Power Mnmzaton Under Constant Throughput Constrant n Wreless etworks wth Beamformng Zhu Han and K.J. Ray Lu, Electrcal and Computer Engneer Department, Unversty of Maryland, College Park. Abstract In mult-access wreless communcaton systems, power control and adaptve modulaton are two mportant means to ncrease spectral effcency, combat wth tme varyng fadng envronment, and reduce co-channel nterference. In ths paper, the overall transmtted power s mnmzed by usng adaptve MQAM modulaton. Each lnk can select a range of dfferent MQAM modulaton accordng to ts current channel condton. Each lnk s tme rage throughput s a constant. The overall network throughput s a constant to ensure the network performance. The scheme can be nterpreted as water fllng each lnk s throughput n tme doman and allocatng overall throughput to dfferent lnks at each tme. From the smulaton results, our schemes reduce up to 80% overall transmtted power at BER =10 3, 66% overall transmtted power at BER =10 6, and ncrease rage spectral effcency by about 1.2 bt/s/hz. Keywords: Power Control, Adaptve Modulaton, Beamformng, Co-channel Interference, Spectral Effcency. I. Introducton The avalable wreless communcaton resources such as power and spectrum are extremely lmted, whle the demand for the servces s growng rapdly. So how to optmally use the rado resources and ncrease the network throughput s therefore of prmary concern n the desgn of future wreless communcaton systems. Two of the mportant detrmental effects to decrease network throughput are the tme varyng nature of the channel and co-channel nterferences. Due to the tme varyng channel, the Sgnalto-Interference-ose-Rato (SIR) at the output of the recever can fluctuate n the order of tens of dbs. Because the co-channel nterferences reduce frequency-reuse dstance, the network throughput for the areas s sgnfcantly decreased. One approach to combat these detrmental effects s to adapt the transmsson power and modulaton level based on the channel condtons. In power control, the transmttng powers are constantly adjusted. Such process mproves the qualty of weak lnks. But at the same tme, t ncreases the co-channel nterferences durng the deep fadng. In adaptve modulaton, the system assgns the modulaton levels wth dfferent spectral effcency to dfferent lnks accordng to ther channel condtons. There are tradeoff and practcal constrants to allocate these resources. So how to optmally manage these resources becomes a hot topc n nowadays wreless research. In tradtonal power control[1], [2], each lnk s transmttng power s selected so that ts SIR s equal to or larger than a fxed and predefned targeted SIR threshold requred to mantan the lnk qualty, whle mnmzng the overall transmttng power of all the lnks. However a lnk wth bad channel response requres too much transmttng power and therefore causes unnecessary co-channel nterference to other lnks. Ths s a major ssue that we wll address here. We concentrate on uplnk stuaton. We ntroduce jont adaptve power and modulaton allocaton scheme usng M-QAM modulaton wth beamformng. The goal of ths paper s to mnmze overall transmttng power. Instead of havng fxed and predefned throughput, each user can select a range of modulaton levels and throughputs accordng to ts channel fadng condton. In [3] and [4], t has been shown that the network throughput can be greatly mproved by usng adaptve modulaton. However n ther papers, the users wth best channel condtons occupy most of transmsson rado resources, whch s very unfar n real communcaton system. In ths paper, the tme rage throughput of each user s mantaned as a constant to ensure the farness. The overall network throughput s kept as a constant at each tme to mantan the network performance. The scheme can be nterpreted as water fllng each user s throughput n tme doman and allocatng network throughput to dfferent lnks at each tme. A varable power and varable rate communcaton system s