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  • Research Article
  • Open Access

Multiple CFOs in OFDM-SDMA Uplink: Interference Analysis and Compensation

EURASIP Journal on Wireless Communications and Networking20092009:909075

https://doi.org/10.1155/2009/909075

  • Received: 1 July 2008
  • Accepted: 11 March 2009
  • Published:

Abstract

In OFDM-based space division multiple access (SDMA) systems, multiple users are served by a multiantenna base station simultaneously on the same frequency resources. In the uplink, each user's signal may be distorted by an independent carrier frequency offset (CFO), which impairs the orthogonality of the subcarrier signals and, if not properly compensated, results in performance degradations. We analyze the influence of multiusers' CFOs on the signal transmission in the OFDM-SDMA uplink and derive suitable bounds for the achievable signal-to-interference conditions. By modifying the signal model suitably, we develop a simple scheme for partial compensation of the CFO distortions. It allows to maintain the subcarrier-wise channel equalization and thus is well suited to be applied for a real-time system implementation. However, as CFOs impair the cyclic structure of the OFDM symbols, our scheme is not able to compensate for the entire distortion. The remaining interference is treated as additional noise, which limits the supported size of the CFOs.

Keywords

  • Orthogonal Frequency Division Multiplex
  • Discrete Fourier Transform
  • Orthogonal Frequency Division Multiplex System
  • Carrier Frequency Offset
  • Orthogonal Frequency Division Multiplex Symbol

1. Introduction

A promising solution to lead wireless communication systems toward high spectral efficiencies is the combination of the orthogonal frequency division multiplexing (OFDM) together with the space-division multiple access (SDMA) technique [1]. In the SDMA uplink, multiple users communicate simultaneously with a multiantenna base station (BS) on the same frequency resources by transmitting their signals on different spatial layers. OFDM is a favored technique for the transmission in frequency-selective channels, as it facilitates the equalization process while at the same time enabling high spectral efficiencies. However, one of its deficiencies is its high sensitivity towards time-variant distortions. In general, these destroy the orthogonality of the single subcarrier signals and give rise to the so-called inter-carrier interference (ICI), limiting the achievable system performance [2, 3]. One source for time-variant distortions is the carrier frequency offset (CFO), owing to a mismatch between the oscillators at the transmitter and receiver sides. While estimation and compensation of CFO distortions in a single user link are fairly easy and conveniently solved [46], coping with different CFOs from multiple users in any OFDM-based multiuser uplink is much more challenging, as all CFOs need to be estimated independently, and the conventional techniques for compensation do not apply.

The influence of CFOs from multiple users in an OFDM-based uplink has been studied extensively in the context of OFDMA systems, where simultaneous access is granted to multiple users by individually assigning distinct sets of subcarriers to them [79]. An overview of existing synchronization techniques together with a sound summary of the general requirements for uplink synchronization can be found in [10]. Estimation of multiple users' CFOs can be performed based on blind techniques exploiting specific properties of the utilized OFDM signals and their statistics [1115] or based on pilot-based techniques [16, 17]. For CFO compensation, the simplest approach is to feed back the estimated CFO to the corresponding user terminal, so that it may adapt its oscillator accordingly or apply a precompensation to its transmit signal [11]. However, the drawback of this feedback approach is that large delays may occur before the CFOs are properly compensated. There also exist some proposals for CFO compensation to be carried out directly at the receiver by adequate signal processing. These approaches are either based on the inversion of a high dimensional matrix representing the ICI-affected channel for a complete OFDMA symbol [18, 19], or they make use of successive interference cancelation techniques [20], which may be performed in an iterative fashion [21]. Unfortunately, all these approaches result in a significant increase of computational complexity compared to common OFDM processing, whose favorable property is to enable an independent subcarrier-wise processing. Although the complexity of the aforementioned approaches based on matrix inversion can be further reduced if specific properties of the signal model are exploited [15, 22, 23], it still remains considerable. A suboptimum solution maintaining the subcarrier-wise signal processing at the receiver is presented in [24]. The user signals are separated first, whereafter they are individually compensated for their user-specific CFO. Although not all ICI can be removed, a satisfactory performance is achieved if the CFOs do not become too large.

