Constrained Joint Bit and Power Allocation for Multicarrier Systems

1 Constrained Joint Bit and Po wer Allocation for Multicarrier Systems Ebrahim Bedeer , Octavia A. Dobre, Mohamed H. Ahmed, and Kareem E. Baddour † Faculty of Engineering and Applied Science, Memorial Uni versity of Ne wfoundland, St. John’ s, NL, Canada † Communications Research Centre, Ottaw a, ON, Canada Email: { e.bedeer , odobre, mhahmed } @mun.ca, kareem.baddour@crc.ca Abstract This paper proposes a nov el low complexity joint bit and po wer suboptimal allocation algorithm for multicarrier systems operating in fading en vironments. The algorithm jointly maximizes the throughput and minimizes the transmitted power , while guaranteeing a target bit error rate (BER) per subcarrier and meeting a constraint on the total transmit power . Simulation results are described that illustrate the performance of the proposed scheme and demonstrate its superiority when compared to the algorithm in [4] with similar or reduced computational complexity . Furthermore, the results show that the performance of the proposed suboptimal algorithm approaches that of an optimal exhausti ve search with significantly lower computational complexity . Index T erms Adaptiv e modulation, bit allocation, joint optimization, multicarrier systems, power allocation. I . I N T RO D U C T I O N Multicarrier modulation is recognized as a robust and e fficient transmission technique, as e videnced by its consideration for di verse communication systems and adoption by sev eral wireless standards [1], [2]. The performance of multicarrier communication systems can be significantly improv ed by dynamically adapting the transmission parameters, such as power , 2 constellation size, symbol rate, coding rate/scheme, or any combination of these, according to the channel conditions or the wireless standard specifications [3]–[10]. Generally speaking, the problem of optimally loading bits and power per subcarrier can be categorized into two main classes: rate maximization (RM) and mar gin maximization (MM) [3]– [7]. For the former , the objectiv e is to maximize the achiev able data rate [3], [4], while for the latter the objectiv e is to maximize the achiev able system margin [6], [7] (i.e., minimizing the total transmit power giv en a target data rate). Most of the prior work has focused on optimizing either the RM or the MM problem separately . Krongold et al. [3] presented a computationally efficient algorithm to maximize the throughput using a look-up table search and the Lagrange multiplier bisection method. The algorithm conv erges faster to the optimal solution when compared to other allocation schemes. In [4], W yglinski et al. proposed an incremental bit loading algorithm to maximize the throughput while guaranteeing a tar get mean BER. The algorithm nearly achie ves the optimal solution given in [5] b ut with lower complexity . On the other hand, Papandreou and Antonakopoulos [6] presented an ef ficient bit loading algorithm to minimize the transmit po wer that con ver ges f aster to the same bit allocation as the discrete optimal bit-filling and bit-remov al methods. The algorithm exploits the differences between the subchannel gain-to- noise ratios in order to determine an initial bit allocation and then performs a multiple bit insertion or remo val loading procedures to achie ve the requested target rate. In [7], Liu and T ang proposed a lo w complexity po wer loading algorithm that aims to minimize the transmit po wer while guaranteeing a target av erage BER. Song et al. [9] proposed an iterative joint bit loading and power allocation algorithm based on statistical channel conditions to meet a target BER, i.e., the algorithm loads bits and po wer per subcarrier based on long-term frequenc y domain channel conditions, rather than instantaneous channel conditions, as in [3]–[7]. The algorithm marginally improv es performance when compared to con ventional multicarrier systems. The authors conclude that their algorithm is not meant to compete