# Partial ML Detection for Frequency-Asynchronous Distributed Alamouti-Coded (FADAC) OFDM

Authors: Bong-seok Kim; Dongjun Na; Kwonhue Choi
Journal: Wireless Communications and Mobile Computing
 (2019)

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## 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.


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## Body

## 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 Section2 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 xb,l denote the lth data symbol of the bth subblock and Xb,kA and Xb,kB denote the Alamouti-coded symbols at the kth subcarrier of the bth 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 fcA and fcB, respectively, and transmitted from each antenna.Figure 1
Structure of (a) TX and (b) RX sides of the distributed Alamouti-coded OFDM.
(a)
(b)The TX signals from two TX antennas are received at RX antenna as shown in Figure1(b). The RX signal is downconverted by two local carrier frequencies, i.e., fcA and fcB, and two downconverted signals denoted by rA and rB are obtained, respectively. The two downconverted signals rA and rB are input to two N-point FFTs in order to demodulate OFDM modulated symbols, and then the two demodulated symbols at the kth subcarrier of the bth subblock of TX antennas A and B denoted by Rb,kA and Rb,kB are obtained from two N-point FFT outputs, respectively. By performing Alamouti decoding on Rb,kA and Rb,kB, the estimated symbol x^l 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. Figure2 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., Xb,kA and Xb,kB are set to(1)Xb,kA=xb,1,ifk=1,−xb,2∗,ifk=2,Xb,kB=xb,2,ifk=1,xb,1∗,ifk=2,where xb,1 and xb,2 denote the two data symbols for the bth subblock and ⋅∗ is the complex conjugate operator.Figure 2
OFDM symbol structure of conventional distributed Alamouti-coded OFDM.In the RX side, two FFTs are performed on the two RX signalsrA and rB by separately synchronizing to two asynchronous TX antenna’s carrier frequencies and time delays, and two FFT outputs Rb,kA and Rb,kB are expressed as(2)Rb,kA=Hb,kAXb,kA+∑β=1N/2∑m=12Q2β−b+m+ε−k⋅Hβ,mBeβ,m−jθτXβ,mB+wb,kA,(3)Rb,kB=Hb,kBXb,kB+∑β=1N/2∑m=12Q2β−b+m−ε−k⋅Hβ,mAeβ,mjθτXβ,mA+wb,kB,where Hb,kA and Hb,kB denote the channel fading coefficients of the kth subcarrier of bth subblock from TX antennas A and B, respectively, and wkA and wkB are AWGN terms and Qx is the ICI coefficient. We assume that Hb,kA and Hb,kB 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, ε=fcB−fcA/fΔ, where fΔ is the subcarrier spacing. The ICI coefficient Qx is given as follows [16]:(4)Qx=sinπxNsinπ/Nxexpjπ1−1Nx.The decoding for the conventional distributed Alaomouti coded OFDM is performed onRb,kA and Rb,kB. The lth normalized decision variable (DV) corresponding to data symbol xb,l denoted by x˜b,l is obtained as follows:(5)x˜b,l=Hb,l∗ARb,lA+Hb,l+1BRb,l+1∗BHb,lA2+Hb,l+1B2,for oddl,Hb,l−1∗BRb,l−1B−Hb,lARb,l∗AHb,lB2+Hb,l+1A2,for evenl.


