On the Capacity of Wireless Multicast Networks

1 On the Capacity of W ireless Multicast Netw orks Seyed Reza Mirghaderi † , Alireza Bayesteh † , a nd Amir K. Kha ndani † Abstract The problem of maximizing the average rate in a mu lticast network subject to a coverage constraint (minim um quality o f service) is stud ied. Assuming th e channe l s tate in formation is available only at the r eceiv er side and single antenna no des, the h ighest expected rate ach iev able by a rand om user in the network, called e xpected typica l rate , is derived in two scenarios: ha rd coverage constraint and soft coverage con straint. In th e fir st case, the coverage is expre ssed in terms of the o utage pro bability , while in the secon d case, the expected rate sho uld satisfy certain minimum requ irement. I t is shown that th e optimum solution in bo th cases (achieving the h ighest expected typical rate for given coverage require ments) is achieved by an infinite layer sup erposition co de for which the op timum power allocation amon g the different layers is deriv ed. For the MISO ca se, a sub optimal codin g scheme is proposed , which is shown to be asymptotically optimal, when the numb er of transmit antennas grows at least logar ithmically with the number of users in the network. I . I N T RO D U C T I O N The widespread applicati on of wireless networks has mot iv ated efficient transmission strategies for diffe rent appli cations. One of t hese applicatio ns is data mult icasting where a group of us ers are interested in receiving the same si gnal, possibl y at differe nt l e vels of resolution. In a wireless mul ticast network, a common source is transm itted to N users throug h a fading channel. In such networks, two criteria are usually stu died as measures of performance: network coverag e (minim um quality of service) and e xpected rate (typical quality of service). In th e first criterion, the objectiv e is to provide all t he users with a mini mum service regardless of their channel qualiti es. In the second criterion , t he a verage data rate observed by a randomly selected user is considered where users with better channel c onditions may recei ve higher d ata rates. An example for such networks is scalable video broadcasting in w hich all the subscribers expect to receiv e a basic signal, wh ile th ose with better channel conditions might enjoy a higher resolution. In [1], the challenges in loss y mu lticasting are studied from an inform ation theoretical po int o f view . In thi s work, for an analog Gaussian source wit h a bandwidth equal to the channel bandwidt h, uncoded † Coding & Signal Transmission Laboratory (ww w .cst.uwaterloo.ca), Dept. of Elec. and Comp. Eng., Uni versity of W aterloo, W aterloo, Ontario, Canada, N2L 3G1, T el: 519-884-85 52, Fax: 519-888-43 38, e-mail: { smirghad, alireza, khandani } @cst.uwaterloo.ca. Financial supports provided by Nortel, and the corresponding matching funds by the Federal gov ernment: Natural Sciences and Engineering Research Council of C anada (NS ERC) and P rovince of Ontario: Ontario Centres of Excellence (OCE) are gratefully ackno wl edged. 2 transmissio n is shown to achie ve t he mini mum a verage end-t o-end disto rtion. The scenario in whi ch the source has a larger bandwidth i s st udied in [2], where diffe rent m ethods of digit al transmissi on are in vestigated. In [3], a different approach to broadcasting, called static broadcasting , is propo sed. It is assumed t hat all the users receiv e t he same amount of data from a commo n source, b ut with different number of channel uses as determined by their respective channel qualiti es. The actual transmission time in thi s scheme depends on the user with the worst channel, and hence, the transmi ssion rate might be very low when the number of users is large. In this work, we consider a wireless mul ticast network in a quasi -static fading en vironm ent with addit iv e Gaussian noise. The obj ectiv e is to maxim ize t he av erage performance, while a coverage constraint is satisfied. A verage performance i s defined as the Qualit y of Service (QoS) observed by a randoml y chosen user (typi cal user), while the cove rage requirement relates to the QoS observed by t he user(s) wi th the worst channel condi tion(s). W e assum e that t he transmi ssion block is large eno ugh to yield reliable communication . Howe ver , averaging over time is not poss ible because of t he delay constraints. In other words, all the sym bols within a transmi ssion block experience the same channel gain. The channel st ate information (CSI) of each user is assum ed to be known only at the correspond ing recei ver . For a channel with the above characteristi cs, t he er godic capacity is not defined, but the outage capacity is defined as the maximum rate decodable with a gi ven probability [4]. In [5], a broadcast approac h for a single user channel with t hese ass umptions is proposed which optimizes th e expected decodable rate. W e apply “mul ticast outage capacity” and “expected minimum rate” definitions to characterize coverage in t he network. Out age capacity is exploited when we ha ve a hard cov erage constraint. In this case, we require that with a given probability , with in each transmission block, a specific amount of data is recei ved by all the users. In the soft coverage cons traint scenario, we relax th e cover age constraint by stat ing it in terms of t he expected minimum rate recei ved by all t he users within each block. For both hard and so ft coverage constraints , another sim ultaneous criterion is the maximization of the expected t ypical rate which is defined as the a verage rate received by a random ly selected user . In general, there is a tradeof f between t hese two criteria. The mi nimum-service criterion has been studied in [6] for a sing le user fading channel, assuming CSI is known at the transmitter . In that work, giv en a service o utage constraint for a real-tim e application, the av erage rate is maximized for a n on real-time appl ication sent on to p of it. An adaptiv e var iable rate code is proposed and shown to be optim um in that scenario. Similarly , a minim um rate constrained capacity measure is defined for broadcast channels in [7]. It is shown that th e mini mum rate region is the ergodic capacity region o f a broadcast channel, with an effecti ve n oise determined