The major difference in OFDM-SDMA systems is that the channel is enhanced by a spatial dimension. To separate the users' signals, knowledge of the SDMA channel per subcarrier is required. With CFO distortions present, solutions to estimate the SDMA channel have been proposed in [25, 26]; joint estimation of SDMA channels and the users' CFOs can be found in [2729]. Contributions [26, 28] also provide approaches to compensate for the CFO distortions at the receiver, which, however, have complexity demands that are similar to the OFDMA techniques mentioned earlier.

The work in this paper was motivated by seeking for a simple receiver-based CFO compensation method for the uplink of an OFDM-SDMA system. Hereby, the subcarrier-wise channel equalization is supposed to be maintained to facilitate implementation in a real-time system. Therefore, we resort to the basic idea from [24] and develop a system concept where the user signals are first separated by common OFDM-SDMA equalization and compensated for their individual CFO distortions afterwards. As this approach is clearly suboptimum, the major focus of our work lies in the proper analysis of the achievable signal conditions with respect to the amount of interference that remains in the system after such compensation. In particular, we derive closed-form expressions characterizing the bounds for the signal-to-interference ratio (SIR) before and after CFO compensation, which are verified by numerical bit-error rate analysis. This way, we obtain insights into the suitability of the approach and reveal the limits of its application range.

Based on our results, it turns out that the proposed CFO compensation concept operates conveniently only if the size of the CFOs present in the system can be kept below a few percent of the subcarrier spacing. Therefore, the approach has to be seen as a technique for fine-synchronization. Correspondingly, a coarse-frequency synchronization of all users' signals has to be ensured. This coarse synchronization can be achieved by a frequency-advance, where terminals precompensate their signals with the CFO estimated in the downlink phase. The concept of frequency-advance was recently realized in a practical system, as reported in [30]. In [31], we already presented the basic idea of this work and initial analytical results. Here we extend the analysis to support linear receivers providing spatial diversity gains, add the case of noncompensated CFOs for illustrative comparison and provide a refined update of the CFO compensation process to be carried out in frequency domain, which facilitates implementation.

The paper is structured as follows: Section 2 introduces the OFDM signal model based on vector notation. As a preparation for analysis of the OFDM-SDMA system, we determine the SIR conditions for a single antenna OFDM link in Section 3. Hereafter, the model is modified to form the basis for the simplified CFO compensation process in OFDM-SDMA systems. In Section 4, we analyze the SIR conditions in the OFDM-SDMA system and derive bounds for the two cases where CFOs are compensated according to the proposed scheme and where they are not. These bounds are verified by simulation results in Section 5.

2. Signal Model

Notation. We use bold capital letters to denote matrices and bold letters for vectors. Scalars are written in italics. and denote conjugate transpose and conjugate operator, respectively. refers to the trace operator. represents a diagonal matrix, whose diagonal is constituted of vector . denotes the expectation operator.

2.1. Vector Notation of OFDM

Consider an OFDM system with a total of subcarriers. The transmission equation for a CFO-distorted single-input single-output (SISO) link is given by
(1)
is the data vector comprising the data symbols constituting the OFDM symbol, is the discrete Fourier transform (DFT) matrix, and and are permutation matrices used to append and cut the cyclic prefix (CP) of length samples. Further, is the Toeplitz channel matrix constituted from the channel impulse response (CIR) where . Finally,
(2)
is the CFO distortion matrix, where the phase rotation factor is defined as , with being the CFO normalized to the subcarrier spacing. For (no CFO), the effective channel
(3)
yields a diagonal matrix, whose elements on the diagonal represent the -point DFT of the CIR . By a few simple transformations, the diagonal matrix can be restored in (1), yielding
(4)
where we introduced
(5)

Note that is unitary thus equals the identity matrix .

2.2. OFDM-SDMA Signal Model

Next the focus is turned to an OFDM-SDMA system, where single-antenna terminals transmit their signals simultaneously to an -antenna base station on the same frequency resource. The users' transmission signals propagate via different paths and will be marked with different spatial signatures, which enable the multiantenna receiver to separate and recover the users' transmission signals.

