Capacity of Fading Broadcast Channels wi Limited-Rate Feedback Rajiv Agarwal and John Cioffi Department of Electrical Engineering Stanford University, CA-94305 Abstract In is paper, we study a fading broadcast channel BC wi perfect channel state information at e receiver CSIR and only a quantized version of it at e transmitter due to limited-rate links for channel feedback at each user. We find an achievable region for e fading BC under is condition using super-position coding and show at it is sumrate optimal. We also derive a closed-form expression for finding channel partitioning, which turns out to be e same in form as at for water-filling of power over time in fading channels. Using e derived closed form expression wi temporal waterfilling of power at e transmitter in an iterative manner, we show numerically at a single iteration is adequate to achieve most of e capacity. Thus e complexity of finding e optimal global maximum or close to optimal local maximum channel partitioning is greatly reduced as compared to using a searchbased k-mean clustering algorim like Lloyd s algorim at requires multiple iterations. I. INTRODUCTION One important scenario for multiuser wireless communications is e broadcast channel where a single transmitter sends independent information to many receivers, for example in e downlink of a cellular system. The capacity region of a fading BC wi perfect channel state information at e transmitter CSIT and e receivers CSIR along wi e capacity-achieving transmission strategy was found in ]. However, knowing e channel perfectly at e transmitter requires perfect channel measurements at e receiver and perfect feedback of e estimates to e transmitter. The second requirement is usually harder to meet an e former in a practical system, especially when e cardinality of channel state space is large and e receivers have limitedrate feedback links to e transmitter. In case of discrete fading distributions is means at if e cardinality of e fading distribution is N < i.e. at any transmission instant e channel can be in one of N possible states wi certain probability distribution, e receiver is allowed to feedback only log M < log N bits to e transmitter for CSIT. In is paper, we study e fading BC wi perfect CSIR and limited CSIT arising from limited-rate channel feedback links at e users. The fading BC wi limited-rate feedback is non-degraded, non-less-noisy and non-more-capable in general and hence its capacity region is unknown ]. Common multiuser transmission techniques for e BC like super-position coding SPC, which requires one of e users to always decode e oer users s message, or Dirty Paper coding DPC, which requires perfect CSIT for one of e users, cannot be used in general. We derive an achievable region for e fading BC wi limited-rate for e symmetric case i.e. when e two users have i.i.d. fading, same log M number of bits for channel feedback and are restricted to have identical channel partitioning, using super-position coding. For each point on e boundary of e achievable region, we find e optimal channel partitioning at e users. We show at e derived rate-region is sum-rate optimal. Thus, we find e sum-capacity of e fading BC wi limited-rate feedback for e symmetric case. Notice at finding e optimal channel partitioning at e users is complex when N and M are large. In 3], e auors derived e capacity of a single-user fading channel wi limited-rate feedback and showed at Lloyd s algorim for k-mean clustering can be applied to find e optimal capacity-achieving channel partitioning. The algorim works in an iterative manner. Lloyd s Algorim: Step : Assume an arbitrary initial channel partitioning into M groups. Step : For a given channel partitioning, find e optimal power allocation for e M groups at minimizes distortion/cost defined to be e negative of ergodic capacity minus e power penalty. 3 Step 3: For a given power allocation for e M groups, find e optimal channel partitioning at minimizes cost. 4 Step 4: Repeat steps and 3 until convergence. In Lloyd s algorim, e complexity of Step is ON using e expression for optimal power allocation waterfilling. The complexity of Step 3 is ONM as for each channel gain value we find assigning it to which of e M groups results in least distortion. So e overall complexity for a single iteration assuming M N is ON. It takes about N iterations 4] for e algorim to converge, where is convergence too can be to a local optima. To avoid convergence to a local optima, e auors in 3] suggest to run Lloyd s algorim from multiple different starting guesses Step. The same technique can be used for e fading BC wi limited-rate feedback as well, wi a modified cost distortion function. Alternatively, we derive a closed-form expression for finding a close-to optimal channel partitioning given a power allocation Step. For e single-user case, e derived We will use e phrase channel partitioning roughout e paper to refer to partitioning of e channel gain values assumed to belong to a discrete set into groups or effectively a quantization of e channel gain values. As seen by numerical examples, e channel partitioning arrived at by using e proposed algorim is eier optimal or very close to it.
