x = x 1 + :::+ x K and the nput covarance matrces are of the form ± = E[x x y ]. 3.2 Dualty Next, we ntroduce the concept of dualty wth the followng t

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1 Sum Power Iteratve Water-fllng for Mult-Antenna Gaussan Broadcast Channels N. Jndal, S. Jafar, S. Vshwanath and A. Goldsmth Dept. of Electrcal Engg. Stanford Unversty, CA, emal: Abstract In ths paper we consder the problem of maxmzng sum rate on a multple-antenna downlnk n whch the base staton and recevers have multpleantennas. The optmum scheme for ths system was recently found to be drty paper codng". Obtanng the optmal transmsson polces of the users when employng ths drty paper codng scheme s a computatonally complex non-convex problem. We use a dualty" to transform ths problem nto a convex multple access problem, and then obtan a smple and fast teratve algorthm that gves us the optmum transmsson polces. 1 Introducton There has been a great nterest n characterzng and computng the capacty regon of downlnk channels n recent years. An achevable regon was found by [6], and ths achevable regon was shown to be sum rate optmal n [6, 2, 9, 10]. Unfortunately, the characterzaton of the regon n [6] leads toa non-convex non-lnear optmzaton problem that s dffcult to solve, and hence obtanng the optmal rates and transmsson polces of each user s computatonally complex. Note that, n the sngle antenna case, although the problem s stll non-convex, t smplfes to only the best user transmttng at any tme nstant. Such a polcy s, however, not the optmal polcy n the multple antenna case. A dualty technque presented n [8, 2] transforms the nonconvex downlnk problem nto a convex sum power uplnk (MAC) problem, whch s much easer to solve. In ths sum power uplnk or sum power MAC problem, the users n the system have a jont power constrant nstead of the ndvdual constrants n the conventonal MAC. As n the case of the conventonal MAC, there exst standard nteror pont convex optmzaton algorthms [11] that solve the sum power MAC problem. A new nteror pont based method has also been found n [12]. However, employng a nteror pont convex optmzaton algorthm to tackle as well structured a problem as sum capacty s neffcent. In ths paper, we explot the structure n ths sum capacty problem to obtan a smple teratve algorthm for calculatng sum capacty. Ths algorthm s nspred by and s very smlar to an teratve algorthm for the conventonal ndvdual power constrant MAC problem by Yu and Coff [1]. Although a rgorous proof of the optmalty of the algorthm for the general case s unknown, ts workng s hghly ntutve and s found to converge n all smulaton results so far. Here, we frst provde an argument that shows that the algorthm ether converges or oscllates between two ponts. Ths paper s structured as follows. In the next secton, we present the system model. In Secton 3, we present some background on drty paper codng and dualty. In Secton 4, we study the Kuhn-Tucker condtons of the problem and present the algorthm. Fnally, we present an analyss of the propertes of ths algorthm n Secton 5 and conclude wth Secton 6. 2 System Model The downlnk channel model consdered s shown n Fgure 1. Note that an uplnk model s depcted alongsde t. Ths s the dual uplnk, where the users n the system have a sum power constrant. The sgnfcance of ths dual uplnk model s explaned n Secton 3 The downlnk and uplnk channel models are as below: y = H x + n ; 8 Downlnk channel model (1) v = =1 H y x + w: Dual uplnk channel model (2) where we assume H 1, H 2,..., H K to be the channel matrces of users 1 through K respectvely on the downlnk, and a transmt power constrant of P. In the next secton we provde some background on