with its counterparts that adapt according the instantaneous channel conditions. In [10], the authors proposed a novel algorithm that jointly maximizes the throughput and minimizes the transmit po wer , while guaranteeing a target average BER. In emerging wireless communication systems, different requirements are needed. For exam- ple, minimizing the transmit power is prioritized when operating in interference-prone shared spectrum en vironments or in proximity to other frequency-adjacent users. On the other hand, maximizing the throughput is fa voured if sufficient guard bands exist to separate users. This 3 moti v ates us to jointly optimize the RM and MM problems, by introducing a weighting factor that reflects the importance of the competing throughput and power objecti ves. In this paper , we propose a suboptimal algorithm that jointly maximizes the throughput and minimizes the total transmit power , subject to constraints on the BER per subcarrier and the total transmit po wer . Limiting the total transmit power reduces the interference to e xisting users, which is crucial in various wireless networks, including cogniti ve radio environments. Moreov er, including the total subcarrier power in the objectiv e function is especially desirable as it minimizes the transmit power even when the power constraint is ineffecti ve, which occurs at smaller signal-to-noise ratios (SNR). Simulation results sho w that the proposed algorithm outperforms W yglinski’ s algorithm [4] with similar or reduced computational complexity . The results also indicate that the proposed suboptimal algorithm’ s performance approaches that of an exhausti ve search for the optimal discrete allocations, with significantly reduced computational complexity . The remainder of the paper is organized as follows. Section II introduces the proposed joint bit and power allocation algorithm. Simulation results are presented in Section III, while conclusions are drawn in Section IV. I I . P RO P O S E D A L G O R I T H M A. Optimization Pr oblem F ormulation A multicarrier communication system decomposes the signal bandwidth into a set of N orthogonal narrowband subcarriers of equal bandwidth. Each subcarrier i transmits b i bits using po wer P i , i = 1 , ..., N . A delay- and error -free feedback channel is assumed to exi st between the transmitter and recei ver for reporting channel state information. In order to minimize the total transmit power and maximize the throughput subject to BER and total po wer constraints, the optimization problem is formulated as Minimize P i N X i =1 P i and Maximize b i N X i =1 b i , subject to BER i ≤ BER th,i , N X i =1 P i ≤ P th , i = 1 , ..., N , (1) 4 where BER i and BER th,i are the BER and threshold value of BER per subcarrier i , respecti vely , and P th is the total transmit power threshold. An approximate e xpression for the BER per subcarrier i in case of M -ary QAM is gi ven by 1 [7] BER i ≈ 0 . 2 exp − 1 . 6 P i (2 b i − 1) |H i | 2 σ 2 n ! , (2) where H i is the channel gain of subcarrier i and σ 2 n is the v ariance of the additi ve white Gaussian noise (A WGN). The multi-objectiv e optimization function in (1) can be rewritten as a linear combination of multiple objectiv e function as follo ws Minimize P i ,b i F ( P , b ) = ( α N X i =1 P i − (1 − α ) N X i =1 b i ) , subject to g j ( P i , b i ) ≤ 0 , i = 1 , ..., N , j = 1 , ..., N + 1 , (3) where α ( 0 < α < 1 ) is a constant whose v alue indicates the relativ e importance of one objectiv e function relativ e to the other , P = [ P 1 , ..., P N ] T and b = [ b 1 , ..., b N ] T are the N -dimensional po wer and bit distribution vectors, respecti vely , with [ . ] T denoting the transpose operation, and g j ( P i , b i ) 2 is the set of N + 1 constraints giv en by g j ( P i , b i ) =            0 . 2 exp  − 1 . 