# Asynchronous Distributed Downlink Beamforming and Power Control in Multi-cell Networks

## Abstract

In this paper, we consider a multi-cell network where every base station (BS) serves multiple users with an antenna array. Each user is associated with only one BS and has a single antenna. Assume that only long-term channel state information (CSI) is available in the system. The objective is to minimize the network downlink transmission power needed to meet the users’ signal-to-interference-plus-noise ratio (SINR) requirements. For this objective, we propose an asynchronous distributed beamforming and power control algorithm which provides the same optimal solution as given by centralized algorithms. To design the algorithm, the power minimization problem is formulated mathematically as a non-convex problem. For distributed implementation, the non-convex problem is cast into the dual decomposition framework. Resorting to the theory about matrix pencil, a novel asynchronous iterative method is proposed for solving the dual of the non-convex problem. The methods for beamforming and power control are obtained by investigating the primal problem. At last, simulation results are provided to demonstrate the convergence and performance of the algorithm.

## 1Introduction

Downlink beamforming within a cell is an efficient technique for balancing the intra-cell interference, which has been studied since 1998 [1]. However, cell edge users may still suffer inter-cell interference in a multi-cell network. By coordinating beamforming and power control among BSs, inter-cell interference can be mitigated [7]. This kind of beamforming is called multi-cell beamforming. Multi-cell beamforming has its applications, like interference management in femto-cell/small-cell networks, or spatial spectrum sharing in cognitive radio networks.

One of the basic beamforming problems is to minimize the total transmission power subject to each user’s SINR constraint which is named the power minimization beamforming problem (PMBP). The problem differs for different kinds of CSI [7]. The PMBP is a convex problem [5] with the instantaneous CSI which appears as a vector. However, the PMBP is non-convex [3] with the long-term CSI. The long-term CSI appears as a spatial correlation matrix whose rank is greater than one.

Centralized algorithms in [1] and [3] are derived for single-cell PMBP with either instantaneous CSI or long-term CSI. Their generalizations are applied to the multi-cell PMBP and provide the optimal solution. However, their distributed implementations are unavailable. The main reason is that they need strict coordination and synchronization among BSs which is impractical with nowadays backhaul network.

In this work, we assume the long-term CSI. Our main contribution is an algorithm that can be implemented asynchronously in a distributed fashion. The algorithm achieves the same optimal solution as given by the centralized algorithms. In the algorithm, a BS updates the beamforming vectors of its users. A BS only communicates with its neighboring BSs for coordination via backhaul networks. Motivated by the concept of “power on demand” [8], we let each user compute its required power and send it to its BS. The BSs allocate the users the required power. The term “asynchronous” has two folds of meanings. First, neither the BSs nor the users need to update simultaneously, or with the same frequency. This property makes the algorithm suitable for distributed implementation. Second, the algorithm converges with outdated coordination information. This implies that the algorithm is robust to the transmission delay and the packet loss of the backhaul network.

To design the algorithm, the PMBP is cast into the dual decomposition framework [9]. A novel asynchronous iterative method is proposed for solving the dual of the PMBP, which relies on the recent progress about the semidefiniteness of a matrix pencil [10]. With the optimal dual variables, a user’s beamforming vector is the null space of a linear combination of matrices which is computed by its BS. The power control algorithm falls in the classical iterative power control framework in [13]. Thus, the users can update the power asynchronously with local information. Besides, we also investigate the feasibility of the multi-cell PMBP with long-term CSI. The algorithm is derived assuming the PMBP being feasible.

### 1.1Related Work

In the literature about PMBP, uplink-downlink duality is a major aspect [1]. The dual of the single-cell PMBP has a similar form to the uplink beamforming problem. Based on the dual of the PMBP, the virtual uplink problem is defined. The virtual uplink problem can be solved with a beamformer-power update method. The method is extended to the multi-cell case in [14] by viewing multiple BSs as an antenna array. The communication requirement may be difficult to meet in practice. Besides, method with limited coordination among BSs is proposed in [15].

