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Resource allocation for maximizing outage throughput in OFDMA systems with finiterate feedback
EURASIP Journal on Wireless Communications and Networking volume 2011, Article number: 56 (2011)
Abstract
Previous works on orthogonal frequency division multiple access (OFDMA) systems with quantized channel state information (CSI) were mainly based on suboptimal quantization methods. In this paper, we consider the performance limit of OFDMA systems with quantized CSI over independent Rayleigh fading channels using the ratedistortion theory. First, we establish a lower bound on the capacity of the feedback channel and build the test channel that achieves this lower bound. Then, with the derived test channel, we characterize the system performance with the outage throughput and formulate the outage throughput maximization problem with quantized channel state information (CSI). To solve this problem in low complexity, we develop a suboptimal algorithm that performs resource allocation in two steps: subcarrier allocation and power allocation. Using this approach, we can numerically evaluate the outage throughput in terms of feedback rate. Numerical results show that this suboptimal algorithm can provide a near optimal performance (with a performance loss of less than 5%) and the outage throughput with a limited feedback rate can be close to that with perfect CSI.
1 Introduction
Orthogonal frequency division multiplexing (OFDM) is a promising technique for the nextgeneration wireless communication systems. OFDM divides the frequencyselective fading channel into N orthogonal flatfading subcarriers to provide a high data rate. Orthogonal frequency division multiple access (OFDMA) adds multiple access to OFDM by allowing a number of users to use different subcarriers. One aim of the OFDMA technique is to find an optimal allocation of resources to users using channel adaptive techniques [1]. It implies that the channel state information (CSI) of users should be known to the base station (BS). However, in the frequency division duplexing (FDD) OFDMA systems, the BS only obtains the quantized CSI. For downlink transmissions, the BS requires the CSI with the minimum distortion to maximize the transmission rate; for the feedback channel, given a feedback rate constraint, the minimum distortion of the downlink CSI can be characterized by the ratedistortion theory [2]. Thus, the maximum throughput of the OFDMA systems will be achieved, if the feedback CSI is optimized in terms of the ratedistortion function (RDF) [2]. However, existing research works, such as [3–5], mainly focused on simple but suboptimal quantization methods, and did not shown the best performance of OFDMA systems.
In this paper, we focus on the performance limit of the OFDMA system with finite feedback rate. As typically done in the literature (e.g., [3–5]), we assume independent Rayleigh downlink channels over subcarriers, i.e., the channel power gain H^{2} on each subcarrier is exponentially distributed. We use the RDF to characterize the lower bound on the required feedback channel's capacity for a given mean quantization error under OFDMA downlink channels [2]. The author in [6] investigated the optimal encoding of the exponential interarrival time of a Poisson process. The RDF of the exponentially distributed time was evaluated with a distortion equal to the absolute error between the quantized arrival time and the actual arrival time. This approach, however, does not result in closedform results. Here, we consider the alternative approach where the quantized channel gain is less than or equal to the actual channel gain. This constraint applies to the situation in which the truncation quantization method is employed, and enables us to derive the analytical expression for RDF. Once the relation between the distortion (mean magnitude error associated with channel quantization) and rate (capacity of feedback channel) has been established, the resource allocation problem with quantized CSI can be formulated under feedback capacity constraints.
We introduce the outage throughput as the performance measure for the downlink throughput. Here, we define the outage throughput as the maximum expected rate of information delivered to users in nonoutage states, where the data rate is lower than the channel capacity. Clearly, the definition of outage throughput is different from that of the ergodic throughput, which is defined as a longterm achievable throughput averaged over all fading blocks [7]. The performance measure of the ergodic throughput is suitable for applications insensitive to delay, but not suitable for delaysensitive applications. For the latter ones, the outage probability has been considered as a valid performance measure [8–10]. It is desirable to minimize the outage probability for the given quantized CSI. However, low outage probability results in low throughput. There exists a tradeoff between minimizing the outage probability and maximizing the throughput. Outage throughput, which can be regarded as a measure of the expected reliably decodable rate at the user side, provides this tradeoff between transmission rate and outage probability [11, 12].