closed-form expression turns out to be e same in form as at for water-filling of power over time in fading channels. For e fading BC, e expression is a modified water-filling due to e fact at users now cause interference to each oer. Using e proposed closed-form expression for finding channel partitioning, we can us start from an intelligent starting point as compared to an arbitrary starting point Step, which may reduce e number of iterations. Indeed, we show numerically at a single iteration is now adequate to achieve capacity e starting point in itself was optimal or most of e capacity e starting point was very close to e optimal. Thus e complexity of finding e optimal global maxima or close to optimal local maxima channel partitioning is greatly reduced when using a search-based k-mean clustering algorim like Lloyd s algorim where it was ON N assuming M N. Furer, when M is large, we propose an alternative iterative water-filling algorim for finding e channel partitioning using e derived closed-form expression, which makes e complexity of Step 3 in e iteration reduce from ONM to ON, now independent of M. The rest of e paper is organized as follows. In Section II we describe e fading broadcast channel wi limited-rate feedback system under study. In Section III, we find an achievable rate region for e fading BC wi limited-rate feedback using SPC and show at it is sum-rate optimal. In Section IV, we derive closed-form expressions for channel partitioning for bo e single-user fading channel and e two-user fading BC. In Section V, numerical results are provided for a discrete fading channel, where we compare e proposed achievable rate region wi a single iteration to e achievable region wi e globally optimum channel partitioning. We also compare e proposed achievable region for e fading BC wi limited-rate feedback to e capacity region wi perfect CSIT upper-bound. Concluding remarks are given in Section VI. II. BROADCAST CHANNEL MODEL Notation: When referring to random variables, small letters will denote a realization of e random variable and capital letters will denote e random variable itself. The notation X denotes e cardinality of e discrete random variable X. We consider a discrete-time fading broadcast channel wi a single transmitter communicating independent information to users 3. The transmitted symbol xi] is composed of independent information sources for e two users, where i represents e time index. The time-varying channel gain of e pa to user k is denoted by h k i], which remains constant during e i channel use and is known to e respective user at all times perfect CSIR. Each receiver has additive Gaussian noise. The received signal of user k is us y k i] = h k i]xi] w k i], k =, 3 All e results can be easily extended to more an two users as are discussed in 5]. where w k i] is white Gaussian noise wi power B, where B is e transmission bandwid. For simplicity, we assume B = Hz roughout is paper. We also define e channel power gain γ k i] = h k i], where e distribution of h k i] induces a distribution on γ k i]. Furer, we assume at Γ i] and Γ i] are discrete random variables for all i, having some joint probability mass distribution. We are interested in e case when Γ and Γ are large and e feedback link has limited rate of log M bits for each user, us e receivers can only feedback log M bits for CSIT. We furer assume at e fading process is stationary and ergodic. Thus fγ i], γ i] = fγ, γ i and is joint distribution fγ, γ is known to e transmitter and e users. Also, e input x has an average power constraint P i.e. Ex ] P. Under ese conditions, we derive an achievable rate region for e fading broadcast channel wi limited-rate feedback. III. ACHIEVABLE RATE REGION When only log M bits of channel knowledge are available at e transmitter per user, from e transmitter s point of view, e channel can be in one of M states called component channels at each transmission instant, assuming Γ and Γ are independent of each oer. For any general Γ and Γ and an arbitrary channel partitioning at e two users, e resulting component channels may not be degraded. One simple example is when Γ = {0, 5, 4, } all wi equal probability partitioned as {0, 5, 4, } and Γ = {, 8, 6, 3} all wi equal probability partitioned as {, 8, 6, 3} and Γ and Γ are independent of each oer 4. Clearly from e point of view of e transmitter, e channel can be one of 4 states - {0, 5,, 8], 0, 5, 6, 3], 4,, 6, 3], 4,,, 8]}, all wi equal probability. Notice at in all component channels bo e users experience fading and e fading gain value is not known to e transmitter. The last component channel is degraded while e rest are non-degraded and hence super-position cannot be used in general 6]. Furermore, since in neier of e component channels, e channel gain is perfectly known to e transmitter for eier of e users, dirty paper coding DPC also cannot be used as a possible transmission strategy. As found in 6], time-division multiplexing and more-capable like transmission are possible transmission strategies in any component channel, but bo are sub-optimum in general. Alough, not much can be said about e capacity of e fading BC wi limited-rate feedback for e general case, if we restrict to identical fading for e two users i.e. Γ = Γ independent of each oer and identical channel partitioning as well, all component channels are degraded. This is easily seen as illustrated in Figure. 