Distributed Power Control in Full Duplex Wireless Networks

Transcription

1 Distributed Power Control in Full Duplex Wireless Networks Yu Wang and Shiwen Mao Departent of Electrical and Coputer Engineering, Auburn University, Auburn, AL Eail: Abstract In this paper, we consider the proble of distributed power control in a full duplex (FD) wireless network consisting of ultiple pairs of nodes, within which each node needs to counicate with its corresponding node. We ai to find the optial transit powers for the FD transitters such that the network-wide capacity is axiized. Based on the high signal-to-interference-plus-noise ratio (SINR) approxiation and a ore general approxiation ethod for logarith functions, we develop effective distributed power control algoriths with the dual decoposition approach. The proposed algoriths are validated with siulation studies. I. INTRODUCTION In this paper, we investigate the proble of distributed power control for a wireless network where the nodes are capable of full duplex (FD) transissions. Although full duplex transission has been used in wireline network for years (e.g., Asyetric Digital Subscriber Line (ADSL) based on echo cancellation), FD transission in wireless networks has becoe feasible only in recent years. Wireless FD systes are ade possible by breakthroughs in self-interference cancellation [1], [2], such as propagation-doain suppression (PDIS) [3], [4], analog-doain interference cancellation, and digital doain interference cancellation (ADIC) [5]. Practical FD systes have been deonstrated that can achieve ore than 70 db [5] or 80 db [6] reduction of self-interference. FD can be incorporated in a wireless network in two ways: (i) point-to-point ode, where two nodes transit to each other siultaneously, and (ii) three-node ode, where a node (e.g., a cellular base station (BS)) siultaneously receives fro a node and transits to another node. FD brings about new challenges to the design of power control algoriths. In a traditional half-duplex (HD) network, if a node increases its power, the signal-to-interference-plus-noise ratio (SINR) at its target receiver can be iproved, but with larger interference to other receivers. In an FD network, an increased power causes not only larger interference to other receivers, but also larger residual self-interference to the node itself and ay even degrade its own SINR. To fully harvest the high potential of FD wireless networks, the power control proble should be carefully addressed with effective algoriths developed. In this paper, we consider an FD wireless network consisting of ultiple node pairs, where the two nodes in each pair transit to each other (i.e., the point-to-point ode). We first consider the sipler case with a single pair of nodes [12]. Taking advantage of the structure of the forulated optial power control proble, we show that the optial solution should be at the boundary of the feasible region. Furtherore, we consider the case of ultiple node pairs in the FD network. Based on the high SINR approxiation, we can transfor the optial power control proble into a convex proble in the high SINR region. We then develop a Dual Decoposition based distributed algorith to find the optial solution to the transfored proble [7]. Finally, we consider the general case and drop the high SINR assuption. Incorporating an iterative approxiation ethod for logarith functions [8], we are able to obtain a convex transforation of the original power control proble and develop a distributed algorith. The proposed algoriths are validated with siulations, where fast convergence and large gain over the traditional HD syste are deonstrated. The reainder of this paper is organized as follows. The syste odel and preliinaries are introduced in Section II. We investigate the case of a single pair of FD nodes in Section III and the general case of an FD network in Section IV. Siulation results are presented in Section V. Section VI reviews related work and Section VII concludes this paper. II. SYSTEM MODEL AND PRELIMINARIES A. Channel and Propagation Models Consider an FD wireless network that