Resource Allocation in MU-OFDM Cognitive Radio Systems with Partial Channel State Information

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Introduction
In performance analyses of wireless communication systems, it is often assumed that perfect channel state information (CSI) is available at the transmitter. This assumption is often not valid due to channel estimation errors and/or feedback delays. To ensure that the system can satisfy target quality of service (QoS) requirements, a careful analysis which takes into account imperfect CSI is required [1].
Cognitive radio (CR) is a relatively new concept for improving the overall utilization of spectrum bands by allowing unlicensed secondary users (also referred to as CR users or CRUs) to access those frequency bands which are not currently being used by licensed primary users (PUs) in a given geographical area. In order to avoid causing unacceptable levels of interference to PUs, CRUs need to sense the radio environment and rapidly adapt their transmission parameter values [2][3][4][5][6].
Orthogonal frequency division multiplexing (OFDM) is a modulation scheme which is attractive for use in a CR system due to its flexibility in allocating resources among CRUs. The problem of optimal allocation of subcarriers, bits, and transmit powers among users in a multiuser-(MU-) OFDM system is a complex combinatorial optimization problem. In order to reduce the computational complexity, the problem is solved in two steps by many suboptimal algorithms [7][8][9][10]: (1) determine the allocation of subcarriers to users and (2) determine the allocation of bits and transmit powers to subcarriers. Resource allocation algorithms for MU-OFDM systems have been studied in [11][12][13][14]. These algorithms are designed for non-CR MU-OFDM systems in which there are no PUs.
In an MU-OFDM CR system, mutual interference between PUs and CRUs needs to be considered. The problem of optimal allocation of subcarriers, bits, and transmit powers among users in an MU-OFDM CR system is more complex. It is commonly assumed that perfect CSI is available at the transmitter [15,16]. As noted earlier, this assumption is often not reasonable. In this paper, we investigate the problem of resource allocation in an MU-OFDM CR system when only partial CSI is available at the CR base station (CRBS). We assume that CSI is acquired perfectly at the CRUs and fed back to the CRBS with a delay of τ d seconds. The channel experiences frequency-selective 2 EURASIP Journal on Wireless Communications and Networking fading. The objective is to maximize the total bit rate while satisfying BER, transmit power, and mutual interference constraints.
The rest of the paper is organized as follows. The system model is described in Section 2. Based on the system model, a constrained multiuser resource allocation problem is formulated in Section 3. A suboptimal algorithm for solving the problem is discussed in Section 4. Simulation results are presented in Section 5 and the main findings are summarized in Section 6.