constructed. From the smulaton results, our scheme reduces about 80% of overall transmttng power and ncreases rage spectral effcency by about 1.2 bt/s/hz. The organzaton of ths paper s as follows: In Secton II, we present system model. In secton III, the optmzaton problem s formulated. The adaptve power and modulaton management s developed. A varable power and varable rate communcaton system s constructed. In Secton IV, we evaluate the performance by usng smulaton study. In Secton V, we h the concluson. II. System Models Consder K co-channel lnks exstng n dstnct cell n wreless networks. Each lnk conssts of a moble and ts assgned base staton. We assume coherent detecton s possble so that t s suffcent to model ths multuser system by an equvalent base band model. Antenna arrays wth P elements are used only at base staton. Each lnk s affected by the multpath slow Raylegh fadng. The maxmum multpath number s L. The propagaton delay s far less than one symbol perod. For uplnk case, the output sgnal at the th base staton antenna array s gven by: x (t) = K L k=1 l=1 ρ k G k α l k P ka k (θ l ) g k (t)s k (t)+n (t) (1) 0-7803-7467-3/02/$17.00 2002 IEEE. 611
Throughput (bt) 12 10 8 6 4 2 0 Fg. 1. MQAM Throughput vs. SIR BER=10E-3 BER=10E-6 Fxed BER=10E-3 0 5 10 15 20 25 30 35 40 SIR (db) Requred SIR of MQAM for Desred BER where αk l s fadng loss, P k s transmtted power, a k (θ l ) s the th base staton array response vector to the sgnal from k th moble at drecton θ l, g k (t) s shapng functon, s k (t) s message symbol, n (t) s thermal nose vector, ρ k s log normal shadow fadng and G k s path loss. We assume slow fadng and defne the mpulse response from the k th moble to the p th element of the th base staton as: h p k = L l=1 αk l ap k (θ l)r pl k, where rpl k ncludes the effect of the transmtter, recever flter, and shapng functon g k (t). The vector form s h k =[h 1 k,...,hp m ]T. We assume the transmtted sgnals from dfferent sources are uncorrelated and zero mean, and the addtve nose s spatally and temporally whte. Let w be the beamformng weght vector. Wthout loss of generosty, let w H h 2 = 1. The th lnk s Sgnal-to-Interference-ose-Rato (SIR) at the beamformer output s: Γ = P ρ G k = P kρ k G k w H h k 2 + w H w (2) It has been shown that Bt Error Rate (BER) of square MQAM wth Gray bt mappng as a functon of receved SIR Γ and constellaton sze M s approxmately gven by Equ. 3. We can calculate the requred SIR for specfc MQAM and requre BER, then power control s appled so that each lnk has the requred SIR. BER MQAM (Γ) 2 ( 1 1 ) ( ) Γ erfc 1.5 log 2 M M M 1 III. Power Mnmzaton wth Adaptve Modulaton A. Problem Defnton We assume the lnks can apply dfferent modulaton levels, accordng to ther channel condtons. The overall network throughput s a constant to mantan the network performance and BER s ensured for each lnk. At each tme, the lnks wth bad channel condtons sacrfce ther throughputs,.e. they only need smaller constellaton sze and lower SIR thresholds; On the other hand, the lnks wth good channel condtons get larger throughputs and hgher SIR thresholds. For each lnk, the tme rage throughput s a constant to ensure the farness. The scheme can be descrbed as water fllng each user s throughput n tme doman and allocatng overall network throughput to dfferent lnks at each tme, accordng to ther channel condtons. Assume each lnk has unt bandwdth. The matrx verson of the optmzaton problem s: mn γ K P (4) =1 Feasble: (I DF)P u K subject to etwork Performance: =1 T (n) =R Farness: lm n=1 T (n)/ = const. where T (n) = log 2 (M (n)), whch s the th user s throughput, R s a constant. P = [P 1,...,P K ] T, u = [u 1,...,u K ] T, u = γ w H w /ρ G, D = dag{γ 1,...,γ K } and { 0 f j = ; [F j ]= ρ jg j w H h j 2 ρ G f j In order to solve the problem above, we dvde the our algorthm n two steps. The frst step s to ensure that lm n=1 T (n)/ = const.. In ths step, the throughput range at dfferent tme that a user can select s determned. Then n the second step, we decde how to manage the powers and modulaton levels wthn each user s throughput ranges at each tme. B. Each User s Throughput Management Instead of havng fxed throughput range [T mn, T max ] for each lnk, we can adaptvely change the throughput range, whch takes nto account of the lnks throughput hstory. Suppose the th lnk can select throughput n (n) T (n) T at tme n and the targeted T mn 0-7803-7467-3/02/$17.00 2002 IEEE. 612