## 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 variablesThe 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, theN total subcarriers are properly divided into the Nb 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, Xb,kA and Xb,kB for 1≤b≤Nb are set as follows:(6)Xb,kA=xb,2k−1,for1≤k≤nc2,−xb,2nc−k+1∗,fornc2+1≤k≤nc,Xb,kB=xb,2k,for1≤k≤nc2,xb,2nc−k+1∗,fornc2+1≤k≤nc.Figure 3
Alamouti-coded block structure of FADAC OFDM.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 rA and rB as in the conventional distributed Alamouti-coded OFDM and two FFT output signals as Rb,kA and Rb,kB are obtained. Then, the kth elements of bth subblock of them Rb,kA and Rb,kA are expressed as(7)Rb,kA=Hb,kAXb,kA+∑β=1Nb∑m=1ncQβ−bnc+m+ε−k⋅Hβ,mBeβ,m−jθτXβ,mB+wb,kA,(8)Rb,kB=Hb,kBXb,kB+∑β=1Nb∑m=1ncQβ−bnc+m−ε−k⋅Hβ,mAeβ,mjθτXβ,mA+wb,kB.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 Δt denotes the TO and we assume that OFDM symbol duration T is assumed to be sufficiently enlarged compared to Δt [17]. The term τ in (7) and (8) denotes the normalized TO, i.e., τ=Δt/T. The terms eβ,m−jθτXβ,mB in (7) and eβ,mjθτXβ,mB in (8) denote the phase rotation of kth subcarrier of bth subblock in frequency domain due to TO τ and θ is calculated as 2πb−1nc+k−1.The Alamouti decoding for FADAC OFDM described in [7] is performed on Rb,kA and Rb,kB as shown in Figure 4 and then, the normalized DVs x˜b,2k−1 and x˜b,2k corresponding to data symbols xb,l and xb,l+1 are obtained. By the Alamouti decoding described in [7], the normalized DVs x˜b,2k−1 and x˜b,2k are obtained as follows:(9)x˜b,2k−1=Hb,k∗ARb,kA+Hb,nc−k+1BRb,nc−k+1∗BHb,kA2+Hb,nc−k+1B2,(10)x˜b,2k=Hb,k∗BRb,kB−Hb,nc−k+1ARb,nc−k+1∗AHb,kB2+Hb,nc−k+1A2.Figure 4
Receiver structure of the proposed scheme.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 as(11)Rb,kA,A=Hb,kA,AXb,kA+∑β=1Nb∑m=1ncQβ−bnc+m+ε−k⋅Hβ,mB,Aeβ,m−jθτXβ,mB+wb,kA,A,Rb,kB,A=Hb,kB,AXb,kB+∑β=1Nb∑m=1ncQβ−bnc+m−ε−k⋅Hβ,mA,Aeβ,mjθτXβ,mA+wb,kB,A,Rb,kA,B=Hb,kA,BXb,kA+∑β=1Nb∑m=1ncQβ−bnc+m+ε−k⋅Hβ,mB,Beβ,m−jθτXβ,mB+wb,kA,B,Rb,kB,B=Hb,kB,BXb,kB+∑β=1Nb∑m=1ncQβ−bnc+m−ε−k⋅Hβ,mA,Beβ,mjθτXβ,mA+wb,kB,B,where Rb,kU,V and wb,kU,V denote FFT output and AWGN signal of the RX signal received at the Vth RX antenna (V∈AandB) and downconverted by the carrier frequency of the Uth TX antenna, i.e., (fcU∈fcAandfcB), and Hb,kU,V denotes the channel coefficient between the Uth TX antenna and the Vth RX antenna. The normalized DVs x˜b,2k−1 and x˜b,2k are obtained as follows [15]:(12)x˜b,2k−1=Hb,k∗A,ARb,kA,A+Hb,nc−k+1B,ARb,nc−k+1∗B,A+Hb,k∗A,BRb,kA,B+Hb,nc−k+1B,BRb,nc−k+1∗B,BHb,kA,A2+Hb,nc−k+1B,A2+Hb,kA,B2+Hb,nc−k+1B,B2,x˜b,2k=Hb,k∗B,ARb,kB,A−Hb,nc−k+1A,ARb,nc−k+1∗A,A+Hb,k∗B,BRb,kB,B−Hb,nc−k+1A,BRb,nc−k+1∗A,BHb,kB,A2+Hb,nc−k+1A,A2+Hb,kB,B2+Hb,nc−k+1A,B2.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 byIb,kA and Ib,kA are generated in order to cancel ICI elements in the FFT output signals Rb,kA and Rb,kB, respectively. The estimated residual ICI terms Ib,kA and Ib,kB are expressed as(13)Ib,kA=∑β=1Nb∑m=1ncQβ−bnc+m+ε−kHβ,mBeβ,m−jθτX^β,mB,Ib,kB=∑β=1Nb∑m=1ncQβ−bnc+m−ε−kHβ,mAeβ,mjθτX^β,mA,where X^b,kA and X^b,kB are the reconstructed (estimated) versions of Xb,kA and Xb,kB based on the Alamouti-decoded data symbol x^l and the transmitter structure in Figure 3. Assume that ε, τ, and θ and channel coefficients Hβ,mA and Hβ,mB are known to the RX side. We update Rb,kA and Rb,kB by Rb,kA−Ib,kA and Rb,kB−Ib,kB which implies the residual ICI cancelled version of Rb,kA and Rb,kB, respectively, and then, we perform Alamouti decoding in (9) and (10) again on the updated Rb,kA and Rb,kB.