- Research
- Open Access
Iterative detection for frequency-asynchronous distributed Alamouti-coded (FADAC) OFDM
- Bong-seok Kim^{1} and
- Kwonhue Choi^{2}Email author
https://doi.org/10.1186/s13638-017-0819-1
© The Author(s) 2017
- Received: 7 June 2016
- Accepted: 1 February 2017
- Published: 23 February 2017
Abstract
We propose a near intercarrier interference (ICI)-free and very low complexity iterative detector for frequency-asynchronous distributed Alamouti-coded (FADAC) orthogonal frequency division multiplexing (OFDM). In the previous cancelation schemes, the entire subcarrier signals from one transmit (TX) antenna are estimated and canceled in the received signal from the other TX antenna and vice versa. However, the reliability of the estimated symbols are revealed to significantly vary across the subcarriers and thus, the poorly estimated symbols lead to the incorrect cancelation. Motivated from this, we first propose a scheme which does not cancel the interfering subcarrier(s) at the half band edges which undergo very high interference in FADAC-OFDM. For further improvement, we propose a so-called selective scheme which instantly measures the reliability of the detected symbols at each iteration and then exclude the unreliable symbols in the estimated interference generation.
Moreover, the proposed scheme has a drastically reduced complexity by converting the cancelation process from the subcarrier domain to the time domain. In accordance with the analysis on the considered reliability measures, the numerical results show that the proposed scheme achieves the near ICI-free level only within three or four iterations for wide ranges of SNR, frequency offset, and delay spread.
Keywords
- Iterative MIMO
- Alamouti
- ICI cancelation
- OFDM
- Distributed antennas
- Frequency offset
1 Introduction
Recently, several studies on Alamouti-coded OFDM (orthogonal frequency division multiplexing) for cooperative systems have been reported. One of the main challenging issues in this area is to mitigate self-interference due to the carrier frequency offset (CFO) between the distributed transmit antennas [1–8]. Very recently in [4], the so-called FADAC-OFDM (frequency-asynchronous distributed Alamouti-coded OFDM) has been proposed and shown to outperform the other existing approaches in [3–5]. In contrast to the conventional distributed Alamouti-coded OFDM, FADAC-OFDM is free from ICI (intercarrier interference) terms from the near subcarriers due to its ICI self-cancelation property only by simple Alamouti-decoding process. Especially, [4] tried to exploit this ICI self-cancelation property even in the selective fading channel by dividing the entire subcarriers into multiple subblocks. However, in the severely frequency-selective fading channels, FADAC-OFDM gets worse due to non-negligible inter-block ICI terms.
Meanwhile, in [5–7], typical types of iterative ICI cancelation schemes for the conventional Alamouti-coded OFDM with the distributed antennas have been proposed. Since the conventional Alamouti-coded OFDM [9] has no ICI self-cancelation property for frequency and timing asynchronous distributed antennas, the accuracy of the initial detection is poor. Thus, a considerable number of iterations of cancelation has to be performed until the performance converges. Moreover, the converged performances are not so impressive. Although in [5] they derived the performance result close to no-CFO case, they assumed the perfect ICI cancelation which has not been justified. The schemes in [6, 7] rapidly break down as the CFO gets larger than 0.5. Moreover, they have high computation overheads because at each iteration, the required number of complex multiplications for the interference reconstruction is 4N ^{2} where N denotes the total number of OFDM subcarriers.
Recently, in [10] and [14], the decision-directed iterative ICI cancelation schemes to distributed multiple input multiple output (MIMO) have been proposed. In [14], the authors considered spatial modulation MIMO as the application system model and they showed the performance results only for the small values of CFO. In [10], we, the authors of this paper, combined a typical decision-directed iterative ICI cancelation scheme to FADAC-OFDM. We achieved better performance compared with [5] even with low complexity due to better initial detection performance of FADAC-OFDM compared to the conventional Alamouti-coded OFDM. This scheme uses all the detected symbols in the interference reconstruction step without any consideration of reliability of detection symbols. However, even in FADAC-OFDM, some of the detection symbols may eventually have relatively high possibility of errors due to severe ICI terms such as inter-block ICI as mentioned before. It is shown that after the first iteration, the performance fairly improves, but from the second iteration, the performance is stuck in the same value. This is because the reconstructed interference term for cancelation is not updated anymore due to the erroneous portion of the constructed interference. Consequently, the performance gap between this scheme [10] and the case of no ICI is still considerable.
