Adaptive Modulation for Multiple Antenna Channels

Transcription

1 Adaptve Modulaton for Multple Antenna Channels June Chul Roh and Bhaskar D. Rao Department of Electrcal and Computer Engneerng Unversty of Calforna, San Dego La Jolla, CA E-mal: Abstract We consder the use of adaptve modulaton scheme for multple transmt and multple receve antenna system wth nstantaneous channel nformaton known to both the recever and the transmtter. We derve an effcent bt allocaton algorthm whch maxmzes the transmsson rate for a gven transmt power. The algorthm s generally a greedy algorthm; however, we derve a suffcent condton for the bt allocaton algorthm to be globally optmal, whch s found to be satsfed n all dgtal modulaton schemes. It s found that when uncoded M-ary QAM s used wth a target symbol error probablty of 1 there s about 9 db gap between the channel capacty and the throughput of adaptve modulaton. I. INTRODUCTION The adaptve modulaton for scalar channels was studed n [1]. The fundamental concept of adaptve modulaton s that the system parameters n the physcal layer are adaptvely changed based on the channel status to ncrease the communcaton lnk qualty, mostly, transmsson rate. Ths paper consders usng the adaptve modulaton method n multple antenna channels. In partcular, we focus on the power allocaton problem over multple spatal channels. In recent years, multple antenna communcaton systems have gathered much attenton for hgh-rate transmsson over wreless channels. Telatar [] showed the nformaton-theoretc capacty of multple-nput multple-output (MIMO) channels wth flat fadng. If the channel state nformaton s known to both the transmtter and the recever, an MIMO channel can be decomposed nto parallel ndependent sngle-nput sngle-output (SISO) channels by employng approprate operatons at the transmtter and the recever. The resultng decomposed channels are characterzed by the channel gan matrx,.e., the gans of the decomposed channels are determned to be the sngular values of the channel gan matrx. After decomposng the channel nto parallel channels, the remanng problem s how to allocate the transmt power over the decomposed channels to maxmze the total transmsson rate. The adaptve modulaton for multple antenna channels s concerned wth adaptaton of modulaton parameters n spatal as well as temporal doman. Ths research was supported by CoRe research grant No. Cor-17 and by a research grant from Ercsson. The power allocaton problem can be equvalently consdered as so-called bt allocaton problem 1 over multple spatal channels f the target lnk qualty s fxed. The bt allocaton problem nvolves solvng an optmzaton problem wth nteger varables, n whch the optmum bt allocaton over the multple channels s determned to mnmze the total transmt power for a gven number of transmsson bts. That s, the cost functon s the total requred transmt power for transmttng the gven number of bts wth a target lnk qualty satsfed. The channel gan matrx characterzes the cost functon. One contrbuton of ths paper s that the bt allocaton problem s formulated as an optmzaton problem, and then an effcent bt allocaton algorthm s derved. The derved algorthm s a greedy algorthm, whch generally may not be the global optmum. However, we derve a suffcent condton for the bt allocaton algorthm to be globally optmal, whch s found to be satsfed n all M-ary dgtal modulaton schemes. We are also nterested n the average transmsson rate that can be affordable wth adaptve modulaton n MIMO systems and how far the average rate s away from the channel capacty. We consder an adaptve modulaton scheme that changes the modulaton order of M-ary QAM, M {,,,..., }, dependng on the channel state. The smulaton results show that the average transmsson rate of the adaptve uncoded M-ary QAM s about 9 db away from the channel capacty when the target symbol error probablty s set to 1. II. SYSTEM MODEL We consder a pont-to-pont flat fadng channel wth multple antennas at both the transmtter and the recever. The number of transmt antennas s denoted by t and the number of receve antennas by r. We consder a lnear dscrete channel model y = Hx + w (1) where x C t 1 s the transmtted sgnal, y C r 1 s the receved sgnal, H C r t s the channel gan matrx, and w C r 1 s the zero-mean complex Gaussan nose wth covarance E{ww } = I r. As n [], we assume that H s a random matrx ndependent of x and w. H s a complex Gaussan 1 The bt allocaton problem has been also studed for mult-carrer communcaton applcatons; there, computatonally effcent suboptmum schemes have been focused because the number of parallel channels s usually large.

