Exploiting Opportunistic Multiuser Detection in Decentralized Multiuser MIMO Systems 1footnote 11footnote 1R. Zhang is with the Institute for Infocomm Research, A*STAR, Singapore (e-mail:rzhang@i2r.a-star.edu.sg). 2footnote 22footnote 2J. M. Cioffi is with the Department of Electrical Engineering, Stanford University, Stanford, USA (e-mail:cioffi@stanford.edu).

# Exploiting Opportunistic Multiuser Detection in Decentralized Multiuser MIMO Systems 111R. Zhang is with the Institute for Infocomm Research, A*STAR, Singapore (e-mail:rzhang@i2r.a-star.edu.sg).222J. M. Cioffi is with the Department of Electrical Engineering, Stanford University, Stanford, USA (e-mail:cioffi@stanford.edu).

Rui Zhang and John M. Cioffi
###### Abstract

This paper studies the design of a decentralized multiuser multi-antenna (MIMO) system for spectrum sharing over a fixed narrow band, where the coexisting users independently update their transmit covariance matrices for individual transmit-rate maximization via an iterative manner. This design problem was usually investigated in the literature by assuming that each user treats the co-channel interference from all the other users as additional (colored) noise at the receiver, i.e., the conventional single-user decoder (SUD) is applied. This paper proposes a new decoding method for the decentralized multiuser MIMO system, whereby each user opportunistically cancels the co-channel interference from some or all of the other users via applying multiuser detection techniques, thus termed opportunistic multiuser detection (OMD). This paper studies the optimal transmit covariance design for users’ iterative maximization of individual transmit rates with the proposed OMD, and demonstrates the resulting capacity gains in decentralized multiuser MIMO systems against the conventional SUD.

{keywords}

Cognitive radio, decentralized multiuser system, MIMO Gaussian interference channel, multiuser detection.

## I Introduction

The Gaussian interference channel is a basic mathematical model that characterizes many real-life communication systems with multiple uncoordinated users sharing a common spectrum to transmit independent information at the same time, such as the digital subscriber line (DSL) network [1], the ad-hoc wireless network [2], and the newly emerging cognitive radio (CR) wireless network [3]. From an information-theoretical perspective, the capacity region of the Gaussian interference channel, which constitutes all the simultaneously achievable rates of the users in the system, is still unknown in general [4], while significant progresses have recently been made on approaching this limit [5], [6]. Capacity-approaching techniques usually require certain cooperations among distributed users for their encoding and decoding. A more pragmatic approach that leads to suboptimal achievable rates of the users in the Gaussian interference channel is to restrict the system to operate in a decentralized manner [7], i.e., allowing only single-user encoding and decoding by treating the co-channel interference from the other users as additional Gaussian noise at each user’s receiver. In such a context, decentralized algorithms for users to allocate their transmit resources such as the power, bit-rate, bandwidth, and antenna beam to optimize individual transmission performance and yet to ensure certain fairness among all the users, become most important.

This paper focuses on a multiuser multiple-input multiple-output (MU-MIMO) wireless system, where multiple distributed links, each equipped with multiple transmit and/or receive antennas, share a common narrow band for transmission in a fully decentralized manner. In such a scenario, the system design reduces to finding a set of transmit covariance matrices for the users subject to their co-channel interference resulting from their simultaneous and uncoordinated transmissions. This design problem has been investigated in a vast number of prior works in the literature, e.g., [8]-[16], by treating the co-channel interference as additional colored noise at each user’s receiver, i.e., the conventional single-user decoder (SUD) for the classic point-to-point MIMO channel is applied. In [8], the authors proposed an algorithm, which is in spirit analogous to the iterative water-filling (IWF) algorithm in [7], for each distributed MIMO link to iteratively update transmit covariance matrix to maximize individual transmit rate. Distributed iterative beamforming (the rank of transmit covariance matrix is restricted to be one) algorithms were also studied in [9] for transmit sum-power minimization given individual user’s quality of service (QoS) constraint in terms of the received signal-to-interference-plus-noise ratio (SINR). The throughput of decentralized MU-MIMO systems has been further analyzed in [10] and [11] for the cases of fading channels and large-size systems, respectively. In [12], [13], centralized strategies were proposed where all users’ transmit covariance matrices are jointly searched to maximize their sum-rate, and numerical algorithms were also proposed to converge to a local sum-rate maxima. Analyzing the decentralized MU-MIMO system via a game theoretical approach has recently been done in [14]-[16].

