#### Abstract

To further enhance frequency-asynchronous distributed Alamouti-coded (FADAC) orthogonal frequency division multiplexing (OFDM), we propose a new scheme which combines the partial maximum likelihood detection (PMLD) to the residual intercarrier interference cancellation (RIC). In order to decrease the performance gap from intercarrier interference- (ICI-) free level after single time iteration of the RIC, the final stage of the proposed scheme performs the PMLD limited to the symbols of less-reliable decision variables. We show that with the practically acceptable candidate symbol set size, a single iteration for RIC is enough to achieve the ICI-free performance. Moreover, the proposed scheme substantially expands the allowable ranges of the three undesirable terms, i.e., the timing and frequency offsets between the transmit antennas and the multipath delay spreads.

#### 1. Introduction

Recently, distributed antenna systems (DASs) have been one of the hottest topics in 5G wireless communication systems [1–3]. In DAS, one of the main challenging issues is to mitigate self-interference due to the carrier frequency offset (FO) between the distributed transmit antennas [4–8]. In the case of DAS, the synchronization is more difficult and challenging due to the distributed nature compared to the conventional synchronization between the transmitter and receiver since each distributed transmit antenna has a different local oscillator and independent Doppler spread. The FO between the distributed transmitters destroys the designed Alamouti code property and degrades the performance. This is more significant in OFDM-modulated Alamouti code because the FO in the OFDM signals generates intercarrier interference (ICI).

In order to overcome this problem, several algorithms have been proposed [4–8]. In [4–8], the so-called frequency reversal space frequency coded OFDM has been proposed. They utilize the ICI self-cancellation property by the frequency reversal structure of the Alamouti-coded symbols with appropriate combining technique at the receiver. However, in severely frequency-selective fading channels, these algorithms get worse because of interblock ICI terms.

Meanwhile, in [9–11], typical structures of interference cancellation algorithms for the conventional Alamouti-coded OFDM with the distributed antennas have been proposed. These algorithms generate the estimated interference terms based on the channel state and FO information and initial detection symbols. In general, ICI terms are cancelled by subtracting the estimated interference terms from the received signal. As the conventional Alamouti-coded OFDM has no ICI self-cancellation property for frequency asynchronous distributed antennas, considerable number of cancellation iterations has to be performed until the performance converges because the accuracy of the initial detection symbol is low. Moreover, the improvement of the converged performance by cancellation is not so impressive. In [12], the Alamouti-coded OFDM scheme for mitigation of ICI due to FO between two relays is proposed. The term of phase drift exists during two consecutive OFDM symbols because this scheme employs the space time code in Alamouti code schemes. In order to mitigate not only the term of phase drift but also ICI term due to FO, the ICI cancellation methods are performed in time domain and frequency domain, respectively. Despite the complicated ICI cancellation methods, this algorithm only achieves near ICI-free performance with small FO.

Meanwhile, in [13], Kim et al. have combined a typical decision-directed iterative ICI cancellation scheme to frequency-asynchronous distributed Alamouti-coded OFDM (FADAC OFDM) shown to outperform the other existing approaches [7]. This approach achieves the considerable improvement compared to the previous ICI cancellation methods due to the far accurate initial detection of FADAC OFDM compared to those in [9–11]. From the second iteration, however, the BER is stuck in the same value and would not converge to the near ICI-free performance. In [14], in order to overcome this drawback, we have proposed a further modified version based on the selective ICI cancellation. Although the adaptive cancellation improves the performance, symbol reliability check for deciding whether or not to cancel is performed by a threshold test, which has a practical problem that the performance is very sensitive to the threshold setting. In addition, to approach the near ICI-free performance, the iterative cancellations more than 3 or 4 times are required.

In this paper, we propose an enhanced scheme for FADAC OFDM to overcome the shortcomings of [9–14]. Compared to [9–14], the proposed scheme achieves better performance even with the single-time ICI cancellation. In order to decrease the performance difference from ICI-free performance after single-time iteration of ICI cancellation, the final stage of the proposed scheme performs the partial maximum likelihood detection (PMLD) limited to the symbols of low-reliability decision variables (DVs). The selection criterion of candidate symbol set for PMLD is simply based on the power of each symbol’s DV normalized by the total average power of the DV. The performance results reveal that the proposed scheme quite significantly enhances the FADAC OFDM in terms of the allowable ranges of the frequency offset and the channel selectivity.