6 C i P i 2 b i − 1  − BER th,i ≤ 0 , j = 1 , ..., N P N i =1 P i − P th ≤ 0 , j = N + 1 (4) where C i = |H i | 2 σ 2 n is the channel-to-noise ratio for subcarrier i . B. Optimization Pr oblem Analysis and Solution The problem in (3) can be solv ed by applying the method of Lagrange multipliers. Accordingly , the inequality constraints in (4) are transformed to equality constraints by adding non-negati ve slack variables, Y 2 j , j = 1, ..., N + 1 [11]. Hence, the constraints are re written as G j ( P i , b i , Y j ) = g j ( P i , b i ) + Y 2 j = 0 , j = 1 , ..., N + 1 , (5) 1 This expression is tight within 1 dB for BER ≤ 10 − 3 . 2 Note that g j ( P i , b i ) is a function of P i and b i for i = j . When j = N + 1 , it is a function of P i , i = 1, ..., N . 5 and further , the Lagrange function L is expressed as L ( P , b , Y , Λ ) = F ( P , b ) + N +1 X j =1 λ j G ( P i , b i , Y j ) , = α N X i =1 P i − (1 − α ) N X i =1 b i + N X i =1 λ i    0 . 2 exp  − 1 . 6 C i P i 2 b i − 1  − BER th,i + Y 2 i    + λ N +1    N X i =1 P i − P th + Y 2 N +1    , (6) where Λ = [ λ 1 , ..., λ N +1 ] T and Y = [ Y 2 1 , ..., Y 2 N +1 ] T are the vectors of Lagrange multipliers and slack variables, respectiv ely . A stationary point can be found when ∇L ( P , b , Y , Λ ) = 0 (where ∇ denotes the gradient), which yields ∂ L ∂ P i = α − 0 . 2 λ i 1 . 6 C i 2 b i − 1 exp  − 1 . 6 C i P i 2 b i − 1  + λ N +1 = 0 , (7) ∂ L ∂ b i = − (1 − α ) + 0 . 2 ln(2) λ i 1 . 6 C i P i 2 b i (2 b i − 1) 2 × exp  − 1 . 6 C i P i 2 b i − 1  = 0 , (8) ∂ L ∂ λ i = 0 . 2 exp  − 1 . 6 C i P i 2 b i − 1  − BER th,i + Y 2 i = 0 , (9) ∂ L ∂ λ N +1 = N X i =1 P i − P th + Y 2 N +1 = 0 , (10) ∂ L ∂ Y i = 2 λ i Y i = 0 , (11) ∂ L ∂ Y N +1 = 2 λ N +1 Y N +1 = 0 . (12) It can be seen that (7) to (12) represent 4 N + 2 equations in the 4 N + 2 unkno wn components of the v ectors P , b , Y , and Λ . Equation (11) implies that either λ i = 0 or Y i = 0, i = 1, ..., N , 6 while (12) implies that either λ N +1 = 0 or Y N +1 = 0. Accordingly , four possible solutions exist, as follows: — Solutions I & II : Choosing λ i = 0, i = 1, ..., N , and Y N +1 or λ N +1 = 0, results in an underdetermined system of N + 1 equations in 3 N + 1 unkno wns; hence, no unique solution can be reached. — Solution III : Choosing Y i = 0, i = 1, ..., N , and Y N +1 = 0, results in 3 N + 1 nonlinear equations in 3 N + 1 unknowns that represent the optimal solution if the total transmit po wer constraint is acti ve. — Solution IV : Choosing Y i = 0, i = 1, ..., N , and λ N +1 = 0, results in 3 N + 1 nonlinear equations in 3 N + 1 unknowns that represent the optimal solution if the total transmit po wer constraint is inacti ve. W e resort to a lo w complex suboptimal solution, which is obtained as follo ws. The constraint on the total transmit po wer in (10) is first relax ed, and the optimal solution of (7) to (9) is found. This provides the initial values for b , P , denoted by b ∗ , P ∗ , to be used with the suboptimal algorithm. Then, the final bit and power distributions are reached in an iterativ e manner to meet the power and BER constraints. The suboptimal algorithm will be presented in Section II- C ; the optimal solution for the initial bit and power distributions while relaxing the po wer constraint is provided below . — Calculation of the initial optimal bit and power distributions, b ∗ , P ∗ : In solution IV , by relaxing the power constraint in (10), we obtain 3 N equations in the 3 N unknowns P , b , and Λ ( λ N +1 = 0) that can be solv ed analytically , as follows. From (7) and (8), we can relate P i and b i as P i = 1 − α α ln(2) (1 − 2 − b i ) , (13) with P i ≥ 0 if and only if b i ≥ 0 . By substituting (13) into (9), one obtains the solution b ∗ i = 1 log(2) log    − 1 − α α ln(2) 1 . 6 C i ln(5 BER th,i )    . (14) Consequently , from (13) one gets P ∗ i = 1 − α α ln(2)    1 −  − 1 − α α ln(2) 1 . 6 C i ln(5 BER th,i )  − 1    . (15) 7 Since (2) is v alid for M -ary QAM, b i should be greater than 2. From (14), to have b i ≥ 2 , the channel-to-noise ratio per subcarrier , C i , must satisfy the condition C i ≥ − 4 1 . 