Reduced-Complexity Power-Efficient Wireless OFDMA using an Equally Probable CSI Quantizer

Transcription

1 Reduced-Coplexity ower-efficient Wireless OFDMA using an Equally robable CSI Quantizer Antonio G. Marques, Fadel F. Digha, Georgios B. Giannais contact author), and F. Javier Raos Dept. of Signal Theory and Counications. Universidad Rey Juan Carlos. Fuenlabrada, Madrid SAIN) Eail: {antonio.garcia.arques, Dept. of Electrical and Coputer Engineering. University of Minnesota. Minneapolis, MN USA) Eail: {georgios, Abstract Eerging applications involving low-cost wireless sensor networs otivate well optiization of ulti-user orthogonal frequency-division ultiple access OFDMA) in the powerliited regie. In this context, the present paper relies on liitedrate feedbac LRF) sent fro the access point to terinals to acquire quantized channel state inforation CSI) in order to iniize the total average transit-power under individual average rate and error probability constraints. Specifically, we introduce two suboptial reduced-coplexity schees to: i) allocate power, rate and subcarriers across users; and ii) design accordingly the channel quantizer. The latter relies on the solution of i) to design equally probable quantization regions per subcarrier and user. Nuerical exaples corroborate the analytical clais and reveal that the power savings achieved by our reduced-coplexity LRF designs are close to those achieved by the optial solution. I. INTRODUCTION Orthogonal frequency-division ultiplexing OFDM) is the ost coon odulation for bandwidth liited wireline and wireless transissions over frequency-selective ultipath channels. OFDM transissions over wireline or slowly fading wireless lins have traditionally relied on deterinistic or perfect -) channel state inforation at the transitters CSIT) to adaptively load power, bits and/or subcarriers so as to either axiize rate capacity) for a prescribed transit-power, or, iniize power subject to instantaneous rate constraints [9]. While the assuptions of -CSI at the transitters and receiver render analysis and design tractable, they ay not be as realistic due to wireless channel variations and estiation errors, feedbac delay, bandwidth liitation, and jaing induced errors [6]. These considerations otivate a liitedrate feedbac LRF) ode, where only quantized Q-) CSIT is available through a typically sall) nuber of bits fed bac fro the receiver to the transitters; see e.g., [10]. Q- CSIT entails a finite nuber of quantization regions describing different clusters of channel realizations [7], [10]. Upon estiating the channel, the receiver feeds bac the index of the region individual uplin channels belong to channel codeword), based on which each terinal adapts its transission paraeters accordingly. This LRF-based ode of operation Wor in this paper was supported by the US ARL under the CTA rogra, Cooperative Agreeent No. DAAD ; by the USDoD ARO grant No. W911NF ; and by the Governent of C.A. Madrid under grant No. -TIC fulfills two requireents: i) the feedbac is pragatically affordable in ost practical wireless lins, and ii) the Q-CIST is robust to channel uncertainties since transitters adapt to a few regions rather than individual channel realizations. Resource allocation in orthogonal frequency-division ultiple access OFDMA) iniizing the transit-power per sybol based on -CSIT was first studied in [9]. Relying on fixed as opposed to adaptive) Q-CSIT, recent wors deal with optiization of power or rate perforance per OFDMA sybol [2], [5]. Different fro these wors, here we jointly adapt power, rate, and subcarrier resources based on Q-CSIT to iniize the average transit-power. Our focus is on allocation algoriths with negligible on-line coputational coplexity. Moreover, we rely on the optial allocation for designing a novel non-iterative channel quantizer that enforces equally probable quantization regions per user and subcarrier. The rest of the paper is organized as follows. After introducing preliinaries on the setup we deal with Section II), for a given quantizer design, we derive suboptial subcarrier, power, and bit OFDMA allocation Section III). Once the allocation is characterized, we capitalize on it for designing a quantizer with equally probable regions Section IV). Nuerical results and coparisons that corroborate our clais are presented Section V), and concluding rears finish this paper Section VI) 1. II. RELIMINARIES AND ROBLEM STATEMENT We consider a wireless OFDMA syste see Fig. 1) with M users, indexed by [1, M], sharing K subcarriers subchannels), indexed by [1, K]. The instantaneous per sybol) power and rate user loads on subcarrier are denoted by p, and r,, respectively. With these as entries we for K M instantaneous power and rate atrices and R, that is [], := p, and [R], := r,. For a given 1 Notation: Lower and upper case boldface letters are used to denote colun) vectors and atrices, respectively; ) T denotes transpose; [ ],l the, l)th entry of a atrix, and [ ] the th entry of a vector; X 0 eans all entries of X are nonnegative; F N stands for the noralized FFT atrix with entries [F N ] n, = e j 2π N n, n, = 0,..., N 1; f X X) denotes the joint probability density function DF) of a atrix X; liewise, f x x) denotes the DF of a scalar x; E X [ ] stands for the expectation operator over X; ) denotes the floor ceiling) operation; I { } is short for the indicator function; i.e., I {x} = 1 if x is true and zero otherwise; and LHSx) denotes the left hand side of equation x).

