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Twoway amplifyandforward relaying with carrier offsets in the absence of CSI: differential modulationbased schemes
EURASIP Journal on Wireless Communications and Networking volume 2014, Article number: 111 (2014)
Abstract
In this paper, differential modulation (DM) schemes, including single differential and double differential, are proposed for amplifyandforward twoway relaying (TWR) networks with unknown channel state information (CSI) and carrier frequency offsets caused by wireless terminals in highspeed vehicles and trains. Most existing work in TWR assumes perfect channel knowledge at all nodes and no carrier offsets. However, accurate CSI can be difficult to obtain for fast varying channels, while increases computational complexity in channel estimation and commonly existing carrier offsets can greatly degrade the system performance. Therefore, we propose the two schemes to remove the effect of unknown frequency offsets for TWR networks, when neither the sources nor the relay has any knowledge of CSI. Simulation results show that the proposed differential modulation schemes are both effective in overcoming the impact of carrier offsets with linear computational complexity in the presence of high mobility.
Introduction
Twoway relaying (TWR) has attracted much interest recently[1–7], where two source terminals communicate with each other through an intermediate relay. Both amplifyandforward (AF) and decodeandforward (DF) relaying schemes under oneway relaying have been extended to TWR[3, 4]. In the DF protocol, the relay first decodes the information transmitted from both sources in the multipleaccess (MA) phase, performs binary network coding to the decoded signal, then broadcasts the networkcoded signal back to the sources in the broadcast (BC) phase. If the relay cannot decode the information correctly, erroneous relaying will cause significant performance degradation. For the AFbased TWR, the relay amplifies the superimposed signal received from the two sources and then broadcasts it back in the BC phase. AFbased TWR is particularly useful in wireless networks, since the wireless channel acts as a natural implementation of network coding by summing the wireless signals over the air. Therefore, we will focus on the AFbased TWR in this paper.
There has been some work investigating TWR using AF[4–6], referred to as analog network coding (ANC). However, most of the existing work assumes that perfect channel state information (CSI) is known at all transmission links. Although in some scenarios, the CSI is likely to be acquired through the use of pilot signals, it may be very difficult to obtain accurate CSI when the channel coefficients vary fast. Moreover, conventional estimation methods do not work for AFbased TWR, although they are effective for DFbased TWR. For example, channel estimation for TWR was studied in[8, 9] for frequencyflat and frequencyselective environments, respectively. These studies showed that AF TWRN systems require very different estimation techniques from conventional pointtopoint systems. Therefore, differential modulation for TWR without the knowledge of CSI is worth being exploited. Differential receivers for TWR were designed in[7, 10, 11]. However, perfect synchronization was assumed in[7, 10], while imperfect synchronization scenario caused by different propagation delay from both sources to the relay due to the distributed nature of all nodes was investigated in[11]. To the best of our knowledge, no work has been reported in TWR with unknown carrier frequency offsets when CSI is not available at all nodes.
In wireless mobile communications, however, Doppler shift is common and inevitable, especially in the highspeed mobile environment. For example,it is anticipated that the thirdgeneration European cellular standards will operate on trains moving as fast as 500 km/h. If the carrier frequency is 2 GHz, the induced Doppler shift may be up to 880 Hz. One technique to mitigate frequency offset is to estimate it at the receiver using a frequency acquisition and tracing circuit and then compensate it with singledifferential modulation, resulting in increased computational complexity in the relay and reduced data rate[12, 13]. Another approach is doubledifferential modulation[14–16], which can effectively handle frequency offsets in the presence of channel fading. A multiple symbol doubledifferential detection based on least squares criteria was proposed in[16], where the system performance was proved to be insensitive to different carrier offsets. However, all the above methods[12–16] are carried out on pointtopoint communication links and cannot be directly applied to TWR with unknown carrier offsets, since the signal received at the relay is a mixture of both source signals, and CSI is not available at all nodes.
Therefore, we investigate both singledifferential detection (SD) and doubledifferential detection (DD) for TWR using AF with unknown carrier offsets in this paper. For SD, a carrier offset estimation and compensation scheme with reduced computational complexity is employed. To further improve the performance of using DD, a fast algorithm of multiplesymbolbased signal detection is proposed. Simulation results show that the proposed SD and DD schemes are both effective in removing the carrier offsets, and the computational complexity remains linear.
