Transmit Beamforming Optimization Design for Broadband Multigroup Multicast System

Spectral efficient transmission techniques are necessary and promising for future broadband wireless communications, where the quality of service (QoS) and/or max-min fair (MMF) of intended users are often considered simultaneously. In this paper, both the QoS problem and the MMF problem are investigated together for transmit beamforming in broadband multigroup multicast channels with frequency-selective fading characters. We first present a basic algorithm by directly using the results in frequencyflat multigroup multicast systems (Karipidis et al., 2008), namely, the approximation algorithms in this paper, for both problems, respectively. Due to high computational consumption nature of the approximation algorithms, two reduced-complexity algorithms for each of the two problems are proposed separately by introducing the time-frequency correlations. In addition, parameters in the new time-frequency formulations, such as the number of optimization matrix variables and the taps of the beamformer with finite impulse response (FIR) structure, can be used to make a reasonable tradeoff between computational burden and system performance. Insights into the relationship between the two problems and some analytical results of the computational complexity of the proposed algorithms are also studied carefully. Numerical simulations indicate the efficiency of the proposed algorithms.


Introduction
Targeting for supporting high throughput and link reliability, multiple-antenna transmission techniques have prevailed in the development of terrestrial wireless communication systems, such as Long Term Evolution Advanced (LTE-A) [1] and future mobile telecommunication networks [2].Not surprisingly, when equipped with multiple antennas at the transmit side, physical-layer multicasting renders its great advantage in spectral efficiency over the other communication mechanisms, especially for some particular applications, including network video service and online gaming.In this regard, transmit beamforming has received enormous attentions in the literature, where perfect channel state information (CSI) is assumed available at both ends and the channel between each transmit antenna and receive antenna appears frequency-flat fading property.
To provide performance assurance to each of the intended receivers in multicast systems, the quality of service (QoS) problem [3] and the max-min fair (MMF) problem [4] are usually formulated and investigated in the literature, where the criteria of minimizing the total transmission power under each user's minimum signal-to-interference-plus-noise ratio (SINR) constraint and maximizing the minimum SINR among all users under the average transmit power constraint are considered, respectively.Due to the NP-hardness of the optimization problems, several efficient algorithms have been proposed to guarantee satisfactory performance in singlegroup multicast scenario.What is more, an analytical result of these two problems was extended to the case of multigroup multicasting in [5], where a solid algorithm based on both semidefinite relaxation (SDR) [6] and Gaussian randomization was proposed to solve the multigroup QoS problem and then an iterative algorithm based on a one-dimensional bisection search [7] was also adopted to handle the MMF problem.To achieve improved performance for the multigroup QoS problem, the authors of [8] proposed an iterative algorithm which solves an approximate second-order cone Multigroup multicast with PACs SDR + Gaussian randomization + bisection [12] programming (SOCP) problem in each iteration.And in [9], an iterative algorithm with low complexity and superior performance was further investigated to cope with the multigroup MMF problem.Recently, contrary to the total transmission power constraints, transmit beamforming under perantenna power constraints (PACs) was introduced in [10][11][12].As an extending work of [5], the weighted multigroup multicast MMF problem with PACs was investigated in [12].
For the sake of clear expression, related works are listed in Table 1.
As aforementioned, the design of transmit beamformer has been primarily studied over multiple-antenna multicasting and frequency-flat fading channels and, to the authors' best knowledge, few works have been dedicated to the case of frequency-selective fading multicasting scenario.Motivated by the potential advantages of multiple-antenna transmission over the frequency-selective fading channels, we are concerned about the QoS and MMF problems in this paper for multigroup multicasting.The main contributions of this paper are listed in the following: (1) For the QoS problem, an approximation algorithm is firstly derived for broadband systems based on the idea of narrowband multigroup multicasting in [5].And two reduced-complexity algorithms are also proposed from the frequency-domain and the timedomain perspectives separately.Some parameters in correlation with the QoS problem are analyzed and computational consumption is comparatively computed to show more insights into the tradeoff between performance and complexity.
(2) For the MMF problem, the corresponding approximation algorithm and its low-complexity modifications are also proposed in a similar way as that of QoS problem.Furthermore, the relationship between the QoS problem and the MMF problem is discussed carefully followed by a complexity analysis.
(3) Simulation experiments demonstrate the effectiveness of the frequency-domain and the time-domain algorithms for both QoS problem and MMF problem.
The main analysis results that the controlled parameters in the proposed algorithms could be used to make a tradeoff between complexity and performance are verified through the numerical examples.
The remainder of this paper is structured as follows.In Section 2, the system model and the QoS problem are introduced briefly.The approximation solution to the QoS problem is derived in Section 3.And Section 4 presents two beamforming algorithms.Section 5 formulates the MMF problem and solves it based on the proposed algorithms.Computational complexity analysis of proposed algorithms is given in Section 6.In Section 7, the performance of the proposed algorithms is evaluated and discussed.Finally, the conclusion is summarized in Section 8.