In order to solve drawbacks of the previous cancelation scheme [10], it is important to carefully decide whether or not to use each of the detected symbols in the interference reconstruction step at each iteration. In other words, we have to use or devise a certain measure to assess the reliability of the detected symbols at each iteration, based on which we have to exclude the unreliable symbols in the interference reconstruction.
To this end, we first propose a deterministic scheme where the fixed number of data symbols at the half band edges are not used in the interference reconstruction because they themselves undergo very high interference in FADAC-OFDM, and thus, they are less reliable. In order to further improve the cancelation performance, we propose a method which instantaneously measures the reliability of the soft detected symbols [12] at each iteration and then exclude the unreliable symbols in the estimated interference generation. We jointly employ two reliability check measures: (1) square error of the decision variable from the corresponding constellation point and (2) the detection consistency for two consecutive iterations.
Apart from the ICI cancelation performance itself, the computational complexity of the scheme should be feasible from the implementation viewpoint. In the proposed scheme, we employ a drastically low complexity structure which attains the complexity reduction in terms of polynomial order.
The remainder of this paper is structured as follows: First, we review FADAC-OFDM in Section 2. In Section 3, we first review our previous iterative cancelation scheme and its problem and then we propose two types of iterative cancelation schemes. In Section 4, we propose a complexity-reduced ICI cancelation structure. In Section 5, we provide the various performance results which support the improved ICI cancelation capability of the proposed schemes.
2 Review on FADAC-OFDM
In this paper, we are considering the iterative ICI cancelation schemes for FADAC-OFDM. In this section, we describe the motivation of FADAC-OFDM and give a self-contained review on the TX and receive (RX) structures of FADAC-OFDM. In addition, we revisit its residual ICI terms for the subsequent sections.
FADAC-OFDM has been proposed for FO-tolerant Alamouti-coded OFDM for frequency-asynchronous distributed antenna systems [4]. FADAC-OFDM employs a frequency reversal structure at the TX side. Then, FADAC-OFDM detects the symbols by performing simple linear combining after two separate DFT operations with local carriers synchronized to each TX antenna. By doing so, FADAC-OFDM cancels the major parts of intra-block ICI terms from neighboring subcarriers, and thus, FADAC-OFDM significantly improves the performance for the distributed antenna systems. However, despite the ICI self-cancelation property of FADAC-OFDM, two kinds of ICI terms, i.e., intra-block ICI and inter-block ICI, still remain non-negligible. This leads us to consider a further cancelation of the remaining ICI terms by using an iterative cancelation scheme which will be introduced in Section 3.
2.1 The system model and the OFDM symbol structure of FADAC-OFDM
where x _{ b,1} and x _{ b,2} denote the two data symbols for the bth subblock.
We assume that the fading is locally flat over the Alamouti-coded block. To justify this assumption, the block size n _{ c } is set smaller than the coherent bandwidth. From Figs. 1 and 2, it is straightforward that as an extreme case, the FADAC-OFDM with n _{ c }=2 is equivalent to CDAC-OFDM.
2.2 RX structure of FADAC-OFDM
where \(w_{b,k}^{(A)}\) and \(w_{b,k}^{(B)}\) are AWGN terms and \(H_{b,k}^{(A)}\) and \(H_{b,k}^{(B)}\) are channel fading coefficients at the kth subcarrier of the bth subblock from TX antennas A and B, respectively, and they are each independent and follow zero mean, unit variance complex Gaussian distribution.
Motivated from this, by using an iterative cancelation step which will be introduced in the next section, we try to cancel further the remaining ICI terms. In Fig. 5 a, by employing iterative cancelation, it is shown that ICI powers at half band edge and band edge significantly decrease compared to the case before iterative cancelation. However, the ICI powers at half subblock edges are still relatively large compared to the middle band. This is because the iterative ICI cancelation is not perfect, and thus, the reason for high interferences at the half subblock edges mentioned above still holds.
