An Efficient Design Method for Vector Broadcast Systems with Common Information

Transcription

1 An Effcent Desgn Method for Vector Broadcast Systems wth Common Informaton R. H. Gohary, T. N. Davdson, and Z.-Q. Luo Department of Electrcal and Computer Engneerng McMaster Unversty Hamlton, Ontaro, Canada Abstract We consder the problem of deterng an optmal transmsson scheme for broadcastng a common message over vector channels, gven (perfect) channel knowledge at both the receve and transmt ends. We provde an effcent method for jontly desgnng a lnear transmtter and and a set of lnear recevers so as to mze a weghted Mean Square Error (WMSE) of the data estmates. The computatonal effcency follows from the convex formulatons that we develop. These formulatons enable utlzaton of hghly effcent nteror pont methods. For dagonal channel matrces, whch appear n multcarrer systems that employ cyclc prefxng, we show that the optmal transmtter s obtaned by subcarrer allocaton and power loadng. The set of mum MSE transcevers for a vector broadcast system s parametrzed by a untary matrx degree of freedom. For the case of dagonal systems, we show how ths untary matrx can be chosen so that the symbol error rate s mzed (over the gven set). Ths optmal untary matrx ensures that for each recever, the subcarrer sgnal-to-nose ratos (SNRs) are all the same. Smulatons ndcate that our desgns can provde sgnfcantly mproved performance over standard desgns. I. INTRODUCTION Several applcatons requre the relable transmsson of a common message from a sngle transmtter to multple recevers. Examples nclude common control sgnals n cellular communcaton systems and (subscrpton based) rado and TV broadcast systems. In such systems, when the channel vares relatvely slowly compared to the transmsson rate, the recevers are able to estmate a suffcently accurate model of the channel wthout substantal sacrfce of the lnk throughput. If the recevers feed ths nformaton back to the transmtter, an optmal transmsson scheme wth respect to a prescrbed measure can be detered for the set of estmated channels. In ths paper, we focus on the desgn of vector transmsson schemes n whch data s transmtted on a block-by-block, rather than symbol-by-symbol, bass. Such schemes are often effectve for transmsson over frequency selectve channels; e.g., Dscrete MultTone modulaton (DMT) [], and Orthogonal Frequency Dvson Multplexng (OFDM) []. However, they also appear naturally n communcaton systems wth multple antennas at the transmtter and recever. For sngleuser vector communcaton systems, there are several mature approaches to transcever desgn (see, for example, [3] for an nsghtful overvew). Typcally, the optmal transmsson scheme nvolves a decomposton of the channel nto a set of parallel subchannels and the allocaton of power and rate to each subchannel. For certan crtera, t may also be x s F H H K ν + y. + ν K y K G G K Fg.. Block dagram for the broadcastng system under consderaton n Secton II. benefcal for the transmtter to lnearly combne the symbols ntended for transmsson over dfferent subchannels (see, for example, [3] [5]). In contrast to the sngle-user case, effectve desgn methods for vector broadcast systems are just begnnng to emerge (e.g., [6] [9]). The goal of ths paper s to contrbute to the development of such methods by provdng a computatonally effcent method for jontly desgnng a lnear transmtter and a set of lnear recevers so as to mze a weghted Mean Square Error (WMSE) measure of the receved data estmates. We show that ths desgn problem can be cast as a convex optmzaton program that can be effcently solved usng Interor Pont (IP) methods []. Furthermore, when the channel matrces are smultaneously dagonalzable as they are n MultCarrer (MC) schemes that employ cyclc prefxng (e.g., DMT/OFDM) optmal transmsson can be obtaned by subcarrer allocaton and power loadng, and the computatonal