2 Fig. 1. Syste bloc diagra. feedbac update, we consider a tie sharing user access per subcarrier; i.e., tie division ultiple access TDMA) 2. This sharing process is described by the K M weight atrix W whose, )th entry w, represents the percentage of tie the th subcarrier is utilized by the th user. Clearly, M =1 w, 1,, and the average power and rate over the transission period between successive feedbac updates is p, w, and r, w, for the th subcarrier of user. Each user s discrete-tie baseband equivalent ipulse response of the corresponding frequency-selective fading channel is h := [h,0,..., h,n ] T, where: N := D,ax /T s denotes the channel order, D,ax the axiu delay spread, T s the sapling period, and N ax := ax [1,M] N,ax. As usual in OFDM, we suppose K N ax. For notational convenience, we collect the M ipulse response vectors in a K M atrix H := [h 1,..., h M ], where the length of each colun is increased to K by padding an appropriate nuber of zeros. Each user applies a K-point inverse fast Fourier transfor I-FFT) to each snapshot of K-sybol streas, and subsequently inserts a cyclic prefix C) of size N ax to obtain a bloc of K + N ax sybols i.e., one OFDM sybol), which are subsequently ultiplexed and digital to analog D/A) converted for transission. These operations along with the corresponding FFT and C reoval at the receiver convert each user s frequency-selective channel to a set of K parallel flat-fading subchannels, each with fading coefficient given by the frequency response of this user s channel evaluated on the corresponding subcarrier. Consider the K M atrix H := 1/ K)F K H, whose th colun coprises the frequency response of user s channel. With the ulti-user channel atrix H acquired via training sybols), the receiver has available a noise-noralized channel power gain atrix G, where [G], := [ H], 2 /σ, 2, 2 Orthogonal access schees other than TDMA are also possible. But as we will see later, the one chosen is not particulary iportant because the optial choice will typically correspond to no sharing; i.e., each subcarrier will be owned by a single user. with σ, 2 denoting the nown variance of the zero-ean additive white Gaussian noise AWGN) at the receiver. We will use g, := [G], to denote the instantaneous noisenoralized channel power gain for the th subchannel of the th user. Liewise, letting Ḡ := E G[G], its generic entry ḡ, := [Ḡ], shall denote the average gain of the, ) subcarrier-user pair. Having practically perfect) nowledge of each G realization, the access point A) allocates subcarriers to users after assigning entries of G to appropriate quantization regions they fall into. Using the indices of these regions, the receiver feeds bac the codeword c = cg) for the users to adapt their transission odes power, rate and subcarriers) fro a finite set of ode triplets. Our wor relies on the following assuptions: as1) Different user channels are uncorrelated; i.e., the coluns of G are uncorrelated. as2) Each user s subchannels are allowed to be correlated, and coplex Gaussian distributed; i.e., g, obeys an exponential DF f g, g, )= 1/ḡ, )exp g, /ḡ, ). as3) Subchannel states regions) reain invariant over at least two consecutive OFDM sybols. as4) The feedbac channel is error-free and incurs negligible delay. as5) Sybols are drawn fro quadrature aplitude odulation QAM) constellations so that the resulting instantaneous BER can be approxiated as κ 1 = 0.2, κ 2 = 1.5) p, κ 2 g, ɛp,, g,, r, ) κ 1 exp 2 r, 1) ). 