Notation
Boldface lowercase letters denote vectors, (·)^{∗} stands for complex conjugate, (·)^{T} represents transpose, (·)^{H} represents conjugate transpose,\mathbb{E}\left\{.\right\} is used for expectation, ∥·∥ denotes the Euclidean vector norm, CN(0,N_{0}) denotes the set of Gaussian distributed complex numbers with the standard variance of N_{0} (i.e., 0.5 N_{0} per dimension), and Re{·} denotes real part.
Singledifferential modulation for bidirectional relay networks under carrier offsets
We consider a network with three nodes including two source nodes, denoted by S_{1} and S_{2}, and one relay node R. A halfduplex system is assumed and all nodes are equipped with one antenna. Information is exchanged between S_{1} and S_{2} with the help of R, which is completed in two phases. In the first phase, the MA phase, both source nodes send the differentially encoded signals to the relay, and in the second phase, the BC phase, the relay broadcasts the superimposed signals back to both source nodes.
Let z_{ i }(k) ∈ Ω,i ∈ {1,2} denote the symbol to be transmitted by source node S_{ i } at discrete symbol time k, where Ω represents a unity power MPSK constellation set. As singledifferential modulation is used, the signal z_{ i }(k) sent by source S_{ i } is given as
In the MA phase, two terminals simultaneously transmit the differentially encoded information to the relay. For simplicity, we assume that the fading coefficients and the carrier offsets keep constant over the frame of length L and change independently from one frame to another[7, 10]. The received signal at the relay at time k is then
where h_{ i },i ∈ {1,2} denotes the complex channel gain with zero mean and unit variance between S_{ i } and R;{\omega}_{i}=2\pi {f}_{\mathrm{d}}^{i}T, T is the symbol interval and{f}_{\mathrm{d}}^{i} is the Doppler shift introduced between S_{ i } and R. n_{r}(k) stands for a zero mean complex Gaussian random variable with variance{\sigma}_{n}^{2}, and P_{ i } denotes the transmit power at source S_{ i }.
In the BC phase, the relay R amplifies y_{r} by a factor α and then broadcasts its conjugate, denoted by{y}_{\mathrm{r}}^{\ast}\left(k\right) back to both S_{1} and S_{2} with transmit power P_{r}. The corresponding signal received by S_{1} at time k, denoted by y_{1}(k), can then be written as
For the decoding simplicity at S_{1}, we can obtain the conjugate of (3) as
where\alpha \phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\left({P}_{1}{\left{h}_{1}\right}^{2}+{P}_{2}{\left{h}_{2}\right}^{2}+{N}_{0}\right)}^{\frac{1}{2}},\mu =\alpha \sqrt{{P}_{1}{P}_{\mathrm{r}}}{\left{h}_{1}\right}^{2},\nu \phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\alpha \sqrt{{P}_{2}{P}_{\mathrm{r}}}{h}_{1}^{\ast}{h}_{2} and the equivalent noise{\stackrel{\u0304}{n}}_{1}\left(k\right)=\alpha \sqrt{{P}_{\mathrm{r}}}{n}_{\mathrm{r}}\left(k\right){h}_{1}^{\ast}{e}^{j{\omega}_{1}k}+{n}_{1}^{\ast}\left(k\right).
Similarly, the received signal at S_{2} can be expressed as
Given that S_{1} and S_{2} are mathematically symmetrical, as shown in (3) and (5), for simplicity, we only discuss the signal detection at S_{1} in the following.