2 x = x 1 + :::+ x K and the nput covarance matrces are of the form ± = E[x x y ]. 3.2 Dualty Next, we ntroduce the concept of dualty wth the followng theorem: Theorem 1 ([2]) The drty paper regon of a MIMO BC channel wth power constrant P s equal to the the capacty regon of the dual MIMO MAC wth sum power constrant P. C drtypaper (P; H) = C unon (P; H y ): Fgure 1: System models of the BC MIMO(left) and the MAC MIMO (rght) channels two mportant concepts that lead to the algorthm - drty paper codng and dualty. 3 Background 3.1 Drty Paper Codng Care and Shama [6] developed an achevable set of rates for the MIMO broadcast channel based on the drty paper codng" result of Costa [5], and hence ths regon s termed the drty paper regon. Ths codng strategy allows a channel wth nterference known at the transmtter to acheve the same data rate as f the nterference dd not exst. Ths translates to the followng codng strategy: The transmtter frst pcks a codeword for recever 1. The transmtter then chooses a codeword for recever 2 wth full (non-causal) knowledge of the codeword ntended for recever 1. Therefore recever 2 does not see the codeword ntended for recever 1 as nterference. Smlarly, the codeword for recever 3 s chosen recever 3 does not see the sgnals ntended for recevers 1 and 2 as nterference. Ths process contnues for all K recevers. Snce the orderng of the users clearly matters n such a procedure, the followng s an achevable set of rates R ß() = 1 2 log ji + H ß()( P j ± ß(j))H y ß() j ji + H ß() ( P j> ± ß(j))H y ß() j = 1;:::;K: (3) The drty-paper regon C drtypaper (P; H) s defned as the unon of all such rate vectors over all covarance matrces ± 1 ;:::;± K Tr(± 1 + :::± K ) = Tr(± x )» P and over all decodng order permutatons (ß(1);:::;ß(K)). The transmtted sgnal s Snce the dual MIMO MAC wth sum power P s, n fact, a convex problem, whle the orgnal drty paper problem s not, ths dualty result s of great use, both computatonally and analytcally. Usng ths dualty, the sum rate of the downlnk has been shown to be achevable by drty-paper codng [2]. The sum rate maxmzaton problem s: P max log ji + H 1 ± 1 H y M 1 j + ± 1 0; =1 Tr(± )»P +log log ji + H 2(± 1 + ± 2 )H y 2 j ji + H 2 ± 1 H y 2 j + ji + H M(± ± M )H y M j ji + H M (± ± M 1 )H y M j : (4) As noted for the general drty paper regon, ths problem s not convex. By usng dualty, however, we get the followng equvalent uplnk sum rate maxmzaton problem: max S P M log ji + 0; =1 Tr(S )»P =1 H y S H j: (5) Ths problem s convex and can be solved usng convex maxmzaton technques whch are polynomal n complexty [11]. 3.3 Iteratve Waterfllng by Yu and Coff The teratve waterfllng algorthm for the conventonal MIMO MAC problem, wth ndvdual power constrants on each user was obtaned by Yu and Coff n [1]. Ths algorthm can also be appled to the sum power MIMO MAC problem, but s however, neffcent, snce t requres a search for covarances over all power splts amongst the K users n the system. Ths teratve waterfllng algorthm, however, forms the bass on whch we develop the sum-power teratve waterfllng algorthm n ths paper.

3 4 The Algorthm We propose a specalzed algorthm whch s found to converge quckly to the optmal covarance matrces. Ths algorthm s based on the same mathematcal quantty as the most common algorthms n fadng and mult-antenna theory - the Karush Kuhn Tucker (KKT) condtons. The KKT condtons were used n [13] to obtan the tme-waterfllng power dstrbuton for sngle antenna pont to pont fadng channels. The space waterfllng results n [3] usng sngular value decomposton n mult-antenna systems, and power allocaton results n [14] for the sngle antenna MAC can also be shown to be connected to KKT condtons. The KKT condtons have also been used to obtan teratve algorthms. For the MIMO MAC, an teratve algorthm was found by [1] that performs sgnfcantly better than convex optmzaton software employed to solve the problem. The algorthm we present here for the MIMO BC s nspred by the MAC algorthm n [1]. In our algorthm, we frst solve the dual sum power MAC problem, and then use the dualty