the mnmum and maxmum throughput respectvely that the th lnk can select. The values are fxed and predefned by the system. In order to track the hstory of T, we defne T md (n) =T md (n 1)+(T (n) T ) β, 0 <β<1, where β s a constant that s depended on how fast the channel changes. If the channel changes fast, β should select a relatvely larger number, so that the SIR range can keep track of the channel changes. If the channel changes slowly, β should select a relatvely smaller number to h smooth effect. The value can be determned by some dynamc programmng method. Here for smplcty, we assume β s a constant. In each teraton, T mn (n), T and T md (n) are updated by the algorthm n Table I. TABLE I Adaptve Algorthm for Throughput Range Intal: T mn mn (0) =,T max max (0) =,T md (0) = T Iteraton: T md (n) =T md (n 1) + β(t (n) T ); T mn (n +1)= mn(max(t T md mn mn max (n)+, ), ); T max (n +1)= max(mn( max T md (n)+t, max mn ), ) When T (n) s contnuously less than T for some tme, the T mn (n) s ncreased to T. Then the next T (n +1) h to select throughput equal to or greater than T, consequently T md (n) stops ncreasng. The same analyss can be appled to T. Snce T mn (n) and T are bounded and they are lnearly modfed by T md (n), the T md (n) s also bounded. Rearrange the T md (n) and sum wth the dfferent tme. We h n=1 T (n) = T md (T () T ) + (5) β Snce T md () s bounded, the second term n the rght hand sde decreases to zero as. So we prove that lm n=1 T(n) = T. C. Optmal Throughput Allocaton for Dfferent Users In ths subsecton, we wll develop adaptve algorthms that fnd the optmal throughput allocaton for dfferent users at each tme. Because there are only lmted numbers of M that each lnk can select, we can use full search to fnd the optmal modulaton allocaton. The constrant s to keep the network throughput as a constant R. Under ths constrant, the overall transmtted powers requred by all combnatons of M are calculated. The modulaton allocaton that generates the lowest overall transmtted power s selected. The optmal adaptve algorthm s gven n Tab.II: The adaptve algorthm A has hgh complexty. When the number of users grows, the complexty s ncreased exponentally, whch s not acceptable n practce. We need TABLE II Adaptve Algorthm A Intal: T 1 (0) = T1...T K (0) = TK P 1 (0),...,P K (0) = any postve feasble const. Adaptve Throughput Allocaton: search all possble T (n) for every lnk subject to K 1 T (n) =R. fnd the combnaton of T (n) that mnmze P sum = K 1 P (T (n)) calculated by the teraton Iteraton: Beamformng: w = arg maxw Γ Power Allocaton Update: γ = SIR for T (n) and desred BER P = DFP + u. Throughput Range Update: Update T md (n), T mn (n), T. to fnd fast algorthm wth low complexty. Frst, n [5], the author fnd the gradents of overall transmtted power to adjust each user s targeted SIR to reduce the overall transmtted power. We need to fnd out whch users contrbute more to the overall transmtted power. If these users can sacrfce ther targeted SIR (throughput) a lttle bt, the overall power s reduced sgnfcantly. We h the th element of gradent g =[g 1...g K ] T of the overall uplnk transmttng power P sum as: ( c w H w + ) j = P jρ j G j w H h j 2 g = = ρ G w H h 2 c P SIR (6) where SIR s the SIR detected at the base staton s antenna dversty output for the th user, c = 1 T (I DF) 1 v, and { 1,f j = ; [ˆv ] j = 0, otherwse. The value of c reflects how severe the co-channel nterferences are. When the co-channel