by the minim um rate requi rements. U sing similar assum ptions for the CSI av ailability , a more general case is considered in [8], where each user specifies its rate constraint s in a triplet o f maxim um rate, mini mum rate, and a 3 so-called shortage probability . In this paper , we use a b roadcast model for an unknown quasi-static fading channel [5] and s how the optimalit y of thi s model in our scenario. The same m odel is used in [9] t o propose a m ultilevel approach for joint so urce-channel cod ing i n a SISO channel (assuming the CSI is not av ailable at the t ransmitter). The broadcast approach in [5] is extended in [10]–[12] for the case t hat the transmitt er has parti al channel state informatio n, and i n [13] for the case of M IMO block fading channel. [14] combines this broadcast approach wit h H ybrid Auto matic Retransmiss ion Request (HARQ) and shows that this approach results in high throughput and low latency i n a point-to-point l ink. References [15], [16] use the broadcast approach in [5] in a two-hop relay network. W e in vestigate t he performance of the proposed scheme in both SISO and MISO cases. The MISO multicast asympt otic capacity lim its are exa mined in [17], when the CSI is a v ailable at t he transm itter . It is shown that the adverse ef fect of ha ving a lar g e numb er of users can be compensated by increasing the number of t ransmit antennas. In this work, we stu dy a sim ilar scenario and derive asymptoti c capacity results for a large nu mber of transmit antennas. The rest of this paper is organized as fol lows: In section II, the system m odel i s int roduced. Section III focuses on th e virtual broadcast model for an u nknown fading channel when the network is delay limited. Sections IV and V di scuss multicast networks when we h a ve a single antenna at the transmi tter and at each recei ver . In sectio n IV , we ev aluate the optimum performance of the network in terms of the achie vable t radeof f between the expected typical rate and the m ulticast outage capacity (hard coverage). Section V studi es a sim ilar problem of computing the opti mum t radeof f, but for a so ft cover age constraint where the expected minimum rate is used as the cov erage criterion. Section VI i n vestigates the MISO case, where we derive asymptotic capacity results for a lar ge number of transmi t antennas. Finally , section VII concludes the paper . Throughout this paper , we represent t he norm o f the vectors by k . k , the conju gate transpose operation by ( . ) † , and the expectation operation by E [ . ] . The notation “ log ” is used for the natural l ogarithm, and rates are expressed i n nat s . W e denote f y ( . ) and F y ( . ) as the probabil ity d ensity function and the cumulat iv e density function of random var iable y , respectively . Notation 1 A is used to define a binary function of x which is equal to 1 if e vent A occurs and 0 ot herwise. F or gi ven functio ns f ( n ) and g ( n ) , f ( n ) = o ( g ( n )) is equiv alent to lim n →∞    f ( n ) g ( n )    = 0 , and f ( n ) = ω ( g ( n )) i s equiv alent to lim n →∞ f ( n ) g ( n ) = ∞ . W e use A ≈ B to denot e the approximate equality between A and B , such that by su bstituti ng A by B the validity of the equations is not comprom ised. 4 I I . S Y S T E M M O D E L In this paper , we consider a wireless network b roadcasting a common message. In the first part, it i s assumed that a singl e-antenna transm itter transmits a common message to N s ingle-antenna recei vers. The recei ved s ignal at the i th receiv er , denoted by y i , can be written as y i = s i x + n i , (1) where x is the transmitt ed sig nal satisfying an a verage power const raint of E [ x 2 ] ≤ P , n i ∼ C N (0 , 1) is the A dditive White Gaussian Noise (A WGN), and s i ∼ C N (0 , 1) is the channel coefficient from the transmitter to the i th receiver . The channel gain h i = | s i | 2 , which is assumed to be constant durin g a transmissio n blo ck, has t he following Cumulative Distribution Function (CDF): F i ( h ) = F ( h ) = 1 − e − h , ∀ i. The typical channel of the m ulticast network is defined as th e channel of a randomly selected (ty pical) user . Since all the channels are i.i.d ., the typical channel gain distribution satis fies F typ ( h ) = F ( h ) = 1 − e − h . (2) Since all the N channels are Gaussian and they receiv e a comm on si gnal, the coverage requirement is determined by the channel wit h the lowest gain h mul = min i ( h i ) , which is called the multi cast channel . Due to t he stati stical i ndependence of the channels , the gain of the m ulticast channel h as the foll owing distribution Pr n min i ( h i ) > h o = ( Pr { h i > h } ) N = e − N h . As a result, we have F mul ( h ) = 1 − Pr n min i ( h i ) > h o = 1 − e − N h . In this paper , we deal with three quality measures defined as follows: • Mu lticast outage capacity , R ǫ , is the rate decodable at th e multi cast channel with probabil ity (1 − ǫ ) . • E xpected multicas t rat e, R mul , is the aver age rate decodable at the multicast channel, i.e. R mul = E [ R ( h ) | h = h mul ] , where R ( h ) is the decodable rate at the channel st ate h . • E xpected t ypical rate , R ave , is the avera ge rate decodable by a rando mly selected user , i.e. R ave = E [ R ( h )] . 5 I I I . B ROA D C A S T M O D E L F O R A N U N K N OW N Q U A S I - S T A T I C F A D I N G C H A N N E L In [5], i t is shown that the expected rate for a receiver with a quasi-s tatic block fading channel, unknown at the transm itter , and a st ringent delay constraint, is equiva lent to a weighted sum rate of a degraded broadcast channel with i nfinite nu mber of virtual receiver s, each correspondi ng to a realization of the channel. In t his paper , we exploit the sam e model in a more general fashion. Noting ou r frequent use of this model, we first stud y it in more details. In thi s work, we assum e a block fading channel for al l users, where t he channel st ate takes values according to a given prob ability densi ty functi on (pdf) (assum ed to be exponential) at the start of each block, stays unchanged durin g the coherence time (block length of the channel), and then chang es independently at the s tart of the subs equent bl ock. The channel state informatio n for each channel is assumed to be ava ilable only at the correspondi ng receive r’ s s ide. In this