A Low-Complexity Zero-forcing CFO Compensation Scheme for OFDMA Uplink Systems

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(1)IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 10, OCTOBER 2008. 3657. Transactions Letters A Low-Complexity Zero-forcing CFO Compensation Scheme for OFDMA Uplink Systems Chao-Yuan Hsu and Wen-Rong Wu. Abstract—Similar to the conventional orthogonal frequencydivision multiplexing (OFDM) system, an OFDM multiple access (OFDMA) system will have a carrier frequency offset (CFO) problem. Since CFOs of all users are different, CFO compensation in the OFDMA uplink system is much more involved. A simple, yet efficient, method is the zero-forcing (ZF) compensation method. However, it involves an inverse of an N × N CFO-induced ICI matrix, where N is the number of subcarriers. Thus, the complexity can become very high when N is large, a case commonly seen in OFDMA systems. In this work, we propose a low-complexity ZF method to overcome the problem. The main idea is to use Newton’s method to solve matrix inversion iteratively. We explore the structure of the CFOinduced ICI matrix and develop a method that can implement Newton’s method with fast Fourier transforms (FFTs). As a result, the required computational complexity is significantly reduced from O(N 3 ) to O(2N log2 N ). Simulations show that, with only three iterations, the proposed method can have similar performance to the direct ZF method. Index Terms—Orthogonal frequency-division multiple access (OFDMA), carrier frequency offset (CFO), intercarrier interference (ICI), fast Fourier transform (FFT), Newton’s method.. O. I. I NTRODUCTION. RTHOGONAL frequency-division multiplexing (OFDM) has been shown to be a successful technique to combat a fading channel effect in wireless communications. Since all subcarrier signals overlap orthogonally in the spectrum, it is considered a bandwidth-efficient scheme. An ideal OFDM system has no intercarrier interference (ICI) problem, and it can be easily developed as a frequencydivision multiple access (FDMA) scheme. An OFDM-based FDMA system is generally referred to as an OFDMA system, which was proposed in IEEE 802.16e-2005 for broadband wireless access. In the presence of CFO, the orthogonal property in an OFDM system is destroyed, and ICI is induced, degrading the system performance significantly [1]. Many works have been reported for CFO estimation and compensation. Different from that in OFDM systems, CFO in OFDMA uplink systems Manuscript received February 13, 2007; revised March 06, 2008; accepted July 22, 2008. The associate editor coordinating the review of this paper and approving it for publication was R. M. Buehrer. The authors are with the Department of Communication Engineering, National Chiao-Tung University, Hsinchu, Taiwan 300, R.O.C. (e-mail: wrwu@faculty.nctu.edu.tw). Digital Object Identifier 10.1109/T-WC.2008.070177. causes not only self-interference, but also multiuser interference (MUI) [2], [3]. Various methods have been proposed to solve the problem. One direct method is to estimate the CFO in the base station, and transmit the information back to mobile stations for CFO correction. Another approach is to transmit redundant information in certain subcarriers such that ICI can be cancelled in the receiver end; this approach is known as self-ICI-cancellation [8]−[13]. Yet another viable approach eliminates the need for extra transmission overhead by compensating for ICI in the receiver. CFO compensation methods for OFDMA uplink systems have been reported [14]−[18], [21]. The simplest one is to compensate for ICI with a time-domain phase-rotation for each user [14]. This approach can suppress self-ICI, but it does not take MUI into account. A post-FFT method was proposed [15] to improve the performance of the phaserotation approach. By combining the parallel interference cancellation (PIC) technique with the method in [15], a more sophisticated scheme was developed [16]. Other PIC-related works can be found in [17], [18]. It can be observed that ICI in a subcarrier mainly comes from neighboring subcarriers. Given this, the ICI matrix is simplified into a banded matrix [19], such that the computational complexity of the zeroforcing (ZF) and minimum-mean-square-error (MMSE) CFOcompensation methods can be reduced. Taking advantage of an interleaved-OFDMA structure, [21] proposes a method that divides the whole system into several smaller subsystems, after which the MMSE method is applied to the subsystems. This method has good performance, and it requires a low computational complexity; however, it is only applicable to an ideal interleaved structure (i.e., uniform subcarrier-spacing for each user). The aforementioned methods were developed for CFO-compensation. CFO estimation methods have also been reported for OFDMA uplink systems [4]−[7]. The ZF method is simple, yet effective for CFO compensation in OFDMA uplink systems. However, because it must invert the ICI matrix, whose dimension equals the number of subcarriers (N ), the complexity can become prohibitively high when N is large, as is commonly found in OFDMA systems. In this work, we propose a low-complexity ZF method to cope with the problem [20]. Using Newton’s method for iterative matrix inversion and exploring the structure of the ICI matrix, we develop a method that is able to implement. c 2008 IEEE 1536-1276/08$25.00 .