nterferences are large, c tells whch user causes more co-channel nterferences to other users. When the co-channel nterferences are small, c c j, j. Snce we only care the drecton of the gradent and do not care the ampltude, we can gnore the value of c when the co-channel nterferences are small. Equ. 6 s very sgnfcant n that t provdes very smple way to fnd the gradent. In ths case, we can measure SIR at each base staton s antenna dversty output and use the feedback channel to get the moble transmtted power value to calculate the drecton of gradent. Consequently the system complexty s reduced greatly. By usng the gradent, we know how to optmally reduce the overall transmtted power. 0-7803-7467-3/02/$17.00 2002 IEEE. 613
ow we ntroduce a greedy algorthm. We compare the gradents of dfferent users. If the user wth larger gradent selects lower code rate,.e. t requres lower targeted SIR threshold, the overall transmtted power s reduced greatly. So frst we fnd whch code rate for the user wth the hghest gradent generates the lowest overall power subject to T = R. When we change the rate of the user wth the hghest gradent, we modfy the rate of the user n the order from the smaller gradent to larger gradent to ensure the constrant. Then we fnd the optmal code rate for the user wth the second hghest gradent, and so on untl the we fnd the rate of last user n the row. Because every user only searches fxed amount of codng rate and only reorderng s needed, ths sub-optmal algorthm has the complexty of only O( 2 log). The algorthm s suboptmal because the optmal code rate or SIR threshold for one user may not be optmal for all the users. Rearrange the user ndex from the largest gradent to the lowest..e. g 1 g 2... g K. The sub-optmal adaptve teratve algorthm at the n th teraton to fnd the modulaton allocaton wth jont power control and beamformng s gven n Tab. III. Fg. 2. Varable Power and Varable Rate System TABLE III Adaptve Algorthm B Intal: T 1 (0) = T1...T K (0) = T1 P 1,P 2...P K = any feasble postve const. Adaptve Throughput Allocaton for =1 to K for T = T mn (n) to T 1.Modfy from T K to T +1 to satsf y the constrant T = Const. (exhaust T K frst) 2.Run teraton 3.F nd the T that generates the lowest power f or ths user end end Iteraton: Beamformng: w = arg maxw Γ Power Allocaton Update: γ = requred SIR for T and desred BER D[n] =dag(γ 1...γ K ); P[n +1]=D[n]F [n]p[n]+u[n]. Throughput Range Update: Update (n), T mn (n), T. T md Wth the adaptve algorthm, we can construct a varable power and varable rate system as shown n Fg. 2. The adaptve algorthm module gets estmaton of users channel responses from the channel estmaton module. Then power control and modulaton nformaton are computed. The power control nformaton and best modulaton method are sent back to mobles. Fg. 3. Smulaton Setup IV. Smulaton Results In order to evaluate the performance of our algorthms, a network wth hexagonal cells s smulated as shown n Fg. 3. The radus of each cell s 1000m. The number n each cell represents the assgned channel number. Two adjacent cells do not share the same channel. There are 50 cells n the networks. One base staton s placed at the center of the each cell. In each cell, one user s placed randomly wth a unform dstrbuton. In the smulaton, we consder three multpaths wth equal power Raylegh fadng wth equal power. The delay spread between the dfferent paths s far less than one symbol duraton. Each base staton has a four element of antenna array. Path loss s due to the decay of the ntensty of a propagatng rado w. In ths paper, we use the two slope path loss model [6] to obtan the rage receved power as a functon of dstance. The rage path loss s gven by: G = C r a (1 + rλ c /(4h b h m )) b (7) where C s a constant, r s the dstance between the moble and the base staton, a s the basc path loss exponent (approxmately two), b s the addtonal path loss component (range from two to sx), h b s the base staton antenna heght, h m s the moble antenna heght, and λ c s the wlength of the carrer frequency. We assume the moble antenna heght s 2m and the base staton antenna heght s 50m. The carrer frequency s 900-MHz. In the urban mcrocell system, the lnk qualty s also 0-7803-7467-3/02/$17.00 2002 IEEE. 614