Since the relay has no knowledge of CSI, we cannot obtain the amplification factor α directly. We may rewrite the received signals at the relay in a vector format as
where Y_{r} = [y_{r}(1),⋯,y_{r}(L)]^{T},n_{r} = [n_{r}(1),⋯,n_{r}(L)]^{T},{\mathbf{S}}_{\mathrm{i}}={\left[{s}_{i}\left(1\right){e}^{\phantom{\rule{0.3em}{0ex}}j{\omega}_{i}},\cdots \phantom{\rule{0.3em}{0ex}},{s}_{i}\left(L\right){e}^{\phantom{\rule{0.3em}{0ex}}j{\omega}_{i}L}\right]}^{T}, i ∈ {1,2}. We therefore have
where\mathbb{E}\left\{{\mathbf{S}}_{1}^{H}\phantom{\rule{0.3em}{0ex}}{\mathbf{S}}_{1}\right\}=\mathbb{E}\phantom{\rule{0.3em}{0ex}}\left\{{\mathbf{S}}_{2}^{H}\phantom{\rule{0.3em}{0ex}}{\mathbf{S}}_{2}\right\}=L,\mathbb{E}\phantom{\rule{0.3em}{0ex}}\left\{{\mathbf{n}}_{\mathrm{r}}^{H}\phantom{\rule{0.3em}{0ex}}{\mathbf{n}}_{\mathrm{r}}\right\}=L{N}_{0},\mathbb{E}\left\{{\mathbf{S}}_{1}^{H}{\mathbf{n}}_{\mathrm{r}}\right\}=\mathbb{E}\left\{{\mathbf{S}}_{2}^{H}{\mathbf{n}}_{\mathrm{r}}\right\}=0 and\mathbb{E}\left\{{\mathbf{S}}_{1}^{H}{\mathbf{S}}_{2}\right\}=\mathbb{E}\left\{{s}_{1}^{\ast}\right.\left(1\right){s}_{2}(1){e}^{\phantom{\rule{0.3em}{0ex}}j\left({\omega}_{2}{\omega}_{1}\right)}+\cdots +\left(\right\}separators="">\n \n \n \n s\n \n \n 1\n \n \n \u2217\n \n \n \n \n L\n \n \n \n \n s\n \n \n 2\n \n \n \n \n L\n \n \n \n \n e\n \n \n \n j\n \n \n \n \n \omega \n \n \n 2\n \n \n \n \n \n \omega \n \n \n 1\n \n \n \n \n L\n \n \n \n. α can thus be approximated at high signaltonoiseratio (SNR) as
Similar to (6), the received signals at source S_{1} can also be rewritten in the vector format as
where{\mathbf{Y}}_{1}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\left[{y}_{\mathrm{r}}\left(1\right),\cdots \phantom{\rule{0.3em}{0ex}},{y}_{\mathrm{r}}\left(L\right)\right]}^{T},{\mathbf{S}}_{1}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\left[{s}_{1}\left(1\right),\cdots \phantom{\rule{0.3em}{0ex}},{s}_{1}\left(L\right)\right]}^{T},{\mathbf{S}}_{2}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\left[{s}_{2}\left(1\right){e}^{\phantom{\rule{0.3em}{0ex}}j\left({\omega}_{2}{\omega}_{1}\right)},\cdots \phantom{\rule{0.3em}{0ex}},{s}_{2}\left(L\right){e}^{\phantom{\rule{0.3em}{0ex}}j\left({\omega}_{2}{\omega}_{1}\right)L}\right]}^{T} and{\stackrel{\u0304}{\mathbf{n}}}_{1}={\left[{\stackrel{\u0304}{n}}_{1}\left(1\right),\cdots \phantom{\rule{0.3em}{0ex}},{\stackrel{\u0304}{n}}_{1}\left(L\right)\right]}^{T}.
It is shown in (3) that the signal received at source S_{1} is a complex superimposed signal; therefore, the application of conventional single or doubledifferential detection on pointtopoint communication link to TWR is not straightforward. It is difficult to decode the expected information z_{2}(k) if we cannot subtract the selfinformation s_{1}(k) from y_{1}(k) when μ is unknown, due to the lack of CSI at S_{1}. Therefore, we propose a threestep approach in the singledifferential detection for TWR with carrier offsets: step 1, the selfinformation of μ s_{1}(k) is subtracted from y_{1}(k), the most important step in the whole detection procedure. Step 2, the carrier frequency offset is estimated and compensated. Step 3, signal z_{2}(k) differentially decoded using the singlesymbol singledifferential detector.