The cited papers on decentralized/centralized designs for the Gaussian MIMO interference channel have all adopted the SUD at each user’s receiver, whereas during the past decade multiuser detection techniques (see, e.g., [17] and references therein) have been thoroughly investigated in the literature, and have been proven in realistic multiuser/MIMO systems to be able to provide substantial performance gains over the conventional SUD. This motivates our work’s investigation of the following question: Considering a decentralized MU-MIMO system where the users iteratively adapt their transmit covariance matrices for individual rate maximization, “Is applying multiuser detection at each user’s receiver able to enhance the system throughput over the conventional SUD?” Note that because of the randomness of channels among the users, as well as their independent rate assignments, at one particular user’s receiver, multiuser detection can be used to cancel the co-channel interference from some/all of its coexisting users only when their received signals are jointly decodable with this particular user’s own received signal. Thus, we refer to this decoding method as opportunistic multiuser detection (OMD). Also note that the OMD in the context of the decentralized MU-MIMO system is analogous to the “successive group decoder (SGD)” in the fading multiple-access channel (MAC) with unknown channel state information (CSI) at the user transmitters (see, e.g., [18] and references therein). With the proposed OMD, this paper derives the optimal transmit covariance matrix for user’s individual transmit-rate maximization at each iteration of transmit adaptation. By simulation, this paper demonstrates the throughput gains of the converged users’ transmit covariance matrices with the proposed OMD over the conventional SUD.

The rest of this paper is organized as follows. Section II presents the system model of the decentralized MU-MIMO system. Section III studies the optimal design of user transmit covariance matrix with the proposed OMD for the special case with two users in the system. Section IV generalizes the results to the case of more than two users. Section V provides the simulation results to demonstrate the throughput gains with the proposed OMD over the SUD. Finally, Section VI concludes the paper.

Notation: Scalars are denoted by lower-case letters, e.g., , and bold-face lower-case letters are used for vectors, e.g., , and bold-face upper-case letters for matrices, e.g., . In addition, , , , and denote the trace, determinant, inverse, and square-root of a square matrix , respectively, and means that is a positive semi-definite matrix [19]. For an arbitrary-size matrix , denotes the conjugate transpose of . denotes a diagonal matrix with as its diagonal elements. and denote the identity matrix and the all-zero vector, respectively. denotes the statistical expectation. The distribution of a circular symmetric complex Gaussian (CSCG) random vector with mean and covariance matrix is denoted by , and stands for “distributed as”. denotes the space of matrices with complex-valued elements. and denote the maximum and minimum between two real numbers, and , respectively, and .

## Ii System Model

This paper considers a distributed MU-MIMO system where users transmit independent information to their corresponding receivers simultaneously over a common narrow band. Each user is equipped with multiple transmit and/or receiver antennas, while for user , , and denote the number of its transmit and receive antennas, respectively. For the time being, it is assumed that perfect time and frequency synchronization with reference to a common clock system have been established for all the users in the system prior to data transmission. We also assume a block-fading model for all the channels involved in the system, and a block-based transmission for all the users over each particular channel fading state. Since the proposed study applies to any channel fading state, for brevity we drop the index of fading state here. The discrete-time baseband signal for the th user transmission is given by

 \boldmath{y}k=\boldmath{H}kk% \boldmath{x}k+K∑j=1,j≠k\boldmath{H}jk% \boldmath{x}j+\boldmath{z}k (1)

where and are the transmitted and received signal vectors for user , respectively, ; denotes the direct-link channel matrix for user , while denotes the cross-link channel matrix from user to user , , ; and is the received noise vector of user .

