Capacity and Optimal Power Allocation for Fading Broadcast Channels with Minimum Rates

Transcription

1 L G R Capaity and Optimal Power Alloation for Fading Broadast Channels with Minimum Rates Nihar Jindal and Andrea Goldsmith Department of Eletrial Engineering Stanford University Abstrat We derive the apaity and optimal power alloation sheme for a multi-user fading broadast hannel in whih minimum rates must be maintained for eah user in all fading states, assuming perfet hannel state information at the transmitter and at all reeivers We show that superposition oding an ahieve the apaity of suh hannels and expliitly haraterize the boundary of the apaity region he optimal power alloation sheme is a two-step proess: We first alloate the minimum power required to ahieve the minimum rates in all fading states, and we then optimally alloate the exess power to maximize the ergodi rates averaged over all fading states in exess of the minimum rate requirements he optimal alloation of the exess power is a multi-level water-filling relative to effetive noise that inorporates the minimum rate onstraints Numerial results are provided for different fading broadast hannel models I INRODUCION he ergodi apaity of fading broadast hannels determines the maximum average rates ahievable in the downlink of a single ell Ergodi apaity is ahieved via multi-level water-filling of power over time and users using superposition oding [1,2 An unfortunate onsequene of the optimal power alloation sheme is that users with poor hannels may reeive no data for large periods of time, depending on the duration of hannel fades Suh a situation may be unaeptable in delayonstrained appliations suh as video transmission With the above motivation in mind, this paper finds the ergodi apaity of a broadast hannel subjet to an average power onstraint and with minimum rate requirements for all users We will show that the optimal power alloation sheme with minimum rates redues to first alloating the minimum power required to meet the minimum rates and then alloating the exess power aording to a multi-level water-filling sheme relative to effetive noise Minimum-rate apaity is essentially a ombination of outage apaity [3 and ergodi apaity: some power is used to maintain the minimum rates in all fading states, similar to outage apaity with zero-outage probability, while the remaining power is used to maximize the average rates in exess of the minimum rates, similar to ergodi apaity Minimum-rate apaity with a peak power onstraint and with outage are extensions of our derivations whih are solved in [6 his paper is organized as follows Setion II desribes the system model Setion III onsiders a single-user fading hannel with a minimum rate onstraint In Setion IV we show that superposition oding is optimal for the fading two-user broadast hannel and find the optimal power alloation sheme using both Lagrangian and greedy tehniques Numerial results are presented in Setion V, followed by our onlusions II SYSEM MODEL For notation, we use boldfae to denote vetors and to denote expetation over the random variable We onsider a flat-fading Gaussian broadast hannel with a single transmitter ommuniating independent information to users over bandwidth he signal intended for user at time is denoted by and has power Eah reeiver has additive white Gaussian noise (AWGN) with noise density he time-varying hannel gain of user is denoted by By inorporating the hannel gain into the noise term as in [2, we define an effetive noise density and get an equivalent form for the reeived signal:! "$# " &(') (1) where ')* +-,/2134* 3 We assume that the noise density vetor $1)9:9)9:13 3 is known to the transmitter and all reeivers at time instant As the noise density vetor inorporates the effets of the hannel gain, we will alternatively refer to 5 as the fading state he transmitter an vary power ; (and therefore the rate) for eah user relative to the noise density vetor, subjet to average power onstraint Using superposition oding with suessive deoding, the rate of user for fading state 5 is given by < 5=>? A@CB*D4EF& GH$IKJL H3M=NOQP RCS; R[Z IKJ*LUWV