(2) 3658. IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 10, OCTOBER 2008. Newton’s method using fast Fourier transforms (FFTs). From simulations, we find that the performance of the proposed method is similar to that of the direct ZF method, while the complexity is reduced from O(N 3 ) to O(2N log2 N ). The rest of this work is organized as follows: Section 2 describes the signal model of an OFDMA uplink system, Section 3 presents the proposed method in detail and analyzes the required complexity, Section 4 presents the simulation results, and finally, Section 5 presents our conclusions. II. S IGNAL M ODEL In an OFDMA system, the available bandwidth is divided into N equally spaced subbands. Each subband has a bandwidth of 1/(N Ts ), where Ts is the symbol sampling period. In such a system with Q users, we assume that each user uses Ns = N/Q subcarriers. For user q, the transmit frequency-domain signal at the kth subcarrier is denoted by x qk , where k ∈ Υq and Υq is the set ofthe subcarrier indices for user q. It is assumed that Υi Υj = ∅ for Q i = j and q=1 Υq = {0, 1, . . . , N − 1}. OFDMA adopts the interleaved subcarrier allocation scheme: in other words, Υq = {q − 1, q − 1 + Q, . . . , q − 1 + ( N Q − 1)Q}. Since the subcarriers assigned to the different users are interleaved throughout the whole bandwidth, this scheme can achieve the maximum frequency diversity. For each user, we assume that the CP (cyclic prefix) length (Ng ) is long enough to prevent intersymbol interference and that the channel is time-invariant in an OFDMA symbol period. Consider a specific OFDMA symbol for user q. The channel output signal after CP removal can be expressed as yq = Hq xq , where yq is the qth user’s N × 1 receive time-domain signal vector, and xq is the qth√ user’s N × 1 time-domain q is the q . Here x symbol vector, i.e., xq = (1/ N )GH x qth user’s N × 1 frequency-domain symbol vector, and G is the N × N normalized DFT matrix with GGH = IN , where IN is an N × N identity matrix. Hq is an N × N circulant channel matrix with the first N × 1 column vector being hq , which is the channel response of xq . Zeros are padded in hq since the channel length is assumed to be smaller q are nonzero only in than Ng . Note that the elements of x designated subcarrier positions and Hq can be decoupled as q ). q G, where H q is an N ×N matrix and H q = diag(h GH H The notation diag(g) indicates√a diagonal matrix with a q = N Ghq . The time-domain diagonal vector g, and h OFDMA√symbol from Q users can be expressed as received q q H q x + v, where Eq = diag(eq ), E H G r = (1/ N ) Q q=1 eq = [Uq0 , . . . , UqN −1 ]T , Uqk = exp{j2πq k/N }, and q is the normalized CFO (with respect to the subcarrier spacing) for user q. Also, v denotes the N × 1 noise vector. After the FFT operation, we are a corresponding frequencyleft with q qy q = GEq GH , , where E E +v domain signal of r= Q q=1 √ qx q q , and v q = H = N Gv. Note that y √ E is qa circulant q = matrix and its first column is e = (1/ N )Ge . Let x = [ [ x0 , . . . , x N −1 ]T and h h0 , . . . , hN −1 ]T be the composite transmit data and channel frequency-response for all users q = [ and h hq0 , . . . , hqN −1 ]T . Then, x k = x qk and hk = hqk if q q k ∈ Υq . Define a diagonal matrix S such that S (j, j) = 1, if j ∈ Υq and Sq (j, j) = 0, otherwise. Thus, we express the. received frequency-domain signal as [19], [21] y + v , r = M. (1). and M q Sq is the x, H = diag(h), = Q E = H where y q=1 CFO-induced ICI matrix. III. P ROPOSED CFO C OMPENSATION M ETHOD From (1), we see that a straightforward method to compen −1 r [21]. sate for ICI is the ZF method given by yZF = M Although the direct ZF method can completely suppress ICI, it needs to invert an N × N matrix. When N is large, the required complexity can become very high. Unfortunately, in most real-world applications, N is large. For IEEE 802.16e, N can be as large as 2048. Here, we propose a low-complexity ZF method to solve the problem. The main idea