1) as6) A realization of each g, gain falls into one of L, disjoint regions {R } L, l=1. Since users are sufficiently separated in space as1) is generally true; as2) corresponds to fading aplitudes adhering to the coonly encountered Rayleigh odel but generalizations are possible; as3) allows each subchannel to vary fro one OFDM sybol to the next so long as the quantization region it falls into reains invariant; error-free feedbac under as4) is guaranteed with sufficiently strong error control codes especially since data rates in the feedbac lin are typically low); the accuracy of as5) is widely accepted; see e.g., [4]; and as6) represents a practical and low coplexity quantization. The ultiate goal in this paper is twofold: G1) design a channel quantizer to obtain c; and G2) given c, find appropriate allocation atrices, R, and W. We want to design, R, W, and {R } L, l=1,, so that the average power is iniized under prescribed average rate r 0 := [ r 0,1,..., r 0,M ] T and average bit error rate BER) ɛ 0 := [ ɛ 0,1,..., ɛ 0,M ] T constraints across users. III. QUANTIZER AND TRANSMISSION MODE DESIGN A. roble Forulation Given as6), let R := {G : g, R } denote the set of atrices G for which g, belongs to the region R. Furtherore, let p and r denote 3 respectively, the 3 The subscript l here will be also written explicitly as lg) in places that this dependence ust be ephasized.

3 instantaneous power and rate loadings of user on subcarrier given that G R. Recall that w, G) 1, and thus the expected power and bit loadings for the G realization over the tie between successive feedbac updates will be p w, G) and r w, G), respectively. Our goal is to iniize the average transit-power E G [p G) w, G)] over all subcarriers and users while satisfying average rate and BER requireents. Specifically, we want the average rate of any user say the th) across all subcarriers to satisfy K =1 E G[r G) w, G)] [ r 0 ]. To enforce the average BER requireent we will have the instantaneous BER stay always below a pre-specified BER 4. Then if ɛ 1 denotes the inverse function involved when solving 1) w.r.t. p, and g in := in{g, g, R } represent the worst channel gain, the power loading p G) := ɛ 1 r G), gg) in, [ ɛ 0] ) 2) will autoatically fulfill the BER requireent. Analytically, the constrained optiization proble we wish to solve is: in RG) 0,WG) 0, where := K M =1 =1 E G[ɛ 1 r G), gg) in, [ ɛ 0] )w, G)] subject to : C1. K =1 E G[r G) w, G)] + [ r 0 ] 0,, C2. M =1 w,g) 1 0,, G, C3. r 0,,, l, C4. w, G) 0,,, G, 3) where through the second constraint we enforce the total utilization of any subcarrier by all users not to exceed one, per G realization. Although the optiization proble in 3) is not convex, defining the auxiliary variable r = r w, with w := G R w, G) f G G)dG we can render it convex. Since w 0 the resultant convex optiization proble using globally convergent interior point algoriths [1]. The final forulation now enjoys both reduced coplexity and global convergence in polynoial tie thans to convexity. The objective in 3) is to iniize the average power over all possible channel realizations. However, the constraints involve different fors of CSI: while C2 and C4 needs to be satisfied per channel realization and thus will change depending on G); C1 is an average requireent and C3 is unique per region, and therefore their values do not depend of the concrete channel realization G. In the following subsection, we will derive the Karush-Kuhn-Tucer KKT) conditions associated with 3). These will lead us not only to the expressions deterining the optial loading variables but will also provide valuable insights about the structure of the power-efficient resource allocation policies. At this point, 4 Although this relaxation ay lead to a suboptial solution, it leads to a less coplex optiization proble that turns out to be convex. Nuerical results will show that solution of the relaxed proble attains coparable perforance to that reached by the optiu solution. a rear is due on another aspect related to power efficiency in OFDMA. Rear 1: Although the pea-to-average-power-ratio AR) plays an iportant role in power battery) consuption of OFDM systes, in 3) we did not ipose AR constraints. The underlying reason is that available digital predistortion schees can be applied to the users OFDM sybols to eet such constraints, see e.g., [8]. B. Optial olicies Let β, r β w, αp, αr, αw, denote the positive Lagrange ultipliers associated with C1-C4, respectively. Specifically, upon defining κ 3, := κ 1 2 lnκ 1 /[ ɛ 0 ] ) and setting the derivative of the dual Lagrangian function in 3) with respect to w.r.t.) the auxiliary variable r equal to zero, yields after tedious but straightforward anipulations the following KKT condition expressed in ters of the original variable 5 r G) = log 2 ) β r + α r )gin G). 