HYX where \^K9 _ is the indiator funtion For simpliity, we assume `ae bd In this paper we impose minimum rate onstraints e< 1)9)9:9)1f< whih must be maintained in all fading states, or <g5hjik< 1l >8E1)9)9:9m1, for all 5 b Clearly, must be in the zero-outage apaity region [3 of the hannel in order for the minimum rates to be ahievable in all fading states III SINGLE USER FADING CHANNEL Before analyzing the broadast hannel, we first find the apaity-ahieving sheme for a single user fading hannel subjet to minimum rate onstraints We must find the optimal power alloation sheme n ( is a salar for the single user senario), or equivalently: onpq n lv (2) GrI w subjet to: n_x Q1y<jzit< { r /1/$17 1 IEEE 1292

2 J { 5 G We denote the minimum power required to ahieve the minimum rate as ~} 2ƒnE and define n as the power alloated to fading state in exess of he total power alloated to fading state then is n~- & n Using standard Lagrangian tehniques, the optimal alloation of exess power is modified water-filling: Q~ ƒtˆ& Š &( / &( zx zœ (3) where is the water-filling level satisfying the redued power onstraint ƒž l_ he speial struture of the apaity formula for Gaussian hannels allows us to treat power alloated to a hannel as an additional soure of noise herefore, the interpretation of this sheme is very simple: First alloate minimum power to eah state hen, treating as additional noise, use the standard water-filling algorithm with effetive noise & rather than and with power onstraint instead of We will soon seen that the multi-user broadast hannel an be interpreted in a similar manner IV WO-USER FADING BROADCAS CHANNEL Now onsider a multi-user fading broadast hannel as desribed in Setion II with We expliitly haraterize the minimum rate apaity for the two-user ase, but these results an be extended to an arbitrary -user broadast hannel For simpliity, we assume 7Œ in all states Later we show how our results extend to the general ase where the ordering of noises differs from state to state Before we find the optimal power alloation sheme, let us first prove that superposition oding ahieves apaity for the broadast hannel with minimum rates Consider a two-user, onstant broadast hannel with minimum rates Our goal then is to find all possible rate pairs < 1f< satisfying minimum rates < and <, or equivalently all rate pairs suh that < i < and < ĩ < Without minimum rates, all possible rate pairs are ahievable by superposition oding he rate pairs satisfying the minimum rates are simply a subset of all rate pairs without minimum rates, and are therefore ahievable by superposition oding Beause the fading broadast hannel an be equivalently viewed as a set of parallel onstant broadast hannels [2, one for eah fading state, superposition oding an be used in eah of these parallel hannels to ahieve apaity Using the time-sharing argument of [2, we also an show that the apaity region is onvex Due to the onvexity of the region, the boundary of the apaity region an be found by the following maximization: onpq š < 5=&Aš < 5hl_ (4) IKJ*L 5h&=;5hl_x < 5hzi < 1y< *5hœit< subjet to: over /jx š xe and šwžeÿƒš Let us now introdue notation similar to that used in Setion III Defining minimum powers 5h} ƒe and 5=~}k 5h& ) ƒte, (4) simplifies to: onpwq w E6& 5h& 5h IKJL (5) & E& =*5h&( 5= 5=&( 5h&A 7 subjet to: n5hl_xt 1y/Qẍ 5hœx n5h where } ƒ w 5=g&ª 5hl_ is the exess power onstraint and n5hz}` 5==&«5h is the exess power alloated to fading state 5 As before, 5h and =5= represent exess power hough the minimum rate apaity of the single-user hannel was found rather easily, the broadast hannel problem is onsiderably more diffiult beause stronger users interfere with weaker users Whenever exess power 5h is alloated to user 1, the interferene seen by user 2 inreases and < dereases herefore, whenever user 1 is alloated exess power some exess power must also be alloated to user 2 to overome the additional interferene As a result, user 1 annot be alloated all of the exess power in a state and 5h is onstrained to /jẍ 5=zẍ n5h In order to solve (5), we deompose the maximization into two steps: 1 Given Q5h for all 5, we must