# Impact of Carrier Frequency Offsets on Block-IFDMA Systems

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## Abstract

Recently, a new multiple access (MA) scheme called block-interleaved frequency division multiple access (B-IFDMA) is under consideration as an MA scheme candidate for 4G wireless applications. In this paper, the two variants of B-IFDMA are considered, the joint- DFT B-IFDMA and the added-signal B-IFDMA, and compared in terms of sensitivity to carrier frequency offsets (CFOs) for both uplink and downlink. CFO gives rise to multiuser interference and self-user interference. We derive analytical expressions for the power of these interferences, and we quantify their detrimental effect through the evaluation of the signal-to-interference-plus-noise ratio (SINR) degradation. We point out that both variants of B-IFDMA are not similarly affected by CFO. Hence, joint-DFT B-IFDMA provides a better robustness to multiuser interference than added-signal B-IFDMA, and so is better suited for the uplink. Then we show by means of numerical results that added-signal B-IFDMA is less sensitive to CFO in the downlink.

### Keywords

Discrete Fourier Transform Multiple Access Carrier Frequency Offset Cyclic Prefix Synchronization Error## 1. Introduction

In the context of the research on beyond 3rd and 4th generation (B3G/4G) mobile radio systems, a novel power-efficient multiple access scheme called block-interleaved frequency multiple access (B-IFDMA) has been proposed as a candidate for nonfrequency-adaptive transmission mode. B-IFDMA is a particular case of discrete Fourier transform (DFT) precoded OFDMA, where the data of the user under consideration is transmitted on blocks of subcarriers that are equidistantly distributed over the total available bandwidth. Hence, it can be viewed as a generalization of DFT precoded OFDMA with interleaved subcarrier allocation, also called IFDMA [1]. Two different variants of B-IFDMA are currently under investigation, the joint-DFT B-IFDMA and the added-signal B-IFDMA [2, 3]. The joint-DFT B-IFDMA signal is based on applying DFT once to all subcarriers assigned to a given user whereas the added-signal B-IFDMA is constructed by applying DFT to groups of subcarriers.

The robustness of B-IFDMA compared to IFDMA to carrier frequency offsets (CFOs) has been discussed in [2] for the uplink. The authors showed that B-IFDMA is expected to be more robust to CFO than IFDMA due to the fact that schemes with interleaved subcarrier allocation are known to be more sensitive to CFO compared to schemes with block allocation. However, it is not clear which variant of B-IFDMA is more robust to CFO. Moreover, to the best of our knowledge, no detailed analysis exists on the sensitivity of B-IFDMA to CFO. The purpose of this paper is to present a comprehensive study of the sensitivity of the joint-DFT and added-signal B-IFDMA to CFO and to compare those two variants in terms of CFO sensitivity.

The effect of CFO on multicarrier schemes has been studied in [4] for OFDM, in [5] for MC-DS-CDMA, and in [6] for MC-CDMA. It was shown that CFO gives rise to signal distortions, yielding interference and power loss which degrades system performance. When this degradation can no longer be tolerated, carrier frequency correction must be applied. For downlink, the CFO is the same for all users. Hence, the carrier frequency can be corrected by using feedback carrier synchronization mechanisms, at the expense of phase jitter [7, 8]. Note that for uplink, since the CFOs associated with different users are different to each other, it is much more difficult to carry out an offset correction [9, 10]. In this paper, we consider both uplink and downlink.

To quantify the performance degradation, we propose to compute the expressions of the signal-to-interference-plus-noise ratio (SINR) degradation for both variants of B-IFDMA. We also provide a detailed analysis of the obtained analytical expressions in order to compare the sensitivity of both variants to CFO. In addition, numerical results illustrate the analysis.

The paper is organized as follows. In Section 2, a system model including the CFO for both variants of B-IFDMA is given. The sensitivity to CFO is investigated in Section 3. Numerical results are presented in Section 4. Section 5 concludes the paper.

## 2. System Model

In this section, a system model including the CFO is given. As added-signal B-IFDMA model can be generated from IFDMA signals [2], here we focus on the joint-DFT B-IFDMA model. The signal model for IFDMA is described in detail in [11]. The model for joint-DFT B-IFDMA is derived as a particular case of general precoded OFDMA system. Although new algorithms for a lower complexity implementation of B-IFDMA based on time-domain signal generation have been proposed in [3], it is more convenient to perform algebra with the general OFDMA transmitter model.

where Open image in new window is a Fourier sequence. Let Open image in new window designate the total number of subcarriers available in the OFDMA system, where Open image in new window is the maximum number of users. Note that Open image in new window will designate the number of active users. Then, the Open image in new window precoded symbols Open image in new window of user Open image in new window are transmitted on blocks of subcarriers that are equidistantly distributed over the Open image in new window subcarriers. Thus, Open image in new window where Open image in new window stands for the number of blocks and Open image in new window the number of subcarriers per block. The Open image in new window th symbol Open image in new window modulates the subcarrier of index Open image in new window , where Open image in new window . This mapping is specific to the joint-DFT B-IFDMA scheme.