Without loss of generality, it is assumed that , and all ’s are independent. We consider a decentralized multiuser system where the users independently encode their transmitted messages and thus ’s are independent over . Since this paper is interested in the information-theoretic limit of each Gaussian MIMO channel involved, it is assumed that , where is the transmit covariance matrix for user .

This paper considers a similar decentralized operation protocol as in [7], [8], [14]-[16], whereby the users in the system take turns to update their transmit covariance matrices for individual rate maximization, with all the other users’ transmit covariance matrices being fixed, until all users’ transmit covariance matrices and their transmit rates get converged. We consider two types of decoding methods at each user’s receiver. One is the conventional SUD, which has been applied in the above cited papers, where the th user decodes its desired message by treating the co-channel interference from all the other users, , as additional colored Gaussian noise . The other decoding method is the newly proposed OMD, whereby each user opportunistically applies multiuser detection to decode some/all of its coexisting users’ messages so as to cancel their resulted interference, provided that these messages are jointly decodable with this user’s own message. In practice, each user in the system is usually interfered with by all the other users, while due to location-dependent shadowing/fading, only a small group of coexisting users who are closest to one particular user and thus correspond to the strongest cross-link channels to this user, will contribute the most to this user’s received co-channel interference. As a result, this user can effectively estimate the transmit rates as well as the cross-link channels of these “strong” interference users, and employ the proposed OMD to suppress their interference at the receiver. Note that the use of OMD instead of SUD still maintains the fully decentralized property of the existing IWF-like operation protocols given in [7], [8], [14]-[16].

## Iii Transmit Covariance Optimization: The Two-User Case

In this section, we present the problem formulation as well as the solution to determine the optimal transmit covariance matrix of each user for individual transmit-rate maximization, when the proposed OMD is employed. For the purpose of exposition, we consider the special case where only two users exist in the system. We will address the general case with more than two users in Section IV. For brevity, only user 1’s transmit adaptation is addressed here, while the developed results apply similarly to user 2.

### Iii-a Problem Formulation

Note that at one particular iteration of user 1 to update its transmission, user 2’s transmit covariance matrix, , and transmit rate, denoted by , are both fixed values. For a given transmit covariance matrix of user 1, , the resultant maximum transmit rate of user 1 can be expressed as

 r1(\boldmath{S}1)=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩log∣∣% \boldmath{I}+\boldmath{H}11\boldmath{S}1% \boldmath{H}H11∣∣r2≤R(a)2log∣∣\boldmath{I}+\boldmath{H}11\boldmath{S}% 1\boldmath{H}H11+\boldmath{H}21\boldmath{% S}2\boldmath{H}H21∣∣−r2R(a)2R(b)2 (2)

where

 R(a)2 =log∣∣\boldmath{I}+(\boldmath{I}+% \boldmath{H}11\boldmath{S}1\boldmath{H}H11)−1\boldmath{H}21\boldmath{S}2\boldmath{H}H21∣∣ (3) R(b)2 =log∣∣\boldmath{I}+\boldmath{H}21% \boldmath{S}2\boldmath{H}H21∣∣. (4)

The above result is illustrated in the following three cases corresponding to the three expressions of in (2) from top to bottom.

• Strong Interference Case: In this case, the received signal from user 2 is decodable at user 1’s receiver with the conventional SUD, by treating user 1’s signal as colored Gaussian noise. This is feasible since given in (3). After decoding user’2 message and thereby canceling its associated interference, user 1 can decode its own message with a maximum rate equal to its own channel capacity. The above decoding method is known as successive decoding (SD) for the standard Gaussian MAC [20].

• Moderate Interference Case: In this case, and thus the received signal from user 2 is not directly decodable by the SUD. However, since given in (4), it is still feasible for user 1 to apply joint decoding (JD) [20] to decode both users’ messages.333Note that SD can also be applied in this case to achieve the same rate for user 1 as JD, if SD is deployed jointly with the “time sharing” [20] or “rate splitting” [21] encoding technique at user 1’s transmitter. Since these techniques require certain cooperations between users, they might not be suitable for the fully decentralized multiuser system considered in this paper. In this case, the rate pair of the two users should lie on the -degree segment of the corresponding MAC capacity region boundary [20], i.e., .