optimally distribute the exess power between the users in eah state: n5= ~} onpwq E6& 5==& 5h IKJ*L (6) & šw^@cb*d =E6& n5=rƒ 5==& 5h 5h&( 5=&Aw subjet to: /Qẍ 5=Ÿx n5h 9 2 After we find 5= for eah 5, we must optimally alloate exess power n5h over all fading states: oqpq GhIKJ*L w = n5h3_ subjet to: w * Q5=_x 9 (7) Equation (6) is a one-dimensional optimization over h 5= and is therefore easily solved n5= is ahieved by the following power distribution: 1) If 5h~ n5h and =*5=6 x±e, then n5hmeƒ m9 2) If i, then 5h² n5= and 5=~-/ 3a) If Ê ³ ³ and n5=fxk µ f 3µ, then 5=7 n5h and 5h¹ n5=)egƒ $9 3b) If Ê ³ ³ and n5h Œº µ f 3 µ, then 5=7 1293

3 µ: Y µ and =;5h n5hÿƒ 5= where µ Y µ ƒ 3 Power alloated to user 1 inreases <, but also leads to a derease in < due to interferene On the other hand, power alloated to user 2 will not lead to as large of an inrease in < beause Œ», but it does not affet < In ase 1, beause š i(š, it is optimal to give user 1 as muh power as allowable while ensuring that user 2 an still maintain < On the other extreme, in ase 2, when š ¼ š, user 2 is given all available power When E³ ³ as in ase 3, the weights are suh that inreasing < is optimal up to the point where ½ µ f 3µ* (ase 3a) Past this point, it is optimal to use the power above = µ Y µ to inrease < (ase 3b) = Now that n5= is known, we must solve (7) We introdue a Lagrangian multiplier to get: onpwq GhIKJL w n5h3l_^ƒa n5=_uƒ m9 (8) F he standard solution to suh a problem satisfies n5h3g, similar to single user water-filling he optimal power alloation sheme is a two-level water-filling sheme where is hosen to satisfy exess power onstraint : 1) If š xtš, then 5h ƒt 5=& ;5h EŸƒ =ƒ 5h& m ƒtem9 2) If i, then 5h~-/ 5h~ ƒt 5=&A 9 3a) If E7xÀ x and Áx±: then 5=6 ƒ 5h& 5=6 EŸƒ hƒt 5=&A m ƒ E$9 3b) If EÂxÀ x and Œª: then 5=- µ Y µ; =5= ƒ &w:3 ƒ µ Y µ 9 (9) (1) (11) (12) After some algebrai manipulation, the state-by-state exess power alloation simplifies to: j5=6 onpwq±ã š ƒt 5=& 3 1 (13) š ƒ 5h&w:3 1w/Ä and 5h and 5= are speified by wn5h3 Extending this sheme to allow states in whih Œe only requires reversing all subsripts in (13) and o simplify this solution, let us first define effetive noises : k k 5h& 5h& 3 ³ 3 i w (14) Let us examine the ³ equations more arefully In Setion III we saw that the effetive noise in a single-user hannel is the sum of the atual noise and power already used in the hannel If we expand 5h&, we find that w& 5= In this expression the first two terms represent noise and interferene, and the last term 5h, represents power already alloated to user 2 It appears that user 1 should only have an effetive noise term of & 5h, but the fator of in is due to the fat that user 1 an only use a fration (speifially ) of the power alloated to it due to the interferene user 1 auses on user 2 Using the effetive noises, the optimal sheme is waterfilling with two water-levels saled by weights š and šw : n5h~ oqpq à š ƒ 1 šw ƒ 1w/ Ä 9 (15) he alloation of exess power to eah fading state is idential to the optimal power alloation [1, 2 used to ahieve ergodi apaity of the broadast hannel with effetive noises and and power he alloation of the exess power between users, however, is not neessarily the same as under ergodi apaity maximization beause under minimum rate onstraints all exess power in a fading state annot be alloated to the stronger user due to the interferene it auses on the weaker user Nonetheless, the state-by-state rates ahieved by eah user, and therefore the average rates, are equal to the ergodi apaity maximization rates he minimum rate apaity is therefore equal to the ergodi apaity of the broadast hannel with effetive noises and and power plus the minimum rates We see that water-level is used for hannels that are filled on and is used for hannels Fig 1 illustrates a four fading state example where š Œžš Note that (15) does not speify the distribution of exess power between the two users If water-filling is done up to, then some power is given to user 1, but power may also be alloated to user 2 aording to = n5h3 Both users are alloated some power in all fading states, due to the minimum