ABSTRACT

In this paper, we consider the frequency offset estimation for Orthogonal Frequency-Division Multiple Access (OFDMA) uplink (UL) transmissions. We first analyze the negative effect of Multi-Access-Interference (MAI) on OFDMA UL, and then propose two interference reduction/elimination methods, i.e., the Reduced-Rank-Projector (RRP) and Shift-Sampling-Projector (SSP) methods, to eliminate/reduce the heavy MAI due to the frequency offsets. Finally, we propose a new training sequence group named the Round-Robin Training Sequence Group (RRTSG), which has a high interference mitigation capabilities for OFDMA UL transmission. The Cramer-Rao Lower Bound (CRLB) for an unbiased frequency offset estimator in a Multiple Access (MA) system is also derived. Numerical results show that the proposed methods are suitable to eliminate/mitigate the effect of the frequency offset on OFDMA UL transmission.

1. Introduction

Orthogonal Frequency-Division Multiple Access (OFDMA) is an effective Multiple Access (MA) scheme that divides the total signal bandwidth into multiple orthogonal subcarrier groups, with each group being allocated to one user [1]. In uplink (UL) transmission, different users have different frequency offsets and result an interference-limited system [2]. Multi-Access Interference (MAI) may appear between neighboring users due to the frequency offsets.

The Bit Error Rate (BER) of Orthogonal FrequencyDivision Multiplex (OFDM) systems impaired by the frequency offset is analyzed in [3]. The effect of the frequency offset on OFDM/OFDMA has been analyzed considerably (see [4-6]). Performance improvement of OFDMA UL transmission through cooperative relaying is studied in [7,8], with either ergodic or outage information rate is analyzed in the presence of the frequency offset.

Although several classical frequency offset estimators [9-13] are proposed, their performance will degrade in an interference-limited environment. Since the frequency offset estimations are imperfect, MAI arises between different users. MAI cancelation in an OFDMA system is discussed in [14], where MAI can be eliminated correctly based on the perfect knowledge of the frequency offset of each user. A conventional estimator, such as that used in [15], is considered as a candidate for the frequency offset estimation for each user. Carrier Frequency Offset and I/Q Imbalances compensations in OFDM Systems by considering subcarrier allocation are studied [16]. Synchronization in OFDMA considering the effect of MAI has been studied [17-22]. OFDMA UL synchronization by exploiting the statistical character of the frequency offset is studied in [23], where the frequency offsets can be modeled as either uniform or Gaussian distributed random variables (RVs). For more references about OFDMA synchronization, please refer to, e.g., [24-26].

In this paper, we consider OFDMA UL frequency offset estimation based on the proposed training sequence. Subcarrier allocation for one user need not be contiguous in the frequency-domain. The MAI introduced by the non-zero frequency offset is analyzed first. The Reduced-Rank-Projector (RRP) and Shift-Sampling-Projector (SSP) methods are proposed to eliminate/reduce MAI due to the frequency offset. A new training sequence group named the Round-Robin Training Sequence Group (RRTSG) is also proposed for interference randomization. The quasi-synchronized transmission of accessing users is considered in this paper; i.e., both the time and frequency of these users have been aligned to the base station. Perfect timing synchronization is considered, and only UL frequency offset estimation is discussed in this paper.

The remainder of this paper is organized as follows. Section 2 proposes the OFDMA UL signal model. MAI analysis is provided in Section 3. Sections 4 and 5 discuss the interference elimination/reduction methods, where the heavy MAI elimination/reduction methods are proposed in Section 4, and an interference randomization method is introduced in Section 5. Section 6 analyzes the proposed schemes and discusses some numerical results. Finally, Section 7 concludes the paper.

Notation: and denote transpose and conjugate transpose of a matrix, respectively. represents the inverse of a matrix. The imaginary unit is., and are the real and imaginary part of $x$, respectively. denotes complex conjugate. A circularly symmetric complex Gaussian variable with mean and variance is denoted by . represents the -th element of vector. represents the -th element of matrix. means “Mod”. The identity matrix is, and the all-zero matrix is. and denote the mean and the variance of, respectively.

2. OFDMA UL Signal Model

In OFDMA, each user is modulated by complex data symbols from a signal constellation, e.g., phase-shift keying (PSK) or quadrature amplitude modulation (QAM). In a frequency-selective fading channel, the received signal at the base station can be represented as

, (1)

where represents the power allocation to each subcarrier of user,

with denoting the channel attenuation at the -th subcarrier, and stands for the subcarriers allocated to user. Note that and

,

(2)

with and representing the initial phase and normalized frequency offset of user, respectively. denotes the Inverse Discrete Fourier Transform (IDFT) matrix for the -th user (where only the subcarriers allocated to user are modulated). is the Discrete Fourier Transform (DFT) length. is the transmit vector of user, and without loss of generality, we assume that are independent and identically distributed (i.i.d.) complex RVs with zero mean and unit variance. represents the time-domain transmit training of user, in (1) is a vector of additive white Gaussian noise (AWGN) with, and represents the additive noise added to user.