Notations.
In the remainder of this paper, boldface uppercase letters and math calligraphy uppercase letters denote matrices, and boldface lowercase letters denote vectors.(⋅) † , E(⋅), and tr(⋅) are the conjugate transpose, the expectation, and the trace operator, respectively.C indicates the set of complex numbers, while ⊗ is Kronecker product.⌊⋅⌋ defines the floor function, A 1/2 is matrix square root function of A, and (⋅) is the impulse function.A ∼ CN(0, I) means that A is circularly symmetric complex Gaussian process with zero mean and unit variance matrix.

System Model and Problem Statement
2.1.System Model.Consider a multigroup multicast system with one transmitter (base station) and  receivers (users).Assume the transmitter has   antenna elements and each receiver is equipped with one antenna.The users are split into 1 ≤  ≤  groups { 1 ,  2 , . . .,   }, each containing    user indices.We assume that each user listens to a single multicast group; that is,   ∩    = 0, where  ̸ =   , and ∪    = {1, . . ., }.A frequency-selective fading channel with  effective paths is supposed between each transmit antenna and receive antenna, and full CSI is available a priori at the transmit side throughout this paper.
With an -tap FIR beamforming filter, the transmitted signal can be written as follows in space-time domain: where w , ∈ C   ×1 and  , denote the beamforming vector and the discrete information sequence in association with the Figure 1: The schematic diagram of a multigroup multicast system.
th tap of FIR beamforming filter for the th group, respectively. stands for the time index.Figure 1 shows the schematic diagram of a multigroup multicast system.Without loss of generality (W.L.O.G.), assume the information sequence is zero mean with unit variance and mutually uncorrelated; that is, E{ ,  ,− } = () ⋅ ().Then for the th user in the th group, the received signal has the form as where H ,,V ∈ C 1×  is the channel impulse response of the Vth path between the transmitter and the th user in the th group and  ,, is an additive Gaussian noise at the th user with zero mean and unit variance.
For brevity purpose, (2) can be represented by transform; that is, where In this regard, the total transmission power of the multigroup multicast system becomes Similarly, the SINR at the th receiver in the th group can be formulated as where ∀ ∈   and ∀ ∈ {0, . . .,  − 1}.To solve this problem, the following discrete-time form is usually adopted; that is, with where  is a sufficiently large positive integer, H ,, = H , ()| = 2/ , and Note that problem P2 is a discrete approximation of problem P1, and its approximate accuracy increases as  approaches infinity.In fact, it is a quadratically constrained quadratic programming (QCQP) problem with nonconvex constraints.Moreover, as a special case of this problem, multigroup multicasting over frequency-flat fading channel (i.e.,  = 1) has been proven to be NP-hard in [5].Therefore, problem P2 is NP-hardness, which motivates us to pursue an approximate solution of it.
Due to the nonconvex nature of problem P3, we drop the  ⋅  rank-one constraints and obtain a semidefinite programming (SDP) variation It is noteworthy that problem P4 can be handled by interior point method (IPM) [13] and the feasible set of this problem is actually a superset of that of problem P3.As a consequence, the optimum objective value of problem P4 is certainly equal or less than that of problem P3.