only consists of two transceivers, denoted as nodes a and b, respectively. The received signals at the two transceivers are { ya P b h ba x b + P a h aa x a + n a y b P a h ab x a + (1) P b h bb x b + n b, where x a and x b are the noralized transitted signals fro node a and b, respectively; P a and P b are the corresponding transit powers; h ab and h ba are the channel gains fro node a to b and fro node b to a, respectively; h aa and h bb are the self-interference channel gain at node a and b, respectively; n a and n b are the theral noises. According to [9], the axiu likelihood channel estiation can be odeled as h ĥ + e h, (2) where ĥ is the estiated channel; e h is the estiation error that is uncorrelated with the real channel h; and h is i.i.d Gaussian with a zero ean and unit variance /15/$ IEEE 1165

2 B. Propagation-Doain Interference Suppression Self-interference cancellation is the enabler of FD systes. Propagation doain interference suppression (PDIS) can draatically reduce the self-interference [10]. Applying PDIS, the self-interference ters, e.g., P a h aa x a, in received signals (1) will be reduced to a fraction κp a h aa x a. Considering both PDIS and channel estiation, the received signals in (1) becoe [12] y a P b ĥ ba x b + κp a (ĥaa + ) e aa haa x a + n a y b P a ĥ ab x a + κp b (ĥbb + ) e bb hbb x b + n b. (3) C. Analog and Digital Interference Cancellation In addition to PDIS, analog and digital interference cancellation (ADIC) can cancel ore known self-interference. For exaple, in the first equation in (3), the first ter P b ĥ ba x b is the expected signal, and the rest ters are interference and noise. However, κp a ĥ aa is already known by node a, since the estiated self-interference channel and PDIS of this channel are known at the node. Thus this part of interference can be eliinated by ADIC, and the reaining interference plus noise at node a is κp b e aa haa x a + n a. Denoting ĥ2 as Ĝ, the received signal power is P bĝba, the reaining interference power after interference cancellation is κp a e aa, and the noise power is N a. Since both κ and e aa are related to self-interference cancellation, we define χ κe aa as the self-interference cancellation coefficient. The SINRs for transceivers a and b can be written as SINR a P bĝba χp a + N a and SINR b P aĝab χp b + N b. (4) We adopt the Shanon forula C B log 2 (1 + SINR) to approxiate the capacity of a channel with bandwidth B. For brevity, we set B 1for all the transitters. The total capacity for the pair of nodes, C ab,is ( ) ( ) C ab log 1+ P bĝba χp a + N a +log 1+ P aĝab χp b + N b. (5) For an FD network with M pair of FD nodes, {a,b },2,,M, the su rate of the entire network, C total, can be written as C total C ab (6) log 1+ +log 1+ Ĝ ba P b χp a + Ĝ al a P al + Ĝ bl a P bl + N a Ĝ ab P a χp b + Ĝ al b P al + Ĝ bl b P bl + N b. Algorith 1: Optial Power Control Algorith for a Single FD Pair 1 Solve C ab P a Pb P ax 0for P a ; 2 Solve C ab P b PaP ax 0for P b ; 3 Substitute (P a,p ax) into (5) to get C ab (1) ; 4 Substitute (P ax,p b ) into (5) to get C ab (2) ; 5 if C ab (1) C ab (2) then 6 Output optial power allocation (P a,p b )(P a,p ax) ; 7 else 8 Output optial power allocation (P a,p b )(P ax,p b ) ; 9 end III. OPTIMAL POWER CONTROL FOR A PAIR OF FD NODES We first consider the sipler case of a pair of FD nodes, i.e., M 1. The su rate of the pair, C ab,isgivenin(5). The power control proble for nodes a and b is ax C ab (7) {P a,p b } s.t. 0 P a P ax and 0 P b P ax. (8) Lea 1. In the optial solution to Proble (7), atleast one node transits at its axiu power P ax. ( )( Proof: Defining Z 1+ P bĝba χp a+n a 1+ PaĜab χp b +N b ), then we have C ab logz and Z>0. Consider the weighted su of the partial derivatives of C ab with respect to P a and P b and with weights 1/P b and 1/P a, respectively. We have C ab 1 + C ab P a P b P b 1 P a ĜbaĜabχ(2N a N b + N a P b χ + N b P a χ) Z(χP a + N a ) 2 (χp b + N b ) 2 + Ĝ ab N b Ĝ ba N a Z(χP b + N b ) 2 + P b Z(χP a + N a ) 2 > 0. (9) P a Therefore, the two partial derivatives cannot be both nonpositive; we have either C ab / P a > 0 or C ab / P b > 0 (or both) for any power allocation in the feasible region defined in (8). We can always increase C ab by increasing the power for the node with a positive partial derivative, until finally hitting the boundary. Thus the optial solution will always be on the boundary of the feasible region, i.e., it will be either (P a,p ax ) or (P ax,p b ). Based on this observation, the algorith for finding the optial power allocation for the FD pair is presented in Algorith 1. In the case of a single pair of FD nodes, increasing the power of one node, say node a, has both positive and negative effects on the su rate: the SINR for node b will be larger since the received signal will be stronger; but the SINR for node a itself will be saller, since the self-interference will also go up. Furtherore, at least one of the nodes should transit at the axiu power. This situation will occur again when power allocation for M>1 pairs of nodes is considered. 1166

3 IV. OPTIMAL POWER CONTROL FOR THE FD NETWORK We next consider the optial power allocation proble for the general case of M>1 pairs of FD nodes. However, the challenge is that proble (6) is not convex in the feasible region, and thus cannot be solved by a convex optiization technique directly. We show how to apply two approxiation ethods to convert this proble into a convex one. A. High SINR Approxiation and Convexity In the high SINR region where SINR 1, wehavelog(1+ SINR) log(sinr). The su rate of the FD network for a given power allocation vector P can be approxiated as C 1 total(p) Ca 1 b (P) [ log Ĝa b +logĝb a +logp a +logp b log χp a + Ĝ al a P al + Ĝ bl a P bl + N a log χp b + Ĝ al b P al + Ĝ bl b P bl + N b. (10) The power control proble for the FD network becoes ax P C1 total(p) (11) s.t. 0 P a P ax and 0 P b P ax,. (12) We show the proble is convex in the high SINR region. Lea 2. Proble (11) is a convex optiization proble. Proof: Let Px logp x, which eans P x e P x, for all x {a, b}, {1, 2,,M}. Consider the proble with variables P [ P a1, P b1,, P am, P bm ]. Taking partial derivative of C 1 total ( P) with respect to P a,itfollowsthat a and b. For a non-zero row vector t, wehave tht T 2 s l t t 2 s l s l kl + N l s l kl + N l k k (t s l ) 2 ( ) t 2 s l s kl + N l k ( l k s kl + N l ) 2 < (t s l ) 2 ( ) t 2 s l s l ( l k s kl + N l ) 2. The inequality is due to the oission of soe negative ters l t2 s l N l /( k s kl + N l ) 2. For each nuerator, letting a t sl and b s l,wehave (t s l ) 2 t2 s l ( s l) 0, for all l, according to the Cauchy- Schwarz Inequality (i.e., (a T b) 2 (a T a)(b T b)). Thus tht T < 0, and Ctotal 1 ( P) is concave. Consider the solution space of P, it is easy to verify that a Ctotal 1 (P) 1 P a a Ctotal 1 ( P). The transforation fro P to P only scales the gradient. Thus Ctotal 1 (P) is also concave. With linear constraints (12), proble (11) is convex. B. Decoposition and Distributed Algorith We next apply the Dual Decoposition technique to develop a distributed power control algorith for the FD network [7]. Rewrite the su rate approxiation (10) as where C 1 total(p) Ca 1 b (P) [ ( ) ( )] Ĝab log P a Ĝba +log P b, (13) χp b + b χp a + a a Ctotal( 1 P) 1 s aa k + (s al a k +s bl a k )+N ak k l l s ab k (s al b k +s bl b k )+N bk, a b Ĝ al a P al + Ĝ bl a P bl + N a Ĝ al b P al + (14) Ĝ bl b P bl + N b where s aa χe P a, sab 0, and s ab l Ĝa b l e P a. Taking derivative again for each nonlinear ter log(χp a + Ĝa l a P al + Ĝb l a P bl + N a ), the Hessian is H a s T a s a diag(s a ), where s a [s a1a,,s bm a ]/( k s ka + N a ). The overall Hessian is l st l s l diag(s l ), where k and l represent either are the received signals for nodes a and b when both of the are idle, i.e., the interference plus noise at nodes a and b fro other nodes, respectively. Proble (11) can be rewritten as axiizing (13) subect to constraints (12) and (14). Define Lagrange ultipliers μ and consider the Lagrangian only including the coupled constrains. 1167