For the system model, the OFDM signal vectors from the users are stacked into one large vector of dimension . Correspondingly, the OFDM reception vectors are stacked into one large vector . Each user may have an individual CFO, resulting in different CFO distortion matrices , which are generated from individual phase factors . For simplicity, let us assume the number of users to be . Based on the signal model in (4), the transmission equation in the OFDM-SDMA system reads
(6)

where each of the single user/receive antenna links is characterized by its own diagonal channel matrix .

2.3. Statistical Channel Model

Within this paper, we will assume Rayleigh-fading conditions for the discrete CIR, meaning that the channel coefficients are drawn independently from complex Gaussian distributions with mean power . For represents the power delay profile (PDP) of the channel, which is assumed to be monotonically decreasing for increasing . Furthermore, we assume the channel to be passive, that is, the sum of the mean powers of all channel coefficients is equal to unity, . To specify suitable bounds within our analysis, we will frequently use a uniform PDP with constant power for all channel taps, which is defined as for all . From these assumptions, it follows for the subcarrier channels that they behave like random variables which are drawn from complex Gaussian distributions with unit power. The correlation between the channels at adjacent subcarrier positions is characterized by the frequency-domain autocorrelation function , where refers to the distance between subcarriers. is obtained from the -point DFT of the PDP, that is,
(7)

In the OFDM-SDMA system, the channels of the single antenna links are characterized by the same statistical properties, but are assumed to be statistically uncorrelated. In particular, we assume all channels to have identical channel length and identical PDP, which may be reasonable for user terminals experiencing non-line-of-sight (NLOS) multipath fading.

3. Analysis of Single-Antenna OFDM Link

4. SIR Analysis in OFDM-SDMA System

Recall the OFDM-SDMA transmission equation from (6). If we want to equalize the effective channel completely, the only viable approach based on linear techniques is to invert the entire channel matrix —which relates to the approach for OFDMA systems conducted in [18, 19]. However, this matrix is of dimension , and hence the complexity of this approach will quickly become infeasible for practical realizations. Although complexity can be reduced by exploiting the block-diagonal band structure of this matrix, it still remains considerably high. Moreover, as CFOs induce phase rotations of the effective subcarrier channels over time, the matrix changes every OFDM symbol and thus has to be recomputed frequently, which increases the complexity for the inversion-based compensation even further.

An equalization approach that maintains the subcarrier-wise signal processing for the equalization and thus requires low complexity demands can be enabled if we alternatively adopt the signal model (17) derived in Section 3.2. Herewith, the compound channel can be written in the structured form:
(30)
The OFDM-SDMA transmission equation then yields
(31)
where , and are the matrices constituting the compound channel matrix above. Evidently, this notation enables the two-step equalization process introduced in the previous section. We first equalize the channel contained in matrix by a subcarrier-wise equalization of the flat-fading SDMA channel and thereby spatially separate the single user signals. The separated user signals may then be compensated individually for their CFO distortions as described in Section 3.4. The entire receiver processing for the simplified CFO compensation in the SDMA system is illustrated in Figure 2.
Figure 2
Figure 2

Receiver processing for simplified signal reconstruction with CFO compensation in the SDMA uplink.

In what follows, we will analyze how the CFO-induced interference will affect spatial diversity gains that can be achieved with a linear multiantenna receiver. As there is some correlation between signal and interference channels, distortion effects from the interference can no longer be expected to be similar to the one of AWGN. In particular, we will analyze the degree of correlation between the channel of the useful signal and the interference channels and derive SIR bounds describing the equivalent situation for AWGN. Analysis will be carried out for the case of no CFO compensation and compensation according to the proposed scheme separately.

4.1. Spatial Diversity Gain

In a brief intermezzo, we derive the basic relations concerning spatial diversity gains that are achievable with linear receivers in case of correlated signals. These relations form the basis for the analysis of the signal conditions in CFO-distorted OFDM-SDMA systems, which will be performed in the succeeding subsections. In particular, we examine here how interference that propagates via a correlated channel will affect the signal conditions at a multiantenna receiver providing spatial diversity gain . Following the notion from [34], the spatial diversity gain can be illustrated by assuming a maximum ratio combining (MRC) receiver that combines the signals from independent receive antennas. Assume a signal with mean power , which is transmitted via independent Rayleigh-fading channels with unit mean power. At each receiving antenna , the signal is distorted by AWGN with power . MRC operation then yields a post-MRC signal-to-noise ratio (SNR) of . The SNR thus is increased by factor compared to the SNR of the signal at a single receive antenna.