6 α ln(2) 1 − α ln(5 BER th,i ) , i = 1 , ..., N . (16) The obtained solution represents a minimum of F ( P , b ) if the KKT conditions are satisfied [11]. Giv en that our stationary point ( b ∗ i , P ∗ i ) in (14) and (15) exists at Y i = 0 , i = 1, ..., N , the KKT conditions can be written as ∂ F ∂ P i + N X ρ =1 λ ρ ∂ g ρ ∂ P i = 0 , (17) ∂ F ∂ b i + N X ρ =1 λ ρ ∂ g ρ ∂ b i = 0 , (18) λ ρ > 0 , ρ = 1 , ..., N . (19) One can easily prov e that these conditions are fulfilled, as follows. — Pr oof of (17)-(19): From (7), one finds λ i = α    0 . 2 1 . 6 C i 2 b i − 1 exp  − 1 . 6 C i P i 2 b i − 1     − 1 , (20) which is positi ve for all values of i , and hence it satisfies (19). Moreover , by substituting (14), (15), and (20) in (17) and (18), one can easily verify that the KKT conditions are satisfied. Note that b ∗ , P ∗ represent an optimal solution when there is no constraint on the total transmit po wer .  C. Pr oposed J oint Bit and P ower Suboptimal Allocation Algorithm The solution ( b ∗ , P ∗ ) given in (14) and (15) is obtained for λ N +1 = 0, which basically means that no constraint on the total transmit po wer is considered for the problem formulated in (3). T o consider such a constraint, we propose a suboptimal algorithm whose idea is as follo ws. The total po wer P N i =1 P i is calculated based on (15) and checked against the threshold v alue P th . If less than the threshold, then the power constraint is met; otherwise, the po wer ∆ P i = P i ( b i ) − P i ( b i − 1) required to reduce the number of bits b i on subcarrier i by one bit is calculated according to (2). The subcarrier , i 0 , with the maximum value of ∆ P i is identified, its allocated bits b i 0 is set to b i 0 − 1 , and its power is reduced by ∆ P i 0 . If the power constraint is not met, the process repeats. The proposed algorithm can be formally stated as follo ws. 8 Proposed Algorithm 1: INPUT The A WGN v ariance σ 2 n , channel gain per subcarrier i ( H i ), target BER per subcarrier i (BER th,i ), and weighting factor α . 2: f or i = 1, ..., N do 3: if C i ≥ − 4 1 . 6 α ln(2) 1 − α ln(5 BER th,i ) then 4: - b ∗ i and P ∗ i are giv en by (14) and (15), respecti vely . 5: - b ∗ i ← Round b ∗ i to the nearest integer . 6: - P ∗ i ← Recalculate P ∗ i according to (2). 7: else 8: - Null the corresponding subcarrier i . 9: end if 10: end f or 11: while P N i =1 P i > P th do 12: f or i = 1, ..., N do 13: - Calculate P i ( b i − 1) corresponding to reducing the number of bits b i on subcarrier i to b i − 1 , according to (2). If b i − 1 < 2 , null the subcarrier i . 14: - Calculate ∆ P i = P i ( b i ) − P i ( b i − 1) . 15: end for 16: - Find subcarrier i 0 with maximum ∆ P i . 17: - Set b i 0 to b i 0 − 1 and P i 0 to P i 0 ( b i 0 ) − ∆ P i 0 . 18: end while 19: OUTPUT The suboptimal b i and P i , i = 1, ..., N . I I I . N U M E R I C A L R E S U L T S This section in vestigates the performance of the proposed algorithm, and compares it with that of the allocation scheme in [4] and the exhausti ve search for the discrete optimal allocation. The computational comple xity of these algorithms is additionally compared. A. Simulation Setup W e consider an orthogonal frequency di vision multiplexing (OFDM) system with a total of N = 128 subcarriers. W ithout loss of generality , the BER constraint per subcarrier , BER th,i , is 9 assumed to be the same for all subcarriers and set to 10 − 4 . The channel impulse response h ( n ) of length N ch is modeled as independent complex Gaussian random v ariables with zero mean and exponential power delay profile [12], i..e, E {| h ( n ) | 2 } = σ 2 h e − n Ξ , n = 0 , 1 , ..., N ch − 1 , where σ 2 h is a constant chosen such that the av erage energy per subcarrier is normalized to unity , i.e., E {|H i | 2 } = 1, and Ξ represents the decay factor . Representativ e results are presented in this section and were obtained by repeating Monte Carlo trials for 10 4 channel realizations with a channel length N ch = 5 taps and decay factor Ξ = 1 5 . B. P erformance of the Pr oposed Algorithm Fig. 1 illustrates the allocated bits and po wers with and without considering the total po wer constraint