We investigate the resource allocation problem to maximize the outage throughput. We show that the algorithm that achieves the optimum could have an exponential time complexity. Thus, to reduce the complexity, we propose a suboptimal algorithm that separates the resource allocation into two steps: subcarrier allocation and power allocation. This suboptimal approach has a linear complexity in the number of users and subcarriers and achieves optimality gaps of less than 5%. With the suboptimal approach, the achieved throughput in the ratedistortion limit is more than twice of the throughput achieved under the thresholdbased quantization approach, when the feedback rate is low.
Notations: Bold letters denote vectors and matrices, and B^{T} denotes the transpose of B. Also, E[·] denotes the statistical expectation, and in particular, E_{ X }[·] denotes that with respect to X.
1.1 Overview
We continue the introduction with a brief review of related work in Section 1.2. Section 2 outlines the downlink channel model and derives the RDF for the downlink CSI. Section 3 presents the expression of outage throughput, formulates the outage throughput maximization problem with quantized CSI, and proposes the resource allocation algorithm that achieves a suboptimal solution. Numerical results are given in Section 4 to illustrate the performance of the outage throughput using the proposed algorithm. Conclusions are drawn in Section 5.
1.2 Related work
In practice, it is difficult for the transmitter to obtain perfect CSI due to feedback delay (for both FDD and time division duplexing (TDD)), channel estimation error (for both FDD and TDD), and quantization error (for FDD) [13]. The impact of imperfect CSI for OFDM systems has been an active research area in recent years. The effect of feedback delay was addressed in [14]. The author considered a minimum square error channel prediction scheme to overcome the detrimental effect of feedback delay and proposed resource allocation algorithms to maximize the downlink throughput. The works in [15–17] focused on the imperfect CSI resulting from channel estimation error and proposed power loading algorithms for the single user OFDM system. Resource allocation with quantized CSI was investigated in [3–5]. The authors in [3] assumed uniform power distribution over subcarriers and derived closedform expressions for the downlink throughput. In [4, 5], the design parameters related to imperfect CSI, such as quantization levels and the feedback period, were optimized to reduce the feedback overhead with a guaranteed system performance for OFDMA systems. However, most previous research works, such as [3–5], were based on suboptimal quantization method. Recently, the authors in [18] proposed OFDMA throughput maximization algorithm under the assumption that quantization for CSI feedback is optimized in terms of the ratedistortion theory point of view. In [18], the feedback of CSI is assumed to be the Gaussian channel gain H. However, in resource allocation for OFDMA systems, we only need the real value of H^{2} instead of the complex value of H. Thus, it could be more efficient to feed back H^{2} than H to minimize the CSI feedback rate. In this paper, we consider the quantization of H^{2}.
The aforementioned research works in [3–5, 14] take the ergodic throughput as the performance measure. For applications insensitive to delay, the ergodic throughput is a suitable performance measure [7]. On the other hand, the outage throughput is more appropriate to characterize the downlink throughput for realtime applications [8]. In this work, we discuss the outage throughput maximization with imperfect CSI.
2 System model
We consider a onecell OFDMA system with N subcarriers (or orthogonal channels) that will be shared by K users. The system model is depicted in Figure 1. We assume that each subcarrier is assigned to one user exclusively and the system employs FDD. It is assumed that each user perfectly estimates the CSI of the downlink channel (from the BS to the user), which is simply referred to as downlink CSI in this paper. Each user quantizes his/her estimated downlink CSI and sends it (actually an index of quantized downlink CSI) to the BS through a dedicated feedback channel. The BS receives the downlink CSI from all users and utilizes this information to assign subcarriers to users and adjust transmit power for each subcarrier.
Denote by H_{ k, n } the channel gain of user k at subcarrier n. Throughout the paper, we assume that the channel gains are independent over subcarriers and the probability density function of the channel power gain α_{ k, n } = H_{ k, n }^{2} is given by
where u(·) denotes the unit step function, and λ_{ k, n } = E[α_{ k, n }]. Here, the channel power gain α_{ k, n } is exponentially distributed, α_{ k, n } ~ exp(λ_{ k, n }), where exp(m) denotes the exponential distribution with mean m. Due to the assumption of independent channels, we may not be able to take the spatial correlation of frequencyselective fading channels. However, if subcarriers are discontinuously allocated to a user, the spatial correlation can be ignored.