Instead of AWGN, we consider an interfering signal with mean power now. The signal at th receive antenna reads
(32)
Let the two signal be uncorrelated, while some correlation between the two channels is assumed. Both variables are assumed to be zero-mean Gaussian variables with variance . The correlation between both variables can be characterized by the correlation coefficient defined as [35]
(33)
where cov stands for the covariance of the two variables given in the parentheses. According to [35, Theorem  10.1], the distribution of conditioned on can be characterized by the two measures:
(34)
Accordingly, can be rewritten as
(35)
where we introduced a new Gaussian variable with zero mean and unit power, which is independent of . Substituting this equation in (32) yields
(36)
MRC operation delivering the spatial diversity gain is carried out by multiplying each received signal with the conjugate channel seen by the useful signal and summing up the signals over all receive antennas: . Within this summation, the signal portions from the first two components in (36), and , which both depend on , add up constructively, yielding a mean power of
(37)
after MRC operation. In contrast to that, the signal portions from the third component in (36), , add up with arbitrary phase, so that the mean power for these signal portions yields after MRC
(38)
Now let, for simplicity, . With the aforementioned results, we obtain for the post-MRC SIR
(39)
clearly revealing that the spatial diversity gain factor is diminished by
(40)

Thus, represents the effective SNR loss factor owing to the channel correlation .

4.2. No Compensation of CFO Distortions

Now we turn our focus back on the signal conditions in the OFDM-SDMA system in case the ICI distortions are not compensated. Consider the signal received at antenna , which, according to (6), is given as
(41)
In case all subcarriers carry signals with identical transmit power, the statistical properties of the ICI are identical for all the elements contained in . Therefore, we carry out the analysis exemplarily for the first element of vector , denoted as . To separate the ICI from the useful signal, we define as the first row vector of matrix , where the first element has been replaced by zero. The transmission equation then yields
(42)
where is the channel coefficient for the first subcarrier extracted from matrix , and is the transmit signal of user at the first subcarrier. We set as the useful signal. An appropriate equalizer is able to remove the signal portion if estimates of the channels can be obtained with sufficient quality. The two scalar products within (42), however, will remain in the system as ICI. The signal structure in the aforementioned equation is now similar to (32), and hence we can use the results from the preceding subsection to determine achievable spatial diversity gains here. Clearly, the two scalar products representing the ICI in (42) are constituted of multiple interfering signals. However, as all elements within vector are assumed i.i.d., each scalar product can be modeled by a single random variable, whose power is constituted from the sum of powers from the single elements in . In particular, we yield for the power of the interfering channel:
(43)
where we used the upper bound for presented in Section 3.1. The covariance between the useful channel and the interference channels is determined by
(44)
As channels from different users are assumed uncorrelated, yields a vector with nonzero entries for only. The elements of the covariance vector can be characterized by the function
(45)
with being the subcarrier correlation function defined in Section 2.3. The total power of the covariance vector is determined as
(46)
which can be read as the power of the covariance of an equivalent random process based on a single random variable. With these results, we can determine a measure representing the correlation between the useful channel and the sum of interference channels, which is calculated equivalently to the correlation coefficient in (33):
(47)

Evaluating this measure for varying reveals that for , suggesting that the useful channel and the interference channels for the ICI generated from are nearly fully correlated. Evidently, this results mainly from the high-frequency correlation of subcarrier channels that is valid for .

Consequently, we can conclude here that if an MRC-like signal combination is performed at the multiantenna receiver, not only the signal portions of the useful signal but also the ones of the interference from will add up fully coherently. In contrast to that, there is no correlation between the useful channel and the interference channels , as the covariance of the corresponding channels yields zero. Consequently, this distortion will behave similarly to AWGN. Resorting to the derivation of the SIR in (39) in the preceding section, we yield for the achievable SIR in an OFDM-SDMA system with spatial diversity gain :
(48)
where we have used the bounds for and from Section 3.1 for the total power of useful signal and ICI, respectively. This result shows that in the OFDM-SDMA system with diversity gain , the ICI power generated by any user is effectively attenuated by factor (i.e., the diversity gain can be realized completely), while the ICI generated from the CFO of user himself is fully preserved (i.e., no diversity gain is achievable). If all users have a CFO of the same size, for all , then the effective reception SIR (referring to the mean power of each user's signal measured at any receive antenna ) for the equivalent AWGN case can be given as
(49)

This result is equivalent to the SIR bound for the single-antenna case (12), reduced by the effective SIR-loss factor .

