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Rajiv Agarwal†, Rath Vannithamby‡ and John M. Cioffi†. †{rajivag ... can be transmitted during feedback is MRfTc, where M is the total number of ..... [10] J. Chen, R. A. Berry, and M. L. Honig, “Performance of limited feedback schemes for ...
Optimal Allocation of Feedback Bits for Downlink OFDMA Systems Rajiv Agarwal† , Rath Vannithamby‡ and John M. Cioffi† † {rajivag,

cioffi}@stanford.edu

Abstract— This paper studies the downlink Orthogonal Frequency Division Multiplexing (OFDM) setup with a single Base Station (BS) serving many users. The BS is assumed to have limited Channel State Information (CSI) obtained by feedback in a Time Division Duplexing (TDD) manner. Given that the feedback rate and the coherence time of the channel are fixed, the question asked in this paper is: how to allocate the feedback resources optimally? Specifically, what is the optimal number of tones grouped as a subchannel, the number of users that feedback for any subchannel and the number of bits used for quantization of CSI? Analytical expressions are derived for the i.i.d. Rayleigh fading case and it is shown that there is a definite hierarchy in the importance of the three design variables. Feedback resources are first allocated to create the maximum number of subchannels possible, then to allow for more users to feedback for any subchannel and lastly to increase the precision of the channel value. Monte-Carlo simulations are performed to verify the accuracy of the derived analytical expressions.

I. I NTRODUCTION Next generation wireless cellular networks will support a variety of Quality of Service (QoS)-sensitive applications like streaming multimedia and high-speed data for downlink users. To support such high-speed wireless services, transmission over a wide band using Orthogonal Frequency Division Multiple Access (OFDMA) is an attractive downlink transmission technique. To utilize limited system resources in an efficient manner, channel-aware resource allocation for the OFDMA downlink has played a key role and attracted much research attention [1]-[5]. One of the major problems in employing a resource allocation scheme in OFDMA networks is the large amount of feedback required to pass to the Base Station (BS), while most existing work require perfect CSI at the transmitter. For example in WiMAX, there are up to M = 2048 tones, so in a network of K = 100 users, the BS needs to collect 204,800 real numbers for CSI through uplink transmission from the users. This information is provided at some fixed feedback rate. Moreover, the frequency of information collocation has to be quicker than that of channel variations, which happen every coherence time of the channel. To address this issue, many works [6]-[9] have analyzed the effects of limited feedback in the downlink OFDMA setting. The coherence time of the channel as well as the feedback rate can be small in certain wireless channel scenarios (e.g. high mobility), making the problem of CSI feedback harder. In this paper, we focus on using the scarce feedback resources optimally. In particular, given a fixed coherence time Tc and a fixed feedback rate Rf per tone, the number of bits that can be transmitted during feedback is M Rf Tc , where M is the total number of tones. Of course, some fraction of the

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coherence time is kept for actual data transmission in the downlink. Hence the number of bits that can be fed back is strictly less than M Rf Tc . For the case of Rayleigh fading, we derive analytical expression for sum-rate in the finite feedback constraint case and maximize it with respect to (w.r.t.) to different design variables. The design variables considered are the number of subchannels1 (multiple tones can be grouped as a subchannel to reduce feedback), the number of users allowed to feedback for any subchannel and the precision of CSI. It is shown that to maximize sum-rate, the optimal split of feedback bits among the design variables is hierarchical in nature. Feedback bits are first allocated to have as many subchannels as possible, then to increase the number of users that feedback on any subchannel, and lastly to increase the precision of CSI. Monte-Carlo simulations confirm the accuracy of the derived analytical results. The work in this paper, thus follows the same idea as in [10] and [11], though in a different setting. Specifically, in [10], the authors also study the case of fixed coherence time and feedback rate, and characterize the asymptotic growth rate of sum-rate when both M and K increase to infinity, with their ratio fixed. Whereas, in this paper, we study the case of finite M and K and additionally, have an extra parameter - precision of CSI. The finite case is more important from a practical system design point of view. In [11], the authors fix the total number of bits allowed for feedback, and study the optimal split of these bits between number of users and precision of CSI for a Multiple Input Multiple Output (MIMO) downlink system. However, we do not fix the number of feedback bits but keep it as a parameter to be optimized. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION Consider a downlink transmission system with K users and M tones where the BS and each user are equipped with a single antenna. It is assumed that the Inter-symbol Interference (ISI) is completely removed by exploiting OFDM techniques. For user u on tone m, the channel value is denoted by hum . We assume that the channel values for the M tones of any user are independent and identically distributed (i.i.d.)2 and are independent across users as well. A zero-mean i.i.d. Gaussian 2 noise with variance σum is added at the receiver part. The 1 In this paper, a “tone” refers to a band of frequencies in the OFDM setting experiencing flat-fading. A “subchannel” is a group of multiple tones. 2 In practice, adjacent tones have correlated fading. The model in this paper corresponds to the case, where there is a certain channel-correlation threshold. Tones that have correlation above this threshold are essentially considered as a single-tone with flat-fading. Whereas, tones that have correlation below this threshold, are assumed to undergo independent fading [12].