Proposition 1. The optimal solution {Q
Proof.In order to prove this proposition, the dual problem of problem P4 is first considered and formulated as where and then we can calculate the rank of Θ , Q , by defining On the other hand, according to Theorem 3.2 in [14], it is readily to verify that the optimal solution to problem P4 satisfies From Proposition 1, it appears that when each group has only one user, that is,    = 1, we have rank(Q , ) ≤ 1, which means problem P4 is actually equivalent to problem P3 in this case, which will obviously result in a frequency-selective fading extension of the work in [3].In general, if the solution of problem P4 meets the rank-one constraint, that is, rank(Q , ) = 1, an eigenvalue decomposition (EVD) of Q , = U , Λ , U † , may help to generate the frequency-domain beamforming vectors, where W , = U , Λ 1/2 , (:, 1).Otherwise, the Gaussian randomization technique [5] is used to obtain candidates of the beamforming vector; that is, {W , V  , },  = 1, 2, . . ., , where  is the maximum number of the randomizations and V  , ∈ C   ×1 ∼ CN(0, I).Note that although these processed candidates satisfy they may still violate SINR constraints.For each candidate, a feasible allocated power should thus be figured out by solving a multigroup multicast power control (MMPC) problem; that is, where  , denotes the power factor for the beamformer W and can be easily solved by basic convex tools if the optimal solution exists.The resulting frequency-domain beamformer can thus be generated by √  , W  , , and the associated objective value (1/) ∑ −1 =0 ∑ −1 =0  ,  , is recorded.When  reaches , the beamformer corresponding to the best candidate with minimum objective value can be selected as the optimal one.The detailed process of the approximation algorithm is summarized in Algorithm 1.
More clearly, the approximation algorithm can be treated as a direct use of the algorithm in [5] on every frequency bin for frequency-selective multigroup multicast system.The higher approximation accuracy is, the larger  may be introduced.For example, an  = 128 or larger is often needed for practical systems, which will lead to extremely high computational complexity.A computing-strong ability is thus required at the transmitter; otherwise we may not figure out the exact beamformers by the approximation algorithm when the channel coefficients change with rapid fluctuation.

Proposed Beamforming Algorithms
To combat the heavy computation burden of the approximation algorithm, new beamforming algorithms which can make tradeoff between performance and complexity are eagerly demanded in these situations.Towards this end, two beamforming algorithms are proposed from perspectives of frequency domain and time-domain, respectively, in this section.and the optimization problem can be converted to P5: min

Beamforming Design in
where ∀ ∈ {0, . . ., / − 1}, ∀ ∈   , and ∀ ∈ {0, . . .,  − 1} in this subsection.Similarly, by dropping the / rank-one constraints, an SDP problem can then be obtained as and the corresponding MMPC problem follows that M2: min where  , denotes the power factor for the beamformer As a matter of convenience, denote P5(, ) as the frequency-domain problem with parameters  and .Here  ≜ [ 0,0 ,  0,1 , . . .,  0, , . . .,  , ].Obviously, the proposed beamforming design in frequency-domain can be treated as a special case of the approximate solution presented in the previous section.In other words, if the parameter  is set to be 1, the frequency-domain QoS problem P5(, 1) is equivalent to the QoS problem P3().