4.3. Compensation of CFO Distortions

Next we consider the case where the CFO distortions are compensated according to the proposed concept. Then interference results from the signal components contained in matrix in (31) only, and the signal received at antenna reads
(50)

Again, we define as the useful signal. The proposed equalization and ICI compensation concept removes the interference from as well as the the ICI induced by , and correspondingly solely the interference from remains in the system. Equivalently to the analysis carried out in the preceding subsection, we will now determine the correlation between useful channels and the channels of the residual interference to specify achievable spatial diversity gains. However, to ease analysis here, we initially focus on the entire channel matrices and to specify the overall statistical properties. Afterwards, we determine the signal conditions per subcarrier signal by averaging over the total subcarriers of the system.

The mean power of the interfering channel per subcarrier amounts to
(51)
For the bound, we used the result from (23). Correspondingly, the mean power of the useful channel yields
(52)
The covariance between useful channel and interfering channel can be characterized by the covariance matrix:
(53)
As is constituted of the channel coefficients related to channel , the covariance matrix will have nonzero entries for only. The corresponding matrix can be determined as follows. Using the definitions of from (14) and from (17), can be written as . For the moment we will exclude the outer DFT matrices and determine the expectation value of the inner matrix product. is a circular Toeplitz matrix based on the channel impulse response , and was shown in Section 3.2 to be a matrix with zero entries except for the submatrix found in its upper right corner. The expectation value of the product of these two components thus yields a matrix with zero entries except for the antidiagonal submatrix in its upper-right corner, whose antidiagonal elements represent partial sums of the channel power weighted by :
(54)
with defined in (16). From the covariance matrix, we can determine the mean power of the covariance between useful and interfering channels per subcarrier signal according to
(55)
where the upper bound is obtained for a uniform PDP. Note that . Similar to (47), we can now determine a measure equivalent to the squared correlation coefficient:
(56)
Assuming again a receiver with spatial diversity gain , we may now determine the SIR for the useful signal after an MRC-like signal combination over independent observations. Resorting to the derivation of the SIR in (39), we yield for the interference from a mean power of , with the interference power according to (25) and the SIR loss factor from (39). As all other interference channels are uncorrelated with the useful channel , the corresponding interference adds up incoherently, yielding a mean power of . Hence, we obtain the post-MRC SIR
(57)
If we have multiple users who all have a constant CFO, that is, for all , the effective reception SIR at any antenna for the equivalent AWGN case can be bounded by
(58)

where we used the bound for from (25), and should be used as specified in (56). This expression is equivalent to the SIR bound found for the single-antenna case in (28) reduced by the effective SIR-loss factor . Note here that the CFO-induced interference scales with the number of parallel SDMA users . In case of full correlation ( ), the SIR-loss factor is identical to , the factor found in case of no CFO compensation in (49). As a major result, we conclude here that the correlated interference from the CFO distortion results in an increase of the effective SIR-loss if a receiver with spatial diversity gain is employed.

5. Simulation Results

In this section we will provide numerical simulations to verify our analytical results found in the previous sections. For the simulations, we assume OFDM signal transmission via a noisy channel, that is, the transmission equation (6) is now given by
(59)

where is a vector consisting of AWGN samples with power . Thus, the mean reception SNR amounts to for the signal of any user at any receive antenna. As we have indicated that the CFO-induced interference can be expected to behave like AWGN, it can be assumed that this interference degrades the interference-free AWGN performance (i.e., no CFO is present) according to the amount of interference power. In particular, if the SNR is equal to the CFO-induced SIR, we can expect that the transmission experiences a performance degradation of 3 dB compared to the interference-free case. (As interference and AWGN are assumed to be independent, their joint distortion can be considered as Gaussian-like with power equal to the sum of powers from the two independent processes.) This basic principle will be used to verify the SIR bounds derived in the preceding sections.