for an example channel realization, SNR = 10 dB and α = 0.5. W ithout considering the total power constraint, it can be seen from the plots in Fig. 1 that when the channel-to-noise ratio per subcarrier , C i , exceeds the v alue in (16), the number of bits and power allocated per subcarrier are non-zero. As expected, (14) yields a non-integer number of allocated bits per subcarrier , which is not suitable for practical implementations. This value is rounded to the nearest integer , as shown in Fig. 1 (b), and the modified value of the allocated power per subcarrier to maintain the same BER th,i is determined using (2). When considering the total power constraint, for the sake of illustration, the po wer threshold is set to half the total transmit po wer with no po wer constraint; subcarriers with maximum ∆ P are identified, and the corresponding bits are reduced by one until the total po wer constraint is met, while guaranteeing the target BER requirement. Fig. 2 depicts the av erage throughput and av erage transmit power as a function of a verage SNR 4 , with and without considering the total power constraint at α = 0.5. W ithout considering the total power constraint and for an a verage SNR ≤ 24 dB, one finds that both the a verage throughput and the av erage transmit power increase as the SNR increases, whereas for an av erage SNR ≥ 24 dB, the transmit po wer saturates, and the throughput continues to increase. This observ ation can be explained as follows. For lower values of the av erage SNR, the corresponding v alues of C i result in the nulling of many subcarriers in (16). By increasing the a verage SNR, the number of used subcarriers increases, resulting in a noticeable increase in the throughput and power . Apparently , for SNR ≥ 24 dB, all subcarriers are used, and our algorithm essentially 4 The average SNR is calculated by averaging the instantaneous SNR values per subcarrier ov er the total number of subcarriers and the total number of channel realizations, respecti vely . 10 20 40 60 80 100 120 0 20 40 60 C i subcarrier index (a) 20 40 60 80 100 120 0 1 2 3 4 subcarrier index (b) b i 20 40 60 80 100 120 0 1 2 subcarrier index (c) P i (mW) 20 40 60 80 100 120 10 −4 10 −3 subcarrier index (d) BER i Channel−to−noise ratio, C i Threshold value given by (16) Continous allocated bits (no power constraint) Discrete allocated bits (no power constraint) Discrete allocated bits (with power constraint) Allocated power (no power constraint) Allocated power (with power constraint) Fig. 1: An example of the allocated bits and power per subcarrier for a giv en channel realization, with and without power constraint, at SNR = 10 dB, α = 0.5. minimizes the av erage transmit power by keeping it constant, while increasing the a verage throughput. By considering a total po wer constraint, P th = 0.1 mW , at lower SNR v alues when the total transmit power is belo w the threshold, the same av erage transmit power and throughput are obtained; ho wever , at higher SNR values, when the total transmit power exceeds the threshold, a small reduction in the av erage throughput is noticed, which emphasizes that the proposed algorithm meets the power constraint while maximizing the throughput, i.e., the throughput does not degrade much when compared to the case of no po wer constraint. Fig. 3 shows the average throughput and av erage transmit power as a function of the weighting factor α , for σ 2 n = 10 − 3 µW , with and without considering the total power constraint. W ithout considering the total power constraint, one can notice that an increase of the weighting factor α yields a decrease of both the average throughput and av erage transmit power . This can be 11 explained as follo ws. By increasing α , more weight is giv en to the transmit power minimization (the minimum transmit po wer is further reduced), whereas less weight is giv en to the throughput maximization (the maximum throughput is reduced), according to the problem formulation. By considering a total po wer constraint, P th = 0.1 