4 Notice at we do not need to consider groups like {0, 4, 5, } because ey do not convey to e transmitter if e channel is in good state or bad. The channel partitioning us is a quantization, where each representative value corresponds to a closed-region and e regions corresponding to different representative values do not overlap.
Fig.. M degraded component channels wi identical fading and identical channel partitioning at e two users. In component channels wi solid line, users have i.i.d. fading, wi dash-dot line, User is better and wi dashed line, User is better. Hencefor in is section we will assume Γ = Γ = Γ and Γ = N. Out of ese M component channels, in M component channels, e two user have i.i.d fading, in MM component channels user is e better user and in e remaining MM component channels user is e better user. In any case, all component channels are degraded and super-position can be used and is capacity achieving as shown in 7]. This can be considered as a generalization of e probabilistic broadcast channel defined in ]. In ], each component channel was a degraded AWGN BC; whereas in our model, each component channel alough degraded, still experiences fading. The transmitter will generate M codebooks corresponding to each component channel wi e optimum input distribution satisfying e power constraint, chooses one of em according based on e available CSIT at any time instant and rely on e ergodicity of e channel to achieve e long-term average rate. Alough e component channels are degraded, e capacity is known only in terms of a mutual information expression wi a maximization over e input distribution because e component channels experience fading and e optimal input distribution depends on e underlying fading process. The optimal input distribution is not known in closed-form even for simple fading processes like i.i.d. Rayleigh fading for bo e users 6]. Hence, in e discussion at follows, we will restrict to Gaussian input. An achievable region can en be easily derived in closed-form by optimal allocation of total power P to e component channels and among e users based on eir priority specified by µ, µ 0, ] s.t. µ µ =. The lagrangian objective function for optimal power allocation is to maximize µ R µ R λp and can be solved in closed-from using utility function as described in detail in 8], 9], where e resulting optimal R, R is e rate-pair on e boundary of e capacity region. The solution essentially involves a two-dimensional waterfilling over time and over e users. The above discussion assumed at e channel partitioning at e users is pre-decided to be identical and is given Figure. If on e oer hand, e channel partitioning can be optimized over as well, e capacity region again is unknown in general. This is because, e optimal channel partitioning at e users for different priorities of e users µ µ need not be identical and so e resulting component channels need not be degraded for a general Γ. This is easy to observe and an example is given in Section V-B. However, e sum-rate optimal point characterized by µ = µ will have identical partitioning for e two users 5 and hence sum-rate capacity is known and is achieved by super-position coding and M codebooks wi optimal power allocation for any general Γ. Alough e capacity region is unknown when channel partitioning can also be optimized over, if we restrict to identical partitioning for e two users for any µ, µ, e BC can be in one of M component channels as already shown in Figure, all of which are degraded. Hence an achievable region can en be proposed using super-position coding, where e optimal identical channel partitioning and power allocation among e component channels and e users needs to be found for any given priorities of e users characterized by µ, µ. For a given priority µ, µ 0, ] s.t. µ µ = of e two users, e objective is to maximize Q I µ µ j µ E j log γ P j N j 0 = ] I µ<µ j µ E j log MM j = µ E j log γ P j γ P j pj ] γ P j µ E j log pj γ P j MM γ P j3 µ E j3 log j 3= γ P j 3 ] µ E j3 log γ P j3 pj 3 subject to e constraint at j P j P j pj 5 Since Γ = Γ = Γ, and µ = µ, e fact at identical partitioning is optimal follows from symmetry and a rigorous proof is skipped.