We consider an OFDM-SDMA system with subcarriers, where single-antenna user terminals are granted simultaneous access. For the bounds to be tight, all subcarriers are occupied with transmission symbols from both users. The channel between each antenna link is modeled as Rayleigh-fading with channel taps and a uniform PDP. The normalized CFO is fixed to . As a performance measure, we use the bit-error rate (BER) that is achieved for an uncoded transmission of uncorrelated 16QAM symbols, averaged over both users. We use a zero forcing (ZF) equalizer to equalize the channel distortions and spatially separate the user signals per subcarrier. The diagonal channel from (31) as well as the CFOs are assumed to be known perfectly at the receiver.

Based on the signal model (17), we first examine the achievable performance for a single-antenna link (SISO). Results are given in Figure 3. The solid bold line shows the achievable BER performance in case no CFO is present. The suggested compensation approach shows a significantly degraded performance. At an SNR equal to the SIR bound (28), which amounts to 19 dB for the given parameter setting, it clearly exhibits a performance loss of 3 dB. This observation thus verifies the bound derived in (28).
Figure 3
Figure 3

BER performance of SISO system distorted by normalized CFO .

The performance curve of the CFO compensated system runs into an error floor for high SNR that corresponds to the BER performance achievable with the CFO-free performance at about 22 dB—which is about 3 dB higher than the SIR bound. The reason for that can be found in the distribution of the interference generated from the distortion terms in in (17). Note that the values in are generated from products of the independent random variables in and the data symbols in , which are all assumed to be Gaussian. The resulting distribution function for the values in is thus in general no longer Gaussian. Instead, we observe that the majority of the values from this distribution is much more concentrated around their mean than in the Gaussian case. Due to this fact, the achieved error floor is significantly lower than it would be if the interference behaved like Gaussian noise with identical power. However, it is worth noting that with increasing and thus with an increasing number of independent variables in , the distribution of the values in approaches the Gaussian case—thanks to the central limit theorem.

If we apply the CFO compensation technique that removes the ICI from the subcarriers in close vicinity only (see Section 3.4), we obtain the performance given by the dashed line for . Interestingly, for the choice of according to given in (27), the CFO compensation accounting only for some of the ICI distortion achieves a slight performance improvement compared to the full CFO compensation. Obviously, this is a benefit related to the correlated interference from in (17), as detailed in Section 3.4.

If we do not compensate for the ICI caused by the CFO but compensate for the CPE only, which corresponds to the case of applying the compensator (29) with , we obtain the performance represented by the uppermost curve. For an SNR equal to the bound in (12), which amounts to 15 dB for the given parameter setting, we clearly observe a performance loss of 3 dB compared to the performance where no CFO is present.

For the 2-user SDMA case, we consider ZF equalization to separate the signals of the different users. In [34] the diversity gain delivered by the ZF receiver has been shown to yield . For our examinations, we consider two cases: a receiver with and antennas, providing a diversity gain of and , respectively. Performance results are shown in Figure 4. The dashed curves refer to , while the solid curves refer to . The curves representing full CFO compensation according to the proposed scheme exhibit a 3 dB performance loss at an SNR equal to the SIR from (58) compared to the curve of CFO-free transmission, which amounts to 16 dB for and 15 dB for , respectively, for the given parameter setting. These losses are highlighted in Figure 4 by the horizontal black lines, clearly verifying the bound derived in (58). As in the SISO case, we observe that we can achieve a slight performance improvement if we use the simplified CFO compensation process based on (29) with . In case we do not compensate the ICI caused by the CFO, we achieve a severely degraded performance, which clearly exhibits a 3 dB performance loss at an SNR of 12 dB for and 10 dB for , respectively, corresponding to the analytical bound (49).
Figure 4
Figure 4

BER performance of 2-user SDMA system distorted by normalized CFO with ZF receiver. Dashed line: diversity gain . Solid line:

In Figure 5 we examine the behavior of the BER when the CFO compensation process based on (29) is applied for different values of the delimiter . We focus on a constant SNR  dB, which reflects the BER of the error floor for . For the selected values of , the subcarrier correlation range from (27) amounts to and , respectively. Interestingly, the corresponding curves exhibit their minimum at and , respectively, which is the nearest integer to . Hence, selecting indeed seems to be a good choice. This result leads us to the conclusion that it definitely suffices to consider only the subcarrier signals in closest vicinity within the CFO compensation via (29).
Figure 5
Figure 5

BER performance at SNR = 20 dB versus delimiter . .