2

| channel gain for user u on tone m is defined as γum , |hσum . 2 um In this paper, we do not consider water-filling of transmit power over subchannels, and focus on the case of equalpower allocation assuming unit power for each tone. In a TDD system, downlink data transmission is preceded by feedback of CSI on the uplink. The feedback and data-transmission periods add up to the coherence time of the channel, denoted by Tc . This is depicted in Figure 1. We assume that for each tone m, γum , follows a block-fading model where γum remains constant for Tc and then changes in an i.i.d. manner to a new realization. Also, let Rf denote the feedback rate per tone in bits per unit time. Feedback Period

Data Transmission Period

u

Coherence Time Tc

Fig. 1.

Partition of Channel Coherence Time Tc in a TDD system.

Perfect Channel State Information at the Transmitter (CSIT) entails that γum is fed back for each tone by all users as a real number. To provide such high precision for CSI, the number of bits required will be large, say breal bits, the total feedback overhead per Tc then is M Kbreal . Given a fixed feedback rate Rf , if too much time is taken for CSI feedback, little time is left for actual data transmission. Hence, feedback load needs to be reduced. Some ways of accomplishing this are: (a) group the tones as subchannels and let a single value denote the channel quality of the subchannel, (b) allow only a few randomly-picked but pre-determined users to feedback for any subchannel and (c) let the channel gain value have a few bits  of precision. Let g be the number of subchannels and so, M is the number of tones in each subchannel. Let g k be the number of users that feedback for any subchannel. Since the users are assumed to have i.i.d. fading for all tones, specifying the actual users that feedback for any subchannel is unimportant and these k users can be any k users among the total K users. Let γu′ s be the subchannel gain for subchannel s (s = 1, 2, . . . , g) for a user u′ (u′ = 1, 2, . . . , k) that feeds back for this subchannel s. There are several approaches discussed in literature for using a single value to denote the channel quality of a group of tones [13]. In this paper, we define γu′ s for a subchannel s as the minimum channel gain for the set of tones that it consists of, i.e. γu′ s =

min

s′ ∈Subchannel−s

γu′ s′ , ∀u′ = 1, 2, . . . , k.

for it, among the k users (u′ = 1, 2, . . . , k) that feedback for subchannel s. Let b be the number of bits used for quantizing each subchannel gain value. Let the operator [x]b denote the value of a real number x quantized using b bits. In this paper, we use an equi-probability quantization approach i.e. the probability density function (pdf) of subchannel-gain γu′ s is partitioned into 2b regions, each having equal probability 2−b . Then [γu′ s ]b denotes the lower threshold for the region in which the actual subchannel gain value γu′ s falls. E [·] denotes expectation w.r.t. to the channel-gain distribution. The optimization problem under consideration can then be denoted as    kgb ∗ Rsum (Θ) = max 1 − k,g,b M R f Tc   h i  ′ ′ M E log 1 + max (2) min γ us ′ ′

(1)

This definition is the same as used in [10] and is intuitive from an information-theoretic framework to ensure error-free transmission on all tones contained in the subchannel. A discussion on the merits and demerits of using this definition for subchannel quality and a few alternative definitions for subchannel quality (and subsequent derivation) is given in [14]. In order to maximize sum-rate Rsum , subchannel s is allocated to the user with the highest subchannel gain γu′ s

s

b

 kgb ≤ M Rf Tc ,    1 ≤ k ≤ K, subject to  1 ≤ g ≤ M,   1 ≤ b.