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 onlyM≪N least-reliable DVs as the target symbol set for PMLD. The selection criterion of candidate symbol set for PMLD is based on γb,2k−1 and γb,2k which are comprised of the power of each symbols’s DV normalized by the total average power of the DVs. More specifically, γb,2k−1 and γb,2k for xb,2k−1 and xb,2k are calculated, respectively as(14)γb,2k−1=Rex˜b,2k−12ERex˜2,γb,2k−1=Rex˜b,2k−12ERex˜2,where x˜ is DV vector, i.e., x˜=x1,1,x1,2,…, Re⋅ denotes the real part of the complex value, and E⋅ denotes the expectation operator. In order to show whether γb,2k−1 and γb,2k reflect the reliability of DV well, the cumulative distribution function (CDF) of γb,2k−1 and γb,2k with N=256, Eb/N0=10 dB, and 20 dB with the normalized FO ε=0.2 and ε=0.5 when each DV is correctly estimated and vice versa is shown in Figure 5.Figure 5
The CDF ofγb,2k−1 and γb,2k with QPSK and Eb/N0=10 dB and 20 dB.
(a)
(b)From Figure5, the probability that the estimated symbols by RIC scheme are incorrect is about 90th percentile when γb,2k−1 and γb,2k 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 γb,2k−1 and γb,2k are small values. Furthermore, the probability that the estimation is incorrect is almost invariant to Eb/N0 and ε. Hence, we conclude that γb,2k−1 and γb,2k are available as the simple indicator of the reliability of the DVs.Based on the observation above, the PMLD algorithm is performed usingγb,2k−1 and γb,2k by the following steps:(i)
Step 1: classification of DVs for PMLD: classify the estimated DV vectorx^ as two vectors, i.e., xPML with length M and xPMLc with length N−M where ac is the complementary set of vector a. The elements of xPML are classified as the symbols corresponding to the M minimum γb,2k−1 and γb,2k. Meanwhile, the rest N−M symbols with high reliability, i.e., xPMLc, 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.,xPML. The detection symbol vector by PMLD x^PML is obtained as follows:(15)x^PML=arg minxPML∈CMrA−gAx^2+rB−gBx^2,where C denotes the constellation point set of the data symbol, rA and rB are FFT output vectors of Rb,kA and Rb,kB, i.e., rA=R1,1A,R1,2A,R1,3A,…,RNb,ncA and rB=R1,1B,R1,2B,R1,3B,…,RNb,ncB, respectively, and gAx^ and gBx^ are the reconstructed versions of rA and rB assuming the transmitted data vector is x^. 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 termrB−gBx^2 (or the term rA−gAx^2). This is because it is intuitive that these two terms are highly redundant due to the fact that both rA and rB are obtained from the common received signal.(2)
We perform the ML search over theJ 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 JM candidate symbol vectors in the space CM and thus the complexity reduction will be substantial for high-order modulation.(3)
As we sequentially scan the candidate symbol vectors in the spaceCM one by one, there is only one element difference between the previous candidate symbol vector and the current symbol vector. Thus, for generating gAx^ (or gBx^) 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., Hb,kAXb,kA and Hb,kBXb,kB, 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, 2NJM additions are required for the proposed PMLD scheme.