is to use an iterative procedure to avoid the direct matrix inversion. In this work, specifically, we use Newton’s method. −1 at the kth iteration. NewLet Wk be the estimate of M ton’s iteration for matrix inversion [22], [23] can be described as Wk+1 = (2IN − Wk M)W k for k = 0, 1, . . . , ∞. Let represent the estimation residual. Newton’s Rk = IN −Wk M ≤ IN − W0 M 2k iteration implies that IN − Wk M < 1, we then have quadratic for all k. If IN − W0 M convergence [24]. From Newton’s iteration, we can also see that matrix-to-matrix multiplications are required and that the computational complexity of Newton’s iteration is even higher than the direct matrix inversion. Thus, direct application of Newton’s method is not feasible. Now, we develop a method to solve the problem. Using Newton’s iteration method, we obtain a sequence of matrices {W0 , W1 , . . . , Wk }. Exploring their structures, we can express Wk as Wk =. k 2 −1. m W0 , ckm (W0 M). (2). m=0. where ckm is the coefficient of the mth summation term in (2). Assign ckm ’s as coefficients of a polynomial function of z, i.e., k gk (z) = ck0 z 0 + ck1 z 1 + . . . + ck2k −1 z 2 −1 . Then, gk+1 (z) can subsequently be derived from gk (z) as gk+1 (z) = 2gk (z) − z[gk (z)]2 , where g0 (z) = 1. Note that (2) is not in the original form of Newton’s iteration. Also note that our final objective is to obtain the CFOr, not the matrix inverse Wk itcompensated result Wk r and sm = self. Multiply (2) by r and let yk = Wk m W0 r, which gives us (W0 M) yk =. k 2 −1. ckm sm .. (3). m=0. From the definition of sm , we obtain the following iterative m , which allows sm to be calculated step: sm+1 = (W0 M)s recursively. With this approach, we have transformed the matrix-to-matrix multiplications in (2) into the matrix-tovector multiplications in (3). To complete our algorithm, we further let W0 be diagonal = Q GEq GH Sq . Thus, we can rewrite and recall that M q=1 Q sm+1 as sm+1 = W0 G[ q=1 Eq (GH Sq sm )]. Note that.

(3) IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 10, OCTOBER 2008. 3659. TABLE I C OMPLEXITY COMPARISON OF THE PROPOSED METHOD , THE BANDED ZF METHOD , AND THE DIRECT ZF METHOD . Complexity Real multiplications. Real divisions Real additions. Proposed method 2(2k −1+Q)N log2 (N )+2(2k − 1)N log2 (N/Q) + [8(2k − 1)Q + 2(2k + 1)]N + 4Q(2S + 1) 2Q 3(2k −1+Q)N log2 (N )+3(2k − 1)N log2 (N/Q) + [6(2k − 1)Q + 2]N + 2Q(3S + 1). Banded ZF method 2QN log 2(N ) + (4B 2 + 14B + 6)N − 83 B 3 − 9B 2 − 19 B 3. Direct ZF method 4 3 N + 5N 2 + 2QN log2 (N ) − 3 1 N 3. (2B + 2)N − B 2 − B 3QN log 2(N ) + (4B 2 + 11B + B 2 − 29 B 3)N − 83 B 3 − 15 2 6. N2 + N 4 3 N + 72 N 2 + 3QN log2 (N ) − 3 11 N 6. calculating sm+1 only involves vector multiplications, IDFTs, and a DFT. It is well-known that DFT/IDFT can be implemented with FFT/IFFT and the computational complexity can be greatly reduced. Thus, the required computational complexity is reduced from O(N 3 ) to O((Q + 1)N log2 N ). Utilizing the interleaved-OFDMA structure, the computational complexity can be further reduced. Let sm = [sm,0 , . . . , sm,N −1 ]T and uqm = Sq sm = [uqm,0 , . . . , uqm,N −1 ]T . From the definition of Sq , uqm,i = sm,i , if i ∈ Υq and uqm,i = 0, otherwise. This is to say that uqm corresponds to an upsampled sequence of the desired elements in sm . The nonzero elements in uqm , denoted by dqm = [sm,q−1 , . . . , sm,q−1+(N/Q−1)×Q ]T , can be obtained by circularly shifting uqm with q − 1 elements and downsampling √ q q the result with a factor of Q. Let dm = (1/ Q)GH Ns dm , where GNs is an Ns × Ns normalized DFT matrix, and q construct an N ×1 vector by duplicating dm Q times, shown as q q aqm = [(dm )T , . . . , (dm )T ]T . We can then express GH Sq sm as GH Sq sm = Cq aqm , where Cq = diag([Zq0 , . . . , ZqN −1 ]T ), and Zqk = exp{j2π(q − 1)k/N }. Note that Cq results from circlurly shifting uqm . As a result, we can implement GH Sq sm by an IDFT with dimension N/Q instead of N . Using this approach, we can reduce