The Open image in new window samples of the transmitted sequence are generated by feeding the mapped symbols to an inverse fast Fourier transform. Then, a cyclic prefix of Open image in new window samples is inserted in order to avoid interference caused by dispersive channel. The transmitter feeds those samples at a rate Open image in new window to a unit energy zero roll-off square root Nyquist filter Open image in new window with respect to the sampling time Open image in new window .

The signal Open image in new window is then transmitted over the dispersive channel from the transmitter of user Open image in new window to the base station with the channel transfer function Open image in new window . The output of the dispersive channel is disturbed by a carrier phase error which linearly increases in time within an OFDM symbol period: Open image in new window , where Open image in new window stands for the CFO for user Open image in new window . Without loss of generality, we assume Open image in new window . We also assume small CFO compared to the bandwidth of the receiver filter Open image in new window .

The base station receives the sum of the signals transmitted by the different users, disturbed by additive white Gaussian noise Open image in new window , with uncorrelated real and imaginary parts, each having a power spectral density Open image in new window . The resulting signal enters the receiver filter, which is matched to the transmitted filter and is sampled at instants Open image in new window assuming perfect timing synchronization.

where Open image in new window is a white complex Gaussian noise with variance Open image in new window and Open image in new window is the contribution of the symbol Open image in new window to the input of the decision device. The next paragraph deals with the computation of the quantity Open image in new window .

is the folded transfer function of the equivalent channel defined in (4) evaluated at the frequencies Open image in new window .

## 3. Impact of Carrier Frequency Offset on B-IFDMA

In this section, we investigate the effect of CFO to the performance of the two B-IFDMA variants, the joint-DFT B-IFDMA and the added-signal B-IFDMA. First, we consider the joint-DFT B-IFDMA signal.

### 3.1. Joint-DFT B-IFDMA

In addition to the interference terms, it follows from (16) that the useful component at the FFT output is reduced compared to the case of a zero CFO. Hence, to keep the power loss within reasonable bounds, the CFO must satisfy Open image in new window which is easy to understand since the IFFT behaves like a bank of filters of bandwidth Open image in new window .

The resulting expression of the degradation for joint-DFT B-IFDMA is obtained by inserting (16), (17), and (18) in (12).

### 3.2. Added-Signal B-IFDMA

The added-signal B-IFDMA model for a given user comes from the superimposing of Open image in new window IFDMA signals, each with Open image in new window subcarriers [2]. These Open image in new window IFDMA signals are mutually shifted by one subcarrier bandwidth.

The resulting expression of the degradation for added-signal B-IFDMA is obtained by inserting (16), (21), and (22) in (12).

### 3.3. Comparison of Sensitivity to CFO for Both Variants of B-IFDMA

In summary, it turns out that for the joint-DFT B-IFDMA, most of the interference comes from the SUI whereas the added-signal B-IFDMA mostly suffers from the MUI. Numerical results are presented in Section 4 to illustrate this analysis.

## 4. Numerical Results

In this section, we present numerical results of SINR degradations due to CFO for the joint-DFT B-IFDMA and added-signal B-IFDMA. We assume the same CFO Open image in new window for all users, that is, Open image in new window for Open image in new window . We also assume that all users exhibit the same energy per symbol with Open image in new window subcarriers assigned to each user. The maximum number of users is Open image in new window and Open image in new window dB.

We also observe that the joint-DFT B-IFDMA is less robust to CFO than added-signal B-IFDMA. For instance, for the same CFO of Open image in new window , the degradation with the joint-DFT B-IFDMA is 1 dB higher than that with the added-signal B-IFDMA.

For the sake of comparison, we plot the degradation obtained for IFDMA systems. The considered IFDMA system has the same number of subcarriers assigned to each user ( Open image in new window ), which are equidistantly distributed over the total bandwidth [11]. As IFDMA can be regarded as a special case of joint-DFT B-IFDMA with Open image in new window , it is straightforward to obtain the degradation expression.

As we observe, the degradation value for IFDMA is very close to that of the added-signal B-IFDMA. Hence, as the added-signal B-IFDMA model is obtained by superimposing IFDMA signals, the behavior of both systems is nearly similar in terms of CFO sensitivity.

## 5. Conclusion

In this paper, the two variants of B-IFDMA, the joint-DFT B-IFDMA and the added-signal B-IFDMA, have been investigated in terms of carrier frequency offset (CFO) sensitivity. CFO gives rise to useful power loss together with interference, leading to performance degradation. To evaluate this performance degradation, we have determined the theoretical expressions of the SINR degradation caused by CFO at the input of the decision device. The results of the analysis have shown a different behavior for both variants of B-IFDMA in terms of CFO sensitivity. Hence, when considering the added-signal B-IFDMA, the multiuser interference contributions are the dominant ones. For the joint-DFT B-IFDMA, the degradation is found to be dominated by self-user interference. As a consequence, it appears that, in terms of sensitivity to CFO, joint-DFT B-IFDMA is better suited than added-signal B-IFDMA for the uplink. Indeed, the effect of multiuser interference is far more complex to be corrected with the uplink case than downlink. Then, the numerical results have shown that the added-signal B-IFDMA is more robust to CFO for the downlink.

## Notes

### acknowledgments

This work has been carried out in the framework of the Campus International sur la Sécurité et l Intermodalité des Transports (CISIT) project and funded by the French Ministry of Research, the Region Nord Pas de Calais, and the European Commission (FEDER funds).

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