Now, we consider the quantization of downlink CSI and determine the capacity of the feedback channel required to deliver the quantized CSI using the ratedistortion theory. From this, we can characterize the minimum distortion of the quantized CSI for a given capacity of the feedback channel.
User k describes his/her knowledge of downlink CSI A_{ k } = (α_{k,1}, ..., α_{ k, N } )^{T} by an index I_{ k } and feeds the index I_{ k } back to the BS. The BS reproduces from the index I_{ k }, where is the quantized description of α_{ k, n }. The quantized power gain is assumed to be not greater than the actual power gain .
To measure the accuracy of the quantized CSI, we introduce the distortion measure function with the magnitude error criterion:
Then, we can define the information RDF of A_{ k } as
where D_{ k } denotes the upper bound of the mean quantization error and I(·;·)denotes the mutual information. By the ratedistortion theory [2], this RDF gives a minimum number of bits for the index I_{ k } that can describe the channel power gain A_{ k } without exceeding the mean quantization error D_{ k }. The RDF of A_{ k } is given by the following theorem:
Theorem 1. Let A_{ k } = (α_{k,1}, ..., α_{ k, N } )^{T} be a vector source with uncorrelated components that are exponentially distributed given by Equation 1. Then,

1.
the RDF of A_{ k } is given by
where θ_{ k } is chosen such that

2.
the test channel that achieves the RDF is given by
where Z_{ k } = (z_{k,1}, ..., z_{ k, N }) is independent of and has uncorrelated components with Z_{ k, n } ~ exp (min{θ_{ k }, λ_{ k, n }}).
Proof: See Appendix Appendix 1.
Remark 1. In downlink throughput maximization with imperfect CSI, we require the probability density function of the actual power gain conditioned on the quantized power gain. By the second part of Theorem 1, for a given , the probability density function of α_{ k, n } is
where v_{ k, n } = min {θ_{ k }, λ_{ k, n }}. Here, the variable v_{ k, n } can be regarded as the mean quantization error for the channel power gain α_{ k, n }.
Remark 2. There are two special cases. By setting θ_{ k } = 0, from Theorem 1, we have D_{ k } = 0, R_{ k }(D_{ k }) = +∞ and z_{ k, n } = 0. In this case, the CSI is perfectly known to the BS. On the other hand, by setting θ_{ k } = +∞, we have and R_{ k }(D_{ k }) = 0, which implies that no CSI is fed back to the BS.
3 Outage throughput maximization with quantized CSI
3.1 Problem formulation
For a given capacity of the feedback channel, we have characterized the distortion in Section 2. With the quantized downlink CSI, the resource allocation can be carried out for a given performance measure. From this, we can formulate the resource allocation with capacity constraints of the feedback channels. Toward this end, in this subsection, we introduce the outage throughput as the performance measure.
Given the quantized CSI, the outage probability on the nth subcarrier to the kth user is defined as
where γ_{ n } is the input signal error ratio (SNR) of the nth subcarrier and R is the transmission rate. From Equation 3, the maximum transmission rate R that can maintain the outage probability ε is
where . Thus, the expected rate of information successfully decoded at user k on subcarrier n is
It is possible to maximize by choosing ε,
Here, the throughput is termed as the outage throughput. Setting , we obtain
where . Substituting Equation 2 yields
The optimal x that maximizes is given by the following theorem:
Theorem 2. There exists a unique globally optimal x that maximizes in Equation 6, which is given by
where W(x) is the LambertW function, which is the solution to the equation W(x)e ^{W(x)}= x.
Proof See Appendix Appendix B.