The duality of the single-cell PMBP with instantaneous CSI is also studied with the Lagrangian theory in [5],[6]. An algorithm that is different from the algorithm in [1] is proposed. The work of [16] is an extension of [6] to the multi-cell case. An efficient distributed multi-cell beamforming algorithm is proposed. In [16], the system is time division duplex, i.e., reciprocal channel. Besides, instantaneous CSI is assumed, and the noise variances at each receiver are the same. In this case, the virtual uplink is the real uplink [4], which is convenient for distributed implementation. However, the algorithm is not amenable to the multi-cell PMBP with long-term CSI. Our work complements the work of [16] by assuming long-term CSI, non-reciprocal channel, and different noise variances at each user.

Another class of methods is based on the convex optimization techniques [3]. The PMBP is relaxed to a convex semidefinite program (SDP). In [3], the relaxation gap is proved to be zero if the PMBP is feasible, which implies strong duality. This conclusion is generalized to the multi-cell case in [7]. With this approach, additional constraints can be considered in problem formulation [18]. However, this method is hard to be implemented distributively.

Other related works are summarized as follows. In [20], the surrogate duality approach is used to maximize the minimum SINR under power constraint. In [21], power minimization of multi-cell multi-cast beamforming system is studied. In [22], max-min weighted SINR problem subject to weighted sum power constraint is studied, where the problem is decouple into multiple sub-problems by introducing a set of slack variables. In [23], [24], beamforming techniques are applied to hierarchical systems for interference reduction. Besides, there are works with other design objectives, such as transmission rate [25], energy-efficiency [26], and robustness[27].

### 1.2Organization

The rest of this paper is organized as follows. In Section 2, the system model and the problem formulation are given. In Section 3, the feasibility of the PMBP is studied. Then, in Section 4, we introduce the computation methods for the solution of the PMBP. In Section 5, we show how to implement the computation methods in an asynchronous distributed manner. The performance is further demonstrated in Section 6 via simulation.

Notation:

Let be a matrix. , and represent the inverse of , the Hermitian transpose of and the transpose of , respectively. Besides, , , , and stand for the determinant of , the rank of , the null space of , and the radius of , respectively. If , is a positive semidefinitepositive definite matrix. denotes an identity matrix with adaptive size. denotes the statistical expectation. When describing vectors’ relation, “”, “”, “”, “” are component-wise.

## 2System Model and Problem Formulation

We consider a multi-cell network with cells indexed by . Each cell has a BS with antennas and users. Within a cell, a user is indexed by . In the system scope, a user is identified by its cell index and its intra-cell index, i.e., . User indicates the user in cell . For later use, “” represents “for all users”.

The cells work in the same spectrum. The channel gain from BS to user is denoted with , . We assume flat-fading channels, i.e., no inter-symbol-interference. The long-term CSI is given as the spatial correlation matrices,

which satisfy and . The long-term CSI assumption implies that the mean SINR is considered.

Let denotes the information signal transmitted from BS , , to its user , . Let be the beamforming vector for . In the remaining of this work, we have as default. The received signal of user is

where is the additive white circularly symmetric Gaussian complex noise with variance . To simplify the notations, as [16] does, we have and . Similar notation schemes are used in the rest of this work.

The SINR of the user is defined as follows,

where is the allocated power of BS for user and . Besides, the notation represents summation over all the users except user . In addition, we denote by the SINR target of user .

The objective is to minimize the total transmission power subject to each user’s SINR constraint. Mathematically, the objective is to find the solution of the following *master problem*,

where we define and . Since the constraints, are default, we omit them in this and the following optimization problems.

## 3Problem Feasibility

The master problem is a mathematical formulation of the multi-cell PMBP. If the master problem is feasible, all users’ SINR targets can be achieved. However, the master problem is not feasible for all sets of SINR targets. In this section, we study the feasibility of the master problem which can be viewed as a multi-cell extension of the work in [28].

The master problem is feasible if its feasible region is non-empty. The feasible region is characterized by , which can be written as,

The norm constraints of the beamforming vectors can be guaranteed by normalization, so they do not impact the feasibility of the master problem.

