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Effect of Common Carrier Frequency Offset at the OFDMA Receiver

The effect of a common carrier frequency offset (CFO) at the OFDMA receiver is explored in this paper. In conventional CFO compensation methods for OFDMA systems, the CFOs are usually compensated for by directly eliminating the intercarrier interference (ICI) caused by the residual CFOs for individual users. Besides the ICI, the multiuser interference can be also introduced at the OFDMA receiver due to different CFO values for the users. From the theoretical and simulated analysis, we find that the mean square error performance of the demodulated output signal is related to a common CFO value which can be adjusted before the DFT. A new common CFO estimation method is then verified by numerical results to demonstrate that a better bit error rate performance can be obtained by using the proposed common CFO correction algorithm compared to conventional CFO compensation methods in OFDMA systems.



ORTHOGONAL FREQUENCY-DIVISION MULTIPLEXING (OFDM) is widely used in modern wireless communications for its good ability in reducing the effect of multipath propagation. When OFDM is used in a multiple access (MA) system, the combination of the frequency division multiple access (FDMA) method has attracted a lot of attention. The OFDM multiple access (OFDMA) technology divides groups of the OFDM subcarriers allocated to different subscribers for simultaneous uplink transmission from subscriber stations (SS) to the base station (BS). A typical system has been proposed for the application of wireless metropolitan area networks (MANs) as the standard IEEE 802.16a [1]. In an OFDMA system, the imperfect synchronization due to a carrier frequency offset (CFO) can introduces intercarrier interference (ICI) among subcarriers and multiple access interference (MAI) among subchannels. Although some methods can be applied to enhance the initial synchronization, the CFO effect is not easy to be completely eliminated since different local oscillators are used at the transmitters. Hence, a CFO tracking loop and MAI reduction process are usually required even though an initial synchronization stage is performed with the residual CFOs within the range of tolerance.

Some approaches to dealing with CFOs in OFDMA systems can be found in the literature [2], [3], [4], [5], [6]. The method of CFO correction by cancelling the estimated offsets before the discrete-time Fourier transform (DFT) for different subscribers can be direct, which is an extension of the method used in single user OFDM [2]. But this method requires multiple DFT blocks and causes MAI due to different offsets among subscribers. In [3], a new method, called the CLJL scheme, was proposed to compensate for the effect of CFOs after the DFT using circular convolution. Although the CLJL scheme reduces the required number of DFT blocks, the multiuser interference (MUI) components still exist in the compensated results. Huang and Letaief [4], called the HL scheme, proposed an iterative interference cancellation scheme to reduce the effect of MUI. The method in [4] can be regarded as a parallel interference cancellation scheme and they showed that only a couple of iterations are required to obtain a satisfying performance. Other methods [5], [6] considered an assumed return path for control information based on the maximum likelihood (ML) estimation of the synchronization parameters.

In those proposals [2], [3], [4], [5], [6], CFOs for different subscribers are estimated and then the corrections are performed independently. However, the residual MAI due to different CFOs degrades the bit error rate (BER) performance even with the use of the MAI cancellation scheme [4]. In this paper, we explore the effect of a common CFO (CCFO) in an OFDMA system. We find that the CCFO value before the DFT affects the ICI and MAI performance for different subscribers. Based on a proper adjustment of CCFO, the minimum mean square error (MSE) performances can be reached in an OFDMA system. The BER performance is compared by using the proposed algorithm with the original CLJL and HL schemes.