Step 1: selfinformation subtraction
Since terminal S_{1} knows its own transmitted signal, μ needs to be estimated before we can subtract the contribution of μ s_{1}(k) from y_{1}(k). We thereby propose a simple estimation method as follows
By taking the expectation of{\mathbf{Y}}_{1}{S}_{1}^{H}, given that s_{1}(k) and s_{2}(k) are independent and have the same distribution, we can approximately obtain
After obtaining the estimation of μ, we can easily subtract the selfinformation of s_{1}(k) as
Step 2: carrier offset estimation
A frequency offset estimation method was introduced in[13], which is effective in removing the impact of carrier frequency offsets, independent of data symbols and channel gains. However, training symbols are required to be transmitted at the beginning of each frame to solve the ambiguous estimation problem. In this case, two training symbols are enough to provide a good estimation of the carrier offsets. Then, the signals received at S_{1} can be rewritten as
where P is the number of training symbols. Define the training symbols as s_{ i }(P) = 1 and P = 2, we have
Since ν is also a complex value, the following transformation is made
where\omega \stackrel{\Delta}{=}{\omega}_{2}{\omega}_{1}. Then, the estimation of ω can be obtained as\widehat{\omega}=\text{arg}\left\{\sum _{l=1}^{0}y\left(l\right)\right\}\in \left(\pi \right.,\left(\right]separators="">\n \n \pi \n \n.
Step 3: singlesymbol singledifferential detection
With the estimation of the carrier frequency offset\widehat{\omega}, the frequency offset effect is compensated, and the received data after compensation can be expressed as
Consider the generalized likelihood ratio test (GLRT) detection of the multiple symbols{\left\{{s}_{2}\left(kn\right)\right\}}_{n=0}^{N}. Define
The GLRT algorithm for detection of{\left\{{s}_{2}\left(kn\right)\right\}}_{n=0}^{N} can be obtained by minimizing the following metric:
Performing the minimization of the metric over ν results in the following decision algorithm:
Let{\left\{{\u015d}_{2}\left(kn\right)\right\}}_{k=0}^{N} be the detection results during the observation length of N for the signal transmitted by S_{2}. Then, by differential decoding, we can recover the z_{2}(kn) as
Doubledifferential modulation bidirectional relay networks under carrier offsets
In this section, we investigate the doubledifferential modulation for TWR. Similar to the singledifferential modulation, the signal s_{ i }(k) sent by source S_{ i } is given as
Same as singledifferential modulation, the signals received at terminal S_{1} can be transformed as
The DD in TWR is divided into two steps. Step 1 is selfinformation elimination, similar to the first step of the singledifferential detection method described in section ‘Singledifferential modulation for bidirectional relay networks under carrier offsets’. Step 2 is the doubledifferential demodulation. The attractive feature of doubledifferential modulation is its insensitivity to unknown frequency offset, so the frequency offset is not necessarily acquired and tracked in step 2. For the second step of DD detection, conventional doubledifferential detector, including symbolbysymbol and multiplesymbol detection can be applied, once the selfinformation μ s_{1}(k) is subtracted from the received signal y_{1}(k). Since the processing of step 1 has been introduced in the above section in detail, we in the next focus on step 2.
Symbolbysymbol doubledifferential detection
From (13), the selfinformation of μ s_{1}(k) can be subtracted at S1 without the need of any CSI; therefore, (13) is equivalent to the DD detection on a direct transmission link[14]. A symbolbysymbol doubledifferential detector is then developed to recover the desired information, as in the following:
Multiplesymbol doubledifferential detection
Even though doubledifferential modulation can eliminate the degradation due to frequency offset, it needs higher SNR power ratio than that of coherent detection, to achieve the same average bit error rate (BER) performance. An attractive approach to mitigate this SNR loss is called multiplesymbol doubledifferential detection[15, 16].
In the absence of noise, we can obtain
which is equivalent to singledifferential detection, and when iterated, it becomes
Here, N denotes the symbol length in the observation. Next, the minimum leastsquare (LS) criterion[16] is applied. By performing the minimization of the metric over\stackrel{\u02c6}{h}{p}_{2}^{\ast}\left(kN+2\right), the following decision can be obtained:
However, (28) has a computational complexity of\frac{\left(\mathit{\text{MN}}3MN+2\right){M}^{N1}}{{\left(M1\right)}^{2}}, which is prohibitively high. Then, a fast algorithm is introduced in the following with a complexity on the order of N log_{2}N independent of the constellation size based on the principle in[17].