effort requred to obtan the optmal transmtter can be consderably reduced. We observe that the mum MSE soluton provdes a untary matrx degree of freedom, and then show how to fnd an optmal rotaton that mzes the average bt error rate for MMSE power-loaded, cyclc prefx based MC broadcast systems. II. MMSE PROBLEM: PROBLEM STATEMENT AND A. Formulaton SOLUTION In the broadcastng scheme n Fgure a sngle transmtter sends the same nformaton vector s to K recevers. Each data block s s assumed to be zero-mean and whte wth dentty covarance matrx. However, our results carry over ŝ ŝ K GLOBECOM /3/$7. 3 IEEE

2 to the colored data case as well, because a whtenng matrx R / s, where R s s the covarance matrx of s, can readly be absorbed nto the precoder F. Each recever n Fgure has a channel matrx H, =,...,K whch s of sze q n, where q s the length of the receved block, y, and n s the length of the transmtted block, x. The desgn problem s to jontly desgn the lnear transmtter F and K lnear recevers G such that the total (weghted) MSE s mzed and the transmtted power remans below a prescrbed level, p. (A related problem s the desgn of lnear transmtters and a sngle lnear recever n a multple access scenaro [].) The receved sgnal vectors y, =,...,K, n Fgure are gven by: y = H Fs + ν, where ν s the zero-mean Gaussan nose assocated wth the th recever, whch has a known covarance matrx R. The equalzer output s ŝ = G H Fs + G ν, =,..., K. () Let e denote the error vector assocated wth the th recever, e = s ŝ. Then the weghted MSE s gven by WMSE = α Tr(E{e e H }), () where the α s are non-negatve weghts assgned to dfferent users dependng on ther relatve prortes and Tr( ) denotes the trace operaton. The covarance matrx of e s E{e e H } =(I G H F)(I G H F) H + G R G H where, as stated earler, the covarance matrces of the sgnal and nose are E{ss H } = I and E{ν ν H } = R respectvely, and the transmtted sgnal and recever nose are uncorrelated;.e., E{sν H } =, =,...,K. The problem of desgnng F and G so as to mze the weghted MSE subject to a bound on the transmtted power can be cast as the followng optmzaton problem: F,G,...,G K α Tr(E{e e H }) (3a) subject to Tr(FF H ) p. (3b) Snce the G s are unconstraned varables, they can be elated from (3) by frst mzng the weghted MSE wth respect to G. Ths results n the MMSE equalzers, G = F H H H ( H FF H H H ) + R = F H H H W, () where W = ( H FF H H H + R ), =,..., K. (5) Substtutng () nto (3a) yelds WMSE = n = n α Tr ( K α F H H H W H F ) α Tr ( K α (W ) R )W = Tr ( α W R. Notng that R and W are postve defnte for all =,...,K, and lettng U = FF H, the optmal MMSE transcever desgn problem can be cast as: U,W,,...,K Tr ( α W R (6a) subject to W (H UH H + R ), (6b) Tr(U) p, U, (6c) (6d) where by X Y, we mean that X Y s postve semdefnte. Usng the Schur complement [], the constrant (6b) can be re-wrtten n a lnear matrx nequalty (LMI) form as: [ ] W I I H UH H, =,..., K. (7) + R Therefore, the formulaton n (6) can be rewrtten as: U,W Tr ( α W R (8a) subject to (7), Tr(U) p, and U (8b) Ths problem s a semdefnte program (SDP) and can be effcently solved usng nteror pont methods. Several convenent mplementatons of these methods are avalable; e.g., []. The arthmetc complexty of these methods s at most O(n 6.5 log(/ɛ)), where ɛ> s the soluton accuracy. Once an optmal U s obtaned, we need to fnd an optmal F such that FF H = U. The set of all WMSE optmal precoders take the form F = FQ, (9) where F s the Cholesky factor of the optmal U and Q s an arbtrary untary matrx. B. Dagonal desgns For a multcarrer system employng cyclc prefxng, the channel matrces H are crculant and hence can be smultaneously dagonalzable usng Dscrete Fourer Transform (DFT) and Inverse Dscrete Fourer Transform (IDFT) matrces. The dagonal elements of the dagonalzed channel matrces are gven by H (k, k) =H (k), where H (k) s the frequency response of user s channel at the kth pont on the DFT grd, ω k =π(k )/n, k =,...,n. By assocatng the DFT and IDFT matrces wth the equalzer and precoder respectvely, we end up wth a dagonal channel matrx. We wll also assume GLOBECOM /3/$7. 3 IEEE

