 Research
 Open Access
Sensitivity analysis of FBMCbased multicellular networks to synchronization errors and HPA nonlinearities
 Brahim Elmaroud^{1}Email authorView ORCID ID profile,
 Ahmed Faqihi^{1, 2} and
 Driss Aboutajdine^{1}
https://doi.org/10.1186/s1363401604410
© The Author(s) 2017
 Received: 4 July 2016
 Accepted: 20 December 2016
 Published: 7 January 2017
Abstract
In this paper, we study the performance of asynchronous and nonlinear FBMCbased multicellular networks. The considered system includes a reference mobile perfectly synchronized with its reference base station (BS) and K interfering BSs. Both synchronization errors and highpower amplifier (HPA) distortions will be considered and a theoretical analysis of the interference signal will be conducted. On the basis of this analysis, we will derive an accurate expression of signaltonoiseplusinterference ratio (SINR) and bit error rate (BER) in the presence of a frequencyselective channel. In order to reduce the computational complexity of the BER expression, we applied an interesting lemma based on the moment generating function of the interference power. Finally, the proposed model is evaluated through computer simulations which show a high sensitivity of the asynchronous FBMCbased multicellular network to HPA nonlinear distortions.
Keywords
 Multicellular networks
 Filter bank multicarrier
 Synchronization errors
 HPA NLD
 Signaltointerferenceplusnoise ratio
 Bit error rate
1 Introduction
Multicarrier modulations can be separated into two main classes: cyclic prefixbased orthogonal frequency division multiplexing (CPOFDM) and filter bankbased multicarrier (FBMC). The first class is the most widely used since it has been adopted by many major communication standards, e.g., DAB, DVBT, IEEE802.11a/g (WiFi), IEEE 802.16 (WiMAX), and LTE. However, OFDM systems still have some drawbacks such as the high side lobes of the rectangular pulse shape (transmission/reception filter) and a loss of the spectral efficiency due to the cyclic prefix extension. Because of the aforementioned weaknesses, researchers are more and more interested by FBMC modulations which employ a frequency welllocalized FIR filter with small side lobes instead of the rectangular pulse shape used in OFDM. This makes FBMC systems more spectral efficient and less sensitive to frequency errors compared to OFDM [1].
When a multicarrier modulation (OFDM or FBMC) is employed, the performance of multicellular networks depends extensively on how well the orthogonality among subcarriers is maintained at the receiver. In an asynchronous system, this orthogonality is destroyed due to synchronization errors which include timing offsets and carrier frequency/phase offsets. The problem of asynchronism in multicarrier systems has been intensively investigated in the literature for OFDM as well as for FBMC systems [2–8]. The sensitivity of OFDM/FBMC systems to synchronization errors has been illustrated in terms of signaltointerference ratio (SIR) in [1, 3, 7] and in terms of bit error rate (BER) in [8, 9], by exploiting the Gaussian approximation of the intercarrier interference. An interesting work in relationship with the scope of this paper is [2] (resp. [5]) where the authors have presented a theoretical analysis of the average error rate of an asynchronous OFDM (resp. FBMC)based multicellular network in the special case of interleaved (resp. block) subcarrier assignment scheme. More recently, Medjahdi et al. [6] have presented an interference modeling based on the so called interference tables [4] for asynchronous OFDM and FBMC systems. This model and the results of [5] will be very useful to accomplish the developments of this paper.
Another serious issue with OFDM and FBMC systems is the fact that the transmitted signal is a sum of a large number of independently modulated subcarriers. Thus, they suffer from high peaktoaverage power ratio (PAPR) which makes the system very sensitive to nonlinear distortion (NLD) caused by nonlinear devices such as highpower amplifiers (HPA). This problem has been largely studied for OFDM systems. The impact of NLD on OFDM signals was presented in [10, 11] and a performance analysis of nonlinear OFDM and OFDMMIMO systems was derived in [12, 13]. As for FBMC systems, the authors in [14] carried out a theoretical analysis of BER performance for nonlinearly amplified FBMC/OQAM signals under additive white Gaussian noise (AWGN) and Rayleigh fading channels. Similarly, [15] has recently derived a closedform expression for the BER of graycoded Mary (QAM or OQAM)based OFDM in the presence of HPA NLD under a frequency flat fading Rayleigh channel.
It is worth noting that there are limited studies that investigate the problem of the joint effect of nonsynchronization and HPA nonlinearities on multicarrier systems. An interesting one is [16], where the authors evaluated the BER of multicarrier DSCDMA downlink systems subject to these impairments in frequencyselective Rayleigh fading channels, assuming QAM modulation. In this paper, we study the joint effect of synchronization errors and HPA nonlinear distortions on FBMCbased multicellular networks. We will derive exact expressions of average error rates for the considered system by carrying out an analytical interference analysis in frequencyselective fading channels. It is worth noting that there are limited studies that investigate the problem of joint effects of nonsynchronization and HPA nonlinearities on multicarrier systems. An interesting one is [17], where the authors analyzed the interference caused by nonlinear power amplifiers together with timing errors on multicarrier OFDM/FBMC transmissions. In this work, we will consider both frequency and timing errors in addition to HPA NLD and we will plot BER curves as functions of SNR for different frequency offset (CFO) ranges and different levels of the nonlinear noise power. Regarding FBMC modulation, a cosine modulated multitone (CMT) system will be considered and the results can be easily extended to staggered modulated multitone (SMT) systems since a CMT signal can be obtained from a SMT one through a simple modulation step [18].
The remainder of this paper is organized as follows. In Section 2, the considered CMTbased multicellular network is introduced and the joint effect of HPA NLD and synchronization errors is described. In Section 3, we present the analytical interference analysis. The BER analysis is carried out in Section 4. Section 5 includes the evaluation of the obtained BER expressions through simulation results. Finally, Section 6 concludes this paper.
2 System model
where R is the cell radius.