4) ln2)κ 3, Because KKT conditions for C3 dictate r αp = 0 [1], for r > 0, we need αr = 0 in 4) and therefore the condition gg) in > ln2)κ 3,β r ) 1 has to be satisfied in order for the optiu rate loading in the region to be nonzero. Intuitively, this condition eliinates fro the optiu allocation set the regions R with very poor channel conditions α r > 0), while for the reaining regions α r = 0. Furtherore, it is worth noting that due to the logarithic expression of ɛ 1 [c.f. 1)], the rate loading is reiniscent of the classical capacity water-filling solution. Before analyzing the optiality condition for w, G), let us define the power cost of user utilizing subcarrier as, G) := 2r G) 1)κ3, gg) in β r+ rg). 5) Supposing that R is active, using 5), and differentiating the Lagrangian of 3) w.r.t. w, G), we find at the optiu, G)f G G)+β w G) α,g) w = 0, G, [1, M]. 6) It is useful to chec three things: i) the LHS6) does not depend explicitly on w, G) but only through the associated ultipliers β w G) and αw, G); ii) the ultiplier βw G) is coon ; and iii) for the sae subcarrier and a given realization G, the power cost, G) is fixed and in general different for each user. Furtherore, for each, the KKT condition corresponding to C4 also dictates w,g)α w,g) = 0, G, [1, M]. 7) roposition 1: In the set {w, G)}M =1, we have w, G) = 1 for a unique user, and w, G) = 0 for. Moreover, the user assigned to 5 Henceforth, x will denote the optial value of x.

4 utilize the th subchannel is the one that satisfies: = arg in {, G)} M =1. roof: Assue that w, G) > 0, then α, w G) = 0 and 6) iplies that β w G) =, G)f G G). If also w, > 0 with G) then α, w = 0, G) and 6) for becoes, G) =, G) G; which is not true and thus w, > 0 is not either G) uniqueness). If now w, = 1 for a G), then β w G) =, G)f GG) and using 6) for we can write α, w G) = [, G), G)]f G G) 0, which is not true if, G) <, G) iniu). Notice that if for a given G the iniu value of {, G)} M =1 is attained by ore than one user, any arbitrary tie sharing of the subcarrier aong the is optiu; or we can siply pic one of the at rando without altering the power cost. Finally, if there exists a realization G such that, G) > 0, then since β w G) 0 it follows fro 6) that α, w G) 0, ; and the optial solution will not allocate this subcarrier to any user. Therefore, introducing a fictitious,0 G) = 0 and G, we can express w, G) in copact for using the indicator function as w,g) = I {=arg in {, G)} M =0 }. 8) Since so far we have obtained conditions that the optial r and w, G) should satisfy, what reains is a condition for p. Although in principle p can be calculated by substituting r and w, G) into 2), recall that to render the optiization proble tractable we had enforced via 2) an instantaneous BER constraint that is a ore restrictive than the original average BER constraint. Capitalizing on this fact, a different approach to obtain the power per region denoted by p + ), given r and w, G), consists of finding p+ to satisfy E G R [ɛp +, g,, r )w,g)] = [ ɛ 0 ] E G R [w,g)]. 