3 n ths secton that the nose covarance matrces are also dagonalzed by the DFT matrx;.e., R (k, j) =, j k and R (k, k) = σ (k). (Ths s a common assumpton n the desgn of sngle-user DMT schemes. See [] for further detals.) If we assume that the optmal matrces W and U are dagonal, we can replace them n the formulaton by vectors w and u that represent the dagonal elements of W and U respectvely. (We wll argue below that the optmal W and U are ndeed dagonal.) Wth these new varables, problem (8) can be cast as n α σ (k)w (k) (a) u,w k= subject to w (k) ( u(k) H (k) + σ (k) ), (b) u(k) p, (c) k u(k), k =,..., n. (d) The constrants n (b) amount to K sets of n Lorentz cones. The optmzaton problem () s a rotated second order cone program that offers sgnfcant computatonal advantage over the SDP n (8). In partcular, the arthmetc complexty of obtanng a soluton to the second order cone program s at most O(n 3.5 log(/ɛ)), where ɛ> s the soluton accuracy. In dervng the above formulaton () we have assumed that the optmal precoder s a dagonal matrx U. We now show that the optmal U s necessarly dagonal. Suppose that the optmal soluton s gven by U, where U s assumed to be not dagonal. Let Ū be the dagonal part of U. Then, Tr(U ) = Tr(Ū ). Therefore, Ū satsfes (6c) and les n the feasble set of (6). Now, for any postve defnte matrx A, Tr(A ), () A j jj wth equalty holdng ff A s dagonal [3]. For a gven U, let A (U) =R / (H U H H + R )R /, =,...,K, where we have assumed that R. If the equalzers G are chosen as n (), then for a gven U WMSE = α Tr(A (U) ). () If we defne A = A (U ), and Ā = A (Ū ), then usng () we have that Tr ( (A ) ) > Tr ( (Ā ) ), =,...,K (3) where the strct nequalty holds because (A ) s nondagonal snce U s non-dagonal. Usng the strct nequaltes n (3) we have that WMSE U=U > WMSE U= Ū. Thus we have a contradcton to our assumpton that U was optmal. Therefore, the optmal precoder F must be such that U = FF H s dagonal. C. Choce of Optmal Rotaton So far, we have shown how to desgn a lnear precoder F that satsfes an MMSE crteron. From an MMSE perspectve, t turned out that the desgn problem amounts to desgnng U = FF H. However, as ponted out n (9), ths soluton offers a untary matrx degree of freedom. That s, n general we can wrte F = FQ, where Q s an arbtrary untary matrx to be desgned. In ths secton, we show how Q can be chosen to essentally mze the average bt error rate (over the class of MMSE recevers). Our development wll nvolve arguments whch parallel those n [], [5] and [3], where mum BER transmtter-recever desgn for the sngle-user case was consdered. Consder a complex valued crcularly symmetrc sgnal (e.g., M ary QAM). The th user equalzed output sgnal block gven by () can be wrtten as where ŝ = Dag(G H F)s + z () z = ( G H F Dag(G H F) ) s + G ν (5) denotes the nterference plus nose term. By modfyng the analyss of MMSE multuser detectors [], [5], t can be shown [5] that as the block sze n ncreases, the ISI term approaches a Gaussan dstrbuted random varable almost surely. Hence, the th user s nose plus nterference term z can be approxmated as a Gaussan process wth zero mean and covarance C = ( G H F Dag(G H F) )( G H F Dag(G H F) ) H + G R G H. Usng ths result, we can compute the asymptotc bt error rate as the block sze grows for sgnals drawn from dfferent constellatons. In the followng, we wll only consder QPSK modulaton. However smlar results can be obtaned for hgher order crcularly symmetrc modulaton schemes. For QPSK, the average bt error rate s gven by P e Kn k= n erfc ( ) [G H F] kk, (6) [C ] kk where, erfc(x) = π x exp( z )dz. The approxmaton n the above expresson follows from the assumpton of Gaussan ISI, and hence the error n the approxmaton decreases almost surely as the block sze, n, grows. Usng the fact that for the MMSE equalzers (), we can show that G ( H FF H H H + R ) G H = G H F, [C ] kk =[G H F] kk ( [G H F] kk ). GLOBECOM /3/$7. 3 IEEE