T is the CMT symbol duration and g(t) is the prototype filter impulse response,

\(s_{n}^{m,k}\) is the transmitted pulseamplitude modulated (PAM) symbol by the kth base station, at the mth subcarrier on the nth time index,

\(\gamma _{m,n}(t)=g(tnT)e^{j\frac {\pi }{2T} t}e^{j\Phi ^{m}(t)}\) and \(\Phi ^{m}(t)=m(\frac {\pi }{T} t+ \frac {\pi }{2})\),

F _{ k } is the set of subcarriers that are assigned to the kth base station. In this paper, we will consider the block subcarrier assignment, described in Fig. 2, for frequency reuse scheme.
where τ _{0}<τ _{1}<…τ _{ L−1}<τ _{max} and τ _{max} is the maximum delay spread of the channel, and {h _{ k,l },∀k,l} are the complex path gains which are assumed mutually independent, where \(E\left [ h_{k,i}, h_{k,i}^{*} \right ]=\gamma _{k,i}\) and \(E\left [ h_{k,i}, h_{k,j}^{*} \right ]=0\), if i≠j. We further assume that the power is normalized such that \(\sum _{l=0}^{L1} \gamma _{k,l}=1, \forall k\).
where \(\phantom {\dot {i}\!}\alpha =\alpha  e^{\phi _{\alpha }}\) is a complex factor.
where η(t) is a zero mean additive noise, uncorrelated to x(t) and with variance \(\sigma _{\eta }^{2}=E\left [ \left \eta \right ^{2} \right ]\).

* is the convolution operator,

K is the total number of neighboring cells,

x _{ k } is the interference signal transmitted by the base station BS _{ k },

n(t) is the additive white Gaussian noise (AWGN) with twosided power spectral density N _{0}/2,

β is the path loss exponent.
3 Interference analysis
3.1 Interference analysis in an AWGN channel
We recall that \(\Phi ^{m}(t)=m(\frac {\pi }{T} t+ \frac {\pi }{2})\).
where ψ=(τ,ε,θ), θ=ϕ+ϕ _{ α } and E _{ s } is the PAM symbol energy.
where we have considered (for simplicity’s sake) that the interference signal is coming from the 0th time index symbol.
where \(\Theta _{m}= \theta +(1\frac {2\tau }{T})\frac {\pi }{2}m\).
Similarly, \({\Gamma _{2}^{q}}(\psi,m,t)\) and \(\Gamma _{3}^{q,q'}(\psi,m,t)\) are, respectively, the primitives of integrals shown in the third and fourth terms of Eq. (13).
with \(\chi (\psi,m)=\left  \left.\Gamma (\psi,m,t) \right _{0}^{4T} \right ^{2}\).
3.2 Interference power in a frequencyselective channel and SINR expression