case, if th ere were n o constraint on the d ecoding delay , coding across different fading blocks would be possibl e, achieving the so-called er godic capacity . Howe ver , i n our model, we impos e a decoding delay constraint which rest ricts the recei ver to decode wi thin a p eriod equal to the length of a fading block. Each receiv er decodes a fraction of the t ransmitted data which is s upported by it s corresponding channel. Hence, for any coding scheme, we hav e a function R ( h ) which determines the data rate d ecoded in channel state h . The time av erage o f the rate decoded b y a given recei ver over infinite num ber of transmission blo cks would be E h [ R ( h )] [5]. Consider an infinite number of virtual recei vers, such that receiv er R X h is experiencing a fading lev el between h and h + d h . W ith these setting s, R X h is recei ving all the d ata recei ved by R X h − d h , i n addition to d R h , where d R h = R ( h ) − R ( h − d h ) . The virtual receive rs int roduce a degraded virtu al br o adcast network in which the rate associated wit h user R X h is d R h . From the degraded nature of the Gaussian broadcast channel [18], it foll ows that d R h ≥ 0 . The original receiver corresponds to the vi rtual receiver R X h with probability η ( h )d h , where η ( h ) i s th e channel gain probability distribution function . W ith this interpretation, for a gi ven coding scheme, th e distinction between dif ferent channels introdu ced in the previous section , namely mul ticast channel and typi cal channel , i s translated to the difference between the probabi lity dis tribution based on which the original receiver is represented by the virtual recei vers. Note that both the mult icast and typical channels correspond to the same common virtual 6 br oadcas t network , and the measures defined in the previous section could be written as follows 1 : R ave = Z ∞ 0 R ( h ) η ( h )d h = − (1 − F ( h )) R ( h )   ∞ 0 + Z ∞ 0 (1 − F ( h ))d R h ( a ) = Z ∞ 0 (1 − F ( h ))d R h , (3) R mul = Z ∞ 0 (1 − F mul ( h ))d R h , (4) R ǫ = R ( h ǫ ) = Z h ǫ 0 d R h , (5) where h ǫ = F − 1 mul ( ǫ ) = − log(1 − ǫ ) N . In the above equation, ( a ) follows from the facts t hat R (0) = 0 and lim h →∞ (1 − F ( h )) R ( h ) = 0 , where the latter i s due to the fact that 1 − F ( h ) = e − h and R ( h ) grows logarithmically with h . A simi lar argument is applied to conclu de (4). In the case of the mul ticast channel in (5), the orig inal recei ver has a channel gain less than h ǫ with probabil ity ǫ , and hence, the hi ghest decodable rate is R ( h ǫ ) , with probabilit y 1 − ǫ . As seen above, the performance measures in thi s work are diffe rent weighted sum rates of the virtual br oadcas t network , resul ting in vectors ( R ǫ , R ave ) and ( R mul , R ave ) for the hard and soft coverage con- straint scenarios, respecti vely . W e refer to the maximu m achiev able r egions for these vectors a s the c apacity r e gion corresponding to each case. Next, we pro vide a definition for the optimality of a performance v ector . Definition 1 The b oundary set B of a closed conve x r e gion R ⊂ R n + is defined as B = { x ∈ R | ∄ x ′ ∈ R n ++ , x + x ′ ∈ R } (6) wher e R + and R ++ ar e the set of non ne gative and strictl y positive r eal numbers, r espectively . W ith the above definition , a performance vector is optim al if it i s in the boundary set of all possible performance vectors. In the following t heorem, we sh ow that t he o ptimal performance vector for each cove rage constraint scenario is achieve d by superpositi on coding in whi ch the rate of the virtual recei ver R X h is giv en by d R h = lo g 1 + hρ ( h )d h 1 + h R ∞ h +d h ρ ( u )d u ! , (7) where ρ ( . ) is t he power dis tribution fun ction s uch that ρ ( h )d h is th e am ount o f p ower allocated to the virtual recei ver corresponding to the channel state h . 1 W ith a small misuse of notation, to simplify the problem formulation, the integration i n the right hand side of all of the t hree equations is computed ov er h , where d R h is expres sed as an explicit function of h throughout the paper . 7 00 00 11 11 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 0000 0000 0000 0000 1111 1111 1111 1111 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 0 1 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 0000 0000 0000 0000 1111 1111 1111 1111 x 2 x 1 B 1 g ( B 1 ) B 2 ⊂ g ( B 1 ) g ( x 1 , x 2 ) = x 1 + x 2 ( √ 2 / 2 , √ 2 / 2) (1 , 0) √ 2 1 0 R 1 (0 , 1) R 2 = g ( R 1 ) Fig. 1. A schematic figure for Lemma 1. Theor em 1 The boundary set of ( W 1 , ..., W k ) , wher e W i = R ∞ 0 w i ( h )d R h ∀ i is a posi tive weighted s um rate of th e underlying virtual br oa dcast channel, is achievable by a sup er- position coding scheme. Pr oof : T o prove t he theorem, we first state and prove the following l emma: Lemma 1 Consider a mapping function g ( . ) from a closed r e gion R 1 ⊂ R n + to R 2 ⊂ R k + such th at g ( x ) = Mx , wher e M ∈ R k + × R n + , such t hat each row of M contains at least one p ositive element. Denote B 1 and B 2 as the boundary sets of the r e gions R 1 and R 2 , r espectively . Then, we have B 2 ⊂ g ( B 1 ) Pr oof : As sume th is is not t rue. Hence, there must exist x 2 ∈ B 2 such that x 2 6∈ g ( B 1 ) and x 1 ∈ R 1 , such that x 2 = g ( x 1 ) . Since x 1 6∈ B 1 , there exists x ′ 1 ∈ B 1 such that x ′ 1 − x 1 ∈ R n ++ . Defining x ′ 2 = g ( x ′ 1 ) ∈ R 2 , we ha ve x ′ 2 − x 2 = M ( x ′ 1 − x 1 ) ⊂ R k ++ , (8) which contradicts the fact that x 2 is in the boundary set of R 2 and the lemma is proved. Figure (1) ill ustrates an example of Lemma 1 w hen n = 2 and k = 1 . The region R 1 in this example is defined b y x 1 , x 2 ≥ 0 , x 2 1 + x 2 2 ≤ 1 and the m apping is defined by g ( x 1 , x 2 ) = x 1 + x 2 . The boundary of R 1 , i .e., B 1 , is the quarter -circle x 1 , x 2 ≥ 0 , x 2 1 + x 2 2 = 1 , whi ch i s mapped to t he l ine segment [1 , √ 2] . The boundary o f R 2 = g ( R 1 ) , i.e., B 2 , which is the line segment [0 , √ 2] , is t he point √ 2 , which is a subset of g ( B 1 ) . W e can arrange the set of t he rates of the virtual broadcast network in a rate vector d R = [d R h ] ∞ 0 and use the resul t of Lemm a 1, when n tends to infinity . In this case, the matrix transform will tend t o k weighted sums of the infinite dimensio nal vector x = d R as fol lows: Mx →  Z ∞ 0 w 1 ( h )d R h , · · · , Z ∞ 0 w k ( h )d R h  . 