For a reference, Fig. 5 b shows the normalized ICI power of FADAC-OFDM with n _{ c }=2 which is equivalent to CDAC-OFDM. With n _{ c }=2, the feature of FADAC-OFDM, i.e., self-cancelation of the intra-block ICI term, is meaningless because there exists only one subcarrier in each half subblock. Thus, the ICI powers over all subcarriers are very high as shown in Fig. 5 b, and the iterative cancelation is not so effective either. This implies that CDAC-OFDM is not suitable for the frequency-asynchronous distributed antenna systems.
3 Combining iterative ICI cancelation schemes to FADAC-OFDM
3.1 The previous iterative ICI cancelation scheme for FADAC-OFDM
where \(R_{b,k}^{(A,i)}\) and \(R_{b,k}^{(B,i)}\) denote the updated versions of \(R_{b,k}^{(A)}\) and \(R_{b,k}^{(B)}\), respectively, at the ith iteration. Finally, at each iteration, we perform the Alamouti combining in (10) and (11) using \(R_{b,k}^{(A,i)}\) and \(R_{b,k}^{(B,i)}\) to obtain the updated detection symbols \(\hat {x}_{b,2k-1}^{(i+1)}\) and \(\hat {x}_{b,2k}^{(i+1)}\), respectively, for the next ((i+1)th) iteration.
In [10], it is shown that due to the good performance of FADAC-OFDM by intra-subblock ICI self-cancelation, this basic iterative scheme for FADAC-OFDM achieves better performance with lower complexity compared with [5]. However, this scheme still has room to be improved. Due to high ICI power at the subband edges shown in Fig.5, the detection symbols at those edges are more likely to be erroneously detected compared to the other detection symbols. The incorrect detection symbols result in the incorrect ICI term reconstruction and thus the incorrect ICI cancelation. As a result, even with increasing iterations, the improvement of performance is limited and the error probability is stuck in a certain point where the non-negligible incorrect contribution to the reconstructed ICI term is not self-corrected by the iterations any more. This will be checked out again in the simulation results.
3.2 The proposed iterative ICI cancelation schemes
In the previous section, we addressed the issue of the previous iterative ICI cancelation in [10], i.e., the drawback of using the entire detection symbols for ICI reconstruction and cancelation. To tackle this issue, we propose two types of selective ICI cancelation schemes.
3.2.1 Scheme I. DS scheme
where E is a set of indices of edge subcarriers, i.e., \(E=\{1, 2, \ldots, M, \frac {n_{c}}2-M+1, \frac {n_{c}}2-M+2,\ldots, \frac {n_{c}}2\}\), and M is the number of data symbols (subcarriers) at each edge to be excluded in the ICI term reconstruction. For example, if M is set to 2 with n _{ c }=16, then set E is equal to {1,2,7,8}. Consequently, 2M(=M pairs of Alamouti code) data symbols are not used, and they are replaced by null data symbols in the ICI term reconstruction. Simply by excluding 2M detection symbols at the edge in each subblock which are severely interfered by inter-block ICIs, we can avoid the performance degradation due to wrong ICI term cancelation. Another merit of this scheme is that it does not need any additional hardware or computations compared to [10].
This scheme excludes the data symbols in the deterministic carrier positions, i.e., predetermined positions based on the average ICI power distribution across the subcarriers as shown in Fig. 5. However, we know from (7) and (8) that the ICI term at each subcarrier contains lots of random variables such as the data symbols in the other subcarriers and their fading coefficients and thus the ICI power at each subcarrier instantaneously varies. This implies that some of the edge subcarriers can eventually undergo rather small instantaneous ICI despite the high average ICI power. As we need the instantaneous reliability of the detection symbols to decide whether or not to use each detection symbol, the proposed DS (deterministically selective) scheme which simply excludes the fixed number of band edge subcarriers still has room to be improved if we can accommodate the instantaneous reliability of the detection symbols.
3.2.2 Scheme II. AS scheme
where \({\hat {x}^{\text {(used)}}}\) denotes the actual value which will be used in the ICI reconstruction and ρ is a threshold value which determines whether or not the detection is sufficiently reliable or not.
Note in Fig. 7 that this criterion possibly misses the correct symbols or possibly uses the incorrect symbols in the ICI reconstruction step. The threshold ρ should be set by considering both of these two possibilities. The optimum value will vary according to the channel parameters, the system parameters, or even the iteration layer. However, in the practical system, it is likely to use, rather, a constant threshold, i.e., global suboptimal setting, and thus, we cannot avoid the performance loss compared to the optimized case.