\(\mathcal {I}(\psi _{k},m,\alpha _{k})\) (Eq. (17)) is the power of the interference caused by the kth base station on the mth subcarrier and corresponding to an error vector ψ _{ k }=(τ _{ k },ε _{ k },θ _{ k }),

H _{ k }(m)^{2} is the power channel gain between the kth interfering base station and the reference receiver.
where \(\sigma _{\hat {\eta }_{0,k}^{0}}^{2}(\varepsilon _{k})=E\left [ \left  \hat {\eta }_{0,k}^{0}(\varepsilon _{k}) \right ^{2} \right ]\) is the variance of the received nonlinear noise \(\hat {\eta }_{0,k}^{0}(\varepsilon _{k}) \).
where B _{ sc } is the bandwidth of 0th subchannel.
In order to get a more explicit expression of the SINR, we suggest in the following subsection to analyze the expression of the received nonlinear noise variance \(\sigma _{\hat {\eta }_{0,k}^{0}}^{2}(\varepsilon _{k})\).
3.3 Variance of the received nonlinear noise
In this subsection, we will simplify the expression of the received nonlinear noise \(\hat {\eta }\) in order to find a closedform expression of its variance \(\sigma _{\hat {\eta }}^{2}\). We note that the BS index will be omitted in this subsection for simplicity’s sake.
where κ is the overlapping factor and G _{ q },q=0,…,κ−1 are the frequency coefficients of the filter.
Furthermore, the nonlinear noise η(t) can be expressed as the sum of his real and imaginary components : η(t)=η _{ r }(t)+j η _{ i }(t).
where \(a(\varepsilon _{k})=8 \frac {T^{2}}{\pi ^{2}} c^{2}(\varepsilon _{k}) (4T\varepsilon _{k}+1)^{2} \sin ^{2}(4\pi T\varepsilon _{k}) \left [1\sin (8\pi T\varepsilon _{k})\right ]\) and \(c(\varepsilon _{k})=2\sum _{q=0}^{3} (1)^{q} \frac {G_{q}}{q^{2}(4T\varepsilon _{k} +1)^{2}}\).
where \(\text {SNR}_{\eta _{k}}=E_{s}/\sigma _{\eta _{k}}^{2}\) and \(b=\frac {1}{d_{0}^{\beta } SNR} =\frac {N_{0} B_{sc}}{d_{0}^{\beta } E_{s}}\).
4 Bit error rate expression
and is SNR the signal to noise ratio.
Our objective is to derive the average BER which can be obtained by averaging out the vector of random variables \(\mathcal {H}\) from Eq. (32). Unfortunately, this average can not be calculated directly because we do not have a closedform expression for the probability density function (pdf) of the SINR. In order to get the BER average with a reduced computational complexity, we refer to the following lemma [2] and we propose to use the results of [5].
Lemma 1
where \(\mathcal {M}_{y}(z)=E_{y}\left [ e^{zy} \right ]\) is the moment generating function (MGF) of y.

A stands for the determinant of the matrix A,

\(\Omega _{k}=\left [\rho _{i,j}\right ]_{(i,j)\in F_{k}\times F_{k}}\) is the correlation matrix of the random variables {H _{ k }(m)^{2},m∈F _{ k }},

L _{ k } denotes the number of subcarriers allocated to the kth BS and \(I_{L_{k}}\) is the L _{ k }×L _{ k } identity matrix,