• Weak Interference Case: In this case, , i.e., the received signal from user 2 is not decodable even without the presence of user 1’s signal. As such, user 1’s receiver has the only option of treating user 2’s signal as colored Gaussian noise and applying the conventional SUD to directly decode user 1’s message, the same as that in the existing IWF-like algorithms (see, e.g., [8], [14]-[16]).

In the above decoding method, multiuser detection is applied in both cases of strong and moderate interferences when , but not in the case of weak interference when . Thus, user 1’s receiver opportunistically applies multiuser detection to decode user 2’s message, either successively (SD) or jointly (JD) with its own message. We thus refer to this decoding method as opportunistic multiuser detection (OMD). From (3) and (4), it follows that . Further more, it is easy to verify that given in (2) with the proposed OMD is in general larger than the achievable rate with the conventional SUD (given by the third expression of in (2) independent of ), for any given set of , and .

With given in (2) for a fixed , we can further maximize user 1’s transmit rate by searching over . Let denote the transmit power constraint of user 1. This problem can be expressed as

 (P1)  max\boldmath{S}1 r1(\boldmath{S}1) s.t. tr(\boldmath{S}1)≤P1,% \boldmath{S}1⪰0

where is given in (2). The optimal solution of in (P1) and the corresponding maximum transmit rate of user 1 are denoted by and , respectively.

### Iii-B Proposed Solution

In this subsection, we study the solution of (P1) for the optimal transmit covariance matrix of user 1, when the proposed OMD is deployed at user 1’ receiver. Note that although the constraints of (P1) are convex, its objective function is not necessarily concave due to the fact that given in (3) is neither convex nor concave function of . As a result, (P1) seems to be non-convex at a first glance. In fact, (P1) is a convex optimization problem after being transformed into a convex form, as will be shown in this subsection. In the following, we will study the solution of (P1) for two cases: and , for which the SUD and the multiuser decoding (MD) (in the form of either SD or JD) should be used to achieve given in (2), respectively.

#### Iii-B1 r2>R(b)2

In this case, the SUD should be applied. Note that is a constant unrelated to . Thus, the optimal that maximizes the third expression of in (2) has the following structure [20]:

 \boldmath{S}SUD1=\boldmath{V}\boldmath{Λ}\boldmath{V}H (5)

where with is obtained from the singular-value decomposition (SVD) of the equivalent channel of user 1 (after the noise whitening) expressed as

 (\boldmath{I}+\boldmath{H}21\boldmath{S}2\boldmath{H}H21)−12\boldmath{H}11=%\boldmath$U$\boldmath{Σ}\boldmath{V}H (6)

with , , , , and with ’s obtained from the standard water-filling solution [20]:

 pi=(μ−1σ2i)+,  i=1,…,T1, (7)

with being a constant to make . The maximum rate of user 1 then becomes

 rSUD1=T1∑i=1log(1+σ2ipi). (8)

#### Iii-B2 r2≤R(b)2

In this case, the MD in the form of either SD or JD should be used. In order to overcome the non-concavity of given in (2) due to , we re-express the first two expressions of in (2) as

 rMD1(\boldmath{S}1)=min(log∣∣\boldmath{I}+\boldmath{H}11\boldmath{S}1%\boldmath$H$H11∣∣,log∣∣\boldmath{I}+% \boldmath{H}11\boldmath{S}1\boldmath{H}H11+\boldmath{H}21\boldmath{S}2\boldmath{H}H21∣∣−r2). (9)

Thus, the maximum achievable rate of user 1 can be obtained as

 rMD1=max\boldmath{S}1:tr(\boldmath{S% }1)≤P1,\boldmath{S}1⪰0rMD1(% \boldmath{S}1). (10)

