Joint optimization of transmit beamforming and receiver selection for cluster-based communications Yating Gao1,2 , Ningbo Zhang1,2 , Guixia Kang1,2 1
Key Laboratory of Universal Wireless Communications, Ministry of Education, Beijing University of Posts and Telecommunications, Beijing 100876, China. 2 Wuxi BUPT Sensory Technology and Industry Institute CO.LTD. E-mail:
[email protected],
[email protected],
[email protected] Abstract—In this paper, we consider a cluster-to-cluster (C2C) multicast scenario where multiple transmitters in a sending cluster cooperatively transmit a common known data packet to multiple receivers of the receiving cluster. The maximization of transmission rate is firstly formulated as a max-min fairness problem given the set of receivers, and an iterative transmit beamforming optimization algorithm is proposed to obtain the optimal transmit beamforming vector. Furthermore, in order to maximize the C2C system throughput, a joint optimization algorithm of transmit beamforming and receiver selection is proposed to search the optimal set of receivers based on the above transmit beamforming algorithm. The simulation results prove that the proposed algorithm has achieved a higher throughput than the conventional schemes in a lower complexity. Index Terms—Cooperative communication, transmit beamforming, receiver selection, cluster-to-cluster multicast, throughput.
I. I NTRODUCTION Over the last decade, rapid developments in Internet of Things have heightened the need for machine-type communication (MTC) service in the communication networks. Since most of the MTC devices are equipped with only one antenna, therefore, there is a growing body of literature [1], [2] that allows several MTC devices to collaborate with each other and perform the cooperative communication. The cluster composed of multiple geographically-close wireless MTC devices can be introduced by the exchange of network control packets to cooperate at the symbol-level and together transmit the message [3], [4]. In this case, a cluster is equivalent to a multipleantenna communication unit, and the cluster-to-cluster (C2C) multicast communication where multiple transmitters of a sending cluster cooperatively transmit a common known data packet to multiple receivers of a receiving cluster becomes a multiple-antenna multicast communication link. This paper focuses on the C2C multicast communication scenario [5], such as multimedia broadcast multicast service, intelligent transport system and disaster emergency system. Considering a MTC device is a single-antenna node, by applying the cooperative beamforming (BF) [6], [7], the cluster head (CH) of sending cluster organizes the cooperative transmission of cluster members to try to increase the number of receivers and maximize the C2C multicast throughput. Therefore, the performance of C2C multicast system depends on the design of transmit BF and the selection of receivers.
Some recent work [8]–[12] has studied the multiple-antenna multicasting by designing the optimal transmit BF vector. Sidiropoulos et.al. [8] put forward the max-min fairness problem that maximizes the minimum received signal-tonoise-ratio (SNR) among all receivers subject to the total transmission power constraint. Due to the NP-hardness of the problem, an semidefinite relaxation (SDR) algorithm for the design of physical-layer transmit BF was proposed, where the original problem was relaxed into a semidefinite program, and then a Gaussian randomization strategy was applied to generate a rank-one approximate solution. Another state-of-the-art algorithm for the problem of interest is the successive linear approximation (SLA) algorithm [9]. At each iteration, the SLA algorithm approximated the original non-convex problem and then solved the resulting convex quadratic program to generate the next iteration. However, the solutions of both SDR and SLA algorithms are sub-optimal, and have high complexity in solving the convex problem. Apart from the above algorithms, some other previous works [10]–[12] on the max-min problem aims at reducing the complexity, but their performance is worse than SDR and SLA algorithms. The transmission performance of C2C cooperative algorithm can be further improved if the receivers are optimally selected from the receiving cluster. The traditional C2C multicast communication generally takes all the cluster members of receiving cluster as the receivers. However, some cluster members with the bad channel states will put a constraint to the minimum received SNR as well as the maximum C2C transmission rate. Preventing these nodes from being the receivers can mitigate the difference in channel states, but it may cut down the C2C multicast throughput. Many existing criteria in channel states have been proposed for guiding the selection of receivers, such as position [13], channel gains [14] and spectral efficiency [15]. How to select an optimal set of receivers to achieve a trade-off between the difference in channel states and C2C multicast throughput is a potential problem. In this paper, we consider a C2C cooperative multicast communication where multiple transmitters of sending cluster share a common known data packet and cooperatively transmit it to multiple receivers of receiving cluster. The inter-cluster throughput optimization problem is formulated and expressed as the combination of the maximization of inter-cluster trans-
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mission rate and the optimization of receiver selection. In order to solve the problem, firstly, given a fixed set of receivers, we analyze the optimal BF vector and propose a transmit BF optimization (TBO) algorithm under the total power constraint. Then, in order to maximize the system throughput, a lowcomplexity joint optimization algorithm (JOA) of transmit BF and receiver selection is designed to search the optimal set of receivers based on the above TBO algorithm.