transformatons n [2] to obtan the downlnk covarances. Before analyzng the dual sum power MAC problem, let us revew the KKT condtons of a multantenna pont to pont system. Ths problem can be wrtten mathematcally as max log ji + (H eff ) y SH eff j (6) fs:tr(s)»p g where H eff s the channel of ths user. The KKT condtons for ths problem are gven by I = H eff (I + (H eff ) y SH eff ) 1 (H eff ) y + Ψ along wth complementary slackness condtons, where Ψ s a slackness varable. Note that these KKT condtons have a deep connecton wth the celebrated space-waterfllng algorthm [3] to obtan the optmum covarance S, wth the nverse of the Lagrangan constant 1= correspondng to the waterlevel. Next, let us focus on the dual sum power MAC problem. It can be wrtten mathematcally as X max log ji + H S y S H j X Tr(S)» P S 0: We can obtan the Lagrangan for the above problem, and dfferentatng t wth respect to S, the KKT condtons are found to be: I = H Z 1=2 (I + Z 1=2 H y S H Z 1=2 ) 1 Z 1=2 H y + Ψ P M where Z = (I + j6= H j y S jh j ). Note that these KKT condtons are very smlar to the KKT condton of the pont to pont channel above. P In fact, for each user, f H eff = M H j (I + 6=j Hy S H ) 1=2, then the KKT condtons are dentcal. Ths observaton was made n [1] to obtan the teratve waterfllng algorthm for the MAC channel wth separate power constrants, and hence dfferent water levels 1=. In our case, we further fnd that the water level 1= s the same for all. Ths nspres the followng sum power teratve waterfllng algorthm: 1. Intalze covarance matrces to zero: S (0) = For teraton l P : Generate effectve channels H eff M j = H j (I + 6=j H y S (l 1)H ) 1=2. 3. Treatng these effectve channels as parallel, nonnterferng channels, obtan the new covarance matrces by waterfllng wth total power P. fs (l)g M =1 = argmax over the set Q 0; =1 =1 log ji + (H eff Tr(Q )» P ) y Q H eff j Ths maxmzaton s equvalent to waterfllng the block dagonal channel wth dagonals equal to Hj eff. 4. Return to Step 2 untl desred accuracy s reached. Any set of covarance matrces that are a fxed pont of ths algorthm can be shown to satsfy the KKT condtons of (5). It s then easy to transform these uplnk covarance matrces to downlnk covarance matrces usng the transformatons specfed n [2]. Ths algorthm s dfferent from that n [1] n that t performs a jont waterfllng on all the users n the system nstead of a user-by-user waterfllng. Note that, to perform user-by-user waterfllng, ndvdual power constrants on the users are essental. Thus, the jont waterfllng algorthm s desgned for the case when there s a jont power constrant on all the users. 5 Analyss of the Algorthm For ths analyss, we assume a smple system wth two users. We use the subscrpt l to denote the teraton number. Let us consder nstead, the followng

4 optmzaton problem: max log ji + H y 1 S 1(l)H 1 + H y 2 S 2(l 1)H 2 j + (7) log ji + H y 1 S 1(l 1)H 1 + H y 2 S 2(l)H 2 j Tr(S 1 (l) + S 2 (l)) = P Tr(S 1 (l 1) + S 2 (l 1)) = P Although each teraton of the sum power teratve algorthm may not ncrease the objectve value of (5), we show that t ncreases the objectve value of the optmzaton problem (7) above. Note that the optmzaton problem n (7) can be rewrtten as: max log ji + H y S 1 1(l)H 1 j S 1 (l);s 2 (l);s 1 (l 1);S 2 (l 1) +logji + (I + H y 1 S 1H 1 ) 1 H y 2 S 2(l 1)H 2 j (8) +logji + (I + H y 2 S 2(l)H 2 ) 1 H y 1 S 1(l 1)H 1 j +logji + H y 2 S 2(l)H 2 j: After the l +1th teraton n the algorthm, we obtan new estmates S 1 (l + 1) and S 2 (l + 1). Substtutng them nto the expresson (8) above, we get max log ji + H y S 1 1(l)H 1 j S 1 (l);s 2 (l);s 1 (l+1);s 2 (l+1) +logji + (I + H1 y S 1 H 1 ) 1 H y 2 S 2(l + 1)H 2 j (9) +logji + (I + H y 2 S 2(l)H 2 ) 1 H y 1 S 1(l + 1)H 1 j +logji + H y 2 S 2(l)H 2 j Note that, by the nature of the algorthm, the expresson n (9) s greater than the expresson n (8). Moreover (9) can be rewrtten