\(D_{k}^{\chi _{d}}\) is a diagonal matrix with diagonal elements \(D_{k}^{\chi _{d}}(i,i)=\chi _{d}(\psi _{k},i)\), i∈F _{ k }.
Equation (34) shows clearly that, in addition to the interference caused by synchronization errors, the BER performance of the studied system is affected by HPA nonlinearities and especially by the nonlinear noise η. A detailed analysis of this expression will be carried out in the next section via computer simulations.
5 Simulations
A _{0} is the HPA input saturation level, υ is the small signal gain, and p is a parameter that controls the smoothness of the transition from the linear region to the saturation region.
Regarding the frequency specifications, they match those of the IEEE 802.11 a/g standards, i.e., a bandwidth of 2 MHz, a subcarrier spacing of 0.3125 MHz and a symbol duration of T=1.6 μs. We recall that the fractional normalized CFO ε _{ N }=2T ε is defined in the range −0.5≤ε _{ N }≤0.5, which means that the CFO ε takes values between −0.16 and 0.16. In the curves presented in this section, ε will be taken in the range [0,0.16] (unless otherwise stated). A numerical computing environment (MATLAB) is used to simulate the interference CMT signals coming from the six interfering base stations. A 4PAM CMT receiver is also simulated and the BER is directly computed by comparing the decoded symbols with the real transmitted symbols from the reference base station. Finally, for the Rayleigh fading channel model, we have considered the pedestrianA model with relative delays [0 110 190 410] ns and corresponding average powers [0 –9.7 –19.2 –22.8] dB [23].
6 Conclusions
In this paper, we have studied jointly the effect of synchronization errors and HPA nonlinear distortions on asynchronous downlink CMT based multicellular networks. A scenario consisting on one reference mobile perfectly synchronized with its reference BS and K interfering BSs was considered and an exact BER expression was derived by carrying out an analytical interference analysis in the presence of a frequencyselective channel. In order to obtain explicit and less complex BER expressions, an interesting lemma based on the moment generating function of the interference power was applied. The obtained theoretical BER expression has been evaluated through simulation results using different evaluation parameters such as the signal to nonlinear noise ratio, the frequency error range, and the input backoff of the power amplifier. We found a perfect match between the simulation and the developed theoretical results. Furthermore, we have highlighted the significant performance degradation caused by the joint effect of HPA nonlinear distortions and synchronization errors.
Declarations
Acknowledgements
The authors would like to thank the associate editor Luca Rugini and the anonymous reviewers for valuable comments and suggestions that have led to improvements in this paper.
Funding
No funding information.
Authors’ contributions
All the authors have contributed to the study conception and design, acquisition of data, analysis and interpretation of data, drafting of manuscript, and critical revision. All authors read and approved the final manuscript.
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 SaeediSourck, Y Wu, JWM Bergmans, S Sadri, B FarhangBoroujeny, Sensitivity analysis of offset {QAM} multicarrier systems to residual carrier frequency and timing offsets. Signal Process.91(7), 1604–1612 (2011).View ArticleMATHGoogle Scholar
 KA Hamdi, YM Shobowale, Interference analysis in downlink ofdm considering imperfect intercell synchronization. Veh. Technol. IEEE Trans.58(7), 3283–3291 (2009). doi:10.1109/TVT.2009.2013959.View ArticleGoogle Scholar
 K Raghunath, A Chockalingam, Sir analysis and interference cancellation in uplink ofdma with large carrier frequency/timing offsets. Wireless Commun. IEEE Trans.8(5), 2202–2208 (2009). doi:10.1109/TWC.2009.071383.View ArticleGoogle Scholar
 Y Medjahdi, M Terre, D Le Ruyet, D Roviras, JA Nossek, L Baltar, in Signal Processing Advances in Wireless Communications, 2009. SPAWC ’09. IEEE 10th Workshop On. Intercell interference analysis for ofdm/fbmc systems (IEEEPerugia, 2009), pp. 598–602. doi:10.1109/SPAWC.2009.5161855.View ArticleGoogle Scholar