We rewrite the above inequalities in the following compact form,

where is a diagonal matrix, is a vector, and is a matrix. In particular, we have

the component of as

and the component of matrix as

The feasibility of the master problem now depends on the inequality . Mathematically, according to the Perron-Frobenius theorem [29], we have the following lemma.

Similar conclusion can be found in [7]. From system perspective, given the users’ SINR targets, captures the interference relationship among the users which mainly depends on the physical environment. If , the lower bound of would be . According to , is in connection with the thermal noise. Since the feasibility of the master problem depends on , to further reveal the physical meaning of , motivated by [4], we have the following proposition.

See Appendix Section 8.

The ratio on the left hand side of inequality implies that the signal to interference ratio should be strictly larger than the target SINR. It shows mathematically that the feasibility mainly depends on the existence of the beamforming vectors that can coordinate the interference.

In summary, Lemma 1 and Proposition 1 can be viewed as further characterizations of the feasible region of the master problem. They can provide practical design insights. The above results remind us that feasibility can be achieved by removing the strongly interfered users, or decreasing some users’ SINR targets. As shown in [31], user scheduling and network-wide congestion control can be employed for feasibility. To focus on the algorithm design, we assume the master problem is feasible in the derivation of the algorithm.

## 4Computational Solutions

In this section, we derive a series of computation methods for solving the mathematical problem . They are the underlying principles of the asynchronous distributed beamforming and power control algorithm. We derive these methods with their asynchronous distributed implementations in mind.

For distributed implementation, we start by employing the Lagrangian dual decomposition method [9]. Furthermore, since strong duality holds for the master problem, which is proved in [7], the optimality of the dual decomposition method is guaranteed. The Lagrangian of the master problem is written as follows,

where and . About the last term of the above equation, we have

Then, equation can be rewritten as,

Furthermore, we define

where and .

Then, the Lagrangian is

According to the Lagrangian theory, the *primal problem* is

Based on equation , the primal problem can be decomposed into sub-primal problems,

With the optimal dual variables, the beamforming and power control can be performed within each BS by solving the corresponding sub-primal problem.

To avoid being trivial, we require, ,

Otherwise, the optimum of the primal problem would be negative infinity.

From and , the *dual problem* is

The dual problem can be centrally solved by CVX [34]. However, a computation method designed for distributed implementation is still unavailable.

If instantaneous CSI is used, inequalities are linear inequalities in essence. When the corresponding dual problem achieves optimality, they are all active, i.e., establishing with equality. Based on this property, an efficient iterative method is derived in [5].

However, since the ranks of the spatial correlation matrices are greater than one, the inequalities in are linear matrix inequalities. This difference makes it challenging to derive a similar iterative algorithm.

### 4.1Methods for Computing Dual Variables

One of our major contributions is an asynchronous iterative method for computing the dual variables. To design the method, we first propose the following lemma.

See Appendix Section 9.

Next, we rewrite the constraints of the dual problem as

Since correlation matrices , we have and . According to Lemma ?, inequalities in are equivalent to

Furthermore, since the right hand side of also contains the dual variables, we define the following functions,

Combining and , we have where . Thus far, the dual problem is equivalent to

where .

To solve the above problem, we propose an *iterative dual computation method* which converges under asynchronous implementation. Briefly, the method is to update the dual variables with . We first propose the following theorem about its synchronous implementation.

See Appendix Section 10.

We employ the asynchronous algorithm model in [9]. Define as the input dual variables used for generating , where

Function is the time index of dual variable used for updating . So, is the value of the dual variable used for updating .

The asynchronous model also captures the case that the dual variables are not updated in every iteration. Denote by the set of time indexes at which dual variable is updated. For example, . We assume that is infinite, and, provided any time , there is such that , for all . The physical meaning of this assumption is that the transmission of the dual variables does not fail in all the iterations.

The convergence of the iterative dual computation method under asynchronous implementation is formally given as the following theorem.

We have proved that is a standard function in Appendix Section 10. The proof of this theorem is also based on this property. With the reasoning similar to [13], this theorem can be proved based on Theorem ? and the conclusions from [9].

In summary, the iterative dual computation method converges to the same optimal point under both synchronous and asynchronous implementations.