8 Setting x ( h ) = d R h , we can conclude th at t he boundary region o f ( W 1 , · · · , W k ) is a subset of the transformation of the boundary set of the rate vector d R , which is achieve d, as sho wn in [19], by superposition coding. In other words, any vector v i n the b oundary set of ( W 1 , · · · , W k ) , is achie ved by a mult i-layer code in which the rate of the virtual receiv er R X h is equal to log 1 + hρ v ( h )d h 1 + h R ∞ h +d h ρ v ( u )d u ! , where t he power distribution function ρ v ( h ) , indexed with v to emph asize its dependence on this vector , satisfies R ∞ 0 ρ v ( u )d u = P . This completes the proof of Theorem 1. Using the above theorem, it easily follows that the optim al performance v ectors ( R ǫ , R ave ) and ( R mul , R ave ) , defined in the sections IV and V , respectively , can be achieved using superposi tion coding. I V . H A R D C OV E R AG E C O N S T R A I N T In this section, we consid er a scenario wh ere a mini mum rate of R ǫ (multicast outage capacity) should be delive red t o all the u sers with a probabili ty of (1 − ǫ ) , where ǫ is the probabil ity of outage. Giv en t his constraint, we want to maxim ize th e average rate receiv ed by a random ly chosen user in the network, i.e. R ave . This includes the rate receiv ed b y a typi cal user , eve n if the user is in o utage. W e can categorize the pos sible states of the virtual broadcast network in two group: (i) h ≤ h ǫ , in which case t he rate d R h contributes to bot h R ǫ and R ave , and (ii) h > h ǫ , in which case the rate d R h contributes onl y to R ave . The problem of maximizing R ave without any constraint on R ǫ is st udied i n [5]. On the other hand, from [4], we know that th e maxim um R ǫ without any con straint o n R ave is achiev ed by a single layer code with power P and rate C ǫ = lo g(1 + h ǫ P ) . (9) In this section, we in vestigate the tradeoff b etween R ǫ and R ave . Setting w 1 ( h ) = 1 { h ≤ h ǫ } and w 2 ( h ) = 1 − F ( h ) , Theorem 1 states that the boundary set of ( R ǫ , R ave ) is achie ved by superpositio n codin g, in which d R h = log  1 + hρ ( h )d h 1+ hI ( h )  = R I ( h )+ ρ ( h )d h I ( h ) h d p 1+ hp , (10) and I ( h ) = R ∞ h +d h ρ ( u )d u . Note that d R h is not necessarily very small since our power allocation function might ha ve som e im pulses. As st ated earlier , we want to jointly optimi ze the weighted sum of these rates according t o t he weighting functions w 1 ( h ) and w 2 ( h ) . The o ptimization is on th e functio n ρ ( h ) , howe ver , in the following we show that it can be simplified to a point optim ization problem . 9 Definition 2 The channel gain-int erfer ence function, s ( p ) is defined as s ( p ) ∆ = sup { h | I ( h ) ≥ p } . (11) In fact, the channel g ain-interfer ence f unction s ( p ) is the in verse of the i nterference function in terms of the channel gain in the poin ts it is in vertible and determin es the channel gain of the virtual receiver experiencing th e interference lev el p . It is evident that s ( p ) i s a decreasing functio n of p . According to (10), we can write the expected typical rate as R ave = R ∞ 0 (1 − F ( h )) d R h = R P 0 g ( p, s ( p ))d p, (12) where g ( x, y ) = (1 − F ( y )) y 1+ xy . In deriving (12), we h a ve used the fact that t he contribution of the virtual recei ver R X h in the expected typical rate, from (3), can be written as (1 − F ( h )) d R h = (1 − F ( h )) R I ( h )+ ρ ( h )d h I ( h ) h d p 1+ hp . Noting that in the interval p ∈ [ I ( h ) , I ( h ) + ρ ( h )d h ] , h can be written as s ( p ) 2 , the contribution of RX h in R ave can be written as R I ( h )+ ρ ( h )d h I ( h ) g ( p, s ( p ))d p . Since R ave is the summ ation of the contributions of all virt ual recei vers, we hav e R ave = R h R I ( h )+ ρ ( h )d h I ( h ) g ( p, s ( p ))d p = R I ( ∞ ) I (0) g ( p, s ( p ))d p = R P 0 g ( p, s ( p ))d p , which is due to the fact that I (0) = P and I ( ∞ ) = 0 . Diffe rentiating g ( x, y ) wi th respect to y , we ob tain ∂ ∂ y g ( x, y ) = 1 − F ( y ) − y f ( y )(1 + xy ) (1 + xy ) 2 . (13) In the case of Rayleigh fading, we ha ve f ( y ) = 1 − F ( y ) = e − y . By studyin g the behavior of ∂ ∂ y g ( x, y ) , it is easy to show that ∂ ∂ y g ( x, y ) | x = p is pos itive for y < I − 1 0 ( p ) and is negati ve for y > I − 1 0 ( p ) , where I 0 ( h ) = (1 − F ( h )) − hf ( h ) h 2 f ( h ) . As a result, arg max( g ( x, y ) | x = p ) = I − 1 0 ( p ) . (14) Note that I 0 ( . ) 3 is i ndeed the i nterference functi on corresponding t o th e o ptimal power allocation in the unconstrained maximizati on of R ave solved in [5 ]. Definition 3 The multicast interfer ence level α for a gi ven channel gain -interfer ence functi on s ( . ) is defined as α ∆ = min { p | s ( p ) ≤ h ǫ } . (15) In fact, α is t he level of the interference o bserved by all the virtual receivers contributing t o R ǫ due to the power allo cated t o the upp er levels not contributing to R ǫ . It follows from d efinitions 2 and 3 that 2 Note that this is tr ue ev en if ρ ( . ) contains an impulse at h . 3 Note that the f unction I 0 ( . ) is monotonically decreasing and as a result, i t is inv erti ble. 10 α = I ( h ǫ ) . Using this definition and the same arguments as in (12), R ǫ in (5) can be written as R ǫ = Z h ǫ 0 d R h = Z h ǫ 0 Z I ( h )+ ρ ( h )d h I ( h ) h d p 1 + hp = Z h ǫ 0 Z I ( h )+ ρ ( h )d h I ( h ) s ( p )d p 1 + s ( p ) p = Z I ( h ǫ ) I (0) s ( p )d p 1 + s ( p ) p = Z P α m ( p, s ( p ))d p, (16) = Z P 0 1 { s ( p ) ≤ h ǫ } m ( p, s ( p ))d p (17) where m ( x, y ) = y 1 + xy , (18) and (17) fol lows from the definition of α in (15). Here, we assum e that h ǫ ≤ 1 . This is not a restrictin g assumption as the solution in the ot her case ( h ǫ > 1 ) follows using a similar approach. Howe ver , this assumption simplifies t he deriv ations as it guarantees I 0 ( h ǫ ) > 0 . O n the ot her h and, th is assu mption is equivalent t o ǫ ≤ 1 − e − N , which covers most of t he cases of interest since we expect the ou tage probability to be much small er t han 1. Using (12) and (16), the probl em is translated to max s ( . ) R ave = max s ( . ) R P 0 g ( p, s ( p ))d p, (19) subject to R ǫ = Z P α m ( p, s ( p ))d p ≥ ζ C ǫ , (20) where C ǫ is defined in (9), and ζ i s a norm alization factor which expresses R ǫ in terms of C ǫ . Since it is not p ossible to