as max log ji + H y 1 S 1(l + 1)H 1 + H y 2 S 2(l)H 2 j +logji + H y 1 S 1(l)H 1 + H y 2 S 2(l + 1)H 2 j Tr(S 1 (l) + S 2 (l)) = P Tr(S 1 (l + 1) + S 2 (l + 1)) = P Thus, the objectve functon n (7) s always ncreasng wth the teraton l n the algorthm. Snce the expresson n (7) s jontly convex n S 1 (l);s 2 (l);s 1 (l 1);S 2 (l 1), and the functon s bounded, the ncrements n the functon must converge to zero. Note that, after the ncrements n the functon reduce to zero, the unqueness of the waterfllng algorthm [3] guarantees that S (l 1) = S (l + 1) for Fgure 2: Convergence to sum rate of the teratve algorthm for dfferent channel realzatons = 1; 2. Thus, the algorthm ether oscllates between two sets of values for S but or t converges to a fxed pont. If S converges to a fxed pont, then we are done. If t does not, we must modfy the algorthm to guarantee convergence. We know, from optmzaton theory, that such a fxed pont for the KKT condtons exst and that ths fxed pont s the optmum pont. We must ntalze the algorthm wth matrces S that are n the neghborhood of ths optmal fxed pont. Ths can be acheved, for example, usng greedy algorthm technques [15]. Ths concludes our analyss of ths algorthm. A plot showng the convergence propertes of ths algorthm for dfferent random channel realzatons n a two transmt antenna, two recevers wth two antennas each s shown n Fgure 2. Note that the algorthm converges wthn 7 teratons n all these cases. 6 Concluson We provde an teratve algorthm to effcently compute the optmal transmt polces correspondng to the sum capacty of broadcast (downlnk) systems. Ths algorthm s based on the Kuhn-Tucker condtons of the dual sum power multple access channel. References [1] W. Yu, W. Rhee, S. Boyd, J. Coff, Iteratve Water-fllng for Vector Multple Access Channels", pp. 322, Proc. IEEE Int. Symp. Inf. Theory, (ISIT), Washngton DC, June 24-29, [2] S. Vshwanath, N. Jndal, and A. Goldsmth, On the Capacty of Multple Input Multple Output

5 Broadcast Channels", Proc. IEEE Int. Conf. Commun. (ICC), New York, Aprl [3] I.E.Telatar, Capacty of Mult-antenna Gaussan Channels", AT&T Labs Tech. Report, [4] G.J. Foschn and M.J. Gans, On Lmts of Wreless Communcatons n a Fadng Envronment when Usng Multple Antennas", Wreless Personal Communcatons, vol 6., , [14] D. N. Tse and S. Hanly, Multacess Fadng Channels I: Polymatrod Structure, Optmal Resource Allocaton and Throughput Capactes", IEEE Trans. Inf. Theory, Vol. 44, Issue 7, pp , Nov [15] Syed A. Jafar and Andrea J. Goldsmth, Optmal Power Allocaton for Multuser Multcellular Multple Antenna Systems", Proc. IEEE Int. Symp. Inf. Theory [5] M. Costa, Wrtng on Drty Paper", IEEE. Trans. Inform. Theory, v. 29, pp , May [6] G. Care and S. Shama, On the Achevable Throughput of a Mult-antenna Gaussan Broadcast Channel", submtted to IEEE Trans. on Inform. Theory, July See also Proc. IEEE Int. Symp. Inf. Theory, ISIT 2001, Washngton D.C., June [7] T. Cover and J. Thomas, Elements of Informaton Theory", J. Wley, NY. [8] N. Jndal, S. Vshwanath and A. Goldsmth, On the Dualty of MAC and BC", submtted to IEEE Trans. Inf. Theory. See also Proc. IEEE Int. Symp. Inf. Theory, June 2002, pp [9] W. Yu and J. Coff, Sum Capacty of Vector Gaussan Broadcast Channels", submtted to IEEE Trans. nf. Theory. See also weyu/publcatons.html. [10] P. Vswanath and D. N. Tse, Sum Capacty of Vector Gaussan Broadcast Channels", submtted to IEEE Trans. nf. Theory. See also dtse/pub.html. [11] S. Boyd and L. Vandenberghe, Introducton to convex optmzaton wth engneerng applcatons," Course Reader, 2001, [12] H. C. Huang, S. Venkatesan and H. Vswanathan, Downlnk Capacty Evaluaton of Cellular Networks wth Known Interference Cancellaton", Proc. DIMACS Workshop on Wreless Commun., Rutgers unv., NJ Oct [13] A. Goldsmth and P.Varaya, Capacty of Fadng Channels wth Channel Sde Informaton", IEEE Trans. Inf. Theory, Vol. 43, No. 6, pp , Nov