On performance of analog network coding in the presence of phase noise
- Amir Ligata^{1}Email author,
- Haris Gacanin^{2} and
- Fumiyuki Adachi^{3}
https://doi.org/10.1186/1687-1499-2013-2
© Ligata et al.; licensee Springer. 2013
Received: 23 March 2012
Accepted: 27 November 2012
Published: 2 January 2013
Abstract
Analog network coding (ANC) as a simple implementation of physical layer network coding based on orthogonal frequency division multiplexing (OFDM) has been proposed to increase the network capacity and reliability of bi-directional link between a pair of users. In ANC protocol, an information between a pair of users is exchanged through two orthogonal time phases (i.e., multiple-access and broadcast phases). On the other hand, the phase noise (PN) introduces phase offset and inter carrier interference (ICI) to the useful signal. Thus, in ANC scheme PN will affect the useful signal during both multiple-access and broadcast phases. In this article, we present a performance analysis of ANC scheme using OFDM in the presence of PN in frequency-selective fading channel. We derive the total composite variance of ANC scheme in the presence of PN to obtain the signal-to-interference-plus-noise ratio (SINR) expression. Then, we evaluate the system’s performance in terms of bit error rate (BER), SINR degradation, and ergodic capacity through both numerical and computer simulations. Computer simulated average BER has been consistent with the numerical results, validating the presented analysis. Our results have shown that the ANC scheme is more sensitive to the PN introduced during the broadcast phase (i.e., at destination) than during the multiple-access phase (i.e., at relay). This is because of the higher ICI to the useful signal and enhanced noise due to the imperfect self-information removal at the destination. In addition, the performance degradation of ANC scheme based on OFDM in the presence of PN is highly expressed for the PN linewidth values up to 20 Hz.
1 Introduction
Although the wireless traffic demands driven by the users requirements continuously grow, available radio resources remain fixed. Intensive effort is being made in order to exploit the limited radio spectrum more efficiently and minimize the mutual interference caused by the multi-user access. Orthogonal frequency division multiplexing (OFDM) is considered to be an attractive transmission technique for broadband communications over a wireless channel. A simple implementation of physical layer network coding, known as analog network coding (ANC) using OFDM, has been proposed to increase the network capacity and reliability of bi-directional communication between a pair of users using the same channel[1–4]. In ANC, an information between a pair of users is exchanged through two orthogonal time phases; (i) in multiple-access phase both users simultaneously transmit towards the relay, and (ii) in broadcast phase the relay broadcast the received signal to both users using amplify-and-forward (AF) protocol. The phase noise (PN) caused by imperfections of the local oscillators rotates all subcarriers in OFDM by a common phase error (CPE) and introduces inter-carrier interference (ICI) to the useful signal which degrades the bit error rate (BER) performance[5–13]. The problem of phase synchronization in relaying based on AF protocol is investigated in[14], and it was shown that the relay’s PN has only limited impact on the system’s performance. Implementation of ANC scheme in frequency-domain was presented in[15], and carrier frequency offset compensation is considered at relay to capture the effect of PN in ANC scheme. In[16], impact of ICI on cooperative SFBC-OFDM is theoretically analyzed and an iterative algorithm for inter-symbol interference (ISI) and ICI cancelation is proposed. It was showed that the two-step cancelation algorithm can efficiently suppress the ICI and ISI. Cooperative STBC-OFDM system with imperfect frequency synchronization is investigated in[17], and an ICI mitigation algorithm was proposed which considerably improves the system’s symbol error rate. In the case of imperfect channel state information (CSI), the performance of ANC scheme in the presence of PN would be further degraded as the noise due to the imperfect-self information removal and CE error would arise[18–20]. We note here that the sampling frequency offset which occurs due to non-synchronized sampling at the transmitter and receiver clock may also degrade the performance of ANC scheme as the ICI arises and the OFDM symbol is rotated[21].
In ANC communication protocol, the transmission is done over two orthogonal phases in a time division multiplexing (TDM) fashion. This way, the PN is introduced at the reception in each transmission phase (i.e., multiple-access and broadcast phases), and consequently may cause CPE and ICI to the useful signal at the receiver end (i.e., relay and destination). Thus, the BER performance of ANC based on OFDM may be degraded. To the best of the authors’ knowledge, an impact of PN on the performance of ANC scheme has not been addressed in the literature.
In this article, we theoretically analyze the impact of PN on the performance of OFDM-based ANC scheme in frequency-selective fading channel. We derive the total composite noise variance of ANC scheme in the presence of PN, while approximating PN as a zero-mean Wiener process to obtain the signal-to-interference-plus-noise ratio (SINR) expression. The BER performance is evaluated by the way of both numerical and computer simulations. The performance of ANC scheme in the presence of PN is further evaluated through numerical simulation of SINR degradation and ergodic capacity. Our simulation results indicate that the performance of ANC scheme is more sensitive to the PN introduced at the reception during the broadcast phase (i.e., destination), since the corresponding PN affects the self-information removal process at the destination.