the maximum transmission power for a single node. The total transmission power for the transmit BF is
II. S YSTEM M ODEL
C = log (1 + γmin ) ,
We consider a C2C multicast scenario. As shown in Fig.1, there are a sending cluster Ct with N nodes and a receiving cluster Cr with M nodes. Each node is equipped with a single omnidirectional antenna. Denote CHT and CHR as the CH of sending cluster and receiving cluster, respectively. The positions of these two CHs are fixed and the clustering radius is set as r, the inter-cluster distance d is set as the distance between CHT and CHR . The remaining nodes are arbitrarily distributed in the circle area around two CH nodes. The channel between the transmitting node n and the receiving node m is modeled as hmn = βmn αmn , where αmn represents the large-scale fading between n and m which is proportional with d−k mn . βmn denotes the complex-valued small-scale fading.
2
Ptot = ∥w∥ .
(3)
Since the transmitters are separated from each other, the channel coefficient vector varies from each others. The maximum achievable C2C transmission rate (symbol/s) can be calculated [16] as (4)
where γmin = min (γ1 , γ2 , · · · , γM ). It can be observed that C depends on the minimum received SNR γmin . Furthermore, let R ⊆ Cr denote the node set of selected receivers, and M ′ is the number of receivers, i.e., |R| = M ′ . Define the inter-cluster throughput as the total number of symbols per second which have successfully arrived at the selected receivers. Therefore, it can be calculated as R = M ′ log (1 + γmin ) .
(5)
In this paper, we impose the sum power constraint (3) and explore the optimal transmit BF vector w as well as receivers set R to maximize the inter-cluster throughput R. The throughput optimization problem can be expressed as max
R
(6) 2
s.t. ∥w∥ = Ptot > 0 R ⊆ Cr . III. P ROPOSED J OINT O PTIMIZATION A LGORITHM In this section, we tackle the challenging problem (6) by decomposing it into two subproblems: One deals with the design of optimal transmit BF vector under the fixed receivers set, and the other aims at selecting an optimal receivers set R from the receiving cluster Cr .
Fig. 1. Cluster-to-cluster multicast communication model.
A. Proposed Transmit BF Optimization Algorithm Assuming that the channel state information (CSI) is available for the transmitters, CHT organizes N cluster members to cooperatively transmit a common [ ] known input signal x to the receiving cluster. Let E |x|2 = 1, the received signal for node m ∈ Cr can be expressed as ym = hm wx + zm ,
(1)
T
where w = [w1 , w2 , · · · , wN ] ∈ CN is the transmit BF vector. hm = [hm1 , hm2 , · · · , hmN ] ∈ CN represents the channel coefficient vector of node m. zm is the complex-valued additive white Gaussian noise (AWGN) with power σ 2 . The received SNR at node m is presented as w † Rm w |hm w| = , σ2 σ2 2
γm = †
(2)
where (·) stands for the complex conjugate transpose operation, and Rm = h† h. We assume that the C2C multicast transmission is performed under the total power constraint Ptot which is also
Suppose that the receivers R = Cr , the object function of (6) is equivalent to maximizing γmin . Therefore, the optimization problem (6) can be rewritten as max
γmin
(7)
2
s.t. ∥w∥ = Ptot > 0. Therefore, the optimal transmit BF vector can be expressed as w = arg max {γmin } .