2 matrx wth..d. entres, each entry havng ndependent real and magnary parts wth zero-mean and varance 1/. And, we assume a power constrant on the transmtted sgnal wth E{x x} P T. A. Channel Decomposton Usng SVD If the channel state nformaton s known at both the transmtter and the recever, the channel can be decomposed nto parallel non-nterferng SISO channels by usng sngular value decomposton (SVD): H = UΣV, where U C r r and V C t t are untary, and Σ R r t contans the sgular values wth σ s the -th sngular value of H. If the transmtter knows the channel matrx H (or V ), x = V s s transmtted where s C t 1 s the modulaton symbol vector. And, at the recever, by pre-multplyng y by U we have ỹ = Σs + w () where ỹ = U y, and w = U w. Snce m rank(h) mn(t, r), () can be rewrtten as { σ s ỹ = + w, 1 m (3) w, otherwse where the subscrpt ndcates the -th element of the correspondng vector. Note that snce U and V are untary matrx, E{s s} = E{x x} and E{ w w } = E{ww }. From (3), we can see that, for a gven channel H, we have m ndependent parallel Gaussan channels wth the -th channel havng a gan σ. Therefore, the demodulaton for the transmtted symbol vector s becomes smple: just demodulate each channel ndependently wth a decson varable ỹ. B. Transmt Beamformng and Maxmal-Rato Combnng We also consder a transmt beamformng and maxmal-rato combnng (MRC) at the recever for comparson. In ths case, the modulaton symbol s now a scalar s and the prncple egenvector of H H s employed as the beamformng weght,.e., x = v 1 s. And, at the recever sde, the weghtng vector s set to h = Hv 1 / Hv 1, whch equals to u 1, the prncple egenvector of HH, then the decson varable s gven by ỹ = h y = σ 1 s + w. () The resultng channel s a sngle Gaussan channel wth a channel gan σ 1 and E{ w } = 1. We place the same nput power constrant, E{ s } P T. C. M-ary QAM We consder M-ary QAM wth M {, 1,,..., }, whch corresponds to transmttng {, 1,,..., } bts per channel use. Snce the decson boundary n the QAM constellaton s not rectangular when M = 3 and 1 as shown n Fgure 1, we use Crag s method [3] to obtan the exact numercal values for the probablty of symbol error. Table I summarzes the requred symbol energy per nose densty, whch s denoted by g(n) for M = n, when the target symbol error probablty (SER) s 1. TABLE I THE REQUIRED SYMBOL ENERGY PER NOISE DENSITY IN db, g(n), OF M -ARY QAM WHEN THE TARGET SYMBOL ERROR PROBABILITY IS 1. M n g(n) III. OPTIMUM BIT ALLOCATION When the channel state nformaton s known to both the transmtter and the recever, after decomposng the channel nto m parallel channels as n the prevous secton, the remanng problem s how to allocate the transmt power over the decomposed channels to maxmze the total transmsson rate. One obvous strategy from nformaton theory s to allocate more power to better channel as n water-fllng []. The water-fllng assumes that there exsts nfntely many contnuous levels of power. But, n the adaptve modulaton wth M-ary modulaton, we need dscrete-valued power to transmt one more addtonal bt wth a gven lnk qualty satsfed. Therefore, the power allocaton n adaptve modulaton for MIMO systems s to fnd the optmum way to allocate the avalable transmt power over multple parallel channels wth each havng ts dscrete levels whch are determned by the modulaton scheme and ts channel gan. Snce the channel gan of the -th decomposed channel s σ, the receved sgnal-to-nose rato (SNR) s gven by σ P, where P s the transmt SNR of the -th channel. Note that snce we assume a normalzed nose wth unt varance, the transmt power on a decomposed channel equals to the transmt SNR. Therefore, f we denote the requred SNR for M = n,.e. n bts transmsson per channel use, by g(n), the requred transmt