10 6 Average Spectral Effcency vs. Overall Transmtted Power 0.55 Power Savng vs. Wndow Sze 0.5 BER=10e-3 BER=10e-6 Overall Transmtted Power 10 5 10 4 fx,1e-6 sub,1e-6 opt,1e-6 fx,1e-3 sub,1e-3 opt,1e-3 Power of Adap. Algr./Power of Fx Algr. 0.45 0.4 0.35 0.3 0.25 0.2 10 3 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Average Spectral Effcency (Bt/s/Hz) 0.15 1 1.5 2 2.5 3 3.5 4 4.5 5 Wndow Sze Fg. 4. Overall Power vs. SpectralEffcency Fg. 5. Power Savng vs. Wndow Sze affected by the shadowng of the lne of sght path from terran, buldngs and trees. The shadowng s generally modeled as lognormal dstrbuton. The probablty densty functon (PDF) s gven by: PDF(ρ) = 1 (log ρ ξ)2 exp{ 2πσρ 2σ 2 },ρ>0 (8) where ξ s related to path loss, σ s shadow standard devaton. For each lnk, 3dB devaton s consdered. In Fg.4, we compare the overall transmtted power as functon of rage spectral effcency for fxed scheme (every lnks use the same MQAM), suboptmal scheme and optmal scheme of BER =10 3 and BER =10 6 at a snapshot. From the curves we can see that our algorthm greatly reduces the overall transmtted power and ncrease the maxmum achevable throughput. The suboptmal algorthm has the performance between the fxed scheme and optmal scheme. In Fg.5 and Fg.6, we show the rage power savng and rage spectral effcency gan as the functon of wndow max T,T mn sze (max( )) for a tme perod of 1000. The power stops decreasng and spectral effcency stops ncreasng as the wndow sze growng. Ths s because of the tme rage throughput constrant for each lnk. The lnk who gets better throughput at ths tme must pay back n the future. The results show that our scheme can reduce up to 80% overall transmtted power when BER =10 3, 66% overall transmtted power when BER = 10 6 and ncrease spectral effcency about 1.2 bt/s/hz. V. Concluson On the whole, by adaptvely managng lnks power and modulaton, we can greatly mprove network performance. Our scheme can be nterpreted as water fllng each lnk s throughput n tme doman and allocatng the network throughput to dfferent lnks at each tme. We propose two sets of adaptve algorthms, whch gves valuable n- Spect ral Effcency Improvement(bt/s/Hz) 1.6 1.4 1.2 1 0.8 0.6 0.4 Spectral Effcency Improvement vs. Wndow Sze BER=10e-3 BER=10e-6 0.2 1 1.5 2 2.5 3 3.5 4 4.5 5 Wndow Sze Fg. 6. SpectralEffcency Gan vs. Wndow Sze sght n the adaptve resource allocaton n mult-user envronments. From our smulaton results, the algorthm reduces up to 80% of the total transmttng power of moble users, whch s very crtcal n terms of battery lfe. The spectral effcency s ncreased about 1.2bt/s/Hz, whch n turn ncreases the whole network throughput. References [1] F.Rashd-Farrokh, L. Tassulas, K.J. R. Lu, Jont optmal power controland beamformng n wreless network usng antenna arrays, IEEE Trans. Commun., vol.46, no.10, pp.1313-1323, Oct. 1998. [2] R. Yates, A framework for uplnk power control n cellular rado systems, IEEE J. Sel. Areas Commun., vol.13, no.7, pp.1341-1348, Sep. 1995. [3] X. Qu and K. Chawla, On the performance of adaptve modulaton n cellular systems, IEEE Trans. Commun., vol.47, no.6, pp.884 895, Jun. 1999. [4] S.T. Chung and A.J. Goldsmth Degrees of freedom n adaptve modulaton: a unfed vew, IEEE Trans. Commun., vol.49, pp.1561-1571, Sep. 2001. [5] Z. Han, K.J.R. Lu, Adaptve SIR threshold allocaton for jont power controland beamformng over wreless network, IEEE VTC., pp.1548-1552, Fall 2001. [6] P. Harley, Short dstance attenuaton measurements at 900MHz and 1.8 GHz usng low antenna heghts for mcrocells, IEEE Trans. Sel. Areas Commun., vol.37, pp.220-222, ov. 1988. 0-7803-7467-3/02/$17.00 2002 IEEE. 615