(8)
w,R=Cr
When M = 1, the C2C communication becomes a multipleinput-single-output (MISO) transmission. Obviously, the maximum received SNR can be obtained by the maximum ratio transmission (MRT) [17], i.e., h† √ wopt = Ptot , h ∈ CN . (9) ∥h∥ When M ≥ 2, the objective function is not convex nor concave [8]. In this case, considering that the only way to maximize γj , j ∈ R is designing w in a MRT fashion similar
to (9), the solution (8) should be within the space spanned by M channel coefficient vectors hj , j = 1,· · ·, M , i.e., the optimal transmit BF wopt is the linear combination of hj , ∑M ∑M λ′j h†j = λj ej , (10) wopt = j=1
j=1
/
where ej = h†j h†j , j = 1,· · ·, M and {λj } , j = 1,· · ·, M satisfies the sum power constraint condition in (7). It can be seen from (10) that any pair of {λj } corresponds to a transmit BF vector. It is hard to obtain a general analytic solution {λj } due to the NP-hardness of problem (7), we propose the following TBO algorithm to approach the optimal solution. In order to maximize γmin , the main idea of proposed algorithm is to, randomly generate {λj } based on (10) firstly, and at each iteration, slowly steer wopt towards the beamforming vector that can improve γmin at that time, ∆
w → c (w + ∆d) ,
(11)
where c is a positive coefficient to impose the sum power constraint and |∆| denotes the step length. Note that the plus or minus sign of ∆ depends on the Euclidean angle between w and d, ⟨w, d⟩R , (12) θ (w, d) = cos−1 ∥w∥ · ∥d∥ where (·)R represents the real part of a complex value. θ (w, d) ∈ (0, π). d stands for the current target direction in which γmin can be improved. When θ (w, d) ≤ π/2, ∆ is positive, otherwise, it is negative. In this case, maximizing γmin is equivalent to searching the optimal d. Therefore, according to the difference of d, the proposed TBO algorithm consists of the following three steps. Step1: Initiate w as a normalized random linear combinations of {ej } by (10). Obtain γmin and, find the number (n) of receivers whose received SNR equals γmin . If n = 1, goto Step2; Otherwise, goto Step3. Step2: Find I = arg mini∈R {γi }, adjust w by the gradient vector of γI which intuitively aims to improve the received SNR of the worst receiver, i.e., γmin can be improved. From (2), the gradient vector of γI with respect to w can be obtained as 2RI w ∇w γ I = = c˜h†I , (13) σ2 where c˜ = 2hσI2w is a constant. Therefore, eI can represents the gradient vector. Let d = eI , update γmin and n. Then, iterate Step2 until n ≥ 2, in which case, if M = 2, algorithm ends, otherwise, goto Step3. Step3: Find the worst two receivers I = arg mini∈R {γi } , J = arg minj∈R/{I} {γj }. Decompose w into three orthogonal vectors y1 , y2 , yw by a Gram-Schmidt process of eI , eJ and w. yw is orthogonal to the space S spanned by y1 and y2 . So we can express w as w = wS + w⊥ ,
(14)
where w⊥ = ∥yw ∥ × yw . Obviously, reducing w⊥ can improve γmin . Therefore, set d = −yw , improve the channel conditions of both I and J by adjusting w towards the space S. Update γmin , n and goto Step2 or Step3 based on n. The above process lasts until γmin stops increasing. Different from the traditional SDR [8], SLA [9] algorithms, where the problem (7) were relaxed as a convex problem firstly, and then the interior point method [18] was applied to obtain the approximate solutions, the proposed TBO algorithm approaches the optimal transmit BF vector by gradually adjusting w and improving the received SNR of the instant channelworst receivers. Therefore, it can be observed from (11) that the complexity of the proposed TBO algorithm depends on the iterative times which is inversely proportional to the step length |∆|, and in the worst case, the adjustment of w on one two-dimensional space requires up to (π/2 |∆|) iterative times, further, the upper bound of total iterative times are M −1 (π/2 |∆|) , and