The first term in (2) is the useful fraction of time available for downlink data transmission. The second term in (2) is the expected value of sum-rate. The variables for optimization are k, g and b, which are all positive integers. In addition, variable g is constrained to be a factor of M , so that there are integer number of tones in any subchannel. The fixed parameters in the objective function (2) are M , K, total number of available feedback bits per tone Rf Tc and the distribution of subchannel-gain. Let Θ denote the set of fixed parameters for ∗ the optimization problem. Clearly, maximum sum-rate Rsum and the optimal values of k, g and b will be a function of Θ. III. O PTIMAL A LLOCATION OF F EEDBACK B ITS In order to solve the optimization problem in Eqn (2) analytically, we consider the case of Rayleigh fading. Further, we upper bound the objective function in Eqn (2) by moving the E [·] inside the log and optimize the upper-bound, which has a closed-form expression. As shown later through Monte-Carlo simulations, the optimal values of k, g and b that maximize the upper-bound are very close to those that maximize the original objective function (2). The new objective can then be written as3      h i  kgb ′ ′ min γ max 1 − log 1 + E max us s′ u′ k,g,b M R f Tc b (3) To evaluate the expected value of the subchannel-gain value, we first consider the case when there is no quantization done. In this case, the expected value has a closed-form expression given as ! k h i X 1 g E max min γu′ s′ = γ¯ (4) u′ s′ i M i=1 3 Hereon, while writing the objective function, we drop the constant term M in (2).

In presence of equi-probability quantization, the above expression is modified4 as   !  b k 2b −1 X 1 g 1 X 2  γ¯ (5) log i M 2b j=1 j i=1 A closed-form expression for the objective is then given as    kgb 1− max k,g,b M R f Tc     ! b  b k 2X −1 X 1 g 1 2   γ¯ log 1 + log b i M 2 j i=1 j=1

(6)

The above expression can be  further simplified by curve P2b −1 Pk 1 b and 21b j=1 log 2j fitting. It is seen that i=1 i 1 can be closely approximated by 1.1 log(k) and 1+e−0.8(b−1.7) respectively for practical values of system-parameters (k = 1 to 200 and b = 1 to 20). Notice that the growth of average subchannel-gain w.r.t. number of users that feedback is logarithmic, whereas w.r.t. to bits of quantization is sigmoidal. This is intuitive because the benefits of increasing the number of bits b tapers off. The final expression for the objective function can now be given as    kgb max 1− k,g,b M R f Tc | {z } st   1 term  g  1 log 1 + γ¯ (1.1 log(k)) M 1 + e−0.8(b−1.7) {z } | 2nd term

(7)

The role of the parameters k, g and b is clearly seen from the above expression. Increasing any of g, k or b decreases the first term in (7) linearly, however g increases the second term in (7) logarithmically, k increases it as log-log (slower) and finally b increases it as log-sigmoidally (slowest). Intuitively, from the rate of growth of the second term w.r.t. to the optimization variables, one can conclude that it is best to increase g because this provides the highest growth to the second term at the same cost as other variables. In other words, grouping independent tones in a subchannel hurts the objective most. Hence, we should have as many subchannels as possible, each with fewer tones. However, g cannot be increased beyond its maximum value M . If Rf Tc is large and more feedback bits are available, once g ∗ hits M , the next variable to increase is k. Finally, once k ∗ reaches its maximum value K, for very large values of Rf Tc , b is increased. Thus, essentially, the optimal way to utilize the feedback bits, essentially falls into three distinct regions. 4 In order to derive the expression in Eqn (5), we move the max ′ operation u inside the [·]b operation in Eqn (3). The accuracy of this approximation is again verified by Monte-Carlo simulations in Section IV.