9) Using this approach we, guarantee that the expected BER averaged over all channel realizations inside the region R satisfies the pre-specified requireent [ ɛ 0 ]. Since p + < p, we will rely on 9) to calculate the final power loading. Note also that, although 9) involves an integration and p + can not be found in closed-for, the onotonicity of the exponential inside the instantaneous BER in 1) allows obtaining p + through line search using e.g., the bisection ethod). The following algorith suarizes the ain steps for finding the optial allocation of R, W and. Algorith 1: Resource Allocation S1.0) Select a sall positive nuber δ, and initialize β r using an arbitrary non-negative vector. S1.1) For each,, l) triplet per iteration: S2.1.1) Use [β r ] to deterine r via 4)α r = 0). S2.1.2) Use [β r ] to deterine w, G) via 8). S1.2) Chec C1 3) ; if LHSC1) < δ[ r 0 ], then go to S1.3); otherwise, increase [β r ] for the users whose LHSC1) > 0; decrease [β r ] for the users whose LHSC1) < 0; and go to S1.1). S1.3) Once R, β r, W are obtained, use 9) to calculate the finally allocated power. The convexity of 3) enables efficient ethods to update β r [1]. For exaple, we can set the initial value of β r equal to any sall nuber and update each coponent [β r ] separately by fixing [β r ], fro the previous iteration. The adaptation of each [β r ] is then perfored using line search until the rate constraint for the th user is tightly satisfied. This siple algorith has guaranteed convergence and facilitates coputation distributed across users. C. Codeword Structure Given the quantizer design, we developed so far resource allocation policies to assign rate, power and subcarriers across users. Once the quantizer and resource allocation strategy are designed, the A quantizes each fading state and feeds bac a codeword that identifies the user-subcarrier assignent and the region index each subchannel falls into per fading realization G. Based on this for of Q-CSIT, each user node is infored about its own subset of subcarriers if any) and relies on the region indices to retrieve the corresponding power and rate levels fro a looup table. The following proposition describes the construction of this codeword. roposition 2: Given the quantizer design and the optial allocation paraeters, R, W β r )) returned by Algorith 1, the A broadcasts to the users the codeword c G) = [c 1G),..., c K G)] specifying the optial resource allocation for the current fading state, where c G) = [ G), l G)]T is deterined as: 1) G) = arg in {, G,, R, β r, {β ɛ } L, l=1 )}M =1 pic randoly any user when ultiple inia occur); and 2) l G) = { l G R, G),l, l = 1,..., L }. The structure of c G) in roposition 2 encodes inforation pertinent to each subcarrier naely, its region and assigned user) which is ore efficient in ters of the nuber of feedbac bits relative to encoding each user s individual inforation, ) yielding a codeword length of K =1 log M 2 =1 L,) bits. We conclude this section by ephasizing that R and thus ) in 3) are involved only in average quantities. Hence, R and are coputed off-line and only the subcarrieruser assignent involved in instantaneous constraints) and the indexing of the corresponding entries of these atrices need to be fed bac on-line. Thus, alost all the coplexity is carried out off-line Algorith 1), while only a light coputation roposition 2) has to be carried out on-line. IV. QUANTIZER DESIGN In the previous section we addressed our objective G2) to derive optiu subcarrier, rate, and power allocation policies

5 assuing the quantization regions, R, are given. In this section, we will address G1) by deriving a quantizer that enforces equally probable quantization regions. A. Equally robable Region Quantizer To design the quantizer, we first solve the optial resource allocation proble supposing CSI is available without quantization i.e., L, = ), and subsequently calculate τ to satisfy τ+1 τ rw, = 1, g, )dg, = τ, L, τ, 1 rw, = 1, g, )dg, /L,, 10) with τ, 1 = 0 and τ, L, +1 =. If the joint probabilities can be coputed, solving 10) yields thresholds {τ } L. l=1 per subcarrier and user that divide the joint probability rw, = 1, g, ) into regions of equal area; hence the ter equally probable region quantizer. Intuitively speaing, this quantizer design tries to axiize the entropy in the feedbac lin. 6 To evaluate rw, = 1, g, ) needed in 10), we apply Bayes rule to re-write it as rw, = 1 g, )f g, g, ) and recall that f g,i g, ) is nown per as2). To calculate rw, = 1 g, ), we will need to first solve the optial resource allocation proble assuing no quantization. Clearly, as L,, we have g in g in 5). Letting, and denote, respectively, the cost indicator and the Lagrange ultiplier when L,, we can write β,g, ) = β ln2) κ 3, β g, g, β log 2 κ 3, ln2)) ). 