System Model
We consider the problem of allocating resources on the downlink of an MU-OFDM CR system with one base station (BS) serving one PU and K CRUs. The basic system model is the same as that described in [15] and is summarized here for the convenience of the reader.
The PU channel is W p Hz wide and the bandwidth of each OFDM subchannel is W s Hz. On either side of the PU channel, there are N/2 OFDM subchannels. The BS has only partial CSI and allocates subcarriers, transmit powers, and bits to the CRUs once every OFDM symbol period. The channel gain of each subcarrier is assumed to be constant during an OFDM symbol duration.
Suppose that P n is the transmit power allocated on subcarrier n and g n is the channel gain of subcarrier n from the BS to the PU. The resulting interference power spilling into the PU channel is given by where represents the interference factor for subcarrier n, d n is the spectral distance between the center frequency of subcarrier n and that of the PU channel, and Φ( f ) denotes the normalized baseband power spectral density (PSD) of each subcarrier.
Let h nk be the channel gain of subcarrier n from the BS to CRU k, and let Φ RR ( f ) be the baseband PSD of the PU signal. The interference power to CRU k on subcarrier n is given by Let P nk denote the transmit power allocated to CRU k on subcarrier n. For QAM modulation, an approximation for the BER on subcarrier n of CRU k is [13] where N 0 is the one-sided noise PSD and S nk is given by (3). Rearranging (4), the maximum number of bits per OFDM symbol period that can be transmitted on this subcarrier is given by where Γ − ln(5BER[n])/1.5 and · denotes the floor function.
Equation (4) shows the relationship between the transmit power and the number of bits loaded on the subcarrier for a given BER requirement when perfect CSI is available at the transmitter. We now establish an analogous relationship when only partial CSI is available.
The imperfect CSI that is available to the BS is modeled as follows. We assume that perfect CSI is available at the receiver. The channel gain, hnk, for subcarrier n and CRU k is the outcome of an independent complex Gaussian random variable, that is, H nk ∼ CN (0, σ 2 h ) [17], corresponding to Rayleigh fading. For clarity, we will denote random variables and their outcomes by uppercase and lowercase letters, respectively.
For notational simplicity, we will use h to denote an arbitrary channel gain. The BS receives the CSI after a feedback delay τ d = dT s , where T s is the OFDM symbol duration. We assume that the noise on the feedback link is negligible. Suppose that h f is the channel gain information that is received at the BS, then h f (t) = h(t − τ d ). From [18], the correlation between H and H f is given by where the correlation coefficient, ρ, is given by In (6) and (7), J 0 (·) denotes the zeroth-order Bessel function of the first kind, f d is the Doppler frequency, E{·} is the expectation operator, and H H f denotes the complex conjugate of H f .
The minimum mean square error (MMSE) estimator of H based on H f = h f is given by [19] From (6), the actual channel gain can be written as [20] follows: EURASIP Journal on Wireless Communications and Networking 3

Formulation of the Multiuser Resource Allocation Problem
Based on the partial CSI available at the BS, we wish to maximize the total CRU transmission rate while maintaining a target BER performance on each subcarrier and satisfying PU interference and total BS CRU transmit power constraints. Let BER[n] denote the average BER on subcarrier n, and let BER 0 represent the prescribed target BER. The optimization problem can be expressed as follows: subject to where P total is the total power budget for all CRUs, I total is the maximum interference power that can be tolerated by the PU, and a nk ∈ {0, 1} is a subcarrier assignment indicator, that is, a nk = 1 if and only if subcarrier n is allocated to CRU k. The term λ k represents the nominal bit rate weight (NBRW) for CRU k, and denotes the total bit rate achieved by CRU k. Constraint (11) ensures that the average BER for each subcarrier is below the given BER target. Constraint (12) states that the total power allocated to all CRUs cannot exceed P total , while constraint (14) ensures that the interference power to the PU is maintained below an acceptable level I total . Constraint (15) results from the assumption that each subcarrier can be assigned to at most one CRU. Constraint (17) ensures that the bit rate achieved by a CRU satisfies a proportional fairness condition. Based on (9), we calculate the average of the righthand side (RHS) of (4), treating h nk as an outcome of an independent complex Gaussian variable. For an arbitrary vector α ∼ CN (μ, Σ), we have [21] the following: where I denotes the identity matrix. Applying (19) to (4), we obtain where From (20), an explicit relationship between minimum transmit power and number of transmitted bits cannot be easily derived. However, since BER[n] in (20) is a monotonically decreasing function of P nk , we obtain the minimum power requirement while satisfying the constraint in (11) by setting BER[n] = BER 0 .
We now derive a simpler, albeit approximate, relationship between the required transmit power, BER, and the number of loaded bits.

Resource Allocation with Partial Csi
Note that the joint subcarrier, bit, and power allocation problem in (10)- (17) belongs to the mixed integer nonlinear programming (MINP) class [23]. For brevity, we use the term "bit allocation" to denote both bit and power allocation.
Since the optimization problem in (10)- (17) is generally computationally complex, we first use a suboptimal algorithm, which is based on a greedy approach, to solve the subcarrier allocation problem in Section 4.1. After subcarriers are allocated to CRUs, we apply a memetic algorithm (MA) to solve the bit allocation problem in Section 4.2.