j P j P j pj j 3 P j 3 P j 3 pj 3 P, where E ] stands for expectation over e channel values in e corresponding component channel. For different values of µ, µ, different ough identical for e two users channel partitioning will be optimal in general. The overall achievable region en is simply e convex hull of all e boundary points since time-sharing is always allowed. An example for is is given in Section V-B. The first term in square brackets in Eqn corresponds to e component channels in which e two users have identical fading channels denoted by e index j. In is case, time division TD multiplexing is optimal and depending on e value of µ and µ, all power is given to only one user in ese component channels denoted by e indicator function I in Eqn. Specifically, if µ µ j en all e power is given to user, else to user, where µ j is e reshold value of µ in e j component channel as shown in Figure. Since for i.i.d. fading R,single-user = R,single-user, µ j =, j. The second and e ird terms in Eqn, correspond to e component channels when user is e better user indexed by j and when user is e better user indexed by j 3 respectively. R µ 4 µ 3 µ µ µ < µ < µ 3 =µ < µ 4 TD Region Fig.. For values of µ < µ e.g. µ, µ, all power is given to User and for values of µ µ e.g. µ 3,µ 4, all power is given to User. In order to maximize Eqn, we need to find e optimal channel partitioning by a k-mean clustering algorim like e Lloyd s algorim along wi e optimal power allocation. As compared to e single-user case as solved in 3], only e cost function changes from R λp to µ R µ R λp = Q λp and e same Lloyd s algorim can be used. However, e complexity remains prohibitively high as discussed in Section I when N is large. In e next section, we derive a closed form expression for finding e channel partitioning for e single-user case and e fading BC wi limited-rate feedback and will show using numerical results at e resulting channel partitioning is very to e optimal value. R IV. OPTIMAL CHANNEL PARTITIONING A. Single-user Channel We first study e single-user fading channel. The received signal in is case is given as y = hx w 3 where w N 0, is AWGN, x is e transmitted signal and h i.i.d. fading coefficients and e corresponding fading gains γ = h are instantiations of e discrete random variable Γ, where Γ = N <. Let e possible values of Γ be represented by γ, γ,..., γ N and e group representatives of e M groups by γ, γ,..., γ M. The objective is to maximize e ergodic capacity given as j= E Γ γ γ j log j γp pj 4 where pj is e probability of e j group, P j is e power assigned for transmission when e channel is in e j group, satisfying j P j pj P. As noted, in order to find ergodic capacity, we need to find bo e optimum channel partitioning into M groups, as well e power assigned to each group. The objective function in Eqn 4 can be upper bounded by log Eγj P j pj 5 j= using Jenson s inequality, where we have used Eγ j to represent E Γ γ γ jγ] for brevity. The upper bound in Eqn 5 wi e constraint j Eγj = Eγ and for given values of P j s has a lot of similarity to e objective function of water-filling of power over time for maximizing ergodic capacity for fading channels as done in 0]. In fact ey are identical, wi e roles of channel gain and allocated power swapped and Eγ a given fixed quantity playing e role of given power constraint P. Hence for a given value of P j s, e channel partitioning using e approximate objective function in Eqn 5 can be solved by a simple Lagrangian formulation in closed-form. The lagrangian is given as L = log Eγj P j pj λ Eγ j pj Eγ N j= 0 j 6 Since is is e familiar water-filling of power maximization problem, it has e classical solution for Eγ j given as Eγ j P = 7 j λ where λ is found by bisection meod ] such at e constraint j Eγj pj = Eγ is satisfied. From Eqn 7, we note at ergodic capacity is maximized by e channel partition in which e value of Eγ j N0 P j s R.H.S. of