11) Because equation 8) establishes that w, G) = I {=arg in {, G)} M }, if, g,) <,0 = 0, =0 then g, > ln2)κ 3, /β is a necessary condition for the user to be active. Taing also into account that channels of different users are uncorrelated [cf. as1)], we can write rw, = 1 g, ) = I {g, > ln2)κ 3, λ r } M µ=1,µ r, <,µ g,). 12) Interestingly, for the active users it holds that, g β,) = I {g, > ln2)κ 3, β } ln2) κ 3, g, β )) g, β log 2 ln2)κ 3, ), and therefore I {g, > ln2)κ 3, β } κ3, g, β ln2) ) κ3, g,, g,) g, = 0, which iplies that, g,) is onotonically decreasing. Therefore, we can find unique channel gains γ,µ, µ, such that,g, ) =,µγ,µ ). 13) It is then clear that, g,),µ g,µ) if g,µ γ,µ, and, g,) >,µ g,µ) if g,µ > γ,µ. And consequently, r, <,µ g,) = rg,µ < γ,µ g, ). 6 The siplicity and efficiency of using an equally probable quantizer were first pointed out in [7] for the basic case of a single user and subcarrier. We here largely expand the scope of this quantizer and apply it to the challenging ultiple-user/ultiple-subcarrier scenario. rw, = 1 g,) fg, g,) rw, = 1, g,) Noralized Subcarrier ower Gain g,/g, Fig. 2. Equally robable Regions Quantizer K = 64, M = 6, L = 5). Excluding the case, g,) > 0, which aounts to w, = 0 and solving 13) w.r.t. γ,µ yields γ,µ g, ) = f W [ κ 3,µ ln2)/β µ 2 κ 3, g, β µ e 1 e ln2)κ3, g, β ) β β µ 14) where f W [x] = y is the real-valued Labert s f W function which solves the equation ye y = x for 1 y 0 and 1/e x 0 [3]. Using 12)-14), we express rw, = 1, g, ) in 10) as e rw, = 1, g, ) = I g, /ḡ, {g, > ln2)κ 3,µ β } ḡ, M ) µ=1,µ 1 e γ,µ g, )/ḡ,µ. 15) Since 15) depends on {βµ } M µ=1, we need to solve the optial allocation proble as L,. The thresholds {τ } L, l=2 are obtained by solving 10) using a line search. Notice that we can also tae advantage of the condition, g,) < 0, by setting τ,,2 κ 3, ln2)/β. An exaple illustrating this quantizer is given in Fig. 2 which depicts rw, = 1 g, ), f g, g, ) and rw, = 1, g, ) versus g, /ḡ, for M = 6, L, = 5, equal average subcarrier gains, and equal rate constraints. The first subplot in this figure, rw, = 1 g, ), reveals that the better the channel the ore liely the corresponding user is to be selected. Coupling this observation with the exponential behavior of f g, g, ) in the second subplot, the bell-ring characteristic of the joint DF, rw, = 1, g, ), results naturally in the third subplot after pair-wise ultiplication of the functions in the first two subplots), where the quantization thresholds and regions) resulting fro 10) are also identified. Rear 2: With the ethod presented in this section, calculation of τ based on 10) has to be executed only once to solve G1)). With the thresholds available, the resource allocation can be easily obtained through Algorith 1 to solve G2)). Nuerical results in the next section will show that this low coplexity non-iterative design exhibits power consuption siilar to that of the -CSIT solution. ],