Thus, for each given transmit power γ_{ n }, quantized power gain and quantization error v_{ k, n }, we can evaluate the outage throughput of the kth user on the nth subcarrier in Equation 5 by Theorem 2. The overall outage throughput conditioned on the quantized CSI is represented as
where ρ_{ k, n } is the subcarrier allocation indicator: if the nth subcarrier is assigned to the kth user, then ρ_{ k, n } = 1; otherwise ρ_{ k, n } = 0. Here, the BS decides γ_{ n } and ρ_{ k, n } with the knowledge of quantized CSI . To emphasize this, we denote the input SNR and the allocation indicator as functions of by and , respectively. The average outage throughput is thus given by
Now, we can formulate the outage throughput maximization under feedback capacity constraints:
where the first constraint is the feedback capacity constraint, the second constraint ensures that each subcarrier is assigned to one user exclusively, and the third constraint is for total transmit power, denoted by γ_{ T }.
By Theorem 1, for each R_{ k } (D_{ k }), there exists a test channel that achieves R_{ k } (D_{ k }). Thus, maximizing the downlink throughput under feedback capacity constraints is equivalent to maximizing the downlink throughput under the corresponding test channel. It can also be observed that to maximize T°, we can maximize the conditional outage throughput for each realization of under the conditional probability density function given in Equation 2. That is,
To make the problem in Equation 10 tractable, we consider a suboptimal solution by breaking the problem into two steps: subcarrier allocation and power allocation. In the first step, subcarriers are assigned to users under the assumption that the transmit power is identical over all subcarriers; in the second step, power is loaded on the subcarriers assigned in the first step.
3.2 Subcarrier allocation
Under the assumption of γ_{ n } = γ_{ T }/N, the optimization problem in Equation 10 reduces to
It implies that the subcarriers should be assigned based on the following criterion:
The above criterion requires to evaluate KN values of the rate given in Equation 5. However, we can simplify this criterion in the case where on subcarrier n, the mean quantization error v_{ k, n } is identical among all users k. We state the following theorem:
Theorem 3. For any given v_{ k, n, }, the throughput defined Equation 5 is monotonically increasing in if in Equation 5 is monotonically increasing in .
Proof By assumption, we have for . Thus,
It follows that
It can be shown that given in Equation 6 is monotonically increasing in . Thus, by Theorem 3, in the case of v_{ k', n } = v_{ k, n } for k ≠ k', the subcarrier allocation reduces to
When a tie occurs, we can select users in random fashion.
3.3 Power allocation
Denote by k_{ n } the selected user on the nth subcarrier, i.e., k_{ n } = arg max _{ k } ρ_{ k, n }. Given the subcarrier allocation, the problem 10 becomes
From the Equations 6 and 7, we can observe that is not concave in γ_{ n }. Hence, the problem 12 is not a convex optimization problem. However, we can employ a dual approach to obtain a suboptimal solution.
The dual problem is
where
where μ is the Lagrangian multiplier of the first constraint in Equation 12. Given μ, the optimal power allocation on the nth subcarrier is
We can use a derivativefree line search method, such as the golden section method to find the γ_{ n } for a given Lagrangian multiplier μ [19].
The Lagrangian dual problem 13 has been shown to be a convex optimization problem in μ [20]. Thus, we can use the bisection method to find the optimal global multiplier μ [19]. The bisection method requires to evaluate the first derivative of g(μ) with respect to μ. Although g(μ) is not continuously differentiable due to the max function, we can use the subgradient instead [21],
where γ_{ n } is obtained from Equation 14.
Using the dual optimization approach, it is possible that the final power allocation may not satisfy . We can multiply the final power allocation on each subcarrier by a constant to arrive a feasible solution.
Complexity: in the first step, assigning subcarriers requires to find the maximum among K users for each subcarrier n, and thereby, the complexity of subcarrier allocation is O(KN). In the power allocation, in each iteration for μ in Equation 13, we need to compute N power allocation values given by Equation 14. Each power allocation value requires a search routine, which is assumed to converge within I_{ γ } iterations. Assuming that I_{ μ } iterations are required to find the optimal μ, the overall complexity of the suboptimal algorithm is O(KN + I_{ μ } I_{ γ }N). Ignoring the constants I_{ μ } and I_{ γ }, the complexity is just O(KN).