4 Algorith 2: Distributed Optial Algorith for M Pair of FD Nodes in the High SINR Region 1 Initialize μ a and μ b to soe non-negative values, for all ; 2 repeat 3 Each FD pair receives power updates fro other nodes ; 4 Each FD pair coputes a (t) and b (t), and solves (17) to update μ a, μ b, ν a,andν b ; 5 Each FD pair solves (15) for {P a,p b } ; 6 Each FD pair distributes {P a,p b } to the entire network ; 7 until convergence; We have L 1 (P, μ) Ctotal(P)+ 1 μ a a Ĝ al a P al Ĝ bl a P bl N a + μ b b Ĝ al b P al Ĝ bl b P bl N b ) (C 1 ab ν a P a ν b P b + (μ a a + μ b b ), where Ca 1 b is the capacity for node pair {a,b }; ν a M l1 μ a l Ĝ aa l + M l1 μ b l Ĝ ab l ; and ν b M l1 μ a l Ĝ ba l + M l1 μ b l Ĝ bb l. The subproble for each pair of nodes {a,b } is ax {P a,p b } C1 (P a,p b )Ca 1 b (P a,p b ) ν a P a ν b P b (15) s.t. 0 P a P ax, 0 P b P ax, (16) The optial solution {P a,p b } can be solved by local Karush-Kuhn-Tucher (KKT) Conditions or using the subgradient ethod. The aster proble is in µ C1 total(μ) C 1 (μ)+ (μ a a + μ b b ) (17) s.t. μ 0. (18) The gradient ethod can be used to solve the aster proble by using a centralized controller or by flooding the power inforation to all other nodes as in [11]. The distributed power control algorith for the FD network is presented in Algorith 2. C. Approxiation in a General Network Note that Algorith 2 is based on the high SINR assuption, which ay not be true in a general FD network. We can exclude low SINR transceivers to enforce this assuption though. In this section, we present an alternative approach to relax the high SINR assuption, with is a ore general approxiation for the power allocation proble. Following the approach in [8], the tightest lower bound for log(1 + x) is α log x + β, which intersects log(1 + x) at x 0 when the coefficient α and β are chosen as follows. α x 0, β log(1+x 0 ) x 0 log x 0. (19) 1+x 0 1+x 0 Therefore we can approxiate the network-wide su rate as Ctotal(P, 2 α, β) Ca 2 b (P, α, β) [ ( ) Ĝba α a log P b + β a + χp a + a ( ) ] Ĝab α b log P a + β b. (20) χp b + b The power control proble for M node pairs becoes ax Ctotal(P, 2 α, β) (21) P s.t. constraints (12) and (14). It can be shown that Ctotal 2 (λ) is also concave, with a siilar approach as in section IV-A. This is because the non-negative and constant scalars α and β do not change the concavity perperty. Hence Ctotal 2 (P) is concave for nonnegative α and β; the ethod of Dual Decoposition can be applied to find the optial solution for given α and β. To further iprove the optiality of the solution, we iteratively update α and β in each step τ as follows, until the optial solution for the original proble is achieved. x a (τ) Ĝb a P b (τ) χpa (τ)+ a (τ) α a (τ) xa (τ) 1+x a (τ) β a (τ) log(1+x a (τ)) xa (τ) 1+x a (τ) log(x a (τ)). (22) Specifically, in each iteration, we use the updated power assignent P(τ) to find a (τ) and b (τ). We then use the to copute x a (τ) as in (22), and use x a (τ) to update α a (τ) and β a (τ). Finally the new power assignent P(τ+1) can be derived based on α(τ) and β(τ), and so forth. The distributed algorith for the general network/sinr case is presented in Algorith 3. V. SIMULATION RESULTS In this section, we validate the proposed algoriths with siulations. We first exaine the case of one pair of FD nodes with unbiased noise. We use an HD network with the sae setting as a benchark; a siplified version of the proposed algorith is used to find the optial powers for the HD only network. Note that in the FD network, it is possible that the optial power for soe nodes are zero, indicating that such nodes operate in the HD ode. In the siulations, we assue log-noral block fading channels, with zero ean and 10 db standard deviation. The path loss exponent is 4. The noise powers are randoly generated around 110 dbw, unless otherwise specified. The results are presented for a tie slot 1168