To complement this, we use another measure to assess the reliability of the detection symbols, i.e., detection consistency between two consecutive iterations. If a certain detection symbol is sufficiently reliable at the ith iteration, the detection result would not change in the (i+1)th iteration. Hence, we treat a detection symbol as the reliable one if its detection result is maintained between two consecutive iterations. This measure well compromises the probability that the first criterion in (24) uses the incorrect symbol(s) in the ICI reconstruction step. For example, when a certain incorrect detection symbol has a small γ, we conclude that it eventually has the small γ and it is unreliable if the detection result in the previous iteration is not equal to the detection result in the current iteration.
where C1 denotes the condition \({\gamma _{b,2k - 1}^{(i)} \le \rho }\) and C2 denotes the condition \({\hat {x}_{b,2k - 1}^{(i)} = \hat {x}_{b,2k - 1}^{(i - 1)}}\). As the similar expressions hold for x _{ b,2k }, we exclude the expressions for x _{ b,2k−1} without loss of generality. To avoid the wrong cancelation, we have to lower \(p_{\text {false}}^{\text {condition}}\), and to avoid missing the correct detection symbols, we have to lower \(p_{\text {miss}}^{\text {condition}}\).
4 Complexity reduction
The hardware structures of the iterative cancelations in [5–7, 10] are basically the same. They all include the calculations for reconstructing the interference term at each FFT outputs expressed in (7) and (8) at each iteration. The interference term in (7) and (8) have 2N complex multiplications. As there are two FFTs with N outputs, the overall required number of complex calculations for interference term reconstruction per iteration is equal to 4N ^{2}.
Meanwhile in the proposed scheme, we modify this complexity-expensive structure into a mathematically equivalent but low complexity structure. Figure 3 shows the receiver structures for the previous iterative cancelation scheme in [10] and the proposed scheme. Instead of performing cancelation at the FFT output stage (subcarrier domain), we can equivalently cancel the interference at the FFT input stage (time domain). Hence, the reconstructed interference corresponds to the time domain version. The reduced computation is intuitive due to the fact that the time domain interference takes the form of just a single sampled vector but it contains the N parallel interfering subcarriers.
In (31) and (32), \({\hat {X}}_{\beta,m}^{(A,i)}\) and \({\hat {X}}_{\beta,m}^{(B,i)}\) denote the estimated versions of \({X}_{\beta,m}^{(A)}\) and \({X}_{\beta,m}^{(B)}\), respectively, at the ith iteration, \( \left [ e^{\frac {-j 2\pi \varepsilon } N }, e^{\frac {-j 2\pi 2\varepsilon } N} \cdots e^{-j 2\pi \varepsilon } \right ]\) and \( \left [ e^{\frac {-j 2\pi \varepsilon } N }, e^{\frac {-j 2\pi 2\varepsilon } N } \cdots e^{-j 2\pi \varepsilon } \right ]\) denote the sampled versions of the residual complex exponentials by CFO, and ⊙ denotes the element-wise multiplication. Note that the term \(\text {IFFT} \left [H_{1,1}^{(B)} {\hat {X}}_{1,1}^{(B,i)}, H_{1,2}^{(B)} {\hat {X}}_{1,3}^{(B,i)}, \cdots, H_{N_{b},n_{c}}^{(B)} {\hat {X}}_{N_{b},n_{c}}^{(B,i)} \right ] \odot \left [ e^{\frac {-j 2\pi \varepsilon } N }, e^{\frac {-j 2\pi 2\varepsilon } N} \cdots e^{-j 2\pi \varepsilon } \right ]\) in (31) corresponds to the reconstructed time domain-sampled signal from TX antenna B to the received signal synchronized to TX antenna A. The similar remark holds for (32).
From (38), we know that \(F_{k}^{(A)}\) is finally equal to (7), and thus, it is proved that the proposed time domain cancelation is equivalent to the previous subcarrier domain cancelation.