This paper is organized as follows. We first provide the system model in Section 2 and the transceiver structure of the conventional distributed antenna Alamouti-coded OFDM with FO in Section 3. In Section 4, the proposed algorithm is addressed. We show that the proposed scheme consists of the three major stages, that is, (1) the initial detection stage using FADAC OFDM, (2) residual ICI cancellation (RIC) algorithm, and (3) PMLD algorithm. Simulation results are shown in Section 5 in order to provide the performance improvement due to the proposed algorithm. Finally, conclusions are drawn in Section 6.

#### 2. The System Model

This section addresses the system model. This paper considers the distributed antenna system that is composed of two transmit (TX) antennas and one receive (RX) antenna as in [4–10]. Figure 1 shows the structure of TX and RX sides of the distributed Alamouti-coded OFDM. In Figure 1(a), let the variable denote the data symbol of the subblock and and denote the Alamouti-coded symbols at the subcarrier of the subblock of TX antennas *A* and *B*, respectively. We will discuss more about how to perform Alamouti encoding and decoding in the next section. In each TX antenna, Alamouti-coded symbols are modulated with OFDM signals with *N* total subcarriers by using *N* point inverse Fourier transform (IFFT). Then, the OFDM modulated signals are upconverted by carrier frequencies of TX antennas *A* and *B* denoted by and , respectively, and transmitted from each antenna.

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The TX signals from two TX antennas are received at RX antenna as shown in Figure 1(b). The RX signal is downconverted by two local carrier frequencies, i.e., and , and two downconverted signals denoted by and are obtained, respectively. The two downconverted signals and are input to two *N*-point FFTs in order to demodulate OFDM modulated symbols, and then the two demodulated symbols at the subcarrier of the subblock of TX antennas *A* and *B* denoted by and are obtained from two *N*-point FFT outputs, respectively. By performing Alamouti decoding on and , the estimated symbol is finally detected.

#### 3. Signal Structure of Conventional Distributed Alamouti-Coded OFDM

In this section, the signal structure of conventional distributed Alamouti-coded OFDM is illustrated. Figure 2 shows the OFDM symbol structure of the conventional distributed Alamouti-coded OFDM [9–11]. In conventional distributed Alamouti-coded OFDM, Alamouti code pairs are mapped to the consecutive subcarriers just like the typical space-frequency Alamouti code structure [15], i.e., and are set towhere and denote the two data symbols for the subblock and is the complex conjugate operator.

In the RX side, two FFTs are performed on the two RX signals and by separately synchronizing to two asynchronous TX antenna’s carrier frequencies and time delays, and two FFT outputs and are expressed aswhere and denote the channel fading coefficients of the subcarrier of subblock from TX antennas *A* and *B*, respectively, and and are AWGN terms and is the ICI coefficient. We assume that and are independent and follow zero mean, unit variance complex Gaussian distribution. The term *ε* in (2) and (3) is the normalized FO between two TX antennas, that is, , where is the subcarrier spacing. The ICI coefficient is given as follows [16]:

The decoding for the conventional distributed Alaomouti coded OFDM is performed on and . The normalized decision variable (DV) corresponding to data symbol denoted by is obtained as follows:

#### 4. The Proposed Scheme

The proposed scheme in the receiver side consists of three major stages as follows:(1)Initial detection stage: Alamouti decoding of FADAC OFDM(2)RIC algorithm using the initial detection of (1)(3)PMLD limited to low-reliability decision variables

The following subsections explain each of these three stages, respectively.

##### 4.1. Initial Detection Stage: FADAC OFDM

In FADAC OFDM, in the TX side, compared to the conventional distributed Alamouti-coded OFDM, the *N* total subcarriers are properly divided into the subblocks according to the selectivity of the channel, and then frequency reversal Alamouti code is applied block by block, i.e., Alamouti pairs are mapped into a “mirror image” symmetric to the center frequency of each subblock as in Figure 3. Hence, and for are set as follows:

The proposed scheme basically employs the combining scheme of the FADAC OFDM in [7] for the initial detection. In the receiver side, two FFTs are performed on and as in the conventional distributed Alamouti-coded OFDM and two FFT output signals as and are obtained. Then, the elements of subblock of them and are expressed as

We further consider the TO (timing offset) between the RX signals between the two TX antennas, which has not been considered in the previous works [13, 14]. Let denotes the TO and we assume that OFDM symbol duration *T* is assumed to be sufficiently enlarged compared to [17]. The term *τ* in (7) and (8) denotes the normalized TO, i.e., . The terms in (7) and in (8) denote the phase rotation of subcarrier of subblock in frequency domain due to TO *τ* and *θ* is calculated as .