power alloation, and therefore superposition oding is neessary in every fading state In reent work on the broadast hannel [2, 4, a greedy interpretation of the optimal power alloation sheme is derived We now use the same approah to find an alternative derivation of the minimum rate apaity o use the greedy approah, we must reformulate our problem slightly We again assume w✠for simpliity and generalize our results later Instead of alloating power to users 1 and 2, we distribute power between two poliies in eah fading state: Poliy 1: Inrease < 5h while ensuring < 5h i-< Denote the power alloated to this poliy as 5= o main- 1294

4 Ç Î š x state 1 state 2 state 3 state 4 Fig 1 Water-filling diagram for two-user hannel with min rates tain < ;5h above the minimum rate, user 2 reeives a fration of 5= : 5=d» 5hmEƒ and 5hÈ 5h Poliy 2: Inrease < ;5h while keeping < 5= onstant Denote the power alloated to this poliy as 5= Beause user 2 does not interfere with user 1, user 2 is given all the power: 5=- 5h and 5h~/ We now perform the optimization in (5) over 5h and 5h By adding the onstant 3ƒŸš < ƒ šw:< to the funtion to be maximized and introduing a Lagrangian, it an be shown that the following is an equivalent maximization: onpwq š YË & šw Ë m_ìƒ G;É IKJ*LlÊ G;É IKJ*L 5h & 5=_ (16) where Ë^ & G;É IKJ*L YÍ É and Ë ; G;É IKJ*L GÉ IKJL IKJL ICJ*L N Due to the onvexity of the problem, we know there exists a ÀŒe/ suh that the solution of the above problem solves the original optimization problem (4) As is done in the standard broadast hannel [2, 4, we deompose (16) into a set of independent optimization problems, one for eah fading state: G;É onpwq IKJ*LlÊ G;É IKJ*L š më^ &šw Ë Ÿƒ 5h& 5= (17) By Lemma 31 of [4, a power alloation sheme is optimal if and only if it is the solution to (17) for every fading states We an find the solution to (17) using the greedy method of [2, 4 as: For =ºE*1f, define utility funtions Î ';r ' &- 5h&A he solution to (17) then is: Ð Ñ Ò onpwq # Ê Î ƒ1y =ºE*1Y Ï9 (18) ';_ NrÓ 'u9 (19) At eah interferene level ', power Ó:Ô is alloated to the poliy with the larger utility funtion Î 'Õ up to the point where both utility funtions are negative he utility funtions are the derivatives of Ë and Ë minus In eah fading state, power Æ is alloated to the better of the two poliies at eah interferene level, or to the poliy leading to the larger marginal inrease of the objetive At the optimal point, the marginal inrease of the objetive funtion in eah fading state will equal he optimal power alloation sheme then is: 1) If š xºš, then Î ';ŒÎ '; for 'i/ so poliy 1 reeives all power herefore, ƒ & 3 2) If i, then Î^'Õ7Œ±Î 'Õ for 'Áiž/ so poliy 2 reeives all power and ƒ &w:3 3) If E-x, then Î '; and Î^*'Õ interset at the point '» ƒ and Î /;niàî /; If Áxª, then the utility funtions are negative at the intersetion point and only poliy 1 reeives power aording to ƒ &A If >Œª, then the funtions interset where the utility funtions are positive and both poliies reeive power: k ƒn 3 and ƒ &w:3 ƒ hese three ases orrespond exatly to (9)-(12) and the sheme an be extended to any ordering of noises by reversing all subsripts he greedy approah gives valuable insight on how power is alloated within eah fading state (by hoosing the poliy with the larger marginal inrease at every interferene level) and also aross fading states (via water-filling level ) V NUMERICAL RESULS In this setion we present numerial results on two-user broadast hannels with symmetri minimum rates and symmetri fading distributions In all plots, the power onstraint is 1 mw and the bandwidth is 1 khz In Fig 2 the apaity region of a two-user hannel with very different noise levels is plotted In one fading state, is 4 db less than, and vie versa in the seond fading state Without minimum rates, apaity is ahieved by alloating most of the power to the better of the two users in eah hannel state When minimum rate onstraints are applied, muh of the power must be alloated to the weaker user in every fading state to meet the minimum rates, leading to a large apaity redution As the differene in the noise levels of the two users