UL synchronization for each user can be performed based on the received training sequence. Without loss of generality, we assume that the frequency offsets of different users are i.i.d. RVs. For an OFDMA UL with users accessing a base station, the Fisher Information Matrix (FIM) is given by

, (3)

where represents the northwestern

sub-matrix of, and is a vector. The CRLB of user is given by

(4)

From Appendix I, for an unbiased estimator, the CRLB is derived as

(5)

where and are defined as in Appendix I, , and are constants specified by the structure of the training sequence used by user and and stand for the Signal-to-Noise-Ratio (SNR) and Signal-to-Interference-Ratio (SIR) of user, respectively. (5) suggests that MAI will degrade the estimation performance of the UL synchronization, and the MAI elimination/reduction is critical in increasing the frequency offset estimation accuracy.

3. Interference Analysis in OFDMA UL

Note that in an OFDMA UL, the subcarrier groups allocated to different users are orthogonal, and this allocation eases the decomposing of the transmitted signal of each user. We consider two types of subcarrier allocation schemes: Block-based and Interleaved, as illustrated in Figures 1(a) and (b).

In an OFDMA UL, the receiver (base station) usually

pre-demodulates the signal of each user, and only signals projected into the signal space of interest will be demodulated. Since the pre-demodulation improves the effective Signal-to-Interference-plus-Noise Ratio (SINR), an optimum CRLB for the -th user is achieved as

(6)

The pre-demodulation of each user’s signal can be performed as

(7)

where and represents the interference from other users, i.e., MAI.

In order to analyze the MAI contributed by users other than user (the user of interest), we first assume that users access the base station, and then we make some presuppositions:

1) andwhere represents the set of subcarriers allocated to user.

2), where represents the cardinality of.

The average SINR of user can be represented as

, (8)

where

is the interference of the subcarrier contributed by the other subcarriers of user (ICI), and

represents the interference of the subcarrier contributed by the users other than.

When, the average SINR of user can be simplified as

(9)

where is a random variable (not necessarily Gaussian) with.

For a system with all the users running at the tracking phase (the frequency offset acquisition has been performed, and the remaining frequency offset is small), it is reasonable to assume that for each is an i.i.d. RV uniformly distributed in, where [23]. The averaged SINR of user k can be further approximated as

(10)

where is used to represent the average received SNR of user.

The SINR reduction due to MAI is shown in Figure 2. A considerable MAI of one user will be contributed by the interfering users because of their non-zero frequency offsets. For example, when, the averaged SINR loss at the objective user may be up to 36.7%.

4. Heavy MAI Elimination

In Section 3, we analyzed the MAI introduced by interfering users. Since a new accessing user’s instantaneous frequency offset may contribute a heavy MAI to its neighbors, the base station should first perform UL synchronization for that user before its totally access. The accessing user will then correct its synchronization errors based on the feedback information from the base station.

In order to illustrate the distribution ofInter-CarrierInterference (ICI) with respect to the distance between the object subcarrier (subcarrier) and the interfering subcarrier (subcarrier), we define

as the “ICI factor” between subcarriers and, and we know that the ICI between subcarrier and is proportional to. The upper figure in Figure 3 shows the distribution of as a function of, from which we know that the ICI factor diminishes as increases. The lower figure in Figure 3 illustrates the concentration of ICI energy, and it shows that most of the ICI energy added to one subcarrier is concentrated within a small range around the subcarrier. For a given ICI energy, the ratio of the concentration range to becomes smaller as increases.

4.1. Heavy MAI Elimination in Block-Based Subcarrier Allocation Scheme

In the Block-based allocation scheme, user is allocated a set of contiguous subcarriers, i.e., , and its marginal subcarriers suffer most of the MAI. If we optimize of user to demodulate only its less-interfered subcarriers (the marginal subcarriers of heavy MAI are left un-demodulated, or in other method, we leave wide enough guard band between neighboring users), most of its MAI can be eliminated. Based on this princeple, we propose a method named the Reduced-RankProjector (RRP) method, i.e.,. In order to maximize the SINR of user k, we have

(11)

where. here is defined as follows: without loss of generality, we first assume that the central frequency of is higher than that of and lower than that of. We also assume that user is interfered by user and user with its leftmost and right-most subcarriers, respectively. can be derived by deleting the left-most columns and the right-most columns from. Here, we also assume that the guard band between user and user is subcarriers, and that between user and user is subcarriers. Therefore,

and

where means the maximum integer part of, and.