The maximization problem in (10) can be explicitly written as

 (P2)  maxr1, \boldmath{% S}1 r1 s.t. r1≤log∣∣\boldmath{I}+\boldmath% {H}11\boldmath{S}1\boldmath{H}H11∣∣ (11) r1≤log∣∣\boldmath{I}+\boldmath% {H}11\boldmath{S}1\boldmath{H}H11+% \boldmath{H}21\boldmath{S}2\boldmath{H}H21∣∣−r2 (12) r1≥0,tr(\boldmath{S}1)≤P1,\boldmath{S}1⪰0. (13)

The optimal solution of in (P2) will be . Note that (P2) is a convex optimization problem since its constraints specify a convex set of . To solve (P2), we apply the standard Lagrange duality method [19]. First, we introduce two non-negative dual variables, and , associated with the two rate constraints (11) and (12), respectively, and write the associated Lagrangian of (P2) as

 L(r1,\boldmath{S}1,μ1,μ2)= r1−μ1(r1−log∣∣\boldmath{I}+% \boldmath{H}11\boldmath{S}1\boldmath{H}H11∣∣) −μ2(r1−log∣∣\boldmath{I}+% \boldmath{H}11\boldmath{S}1\boldmath{H}H11+\boldmath{H}21\boldmath{S}2\boldmath{H}H21∣∣+r2) (14)

By reordering the terms in (III-B2), we obtain

 L(r1,\boldmath{S}1,μ1,μ2)= (1−μ1−μ2)r1+μ1log∣∣\boldmath{I}+\boldmath{H}11\boldmath{S}1\boldmath{H}H11∣∣ +μ2log∣∣\boldmath{I}+\boldmath{H}11\boldmath{S}1\boldmath{H}H11+\boldmath{H}21\boldmath{S}2\boldmath{H}H21∣∣+μ2r2. (15)

The Lagrange dual function of (P2) is then defined as

 g(μ1,μ2)=max(r1,\boldmath{S}1)∈AL(r1,\boldmath{S}1,μ1,μ2) (16)

where the set specifies the remaining constraints of (P2) given in (13). The dual problem of (P2), of which the optimal value is the same as that of (P2),444It can be easily checked that the Slater’s condition holds for (P2) and thus the duality gap for (P2) is zero [19]. is defined as

 (P2-D)  minμ1≥0,μ2≥0g(μ1,μ2). (17)

Let and denote the optimal solutions of (P2). Let and denote the optimal dual solutions of the dual problem (P2-D). Next, we will present a key relationship between and as follows.

###### Lemma iii.1

In problem (P2-D), the optimal solutions satisfy that .

{proof}

See Appendix A.

Given Lemma III.1, without loss of generality, we can replace by in (III-B2). Thus, the maximization problem in (16) can be equivalently rewritten as (by discarding the constant term )

 (P3)  max\boldmath{S}1 μ1log∣∣\boldmath{I}+\boldmath{H}11\boldmath{S}1\boldmath{H}H11∣∣+(1−μ1)log∣∣\boldmath{I}+\boldmath{H}11\boldmath% {S}1\boldmath{H}H11+\boldmath{H}21% \boldmath{S}2\boldmath{H}H21∣∣ s.t. tr(\boldmath{S}1)≤P1,% \boldmath{S}1⪰0. (18)

Further more, the dual problem (17) now only needs to minimize (since ) over . Then, there are the following three cases in which takes different values.

• : In this case, . From the Karush-Kuhn-Tucker (KKT) optimality conditions [19] of (P2), it is known that the constraint (11) is inactive while the constraint (12) is active. This suggests that JD instead of SD is optimal. Furthermore, from (P3), with , it follows that , denoted by , maximizes the sum-rate, , from which we can show that

 \boldmath{S}JD1=\boldmath{S}SUD1 (19)

where is given in (5), i.e., the optimal transmit covariance matrix is the same for both cases of SUD and JD. However, the optimal in this case with JD, denoted by , is equal to

 rJD1=rSUD1+R(b)2−r2 (20)

where is given in (8). Finally, we need to check the condition under which this case holds. Since the constraint (11) should be inactive, it follows that

 rJD1

From (20) and (21), it can be shown that the case of interest holds when

 r2>log∣∣\boldmath{I}+(\boldmath{I}+\boldmath{H}11\boldmath{S}JD1\boldmath{H}H11)−1\boldmath{H}21\boldmath{S}2\boldmath{H}H21∣∣≜¯R(a)2. (22)

Note that can also be obtained from given in (3) by letting .