(26)
can be rewritten as
{\u0177}_{1}\left(k\right)={\stackrel{\u0304}{y}}_{1}\left(k\right){\stackrel{\u0304}{y}}_{1}^{\ast}\left(k1\right)={\left\nu \right}^{2}{e}^{\phantom{\rule{0.3em}{0ex}}j\left({\omega}_{2}{\omega}_{1}\right)}{p}_{2}\left(k\right)(29)
With the theorem[17] that the vector Z_{2} maximizes\mathrm{p}\left({\u0177}_{1}{\mathbf{Z}}_{2}\right) if and only if the vector P_{2} maximizes\mathrm{p}\left({\u0177}_{1}{\mathbf{P}}_{2}\right),{\widehat{\mathbf{P}}}_{2} which maximizes the following is then selected:
where Z_{2} = [z_{2}(k + 1),⋯,z_{2}(k + N)]^{T},P_{2} = [p_{2}(k + 1),⋯,p_{2}(k + N)]^{T},k ≥ 0.
If{\widehat{\mathbf{P}}}_{2}={\mathbf{P}}_{2}, following the corollary[17], for any k, l, with 1 ≤ k, 1 ≤ N, we have
For any k,k = 1,⋯,N and any{\widehat{p}}_{2}\left(k\right),{\u0177}_{1}\left(k\right){\widehat{p}}_{2}\left(k\right) is termed as a remodulation of{\u0177}_{1}\left(k\right). Therefore, it is sufficient to consider only those sets of remodulations of{\u0177}_{1}\left(k\right),k=1,\cdots \phantom{\rule{0.3em}{0ex}},N, which contain the remodulations within\frac{2\pi}{M}. Let{\stackrel{\u0304}{\mathbf{P}}}_{2} be the unique P_{2}, which satisfiesarg\left({\u0177}_{1}\left(k\right){\stackrel{\u0304}{p}}_{2}\left(k\right)\right)\in \left(0,\frac{2\pi}{M}\right]. For simplicity, we define{d}_{k}={\u0177}_{1}\left(k\right){\stackrel{\u0304}{p}}_{2}\left(k\right) and then list the arg{d_{ k }} ordering from the largest to the smallest. Define the function k(i) where the value of k(i) denotes the subscript k of d_{k(i)} and i represents the i th position in the list. To get all the possible remodulations of{\u0177}_{1}\left(k\right),k=1,\cdots \phantom{\rule{0.3em}{0ex}},N, let the list going clockwise around the circle at the interval of\frac{2\pi}{M}. Let q_{ i } = d_{k(i)}, i = 1,⋯,N for m = 1,⋯,M  1 and then for\mathit{\text{mN}}<i\le \left(m+1\right)N,{q}_{i}={e}^{\phantom{\rule{0.3em}{0ex}}j\frac{2\pi}{M}}{q}_{i\mathit{\text{mN}}}.
To maximize (30), it is sufficient to obtain the starting position as
Note that the magnitudes in (32) are periodic in N, resulting in Mfold ambiguity in (30), which will not affect differential decoding. Thus, only the following is required to be obtained:
and hence, the algorithm has the complexity on the order of N log_{2}N. Then, vector{\widehat{\mathbf{P}}}_{2} can be obtained, where
By reordering the elements{\widehat{p}}_{2,\phantom{\rule{0.3em}{0ex}}k\left(i\right)},i=1,\cdots \phantom{\rule{0.3em}{0ex}},N in the order of the subscript value (i), we can get the vector{\widehat{p}}_{2}\left(k\right),k=1,\cdots \phantom{\rule{0.3em}{0ex}},N. For differential decoding, z_{2}(k) can be recovered as
Simulation results
In this section, we present some simulation results for the proposed SD and DD schemes for TWR using AF with different Doppler shifts corresponding to different relative velocities between the relay R and the terminal S_{ i },i ∈ {1,2}. We choose the carrier frequency 2 GHz and the symbol interval T = 100u s. Three different normalized Doppler frequencies have been selected, f_{ d }T = 0.12, f_{ d }T = 0.24 and f_{ d }T = 0.36, corresponding to a mobile terminal moving at speeds of 100, 200, and 300 Km/h, respectively. We also plot the performance of the analog network coding scheme with differential modulation (ANCDM)[7] with no frequency offset for comparison. For simplicity, it is assumed that P_{1} = P_{2} = P_{ r } = 1, both source nodes and the relay have the same noise variance N_{0}, and the variance of complex channel coefficient is set to 1 for all links. All simulations are performed with BPSK modulation and the length of the frame is set to 100.