Figure 4 illustrates the frequency offset estimation with the help of the RRP method. Three frequency-domain neighbors, i.e., user, user and user, have instantaneously large frequency offsets. Since user suffers the highest MAI, we perform a synchronization for user and user first. At that time, the signals of user and user are demodulated by using projectors and, respectively. Using the estimation results and, and can be derived to optimize. After that, is used to calculate and, and then and are optimized. The optimized and then can then be used to further improve the performance of and 1.

4.2. Heavy MAI Elimination in Interleaved Subcarrier Allocation Scheme

In Interleaved subcarrier allocation scheme (as illustrated in Figure 1(b)), there is no enough guard band to separate the different users, and the frequency offset immune capability will be degraded as a result. In this case, without loss of generality, we assume user to be the user of interest and that user acts as an interfering user to it. We also assume that the frequency offset estimation result of user is, where and represent the integer part and the fractional part of, respectively. Here, we assume that has already been corrected and that MAI is contributed only by. In order to eliminate this kind of MAI, we propose a method named the Shift-Sampling-Projector (SSP) method. We first perform the frequency offset estimation for user to obtain. The Shift-Sampling-Projector for user can be represented as

where

and

as illustrated in Figure 5. Note that in Figure 5 is given by (2) with being replaced by. If is accurate enough, the interference from user can be eliminated. The estimation result, i.e., , then will be used to optimize in order to further increase the accuracy of. By using the SSP method, the variance error of will satisfy

.

5. Frequency Offset Estimation with MAI Reduction/Randomization

As mentioned earlier, instantaneously heavy MAI introduced by frequency-domain neighboring users should be eliminated before a user totally access the base station. In this section, we first discuss interference elimination by using the Successive Interference Cancellation (SIC) method, and then propose a frequency offset estimation algorithm based on an interference reduction scheme.

5.1. SIC with MAI Elimination

Let us assume that users access a base station, and that are perfectly known at the receiver (base station). Based on the joint PDF of and, i.e., , (as well as) can be jointly estimated as

(12)

(16) provides a feasible way to estimate the signal/frequency offset of each user: the signals of different users can be estimated one by one based on. After performing a synchronization for the first users, user will estimate based on. In theory, if are accurate enough, can be estimated successfully2. However, if one previous user has a large estimation error, it will propagate to; i.e., the estimation accuracy of depends on the estimation accuracy of v. For a large, estimation performance by using the SIC method will degrade because of the error propagation. Figure 6 shows the SIC-based frequency offset estimation performance. In this figure, a total of 16 users access a base station simultaneously. In order to reduce the error propagation in SIC, more than one iteration is required when performing MAI elimination, and the algorithm converges after 5 iterations.

5.2. RRTSG for MAI Randomization

The SIC-based method aforementioned requires both CSI and initial phase information, and in addition, more interfering users imply more iterations being required. Therefore, SIC is inefficient in a multiuser environment. Some differential algorithms that are independent of the initial phase and the wireless channel provide a new way for a solution [9,15]. In these differential algorithms, the training sequence contains repetition information, and the frequency offset can be estimated at the receiver by estimating the phase rotations between these repetitions. However, these conventional algorithms were originally proposed for DL synchronization, and they are not immune to MAI.

In this section, a new training sequence group named the Round-Robin Training Sequence Group (RRTSG) is proposed to perform OFDMA UL synchronization. A total of training sequences are included in RRTSG, with each training sequence being composed of two

training symbols, where the second symbol is generated by copying data from the first one (but at a different order). When generating the second training symbol of the -th () training sequence, we simply copy data from the already generated first symbol with its last $d$ samples shifted to the front. After padding each symbol with its corresponding CP, the -th training sequence is generated as:

(13)

where represents the CP length, and

is a pre-coding matrix for user in generating the -th training sequence.

RRTSG can randomize MAI when performing UL synchronization, and an unbiased frequency offset estimator is always achieved, provided that each user is allocated a unique training sequence. A Best Linear Unbiased Estimator (BLUE) of based on the -th training sequence is provided by:

(14)

where

, ,

and represents the element of. We can easily prove that the BLUE estimator provided by (18) is conditionally unbiased, and that its CRLB is found to be

(15)

Among all the training sequences in the proposed RRTSG, the minimum CRLB is achieved when.

In order to make the estimator work properly,

should be satisfied for each, and this requires the normalized frequency offset estimation range of the -th estimator to be limited within

.

6. Numerical Results

In our simulation, an OFDMA UL transmission with a bandwidth of 10 MHz and DFT length of 1024 is considered. A length-64 cyclic-prefix is padded to the front of each symbol or training sequence. Wireless channel parameters are defined in Tables 1 and 2.