4 Substtutng nto the rght hand sde of (6), we get P e erfc (( [ (Dag(G H F)) I ] ) / ) Kn kk,k (7) Applyng the fact that erfc( x ) s a convex functon of x for all x 3 [5], we observe that f [G H F] rr 3, =,...,K and r =,..., n, (8) then the probablty of error defned n (7) s a convex functon of the dagonal entres of G H F. Hence, a tght lower bound on the average error probablty can be obtaned by applyng Jensen s nequalty. That s, P e K = K n r [( Dag(G H F) ) ] I n Tr( Dag(G H F) ). kk (9) The lower bound n (9) s achevable f for each [,K], the followng condton holds: [G H F] rr =[G H F] ll, r, l [,n]. () We now desgn a precoder that mzes (9). As n (9), F = FQ. Hence, we can wrte G H F = F H H H ( H F FH H H ) H + R F = Q H Γ Q, where Γ = F ( ) H H H H F FH H H H + R F s postve defnte and does not depend on Q. We now restrct our attenton to the moderate to hgh SNR regon satsfyng (8) and wll show how to fnd a untary matrx Q that not only mzes the lower bound n (9) but also acheves t. We start by observng that erfc( ) s a monotoncally decreasng functon of ts argument and hence mzng (9) amounts to solvng the followng optmzaton problem, QQ H =I It can be shown that [5], Tr ( Dag(Q H Γ Q) ), Tr ( Dag(QΓ Q H ) ) n [,K]. Tr(Γ ), () wth the lower bound acheved f and only f the dagonal elements of Q H Γ Q are all equal. One choce of Q that satsfes ths requrement s Q = X V, where X s the matrx whose columns are the egenvectors of Γ and V s the normalzed DFT matrx. Snce we have a sngle precoder to optmze wth respect to the transmsson over K channels, our argument wll only hold when X are dentcal for all [,K]. That s, all Γ s share the same egenvectors. A specal case where ths happens naturally s n cyclc prefxed multcarrer systems. In that case, the MMSE soluton mples dagonal Γ s. Hence, X = I [,K] and Q = V. Therefore, wth Q n (9) chosen to be the DFT matrx, the lower bound n (), and hence that n (9), are acheved. The resultng mzed bt error rate s gven by P e = K n [ Tr ( )] Γ. () Interestngly, choosng Q = V results n each recever seeng dentcal SNRs on all ts subcarrers. Our smulatons n the next secton ndcate that our choce of Q can generate sgnfcant SNR gans. III. NUMERICAL RESULTS In the followng examples, we consder a two user broadcast system whch employs cyclc prefx based multcarrer modulaton wth 3 subcarrers. The nose at each recever s assumed to be whte wth a common varance;.e., R = σ I. Hence, each user has the same block SNR, p/σ. In each of the followng examples the weghts α n () are chosen to be equal. () In ths example, each user s channel s a three-tap FIR flter. The frequency responses of these flters are plotted n Fgure, along wth the optmal power allocaton. Ths fgure shows that optmal power loadng, n the MMSE sense, at an SNR of db, assgns power n a way smlar to the water-fllng prncple. That s, subcarrers that see better channels are assgned hgher powers and vce versa. () In Fgure 3, we consder the same channels as n Fgure. We compare the bt error rate performance when optmal and unform power loadngs are used. Frst, we remark that smulaton results agree wth analyss. Ths supports the fact that the ISI can be modeled accurately by a Gaussan random varable even for the rather small block length n = 3. When the untary matrx degree of freedom s not exploted, we observe that for a bt error rate of, a gan of about 3 db s obtaned va optmal power loadng. The potental performance mprovement from employng a proper rotaton va a DFT matrx at the precoder s also llustrated. The optmal rotaton ntroduces an addtonal gan of about 8 db at a bt error rate of. When the optmal rotaton s employed wth optmal power loadng, a gan of about 3 db s acheved over the case of optmal rotaton and unform power loadng. IV. CONCLUSIONS In ths paper we have addressed the problem of optmal transmtter and recever desgns n a weghted MMSE sense for vector broadcast systems wth a common message. A convex formulaton was derved for general channel matrces, and an alternatve, sgnfcantly smplfed, formulaton was presented for cyclc prefx based multcarrer modulaton schemes. For those applcatons, t was shown that the optmum lnear GLOBECOM /3/$7. 3 IEEE