The Alamouti decoding for FADAC OFDM described in [7] is performed on and as shown in Figure 4 and then, the normalized DVs and corresponding to data symbols and are obtained. By the Alamouti decoding described in [7], the normalized DVs and are obtained as follows:

For more details on the Alamouti decoding of FADAC OFDM, refer to [7].

###### 4.1.1. Extension of FADAC OFDM to Two RX Antennas System

Meanwhile, in order to meet the requirement of 5G standard, we consider FADAC OFDM with two TX and two RX antennas. The overall process of this system is almost the same as in the previous system, i.e., two TX and one RX antennas, except that the number of channels and the number of FFT outputs increase as the RX signal increases. Hence, four FFT outputs are expressed aswhere and denote FFT output and AWGN signal of the RX signal received at the *V*th RX antenna () and downconverted by the carrier frequency of the TX antenna, i.e., (), and denotes the channel coefficient between the TX antenna and the *V*th RX antenna. The normalized DVs and are obtained as follows [15]:

According to increase of the number of RX antennas, 3 dB gain is obtained compared with two TX antennas and one RX antenna, as is well known [15].

##### 4.2. RIC Algorithm

After the initial detection, the RIC algorithm is performed. First, the estimated residual ICI terms denoted by and are generated in order to cancel ICI elements in the FFT output signals and , respectively. The estimated residual ICI terms and are expressed aswhere and are the reconstructed (estimated) versions of and based on the Alamouti-decoded data symbol and the transmitter structure in Figure 3. Assume that *ε*, *τ*, and *θ* and channel coefficients and are known to the RX side. We update and by and which implies the residual ICI cancelled version of and , respectively, and then, we perform Alamouti decoding in (9) and (10) again on the updated and .

Unlike the previous iterative RIC schemes to FADAC OFDM in [13, 14], the proposed scheme performs the RIC just once for less computations and less demodulation latency which is a critical issue in 5G systems. In order to reduce this latency, RIC algorithm is required for just a single time. However, applying RIC algorithm for just a single time cannot achieve the performance of ICI-free level. To ameliorate this, we perform PMLD instead of the complicated successive interference cancellation algorithm in [14].

##### 4.3. PMLD Limited to Less-Reliable Decision Variables

After single-time RIC stage, the final stage of the proposed scheme performs PMLD limited to the symbols of less-reliable DVs in order to enhance the performance. In order to reduce the computation load, we select only least-reliable DVs as the target symbol set for PMLD. The selection criterion of candidate symbol set for PMLD is based on and which are comprised of the power of each symbols’s DV normalized by the total average power of the DVs. More specifically, and for and are calculated, respectively aswhere is DV vector, i.e., , denotes the real part of the complex value, and denotes the expectation operator. In order to show whether and reflect the reliability of DV well, the cumulative distribution function (CDF) of and with , dB, and 20 dB with the normalized FO and when each DV is correctly estimated and vice versa is shown in Figure 5.

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From Figure 5, the probability that the estimated symbols by RIC scheme are incorrect is about 90th percentile when and are less than 0.5. On the other hand, the probability that the estimated symbol by the RIC scheme is correct is less than 15th percentile with the same condition. This implies that almost of the estimation errors occur when and are small values. Furthermore, the probability that the estimation is incorrect is almost invariant to and *ε*. Hence, we conclude that and are available as the simple indicator of the reliability of the DVs.

Based on the observation above, the PMLD algorithm is performed using and by the following steps:(i)Step 1: classification of DVs for PMLD: classify the estimated DV vector as two vectors, i.e., with length *M* and with length where is the complementary set of vector . The elements of are classified as the symbols corresponding to the *M* minimum and . Meanwhile, the rest symbols with high reliability, i.e., , are determined as final detection symbol in this step.(ii)Step 2: PMLD: in this step, the PMLD is performed only for the DVs with low reliability, i.e., . The detection symbol vector by PMLD is obtained as follows:where *C* denotes the constellation point set of the data symbol, and are FFT output vectors of and , i.e., and , respectively, and and are the reconstructed versions of and assuming the transmitted data vector is . Note that the computation complexity for (15) is impractically high. Hence, we employ the following three techniques to get the reasonable complexity.(1)We neglect the term (or the term ). This is because it is intuitive that these two terms are highly redundant due to the fact that both and are obtained from the common received signal.(2)We perform the ML search over the *J* closest constellation points to the initial detection point rather than exhaustive search over all the constellation point in the set *C.* Hence, we scan only candidate symbol vectors in the space and thus the complexity reduction will be substantial for high-order modulation.(3)As we sequentially scan the candidate symbol vectors in the space one by one, there is only one element difference between the previous candidate symbol vector and the current symbol vector. Thus, for generating (or ) in (15), we only need to generate the newly added signal component and then recursively replace the previously added component. Specifically, to update the signal parts in (7) and (8), i.e., and , only two sign changes are required and to update the ICI terms, *N* subtractions and additions are required because we have already reconstructed the ICI terms in the RIC stage and we will use them as the initial values for the recursive updates. Overall, neglecting sign changes that are negligible, additions are required for the proposed PMLD scheme.