This article is organized as follows. Section 2. gives an overview of the network model. The performance analysis of ANC in the presence of PN is presented in Section 3. Numerical results and discussion are given in Section 4. Section 5. concludes the article.
2 Network model
In this section, first we introduce a PN model and then the ANC network model in the presence of PN is presented. In our analysis, T_{ c } discrete-time representation is used, with T_{ c } being the fast Fourier transform (FFT) sampling period.
2.1 PN model
for n = 0,…,N_{ c } − 1. PN affects the received signal by attenuating and rotating the useful signal, and its impact on the OFDM system is reflected through CPE and ICI[8]. Thus, in the presence of PN in OFDM-based system all subcarriers of the signal are rotated by a CPE, while ICI causes a blurring of the constellation like thermal noise.
2.2 ANC network model
During the first time phase, both users simultaneously transmit its information towards the relay (multiple-access phase), while in the second time phase relay broadcasts the received signal using AF protocol (broadcast phase).
2.2.1 Multiple-access phase
During the multiple-access phase, both users simultaneously transmit its information towards relay, where the j th (j = 0,1) user’s information is given by {d_{ j }(n); n = 0,…,N_{ c }−1} with E [ |d_{ j }(n)|^{2} ] = 1, and E[·] being ensemble average operation. The time-domain signal {s_{ j }(t); t = 0,…,N_{ c }−1} denotes the IFFT of the j th user’s data-modulated symbol sequence {d_{ j }(n)}. Then, a guard interval (GI) length of N_{ g }(N_{ g } > L) is inserted, where L denotes the number of channel paths. Finally, the signal is transmitted over a frequency-selective fading channel.
Wireless channel is characterized with discrete-time channel impulse response${h}_{m,j}\left(t\right)=\sum _{l=0}^{L-1}{h}_{l,m,j}\delta (t-{\tau}_{l})$, where L, h_{l,m,j}, τ_{ l } and δ(t), respectively, denote the number of paths, the path gain between the j th terminal and the relay during m th (m = 0,1)stage, the l th path normalized by IFFT sampling period, and the Dirac delta function. {h_{l,m,j}; l = 0,…,L − 1} are zero-mean independent complex variables with variance E[|h_{l,m,j}|^{2}] = 1 / L. Without loss of generality, we assume τ_{0} = 0 < τ_{1} < ⋯ < τ_{L−1}, where the l th path delay is given by τ_{ l } = l△, with △(≥ 1) being the time delay separation between adjacent paths.
where α_{ r }(n) denotes the n th component of the PN at the relay defined with (2).
2.2.2 Broadcast phase
for n = 0,…,N_{ c } − 1.
where the bar over the expression signifies the unitary complement operation (i.e., ’NOT’ operation) that performs logical negation of the value under the bar, with$\stackrel{\u0304}{j}\in \{0,1\}$.
Finally, one tap frequency-domain equalization (FDE) is applied to combat the effects of frequency-selective channel as${\widehat{d}}_{j}\left(n\right)={\stackrel{~}{R}}_{j}\left(n\right){w}_{j}\left(n\right)$ for n = 0,…,N_{ c } − 1, where the equalization weight is given by${w}_{j}\left(n\right)={H}_{0,\overline{j}}^{\ast}\left(n\right){H}_{1,j}^{\ast}\left(n\right)/\left|{H}_{0,\overline{j}}\right(n\left){H}_{1,j}\right(n){|}^{2}$[4].
3 Performance analysis
In this section, we present derivation of the composite noise variance, and then expression for the SINR degradation is given. We note in this article that no path loss and shadowing are assumed. However, the general conclusion remains irrespective of the assumption regarding the path loss and shadowing. Moreover, we assume perfect knowledge of the CSI. The performance of ANC scheme with imperfect CSI in the presence of PN would be degraded. The performance of ANC with imperfect CSI has been considered in[18–20], while the joint analysis of PN and imperfect CSI would be difficult if not impossible to derive.
3.1 Composite noise variance
for n = 0,…,N_{ c } − 1, with${\stackrel{\u02c6}{H}}_{j}\left(n\right)={H}_{0,\overline{j}}\left(n\right){H}_{1,j}\left(n\right){w}_{j}\left(n\right)$ and${\stackrel{\u02c6}{H}}_{1,j}\left(n\right)={H}_{1,j}\left(n\right){w}_{j}\left(n\right)$. It can be seen from (11) that due to PN introduced at the relay and j th user, perfect self-information removal is not possible.
We assume that the modulated data-symbols, the PN, the channel coefficients, and AWGN are independent of each other. Consequently, the terms given by (11) are independent random variables. Thus, the variance of the composite noise (i.e., the sum of the noise due to imperfect self-information removal, the ICI at relay, the ICI at destination, and the AWGN) is represented by$2{\sigma}^{2}=E\left[\right|{\xi}_{j}\left(n\right){|}^{2}]+E[\left|{\mathcal{C}}_{r}\right(n\left){|}^{2}\right]+E\left[\right|{\mathcal{C}}_{j}\left(n\right){|}^{2}]+E[\left|\mathcal{N}\right(n\left){|}^{2}\right]$.