5 H (jω) Frequency Index H (jω) Frequency Index.6 Power Loads.. SNR=dB Subcarrer Index Fg.. Optmal power loadng n the MMSE sense at a block SNR of db. The frequency responses of the two users channels are shown. BER 3 Analyss: MMSE wth DFT Analyss: MMSE wthout DFT Analyss: Unform Loadng wth DFT Analyss: Unform Loadng wthout DFT Smulaton: MMSE wthout DFT Smulaton: MMSE wth DFT Smulaton: Unform Loadng wthout DFT Smulaton: Unform Loadng wth DFT Block SNR Fg. 3. A comparson of the bt error rate performance of varous systems: Analytcal and smulaton results. REFERENCES [] P. S. Chow, J. Coff, and J. Bngham, A practcal dscrete multtone transcever loadng algorthm for data transmsson over spectrally shaped channels, IEEE Trans. Commun., vol. 3, no., pp , 995. [] Z. Wang and G. Gannaks, Wreless multcarrer communcatons: where Fourer meets Shannon, IEEE Sgnal Processng Magazne, vol. 7, pp. 9 8, May. [3] D. P. Palomar, J. M. Coff, and M. A. Lagunas, Jont Tx-Rx beamforg desgn for multcarrer MIMO channels: A unfed framework for convex optmzaton, IEEE Trans. Sgnal Processng, vol. 5, pp. 38, Sept. 3. [] Y. Dng, T. N. Davdson, Z.-Q. Luo, and K. M. Wong, Mnmum BER block precoders for zero-forcng equalzaton, IEEE Trans. Sgnal Processng, vol. 5, pp. 3, Sept. 3. [5] S. S. Chan, T. N. Davdson, and K. M. Wong, Asymptotcally mum bt error rate block precoders for mum mean square error equalzaton, n Proceedngs of the IEEE Sgnal Processng Workshop on Sensor Array and Multchannel Sgnal Processng, (Roslyn, VA, USA), Aug.. [6] A. J. Goldsmth and M. Effros, The capacty regon of broadcast channels wth ntersymbol nterference and colored Gaussan nose, IEEE Trans. Informat. Theory, vol. 7, pp. 9, Jan.. [7] W. Yu and J. Coff, Sum capacty of Gaussan vector broadcast channels, Mar.. Submtted to the IEEE Transactons on Informat. Theory. Also avalable at weyu/publcatons.html. [8] D. N. Tse, Optmal power allocaton over parallel gaussan broadcast channels, n Proceedngs of Internatonal Symposum on Informaton Theory, (Ulm, Germany), p. 7, June 997. A longer verson s avalable on lne at dtse/pub. html. [9] J. Zhang, A. Sayeed, and B. V. Veen, Optmal transcever desgn for selectve wreless broadcast wth channel state nformaton, n Proceedngs of the Internatonal Conference on Acoustcs, Speech and Sgnal Processng, (Orlando, FL), pp , May. [] S. Boyd and L. Vandenberghe, Introducton to convex optmzaton wth engneerng applcatons,. Course Notes, Stanford Unversty. [] Z.-Q. Luo, T. N. Davdson, G. B. Gannaks, and K. M. Wong, Transcever optmzaton for multple access through ISI channels. IEEE Transactons on Sgnal Processng, To appear. [] J. F. Strum, Usng SeDuM., a MATLAB toolbox for optmzaton over symmetrc cones, Optmzaton Methods and Software, vol., pp , 999. [3] F. A. Graybll, Matrces wth Applcatons n Statstcs. Belmont, CA: Wadsworth, second ed., 983. [] H. V. Poor and S. Verdu, Probablty of error n MMSE multuser detecton, IEEE Trans. Informat. Theory, vol. 3, pp , May 997. [5] J. Zhang, E. K. P. Chong, and D. N. C. Tse, Output MAI dstrbutons of lnear MMSE multuser recevers n DS-CDMA systems, IEEE Trans. Informat. Theory, vol. 7, pp. 8, Mar.. precoder problems performs subcarrer allocaton followed by power loadng. For moderate to hgh SNR, we have also shown how to optmally explot the untary matrx degree of freedom provded by the MMSE soluton. GLOBECOM /3/$7. 3 IEEE