive entries. Set . Then, we obtain . Thus, each reduces to the Kronecker model, and the considered channel model reduces to that in . Many channel models are already included in the channel models of  and . See the references for more details.
Ii-D Problem Formulation
Let denote . In this paper, we are interested in computing the ergodic input-output mutual information of the channel and deriving the sum-rate capacity achieving input covariance matrices. In particular, we consider the large-system regime where and are fixed but and go to infinity with ratios such that
We first consider the problem of computing the ergodic input-output mutual information. For simplicity, we assume . The results for general precoders can then be obtained by replacing with . Let denote the ergodic input-output mutual information of the channel and denote the channel Gram matrix . Under the assumption that the transmitted vector is a Gaussian random vector having an identity covariance matrix and the receiver at the BS has perfect channel state information (CSI), is given by 
Furthermore, we have . For the considered channel model, an exact expression of is intractable. Instead, our goal is to find an approximation of . From Section II-A and Section II-B, we know that the Shannon transform can be obtained from the Cauchy transform and the free deterministic equivalent method can be used to derive the approximation of . Thus, the problem becomes to construct the free deterministic equivalent of , and to derive the Cauchy transform and the Shannon transform . This problem will be treated in Sections III-A to III-C.
To derive the sum-rate capacity achieving input covariance matrices, we then consider the problem of maximizing the ergodic input-output mutual information . Since , the problem can be formulated as
where the constraint set is defined by
We assume that the UEs have no CSI, and that each is fed back from the BS to the -th UE. Moreover, we assume that all are computed from the deterministic matrices and .
Since is an expected value of the input-output mutual information, the optimization problem in (37) is a stochastic programming problem. As mentioned in  and , it is also a convex optimization problem, and thus can be solved by using approaches based on convex optimization with Monte-Carlo methods . More specifically, it can be solved by the Vu-Paulraj algorithm , which was developed from the barrier method  with the gradients and Hessians provided by Monte-Carlo methods.
However, the computational complexity of the aforementioned method is very high . Thus, new approaches are needed. Since the approximation of will be obtained, we can use it as the objective function. Thus, the optimization problem can be reformulated as
The above problem will be solved in Section III-D.
Iii Main Results
In this section, we present the free deterministic equivalent of , the deterministic equivalents of the Cauchy transform and the Shannon transform . We also present the results for the problem of maximizing the approximate ergodic input-output mutual information .
Iii-a Free Deterministic Equivalent of
In , independent rectangular random matrices are found to be asymptotically free over a subalgebra when they are embedded in a larger square matrix space. Motivated by this, we embed in the larger matrix space . Let be the matrix defined by
where is defined by
Then, can be rewritten as
where is defined by
Recall that . Inspired by , we rewrite as
where and are defined by
The free deterministic equivalents of and are constructed as follows. Let be a unital algebra, be a non-commutative probability space and be a family of selfadjoint matrices. The entries are centered semicircular elements, and the entries , are centered circular elements. The variance of the entry is given by . Moreover, the entries on and above the diagonal of are free, and the entries from different are also free. Thus, we also have , where , , and .
Let denote , where . Based on the definitions of , we have that both the upper-left block matrix and the lower-right block matrix of are equal to zero matrices. Thus, can be rewritten as (14), where denotes the upper-right block matrix of . For fixed , we define the map by . Then, we have that
where . Let