In order to calculate the variance of the composite noise given by (12), we have to evaluate the variance of PN term, the ICI at the relay and ICI at the destination.
3.1.1 PN term
where T_{ s } (= N_{ c }T_{ c }) denotes the symbol length duration.
3.1.2 ICI at the relay
3.1.3 ICI at the destination
We observe from (17) that the impact of the PN on the composite noise variance can be seen through noise due to imperfect self-information removal, the ICI to the useful signal at the relay and at the destination and the AWGN, in that order.
3.2 SINR degradation
By increasing the transmitted power, the total composite noise increases due to the ICI at relay given by (15). In the expression for the ICI at destination given by (16), the first term is caused by a CPE term at relay while the second and the third terms are due to the ICI at relay and AWGN enhanced by energy of a PN term. In the high SNR region, the first and the second terms in the above expression drastically increase causing further performance degradation due to the PN and consequently, limits the system’s performance. By analyzing (16) and (15), we observe that the ICI at destination includes the ICI at relay just enhanced by the energy of a PN term. Thus, the ICI at the relay has the highest impact on the total composite noise variance given by (17). The SINR degradation given by (20) directly depends on the composite noise variance. Due to that, among all terms in composite noise variance, the ICI at the destination has highest impact on the SINR degradation.
3.3 BER
Theoretical average BER is evaluated using Monte Carlo numerical computation method as follows. First, the channel impulse response of each link for both phases is generated to obtain the corresponding channel gains {H_{m,j}(n)}. The equalization weights {w_{ j }(n)} are computed, and the conditional SINR expression given by (18) is computed. Then, the average BER is computed for the given set of channel gains {H_{m,j}(n)} and this procedure is repeated for large number of times.
3.4 Ergodic channel capacity
Next, we evaluate the ergodic channel capacity in the presence of PN for OFDM-based ANC scheme. Ergodic capacity represents an ensemble average of the information rate over the channel distribution, and it is important since it defines a transmit rate irrespective of the channel realization.
where C(E_{ s } / N_{0},{H_{m,j}(n)},β_{ o }) is the conditional channel capacity and f_{ h } = f_{ h }H_{m,j}(n)] denotes the Rayleigh probability density function normalized such that$\sum _{l=0}^{L-1}E\left[\right|{h}_{l,m,j}{|}^{2}]=1$. To date, the solution to the integral in (23) has not been found in the closed-form, so we resort to the numerical computational methods.
The evaluation of the ergodic capacity is done by using the Monte Carlo numerical computational method as follows. First, a set of path gains {h_{m,j}; l = 0,…,L − 1} is generated to obtain the channel gains {H_{m,j}(n);n = 0,…,N_{ c } − 1}. This is followed by the computation of the equalization weights {w_{ j }(n);n = 0,…,N_{ c } − 1}, and afterwards the SINR expression given by (18) is computed. Finally, the Ergodic capacity is evaluated by averaging (24) a sufficient number of times for different realization of the channel.
4 Simulation results and discussion
Numerical parameters
Data modulation | QPSK | |
---|---|---|
Transmitter | Block (IFFT) size | N_{ c } = 256 |
GI | N_{ g } = 32 | |
Channel | L = 16-path frequency-selective block | |
Rayleigh fading | ||
FDE | ZF | |
Receiver | Channel estimation | Ideal |
4.1 SINR degradation
4.2 BER
4.3 Ergodic capacity
We underline that in this study, we have assumed perfect CSI. Imperfect CSI would further degrade the performance of ANC in the presence of PN, but it has been left out as interesting future study.
Based on the presented results, we conclude that implementation of the algorithms for mitigating the PN effects in ANC scheme should be done at the destination. Presented analysis should facilitate the design of such algorithms.
5 Conclusion
In this article, we theoretically analyze the impact of PN on the performance of ANC scheme using OFDM access in frequency-selective fading channel. Due to the property of ANC protocol, the PN is introduced during both, multiple-access phase (i.e., at relay) and broadcast phase (i.e., at destination). We derive the total composite noise variance of the ANC scheme in the presence of PN while approximating the PN as a zero-mean Wiener process to obtain the SINR expression. The average BER performance of ANC scheme is evaluated by the way of both, numerical and computer simulations. The performance is further evaluated through numerical simulation of the SINR degradation and ergodic channel capacity. Our results have shown that the performance of ANC scheme in terms of the average BER and ergodic capacity is more sensitive to the PN introduced during the broadcast phase, than during the multiple-access phase. This is because the PN at the destination introduces higher ICI to the useful signal and increases noise due to imperfect self-information removal leading to a inferior performance. Moreover, the SINR degradation increases as the transmitted power increases, which corresponds to the higher CPE and ICI contribution in the total composite noise. In addition, we observed rapid performance degradation of ANC scheme due to PN for the PN linewidth values up to 20 Hz. Using analysis presented in this article, one may estimate the impact of phase noise during the first phase and accordingly design an iterative algorithm at the destination.
Declarations
Acknowledgements
This study was supported in part by the 2010 KDDI Foundation Research Grant Program.
Authors’ Affiliations
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