The number of multiplications required in the operations for the interference term reconstruction at each branch per iteration of the proposed scheme
Operation | Number of multiplications |
---|---|
IFFT input vector generation | N |
IFFT for time domain conversion | \(\frac {N}{2} {\log }_{2} N\) |
N-point complex sinusoid multiplication | N |
FFT for the original OFDM demodulation | \(\frac {N}{2} {\log }_{2} N\) |
Total | 4N+2N log2N |
Comparison of the number of multiplications between the previous and the proposed schemes according to N
N | Previous (frequency domain) | Proposed (time domain) | Proposed/previous |
---|---|---|---|
64 | 4096 | 1024 | 0.25 |
128 | 16,384 | 2304 | 0.14 |
256 | 65,536 | 5120 | 0.07 |
512 | 262,144 | 11,264 | 0.04 |
1024 | 1,048,576 | 24,576 | 0.02 |
5 Simulation results
In this section, we provide the simulation results to evaluate the performance of the proposed scheme. Commonly, we set N=256. Regarding multi-path profile for generating \(H_{b,k}^{(A)}\) and \(H_{b,k}^{(B)}\), the number of multi-paths is 8 and their delays are distributed uniformly in [0 T _{max}] where T _{max} is the maximum delay spread. The guard interval is set to be larger than T _{max}. The subcarrier spacing (= 1/T where T = OFDM symbol duration prior to the guard time insertion) is set to 15 kHz by referring to the Long-Term Evolution (LTE) standard. For the proposed AS (adaptive selective) scheme, the threshold value ρ is set to 0.4 regardless of the iteration number and the other parameters.
As a baseline for the performance comparison, CDAC-OFDM with the iterative ICI cancelation using the entire detection symbols in the OFDM frame is included. Note that the iterative ICI cancelation results in almost no improvement to CDAC-OFDM. This is because the initial detection performance of CDAC-OFDM under frequency-asynchronous environment is poor, and thus, the ICI cancelation based on unreliable initial detection does not work properly. On the other hand, the iterative ICI cancelation works better for the case when it is applied to FADAC-OFDM which has a superb initial detection performance. However, the performance gain of ICI cancelation scheme in [10] is still not so significant. This scheme uses all the detected symbols in the interference reconstruction step without any consideration of the reliability of the detection symbols. Even in FADAC-OFDM, some of the detection symbols may eventually have relatively high possibility of errors due to severe ICI terms such as inter-block ICI as mentioned before. It is shown that after the first iteration, the performance fairly improves but from the second iteration, the performance is stuck in the same value. This is because the reconstructed interference term for cancelation is not updated anymore due to the erroneous portion of the constructed interference. Consequently, the performance gap between the scheme in [10] and the case of no ICI is still significant.
Note that two proposed schemes in this paper achieve significantly improved performance compared to the scheme in [10]. In the first iteration, the proposed DS scheme with M=1 achieves a substantially decreased BER compared to the scheme in [10]. This implies that simply excluding the band edge subcarriers can efficiently avoid the erroneous ICI reconstruction. This results in the significant improvement by the canceling ICI from the rest of the subcarriers in the first iteration. However, the band edge subcarriers will not be canceled in the remaining iterations as well, and the BER converges to a still significantly higher level than that of the ICI-free case.
Meanwhile, the proposed AS scheme has significantly improved performance compared to the proposed DS scheme. Only within three or four iterations, the proposed AS scheme approaches nearly ICI-free level. This means that adaptively selecting the detection symbols for ICI reconstruction works properly, and as the iteration goes on, even the band edge subcarriers having high ICI power are gradually canceled. Despite the inferior performance of the proposed DS scheme to the proposed AS scheme, the proposed DS scheme has a merit that it is easy to implement and needs nearly no complexity overhead.
We can further improve the performances of the proposed schemes by more carefully optimizing or adaptively changing the system parameters such as the subblock size of FADAC-OFDM n _{ c }, M for the proposed DS scheme, or ρ for the proposed AS scheme. However, we do not cover this case because the main point of this paper is to make sure of the improved performance of the proposed schemes even with suboptimal parameters. In addition, adaptively changing the system parameters is practically a burden in terms of system implementation.
6 Conclusions
We proposed an enhanced iterative ICI cancelation scheme distributed Alamouti-coded OFDM both in terms of the performance and the complexity. By avoiding the incorrect cancelation due to incorrect symbols, the proposed scheme achieves better performance than other ICI cancelation schemes. Only within three or four iterations, the proposed scheme achieves near ICI-free performance by instantaneously reflecting the reliability of detection symbols at each iteration. As for complexity, by converting the ICI cancelation with functional equivalence, the proposed scheme has a drastically reduced computational complexity. The performance results shown in this paper sufficiently appeal as a promising solution for the current and future cooperative transmit antenna systems using OFDM waveform and Alamouti code. The proposed scheme will be further improved by combining with some sophisticated schemes, such as the adaptively selective cancelation based on the soft decision feedback [12]. We leave this as one of our future works.