dereases, the differene between the minimum-rate apaity region and the ergodi apaity region dereases he apaity region of a hannel where and differ by 2 db in eah fading state is plotted in Fig 3 Minimum rate onstraints fore power to be alloated to both users in every state, but beause the poorer hannel is only 2 db weaker than the stronger hannel, as opposed to the 4 db in the first example, alloating power to the weaker user is not quite as sub-optimal herefore, the differene between the ergodi and minimum rate apaity regions is not as large as in the first example In the final two plots, results for more realisti hannel models are presented Independent fading is assumed for both reeivers In Fig 4, Riian fading with Ö `E is modeled his is not as severe as Rayleigh fading, but the power of the 1295

5 1 1 Ergodi Capaity Min Rate = 3 kbps Min Rate = 34 kbps Zero Outage Capaity 35 3 Ergodi Capaity Min Rate = 5 kbps Min Rate = 1 kbps Zero Outage Capaity B) = (1 mw, 1e 4 mw) wp 1/2 B) = (1e 4 mw, 1 mw) wp 1/2 1 5 E[1/ B) = E[1/(n 2 B) = 1 mw Fig 2 Capaity of symmetri hannel with 4 db differene in SNR Fig 4 Riian fading withú Û Ü, Average SNR = 1 db B) = (1 mw, 1 mw) wp 1/2 B) = (1 mw, 1 mw) wp 1/2 Ergodi Capaity Min Rate = 3 kbps Min Rate = 5 kbps Zero Outage Capaity Fig 3 Capaity of symmetri hannel with 2 db differene in SNR E[1/ B) = E[1/(n 2 B) = 1 mw 1 Ergodi Capaity Min Rate = 1 kbps Min Rate = 15 kbps Zero Outage Capaity Fig 5 Riian fading withú Û Ý, Average SNR = 1 db multipath omponent equals the power of the line of sight omponent he hannels of the two users often vary quite signifiantly, and, as expeted by our earlier results, minimum rates redue apaity signifiantly In Fig 5, Riian fading with ÖØ«Ù is modeled Beause the line of sight omponent is quite large, both users generally have strong hannels As a result, minimum rates do not redue apaity signifiantly In eah of the plots, the zero-outage apaity region is also shown Notie that the differene between the minimum-rate and ergodi apaity regions is roughly proportional to the differene between the zero-outage and ergodi apaity regions his relationship is due to the fat that minimum-rate apaity is a ombination of zero-outage and ergodi apaity VI CONCLUSION We have obtained the apaity region of a multi-user fading broadast hannel with minimum rates We found that the minimum-rate apaity region is ahievable by superposition oding with suessive deoding and we derived the optimal power alloation sheme By using minimum power and effetive noise terms, we saw that the minimum rate problem deomposes into two independent problems: a zero-outage apaity problem (ie minimizing the power needed to ahieve the minimum rates), and an ergodi apaity problem (ie max- imizing the ergodi apaity of the broadast hannel with the effetive noise terms and the exess power onstraint) We also derived a greedy power alloation sheme to give additional insight into the optimal power alloation sheme Finally, by analyzing our numerial results, we determined that severely fading hannels, ie hannels with wide-ranging noise levels, inur a large apaity redution due to minimum rate onstraints, while benign fading environments are able to support large minimum rates with little apaity redution REFERENCES [1 D Hughes-Hartog, he apaity of the degraded spetral Gaussian broadast hannel, PhD Dissertation, Stanford University, 1975 [2 L Li and A Goldsmith, Capaity and optimal resoure alloation for fading broadast hannels: Part I: Ergodi apaity, IEEE rans Inform heory, pp , Marh 1 [3 L Li and A Goldsmith, Capaity and optimal resoure alloation for fading broadast hannels: Part II: Outage apaity, IEEE rans Inform heory, pp , Marh 1 [4 DN se, Optimal power alloation over parallel Gaussian broadast hannels, Proeedings of ISI, Ulm, Germany, June 1997, p 27 [5 A Goldsmith and M Effros, he apaity region of broadast hannels with intersymbol interferene and olored Gaussian noise, IEEE rans Inform heory, vol 47, pp , Jan 1 [6 N Jindal and A Goldsmith, Capaity and optimal power alloation for fading broadast hannels with minimum rates, in preparation for journal submission 1296