achieve values of R ǫ above C ǫ , we can restrict ourselves to the values of ζ between 0 and 1 . Note t hat the maxim ization in (19) is over all decreasing positive functions s ( . ) . Also note that α in (20) is the multicast interfer ence level defined in (15) and depends on the function s ( . ) . From (18), we note that for any chosen x , m ( x, y ) is an increasing funct ion of y . Hence, noting t he definition of α in (15), we can write R ǫ ≤ Z P α m ( p, h ǫ )d p = log  1 + h ǫ P 1 + h ǫ α  = C ǫ − log (1 + h ǫ α ) . (21) 11 Therefore, following (9), (20) and (21), we hav e ζ C ǫ ≤ C ǫ − log (1 + h ǫ α ) ⇒ α ≤ e (1 − ζ ) C ǫ − 1 h ǫ . (22) Lemma 2 F or th e optimizer of (19), we have α ≤ I 0 ( h ǫ ) . Pr oof : Assume α > I 0 ( h ǫ ) . Denote the optimi zer function by s ∗ ( . ) and its resultin g multicast outage capacity and expected typical rate by R ∗ ǫ and R ∗ ave , respectiv ely . Also, define ˆ s ( p ) as ˆ s ( p ) =          I − 1 0 ( p ) p < I 0 ( h ǫ ) h ǫ I 0 ( h ǫ ) ≤ p ≤ α s ∗ ( p ) p > α , ( 23) and i ts result ing multi cast outage capacity and expected typical rate by ˆ R ǫ and ˆ R ave , respectively . W e can write ˆ R ǫ − R ∗ ǫ ( 17 ) = Z P 0  1 { ˆ s ( p ) ≤ h ǫ } m ( p, ˆ s ( p )) − 1 { s ∗ ( p ) ≤ h ǫ } m ( p, s ∗ ( p ))  d p ( a ) = Z α I 0 ( h ǫ ) m ( p, ˆ s ( p ))d p > 0 , (24) where ( a ) follows from the facts that (i) ˆ s ( p ) ≤ h ǫ for p ≥ I 0 ( h ǫ ) , (ii) s ∗ ( p ) ≤ h ǫ for p ≥ α (from the definitio n of α in (15)), and (iii) for p ≥ α , ˆ s ( p ) = s ∗ ( p ) (from (23)). (24) implies t hat ˆ R ǫ > R ∗ ǫ . Moreover , we have ˆ R ave − R ∗ ave = Z P 0 ( g ( p, ˆ s ( p )) − g ( p, s ∗ ( p )))d p (25) ( a ) = Z I 0 ( h ǫ ) 0  g ( p, I − 1 0 ( p )) − g ( p, s ∗ ( p ))  d p + Z α I 0 ( h ǫ ) ( g ( p, h ǫ ) − g ( p, s ∗ ( p )))d p > 0 , where ( a ) follows from the fact that for p ≥ α , ˆ s ( p ) = s ∗ ( p ) . In th e above inequality , the positivity of the first term in the left hand side is conclud ed from (14 ), and t he positivity of t he second term is concluded from the fact that s ∗ ( p ) > h ǫ for p ≤ α (from t he definition of α in (15)), and also (13) which im plies that g ( x, y ) | x = p is decreasing for y > I − 1 0 ( p ) . Th erefore, ˆ R ave > R ∗ ave and ˆ R ǫ > R ∗ ǫ , which contradict our assumption of optim ality of s ∗ ( . ) and the lem ma is prov ed. The above lem ma states the fact that, app lying the multicast outage const raint, more power wil l be allocated to the channel gains lower than t he o utage thresho ld, com pared to the unconstrained scenario studied in [5]. 12 Lemma 3 Given α , the optimiz er of (19) is given by s α ( p ) =          I − 1 0 ( p ) p < α h ǫ α ≤ p ≤ I hc λ ( h ǫ ) I hc λ − 1 ( p ) p > I hc λ ( h ǫ ) , (26) wher e I hc λ ( h ) = ( λ +1 − F ( h )) − hf ( h ) h 2 f ( h ) in which the superscript ( . ) hc stands fo r the “har d coverage” const raint scenario, I hc λ − 1 ( . ) r epre sents the in verse of t he function I hc λ ( . ) , and λ = ( 0 , if R P α m ( p, s α ( p ))d p > ζ C ǫ arg ( R P α m ( p, s α ( p ))d p = ζ C ǫ ) , otherw ise . Pr oof : As observed from (20), the value of s ( p ) in the range 0 ≤ p ≤ α dose not affect the mul ticast constraint. Hence, (19) can be written as max s ( . ) R ave = max s ( . ) Z α 0 g ( p, s ( p ))d p (27) + max s ( . ) , R P α m ( p,s ( p ))d p ≥ ζ C ǫ Z P α g ( p, s ( p ))d p Note that as menti oned earlier , all t he maximization s are performed over positive decreasing functions . Moreover , s ince s ( α ) = h ǫ (due to th e definition of α in (15 )), the solu tion of the above m aximization problem must sati sfy s ( p ) ≥ h ǫ in t he interval p ∈ [0 , α ] and s ( p ) ≤ h ǫ elsewhere . (14) impli es that the first term in (27) can be upp er -bounded as follows: max s ( . ) Z α 0 g ( p, s ( p ))d p ≤ Z α 0 g ( p, I − 1 0 ( p ))d p. (28) Moreover , writing K.K.T . condi tion for the second term of (27), the maximizati on problem will be translated to max s ( . ) Z P α T λ ( p, s ( p ))d p, (29) where T λ ( x, y ) = g ( x, y ) + λm ( x, y ) . λ is 0 , if the outage cons traint is not lim iting; otherwise, its value can be computed using th e ou tage constraint R P α m ( p, s ( p ))d p = ζ C ǫ . Differentiating the function T λ ( x, y ) with respect to y , we obtain ∂ ∂ y T λ ( x, y ) = λ + 1 − F ( y ) − y f ( y )(1 + xy ) (1 + xy ) 2 . (30) In the case of Rayleigh fading, by s tudying the behavior of ∂ ∂ y T λ ( x, y | x = p ) , it follows that ∂ ∂ y T λ ( x, y | x = p ) > 0 for y < I hc λ − 1 ( p ) and ∂ ∂ y T λ ( x, y | x = p ) < 0 for y > I hc λ − 1 ( p ) , where I hc λ ( h ) = λ +1 − F ( h − h f ( h ) h 2 f ( h ) 4 . As a result, arg max T λ ( x, y ) | x = p = I hc λ − 1 ( p ) . (31) 4 Note that the function I hc λ ( . ) is monoton ically decreasing for h < h ǫ (which is the region of interest here), and as a result, it is in vertible. 13 Note that as I 0 ( h ) ≤ I λ ( h ) , ∀ h , and α ≤ I 0 ( h ǫ ) (from Lemm a 2 ), it foll ows t hat α ≤ I hc λ ( h ǫ ) . Hence, the in tegral R P α T λ ( p, s ( p ))d p can be written as the summation of the two integrals R I hc λ ( h ǫ ) α T λ ( p, s ( p ))d p and R P I hc λ ( h ǫ ) T λ ( p, s ( p ))d p . First, we note that in the whole interval of [ α, P ] , we ha ve s ( p ) ≤ h ǫ (due to th e definition of α ). Moreover , as I hc λ − 1 ( . ) is a d ecreasing fun ction (in the i nterval [ α, P ] ), we have h ǫ ≤ I hc λ − 1 ( p ) , in the interval [ α , I hc λ ( h ǫ )] . Combining thi s with the fact that s ( p ) ≤ h ǫ in [ α, I hc λ ( h ǫ )] implies t hat s ( p ) ≤ I hc λ − 1 ( p ) in this in terva l. As seen before, T λ ( x, y ) is an i ncreasing functi on of y for y < I hc λ − 1 ( x ) . Consequently , Z I hc λ ( h ǫ ) α T λ ( p, s ( p ))d p ≤ Z I hc λ ( h ǫ ) α T λ ( p, h ǫ )d p. (32) Moreover , using (31), we have Z P I hc λ ( h ǫ ) T λ ( p, s ( p ))d p ≤ Z P I hc λ ( h ǫ ) T λ  p, I hc λ − 1 ( p )  d p. (33) Hence, for any functi on s ( p ) such that s ( p ) ≤ h ǫ for p > α , we can writ e Z P α T λ ( p, s ( p ))d p ≤ Z P α T λ ( p, s α ( p ))d p. (34) Combining the above equation with (28) yi elds t hat max s ( . ) R ave ≤ Z P 0 g ( p, s α ( p ))d p. (35) T o com plete th e proof, it is sufficient to show that s α ( . ) satis fies t he con ditions m entioned earlier; i.e., s α ( . ) is a decreasing funct ion and s α ( α ) = h ǫ . The l atter is obvious from th e definition of s α ( . ) in (26). For showing the former , we first note that I − 1 0 ( . ) and I hc λ − 1 ( . ) are decreasing funct ions i n the intervals [0 , α ] and [ I hc λ ( h ǫ ) , P ] , respectiv ely . Moreover , from Lemma 2, we hav e α ≤ I 0 ( h ǫ ) which impl ies t hat I − 1 0 ( α ) ≥ h ǫ . This shows that s α ( . ) is decreasing in t he who le interval [0 , P ] , which completes the proof of the lemma. An interesting consequence of L emma 3 is that the prob lem of maximization ove r the fun ction s ( . ) is simplified to the point optim ization problem ov er the value of α . Theor em 2 The capacity r e g ion of a Rayleigh fading mult icast network with a ha r d cover age constraint is given by R ave ≤ max 0 ≤ α ≤ min „ e (1 − ζ ) C ǫ − 1 h ǫ ,I 0 ( h ǫ ) « Z P 0 g ( p, s α ( p ))d p, (36) R ǫ ≤ ζ C ǫ , (37) wher e ζ changes fr om 0 to 1 . 