usually are much less, especially when w is initiated as ej , j = arg minj∈R {∥hj ∥}, the normalized channel coefficient vector of the initial channel-worst receiver. Moreover, the smaller the value of |∆| is set, the higher the accuracy of the result will be. B. Proposed Joint Optimization with the receiver selection The above TBO algorithm shows that when the number of receivers M ′ decreases, the optimal γmin will increase which improves the maximum transmission rate C, but it may also lower the inter-cluster throughput R as shown in (5). In this section, in order to maximize R, we propose a JOA to achieve the joint optimization of transmit BF and the set of receivers R by combining the proposed TBO algorithm and the optimization of receiver selection. The maximum γmin under a given R can be obtained from the above TBO algorithm. That is, different R corresponds to the specified wopt , and maximizing R is equivalent to optimizing R. Therefore, the problem (6) can be expressed as {R∗ , wopt } = arg max {R (R, w)} .
(15)
R⊆Cr ,w∈CN
(15) can be solved by combining the proposed TBO plus exhaustive search algorithm (ESA) [13]. However, the complexity of ESA exponentially increases with the receiving cluster size M since the number of subsets of Cr is 2M − 1. When M is large, ESA tends to be inefficient. In order to solve this set optimization problem, the following JOA is proposed to obtain the optimal R from the receiving cluster Cr . For a given R, it can be observed from the above TBO algorithm that wopt is obtained by continuously adjusting w to improve the received SNR of the instant channel-worst receivers. From (2) we can see that the received SNR γj at node j ∈ R is equivalent to the scalar projection of hj onto w, therefore, based on the Euclidean angle (12) between ej
and the final wopt , ϕj can be used to measure the channel condition of receiver j, [ ] { θ (wopt , ej ) , θ (wopt , ej ) ∈ ( 0, π2 ] ϕj = . (16) π − θ (wopt , ej ) , θ (wopt , ej ) ∈ π2 , π {ϕj } indicates the constraints on the minimum received SNR by receiver j, and a smaller ϕj represents the stronger constraint. Therefore, removing the receiver with the worst channel condition from R becomes the most efficient way to release the potential of improving γmin and C, further, it is possible to improve the inter-cluster throughput R. On the basis of above analysis, the pseudocode of the proposed iterative JOA procedure for searching the optimal receivers is shown in Algorithm 1. In Algorithm 1, R is initiated as Cr . The optimal wopt can be obtained by the proposed TBO algorithm, and then remove the receiver with the minimum ϕj , j ∈ R. Lines 2-11 show the iterative search process for the optimal receivers by removing the receiver with the worst channel condition from R. Compared with the ESA, the proposed receiver selection algorithm has greatly reduced the searching times from 2M − 1 to M . IV. S IMULATION R ESULTS In this section, simulation results are provided to investigate the performance of proposed JOA. Under the system setup in Fig.1, we simulate a C2C multicast communication scenario where CHT and CHR are fixed, then N−1 and M−1 nodes are randomly distributed in a circular area around two CHs, respectively. The distance between CHT and CHR is taken as the inter-cluster distance. The channel hij follows i.i.d. Rayleigh 2 −k fading [19], specifically, |hmn | = 10−4 ρ2mn (dmn ) , where ρmn follows the standard Rayleigh fading, k = 3.5. The transmit sum power Ptot is fixed as 20dBm, the noise power is set to −70dBm, the clustering radius r is set as 5m. Moreover, in order to reduce the iterative times of proposed TBO algorithm, w is initiated as ej , j = arg minj∈R {∥hj ∥}, the normalized channel coefficient vector of the initial channelworst receiver. The simulation results are averaged over node locations and channel realizations.