Subcarrier Allocation.
From (17), it can be seen that the subcarrier allocation depends not only on the channel gains, but also on the number of bits allocated to each subcarrier. Moreover, allocation of subcarriers close to the PU band should be avoided in order to reduce the interference power to the PU to a tolerable level. Therefore, we use a threshold scheme to select subcarriers for CRUs. Suppose that N subcarriers are available for allocating to CRUs. We assume equal transmit power for each subcarrier. Let If a subcarrier is assigned to CRU k, the maximum number of bits which can be loaded on the subcarrier is given by b k = min log 2 1 + Ψ k P total N , log 2 1 + Ψ k I total NIF , ∀k = 1, 2, . . . , K.
Using (26)-(28), we can determine the number of subcarriers assigned to each CRU as follows. Let m k be the number of subcarriers allocated to CRU k. Assuming that the same number of bits is loaded on every subcarrier assigned to a given CRU, the objective in (10) is equivalent to finding a set of {m 1 , m 2 , . . . , m K } subcarriers to maximize subject to m 1 b 1 : m 2 b 2 : · · · : m K b K = λ 1 : λ 2 : · · · : λ K , where P is the total transmit power allocated to all subcarriers and I is the total interference power experienced by the PU due to CRU signals. The subcarrier allocation problem Algorithm: SA for n = 1 to number of subcarriers do find k * ∈ {1, 2, . . . , K} which maximizes (|H nk | 2 + σ 2 )/(Γ (N 0 W s + S nk )); Using (25), calculate the number of bits loaded on Subcarrier n as b nk * with P nk * = P total /N; initialize N to 0; if b nk * > 2then subcarrier n is available; increment N by 1; else subcarrier n is not available; end if end for For each k ∈ {1, 2, . . . , K}, initialize the number, m k , of subcarriers allocated to CRU k to 0 calculate b k using (28); for n = 1 to N do find the value, η, of k ∈ {1, 2, . . . , K} which minimizes m k b k /λ k ; allocate subcarrier n to CRU η; increment m η by one. in (29)-(32) can be solved using the SA algorithm proposed in [24]. Note that we need to make use of (24) in the SA algorithm if only partial CSI is available. A pseudocode listing for the SA algorithm is shown in Pseudocode 1. The algorithm has a relatively low computational complexity O(KN). After subcarriers are allocated to CRUs, we then determine the number, b n , of bits allocated to subcarrier n.

Bit Allocation.
Memetic algorithm (MAs) are evolutionary algorithms which have been shown to be more efficient than standard genetic algorithms (GAs) for many combinatorial optimization problems [25][26][27]. Using (24), the bit allocation problem can be solved using the MA algorithm proposed in [24]. It should be noted that the chosen genetic operators and local search methods greatly influence the performance of MAs. The selection of these parameters for the given optimization problem is based on the results in [24]. A pseudocode listing of the proposed memetic algorithm is shown in Pseudocode 2.
Let x i be the chromosome of member i in a population, expressed as ∀i = 1, 2, . . . , pop size, (33) where pop size denotes the population size. A brief description of the MA algorithm in [24] is now provided.
(1) The selectForV ariation function selects a set, S = {s 1 , s 2 , . . . , s pop size }, of chromosomes from P in a roulette wheel fashion, that is, selection with replacement.
(2) Crossover: suppose that S = {y 1 , y 2 , . . . , y pop size }. Let P cross denote the crossover probability, and let u i , i = 1, 2, . . . , pop size denote the outcome of an independent random variable which is uniformly distributed in [0, 1], then y i is selected as a candidate for crossover if and only if u i ≤ P cross , i = 1, 2, . . . , pop size. Suppose that we have n c such candidates, we then form n c /2 disjoint pairs of candidates (parents).
(3) Mutation: let P mutation denote the mutation probability. For each chromosome in S, we generate u i , i = 1, 2, . . . , N, where u i denotes the outcome of an independent random variable which is uniformly distributed in [0, 1]. Then for each component i for which u i ≤ P mutation , we substitute the value with a randomly chosen admissible value.
(4) Selection of surviving chromosomes: we select the pop size chromosomes of parents and offsprings with the best fitness values as input for the next generation.