Proposed Common CFO Correction Scheme

Consider an N-point OFDMA system with P subscribers, and each subscriber is allocated N/P subcarriers. Let ∊i, i = 1,2,…, P, denote the residual CFO for subscriber i and ∊C the proposed CCFO to be corrected at the base station as shown in Fig. 1. Suppose y(i)n, n = 0, 1, …, N−1 is the ith subscriber's symbol after passing through the channel. The received symbol information at the base station consists of P subscribers' symbols along with their equivalent CFO effects and noises, which is given byFormula TeX Source $$r_n = \sum^P_{i=1}y^{(i)}_ne^{j 2\pi(\varepsilon_i+\varepsilon_C)n/N}+Z_n, n=0,1,\ldots,N-1\eqno{\hbox{(1)}}$$where Zn is the additive white Gaussian noise (AWGN). After removing the guard-interval, the vector form of the DFT output signal Rk, where k is the subcarrier index and 0 ≤ kN − 1, can be expressed asFormula TeX Source $$\eqalignno{{\bf R}(\varepsilon_C) &= {\rm DFT}_{\rm N}({\bf r}(\varepsilon_C))\cr&=\sum^{\rm P}_{i=1}{\bf Y}^{({\rm i})}\otimes {\bf C}^{(i)}+{\bf Z}\cr&={\bf Y}^{(m)}\otimes {\bf C}^{(m)}+\sum^P_{{i=1}\atop {i\ne m}}{\bf Y}^{(i)}\otimes {\bf C}^{(i)}+{\bf Z}&\hbox{(2)}}$$where ⊗ denotes the circular convolution operation, R(∊C) = [R0, R1,…, RN−1]T,r(∊C) = [r0, r1,…, rN−1]T, the vector Y(i) contains N-point signals of the ith subscriber, Y(i)k and Y(i)k = DFTN(y(i)n),C(i) is a column vector containing N-point values of the equivalent CFO effects, C(i)k and C(i)k = DFNN(ej2 π(∊i+ ∊C)n/N)/N, and the column vector Z contains the N-point DFT results of AWGN, Zk, and Zk = DFTN(zn). In (2), the first term is the mth subscriber's received signal and the second term is MUI due to the CLJL scheme. If the MUI can be ignored and the AWGN power is small compared with the signal power, we can approximate the mth subscriber's received signal asFormula TeX Source $$\eqalignno{\hat{\bf R}^{(m)}(\varepsilon_C) &= {\bf Y}^{(m)}\otimes {\bf C}^{(m)}\cr&\approx {\bf A}^{({\rm m})} {\bf R}(\varepsilon_C)&\hbox{(3)}}$$where A(m) is a diagonal matrix with the diagonal elements defined as A(m)(i + 1, i + 1) = 1 for i ∊ Ωm and 0 for i ∉ Ωm, where Ωm is the set of subcarriers allocated to the mth subscriber. Here, A(m) acts as the filter that keeps most of the mth subscriber's output power. From (3), we can restore the mth subscriber's signal Y(m) from Formula by removing the circular convolution operation in the following equation:Formula TeX Source $$\eqalignno{\hat{\bf Y}^{(m)}(\varepsilon_C) &={\bf A}^{({\rm m})} \left(\hat{\bf R}^{({m})}(\varepsilon_C)\otimes {\bf C}^{'({m})}\right)\cr&= {\bf A}^{({\rm m})}\left({\bf A}^{({\rm m})}{\bf R}(\varepsilon_C)\otimes {\bf C}^{'({m})}\right)&\hbox{(4)}}$$where C′ (m) denotes the inverse of C(m), which has components C′ (m)k and C′ (m)k = DNTN(ej2 π(∊m+ ∊C)n/N)/N.

Figure 1
Fig. 1. The common CFO correction scheme at the OFDMA receiver.