5 Algorith 3: Distributed Optial Algorith for M Pair of FD Nodes in the General Case 1 Set τ 0,andα(0) 1, β(0) 0 ; 2 repeat 3 Initialize λ a and λ b to soe non-negative values, and set a 0and b 0,forall ; 4 repeat 5 Each FD pair receives updated powers fro other nodes ; 6 Each FD pair coputes a and b, and solves the aster proble to update μ a, μ b and ν a, ν b ; 7 Each FD pair solves the subproble for {P a,p b } ; 8 Each FD pair distributes {P a,p b } to the entire network ; 9 until convergence; 10 Each FD pair updates α a (τ), α b (τ), β a (τ), andβ b (τ) as in (22) ; 11 τ τ +1; 12 until convegence; Throughput (bit/sec/hz) Fig HD only FD HD Average noise power (dbw) Optial throughput of a single pair of FD nodes versus noise level. Throughput (bit/sec/hz) HD only FD HD χ (db) Fig. 1. Optial throughput of a single pair of FD nodes versus χ. within which the channel gains do not vary. The peak power is set as P ax 5W. We first deonstrate the ipact of the self-interference coefficient χ and noise on FD transissions in Figs. 1 and 2. In Fig. 1, we plot the noralized throughput as a function of χ. It can be seen that, as χ is increased, the noralized throughput of the FD network converges to that of the HD network. Particularly, when χ is less than 45 db, FD achieves significant throughput gain over HD; the ore effective the self-interference cancellation, the higher the throughput gain. In Fig. 2, we fix χ 80 db and plot the noralized syste throughput for increasing noise levels. In this siulation, the FD thoughput is always higher than that of HD for the entire range of noise levels siulated. As the noise level is decreased, especially when the average noise power is lower than 120 db, the FD network throughput approaches that of the HD network. This is because when the noise is extreely low, if one of the nodes, say node b, has P b 0, the denoinator of SINR b (see (4)) will be reduced close to zero. The throughput of node b will be draatically increased, which doinates the reduction in the throughput of node a. Thus the pair will work in the HD ode. Next, we deonstrate the perforance of the proposed algoriths for an FD network of M 4 node pairs. The general approxiation algorith shown in Algorith 3 is used to obtain the results shown in Figs. 3 and 4. The convergence of the eight transit powers are plotted in Fig. 3. With the proposed algorith, all the transit powers converge to the optial power allocation after several iterations. After convergence, node 2 transits at the axiu power P ax, nodes 3, 4, 7, and 8 are not allowed to transit (with transit power zero), and nodes 1, 5, and 6 assues soe transit power lower than P ax. This is consistent with the single pair of node case in Lea 1 (i.e., at least one node transits at the peak power). Aong the four node pairs, two of the operates in the FD ode (1 2 and 5 6), while two of the are turned off (3 4 and 7 8), in order to achieve a larger su rate. As the channels vary over tie slots, the node pairs with power zero in this tie slot will get their turns to transit in soe future tie slots. The convergence of the su rate is presented in Fig. 4. We find that the su rate also converges after several iterations. With Algorith 3, the su rate is always non-decreasing across the iterations. The optial su rate of the FD network is about 1.6 ties of that of the HD network, deonstrating the benefits of FD transissions and distributed power control. VI. RELATED WORK We briefly review related work on FD in this section. There have been several seinal works on both the theoretic aspect (such as deriving the capacity region [1]) and practical design of FD wireless networks [2]. A practical design and ipleentation has been reported in [2], including the antenna structure, several kinds of self-interference cancellation techniques, and MAC layer design. The authors showed that up to 73dB 1169