## 4.1. Initial Detection Stage: FADAC OFDM

In FADAC OFDM, in the TX side, compared to the conventional distributed Alamouti-coded OFDM, theN total subcarriers are properly divided into the Nb 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, Xb,kA and Xb,kB for 1≤b≤Nb are set as follows:(6)Xb,kA=xb,2k−1,for1≤k≤nc2,−xb,2nc−k+1∗,fornc2+1≤k≤nc,Xb,kB=xb,2k,for1≤k≤nc2,xb,2nc−k+1∗,fornc2+1≤k≤nc.Figure 3
Alamouti-coded block structure of FADAC OFDM.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 rA and rB as in the conventional distributed Alamouti-coded OFDM and two FFT output signals as Rb,kA and Rb,kB are obtained. Then, the kth elements of bth subblock of them Rb,kA and Rb,kA are expressed as(7)Rb,kA=Hb,kAXb,kA+∑β=1Nb∑m=1ncQβ−bnc+m+ε−k⋅Hβ,mBeβ,m−jθτXβ,mB+wb,kA,(8)Rb,kB=Hb,kBXb,kB+∑β=1Nb∑m=1ncQβ−bnc+m−ε−k⋅Hβ,mAeβ,mjθτXβ,mA+wb,kB.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 Δt denotes the TO and we assume that OFDM symbol duration T is assumed to be sufficiently enlarged compared to Δt [17]. The term τ in (7) and (8) denotes the normalized TO, i.e., τ=Δt/T. The terms eβ,m−jθτXβ,mB in (7) and eβ,mjθτXβ,mB in (8) denote the phase rotation of kth subcarrier of bth subblock in frequency domain due to TO τ and θ is calculated as 2πb−1nc+k−1.The Alamouti decoding for FADAC OFDM described in [7] is performed on Rb,kA and Rb,kB as shown in Figure 4 and then, the normalized DVs x˜b,2k−1 and x˜b,2k corresponding to data symbols xb,l and xb,l+1 are obtained. By the Alamouti decoding described in [7], the normalized DVs x˜b,2k−1 and x˜b,2k are obtained as follows:(9)x˜b,2k−1=Hb,k∗ARb,kA+Hb,nc−k+1BRb,nc−k+1∗BHb,kA2+Hb,nc−k+1B2,(10)x˜b,2k=Hb,k∗BRb,kB−Hb,nc−k+1ARb,nc−k+1∗AHb,kB2+Hb,nc−k+1A2.Figure 4
Receiver structure of the proposed scheme.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 as(11)Rb,kA,A=Hb,kA,AXb,kA+∑β=1Nb∑m=1ncQβ−bnc+m+ε−k⋅Hβ,mB,Aeβ,m−jθτXβ,mB+wb,kA,A,Rb,kB,A=Hb,kB,AXb,kB+∑β=1Nb∑m=1ncQβ−bnc+m−ε−k⋅Hβ,mA,Aeβ,mjθτXβ,mA+wb,kB,A,Rb,kA,B=Hb,kA,BXb,kA+∑β=1Nb∑m=1ncQβ−bnc+m+ε−k⋅Hβ,mB,Beβ,m−jθτXβ,mB+wb,kA,B,Rb,kB,B=Hb,kB,BXb,kB+∑β=1Nb∑m=1ncQβ−bnc+m−ε−k⋅Hβ,mA,Beβ,mjθτXβ,mA+wb,kB,B,where Rb,kU,V and wb,kU,V denote FFT output and AWGN signal of the RX signal received at the Vth RX antenna (V∈AandB) and downconverted by the carrier frequency of the Uth TX antenna, i.e., (fcU∈fcAandfcB), and Hb,kU,V denotes the channel coefficient between the Uth TX antenna and the Vth RX antenna. The normalized DVs x˜b,2k−1 and x˜b,2k are obtained as follows [15]:(12)x˜b,2k−1=Hb,k∗A,ARb,kA,A+Hb,nc−k+1B,ARb,nc−k+1∗B,A+Hb,k∗A,BRb,kA,B+Hb,nc−k+1B,BRb,nc−k+1∗B,BHb,kA,A2+Hb,nc−k+1B,A2+Hb,kA,B2+Hb,nc−k+1B,B2,x˜b,2k=Hb,k∗B,ARb,kB,A−Hb,nc−k+1A,ARb,nc−k+1∗A,A+Hb,k∗B,BRb,kB,B−Hb,nc−k+1A,BRb,nc−k+1∗A,BHb,kB,A2+Hb,nc−k+1A,A2+Hb,kB,B2+Hb,nc−k+1A,B2.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].