### 4.2Methods for Computing Primal Variables

We derive the methods for computing the primal variables in this section.

#### Beamforming

Thanks to the dual decomposition, each cell’s beamforming problem realtes to one sub-primal problem. The optimality conditions for the th cell’s sub-primal problem are

which are equivalent to

Furthermore, define

which is Hermitian. If the matrix is full rank, the null space will be a zero vector. From the computation of the dual variables, we can see that the matrix has an eigenvalue to be . Thus, is not full rank and has a non-zero null space.

It requires that the beamforming of each cell can only be performed after the dual variables are optimal. The beamforming could be delayed for both numerical reasons, and practical reasons, like transmission delay. Our method enables beamforming to be performed at any time.

When is full rank, we have the eigendecomposition of it as , where is a diagonal matrix and is a unitary matrix. Let

Define a diagonal matrix whose components are the same as except . The beamforming vector can be computed as,

#### Power Control

Going back to the master problem with the optimal beamforming vectors, the master problem reduces to a typical power control problem [13],

where

with power gains .

Furthermore, let . Since , the solution of problem can be computed with the following iterative method,

Moreover, this iterative power control method can be implemented asynchronously in a distributed fashion as given in [13].

## 5Asynchronous Distributed Beamforming and Power Control

In this section, we introduce the asynchronous distributed beamforming and power control algorithm which is based on the above computation methods. For ease of demonstrating the features of the algorithm, we describe the algorithm in the form of threads which are given in Table ?. The threads can work in parallel without strict coordination. Besides, with the proposed algorithm, the information exchange among BSs is illustrated in Figure 1.

CSI thread:

User :

estimate and , if ;

send them to BS .BS :

receive and , , from user ;

send () to BS ; receive from BS .The period of updating CSI: .

Dual computation (DC) thread:

BS :

receive from BS when ;

update dual variable with , ;

send to BS , if is reported by user .The period between two iterations: .

Beamforming (BF) thread:

BS :

fetch the CSI from the CSI thread and the dual variables from the DC thread;

update with equation .The period of beamforming: .

Power control (PC) thread:

User :

update power with ;

feedback the demanded power to BS .BS :

transmit to user with required power.The period of power control: .

The CSI thread is used to collect the long-term CSI. Each user can estimate the long-term CSI, from its associated BS to it, and, from the neighboring BSs to it. This can be done through pilot channels. The users send the long-term CSI to their BSs. When a BS interferes with a user of another cell, or its user is interfered with by other BSs, it exchanges the long-term CSI with the related neighboring BSs. Comparing to the algorithms using instantaneous CSI, the communication overheads here are relatively low. The larger is, the smaller the time average communication overhead is. Moreover, we assume , i.e., the long-term CSI can be viewed as fixed for the other threads.

The DC thread is the key to the asynchronous implementation of the algorithm. Each BS runs a DC thread which updates the dual variables of its users. Mathematically, the update of a dual variable needs all the dual variables. Indeed, in addition to its own dual variables, a BS only needs to know the dual variables of the users interfered by it. The BSs communicate via the backhaul network. In practice, transmission delay and packet loss could happen, which means that some BSs use outdated information. According to Theorem ?, the BSs can still compute the optimal dual variables in this situation. Furthermore, the BSs do not need to update the dual variables simultaneously, nor with the same frequency. As to the computational complexity, each BS needs to compute the generalized eigenvalues. If using the QZ method [35], the computational complexity of a BS per iteration is about .

Each BS runs a BF thread to compute the beamforming vectors of its users. With the beamforming computation method in , the update frequency of the BF thread does not need to coordinate with the DC thread. The needed dual variables in the computation are fetched from the DC thread. The main computation load of the BF thread is to compute the eigenvalues and the null spaces of matrices with size .

The PC thread runs at each user. Each user computes the required power with equation based on the local information: the total received signal power and the power gain between it and its BS. Each user sends the demanded power to its BS and the BS allocates the user the required power. This kind of power control can be understood as “power on demand” [8]. Besides, this iterative power control method falls in the iterative power control framework in [13]. Thus, the users can update the power asynchronously and distributively. Besides, the computational complexity for each user is low.