Eqn 7 for e M groups j =,,..., M are all equal or are as close as possible 6. One way to use e result in Eqn 7 is to find a good initial guess of channel partitioning when using e iterative Lloyd s algorim. Furer, when M is large and e channel gain values in any of e M groups are closely spaced, e approximation of e objective function in Eqn 4 by Eqn 5 is fairly accurate. In is case, we can have a less complex iterative water-filling algorim to find e channel partitioning which goes as follows. Iterative Water-filling for single-user: Step : Assume equal power allocation i.e. P j = P, j =,,..., M Step : Find e channel partitioning using Eqn 7 and denote its solution by. Let e channel gain values taken by Γ be γ γ... γ N. Now given, in order to find M groups, we start wi γ and keep on adding more channel values to e first group as P γ N p γ p... γ i= pi N p N, where N is an integer such at e average channel gain value in e first group Eγ when substituted in e R.H.S. of Eqn 7 is closest to λ. Here p i denotes e probability of channel gain γ i, i =,,..., N obtained from e channel gain distribution. We continue to do is for all e groups e.g for group, we start accumulating channel gain values from γ N onwards. Clearly e complexity of finding e channel partitioning given a power allocation is reduced from ONM in Step 3 of Lloyd s algorim Section I to ON as it is e same as water-filling of power Step of Lloyd s algorim. 3 Step 3: For e channel partitioning resulting from Step, we find e optimal power allocation for all e groups same as Step of Lloyd s algorim. 4 Step 4: Go back to Step and iterate till e ergodic capacity increases. In Section V-A, we show at e channel partitioning obtained by e proposed iterative water-filling algorim in a single iteration is optimal capacity-achieving or very close to it for e single-user case. We now find e corresponding closed-form solution for e fading BC wi limited-rate feedback. B. Two-user Broadcast Channel Consider e following approximation of Eqn following e same idea as in e single user case. We replace e channel gain values in e i group i.e. γ Ni, γ Ni,..., γ Ni N i by e average channel value in e group i.e. E Γ γ γ iγ], i =,,..., M and en all e component channels become degraded AWGN channels. The approximated objective now is to maximize 6 Since wi discrete fading distributions, ere may not exist a channel partition in which e R.H.S. of Eqn 7 for all e groups in e partition are equal to λ. Q, which is given as Q I µ µ log Eγi P i,i i = ] I µ< µ log Eγi P i,i pi pi M i = j =i µ log M i 3 = j =i 3 µ log µ log Eγi P i,j Eγ j P i,j Eγ j P i,j µ log Eγi 3 P j,i 3 ] pi pj Eγ j P j,i3 Eγ j P j,i 3 ] pi 3 pj 8 where in going from Eqn to Eqn 8 we have re-indexed e allocated powers in each group using double index to emphasize at power allocation to a component channel is a function of e group pair. The first and e second index refer to e current channel partition seen by user and user respectively. Also we used e fact at e fading distributions for e two users are independent to write e probabilities of e component channels. Notice at when e channel values in a group are close by or equivalently e number of groups M is close to N, e approximation of Eqn by Eqn 8 is fairly accurate. In order to find e channel grouping for e approximate objective function in Eqn 8, we can form e Lagrangian as follows M J = Q λ Eγ i pi Eγ 9 i= since Γ = Γ = Γ. It is easy to see at J is concave in any Eγ i k, k =, ; i =,,..., M. Assuming µ >, we7 can solve for e optimim value of Eγ to get λ = µ P, µ M j = µ M j = Eγ P,j P j, p Eγ Eγ pj pj 0 Notice at e equation is a function of Eγ alone and can be solved to find Eγ for given values of pi s, P,j s, P j, s and λ. 7 Notice at we can assume µ < w.l.o.g. because e two users have identical fading Γ = Γ = Γ and e case when µ > follows from symmetry because e capacity region is symmetric.