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The minimum received SNR (γmin ) (dB)
Algorithm 1 Proposed Joint Optimization Algorithm Input: Ct , Cr , ei , i ∈ Cr Output: R, wopt , Rmax 1: R ← Cr , M ′ ← M, Rmax ← 0 2: while M ′ > 0 do 3: Get γmin , wopt by TBO algorithm as Sect.3.A 4: Calculate R by (5) 5: if R ≥ Rmax then 6: Rmax ← R, record R, wopt 7: end if 8: Calculate ϕi , i ∈ R by (16) 9: R ← R/{I}, I = arg mini∈R {ϕi } 10: M ′ ← M ′ −1 11: end while 12: return {R, wopt , Rmax }
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Fig. 2. The convergency time versus the different step length |∆|.
It can be seen from the proposed TBO algorithm that the complexity depends on the selection of step length |∆|. Therefore, we firstly investigate the impact of |∆| on the convergency time which is measured with the iterative times. Set the cluster sizes as N × M = 5 × 5, and the intercluster distance d = 20m. Three different values of |∆| = {0.1, 0.01, 0.001} are simulated respectively. Fig.2 shows the convergency time with the different step length. When |∆| is set as {0.1, 0.01, 0.001}, the converged minimum received SNR γmin is {3.3770, 3.9297, 4.0083}(dB), and the corresponding iterative times is {117, 3596, 12084}, respectively. This proves that the larger |∆| is, the more accurate the maximum γmin can be obtained. Meanwhile, the actual iterative times is far less than the analyzed upper bound. Moreover, Fig.2 shows that when |∆| drops from 0.1 to 0.01, the iterative times has increased to 30 times, and the achieved maximum γmin has increases by 16.4%. Correspondingly, when |∆| changes from 0.01 to 0.001, the increases of iterative times and achieved maximum γmin becomes 3.36 and 2%, which shows that there has been an exponential increase in the iterative times when |∆| drops. Therefore, to achieve a proper trade-off between the optimality of the solution and the computational complexity, |∆| is set as 0.01 in the following simulations. Then, in order to show effectiveness of the proposed TBO algorithm, the traditional SDR and SLA algorithms are performed to find the optimal transmitted BF vector and provide the baseline performance. Furthermore, by comparing the proposed JOA with TBO algorithm to prove the necessity in optimizing the receiver selection. Meanwhile, ’TBO+ESA’, the combination of proposed TBO algorithm and the traditional ESA, is also simulated to prove the effectiveness of the proposed JOA. The second simulation is designed to demonstrate the effect of varied cluster size on the performance of C2C communications. In this experiment, the inter-cluster distance is set as {20, 30}m. Fig.3 plots the maximum transmission rate C versus the
22 Prop. TBO d=20m Trad. SLA d=20m Trad. SDR d=20m Prop. TBO d=30m Trad. SLA d=30m Trad. SDR d=30m
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Fig. 3. The maximum transmission rate versus the sending cluster size with the inter-cluster distance d = {20, 30}m, M = 5.
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Fig. 4. The maximum transmission rate versus the receiving cluster size with the inter-cluster distance d = {20, 30}m, N = 5.
sending cluster size N with the receiving cluster size M fixed as 5. As we can see, C increases with the increase of sending cluster size due to the increased gains from the transmit diversity. Meanwhile, Fig.3 also shows that the proposed TBO algorithm always outperforms the traditional SDR and SLA algorithms, and the gap between TBO algorithm and compare objects becomes bigger as N increases, this proves that the proposed TBO algorithm has obtained a better transmit BF vector on the problem (6). Moreover, when d increases from 20m to 30m, the distance between any pair of transmitterreceiver nodes increases, therefore, the the maximum transmission rate C drops. Fig.4 shows the maximum transmission rate C versus the receiving cluster size M with the sending cluster size N fixed as 5. It can be observed from Fig.4 that C drops with the increase of receiving cluster size M , which is because the increase of M exacerbates the difference of channel conditions and lowers the minimum received SNR. Moreover, Fig.4 presents that the proposed TBO algorithm always outperforms the traditional algorithms in both cases of d, it shows the effectiveness of
Fig. 5. The inter-cluster throughput versus the receiving cluster size with the inter-cluster distance d = {20, 30}m, N = 5.