Results
In this section, performance results for the proposed algorithm described in Section 4 are presented. In the simulation, the parameters of the MA algorithm were chosen as follows: population size, pop size = 40; number of generations = 20; crossover probability, P cross = 0.05; mutation probability, P mutation = 0.7. We consider a system with one PU and K = 4 CRUs. The total available bandwidth for CRUs is 5 MHz and supports 16 subcarriers with W s = 0.3125 MHz. We assume that W p = W s and an OFDM symbol duration, T s of 4 μs. In order to understand the impact of the fair bit rate constraint in (17) on the total bit rate, three cases of user bit rate requirements with λ = [1 1 1 1], [1 1 1 4], [1 1 1 8] were considered. In addition, three cases of partial CSI with ρ = 1, 0.9 and 0.7 were studied. It is assumed that the subcarrier gains h nk and g k , for n ∈ {1, 2, . . . , N}, k ∈ {1, 2, . . . , K} are outcomes of independent identically distributed (i.i.d.) Rayleigh-distributed random variables (rvs) with mean square value E(|H nk | 2 ) = E(|G k | 2 ) = 1. The additive white Gaussian noise (AWGN) PSD, N 0 , was set to 10 −8 W/Hz. The PSD, Φ RR ( f ), of the PU signal was assumed to be that of an elliptically filtered white noise process. The total CRU bit rate, R s , results were obtained by averaging over 10,000 channel realizations. The 95% confidence intervals for the simulated R s results are within ±1% of the average values shown. Figure 1 shows the average total bit rate, R s , as a function of the total CRU transmit power, P total , for ρ = 0.7, 0.9, and 1 with λ = [1 1 1 1], I total = 0.02 W, and a PU transmit power, P m , of 5 W. As expected, the average total bit rate increases with the maximum transmit power budget P total . It can be seen that the average total bit rate, R s , varies greatly with ρ.  For example, at P total = 5 W, R s increases by a factor of 2 as ρ increases from 0.7 to 0.9. This illustrates the big impact that inaccurate CSI may have on system performance. The R s curves level off as P total increases due to the fixed value of the maximum interference power that can be tolerated by the PU.
Corresponding results for λ = [1 1 1 4] and λ = [1 1 1 8] are plotted in Figures 2 and 3, respectively. The average total bit rate, R s , decreases as the NBRW distribution becomes less uniform; the reduction tends to increase with P total .   Figure 4 shows R s as a function of P total for three different cases of λ with ρ = 0.9, I total = 0.02 W, and P m = 5 W. As to be expected, R s increases with P total . It can be seen that R s for λ = [1 1 1 1] is larger than for λ = [1 1 1 4], and R s for λ = [1 1 1 4] is larger than for λ = [1 1 1 8]. When the bit rate requirements for CRUs become less uniform, R s decreases due to a decrease in the benefits of user diversity. With P total = 15 W, R s increases by about 30% when λ changes from [1 1 1 8] to [1 1 1 1]. Results for ρ = 0.7 are shown in Figure 5 and are qualitatively similar to those in Figure 4.  The average total bit rate, R s , is plotted as a function of the maximum PU tolerable interference power, I total , with P total = 25 W and P m = 5 W, for ρ = 0.9 and 0.7 in Figures  6 and 7, respectively. As expected, R s increases with I total and decreases as the CRU bit rate requirements become less uniform. The R s curves level off as I total increases due to the fixed value of the total CRU transmit power, P total .

Conclusion
The assumption of perfect CSI being available at the transmitter is often unreasonable in a wireless communication system. In this paper, we studied an MU-OFDM CR system in which the available partial CSI is due to a delay in the feedback channel. The effect of partial CSI on the BER was investigated; a relationship between transmit power, number of bits loaded, and BER was derived. This relationship was used to study the performance of a resource allocation scheme when only partial CSI is available. It is found that the performance varies greatly with the quality of the partial CSI.