Next, we solve for Eγ to get λ = µ µ M j =3 µ M j =3 P, P,j P j, Eγ p Eγ Eγ Eγ P, Eγ P, pj pj µ P, p Eγ µ P, p Eγ P, P, P, P, Notice at e equation is a function of Eγ alone and can be solved to find Eγ for a given value of pi s, power allocations and λ. Similarly we can find e value of e oer Eγ i s, i = 3,..., M. The value of λ is found by bisection meod ] to satisfy e constraint M i= Eγi pi = Eγ. Once again, one way to use e result in Eqn 0, Eqn is to find a good initial guess of channel partitioning when using e iterative Lloyd s algorim. Furer, when e N and M are bo large e approximation of e objective function in Eqn by Eqn 8 is fairly accurate. In is case, we can have a less complex iterative water-filling algorim to find e channel partitioning which goes as follows. Iterative Waterfilling for Fading BC: Step : Initially, set all power allocations to be equal i.e. irrespective of e component channel, we give equal power 0.5P to user and user for e second and ird terms when one user is better and total power P to one of e user for e first term when bo users have i.i.d. fading in Eqn 8. Given is Eqn 0 is simplified as and Eγ as λ = µ P Eγ 0.5P Eγ λ = µ 0.5P p Eγ 0.5P Eγ p p p 0.5P p p Eγ 0.5P Eγ P 3 Step : Let e channel gain values taken by Γ be γ γ... γ N. Now given Eqn, in order to find Eγ, we start wi γ and keep on adding more channel values to e first group as P γ N p γ p... γ i= pi N p N, where N is an integer such at e resulting Eγ and p values when substituted in Eqn is e closest to λ. Next we find, Eγ. Again, given a value of λ and e already calculated value of p, we keep on accumulating channel gain values from γ N into e second group till γ N N, where N is an integer such at e resulting Eγ and p values when substituted in Eqn 3 is e closest to λ. It is easy to see at we can continue in is manner to get all e Eγ i s i =,,..., M. 3 Step 3: For e given channel partitioning, e optimal power allocation can be found using two-dimensional water filling as detailed in 8]. 4 Step 4: Go back to Step and iterate till µ R µ R increases. Notice at in e second and later iterations, we solve Eqn 0, Eqn, etc. in Step using e channel-group probabilities pi s, i =,..., M of e previous iteration Step of previous iteration to find e Eγ i s. In Section V-B, we show at e channel partitioning obtained by e proposed iterative water-filling algorim in a single iteration is optimal capacity-achieving or very close to it for e fading BC wi limited-rate feedback. V. NUMERICAL RESULTS As argued in Section IV, e result of Eqn 7 or Eqn 0, Eqn can be used eier for a an initial good guess for Lloyd s algorim which works for any value of N, M and channel gain values, to reduce e number of iterations or b in a low-complexity iterative water-filling algorim which is accurate when e channel gain values in any group are close typical when N and M are large. In is section, we show e results for a only i.e. we show at e initial guess for channel partitioning using e derived closed-form expressions is eier equal or very close to e optimal. The results for b is work under progress. In all e examples, we assume a total power P = 0 and AWGN noise variance = and one bit of allowed feedback per user. Throughout is section, for any given N, we list e possible partitions by adding more channel gain values to e first group one by one and refer to em as e first, second,..., last partition in e same order. This way, we exhaust all possible quantization boundaries for e discrete fading distribution. A. Single-user Channel In e first example, we let Γ take values {4, 3,, } wi probabilities {0., 0., 0.3, 0.4} respectively. The receiver needs bits to feedback e channel completely. If only one bit of feedback is allowed, we need to find e optimal