14 Pr oof : The proof follows from Lemm a 2 , Lemma 3, and inequality (22). Corollary 1 F or an y outage probability ǫ > 0 such that h ǫ ≤ I − 1 0 ( P ) , the capacity re gion of a Rayleigh fading multicast network, i.e., ( R ǫ , R ave ) , is given by R ǫ ≤ log  1 + h ǫ β P 1 + h ǫ (1 − β ) P  , (38 ) R ave ≤ 2( E 1 ( θ ( β )) − E 1 (1)) − ( e − θ ( β ) − e − 1 ) + e − h ǫ log  1 + h ǫ β P 1 + h ǫ (1 − β ) P  , (39) wher e β changes fr om 0 to 1 , θ ( β ) = 2 1+ √ 1+4(1 − β ) P , and E 1 ( x ) , R ∞ x e − t t d t . Pr oof : Since h ǫ ≤ I − 1 0 ( P ) , it follows that I 0 ( h ǫ ) ≥ P . Noti ng t hat I 0 ( h ) ≤ I hc λ ( h ) , ∀ h, λ ≥ 0 , it follows that I hc λ ( h ǫ ) > P for any λ ≥ 0 . Therefore, (26) can be written as s α ( p ) =    I − 1 0 ( p ) p < α h ǫ α ≤ p ≤ P . (40) In t his case, th e so lution to the maximi zation problem (36) i s α = e (1 − ζ ) C ǫ − 1 h ǫ . D efining β , 1 − e (1 − ζ ) C ǫ − 1 h ǫ P , first we note that 0 ≤ β ≤ 1 . Moreover , (37) can be written as R ǫ ≤ ζ C ǫ = C ǫ − log (1 + h ǫ (1 − β ) P ) = log  1 + h ǫ β P 1 + h ǫ (1 − β ) P  . (41) The interference function corresponding to s α ( p ) in (40) can be expressed as I ( h ) = αU  I − 1 0 ( α ) − h  + β P U ( h ǫ − h ) + I 0 ( h ) U  h − I − 1 0 ( α )  , (42) where U ( . ) denotes t he uni t step functio n and α = (1 − β ) P . Di ff erentiating I ( h ) wi th respect t o h results in the following power all ocation function: ρ ( h ) = ( P − α ) δ ( h − h ǫ ) + ρ 0 ( h ) , where ρ 0 ( h ) =    2 h 3 − 1 h 2 I − 1 0 ( α ) < h < 1 0 else 15 is the po wer allocation funct ion in t he unconst raint problem [5]. Using (3) and (10), the expected t ypical rate can be written as R ave = Z ∞ 0 (1 − F ( h )) d R h = e − h ǫ log  1 + h ǫ β P 1 + h ǫ (1 − β ) P  + Z 1 I − 1 0 ( α ) e − h hρ 0 ( h ) 1 + hI 0 ( h ) d h = e − h ǫ log  1 + h ǫ β P 1 + h ǫ (1 − β ) P  + Z 1 I − 1 0 ( α ) e − h h  2 h 3 − 1 h 2  1 + h ( 1 h 2 − 1 h ) d h = e − h ǫ log  1 + h ǫ β P 1 + h ǫ (1 − β ) P  + Z 1 I − 1 0 ( α ) 2 e − h h d h − Z 1 I − 1 0 ( α ) e − h d h. (43) Noting that I − 1 0 ( α ) = θ ( α ) = 2 1+ √ 1+4 α and α = (1 − β ) P com pletes the proof of the corollary . An interesti ng conclusion of Corollary 1 is that, t he expected typical rate is maximized when the multicast m inimum rate i s provided in a singl e layer . In the case we have n o multicast constraint, it i s shown in [5] that a multi-layer code wit h a small rate in each layer is optim al in terms of m aximizing the expected rate. Howev er , when we are constrained to di stribute a fraction o f the av ailable power to a set of low channel gains [0 , h ǫ ] (coverage constraint), the optim um solution allo cates all the power to th e highest gain, i.e. h ǫ . Note that th e ass umption h ǫ ≤ I − 1 0 ( P ) is not difficult to satis fy , si nce t he outage probability ǫ is us ually small. Moreove r , as h ǫ = − log(1 − ǫ ) N , the value of h ǫ decreases si gnificantly with the number of users. For example, for N = 5 and P = 100 , the ou tage probabil ity ǫ could be as hi gh as 0 . 38 in order t o have h ǫ ≤ I − 1 0 ( P ) . Figure (2) shows the capacity region of thi s network when ǫ = 0 . 01 . It is evident that due to the h ard coverage constraint , the multi cast outage capacity is in general very small i n comparison wi th the expected t ypical rate. V . S O F T C OV E R AG E C O N S T R A I N T In the pre vious section, we ob served th at a strict coverage constraint can result i n very small values for the multicast outage capacity . W e can relax the coverage requirement b y relying on the ex pected mult icast rate , i.e. R mul . This results in the performance vector ( R mul , R ave ) and it s corresponding capacity region. According to Th eorem 1, the optim ality of superposit ion coding is conclu ded for th is performance vector . 16 0 0.05 0.1 0.15 0.2 0 0.5 1 1.5 2 2.5 3 R ε (nats/symbol) R average (nats/symbol) Fig. 2. Hard cov erage constraint: multicast outage capacity vs. exp ected typical rate for P = 100 and N = 5 . Theor em 3 The capacity r e g ion of a Rayleigh fad ing multicast network with soft covera ge con straint is given by R ave = Z ∞ 0 e − u uρ sc γ ( u )d u 1 + uI sc γ ( u ) (44) R mul = Z ∞ 0 e − N u uρ sc γ ( u )d u 1 + uI sc γ ( u ) , (45) wher e I sc γ ( h ) =          P if h < h 0 e − h (1 − h )+ γ e − N h (1 − N h ) h 2 ( e − h + γ N e − N h ) h 0 < h < h 1 0 h > h 1 , (46) in which the su perscript ( . ) sc stands fo r the “soft cover age” constraint scenari o, ρ sc γ ( h ) = − ∂ I sc γ ( h ) ∂ h , and h 0 and h 1 ar e r eal numbers, such t hat e − h 0 (1 − h 0 )+ γ e − N h 0 (1 − N h 0 ) h 2 0 ( e − h 0 + γ N e − N h 0 ) = P , e − h 1 (1 − h 1 )+ γ e − N h 1 (1 − N h 1 ) h 2 1 ( e − h 1 + γ N e − N h 1 ) = 0 , for differ ent values of γ ≥ 0 . Pr oof : I f we set w 1 ( h ) = 1 − F typ ( h ) and w 2 ( h ) = 1 − F mul ( h ) , Theorem 1 states that we should search for an infinite layer superposi tion code. Considering ρ ( h )d h as the power of the layer asso ciated with the channel gain h , the corresponding rate is 5 d R h = log  1 + hρ ( h )d h 1 + hI ( h )  = hρ ( h )d h 1 + hI ( h ) , (47) 5 Note that here, unlike the hard cove rage constraint scenario, t he power distribution function does not hav e any impulses. In fact, the optimization problem in the soft cove rage constraint scenario, as seen in the proof, is similar to that of t he unconstrained scenario [5] for which the optimal po wer allocation function has been proved to have no impulses. 