power wth a channel gan σ s gven by Γ (n) = g(n) σ, n = 1,..., n max. () Consder that we transmt n N {, 1,,..., n max } bts over the -th channel, 1 m, we call ths a bt allocaton n = (n 1,..., n m ) N m. Then, the total transmt power, the cost functon n ths optmzaton problem, s gven by f(n) = f(n 1,..., n m ) = Γ (n ) = a g(n ) () where a = 1/σ and wthout loss of generalty we assume that σ 1 σ... σ m >, so < a 1 a... a m. For smpler notaton, we defne n g(n) g(n 1), and n () 1 as a vector wth same elements of n except -th element reduced by one. Snce we are dealng wth a non-negatve nteger n N, t s more useful to defne n () 1 ( n 1,..., n 1, [n 1] +, n +1,..., n m ). (7)

3 where [x] + max{, x}. Theorem 1: Consder two bt allocatons, n and k, wth n k for some and the total transmt powers wth two bt allocatons satsfyng f(n) f(k). Then, wth a suffcent condton f(n () 1 ) f(k () 1 ) () < k n. (9) Proof: Snce f(n) f(k), the hypothess on the total transmt powers can be rewrtten as a j [g(n j ) g(k j )] After subtractng the -th term from both sdes, we have a j [g(n j ) g(k j )] a [g(k ) g(n )] j From the above, we can show the nequalty (). f(n () 1 ) f(k () 1 ) = a [g(n 1) g(k 1)] + a j [g(n j ) g(k j )] j a [g(n 1) g(k 1)] + a [g(k ) g(n )] = a {[g(k ) g(k 1)] [g(n ) g(n 1)]} = a ( k n ) The last nequalty s satsfed under the suffcent condton (9). Hence, the theorem was proved. Example 1: Assume m =, N = {, 1,..., }, and bt allocaton n = (7, ) s the optmal n transmttng 1 bts per channel user (see Fgure ). That s, f(7, ) = Applyng Theorem 1, mn f(k 1, k ). k 1,k N k 1 +k =1 f(, ) f(, ), f(, 7), f(3, ) and f(7, ) f(, 3) Therefore, mn f(k 1, k ) = mn{f(, ), f(7, )}. k 1,k N k 1 +k =11 We want to generalze the dea n the above Example and derve an effcent bt allocaton algorthm based on Theorem 1. Before dong that, we need a set of defntons for the subsets of all possble bt allocatons, N m. S N {n : n = N and n N, = 1,..., m} whch s the set of all bt allocatons that are correspondng to N bts transmsson per channel use. We also defne a subset of S N wth a parameter a S N as set of all the vectors n S N wth the -th element less than the -th element of a,.e., D N (a) { n : n S N and n a }, = 1,..., m. We can see that S N = m DN (a), for some a SN. And, we also defne D N () 1 (a) { n () 1 : n S N (a) }. Then, from the above defntons we can easly see the followng Lemma. Lemma 1: For some vector a S N, S N 1 = m D() 1 N (a). (1) We can dvde the mnmzng problem over a varable space, say S, nto two stages: Frst, do m number of mnmzatons over the subset varable spaces, say D, = 1,..., m, where S = m D ; and secondly, fnd the fnal mnmzng pont among the m mnmzng ponts obtaned from the frst step. The followng Lemma summarzes ths dea. Lemma : Suppose the varable space for the optmzaton, S = m D, and n s the mnmzng pont over the subset space D,.e., f(n ) = mn n D f(n). Then, the mnmzng pont n over the whole space S s gven by f(n ) = mn n S f(n) = mn f(n ). (11) Lemma 3: Suppose that a s the mnmzng pont n S N,.e., f(a) = mn n S N f(n) and that g(n) satsfes the followng condton, < 1... nmax. (1) Then, a () 1 s the the mnmzng pont n D N () 1 (a) SN 1. That s, f(a () 1 ) = mn f(n) for = 1,..., m. (13) n D N () 1 (a) Proof: Snce f(a) = mn n S N f(n), f(a) f(k) for all k S N. By Theorem 1, t mples that f(a () 1 ) f(k () 1 ) for all k S N. (1) Snce we defned D() 1 N (a) = { k () 1 : k S N (a)}, (1) s equvalent to (13). From Lemma 1, and 3, we can derve the followng Theorem whch s drectly related to the optmum bt allocaton algorthm. Theorem : If a s the bt allocaton wth N bts transmsson that mnmzes the transmt power,.e., f(a) = mn n S N f(n), then the optmum bt allocaton b wth N 1 bts transmsson s gven by f(b) mn n S N 1 f(n) = mn f(a () 1 ). (1)