mW , the same a verage throughput and po wer are obtained if the total transmit power is less than P th , while the average throughput and po wer saturate if the total transmit power exceeds P th . Note that this is different from Fig. 2, where the av erage throughput increases while the transmit power is kept constant, which is due the increase of the average SNR value. Fig. 3 illustrates the benefit of introducing such a weighting factor in our problem formulation to tune the av erage throughput and transmit po wer lev els as needed by the wireless communication system. In Fig. 4, the a verage throughput and av erage transmit power are plotted as a function of the po wer threshold P th , at α = 0.5 and σ 2 n = 10 − 3 µW . It can be noticed that the av erage throughput increases as P th increases, and saturates for higher values of P th ; moreo ver , the av erage transmit power increases linearly with P th , while it saturates for higher v alues of P th . This can be e xplained, as for lo wer values of P th , the total transmit power is restricted by this threshold value, while increasing this threshold value results in a corresponding increase in both the av erage throughput and total transmit power . For higher v alues of P th , the total transmit po wer is alw ays less than the threshold v alue, and, thus, it is as if the constraint on the total transmit power is actually relaxed. In this case, the proposed algorithm essentially minimizes the transmit po wer by keeping it constant; consequently , the a verage throughput remains constant for the same noise variance as for the previous scenario. C. P erformance and Complexity Comparison In Fig. 5, the throughput achieved by the proposed algorithm is compared to that obtained by W yglinski’ s algorithm [4] for the same operating conditions, with and without considering the total power constraint. For a fair comparison, the uniform power allocation used by the allocation scheme in [4] is computed by dividing the a verage transmit po wer allocated by our algorithm by the total number of subcarriers. As shown in Fig. 5, the proposed algorithm provides a significantly higher throughput than the scheme in [4] for lo w average SNR v alues. This result demonstrates that optimal allocation of transmit power is crucial for lo w po wer budgets. T o characterize the gap between the proposed suboptimal algorithm and the optimal solution, Fig. 6 compares v alues of the objecti ve function achie ved with the proposed suboptimal algorithm 12 0 5 10 15 20 25 30 35 40 45 10 1 10 2 10 3 Average throughput (bits/OFDM sybmol) Average SNR (dB) 0 5 10 15 20 25 30 35 40 45 10 −2 10 −1 10 0 Average transmit power (mW) no p owe r con tr ain t P th =0 . 1m W Average throughput Average transmit power Fig. 2: A verage throughput and average transmit power as a function of av erage SNR, with and without po wer constraint, at α = 0.5. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10 2 10 3 10 4 Average throughput (bits/OFDM sybmol) α 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10 −3 10 −2 10 −1 10 0 Average transmit power (mW) no p o w e r c ons t r a i n t P th =0 . 1 m W no p owe r cons tr aint P th =0 . 1m W Average transmit power Average throughput Fig. 3: A verage throughput and average transmit power as a function of α , with and without power constraint, at σ 2 n = 10 − 3 µW . 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25 10 2 10 3 Average throughput (bits/OFDM sybmol) P th (mW ) 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Average transmit power (mW) Average throughput Average transmit power Fig. 4: A verage throughput and average transmit power as a function of the power constraint P th , at α = 0.5 and σ 2 n = 10 − 3 µW . 13 0 5 10 15 20 25 30 35 10 0 10 1 10 2 10 3 Average SNR (dB) Average throughput (bits/OFDM sybmol) dat a 1 P r op os e d no p owe r cons tr aint P th =0 . 