Effect and Estimation of CCFO

A. Effect of CCFO

Suppose X(m)k and H(m)k are the transmitted signal and the channel response of the mth subscriber at the kth subcarrier, respectively. From (2), after some mathematical manipulation we have [7]Formula TeX Source $$R_k = X^{(m)}_k H^{(m)}_k + ICI^{(m)}_k + MAI^{(m)}_k + Z_k\eqno{\hbox{(5)}}$$andFormula TeX Source $$\eqalignno{ICI^{(m)}_k &= \sum_{j\in\Omega_m}\left({\bf X}^{({m})}_j\cdot {\bf H}^{({m})}_j\right)\cdot D\left({2\pi(\varepsilon_m +\varepsilon_C-j+k)\over N}\right)\cr&\qquad \qquad \qquad \qquad \qquad \cdot e^{j\pi (\varepsilon_m+\varepsilon_C-j+k)(N-1)\over N}-X^{(m)}_kH^{(m)}_k&\hbox{(6)}\cr MAI^{(m)}_k &= \sum^P_{{i=1}\atop {i\ne m}}\sum_{j\in\Omega_i}\left({\bf X}^{({i})}_j\cdot {\bf H}^{({i})}_j\right)\cdot D\left({2\pi(\varepsilon_i +\varepsilon_C-j+k)\over N}\right)\cr&\qquad \qquad \qquad \qquad \qquad \cdot e^{j\pi (\varepsilon_i+\varepsilon_C-j+k)(N-1)\over N}&\hbox{(7)}}$$where D(x) = sin (Nx/2)/N sin (x/2). In (6) and (7), we notice that ICI and MAI are both influenced by the CCFO. To explore the effect of CCFO, we simply consider the case that the expectations of X(i)k and H(i)k are normalized. Then, the powers of ICI and MAI areFormula TeX Source $$P_{ICI^{(m)}_k} \cong \sum_{j\in\Omega_m}{\sin^2(\pi(\varepsilon_m+\varepsilon_C-j+k))\over N^2\cdot\sin^2\left(\pi\left({\varepsilon_m+\varepsilon_C-j+k\over N}\right)\right)}-1\eqno{\hbox{(8)}}$$ andFormula TeX Source $$P_{MAI^{(m)}_k} \cong \sum^P_{{i=1}\atop {i\ne m}}\sum_{j\in\Omega_i}{\sin^2(\pi(\varepsilon_i+\varepsilon_C-j+k))\over N^2\cdot\sin^2\left(\pi\left({\varepsilon_i+\varepsilon_C-j+k\over N}\right)\right)},\eqno{\hbox{(9)}}$$ respectively.

B. Estimation of CCFO

From (4), we can obtain the estimate of transmitted signals in relation to ∊C for the mth subscriberFormula TeX Source $$\hat{X}^{(m)}_k (\varepsilon_C) = \hat{Y}^{(m)}_k (\varepsilon_C)/H^{(m)}_k\eqno{\hbox{(10)}}$$where H(m)k denotes the estimated subchannel allocated by the mth subscriber. If the transmitted signal is X(m)k the MSE performance can be obtained byFormula TeX Source $$\zeta (\varepsilon_C) = \sum^P_{m=1}\sum_{k\in \Omega_m}\left\vert X^{(m)}_k - \hat{X}^{(m)}_k (\varepsilon_C)\right\vert^2\eqno{\hbox{(11)}}$$Here, we adopt the MSE criterion to find the optimum value of CCFO by minimizing the MSE performance.

By the steepest descent approach, the minimization of the MSE performance can be obtained by calculating the following recursion:Formula TeX Source $$\hat{\varepsilon}_C(n+1) = \hat{\varepsilon}_C(n) - \mu \cdot \nabla_{\varepsilon_C}\zeta(\varepsilon_C)\eqno{\hbox{(12)}}$$where μ is a step size which controls the convergence rate and the steady-state estimation accuracy. However, it is difficult to obtain the exact formulation of the divergence operation ∇C ξ (∊C) we approach the derivative asFormula TeX Source $$\nabla_{\varepsilon_C}\zeta (\varepsilon_C) = \lim_{\Delta\varepsilon \to 0}{\zeta(\varepsilon_C + \Delta \varepsilon_C)-\zeta(\varepsilon_C)\over \Delta \varepsilon}\eqno{\hbox{(13)}}$$where Δ ∊ is chosen as a small value approaching zero, for example, Δ ∊ = 10−7 in our numerical implementation.

It is worth to note that ξ (∊C) is not a quadratic function of ∊C, and hence, ξ(∊C) may have some local minimums with respect to ∊C as shown in Fig. 2. Some routine is required to search for the initial value at the region of the global minimum in order to guarantee optimal convergence. A simple method is to divide the whole searching region into several sub-regions as possible initial points Formula at first. Then, we can determine the optimal CCFO initial value by finding the minimum value of their MSE metrics from (12). Since this method is proposed for improving performance after coarse synchronization and channel estimation have been achieved by the preamble, pilots or decision feedback data can be used to avoid knowing X(m)k as required in (11).

Figure 2
Fig. 2. Global optimum of the initial CCFO value.