• : In this case, . From the KKT optimality conditions of (P2), it is known that the constraint (11) is active while the constraint (12) is inactive. This suggests that SD instead of JD is optimal. Furthermore, from (P3), with , it follows that , denoted by , maximizes user 1’s own channel capacity (without the presence of user 2), , from which we can easily show that [20]

 \boldmath{S}SD1=\boldmath{V}1\boldmath{Λ}1\boldmath{V}H1 (23)

where is obtained from the SVD of the direct-link channel of user 1 expressed as , with , , , , and with ’s obtained from the standard water-filling solution [20]:

 qi=(ν−1γ2i)+,  i=1,…,T1, (24)

with being a constant to make . The optimal in this case with SD, denoted by , then becomes

 rSD1=T1∑i=1log(1+γ2iqi). (25)

Similarly like the previous case, we can show that this case holds when

 r2

At last, we have the following lemma.

###### Lemma iii.2

For defined in (22) and defined in (26), it holds that .

{proof}

See Appendix B.

• : In this case, , and from the KKT optimality conditions of (P2), it is known that both the constraints (11) and (12) are active. This suggests that , i.e., SD is optimal. However, the optimal solution of (P2), or that of (P3) with , denoted by , in general does not have any closed-form expression, and thus needs to be obtained by a numerical search. Since (P3) is convex, the interior-point method [19] can be used to efficiently obtain its solution for a given . Let denote the optimal solution of (P3) for a given . Then, can be efficiently found by a simple bisection search based upon the sub-gradient [19] of , which can be shown from (III-B2) (with ) to be

 log∣∣∣\boldmath{I}+(\boldmath{I}+\boldmath{H}11\boldmath{S}⋆1(μ1)\boldmath{H}H11)−1\boldmath{H}21\boldmath{S}2% \boldmath{H}H21∣∣∣−r2. (27)

Once converges to , the corresponding becomes the optimal . The optimal in this case with SD, denoted by , is then expressed as

 ~rSD1=log∣∣∣\boldmath{I}+\boldmath{H}11~\boldmath{S}SD1\boldmath{H}H11∣∣∣. (28)

Similarly like the previous two cases and using Lemma III.2, we can show that this case holds when

 ^R(a)2≤r2≤¯R(a)2. (29)

#### Iii-B3 Combing r2>R(b)2 and r2≤R(b)2

To summarize, the following theorem is obtained for the optimal solution of (P1).

###### Theorem iii.1

For a given set of and of user 2, the optimal transmit covariance matrix of user 1 and the maximum transmit rate of user 1 with the proposed OMD are given as follows:

 \boldmath{S}OMD1=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩\boldmath{S}SD1,0R(b)2, (30)
 rOMD1=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩rSD1,0R(b)2. (31)

The corresponding optimal decoding methods at user 1’s receiver are (from top to bottom) SD, SD, JD, and SUD, respectively.

In Fig. 1, we show in (31) as a function of for some fixed . The rate gain of for OMD over for SUD is clearly shown when . There are three pentagon-shape capacity regions shown in the figure, which are , , and , respectively, where denotes the capacity region of a two-user Gaussian MIMO-MAC with user 1’s and user 2’s transmitters transmitting to user 1’s receiver, and , denoting the transmit covariance matrices of user 1 and user 2, respectively. More specifically, can be expressed as [20]

 (32)

Note that in Fig. 1, the sold line consisting of different rate pairs of constitute the boundary rate pairs of the aforementioned capacity regions. Also note that there is a curved part of this rate-pair line in the case of , where is equal to and is achievable by , which is the solution of problem (P3) for some given , .