#### 5. Simulation Results

The FADAC OFDM in [6] and its modified versions in [13, 14] has been already intensively compared to the previous distributed Alamouti-coded OFDM schemes such as the ones in [5] or [9]. Hence, in this paper, we focus on comparing the proposed scheme with the basic FADAC OFDM [6] and the FADAC OFDM with SIC in [14]. We exclude the FADAC OFDM with SIC in [13] for compactness as its upgraded version in [14] works far better. The main benefit of the proposed scheme lies in the feature that it needs RIC just once. Hence, for comparing on the same basis, the number of SIC iterations is commonly set to 1.

Figure 6 shows the BER performances of the schemes in comparison over the frequency offset and the timing offset plane for various cases of delay spread with = 20 dB. We commonly set *N* = 256 and use QPSK constellation. For the PMLD in the proposed scheme, we set and *M* = 4 which needs a marginal computation overhead compared to the computations for the interference reconstruction in the RIC stage. Regarding the multipath profile for generating and , the number of multipaths is 8 and their delay are distributed uniformly in . The guard interval is set to be larger than . The subcarrier spacing is set to 15 kHz which is taken from long-term evolution (LTE) standard [18]. The subblock size which varies according to delay spread is taken from the results in [7].

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We use the compact labels to each scheme in the legend: “(1)” for FADAC OFDM as it only uses the stage (1) in Figure 4, “(1) + (2)” for FADAC OFDM with SIC as it corresponds to the stages (1) and (2) in Figure 4, and “(1) + (2) + (3)” for the proposed scheme. For FADAC OFDM with SIC “(1) + (2),” the adaptive selection (AS) mode for interference cancellation in [14] is employed and thus, the RIC stage outperforms that of the proposed scheme although we specify it as “(1) + (2)” for easy expression. The conventional space frequency Alamouti-coded OFDM with the label is also included for reference.

Note that although the FADAC OFDM with SIC “(1) + (2)” achieves the substantially improved performance compared to the basic FADAC OFDM, it still degrades significantly for larger FOs. Meanwhile, except the severely selective fading (say the case with ), the proposed scheme achieves near ICI-free performance for wide range of FO. This confirms that the PMLD in the final stage of the proposed scheme well fixes the errors which were not fixed in the RIC stage.

Also note that the main factor affecting the performance is FO between the two TX antennas while the BERs are almost irrespective of the timing offset (TO). Hence, for clearer view, the BER curves are plotted in two-dimension as a function of FO for two cases of fading channel selectivity. As a mildly selective fading case, we set in Figure 7 and as a severely selective fading case, we set in Figure 8. It is remarkable that the proposed scheme significantly extends the FO range of ICI-free level even with the severe selective fading channel, i.e, up to FO = 0.4 for . In addition, even in the small range of FO, there exist the considerable performance gaps between the proposed scheme and the other two schemes.

#### 6. Conclusions

We propose a new scheme which includes RIC and PMLD in the FADAC OFDM for frequency and timing asynchronous distributed antenna systems. From simulation results, we show that additional improvement by RIC and PMLD can be dramatically accelerated with acceptable complexity. The proposed scheme achieves near ICI-free performance with wider range of FO and TO. Moreover, the proposed scheme achieves near ICI-free performance with relatively large FOs in quite selective fading channel.

#### Data Availability

The matlab data used to support the findings of this study are included within the supplementary information file.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work was supported by the DGIST R&D Program of the Ministry of Science, ICT and Future Planning, Korea (19-IT-01), the Brain Korea 21 Plus Program (No. 22A20130012814) funded by the National Research Foundation of Korea, (NRF) and the 2018 Yeungnam University Research Grant.

#### Supplementary Materials

(1) “Partial ML FADAC mfile.zip” file is source code for simulations; (2) “figures.zip” file is figure file (.eps) for latex format.* (Supplementary Materials)*