## 4.1.1. 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 as(11)Rb,kA,A=Hb,kA,AXb,kA+∑β=1Nb∑m=1ncQβ−bnc+m+ε−k⋅Hβ,mB,Aeβ,m−jθτXβ,mB+wb,kA,A,Rb,kB,A=Hb,kB,AXb,kB+∑β=1Nb∑m=1ncQβ−bnc+m−ε−k⋅Hβ,mA,Aeβ,mjθτXβ,mA+wb,kB,A,Rb,kA,B=Hb,kA,BXb,kA+∑β=1Nb∑m=1ncQβ−bnc+m+ε−k⋅Hβ,mB,Beβ,m−jθτXβ,mB+wb,kA,B,Rb,kB,B=Hb,kB,BXb,kB+∑β=1Nb∑m=1ncQβ−bnc+m−ε−k⋅Hβ,mA,Beβ,mjθτXβ,mA+wb,kB,B,where Rb,kU,V and wb,kU,V denote FFT output and AWGN signal of the RX signal received at the Vth RX antenna (V∈AandB) and downconverted by the carrier frequency of the Uth TX antenna, i.e., (fcU∈fcAandfcB), and Hb,kU,V denotes the channel coefficient between the Uth TX antenna and the Vth RX antenna. The normalized DVs x˜b,2k−1 and x˜b,2k are obtained as follows [15]:(12)x˜b,2k−1=Hb,k∗A,ARb,kA,A+Hb,nc−k+1B,ARb,nc−k+1∗B,A+Hb,k∗A,BRb,kA,B+Hb,nc−k+1B,BRb,nc−k+1∗B,BHb,kA,A2+Hb,nc−k+1B,A2+Hb,kA,B2+Hb,nc−k+1B,B2,x˜b,2k=Hb,k∗B,ARb,kB,A−Hb,nc−k+1A,ARb,nc−k+1∗A,A+Hb,k∗B,BRb,kB,B−Hb,nc−k+1A,BRb,nc−k+1∗A,BHb,kB,A2+Hb,nc−k+1A,A2+Hb,kB,B2+Hb,nc−k+1A,B2.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 byIb,kA and Ib,kA are generated in order to cancel ICI elements in the FFT output signals Rb,kA and Rb,kB, respectively. The estimated residual ICI terms Ib,kA and Ib,kB are expressed as(13)Ib,kA=∑β=1Nb∑m=1ncQβ−bnc+m+ε−kHβ,mBeβ,m−jθτX^β,mB,Ib,kB=∑β=1Nb∑m=1ncQβ−bnc+m−ε−kHβ,mAeβ,mjθτX^β,mA,where X^b,kA and X^b,kB are the reconstructed (estimated) versions of Xb,kA and Xb,kB based on the Alamouti-decoded data symbol x^l and the transmitter structure in Figure 3. Assume that ε, τ, and θ and channel coefficients Hβ,mA and Hβ,mB are known to the RX side. We update Rb,kA and Rb,kB by Rb,kA−Ib,kA and Rb,kB−Ib,kB which implies the residual ICI cancelled version of Rb,kA and Rb,kB, respectively, and then, we perform Alamouti decoding in (9) and (10) again on the updated Rb,kA and Rb,kB.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 onlyM≪N least-reliable DVs as the target symbol set for PMLD. The selection criterion of candidate symbol set for PMLD is based on γb,2k−1 and γb,2k which are comprised of the power of each symbols’s DV normalized by the total average power of the DVs. More specifically, γb,2k−1 and γb,2k for xb,2k−1 and xb,2k are calculated, respectively as(14)γb,2k−1=Rex˜b,2k−12ERex˜2,γb,2k−1=Rex˜b,2k−12ERex˜2,where x˜ is DV vector, i.e., x˜=x1,1,x1,2,…, Re⋅ denotes the real part of the complex value, and E⋅ denotes the expectation operator. In order to show whether γb,2k−1 and γb,2k reflect the reliability of DV well, the cumulative distribution function (CDF) of γb,2k−1 and γb,2k with N=256, Eb/N0=10 dB, and 20 dB with the normalized FO ε=0.2 and ε=0.5 when each DV is correctly estimated and vice versa is shown in Figure 5.Figure 5
The CDF ofγb,2k−1 and γb,2k with QPSK and Eb/N0=10 dB and 20 dB.