TBO algorithm in solving the max-min fairness problem (6). It is noted that both Fig.3 and Fig.4 show that the traditional SLA algorithm obtains relatively close performance to the proposed TBO algorithm, but with much more computational burden due to iteratively solving a series of convex problems. Furthermore, Fig.5 shows the throughput R versus the receiving cluster size M with the sending cluster size N fixed as 5. We can observe from the figure that R grows with the increase of receiving cluster size M . The gap between the proposed JOA and TBO represents the gains from the optimization of receiver selection, and it becomes bigger and bigger as the receiving cluster size M increases, which is because the larger M results in the greater difference on the channel conditions of receivers, so that the constraints on the maximum transmission rate by the channel-worst receivers become more obvious. In this case, the optimization of the receiver selection becomes more necessary. Besides, the comparison of proposed JOA and ’TBO+ESA’ proves that the results of the proposed receiver selection basically matches with ESA, with much less computational complexity. The third simulation is designed to demonstrate the effect of varied inter-cluster distance d on the performance of C2C communications. In this experiment, d ranges from 15 to 50m, and the cluster sizes are set as {5 × 5, 8 × 8}. Fig.6 shows the maximum transmission rate C versus the inter-cluster distance d. It can be observed from Fig.6 that C drops with the increase of the inter-cluster distance because of the increased path loss. Meanwhile, the achieved C of proposed TBO algorithm is always higher than the compare objects which proves the effectiveness of the proposed algorithm. Besides, Fig.6 also shows that when the cluster sizes change from 5 × 5 to 8 × 8, C with the traditional SDR algorithm decreases while that of the others grows. The reason for this result is as follows: the increased sending cluster size can increase the transmission diversity which can improve the performance, meanwhile, the increased receiving cluster size exacerbates the differentiation in channel conditions of receivers which will pull back the gains. In contrast with the
and the joint optimization algorithm for transmit BF and receiver selection is proposed to obtain the optimal receivers set and maximize the inter-cluster throughput. Simulation results showed that the proposed TBO algorithm has effectively improved the maximum transmission rate compared with the traditional SDR and SLA algorithms. Further simulations have been performed to prove the necessity and effectiveness of the proposed joint optimization.
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Fig. 6. The maximum transmission rate versus the inter-cluster distance with the cluster sizes 5 × 5, 8 × 8.
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ACKNOWLEDGMENTS This work was supported by the National Science and Technology Major Project of China (No.2017ZX03001022, 2016ZX03001012).
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Fig. 7. The inter-cluster throughput versus the inter-cluster distance with the cluster sizes 5 × 5, 8 × 8.
SDR algorithm, the obtained BF vectors for both TBO and SLA algorithms have effectively reduced differentiation so that C has achieved an increase. Fig.7 plots the throughput versus the inter-cluster distance d. As we can see, similar to the results of Fig.6, the throughput R drops with the increase of d, and the proposed JOA outperforms the TBO algorithm which proves the necessity in the optimization of receiver selection. Moreover, the performance of proposed JOA basically matches with the results of ’TBO+ESA’ which proves the effectiveness of proposed receiver selection algorithm. V. C ONCLUSION This paper proposed a JOA for transmit BF and receiver selection to solve the throughput optimization problem in a C2C multicast communication scenario. By formulating the transmit BF optimization problem as a max-min fairness problem, we proposed the TBO algorithm to find the optimal transmit BF vector and maximize the inter-cluster transmission rate with the fixed receivers set. Furthermore, the receiver selection is also introduced into the optimization problem
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