Resource allocation for maximizing outage throughput in OFDMA systems with finite-rate feedback

Orthogonal frequency division multiplexing (OFDM) is a promising technique for the next-generation wireless communication systems. OFDM divides the frequency-selective fading channel into N orthogonal flat-fading 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 rate-distortion theory [2]. Thus, the maximum throughput of the OFDMA systems will be achieved, if the feedback CSI is optimized in terms of the rate-distortion 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 inter-arrival 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 closed-form 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 non-outage 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 long-term 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 delay-sensitive 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 rate-distortion limit is more than twice of the throughput achieved under the threshold-based 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.

We consider a one-cell 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

(1)

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 frequency-selective 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 rate-distortion 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 rate-distortion 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:

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

(2)

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.

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 full-searching algorithm. This full-searching 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 rate-distortion limit using the proposed suboptimal algorithm with the threshold-based quantization method considered in [4, 22]. In the threshold-based 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 T₀ = 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

(15)

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 water-filling method [23]. This method gives the maximum throughput when [23].

Figure 3 shows the rate-distortion 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 rate-distortion limit is about 50-80% of the threshold-based 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 threshold-based 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 rate-distortion limit is more than twice of the threshold-based 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 rate-distortion 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.

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 rate-distortion 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 low-complexity 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 rate-distortion limit outperforms that with the threshold-based quantization method, and with limited feedback rate, the system performance can be close to that with perfect CSI.

First, we show that the exponential distribution maximizes the entropy over all distributions with non-negative support.

Lemma 1. Let the non-negative 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 non-negative random variable x satisfying . Let y be an exponentially distributed random variable with the Probability Density Function g(y) = exp (-y/m)/m. Then,

(A.1)

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 one-dimensional 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

(A.2)

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

(A.3)

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.

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 non-negative 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.