17 where I ( h ) = Z ∞ h d uρ ( u ) , and I ( 0 ) = P . Using the above expression, the rate corresponding to the fading lev el h is R ( h ) = Z h 0 uρ ( u )d u 1 + uI ( u ) . Follo wing the definit ions, we hav e R mul = Z ∞ 0 (1 − F mul ( u ))d R ( u ) = Z ∞ 0 e − N u uρ ( u )d u 1 + uI ( u ) , (48) R ave = Z ∞ 0 (1 − F typ ( u ))d R ( u ) = Z ∞ 0 e − u uρ ( u )d u 1 + uI ( u ) . (49) The problem is that given R mul = r , what is the maximum achiev able R ave . In other words, R ave = max I ( u ) Z ∞ 0 e − u uρ ( u )d u 1 + uI ( u ) , (50) subject to: Z ∞ 0 e − N u uρ ( u )d u 1 + uI ( u ) = r, (51) I ( 0 ) = P , and I ( ∞ ) = 0 . Equiv al ently , t o deriv e the capacity region ( R mul , R ave ) , it is sufficient to sol ve t he following opt imization problem: 6 max I ( . ) R ave + γ R mul , (52) subject to I ( 0 ) = P , I ( ∞ ) = 0 , 6 Note that as we are allowed to user t ime-sharing, the capacity region is con vex. As a result, the capacity region can be characterized as (52). 18 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 R mul (nats/symbol) R ave (nats/Symbol) Two Layers Infinite Layers Fig. 3. Soft cov erage constraint: expec ted minimum rate vs. expected t ypical rate for P = 100 and N = 5 for all values of γ ≥ 0 . T o solve the above optimizati on problem, we define S ( x, I ( x ) , I ′ ( x ) , γ ) as follows: S ( x, I ( x ) , I ′ ( x ) , γ ) = e − x xI ′ ( x ) 1 + xI ( x ) + γ e − N x xI ′ ( x ) 1 + xI ( x ) . (53) Note that I ′ ( x ) = − ρ ( x ) . The necessary condi tion for I ( x ) to maximize (50) with t he cons traint (51) is the zero functional v ariation [20] of S ( x, I ( x ) , I ′ ( x ) , γ ) , ∂ ∂ I S − d d x ∂ ∂ I ′ S = 0 , (54) where ∂ ∂ I S = ( e − x + γ e − N x ) x 2 I ′ ( x ) (1+ xI ( x )) 2 , ∂ ∂ I ′ S = ( e − x + γ e − N x ) − x 1+ xI ( x ) , d d x ∂ ∂ I ′ S = x ( e − x + γ N e − N x ) 1+ xI ( x ) + ( e − x + γ e − N x ) x 2 I ′ ( x ) − 1 (1+ xI ( x )) 2 . Therefore, (54) simplifies to a linear equation whi ch leads t o the opti mum int erference funct ion g iv en in (46). Figure (3) shows th e capacity region for N = 5 and P = 100 . It is obs erved that t he maximum R ave is achieved for R mul ≤ 1 . 05 . It is shown in [21] that a good fraction of the highest expected rate with infinite layers is achieved by two layers. Figure (3) s hows that this i s true for ou r multicast network as well. Furtherm ore, we observe that t he two-layer rate region gets closer to t he capacity region for hi gher values of R mul . 19 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 500 1000 1500 2000 2500 3000 3500 h ρ (h) Rmul=1.4 No Multicast Constraint Fig. 4. Power distribution function ρ ( h ) for no multicast requirement and for R mul = 1 . 4 W ithou t the m ulticast const raint, th e power dist ribution function to maximize th e average rate would be [5], ρ 0 ( h ) =    2 h 3 − 1 h 2 s 0 < h < 1 0 else , where s 0 = 2 1 + √ 1 + 4 P . This function is depicted in figure (4), and is compared with the case corresponding t o a multicast requirement of R mul = 1 . 4 . As shown in the figure, the covera ge requirement has s hifted the power distribution to sm aller channel gains to provide service to u sers with worse chann el conditions . V I . E X T E N S I O N T O M I S O In the case t hat there are m ultiple ( M > 1 ) antennas at the transm itter , one can adopt the broadcast approach proposed in [5]. In this approach, the recei ver with unknown quasi-st atic fading MISO channel, denoted by h , is modeled as a con tinuum of recei vers each associated w ith a channel realization. These recei vers are ordered in a degraded fashion. Howe ver , since MIM O Broadcast Channel (MIMO-BC) is inherently non-degraded [22 ], this approach dose not necessarily lead to the o ptimum performance. Assuming a single antenn a at each receiver , the ordering of the virtual receiv ers in this approach is based on their normalized channel norm, i.e., k hh † k M . Hence, the rate of t he virtu al recei ver at th e fading lev el k hh † k M equals R  k hh † k M  = log  1 + P S k hh † k M 1+ P I k hh † k M  , where P S and P I are the decodable and undecodable signal power l e vels, respecti vely . 20 Now , assum e there are N users in the network, all receiving a common so urce through an infinit e-layer code. Same as before, we would like t o design the cod e to maximi ze the aver age rate observed by a typical user , while providing a given covera ge constraint for all the users. For this p urpose, we sho uld provide the m inimum rate to the worst user in the degraded broadcast m odel, i.e., the user with the lowest channel norm. The normali zed channel norm of user i , denot ed by k h i h † i k M , is a scaled χ 2 random variable with 2 M degrees of freedom, with the following CDF: F typ ( h ) = F k hh † k M ( h ) = 1 − Γ( M , M h ) Γ( M ) , (55) where Γ( . ) is the Gamma function , and Γ( ., . ) is the upper incomp lete Gamm a function [23]. Since the users’ channels are statisticall y independent, the distribution of the norm of th e worst channel can be computed as Pr ( min i k h i h † i k M > h ) = Pr ( k h i h † i k M > h )! N =  Γ( M , M h ) Γ( M )  N . Hence, the CDF for the worst channel norm is F mul ( h ) = 1 −  Γ( M , M h ) Γ( M )  N . (56 ) Here, we j ust con sider th e soft coverage constraint s cenario. Follo wing the same approach as in section IV , we obtain R mul = Z ∞ 0 (1 − F mul ( u ))d R ( u ) = Z ∞ 0  Γ( M , M u ) Γ( M )  N uρ ( u )d u 1 + u I ( u ) , (57) R ave = Z ∞ 0 (1 − F typ ( u ))d R ( u ) = Z ∞ 0 Γ( M , M u ) Γ( M ) uρ ( u )d u 1 + u I ( u ) , (58 ) where ρ ( u ) and I ( u ) are the corresponding power a llocation and interference power functions, respectively . T o characterize the achieva ble rate region ( R ave , R mul ) , we need to s olve the following op timization problem for all γ ≥ 0 : max I ( . ) R ave + γ R mul , (59) subject to I ( 0 ) = P , I ( ∞ ) = 0 . 