1) For small values of Rf Tc , hereby denoted as region 1, k ∗ and b∗ take their minimum possible values and only g ∗ is increased until it meets its maximum value M . 2) For large values of Rf Tc , hereby denoted as region 2 where g ∗ has reached its maximum value, b∗ is still fixed at its minimum value and only k ∗ is increased until it meets its maximum value K. 3) Finally, for very large values of Rf Tc , hereby denoted as region 3 where both g ∗ and k ∗ have hit their maximum values, b∗ is increased. This intuition is correct as verified by Monte-Carlo simulations in Section IV, with a small difference in region 2. In Section IV, it seen that both k ∗ and b∗ increase in region 2. This behavior can be explained as follows. At the end of region 1, both k ∗ and b∗ are small. Now, for small values of (b ≤ 5), the  sigmoidal function is similar to log i.e.  bP 2b −1 2b 1 log is very close to 0.47 log (1.2b + 1). So, b j=1 j 2 at the end of region 1, both k and b increase the second term of the objective function (7) in a similar manner. Hence, in region 2, both k ∗ and b∗ increase. The rate of growth of k ∗ , however, is higher than that of b∗ because the sigmoidal 1 is always less than 1, whereas there is function 1+e−0.8(b−1.7) no such upper-bound on 1.1 log(k). Eqn (7) has two-fold advantage. First, notice that the original objective (2) is hard to solve (solved by extensive Monte-Carlo simulations in this paper), and rewriting it as (7) gives an easy explanation for the optimal solution of the original objective. Second, if we relax k, g and b to be real numbers, it can be seen that the (simplified version of the) objective function (7) is convex in all k, g and b and unique optimal values for all the three exist. From the above intuition, in the three different regions, only one variable changes while the others are fixed. This enables us to easily derive closedform expressions for the dependence of g ∗ , k ∗ and b∗ and ∗ on different elements of Θ using (7), the dependence of Rsum as done in the next section. For the case of Rayleigh fading, the fixed parameter set Θ = {Rf Tc , M, K, γ¯ }. As shown in Section IV, the derived expressions closely follow the results found from Monte-Carlo simulations. A. Derivation for g ∗ (Θ) From the above discussion, when feedback bits are scarce, the best way to utilize them is to keep k and b fixed at their minimum values and optimize g. To find the dependence of g on Rf Tc , the expression in (7) is simplified as max (1 − αg) log (1 + βg) g

(8)

where α and β are positive constants that contain the fixed (8), we get g ∗ log(g ∗ ) ∝ α1 parameters and Rf Tc ∝ α1 . Solving √ ∗ and so approximately g ∝ Rf Tc or increases with it sublinearly. From convexity theory, the optimal value of g ∗ before it has saturated is independent of M , however, the value at which g ∗ saturates, of course, increases linearly with M as seen by writing (7) as   g  g  log 1 + β (9) max 1 − α g M M

1

for some positive constants α and β. To find the dependence on γ¯ , we consider the expression (8) again and notice that ∗ γ¯ ∝ β. Solving (8), we get 1−2αg αg 2 ∝ β and so g decreases with increasing γ¯ in a quadratic manner. The intuition behind this is, when the channel is stronger, the second term in (7) increases at a faster rate and hence the optimal is attained at a smaller value of g. The optimal value of g is independent of K, the last element of Θ, because g ∗ saturates to its maximum value much before k starts to increase.

(log(Rf Tc )) rb respectively, k ∼ 2.1 and rb ∼ 1.3.   where 1r ∗ Hence, Rsum ∝ log log (Rf Tc ) r , where r > 0 is a positive constant. Since, in region 3, b∗ ∝ log(Rf Tc ), we ∗ derive that Rsum ∝ log log(Rf Tc ). Overall, as seen in Section ∗ IV, in regions 2 and 3, Rsum is seen to closely follow the log log(Rf Tc ) asymptote.

B. Derivation for k ∗ (Θ) and b∗ (Θ)

In this section, we perform Monte-Carlo simulations to solve the original objective function (2) and evaluate the proposed simplified version of the objective function (7) numerically. Results from Monte-Carlo simulations show that ∗ the dependence of the variables g ∗ , k ∗ , b∗ and Rsum on Θ = {Rf Tc , M, K, γ¯ } is exactly the same as derived in Section III. The curves corresponding to Monte-Carlo simulations, which solve (2), are skipped and can be found in [14], and only those obtained by numerically solving (7) are shown because the two were found to coincide. System parameters used are M = 256, K = 200 and Tc is varied from 10−3 to 105 . Since, Rf Tc characterizes available feedback bits as a whole, Rf is fixed at Rf = 40 and only Tc is varied. As argued in Section III, there are three distinct regions of Tc . For values of Tc < 100 , both k ∗ and b∗ are pinned to their minimum value 1, and only m∗ increases until it reaches its maximum value M = 256. In the region 100 < Tc < 103 , both k ∗ and b∗ increase, however, the former increases at a faster rate and hits its maximum value K = 200 and the latter increases slowly only until the value b∗ = 5. In the third region, Tc > 103 , once both g ∗ and k ∗ saturate, b∗ increases logarithmically with Tc . From figures 2-4, we see that the dependency of g ∗ , k ∗ and ∗ b as derived in Sections III-A and III-B is fairly accurate as shown by the asymptotes (the dashed black curve). For example, in Figure 4, b∗ increases as log Tc for Tc > 103 . Another observation made from figures 2-4 is the fact that all g ∗ , k ∗ and b∗ decrease with increasing γ¯ , however their dependence on γ¯ is weak. Similar observation can be made ∗ increases logarithmically with γ¯ and once from Figure 5. Rsum ∗ ∗ g has saturated at its maximum value (for Tc > 100 ), Rsum increases as log log Tc .