Beamforming Design in Time-Domain.
Besides reducing the number of optimization variables in frequency-domain immediately, an alternative way can also benefit the complexity reduction by cutting down the number of the FIR filter taps from time-domain perspective.Assume the timedomain FIR filter has  taps, where  is far less than ; the transmitted signal in (1) changes to Define which can be converted into an equation in Kronecker form . . .
with K  ∈ C   ×  and W  ∈ C   ×1 .In the sequel, the total transmission power is reformulated accordingly as By defining Q  = W  W †  , the QoS problem thus becomes P7: min with and ∀ ∈   , ∀ ∈ {0, . . .,  − 1} in this subsection.Similar to previous two algorithms, an SDP problem is obtained after dropping  rank-one constraints: P8: min and the corresponding MMPC problem can be represented as where   is the power factor for the beamformer W It can be confirmed that {Q exp  } −1 =0 is also a feasible solution to problem P7(,  2 ) by substituting it.Proposition 5. Assume P7(,  1 ) is feasible with a fixed set of channel vectors, SINR constraints, and noise powers with optimal value  3 .If  2 is greater than  1 , the optimal value of problem P7(,  2 ), defined as  4 , is less than or equal to  3 ; that is,  4 ≤  3 , and the equality holds up if and only if the solutions of these two problems are the same.
The basic idea of the proof process is similar to that of Proposition 3 and thus omitted here.We can replace , and then the result can be reached.

Max-Min Fair Problem
In addition to the QoS problem, another problem always considered in a multigroup multicast system is the MMF problem.The original problem of maximizing the minimum SINR of all users under the total transmission power constraint can be written as In fact, this problem contains a special case with multicast over frequency-flat fading channel ( = 1), which has been proven to be NP-hard in [5]; therefore problem Q1 is also NPhard.By virtue of the idea for solving QoS problem, it can be relaxed by dropping the rank constraints However, contrary to the QoS problem P4, problem Q2 cannot be transformed into an SDP problem due to the existence of  nonlinear inequality constraints.The causes of these nonlinear inequality constraints is that the SINR target  for all users is no longer a constant but a variable in the MMF problem.
Fortunately, problem Q2 can be relaxed and its  inequality constraints can be changed into linear constraints for a given .Thus the bisection search method [5,15] can be used to deal with this problem.Note that after getting some beamforming candidates, an MMPC problem is considered here where all variables have been defined in Section 3. Due to the variation property of , problem M4 cannot be solved as an equivalent LP.Therefore, we continue to rely on bisection search method to solve this problem.Alike the QoS problem, the reduced-complexity MMF problem can also be considered both in frequency domain and in time-domain, where the frequency-domain version can be formulated as

Q3:
max Drop the rank constraints and we can obtain the relaxed problem as follows: ≤ , ∀, ∀, Q6: max before and after rank relaxation, respectively.To solve problems Q3 and Q5, the same idea can be found when solving problem Q1.To illustrate the procedure of the proposed algorithms, a general solving framework for QoS and MMF problems is shown in Figure 2. Denote Q1(, ), Q3(, , ), and Q5(, , ) as for problem Q1, problem Q3, and problem Q5, respectively, with particular parameters , , , and .It can be seen that if the parameter  is set to be 1, the frequency-domain MMF problem Q3(, , ) is equivalent to the MMF problem Q1(, ).Also, the time-domain MMF problem Q5(, , ) is equivalent to problem Q1(, ) too, if  = .
The following analytical results demonstrate the relationship between the MMF problems with different parameters.
is the optimal solution to the frequency-domain MMF problem Q3(, ,  1 ) with optimal value  1 and is the optimal solution to problem Q3(, ,  2 ) with optimal value  2 .The sufficient condition for  2 ≥  1 is that  1 is divisible by  2 , and the equality holds up if and only if the solutions of these two problems are the same.
=0 is the optimal solution to the time-domain MMF problem Q5(, ,  1 ) with optimal value  3 and {Q opt2  } −1 =0 is the optimal solution to problem Q5(, ,  2 ) with optimal value  4 .The sufficient condition for  4 ≥  3 is that  2 is greater than  1 and the equality holds up if and only if the solutions of these two problems are the same.Furthermore, from the problem formulation, it appears that the MMF problems are always feasible, while things could be different for the QoS problem.The relationship between QoS and MMF problems for narrowband multigroup multicast case is discussed in [5].Results therein can be also extended to the broadband multigroup multicast case (i.e.,  > 1).For completeness, several valuable conclusions are drawn here.Proposition 8.For a fixed set of channel vectors and noise powers, the QoS problem P3 is parameterized by , where  = [ 0,0 ,  0,1 , . . .,  0, , . . .,  , ].Then it can be represented as P3().Likewise, the MMF problem Q1 is parameterized by  and , that is, Q1(, ).The QoS problem P3 and the MMF problem Q1 have the relationship as Proof.Define {Q opt , } as the optimal solution to problem Q1(, ), and its corresponding optimal value is  opt .It is easy to verify that {Q opt , } is a feasible solution to problem P3( opt ) and the corresponding optimal value is .Assume there is a feasible solution {Q fea , } to problem P3( opt ) and  fea is the associated optimal value which satisfies  fea < , and we can distribute the power  −  fea to all {Q fea , } evenly to obtain a larger optimal value  fea than  opt under the same power constraint.It contradicts the optimality of {Q opt , } for problem Q1(, ) which means that the assumption of {Q fea , } is wrong and (36) has been proved.
In order to prove (37), a similar process could be applied.Define {Q  opt , } as the optimal solution and   opt as the associated optimal value to problem P3() (