In summary, the DC thread is the key to the asynchronous implementation of the algorithm. As long as the optimal dual variables are achieved, the optimal beamforming and power control can be realized by the BSs and the users asynchronously. In other words, if the iterative dual computation method could not converge under asynchronous implementation, the beamforming and power control algorithm could not converge asynchronously either. With Theorem ?, we conclude that the asynchronous distributed beamforming and power control algorithm can provide the optimal solution to the master problem.

## 6Simulation

In this section, we demonstrate the performance of the proposed beamforming and power control algorithm via simulation. The downlink channel model captures the path loss and the spatial correlation. Let , , where is the distance from BS to user and . According to [3], the spatial correlation matrix is approximated by, ,

where means that user locates at relative to the array broadside of BS . Each user is surrounded by local scatters corresponding to a spread angle of . Without loss of generality, the SINR targets of the users are assumed to be the same. The other simulation settings are given as follows, if not specified. Let . The distance between two BSs is . The users are randomly located within their cells.

As pointed above, the iterative dual computation method is the key to the asynchronous implementation of the whole algorithm. The asynchronous implementation here is to let each dual variable has a positive probability being not updated. The probability is denoted by . This setup simulates the cases that the BSs do not update simultaneously and that packet loss or delay happens.

Fig. ? shows the convergence of the iterative dual computation method under asynchronous implementation with . In case 1, we set resulting in increasing sequences. In case 2, we set resulting in decreasing sequences. To show the effect of the asynchronous implementation, the convergence of the iterative dual computation method under synchronous implementation is illustrated in Fig. ?. Comparing Fig. ? and Fig. ?, we notice that the method under asynchronous implementation has a slower convergence speed. Besides, Figure 2 shows the convergence of the iterative computation method with where only the dual variables of each cell’s first user are shown.

To further confirm this observation, we study the influence of the asynchronous implementation with . The four BSs are put at the four corners of a square. Table 1 shows the average iteration numbers needed for the iterative dual computation method to converge under different values of . The convergence criteria is . We can see that, as increases, the average number of iterations also increases.

0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | |
---|---|---|---|---|---|---|

Num. of Iter. | 24 | 28 | 33 | 39 | 46 | 56 |

Next, we show the convergence of the asynchronous distributed beamforming and power control algorithm with . In [7], the authors prove that the optimum of the SDP-relaxed master problem is the same as the master problem as long as the master problem is feasible. Based on this conclusion, we use CVX to compute the optimum of the SDP-relaxed master problem as a benchmark. It is plotted as the black dot curve in Figure 3. Furthermore, we assume . Specifically, the CSI is assumed to be fixed. In each iteration, the BSs update the beamforming vectors once and the users update the power allocations once. The dual variables are updated according to the probability in each iteration. Figure 3 shows that the beamforming and power control algorithm converges to the benchmark under different . Besides, Figure 3 shows that when is large, the total transmission power could have large fluctuations. When , the algorithm converges quickly. This manifestation reminds us that in practice reliable backhaul networks are important for the performance of the algorithm.

If a BS computes the beamforming vectors and power allocations treating the inter-cell interference as noise, we call this kind of beamforming the single-cell beamforming. The performance gain of multi-cell beamforming over single-cell beamforming is illustrated in Figure 4. Each curve in Figure 4 is plotted via -time averaging. Only the setup, with which the master problem is feasible, is used.

In both the cases and , the averaged power consumption of the multi-cell beamforming is less than that of the single-cell beamforming. Furthermore, the power consumption of the single-cell beamforming increases more quickly than the multi-cell beamforming. It is because the single-cell beamforming only coordinates intra-cell interference. When the SINR target increases, a BS needs to increase power to increase the SINR. When all BSs do so, the inter-cell interference received by the users will increase. As a result, the BSs need to further increase power. However, when using multi-cell beamforming, inter-cell interference is coordinated. The BSs increase power mainly for increasing the users’ SINRs.