21 Defining S ( x, I ( x ) , I ′ ( x ) , γ ) as S ( x, I ( x ) , I ′ ( x ) , γ ) = Γ( M , M x ) Γ( M ) xI ′ ( x ) 1 + xI ( x ) + γ  Γ( M , M x ) Γ( M )  N xI ′ ( x ) 1 + xI ( x ) , and setting its functional variation equal to zero to maximize the av erage rate, and defining w ( x ) , Γ( M , M x ) Γ( M ) + γ  Γ( M , M x ) Γ( M )  N , (60) similar to (54), we obtain x 1 + xI ( x ) w ′ ( x ) + w ( x ) (1 + xI ( x )) 2 = 0 , (61) which implies that I ( x ) = − w ( x ) x 2 w ′ ( x ) − 1 x . (62) Noting that d d x Γ( M , x ) = − x M − 1 e − x , we can write w ′ ( x ) = − M M x M − 1 e − M x Γ( M ) 1 + γ N  Γ( M , M x ) Γ( M )  N − 1 ! . (63) Substitutin g in (62) yi elds the optimizer I ( h ) as I ( h ) =          P if h < h 0 µ ( h ) h 0 < h < h 1 0 h > h 1 , (64) where µ ( h ) , Γ( M , M h ) + γ Γ( M ,M h ) N Γ( M ) N − 1 M M h M +1 e − M h  1 + γ N  Γ( M ,M h ) Γ( M )  N − 1  − 1 h , and h 0 , h 1 are the solution s of the foll owing equations: µ ( h 0 ) = P , µ ( h 1 ) = 0 , respectiv ely . The achiev able rate region is shown in figure (5) for different values of N and for M = 2 . From t his figure, w e can see that as the number of users decreases, the proposed achiev able rate region expands. It is also evident by comparing the regions for the MISO and SISO cases with N = 5 us ers (figures (3) and (5)) that using m ultiple antennas improves the achie v able rate. Howe ver , its ef fect is more consi derable for the multi cast channel as compared to the typical chann el. This prom inent gain for the multi cast channel 22 1.4 1.6 1.8 2 2.2 2.4 2.6 2.7 2.75 2.8 2.85 2.9 2.95 3 3.05 3.1 3.15 3.2 R mul (nats/symbol) R ave (nats/symbol) N=3 N=5 N=7 Fig. 5. Soft cov erage constraint: MISO expected minimum rate vs. expec ted typical rate for different number of users, M = 2 and P = 100 is due to the fact that we are using m ultiple i ndependent paths to con ve y data (higher div ersity order), so the probability of having very low channel gains significantly d ecreases. In fact, we wil l show that one can compensate the ef fect of having a large numb er of u sers by in creasing the number of transm it antennas. More specifically , if bot h N and M tend to i nfinity and M gro ws fast enough with respect to N , we show i n the next theorem that the multicast rate can reach the expected ty pical rate and our scheme giv es the optimal solution. Theor em 4 F or l ar ge values of M and N , the pr oposed infinite layer superposit ion coding will pr ovide R mul such th at R mul ≥ R opt − σ, (65) if M > 2 P 2 log( N ) + ω (1) (1 + P ) 2 σ 2 , (66) wher e R opt is th e highest achievable average rate for a typical user in the network, σ is an ar bitrarily small positive number , and ω (1) denotes any f unction of N which tends to infinity as N → ∞ . Pr oof : First, we derive an upper bound on the achie v able average rate for a typical user by assuming no st ringent delay constraint, meaning that the transmissi on bl ock can be chosen as l ar ge as the fading block. In this case, the channel h as an er godic behavior , and hence, the ergodic capacity is defined and is shown to be C er g = E  log  1 + k hh † k M P  . (67) As a result, R opt ≤ C er g . (68) 23 Using the conca vity of log function, and having the fact that E h k hh † k M i = 1 , we ha ve C er g ≤ log (1 + P ) . (69) W e will show that our scheme provides a mul ticast rate arbit rarily close to th is up per bound, if we use enough trans mit ant ennas. Since this upper b ound is larger than the expected typical rate, the t heorem will be proved. Using a singl e-layer code wi th power P 7 and rate R σ , where R σ = log (1 + P (1 − σ ′ )) , (70) and σ ′ = (1 + P ) σ P , the expected m ulticast rate in our network will be R mul = Pr  k hh † k mul M > 1 − σ ′  R σ , (71) where k hh † k mul = min i k h i h † i k . Regarding the central l imit t heorem [23], the dis tribution of k hh † k M , where 1 M k hh † k = h 1 2 + h 2 2 + ... + h M 2 M , (72) and { h m } M m =1 ’ s are independent Rayleigh distributions with unit variance and uni t mean, approaches to a Gaussian distribution with the CDF F ( h ) = Q h − 1 1 √ M ! . (73) Consequently , the CDF of the multi cast channel wi ll be F mul ( h ) ≈ 1 − Q h − 1 1 √ M ! N . (74) Using the above equation, (71) can be writt en as R mul ≈ Q ( − √ M σ ′ ) N R σ = h 1 − Q ( √ M σ ′ ) i N R σ . (75) Assuming M is large enough to have √ M σ ′ >> 1 , and consequently Q ( √ M σ ′ ) << 1 , we can rewrite the above equation as R mul ≈ e − N Q ( √ M σ ′ ) R σ . (76) 7 Note that as t he single-layer coding i s a special case of superposition coding, our proposed scheme outperforms the single-layer coding scheme. 24 Now , using the approxi mation Q ( x ) ≈ 1 √ 2 π x e − x 2 2 (77) for large values of x , we can write Q ( √ M σ ′ ) ≤ e − M σ ′ 2 2 . (78) Therefore, ha ving M = 2 log( N ) + ω (1) σ ′ 2 , (79) incurs N Q ( √ M σ ′ ) = o (1) , and as a resul t, lim N →∞ R mul − R σ = 0 . (80) Moreover , assumi ng σ ≪ 1 , (70) can be written as, R σ ≈ log(1 + P ) − P σ ′ 1 + P ≥ C er g − σ, (81) where t he second l ine result s from (69). Comb ining (68), (80), and (81), the result of Theorem 4 easily follows. Theorem 4 sim ply implies that as the number of transmit ant ennas grows at l east log arithmically wi th the num ber of us ers, the gain of the worst channel conv er ges to the gain of the typical channel in the network, with probability one. In other words, increasing the number of transmit antennas provides f airness in the system, such that all users almos t get the same quality of s ervice. This fac t i s also noticed by [24] in the context of MIMO-BC. V I I . C O N C L U S I O N In this paper , we hav e considered a m ulticast channel, where a com mon data is transmitted from a source to severa l users. It is assumed that a m inimum service m ust be provided for all th e users. For this setup, we have optimized the av erage service receive d by a typical user in the network. T wo scenarios are considered for the coverage constraint. In the case of hard coverage const raint, th e m inimum mul ticast requirement is stated in terms of the minimum rate (multicast out age capacity) recei ved by all the users in a single t ransmission b lock. For sm all enough outage probabili ties, it is shown that the capacity region is achie ved by providing the required multi cast rate in a single l ayer code, and design ing an infinite-layer code as in [5], on top of it. In the case of soft coverage const raint, the m ulticast requirement is expressed in terms of the expected mi nimum rate receiv ed by all the users. An infinite layer superposition codi ng is shown t o achieve the capacity region in this scenario. W e have also proposed a subo ptimal cod ing scheme for the MISO multi cast channel. This schem e is shown t o be asymptoti cally optim al, when the number of transmit antennas grows at least l ogarithmically with the number of users. 25 R E F E R E N C E S [1] S. Sesia, G. Caire, and G . 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