4 Numerical results
We present several numerical results to demonstrate the performance of OFDMA systems using the proposed algorithms. We assume an OFDMA system with the average channel power gain E[α_{ k, n }] = 1. Furthermore, the feedback capacities of all users are assumed to be identical. That is, C_{ K } = C_{ K' }for all k ≠ k'. By Theorem 1, it implies that the mean quantization errors of all users on each subcarrier n are identical, v_{ k, n } = v_{ k', n }
First, for the problem 10, we compare the proposed suboptimal algorithm with a fullsearching algorithm. This fullsearching algorithm considers all possible subcarrier allocations, and for each subcarrier allocation, it assigns transmit power based on the dual optimization approach as proposed in Section 3.3 without projecting the final power allocation back to the feasible region. Thus, this algorithm gives an upper bound on the optimal solution to the problem in 10 [20].
Figure 2 plots both the suboptimal results and the upper bound of the optimal results for an OFDMA system with N = 8 subcarriers and K = 2 users. In Figure 2, as the capacity of the feedback channel increases from C_{ k } = 1.6 bps/Hz to C_{ k } = 64 bps/Hz, the performance gap between the suboptimum and the upper bound of the optimum gets larger. However, in both scenarios, the difference between the optimum and suboptimum is within 5%.
Next, we consider an OFDMA system with N = 1,024 subcarriers and K = 8 users. We compare the outage throughput achieved in the ratedistortion limit using the proposed suboptimal algorithm with the thresholdbased quantization method considered in [4, 22]. In the thresholdbased quantization method, the channel power gain α_{ k, n } on each subcarrier n of each user k is quantized in intervals with thresholds T_{ q } with q = 0, ..., W, where T_{0} = 0, T_{ W } = + ∞, and N_{ Q } is the number of quantization bits per subcarrier. Here, we assume that all users have identical N_{ Q } on all subcarriers. The thresholds T_{ q } for q = 1, ..., W  1 are determined by partitioning the probability density function of α_{ k, n } into W equiprobable intervals. It implies that T_{ q } = F^{1}(q/W), where F(·)is the cumulative density function (cdf) of α_{ k, n }. The decoded channel power gain at the BS side is assumed to be
Then, the BS assigns subcarriers and transmit power with the knowledge of the power gain : the user with the highest power gain is chosen on each subcarrier, and the transmit power on each subcarrier is determined using the waterfilling method [23]. This method gives the maximum throughput when [23].
Figure 3 shows the ratedistortion curves for the two schemes. In this figure, for a wide range of the average distortion, the required capacity of the feedback channel in the ratedistortion limit is about 5080% of the thresholdbased quantization scheme. However, when the capacity of the feedback channel is zero (no CSI is fed back to the BS), both schemes result in the average distortion of NE[α_{ k, n } ] = 1,024.
Figure 4 depicts the outage throughput in terms of the capacity of the feedback channel. When no CSI is available at the BS, according to Sections 3.2 and 3.3, the proposed scheme tends to assign subcarriers randomly to users and allocate equal transmit power γ_{ n } on each subcarrier n. In this case, the outage throughput is N max _{ x } log(1+xγ_{ T }/N)Pr(α_{ k, n } ≥ x). For the thresholdbased method, since the decoded power gain is equal to the knowledge of the lower bound on the actual power gain as given by Equation 15, the BS can only set . In this case, no signal is transmitted on subcarriers. At C_{ k } < 400 bps/Hz, the achieved outage throughput in the ratedistortion limit is more than twice of the thresholdbased method. The difference between the two schemes decreases for larger capacity of the feedback channel. When the feedback channel's capacity of each user reaches 6,144 bps/Hz, the throughput is saturated regardless of any type of the schemes (could happen when the perfect CSI is available at the BS). It can also be noted that at γ_{ T }/N = 30 dB and C_{ k } = 1,024 bps/Hz, the performance gap between the outage throughput in the ratedistortion limit and that in the perfect CSI case is within 6%. Thus, it implies that with limited feedback rate, the system performance can be close to that of the perfect CSI one.