Simulation Results

The OFDMA parameters are set as follows: the FFT size N = 512, the length of guard-interval is 64, the number of subscribers P = 4 with block subcarrier allocation, eight pilots are used for each subscriber, and the Gray-coded 16-QAM signals are transmitted. The power profile of the multipath channel impulse response is assumed to be exponentially decaying with the characteristics E{| h(n)|2} = exp (−n/5),n = 0, 1, …, 11. For the purpose of estimating coarse CFOs and channel responses regarding the four subscribers, two training symbols as the preamble lead in the advance of the transmitted data. To avoid the interference among subscribers in using the preambles for initial estimation of CFOs and channel responses [4], the preambles are assigned on non-overlapped symbol instant slots for simplicity.

We randomly choose two sets of CFO values to see the influence of CCFO correction in this OFDMA system: One is with large CFO values for the four subscribers and denoted by CFO1 = [−0.1 0.3 0.25 −0.15], the other is with smaller values and denoted by CFO2 = [0.1 −0.1 −0.05 0.05]. After the coarse CFOs and channel responses are obtained from preambles, we search for the optimal initial value for CCFO estimation based on the first 30 decision feedback data symbols. Note that in these 30 data symbols, the CCFO value is assumed to be zero and the performance is equivalent to the conventional methods without using CCFO correction. Then, the initial CCFO is used in (12) for the convergence of optimal estimation.

In Fig. 3, we show the MAI power with respect to the CCFO value for the four subscribers in the CFO1 case. That is, for each subscriber we consider the interferences caused from others. It is obvious that for the four users, the optimum CCFO values is to eliminate the ICI effect, i.e., the CCFO values are supposed to be −0.1, 0.3, 0.25, and −0.15. However, the MAI effect caused by others may result in a different CCFO value to achieve a minimum MAI power. If we consider the overall MSE performance including the effect of ICI and MAI, we can plot the MSE results for different CCFO values with comparing the CLJL and the HL schemes in Fig. 4. From Fig. 4 for the CFO1 case, the optimum CCFO value for the CLJL scheme is at about zero and the value for the HL scheme is at about −0.1. Since the HL scheme can reduce the MAI effect, the MSE performance by using HL is better than that by using CLJL.

Figure 3
Fig. 3. MAI powers v.s. CCFO values for the CFO1 case.
Figure 4
Fig. 4. MSE performance v.s. CCFO values by using CLJL and HL schemes for the CFO1 case.

We show the BER performance for different schemes in Figs. 5 and 6 for different CFO compensation schemes. The direct method [2] and the CLJL method only compensate for the effect of CFO, the MAI effect is not dealt with. Thus, the HL method with cancelling MAI has better result than the two methods. However, the proposed method can search for the optimal MSE performance with the aid of adjusting the CCFO. Hence, the proposed method with the HL's MAI cancellation scheme outperforms others. In Fig. 5, the large CFO case (CFO1) is compared such that the significant performance difference can be noticed. However, in Fig. 6 the smaller CFO case (CFO2) is compared, and we can also observe the performance improvement by using the proposed method.

Figure 5
Fig. 5. BER comparison of different CFO compensation methods in OFDMA for overall performance with CFO1.
Figure 6
Fig. 6. BER comparison of different CFO compensation methods in OFDMA for overall performance with CFO2.


In conventional methods, the CFO effect can be compensated for by the CLJL scheme at the base station. In this paper, we show that the minimum MSE performance of the demodulated signals is related to a CCFO value which should be corrected in advance of the DFT operation at the OFDMA receiver. We explore the property of the CCFO and propose a new algorithm for CCFO estimation and correction. The performance of the proposed method with the HL scheme is superior to that of others in our simulations.


Dah-Chung Chang, Yen-Heng Lai, and Yi-Chia Hsu are with the Department of Communication Engineering, National Central University, No. 300, Jhongda Rd., Jhongli, Taoyuan 320, Taiwan E-mail:

This work was supported in part by the National Science Council of Taiwan under the contract NSC 96-2219-E-008-003 and NSC 97-2221-E-008-004.


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Dah-Chung Chang

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Yen-Heng Lai

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Yi-Chia Hsu

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