channel partitioning. There are ree possible channel partitions - {4, 3,, }, {4, 3,, } and {4, 3,, }. It can be found at e last partition {4, 3,, } is optimal and has an ergodic capacity of.0608 nats/-d symbol. Using e proposed iterative waterfilling algorim for single-user Section IV, in e first iteration wi equal power allocation we find at e last partition has e least difference between Eγ j N0 P s, j =,, hence is e optimal group. Thus a single iteration gives e optimal solution. In e second example, we let Γ take values {,.9,.8,.7,.6,.5,.4,.3,.,.} wi probabilities {.,.0,.03,.5,.,.5,.04,.3,.05,.0} respectively. Now ere are 9 possible channel partitions 8. It can be found at e 7 partition is optimal and has an ergodic capacity of.793 nats/-d symbol. Alternatively using e proposed algorim, we find at e 6 partition is optimal in e first iteration, for which e ergodic capacity is.793 nats/-d symbol. Thus in a single-iteration we get almost all e capacity 9. If we compute e value Eγ j N0 P j s for all e 9 possible partitions wi eir optimal power allocation instead of equal power allocation which is our first guess, we find at e 7 partition indeed has e least value of e difference between e Eγ j P j s j =, ; hence e optimality condition in Eqn 7 indeed holds true. Thus, e proposed iterative waterfilling algorim brings us pretty close to ergodic capacity in a single iteration, which is very important especially when N and M are large. Also, in all e examples above, e optimal channel partitioning satisfies e optimality condition 7 and so eventually e algorim converges to e optimal channel partitioning, if not stuck locally due to e discrete fading distribution. In general, e optimal channel partitioning need not satisfy Eqn 7, especially when e channel gain values in a group are far apart and e approximation is bad or equivalently a coarse quantization of e continuous-distributed channel gains has already been done. B. Two-user Broadcast Channel We consider e two users to have e same fading distribution as considered in e single user channel in e first example. Specifically, Γ and Γ are distributed i.i.d. as Γ in e first example in Section V-A. In Figure 3, we show e achievable regions 0 wi e ree possible channel partitioning. In e first iteration, we use Eqn and Eqn 3 and simply find e partition for which e value of e R.H.S. of equations and 3 have e least difference. For e first partition, e values of e R.H.S. of equations and 3 are 0.4938 and.6667 respectively and e difference is 9.87. For e second partition, e difference 8 In general for a given N, M, ere are ` N N M possible channel partitions. 9 This improvement should be compared to e ergodic capacity for a bad initial channel partitioning guess which is.7908 nats/-d symbol. 0 Rate pairs in e achievable region are calculated and plotted only for µ =.,.5,.4 and.49 in all e figures. R Fig. 3..4. 0.8 0.6 0.4 0. Achievable Region for st partition Achievable Region for nd partition Achievable Region for 3 rd partition Perfect CSIT capacity region 0 0 0. 0.4 0.6 0.8..4 R Achievable regions wi all possible identical channel partitioning. is.036 and for e ird partition it is.667. Clearly, e ird partition is optimal and we find it in a single iteration. We also plot e perfect CSIT outer bound. Notice at wi half e amount of feedback and a single iteration, we come pretty close to e outer bound wi perfect CSIT. Fig. 4. Achievable regions wi all possible identical channel partitioning. Different channel partitioning are optimal for different values of µ zoomin view In Figure 3, e achievable region for e ird partition contains e one for e second partition which in turn contains e one wi e first partition. There are no intersections and a single partition ird partition is optimal for all values of µ. As already noted in Section III, is is not true in general, and for different values µ, different channel partitioning may be optimal. To illustrate is, we consider anoer example, in which Γ and Γ are distributed i.i.d. as Γ as in example 3 in Section V-A i.e. Γ takes values {,.9,.8,.7,.6,.5,.4,.3,.,.} wi probabilities {.,.0,.03,.5,.,.5,.04,.3,.05,.0} respectively.