## Iv Extension to More Than Two Users

In this section, we extend the results obtained for the two-user MIMO system to the general MU-MIMO system with more than two users, i.e., . Due to the symmetry, we consider only user 1’s transmit optimization over to maximize transmit rate , with all the other users’ transmit rates, , and transmit covariance matrices, , being fixed.

To apply OMD at user 1’s receiver, we need to first identify the group of users whose signals are (jointly or successively) decodable at user 1’s receiver without the presence of user 1’s own received signal. We thus have the following definitions:

###### Definition iv.1

A set , , is called a decodable user set for user 1, if the received signals at user 1’s receiver due to the users in are decodable without the presence of user 1’s own received signal, by treating the received signals from the other users in as colored Gaussian noise, where denotes the complementary set of , i.e., and . More specifically, the transmit rates of users in must satisfy [20]

 ∑i∈Jri≤log∣∣ ∣ ∣∣\boldmath{I}+⎛⎜⎝\boldmath{I}+∑k∈¯¯¯¯¯U1% \boldmath{H}k1\boldmath{S}k\boldmath{H}Hk1⎞⎟⎠−1∑i∈J\boldmath{H}i1\boldmath{S}i\boldmath{H}Hi1∣∣ ∣ ∣∣,∀J⊆U1. (33)
###### Definition iv.2

A set is called an optimal decodable user set for user 1, if is a decodable user set for user 1, and among all possible decodable user sets for user 1, has the largest size.

Next, we have the following important proposition:

###### Proposition iv.1

The set is unique. Furthermore, for any decodable user set for user 1, , it holds that .

{proof}

See Appendix C.

For conciseness, we show the algorithm to find the unique set for user 1, , in Appendix D.

From Proposition IV.1, it follows that the optimal decoding strategy for user 1’s receiver is applying OMD to the users in the set (it may be possible that ), while taking the users in the set as additional colored Gaussian noise. For an arbitrary set , let denote the size of . Note that to make the OMD feasible, the rate of user 1, , and the rates of users in must be jointly in the capacity region of the corresponding -user Gaussian MIMO-MAC for a given set of user transmit covariance matrices and the receiver noise covariance matrix, , which, similar to (32), can be defined as

 CMAC(U∗1)≜ {(r1,{ri}i∈U∗1):∑i∈Jri≤log∣∣ ∣∣\boldmath{I}+\boldmath{Φ}−1∑i∈J\boldmath{H}i1\boldmath{S}i\boldmath{H}Hi1∣∣ ∣∣,∀J⊆{1}⋃U∗1}. (34)

Note that in (34), the rate inequalities involving subsets ’s containing users solely from all hold due to the definition of . Therefore, in order to find the optimal for user 1 to maximize , with fixed ’s and ’s, , it is sufficient to consider the following optimization problem:

 (P4)  max\boldmath{S}1,r1 r1 s.t. r1+∑i∈Jri≤log∣∣ ∣∣% \boldmath{I}+\boldmath{Φ}−1(\boldmath{H}11\boldmath{S}1\boldmath{H}H11+∑i∈J\boldmath{H}i1\boldmath{S}i\boldmath{H}Hi1)∣∣ ∣∣,∀J⊆U∗1 (35) r1≥0,tr(\boldmath{S}1)≤P1,\boldmath{S}1⪰0 (36)

Problem (P4) is convex in terms of and since its constraints specify a convex set of . Similarly like for problem (P2), we introduce a set of non-negative dual variables, ’s, , each associated with one corresponding constraint in (35) for a particular subsect (including ) denoted by , and obtain an equivalent problem for the optimization over for a given set of fixed ’s, which is expressed as

 (P5)  max\boldmath{S}1 2|U∗1|∑n=1μnlog∣∣ ∣∣\boldmath{I}+\boldmath{Φ}−1(\boldmath{H}11\boldmath{S}1\boldmath{H}H11+∑i∈Jn\boldmath{H}i1\boldmath{S}i% \boldmath{H}Hi1)∣∣ ∣∣ s.t. tr(\boldmath{S}