the complexity further Q q q q q with sm+1 = W0 G q=1 E C am . Note that am is q q a column vector and that both C and E are diagonal matrices. As assumed, W0 is a diagonal matrix. As a result, this approach only involves one FFT of size N , Q IFFTs of size N/Q, and several vector operations. Thus, the computational complexity of the ZF method can be reduced from O(N 3 ) to O(N log2 (N 2 /Q)). The final task is to determine W0 . A good initial value can reduce the number of iterations significantly. Letting W0 = diag([w0 , w1 , . . . , wN −1 ]T ), we propose a minimumFrobenius-norm criterion to obtain optimum initial values. The 2, criterion is shown as Wopt,0 = arg min W IN − W0 M F 0 where RF denotes the Frobenius norm of R. The optimum initial values can be obtained by setting ∂{IN − 2 }/∂w∗ = 0. Thus, we can express the optimal initial W0 M F k −1 value wopt,k as wopt,k = m ∗k,k / N k,j |2 , where m i,j = j=0 |m M(i, j). For further complexity reduction, we approximate wopt,k as wopt,k ≈ m ∗k,k / j=<k−S:k+S,N > |m k,j |2 , where S is the number of ICI terms considered (0 ≤ S ≤ N/2 − 1), and < i : j, N > denotes a sequence of {i − N i/N , i + 1 − N (i + 1)/N , . . . , j − N j/N } (i and j are integers and i ≤ j). The approximation is based on the fact that the ICI in a subcarrier mainly comes from neighboring subcarriers.. For the direct ZF method, we apply Gaussian elimination [25] to implement the matrix inversion. Finally, for signal detection, we apply a one-tap frequency domain channel equalizer to each subcarrier. The result can be expressed as −1 y, where x is the estimate of x while y is the CFOx=H . The complexity comparison for the proposed compensated y and existing methods is shown in Table I. For convenience, the methods in [15] and [19] are referred to as the CLJL and banded ZF methods, respectively. In Table I, B is the ICI matrix bandwidth for the banded ZF method. IV. S IMULATIONS In this section, we present simulation results to evaluate the performance of the proposed method. Here, we use an interleaved-OFDMA uplink system with N = 2048, Q = 16, and Ng = 128. The modulation scheme is 16-QAM. The channel length, L, is set to 127 for all users, and the power delay profile for the qth user by the exponential is described 2 −αq m = e−αq l / L−1 e , where l is the tap function σq,l m=0 index and αq is a parameter of the function. Here, we let {α1 , α2 , . . . , αQ } = {0, 0.2, 0.4, . . . , 3}. Each channel tap fades independently, and it has a Rayleigh distribution. The averaged bit-error-rate (BER) is adopted as the performance index, and CFOs for all users are set to {0.1, -0.2, -0.05, 0.2, -0.3, 0, -0.1, 0.4, -0.3, 0.05, 0, -0.1, 0.05, -0.1, 0.3, 0.15}. It is found that the performance of the proposed method with S = 2 is almost the same as that with S = 1023. Thus, in the following simulations, we only consider the condition S = 2. The performances of five methods, namely, the conventional, CLJL, direct ZF, banded ZF, and proposed methods, are compared in our simulations. The conventional method is that described in [14]. Figure 1 shows the simulation results. From this figure, we find that the conventional and CLJL methods both have a serious error floor phenomenon. This is because compensation of the qth user’s CFO may cause other users’ CFOs to become enlarged, contributing to an increased MUI. The performance of the proposed method with three iterations can approach that of the direct ZF method. The complexity of the banded ZF method depends greatly on its band dimension, denoted as B. For a fair comparison, we let B be 16 for the banded ZF method. In this case, the complexities of the banded ZF method and the proposed method (k = 3) are roughly equal. Figure 1 shows that the proposed method performs much better than the banded ZF method (B = 16). To see the impact of the CFO magnitude, we consider a scenario in which the Qth user’s CFO is increased from 0 to 0.5. The CFOs of the other (Q − 1) users remain the same as.