5 Conclusions
In this paper, we investigated the outage throughput maximization for an OFDMA system with finite feedback rate over independent Rayleigh fading channels. First, we derived the RDF for the downlink CSI. This RDF gives a lower bound on the capacity of the feedback channel according to the ratedistortion theory. Meanwhile, we obtained the test channel that achieves this RDF. The derived test channel enabled us to formulate the resource allocation problem that maximizes the outage throughput with capacity constraints of feedback channels. For this problem, we proposed a lowcomplexity suboptimal algorithm. This algorithm divides the problem into two subproblems, namely subcarrier and power allocation problems. Through numerical results, we found that the proposed suboptimal algorithm has performance close to the optimum. We also observed that the outage throughput in the ratedistortion limit outperforms that with the thresholdbased quantization method, and with limited feedback rate, the system performance can be close to that with perfect CSI.
Appendix A Proof of Theorem 1
First, we show that the exponential distribution maximizes the entropy over all distributions with nonnegative support.
Lemma 1. Let the nonnegative random variable x have the mean E[x] = m. Then, the differential entropy of x is upper bounded by, and the equality is achieved iff x is exponentially distributed, x ~exp(m).
Proof Let f(x) be the probability density function of a nonnegative random variable x satisfying . Let y be an exponentially distributed random variable with the Probability Density Function g(y) = exp (y/m)/m. Then,
where (Appendix A.1a) follows from , and (Appendix A.1b) follows from the concavity of the function log.
Then, we derive the RDF for an onedimensional exponentially distributed source x ~ exp(m).
Lemma 2. Define the RDF of an exponentially distributed source x ~ exp(m) as
where is the quantized description of x. Then, the RDF is given by
and the test channel that achieves this RDF is
where z is independent of with z ~ exp(min{D, m}).
Proof In the case D > m , the quantizer need not transmit any information since the the decoded information can be chosen as
This ensures that the constraints and are satisfied. In this case, and z ~ exp(m). Henceforth, we assume 0 ≤ D ≤ m. We observe that
where (Appendix A.2a) follows from the fact that conditioning reduces entropy, and (Appendix A.2b) follows from Lemma 1. The equality in (Appendix A.2a) is achieved iff independent of , and the equality in (Appendix A.2b) is achieved iff z ~ exp(D).
Now, we are able to prove Theorem 1.
Proof [Proof of Theorem 1] We have
where , (Appendix A.3a) follows from the fact that the components of A_{ k } are uncorrelated, (Appendix A.3b) from the fact that conditioning reduces entropy, and (Appendix A.3c) follows from Lemma 2. The equality (Appendix A.3c) is achieved iff with z_{ k, n } ~ exp(min{λ_{ k, n }, D_{ k, n }}) is independent of , and the equality in (Appendix A.3b) is achieved iff . From this, it also implies that Z_{ k } = (z_{k,1}, ..., z_{ k, N })^{T} has uncorrelated components.
The problem of finding the RDF of A_{ k } now reduces to
The Lagrangian of the problem is
where μ is the Lagrangian multiplier. We can find the optimal D_{ k, n } that minimizes L by differentiating L with respect to D_{ k, n },
Thus, we conclude the optimal D_{ k, n } is
where θ = log e/μ.. Recalling the constraint ∑_{ n } D_{ k, n } = D_{ k }, we obtain the result of the Theorem 1.
Appendix B Proof of Theorem 2
Proof First, we show that ln in Equation 6 is concave in x ∈ (0, + ∞). From Equation 6, we express ln as
Since log(1 + xγ_{ n }) is concave in x and log(1 + xγ_{ n }) > 0 for x > 0, γ_{ n } ≥ 0, lnlog(1 + xγ_{ n }) is concave in x for i > 0, γ_{ n } ≥ 0 [[20], p.86]. Since nonnegative weighted sum and pointwise infimum preserve the concavity [[20], Section 3.2], ln is concave in x.
Also, note that in Equation 6 satisfies , and . Thus, there exists a globally unique x that maximizes .
Differentiating with respect to x for and setting equal to zero, we have
That is,
For , is maximized when . Thus, we have the solution in 7.
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Acknowledgements
This work has been supported by the China Postdoctoral Science Foundation and the China National 973 project under the grant No. 2009CB320403.
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Keywords
 Orthogonal frequency division multiple access (OFDMA)
 limited feedback
 quantized channel information
 ratedistortion
 resource allocation
 twostep suboptimal algorithm