1m W Wyglinski [4] Proposed algorithm Fig. 5: A verage throughput as a function of av erage SNR for the proposed algorithm and W yglinski’ s algorithm in [4], with and without power constraint, at α = 0.5. Fig. 6: Objective function for the proposed suboptimal algorithm and the exhaustiv e search when P th = 5 µ W , α = 0.5 and N = 8. and the optimal exhausti ve search. Note that the latter finds the discretized optimal allocation for the problem in (3). Results are presented for P th = 5 µ W , α = 0.5 and N = 8; a small number of subcarriers is chosen, such that the e xhausti ve search is feasible. As can be seen in Fig. 6, the proposed suboptimal algorithm approaches the optimal results of the e xhausti ve search. Based on the algorithm description in Section II- C , one can show that the worst case compu- tational complexity of the proposed algorithm is of O ( N 2 ) if the po wer constraint is effecti ve, whereas it is of O ( N ) if the po wer constraint is inef fecti ve, which is similar to or lo wer than the O ( N 2 ) of the W yglinski’ s algorithm, and significantly lower than O ( N !) of the e xhausti ve search. 14 I V . C O N C L U S I O N This paper proposed a novel suboptimal algorithm that jointly maximizes the total throughput and minimizes the total transmit power , with constraints on the BER per subcarrier and the total transmit power , for multicarrier communication systems. Simulation results demonstrated that the proposed algorithm outperforms the algorithm in [4] under the same operating conditions, with similar or reduced computational effort. Additionally , it was sho wn that its performance approaches that of an exhausti ve search with significantly lo wer complexity . A C K N O W L E D G M E N T The authors would like to thank Dr . How ard Heys for his help on the analysis of the compu- tational complexity of the recurrence relation in the proposed algorithm. This work has been supported in part by the Communications Research Centre, Canada. R E F E R E N C E S [1] K. Fazel and S. Kaiser, Multi-carrier and Spread Spectrum Systems: fr om OFDM and MC-CDMA to LTE and W iMAX . John W iley & Sons Inc, 2008. [2] H. Mahmoud, T . Y ucek, and H. Arslan, “OFDM for cognitiv e radio: merits and challenges, ” IEEE W ireless Commun. Mag. , vol. 16, no. 2, pp. 6–15, Apr . 2009. [3] B. Krongold, K. Ramchandran, and D. Jones, “Computationally ef ficient optimal power allocation algorithms for multicarrier communication systems, ” IEEE T rans. Commun. , vol. 48, no. 1, pp. 23–27, Jan. 2000. [4] A. W yglinski, F . Labeau, and P . Kabal, “Bit loading with BER-constraint for multicarrier systems, ” IEEE T rans. W ir eless Commun. , vol. 4, no. 4, pp. 1383–1387, Jul. 2005. [5] B. Fox, “Discrete optimization via marginal analysis, ” Management Science , vol. 13, no. 3, pp. 210–216, Nov . 1966. [6] N. Papandreou and T . Antonakopoulos, “ A ne w computationally efficient discrete bit-loading algorithm for DMT applications, ” IEEE T rans. Commun. , vol. 53, no. 5, pp. 785–789, May 2005. [7] K. Liu and B. T ang, “ Adaptiv e power loading based on unequal-BER strategy for OFDM systems, ” IEEE Commun. Lett. , vol. 13, no. 7, pp. 474–476, Jul. 2009. [8] E. Bedeer , M. Marey , O. Dobre, and K. Baddour, “ Adaptive bit allocation for OFDM cognitiv e radio systems with imperfect channel estimation, ” in Pr oc. IEEE Radio and W ireless Symposium , Jan. 2012, pp. 359 – 362. [9] Z. Song, K. Zhang, and Y . Guan, “Joint bit-loading and power -allocation for OFDM systems based on statistical frequency- domain fading model, ” in Pr oc. IEEE V ehicular T echnology Conference (VTC)-F all , Sep. 2002, pp. 724–728. [10] E. Bedeer, O. A. Dobre, M. H. Ahmed, and K. Baddour , “Joint optimization of bit and power allocation for multicarrier systems with average BER constraint, ” accepted Pr oc. IEEE V ehicular T echnology Conference (VTC)-F all , 2012. [11] I. Griv a, S. Nash, and A. Sofer , Linear and Nonlinear Optimization . Society for Industrial Mathematics, 2009. [12] H. Lin, X. W ang, and K. Y amashita, “ A lo w-complexity carrier frequency offset estimator independent of DC offset, ” IEEE Commun. Lett. , vol. 12, no. 7, pp. 520–522, Jul. 2008.