In a similar manner, we derive that, in region 2 when g ∗ has saturated to its maximum value M , k ∗ log(k ∗ ) log log(k ∗ ) ∝ 1 Rf Tc and so approximately k ∗ ∝ (Rf Tc ) rk , where rk > 2. Using numerical results in Section IV, it is found that rk ∼ 2.1. In region 2, the dependence of b∗ on Rf Tc is harder to derive as k ∗ changes along with b∗ in region 2. Through numerical 1 results in Section IV, it is seen that b∗ ∝ (log(Rf Tc )) rb , where rb ∼ 1.3. Once k ∗ has saturated at its maximum value K, the only variable left is b which may not have an upper-bound in general and depends on the method used for channel measurement (and thereby its accuracy) at the receivers. This is the third region of Rf Tc . For this region, we derive b∗ ∝ log(Rf Tc ). For the dependence on γ¯ , in regions 2 and 3, when k ∗ and ∗ b are not pinned to their minimum values and are allowed to take non-minimum values, both k ∗ and b∗ decrease with γ¯ by the same argument as given for g ∗ . This can be seen from the expression in (7), rewritten as max (1 − αkgb) log (1 + β¯ γ g log(k) log(νb + µ))

(10)

k,g,b

where α, β, ν and µ are some positive constants. The rate of decrease of g ∗ was shown to be quadratic w.r.t. to increasing γ¯ in Section III-A, hence from (10), the rate of decrease of k ∗ is slower than that of g ∗ , and the rate of decrease of b∗ is slower than that of k ∗ . The dependence of k ∗ and b∗ on M and K is straightforward and so is skipped. ∗ C. Derivation for Rsum (Θ) ∗ The dependence of Rsum on M , γ¯ and K is straightforward and skipped due to space constraints. The interesting case is ∗ on Rf Tc . Notice that to find the dependence of Rsum   ∗ ∗ ∗ g k b ∗ Rsum ∝ 1−α R f Tc   1 log 1 + βg ∗ log(k ∗ ) (11) 1 + e−0.8(b∗ −1.7)

where α and β are some positive constants. Now, as derived in Sections III-A and III-B, in region 1, k ∗ and b∗ 1 ∗ ∗ are  fixed and 1g ∝ (Rf Tc ) 21. Hence, essentially Rsum ∝ 1 − c(Rf Tc )− 2 log (Rf Tc ) 2 , for some constant c. In regions 2 and 3, Rf Tc is large enough, so the first term  g ∗ k ∗ b∗ in (11) can be neglected. In region 2, g ∗ 1 − α Rf Tc 1

is fixed at M , but k ∗ and b∗ increase as (Rf Tc ) rk and

IV. M ONTE -C ARLO S IMULATIONS AND N UMERICAL R ESULTS

V. S UMMARY AND D ISCUSSION In this paper, we studied the downlink OFDMA system with a finite channel coherence time and finite feedback rate and found the optimal number of bits used for feedback and its optimal split between different system design parameters, namely, the number of subchannels, the number of users that feedback for any subchannel and the number of bits used for CSI precision, which was found to be hierarchical in the same order. To show that the behavior of the optimal solution is not a special case, true only when sub-channel gain is defined as in Eqn (1); we simulated system performance for different definitions of sub-channel gain, by replacing the min in (1) by max and mean for both opportunistic [15] and sequential feedback schemes in [14]. In all cases, we found that the same intuition for the optimal split of feedback bits holds. This leads

3000

300

γavg=1

γavg=1

γavg=10

γavg=10 250

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R*sum

g*

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(Tc)1/2 asymptote

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Fig. 4.

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∗ Dependency of Rsum on Tc for different values of γ ¯.

R EFERENCES

γavg=100 γavg=10000

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to our conjecture that irrespective of the feedback scheme employed, the results derived in this paper are applicable.

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Fig. 3.

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Fig. 2.

M=256 K=200 Rf=40

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