4 Proof: For the optmzaton problem over S N 1, we consder the subsets D() 1 N SN 1, 1 m, as n Lemma 1, and apply two steps optmzaton stated n Lemma, then we can easly arrve the result by Lemma (3). From the above Theorem, we derve an effcent bt allocaton algorthm as s summarzed as follows: 1) Start wth N = m n max ; and n = (n max,..., n max ). ) N N 1; = arg mn 1 m f(n () 1 ), then n n ( ) 1. 3) Repeat Step untl f(n ) P T. Otherwse, stop. where n ( ) 1 as a vector wth same elements of n except - th element reduced by one as defned n (7). Snce the cost functon, the total transmt power, has a form of f(n) = m a g(n ), by notcng the fact f(n () 1 ) = f(n ) a n, the second step n the above bt allocaton algorthm s equvalent to ) N N 1; = arg max 1 m a n, then n n ( ) 1, f(n ) f(n ) max a n. The above bt algorthm always gves the bt allocaton that maxmzes the total number of transmsson bts for a gven power constrant, snce the resultng bt allocaton n for each N always provdes the mnmum transmt power. The algorthm s, n general, a greedy algorthm whch may not be the global optmum, snce t follows the best way at each stage wthout care of the prevous path, regardless of the characterstcs of the cost functon. In ths secton, we have shown that the derved bt allocaton algorthm provdes the globally optmum pont under a suffcent condton (1), whch s satsfed for all M-ary dgtal modulaton schemes. One can also develop, wth the same prncple, an equvalent bt allocaton algorthm that goes the reverse drecton: 1) Start wth N = ; and n = (,..., ). ) N N + 1; n tmp n ; = arg mn 1 m a n +1, then n n ( )+1, f(n ) f(n ) + mn a n +1. 3) Repeat Step untl f(n ) > P T. Otherwse, stop and n n tmp. where n ()+1 (n 1,..., n 1, n + 1, n +1,..., n m ). IV. THROUGHPUT OF ADAPTIVE M -ARY QAM The ergodc channel capacty of MIMO channels wth perfect channel state nformaton known to both the transmtter and the recever s gven by C(P T ) = E H {C(P T ; H)} (1) where E H { } s the expectaton over the random channel matrx H, and C(P T ; H) s the condtonal capacty for a gven channel realzaton H, whch s gven by [] C(P T ; H) = [ log (νσ ) ] + (17) and ν s the water-fllng level satsfyng [ ] ν 1/σ + = PT. (1) The cost functon of () s determned by the channel matrx H, or more precsely by {σ 1,..., σ m }; and, for a channel realzaton, the optmum bt allocaton can be solved wth the algorthm descrbed n the prevous secton. Then, the maxmum number of transmsson bts, let us say R(P T ; H) = R(σ 1,..., σ m ; P T ), s smply the sum of all the elements of the bt allocaton,.e., R(P T ; H) = n, where n s the -th element of the optmum bt allocaton n. Therefore, for a gven transmt power constrant P T, the average bts per channel use, that s offered by adaptve modulaton, can be expressed as, R(P T ) = E H {R(P T ; H)} = R(σ 1,..., σ m ; P T )p(σ 1,..., σ m ) dσ 1 dσ m (19) where p(σ 1,..., σ m ) s the jont probablty densty functon of {σ }, whch can be found n lterature, e.g., []. However, unfortunately, t s not easy to fnd any closed-form expresson for R(σ 1,..., σ m ; P T ). Instead, we resort to the solutons from the bt allocaton algorthm. We ran smulatons to evaluate the throughput of adaptve modulaton. We consdered M-ary QAM wth requred SER of 1, and used the requred transmt power (or transmt SNR) for each transmsson bt, g(n), n =, 1,...,, that are shown n the Table I n Secton II. Fgures 3 shows the results when t = {1,, 3, } and r =. We can see that the throughput of adaptve modulaton s away 9 1 db from the channel capacty calculated from (1). One apparent observaton s that as P T ncrease R(P T ) m n max, because at hgh P T all m channels are utlzed wth n max bts