 Y Medjahdi, M Terre, D Le Ruyet, D Roviras, A Dziri, Performance analysis in the downlink of asynchronous ofdm/fbmc based multicellular networks. Wireless Commun. IEEE Trans. 10(8), 2630–2639 (2011). doi:10.1109/TWC.2011.061311.101112.View ArticleGoogle Scholar
 Y Medjahdi, Terre, Ḿ, DL Ruyet, D Roviras, Interference tables: a useful model for interference analysis in asynchronous multicarrier transmission. EURASIP J. Adv. Signal Process.2014(1), 1–17 (2014). doi:10.1186/16876180201454.View ArticleGoogle Scholar
 SK Hashemizadeh, MJ Omidi, H SaeediSourck, B FarhangBoroujeny, Sensitivity analysis of ofdma and scfdma uplink systems to carrier frequency offset. Wirel. Pers. Commun.80(4), 1381–1404 (2014). doi:10.1007/s1127701420890.View ArticleGoogle Scholar
 B Aziz, F Elbahhar, in Communications (ICC), 2015 IEEE International Conference On. Impact of frequency synchronization errors on ber performance of mbofdm uwb in nakagami channels (IEEELondon, 2015), pp. 2698–2703. doi:10.1109/ICC.2015.7248733.View ArticleGoogle Scholar
 L Rugini, P Banelli, Ber of ofdm systems impaired by carrier frequency offset in multipath fading channels. IEEE Trans. Wirel. Commun.4(5), 2279–2288 (2005). doi:10.1109/TWC.2005.853884.View ArticleGoogle Scholar
 D Dardari, V Tralli, A Vaccari, A theoretical characterization of nonlinear distortion effects in ofdm systems. IEEE Trans. Commun.48(10), 1755–1764 (2000). doi:10.1109/26.871400.View ArticleGoogle Scholar
 T Araujo, R Dinis, On the accuracy of the gaussian approximation for the evaluation of nonlinear effects in ofdm signals. Comm. IEEE Trans.60(2), 346–351 (2012). doi:10.1109/TCOMM.2011.102011.110151.View ArticleGoogle Scholar
 L Yiming, M O’Droma, J Ye, A practical analysis of performance optimization in ostbc based nonlinear mimoofdm systems. IEEE Trans. Commun.62(3), 930–938 (2014). doi:10.1109/TCOMM.2014.010414.130533.View ArticleGoogle Scholar
 L Yiming, M O’Droma, A novel decomposition analysis of nonlinear distortion in ofdm transmitter systems. IEEE Trans. Signal Process.63(19), 5264–5273 (2015). doi:10.1109/TSP.2015.2451109.View ArticleMathSciNetGoogle Scholar
 H Bouhadda, et al., Theoretical analysis of ber performance of nonlinearly amplified fbmc/oqam and ofdm signals. EURASIP J. Adv. Sig. Proc. 2014(1) (2014). doi:10.1186/16876180201460.
 R Zayani, H Shaiek, D Roviras, Y Medjahdi, Closedform ber expression for (qam or oqam)based ofdm system with hpa nonlinearity over rayleigh fading channel. Wirel. Commun. Lett. IEEE. 4(1), 38–41 (2015). doi:10.1109/LWC.2014.2365023.View ArticleGoogle Scholar
 L Rugini, P Banelli, Joint impact of frequency synchronization errors and intermodulation distortion on the performance of multicarrier dscdma systems. EURASIP J. Adv. Signal Process.2005(5), 1–13 (2005). doi:10.1155/ASP.2005.730.View ArticleMATHGoogle Scholar
 M KhodjetKesba, C Saber, D Roviras, Y Medjahdi, in Vehicular Technology Conference (VTC) Spring, 2011 IEEE. Multicarrier interference evaluation with jointly nonlinear amplification and timing errors (IEEEBudapest, 2011), pp. 1–5. doi:10.1109/VETECS.2011.5956177.View ArticleGoogle Scholar
 B FarhangBoroujeny, GCH Yuen, Cosine modulated and offset qam filter bank multicarrier techniques: a continuoustime prospect. EURASIP J. Appl. Sig. Pro.ID 165654:, 16 (2010).Google Scholar
 PM Shankar, Introduction to Wireless Systems, 1st edn (Wiley, New York, 2001).Google Scholar
 MG Bellanger, in Proceedings. ICASSP ’01, 4. Specification and design of a prototype filter for filter bank based multicarrier transmission (IEEESalt Lake City, 2001), pp. 2417–24204. doi:10.1109/ICASSP.2001.940488.Google Scholar
 JG Proakis, Digital Communications, 4th edn (McGrawHill, NY, 2001).MATHGoogle Scholar
 KA Hamdi, A useful technique for interference analysis in nakagami fading. Commun. IEEE Trans.55(6), 1120–1124 (2007). doi:10.1109/TCOMM.2007.898823.View ArticleGoogle Scholar
 R ITUR.M.1225, Guidelines for evaluation of radio transmission technologies for imt2000, Technical report, ITU (1997). https://www.itu.int/dms_pubrec/itur/rec/m/RRECM.12250199702I!!PDFE.pdf.