R.8.6.4. 0.8 0.6 0.4 0. Achievable Region single iteration Perfect CSIT capacity region Convex hull achievable region TD region 0 0 0. 0.4 0.6 0.8..4.6.8 R Fig. 5. Comparison of e proposed achievable region in a single iteration wi at obtained for e globally optimal channel partitioning. In Figure 4, we see at for µ = 0., 6 partition is optimal, for µ = 0.5, 9 last partition is optimal and for µ = 0.4 9, 8 and 6 partitions are optimal since e tangent touches e achievable regions for all e ree partitions, whereas for µ = 0.49, 7 partition is optimal. Alternatively, we use Eqn 0 and Eqn to find an approximation to e best partition in a single iteration. In Figure 5, we show e perfect CSIT outer bound, e time-division TD inner bound region, e achievable region after taking a convex hull of all possible channel partitions which is e best at can be done wi SPC and identical channel partitioning and e achievable region wi e derived closed-form expression in a single iteration. Notice at in a single iteration, e channel partitioning given by Eqn 0 and Eqn is independent of µ, µ and hence consists of only e achievable region wi e 6 partition. As seen in Figure 5, e achievable region using e closedfrom expression in a single iteration is indistinguishable from e convex hull region of all possible channel partitioning and also close to e outer bound wi perfect CSIT which needs twice as much feedback. for e globally optimum channel partitioning. REFERENCES ] A. Goldsmi and L. Li. Capacity and optimal resource allocation for fading broadcast channels: Part I ergodic capacity. IEEE Trans. Info. Th., 47:083 0, Mar 00. ] T. Cover and J. Thomas. Elements of Information Theory. Wiley & Sons, Inc., 99. 3] Y. Liu V. K. N. Lau and T. Chen. Capacity of memoryless channels and block-fading channels wi designable cardinality-constrained channel state feedback. IEEE Trans. on Information Theory, 509:038 049, Sep. 004. 4] Sergei Vassilvitskii David Aur. How slow is e k-means meod? SoCG, 006. 5] J. Cioffi R. Agarwal. Capacity of fading broadcast channels wi imperfect csit. Technical report, Journal Version, in preparation, 006. 6] D. Tuninetti S. Shamai. Fading gaussian broadcast channels wi state information at e receivers. In DIMACS Workshop on Sig. Pro. for Wireless Comm., 003. 7] A. El Gamal. Capacity of e product and sum of two unmatched broadcast channels. Problemy Peredachi Informatsii, 6:3 3, Jan- March 980. 8] R. Agarwal J. Cioffi. Capacity of fading broadcast channels wi one-sided feedback. In Asilomar Conference on Signals, Systems, and Computers submitted, 006. 9] David Tse. Optimal power allocation over parallel gaussian broadcast channels. In Proceedings of Int. Symp. on Information Theory, page 7, June 997. 0] A. J. Goldsmi and P. P. Varaiya. Capacity of fading channels wi channel side information. IEEE Transactions on Information Theory, 43:986 99, Nov. 997. ] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, March 004. VI. CONCLUSION In is paper, we found an achievable rate region for e fading BC wi limited-rate feedback using SPC and showed at it is sum-rate optimal. We also derived closed-form expressions for e optimal channel partitioning for bo e single-user fading channel and e two-user fading BC. The derived closed-form expression can be used to find a good initial guess when using Lloyd s iterative algorim to find e capacity region. Furer, when N and M are large and e channel gain values in any group are close, we propose an iterative waterfilling algorim for finding e optimum channel partitioning which is much simpler as compared to k-mean clustering algorims like Lloyd s algorim. Using numerical results, we showed at e achievable region found in a single iteration is identical or very close to at