transmsson on each channel. Before R(P T ) s saturated, the throughput s almost parallel to C(P T ). One obvous way to reduce the gap to the channel capacty s to employ channel codes. If the perfect channel s known to both sdes so that the MIMO channel can be decomposed nto non-nterferng parallel channels, there s no need to use vector channel codng that needs hgh complexty, and conventonal scalar channel codng for each channel s suffcent. The throughput of adaptve modulaton wth ndependent scalar codngs can be obtaned usng a smlar approach, because we separated the power allocaton problem n the MIMO channel from the modulaton ssues. That s, once the requred transmt power {g(n)} that accounts for the effect of channel codng s

5 obtaned, we can obtan the throughput by just substtutng the new {g(n)} n the bt allocaton algorthm. The beamformng strategy descrbed n Secton II needs less nformaton about the channel (only v 1 and σ 1 are necessary at the transmtter). In Fgure, we compared the performance of the transmt beamformng wth the channel decomposton method usng SVD. When P T s low, the transmt beamformng scheme shows a comparable performance to the channel decomposton method. Ths s because at low transmt power only the best spatal channel s used most of the tme. Therefore, the two schemes gve lttle dfference n performance. But, as P T ncreases, the other remanng channels are startng to be utlzed; hence, the channel decomposton method outperforms the transmt beamformng counterpart. V. CONCLUSION We consder the use of adaptve modulaton scheme for MIMO channels wth nstantaneous channel nformaton known to both the recever and transmtter. An effcent bt allocaton algorthm s derved and ts optmalty s proved under a suffcent condton, whch s satsfed n all M-ary modulaton schemes. It s found that when M-ary QAM s used wth a target symbol error probablty of 1 there s about 9 db gap between the channel capacty and the throughput of adaptve modulaton. And, t s also shown that a transmt beamformng scheme, whch needs less channel nformaton, gves a comparable performance when the power constrant s low. Relaxng the assumpton of perfect channel nformaton at both communcaton sdes needs further studes. REFERENCES [1] A. J. Goldsmth and S.-G. Chua, Varable-rate varable-power MQAM for fadng channels, IEEE Trans. Commun., vol., no. 1, pp , [] I. E. Telatar, Capacty of mult-antenna Gaussan channels, AT&T Bell Labs Tech. Memo., 199. [3] J. W. Crag, A new, smple and exact result for calculatng the probablty of error for two-dmensonal sgnal constellatons, n Proc. IEEE MIL- COM, pp. 71 7, [] T. M. Cover and J. A. Thomas, Elements of Informaton Theory. New York: Wlly, M = M = (a) M = 3 (b) M = 1 Fg. 1. Sgnal constellaton of M-ary QAM (M = 3, 1). f(n 1, n ) Optmum Bt Allocatons n n 1 n 1 + n = 1 n + n = 11 1 Fg.. Example of bt allocaton algorthm (m =, n max = ). Average Bts per Channel Use Throughput of Adaptve M ary QAM: r = Channel Capacty (t = 1,, 3, from the bottom) Adaptve Modulaton (t = 1,, 3, from the bottom) Transmt Power Constrant, P T (db) Fg. 3. Average bts per channel use of M-ary QAM adaptve modulaton when r = fxed. The sold lnes are for adaptve modulaton wth t = 1,, 3, (from the bottom), and the dashed lnes for the channel capactes wth r = and t = 1,, 3, (from the bottom). Average Bts per Channel Use Throughput of Adaptve M ary QAM: r = Channel Decomposton (t = 1,, 3, from the bottom) Transmt Beamformng (t = 1,, 3, from the bottom) Transmt Power Constrant, P T (db) Fg.. Average bts per channel use of M-ary QAM adaptve modulaton when r = fxed. The sold lnes are for channel decomposton method, and the dashed lnes for the transmt beamformng scheme, wth t = 1,, 3, (from the bottom).