The BER performance of estimating μ as described in (11), and (12) is presented in Figure1a,b with random Doppler shift. For comparison, we also included the Genieaided result by assuming that μ is perfectly known by the source such that traditional differential decoding without carrier offsets can be performed both for SD and DD. It is shown that there is almost no performance loss using the estimation method with the existence of carrier offsets, which clearly justifies the robustness of the proposed schemes.
In Figure2, the BER of the proposed SD for TWR is compared with that of the ANCDM[7] with different Doppler shifts. It can be observed that the proposed SD scheme based on Doppler shift estimation and compensation nearly has the same performance under different Doppler shifts. It is about 3 dB inferior to ANCDM[7] without Doppler shift. However, ANCDM[7] is shown to experience high error floor under the Doppler shift.
In Figure3, the BER of the proposed multiplesymbol doubledifferential detection (MSDD) for TWR is compared with that of the ANCDM[7] with different Doppler shifts. It can be observed that the proposed MSDD scheme nearly has the same performance under different Doppler shifts; it nearly has the same performance as the ANCDM[7] without Doppler shift at high SNR. However, ANCDM[7] is shown to experience high error floor under the doppler shift.
In Figure4, the BER of the proposed DD scheme is compared with that of ANCDM[7] under random Doppler shift. It can be observed that the performance improves significantly with the increasing of the observation length N, approaching a limit about 0.5 dB away from the performance of ANCDM[7] under no Doppler shift with N = 64. However, ANCDM[7] under random Doppler shift can not work.
Figure5 compares the BER performance between the proposed SD scheme and DD scheme. Liu et al.[13] shows the performance of multiplesymbol singledifferential (MSSD) detection degrades with the increasing of the observation length N because of the inaccurate Doppler shift estimation caused by the short training symbols; therefore, for singledifferential modulation in TWR, singlesymbol detection is preferred. It is shown that the BER performance of DD with N = 64 is about 2 dB superior to that of SD with P = 2 with random carrier frequency offsets.
Next, the computational complexity of the two proposed methods is compared. The computational complexity of the proposed DD with multiplesymbol detection using fast algorithm is O(N log_{2}N), which is independent of the constellation size M, while that of the SD is O(M), which are both linear. It is also demonstrated that singledifferential detector using the frequency offset estimation needs extra training symbols, which decreases the transmit rate, while doubledifferential detector has its insensitivity to unknown frequency offsets, allowing the hardware implementation to be easy, without the need of complicated frequency offset acquisition and tracking circuitry. Its inherent SNR loss can be greatly minimized by using the multiplesymbol detection.
Conclusion
In this paper, we have proposed two differential modulation schemes to effectively attenuate the degrading effects on performance due to the Doppler shifts in TWR using ANC, when neither the sources nor the relay has any knowledge of CSI. The simulation results indicate that the proposed algorithms can effectively remove the impact of Doppler shift in the presence of channel fading with low computational complexity in highspeed mobile environment.
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Acknowledgements
This work was supported in part by the National Natural Science Foundation of China under Grant No. 60972055, No. 61132003, and No. 61171086 by Shanghai Natural Science Foundation under Grant No. 14ZR1415100 and by the Shanghai Leading Academic Discipline Project under No. S30108. WT’s work is supported by The Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, Innovation Program of Shanghai Municipal Education Commission (14ZZ096), Specialized Research Fund for the Doctoral Program of Higher Education (20133108120015) and Innovation Fund of Shanghai University.
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Wu Zhuo, Li Guangxiang and Wang Tao contributed equally to this work.
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Zhuo, W., Guangxiang, L. & Tao, W. Twoway amplifyandforward relaying with carrier offsets in the absence of CSI: differential modulationbased schemes. J Wireless Com Network 2014, 111 (2014). https://doi.org/10.1186/168714992014111
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DOI: https://doi.org/10.1186/168714992014111
Keywords
 Bidirectional relay communication
 Amplifyandforward
 Differential
 Carrier offsets