6 Power (W) 5 x Steps Fig. 3. Convergence perforance of the proposed algorith for four pairs of FD nodes. Throughput (bit/sec/hz) Steps Fig. 4. P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 FD HD MaxHD Convergence of the su rate of four pairs of FD nodes. suppression of self-interference for a 10MHz OFDM signal and at least 45dB reduction at a 40MHz channel are achievable. Various self-interference cancellation schees have been investigated in prior works [3] [5], such as PDIS, Analog Self-Interference Cancellation, which reduces the self-signal by the design of distance between transitting antenna and receiving antenna, and Digital Self-Interference Cancellation, which cancels the self-signal by subtracting the transitted signal fro the receiving signal. In [12], the authors studied power allocation for a single pair of FD nodes equipped with MIMO. It was assued that the power for each single antenna at the sae node were equal. The two transit powers for the FD pair were then optiized to achieve the axiu throughput. In [6], the details of self-interference cancellation was reviewed and the paper was then focused on antenna design, including the arrangeent of the position of transitting and receiving antennas. In a recent work [13], we studied power control for an underlay cognitive ratio network with FD transissions, and developed a centralized schee with a control-theoretic approach. In another recent work [14], we investigated incorporation of FD transissions in cognitive fetocell networks. VII. CONCLUSION In this paper, we developed distributed algoriths for optial power control in FD wireless networks. For the case of a single pair of nodes, we presented a siple algorith that can copute the optial powers. For the case of ultiple pair of FD nodes, we first developed a distributed algorith by applying the high SINR approxiation, and then proposed a distributed algorith based on an iterative approxiation ethod for the logarith function. The proposed algoriths are validated with siulation studies. ACKNOWLEDGMENT This work is supported in part by the US National Science Foundation (NSF) under Grants CNS and CNS , and through the NSF Broadband Wireless Access & Applications Center (BWAC) site at Auburn University. REFERENCES [1] V. R. Cadabe and S. A. Jafar. Degrees of freedo of wireless networks with relays, feedback, cooperation, and full duplex operation, IEEE Trans. Inf. Theory, vol.55, no.5, pp , Apr [2] M. Jain, J. I. Choi, T. Ki, D. Bharadia, S. Seth, K. Srinivasan, and P. Sinha, Practical, real-tie, full duplex wireless, In Proc. ACM MobiCo 11, Las Vegas, NV, Sept. 2011, pp [3] E. Ahed, A. M. Eltawil, and A. Sabharwal, Rate gain region and design tradeoffs for full-duplex wireless counications, IEEE Trans. Wireless Coun., to appear. [4] B. P. Day, A. R. Margetts, D. R. Bliss, and P. Schniter, Full-duplex bidirectional MIMO: Achievable rates under liited dynaic range, IEEE Trans. Signal Process., vol.60, no.7, pp , July2012. [5] M. Duarte and A. Sabharwal.. Full-duplex wireless counications using off-the-shelf radios: Feasibility and first results. In Proc. IEEE ASILOMAR 10, Pacific Grove, CA, Nov. 2010, pp [6] A. K. Khandani, Full-duplex (two-way) wireless: Antenna design and signal processing, [online] Available: ca/reports/antenna design.pdf, [7] D. P. Paloar and M. Chiang, A tutorial on decoposition ethods for network utility axiization, IEEE J. Sel. Areas Coun., vol.24, no.8, pp , Sept [8] J. Papandriopoulos and J. S. Evans, Low-coplexity distributed algoriths for spectru balancing in ulti-user DSL networks, In Proc. IEEE ICC 06, Istanbul, Turkey, June 2006, pp [9] C.Wang,E.K.Au,R.D.Murch,W.H.Mow,R.S.Cheng,andV.Lau, On the perforance of the MIMO zero-forcing receiver in the presence of channel estiation error, IEEE Trans. Wireless Coun., vol.6, no.3, pp , Mar [10] J. I. Choi, M. Jain, M,K. Srinivasan, P. Levis, and S. Katti, Achieving single channel, full duplex wireless counication, In Proc. ACM MobiCo/MobiHoc 10, Chicago, IL, Sept. 2010, pp [11] M. Chiang, Balancing transport and physical layers in wireless ultihop networks: Jointly optial congestion control and power control, IEEE J. Sel. Areas Coun., vol.23, no.1, pp , Jan [12] W. Cheng, X. Zhang, and H. Zhang, Optial dynaic power control for full-duplex bidirectional-channel based wireless networks, In Proc. IEEE INFOCOM 13, Turin, Italy, Apr. 2013, pp [13] N. Tang, S. Mao, and S. Kopella, Power control in full duplex underlay cognitive radio networks: A control theoretic approach, in Proc. IEEE MILCOM 14, Baltiore, MA, Oct [14] M. Feng, S. Mao, and T. Jiang, Duplex ode selection and channel allocation for full-duplex cognitive fetocell networks, in Proc. IEEE WCNC 2015, New Orleans, LA, Mar