Declarations
Acknowledgements
This work was supported in part by the DGIST R&D Program of the Ministry of Science, ICT and Future Planning, Korea (17-IT-01), Basic Science Research Program through the National Research Foundation (2015R1D1A3A01015970) funded by the Ministry of Education, and the Information Technology Research Center support program (IITP-2016-R2718-16-0035) supervised by the Institute for Information & Communications Technology Promotion funded by the Ministry of Science, ICT and Future Planning, Korea.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- H Wang, XG Xia, Distributed space-frequency codes for cooperative communication systems with multiple carrier frequency offsets. IEEE Trans. Wirel. Commun. 8(2), 1045–1055 (2009).View ArticleGoogle Scholar
- Z Li, XG Xia, An Alamouti coded OFDM transmission for cooperative systems robust to both timing errors and frequency offsets. IEEE Trans. Wirel. Commun. 7(5), 1839–1844 (2008).View ArticleGoogle Scholar
- K Choi, Inter-carrier interference-free Alamouti-coded OFDM for cooperative systems with frequency offsets in non-selective fading environments. IET Commun. 5(15), 2125–2129 (2011).MathSciNetView ArticleMATHGoogle Scholar
- B Kim, K Choi, FADAC-OFDM: frequency asynchronous distributed Alamouti-coded OFDM. IEEE Trans. Vehi. Tech. 64(2), 466–480 (2015).View ArticleGoogle Scholar
- Y Zhang, J Zhang, in Proc. IEEE WCNC 2009. Multiple CFOs compensation and BER analysis for cooperative communication systems, (2009), pp. 1–6.Google Scholar
- T Lu, H Lin, T Sang, in Proc. IEEE ISPMRC 2010. An SFBC-OFDM receiver to combat multiple carrier frequency offsets in cooperative communications, (2010).Google Scholar
- J Lee, H Lin, T Sang, in Proc. IEEE ISCAS 2012. An SFBC-OFDM receiver with MLSE equalizer to combat multiple carrier frequency offsets, (2012).Google Scholar
- Y Yao, X Dong, Multiple CFO mitigation in amplify-and-forward cooperative OFDM transmission. IEEE Trans. on Commun. 12(60), 3844–3854 (2012).View ArticleGoogle Scholar
- K Lee, D Williams, in Proc. IEEE GLOBECOM ’00. A space-frequency transmitter diversity technique for OFDM systems, (2000), pp. 1473–1477.Google Scholar
- B Kim, J Lee, D Jeong, K Choi, in Proc. ITNG 2014. Combining successive ICI cancelation to ICI suppressed Alamouti coded OFDM for frequency asynchronous distributed antenna systems, (2014).Google Scholar
- OE Agazzi, N Seshardsi, On the use of tentative decisions to cancel intersymbol interference and nonlinear distortion (with application to magnetic recording channels). IEEE Trans. Inf. Theory. 2(43), 394–408 (1997).View ArticleMATHGoogle Scholar
- SH Muller, WH Gerstacker, JB Huber, in Proc. GLOBECOM ’96. Reduced-state soft-output trellis-equalization incorporating soft feedback, (1996), pp. 95–100.Google Scholar
- Y Chen, C Tellambura, Performance analysis of maximum ratio transmission with imperfect channel estimation. IEEE Commun. Lett. 4(9), 322–324 (2005).View ArticleGoogle Scholar
- B Zhou, Y Xiao, P Yang, S Li, in Proc. WiCOM, 2011. An iterative CFO compensation algorithm for distributed spatial modulation OFDM systems, (2011).Google Scholar
- SM Alamouti, A simple transmitter diversity scheme for wireless communications. IEEE J. Select. Areas Commun. 16(8), 1451–1458 (1998).View ArticleGoogle Scholar
- P Dharmawansa, N Rajatheva, H Minn, An exact error probability analysis of OFDM systems with frequency offset. IEEE Trans.Commun. 57(1), 26–31 (2009).View ArticleGoogle Scholar