Figure 7 illustrates the performance improvement achieved in RRP by eliminating the instantaneously heavy MAI. In this simulation, we assume that 10 users access a base station simultaneously, and users #1, #2 and #3 are assumed to have instantaneously large frequency offsets. The frequency offsets for the other 6 users are assumed to have already been estimated and corrected, and their residual frequency offsets are assumed to be uniformly distributed within (–0.01, 0.01). This simulation parameters for users #1, #2 and #3 are presented in Table 1. User #2 is assumed to be the user of interest. The frequency offset estimation can be improved by using the RRP method. At low SNR, a variance error of less than (or) can be obtained if we reduce the demodulated subcarrier range to [–22, 23] (or [–20, 21]).

Although the RRP method can achieve a considerable performance improvement with respect to MAI elimination, this method is not applicable in the Interleaved

subcarrier allocation scheme, which relies on the SSP method. Table 3 illustrates the parameters of this simulation. The simulation results are illustrated in Figure 8. Without using SSP, the variance error will always be higher than, no matter how high the SNR is. When the SSP method is applied, a considerable performance improvement can be obtained, even with imperfect frequency offset estimation in the interfering users. For example, at an SNR of 16dB, when the SSP method is applied with a 30% compensation error being assumed in the interfering user, a variance error smaller than can be obtained. The variance error can be further reduced to about if the compensation error is reduced to 20%.

When the number of accessing users is large, MAI elimination is impractical to perform. As a feasible MAI reduction scheme, MAI randomization is usually utilized. The proposed RRTSG has a high ability in performing MAI randomization in OFDMA UL transmission. Figure 9 illustrates the performance comparison between Moose’s algorithm and the proposed RRTSG-based algorithm. The simulated environment is defined by Table 2. The simulation results show that the proposed RRTSG-based algorithm outperforms Moose’s algorithm with respect to interference mitigation capability. Without using any heavy MAI elimination method (e.g., RRP or SSP), a biased frequency offset estimator is obtained by using Moose’s algorithm, so that its variance error is much larger than that of the proposed RRTSG-based algorithm. For example, a variance error of can be achieved at a high SNR by using the proposed RRTSG-based algorithm, as compared to the variance error of obtained in Moose’s algorithm. The proposed MAI elimination methods proposed can improve the performance of both algorithms considerably. When most of the heavy MAI is eliminated by using RRP, e.g., the subcarriers ranging from & [26,55] are demodulated, a variance error of about is achieved by using the proposed RRTSG-based algorithm at an SNR

of 15 dB. This error is still much smaller than that achieved by using Moose’s algorithm, i.e.,.

7. Conclusion

This paper discussed OFDMA UL frequency offset estimation and showed that the performance improvement in estimation benefits from MAI elimination/reduction results in improved estimation accuracy. Two MAI elimination/reduction methods, i.e., the RRP method and the SSP method, were proposed for a Block-based subcarrier allocation scheme and interleaved subcarrier allocation scheme, respectively. In order to randomize the MAI contributed by the other synchronized users (because of their non-zero residual frequency offsets), a new training sequence group named the RRTSG was also proposed. Numerical results proved that the proposed RRTSGbased algorithm outperforms the conventional algorithm considerably in terms of frequency offset estimation accuracy because of its MAI randomization/mitigation capability.

REFERENCES

Appendix

Derive the Fim

From [15], the -th element of FIM can be represented as

(16)

where and

Note that

(17)

where

represents the covariance matrix of user, and

is the MAI matrix of user contributed by user. For the non-zero frequency offsets, we have. is decomposed as

(18)

where is a Unitary matrix,

with representing the -th eigenvalue of, and

with representing the -th eigenvalue of. Therefore, we also have

(19)

From the above discussion,

(20)

when, the -th element in FIM can be represented as

(21)

and the -th element in FIM is given by

(22)

NOTES

^*This work was supported by the National Basic Research Program of China (2012CB315905), the National Natural Science Foundation of China under Grand No. 61172050, National Key Projects (2012ZX 03001029-005 and 2012ZX03001032-003), Beijing Science and Technology Program under Grand No. 2009ZX03006-004, Chang Jiang Scholars Program of the Ministry of Education of China, and Program for New Century Excellent Talents in University.

^#Corresponding author.

¹This kind of operation can be applied in the scenario of multiple users transmitting and receiving jointly, e.g., multiuser Multiple-Input Multiple-Output (MIMO).

²SIC algorithm requires a knowledge of channel state information (CSI) at the base station, and the CSI information can be obtained through channel estimation (please refer to, e.g., [26]), which is beyond the scope of this paper.