(4) 3660. BER. 10. 10. 10. 10. 0. 10. −2. Without CFO Direct ZF Conventional CLJL Banded ZF (B=16) Proposed (S=2, k=1) Proposed (S=2, k=2) Proposed (S=2, k=3). −3. −4. 0. 5. 10. 25. 30. 35. −1. BER of the first (Q−1) users. 10. 10. 10. −2. −3. 10 −15. Direct ZF Banded ZF (B = 16) Proposed (S = 2, k = 3). −3. −4. 0.1. 0.2 0.3 CFO of the Qth user. 0.4. −10. −5 0 5 Near far power ratio (dB). 10. 15. Fig. 3. BER performance comparison for the proposed method (k = 3) and the direct ZF method in the near-far scenario (16-QAM modulation, and CFOs = {0.1, -0.2, -0.05, 0.2, -0.3, 0, -0.1, 0.4, -0.3, 0.05, 0, -0.1, 0.05, -0.1, 0.3, 0.15}).. Methods. −2. 0. Direct ZF (SNR=25 dB) Proposed (S=2, k=3, SNR=25 dB) Direct ZF (SNR=35 dB) Proposed (S=2, k=3, SNR=35 dB). TABLE II C OMPLEXITY COMPARISON OF THE DIRECT ZF METHOD , THE BANDED ZF METHOD , AND THE PROPOSED METHOD WHEN N = 2048 AND Q = 16.. Direct ZF (SNR=25 dB) Proposed (S=2, k=3, SNR=25 dB) Direct ZF (SNR=35 dB) Proposed (S=2, k=3, SNR=35 dB) 10. 10. −4. 15 20 SNR (dB). Fig. 1. BER performance comparison for the conventional, CLJL, proposed, and direct ZF methods (16-QAM modulation, and CFOs = {0.1, -0.2, -0.05, 0.2, -0.3, 0, -0.1, 0.4, -0.3, 0.05, 0, -0.1, 0.05, -0.1, 0.3, 0.15}). 10. −1. −1. BER of the first (Q−1) users. 10. IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 10, OCTOBER 2008. 0.5. Fig. 2. BER performance comparison for the proposed and direct ZF methods (16-QAM modulation, CFOs of the first (Q-1) users = {0.1, -0.2, -0.05, 0.2, -0.3, 0, -0.1, 0.4, -0.3, 0.05, 0, -0.1, 0.05, -0.1, 0.3}, and the CFO of the Qth user increases from 0 to 0.5).. those in Fig. 1. Fig. 2 shows the averaged BER of the first (Q− 1) users. From this figure, we can see that the proposed method (k = 3) is largely unaffected by increasing the CFO of the Qth user. Also, the proposed method has the same performance as the direct ZF method. Since OFDMA is a multiuser system, the near-far phenomenon may occur. To see the impact of the phenomenon, we consider a scenario in which the powers of the first (Q − 1) users are equal, while that of the Qth user is varied. The power of the Qth user over that of one of the remaining users is defined as the near-far power ratio κ, which ranges from -15 dB to 15 dB. Here, CFOs are set as shown in Fig. 1. Figure 3 shows the averaged BER of the first (Q − 1) users. From this figure, we see that the near-far effect does affect BER, slightly. For κ ≤ 5 dB, there is almost no performance degradation. We also find that the proposed and direct ZF methods have similar performance regardless of the value of κ.. Real multiplications 11474937856 3275760. Real divisions. Real additions. 4196352 69360. 11469003776 3532168. 3109184 (0.000271, 0.949150). 32 (0.000008, 0.000461). 3236064 (0.000282, 0.916170). Table II shows the computational complexities of the direct ZF, banded ZF, and proposed algorithms. In the table, the two numbers inside each set of parentheses (in the forth row) are the ratios of the number of operations (indicated by each column) required for the proposed method to those of the direct ZF and banded ZF methods, respectively. From this table, we can see that the real multiplications/additions/divisions required for the proposed method are 0.000271/0.000282/0.000008 times those for the direct ZF method. It is apparent that the proposed method requires a much lower complexity. Although the banded ZF method can have low complexity, its performance is not satisfactory. At a similar complexity, the proposed method outperforms the banded ZF method. V. C ONCLUSIONS In this work, we propose a low-complexity CFOcompensation method for an interleaved-OFDMA uplink system. The proposed method is an efficient implementation of the ZF method. Using Newton’s iteration for matrix inversion and exploring the structure inherent in the CFO-induced ICI matrix, we develop a method that can be implemented with FFTs. As a result, the complexity can be reduced to O(2N log2 N ). Since the FFT/IFFT module is already available in OFDMA transceivers, implementation of the proposed.

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