6 TABLE I TOTAL AVERAGE TRANSMIT-OWER IN db W ) FOR -CSIT, Q-CSIT-E AND Q-CSIT-UN SCHEMES. CASE Q-CSIT-UN Q-CSIT-E -CSIT Reference Case ɛ 0 = r = [30, 30, 30] T K = M = SNR = SNR = TABLE II TOTAL AVERAGE TRANSMIT-OWER IN db W ) AS L, VARIES. L, CSIT) V. NUMERICAL EXAMLES To nuerically test our power-efficient designs, we consider an adaptive OFDMA syste with M = 3 users, K = 64 subcarriers, noise power per user and subcarrier at 0 db W, L, = 5 regions i.e., 4 active regions) per subcarrier, and [ ɛ 0 ] = ɛ 0 = The average signal-to-noise-ratio SNR) considered was 3 db; and three uncorrelated Rayleigh taps were siulated per user. Test Case 1 Coparison of allocation schees): For coparison purposes we will contrast the power consuption of our proposed quantization and allocation schee denoted by Q-CSIT-E) with the power consuption of the allocation based on -CSIT that represents a lower bound) as well as with the power consuption of a heuristic schee based on Q-CSIT with unifor quantization and subcarrier allocation but optiu rate and power loading denoted by Q-CSIT- UN). Nuerical results assessing the perforance of -CSIT, Q-CSIT-E and Q-CSIT-UN schees over a wide range of paraeter values are suarized in Table I, where the reference case is K = 64, M = 3, r 0 = [60, 60, 60] T, SNR = 3 db; and the colun case entails only a single variation w.r.t. the reference case. The striing observation here is the alost equivalent perforance of Q-CSIT-E and -CSIT schees. These results certify the usefulness of the proposed quantizer based on equally probable regions and validates the optiality of Algorith 1. Moreover, the power loss of at least 6 db with respect to Q-CSIT-UN shows the iportant role of subcarrier allocation and quantization in ters of iniizing the transit-power. Test Case 2 Nuber of quantization regions): Table II lists the average transit-power when varying the nuber of regions per subcarrier for r 0 = [20, 40, 60] T. Notice that usually the nuber of active regions is equal to L, 1, since the optiu solution yields one outage region for the cases when the channel gain is very poor). Siulation results deonstrate that our proposed equally probable quantization and Algorith 1 used together lead to a power loss no greater than 3-5 db with respect to the -CSIT case L, = ). Moreover, the resulting power gap shrins as the nuber of regions increases reaching a power loss of approxiately only 0.5 db in the case of four active regions. VI. CONCLUDING SUMMARY AND FUTURE RESEARCH Based on Q-CSIT, we devised a power efficient OFDMA schee under prescribed individual average rate and BER constraints. In this setup, an access point quantizes the subcarrier gains and feeds bac to the users a codeword conveying the optiu power, rate, and subcarrier policy. The resulting nearoptial transceivers are attractive because they only incur a power loss as sall as 1 db relative to the benchar design based on perfect CSIT which requires feedbac inforation which ay be unrealistic in wireless systes. Our novel design separates resource allocation fro channel quantization. The first was obtained as the solution of a convex constrained optiization proble, while the second was designed to ensure equally probable regions per user and subcarrier. We ended up with a lightweight resource allocation protocol where both rate and power are available at the transitter through a looup table and only the subcarrier assignent needs be deterined on-line. To build on the presented fraewor, future directions will exploit the possible correlation across subcarriers to group subcarriers and then index each group. 7 REFERENCES [1] D.. Bertseas, Nonlinear rograing. Athena Scientific, [2] M. Cho, W. Seo, Y. Ki, D. Hong, A Joint Feedbac Reduction Schee Using Delta Modulation for Dynaic Channel Allocation in OFDMA Systes, roc. of IEEE Int. Syposiu on ersonal, Indoor and Mobile Radio Coun., Berlin, Gerany, vol. 4, pp , Sep [3] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, On the Labert W function, Advances in Coputational Matheatics, pp. 5: , [4] A. J. Goldsith and S. -G. Chua, Variable-rate variable-power M-QAM for fading channels, IEEE Trans. on Coun., vol. 45, pp , Oct [5] J. H. Kwon, D. Rhee, I. M. Byun, Y. Whang, K. S. Ki, Adaptive Modulation Technique with artial CQI for Multiuser OFDMA Systes, roc. of Int. Conf. on Advanced Coun. Technology, hoenix ar, Korea, vol. 2, pp , Feb [6] A. Lapidoth and S. 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