To illustrate the performance degradation caused by the incomplete compensation of the CFO effects in the OFDM-SDMA system, we specify the effective SNR loss SNR based on the ratio of the interference power bound from (58) and the AWGN power as done in [8], which yields (in dB)
(60)
The numerical evaluation of the effective SNR loss for various CFO sizes is depicted versus the SNR in Figure 6; the corresponding parameter setting is specified in its caption. In accordance with the observations drawn from Figure 4, where evaluations where based on a CFO of size , the corresponding curve indicates here a 3 dB SNR loss at an SNR 16 dB. For comparison, we also added the SNR loss for the case of no ICI compensation (dashed curves), where we used the interference power bound from (49). Although we observe that the proposed CFO compensation is able to reduce the SIR loss significantly, it still increases steeply for increasing CFO size . If the CFO amounts to 20% of the subcarrier spacing, the performance of the system is degraded by 3 dB already at an SNR level of about 10 dB.
Figure 6
Figure 6

SNR loss after CFO compensation versus SNR for CFOs of different size . Solid line: ICI compensation. Dashed line: no compensation. .

These results show that the system's sensitivity toward CFO errors is still very high, and hence we conclude that with the suggested approach, we can conveniently compensate CFOs of small size only. Thus, the method is suitable for a fine-frequency synchronization only, and hence it has to rely on a coarse synchronization, which has to be established in advance. In a practical system, such a coarse synchronization can be achieved if the terminals use their frequency estimates obtained during the preceding downlink phase for a proper frequency precompensation of their transmit signals. We denote this as frequency advance, which has been the basic concept for our real-time system implementation that has been reported in [30]. It is worth noting that the analysis presented in this paper and in particular the derived bounds for the SIRs served as an important guideline in preparing the experiments that have been summarized in that reference, which have shown that a convenient system operation in a practical setup can be achieved.

Finally, note that if the CFOs are kept small, the signal degradation from ICI is limited, and thus common pilot-based channel estimation techniques can still be used to obtain channel estimates of sufficient quality. The more pilots available in one OFDM symbol can be used for that channel estimation, the better the ICI can be suppressed, as the ICI behaves similar to AWGN. Moreover, the CFOs of the single users can be obtained from observing the phase drift of the estimated subcarrier channels over several successive OFDM symbols. With (10), the ICI coefficients can then be determined, which can finally be applied in (29) for proper ICI compensation of the single users' signals.

6. Conclusion

We have investigated OFDM-SDMA uplink transmission in the presence of multiple users' CFOs. We modified the common signal model suitably to enable a subcarrier-wise SDMA equalization followed by a user-specific CFO compensation, yielding a simple equalization process ready to be applied in practice. However, as CFOs violate the periodic structure of the OFDM signals, some interference remains in the system after CFO compensation, which cannot be compensated as long as simple frequency domain processing is targeted. The SIR conditions in OFDM-SDMA systems have been analyzed if CFOs are compensated according to the proposed scheme as well as if they are not. We derived suitable upper bounds for the SIRs depending on the system parameters, which have been verified by numerical simulations. To enable a convenient operation of the proposed scheme, we conclude from the results that the users' CFOs should not exceed values that are much larger than a few percent of the OFDM subcarrier spacing, which classifies this scheme as a technique for fine frequency synchronization. Correspondingly, coarse-frequency synchronization has to be ensured, which can easily be established if the CFO estimates from the downlink are used in the uplink for a proper predistortion of each user's transmit signal, as suggested also in [10] and practically realized in [30]. Together with this concept, the proposed scheme can be regarded as a convenient solution to synchronize the OFDM-SDMA uplink. Note that this concept based on coarse synchronization also enables to estimate user channels based on common pilot-based channel estimation techniques. Suitable estimates of the users' CFOs can then be obtained from the phase drift of the estimated channels observed over several consecutive OFDM symbols.

Declarations

Authors’ Affiliations

(1)
Fraunhofer Institute for Telecommunications, Heinrich-Hertz-Institut, Einsteinufer 37, 10587 Berlin, Germany

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