Besides, the increase of the total power consumption implies that certain BS’s power consumption must increase. In practice, the maximum transmission power of a BS is constrained. Considering each BS’s power constraint, the method used in [6] can be used to find the minimum power consumption. Briefly, the power minimization problem can be decomposed into two levels by decomposing the BSs’ power constraints. Our method here can be used to solve the lower level problem.

## 7Conclusion

In this paper, our main contribution is an asynchronous distributed beamforming and power control algorithm for the multi-cell networks. The algorithm is derived via investigating a multi-cell PMBP which assumes long-term CSI, non-reciprocal channel, and different noise variances at each user. Via exploring the special problem structure, we are able to propose a novel asynchronous iterative dual computation method to compute the dual of the multi-cell PMBP. Besides, beamforming and power control could be done within each cell by the BSs and the users asynchronously in a distributed fashion. At last, the convergence and performance of the algorithm are also demonstrated via simulation.

## 8Proof of Proposition

We first introduce the definitions of Z-matrix, P-matrix, and K-matrix for the proof of this proposition.

The K-matrix is related to the matrix radius according to the following lemma.

According to Lemma ?, we need to prove that the inequalities is the necessary and sufficient condition for . Since where is a K-matrix, we conclude that is a K-matrix according to Lemma ?.

Furthermore, is a P-matrix whose components are

According to the definition of P-matrix, we have

where is the component of vector corresponding to user .

If inequalities hold for all , we obtain which are equivalent to .

## 9Proof of Lemma

To prove this lemma, we resort to the theoretical results about the *semidefiniteness of a matrix pencil* [35]. We first introduce some necessary terminologies about matrix pencils, i.e., the following two definitions, which can be found in [35]. The eigenvalues of a matrix pencil are defined as follows.

With the above definitions, we start the formal proof of Lemma ?. More strict proof of similar results in the general case can be found in the study about semidefiniteness intervals of matrix pencil [11],[12]. Our proof is given as follows.

### 9.1Proof of Existence and Finiteness of :

According to [10], we have

Since by assumption, the minimum of exists and is finite due to convexity.

### 9.2Proof of :

Since is positive semidefinite, we have

Since , we have and .

If , then by assumption.

If , then based on .

Furthermore, is equivalent to

The value of is equivalent to the optimal value of the following problem

because we can always scale such that . According to [10], the minimizing vector is the eigenvector of matrix pencil , whereas the minimum is the corresponding eigenvalue. Thus, we can conclude that , i.e., being smaller or equal to the minimum non-negative eigenvalue of matrix pencil .

### 9.3Proof of :

This proof is based on the determinant of the matrix pencil .

Let be the eigenvalues of matrix given . We have

In addition, we have the following lemma.

Let . To prove the continuity is to prove

Let represents the th eigenvalue of matrix , assuming . We have

The matrix can be viewed as a perturbation to matrix .

Next, we resort to the *matrix perturbation theory*. Since are symmetric matrices, according to the Hoffman-Wielandt Theorem [37], we have

where represents Frobenius matrix norm. We can see that

In addition, since is finite, . Therefore, we can conclude that

That is, This completes the proof.

Moreover, based on Definition ?, we have

With , we have

It is easy to see that if , and all the eigenvalues are positive. Let increase continuously from . From the polynomial in , we can see that the determinant keeps to be positive until where . With Lemma ?, it implies that no changes sign during the process. So, we can conclude that if , then .

## 10Proof of Theorem

The proof of this theorem is motivated by the work of [13]. We need to prove is a standard function as defined in [13]. That is, for all , the following properties are satisfied:

Positivity: If , ,

Monotonicity: If , ,

Scalability: .

If is a standard function, then the remaining proof is identical to the proof in [13]. Our contribution here is to prove that the specific function is standard.

### 10.1Proof of Positivity:

The dual variables are greater than or equal to zero trivially. Since , are the non-negative eigenvalues, we have .

### 10.2Proof of Monotonicity:

To prove the monotonicity, we first propose the following lemma,

Let and be the corresponding generalized eigenvector. Similarly, let and be the corresponding generalized eigenvector.

From equation , we have

while

Since minimize subject to , we have