Complexity Analysis
First of all, the computational complexities of solving the QoS problems are discussed in this section.For the approximation algorithm derived in Section 3, the SDP problem P4 has  matrix variables with size   ×  t and  linear inequality constraints.Based on the results in [13], it takes O( 0.5  0.5  0.5  log(1/)) iterations, and each iteration needs O( 3  3  6  +  2  ) arithmetic operations. is the accuracy of the solution here.When solving MMPC problem M1 it takes O( 0.5  0.5 log(1/)) iterations, and each iteration needs O( 3  3 + ) arithmetic operations.Assuming the parameter  is same for all algorithms for the sake of simplicity, the total computational complexity for solving the QOS problem P2 is thus O(( 3.5  3.5  6.5  +  1.5  1.5  2.5  +  3.5  3.5 +  1.5  1.5 )log(1/)).
For the time-domain beamforming algorithm, the SDP problem P8 has  matrix variables with size   ×   and  linear inequality constraints.According to above results, it takes O( 0.5  0.5  0.5  log(1/)) iterations, and each iteration needs O( 3  6  6  + 2  2  ) arithmetic operations.When solving MMPC problem M3 it takes O( 0.5 log(1/)) iterations, and each iteration needs O( 3 + ) arithmetic operations.Therefore the total computational complexity From the relaxation process we can see that the feasible set of the relaxed SDP problems is indeed a superset of one of the original QCQP problems, which leads to the following conclusions: If the relaxed problems are not feasible, the original ones are not feasible either.Rather, if the relaxed problems are feasible, the original ones may be not feasible.Therefore, the feasibility of the SDP problems is a necessary condition for the validity of proposed algorithms.In this subsection, the feasibility of the SDP problems is evaluated under conditions of different number of multigroups , transmit antennas   , and users .All users are equally distributed into the groups, which means that each group has / users in the simulation.
Figure 4 shows the feasibility of SDP problems in different cases.Because the approximation algorithm works well in all situations, which has all 100 percent feasibility for all SINR constraints, we just use one red line marked "approximation" to present it.By comparing the line pairs, some conclusions can be obtained.For example, with all other things being equal, we can get that the frequency-domain SDP problem with less users is feasible with higher probability by comparing the " = 2   = 6  = 8  = 8" line and " = 2   = 6  = 12  = 8" line (for time-domain SDP problem we can use " = 2   = 6  = 8  = 4" line and " = 2   = 6  = 12  = 4" line).The reason lies in that the more the users are, the more the interuser interference exists in multigroup system.By comparing different line pairs, results can be concluded as follows: (i) The frequency-domain (or time-domain) SDP problem with less users is feasible with higher probability (just mentioned above).

Figure 2 :
Figure 2: Block-diagram of the solving framework.

Figure 3 :
Figure 3: The total arithmetic operations for QoS problems.

Table 1 :
List of related works.