(a)
(b)From Figure5, the probability that the estimated symbols by RIC scheme are incorrect is about 90th percentile when γb,2k−1 and γb,2k 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 γb,2k−1 and γb,2k are small values. Furthermore, the probability that the estimation is incorrect is almost invariant to Eb/N0 and ε. Hence, we conclude that γb,2k−1 and γb,2k are available as the simple indicator of the reliability of the DVs.Based on the observation above, the PMLD algorithm is performed usingγb,2k−1 and γb,2k by the following steps:(i)
Step 1: classification of DVs for PMLD: classify the estimated DV vectorx^ as two vectors, i.e., xPML with length M and xPMLc with length N−M where ac is the complementary set of vector a. The elements of xPML are classified as the symbols corresponding to the M minimum γb,2k−1 and γb,2k. Meanwhile, the rest N−M symbols with high reliability, i.e., xPMLc, 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.,xPML. The detection symbol vector by PMLD x^PML is obtained as follows:(15)x^PML=arg minxPML∈CMrA−gAx^2+rB−gBx^2,where C denotes the constellation point set of the data symbol, rA and rB are FFT output vectors of Rb,kA and Rb,kB, i.e., rA=R1,1A,R1,2A,R1,3A,…,RNb,ncA and rB=R1,1B,R1,2B,R1,3B,…,RNb,ncB, respectively, and gAx^ and gBx^ are the reconstructed versions of rA and rB assuming the transmitted data vector is x^. 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 termrB−gBx^2 (or the term rA−gAx^2). This is because it is intuitive that these two terms are highly redundant due to the fact that both rA and rB are obtained from the common received signal.(2)
We perform the ML search over theJ 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 JM candidate symbol vectors in the space CM and thus the complexity reduction will be substantial for high-order modulation.(3)
As we sequentially scan the candidate symbol vectors in the spaceCM one by one, there is only one element difference between the previous candidate symbol vector and the current symbol vector. Thus, for generating gAx^ (or gBx^) 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., Hb,kAXb,kA and Hb,kBXb,kB, 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, 2NJM 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.Figure6 shows the BER performances of the schemes in comparison over the frequency offset and the timing offset plane for various cases of delay spread Tmax with Eb/N0 = 20 dB. We commonly set N = 256 and use QPSK constellation. For the PMLD in the proposed scheme, we set J=2 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 Hb,kA and Hb,kB, the number of multipaths is 8 and their delay are distributed uniformly in 0Tmax. The guard interval is set to be larger than Tmax+τ. The subcarrier spacing is set to 15 kHz which is taken from long-term evolution (LTE) standard [18]. The subblock size nc which varies according to delay spread is taken from the results in [7].Figure 6
The BERs of FADAC OFDM “(1),” FADAC OFDM with SIC “(1) + (2),” the proposed scheme “(1) + (2) + (3)” withEb/N0=20 dB, N=256, and QPSK: (a)Tmax=T/10, (b)Tmax=T/27, (c)Tmax=T/50, and (d)Tmax=T/100.
(a)
(b)
(c)
(d)We use the compact labels to each scheme in the legend: “(1)” for FADAC OFDM as it only uses the stage (1) in Figure4, “(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 nc 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 withTmax=T/10), 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 setTmax=T/50 in Figure 7 and as a severely selective fading case, we set Tmax=T/10 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 Tmax=T/10. In addition, even in the small range of FO, there exist the considerable performance gaps between the proposed scheme and the other two schemes.Figure 7
The BERs of FADAC OFDM “(1),” FADAC OFDM with SIC “(1) + (2),” the proposed scheme “(1) + (2) + (3)”withEb/N0=20dB, N=256, QPSK, and Tmax=T/50.Figure 8
The BERs of FADAC OFDM “(1),” FADAC OFDM with SIC “(1) + (2),” the proposed scheme “(1) + (2) + (3)” withEb/N0=20dB, N=256, QPSK, and Tmax=T/10.


## 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.



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*Source: 4319802-2019-09-30.xml*