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Interference tables: a useful model for interference analysis in asynchronous multicarrier transmission
EURASIP Journal on Advances in Signal Processing volume 2014, Article number: 54 (2014)
Abstract
In this paper, we investigate the impact of timing asynchronism on the performance of multicarrier techniques in a spectrum coexistence context. Two multicarrier schemes are considered: cyclic prefixbased orthogonal frequency division multiplexing (CPOFDM) with a rectangular pulse shape and filter bankbased multicarrier (FBMC) with physical layer for dynamic spectrum access and cognitive radio (PHYDYAS) and isotropic orthogonal transform algorithm (IOTA) waveforms. First, we present the general concept of the socalled power spectral density (PSD)based interference tables which are commonly used for multicarrier interference characterization in spectrum sharing context. After highlighting the limits of this approach, we propose a new family of interference tables called ‘instantaneous interference tables’. The proposed tables give the interference power caused by a given interfering subcarrier on a victim one, not only as a function of the spectral distance separating both subcarriers but also with respect to the timing misalignment between the subcarrier holders. In contrast to the PSDbased interference tables, the accuracy of the proposed tables has been validated through different simulation results. Furthermore, due to the better frequency localization of both PHYDYAS and IOTA waveforms, FBMC technique is demonstrated to be more robust to timing asynchronism compared to OFDM one. Such a result makes FBMC a potential candidate for the physical layer of future cognitive radio systems.
1 Introduction
Nowadays, we witness a continuous evolution of applications for wireless communications requiring higher and higher spectral resources. In order to overcome the problem of spectrum scarcity resulting from conventional static spectrum allocation, there is a growing interest in the design and the development of cognitive radio technology [1]. The concept of cognitive radio is based on opportunistic access to the available frequency resources. It offers to future communication systems the ability to dynamically and locally adapt their operating spectrum by selecting it from a wide range of possible frequencies.
Multicarrier techniques are promising and potential candidates offering flexible access to these new spectrum opportunities [2]. Indeed, orthogonal frequency division multiplexing (OFDM), which is the most commonly used multicarrier technique, has been adopted in IEEE 802.22 standard for unlicensed wireless regional area network (WRAN) using cognitive communications on the unused TV bands [3]. Unfortunately, OFDM presents some weaknesses. In fact, the redundancy, caused by the insertion of the cyclic prefix mandatory part of the transmitted OFDM symbol, reduces the useful data rate. Furthermore, it has a limited frequency resolution due to the large sidelobes generated by the rectangular pulse shape frequency response. These shortcomings have stimulated the research for an alternative scheme that can overcome these problems.
In the last few years, a number of papers, e.g., [4–7] have focused on an enhanced physical layer based on the filter bank processing called filter bankbased multicarrier (FBMC) technique which can offer a number of advantages compared to CPOFDM systems such as the improved spectral efficiency by not using a redundant CP and by having much better control of outofband emission, thanks to the timefrequency localized shaping pulses [8, 9]. In the literature, we find two typical waveforms that are used in filter bank systems: the isotropic orthogonal transform algorithm (IOTA) [10] and the reference physical layer for dynamic spectrum access and cognitive radio (PHYDYAS) [11] prototype filter [12].
However, due to various factors, e.g., the propagation delays and the spatial distribution of users, timing asynchronism is considered as one of the most challenging issues in spectrum coexistence contexts. Indeed, the timing asynchronism between coexisting systems can harmfully affect the performance by causing the socalled asynchronous interference. Consequently, it is relevant to evaluate the impact of this asynchronism on the system performance.
Interference modeling is an important problem, with numerous applications to the analysis and design of multiuser communication systems, as well as the development of interference mitigation techniques. This problem has been intensively investigated in the literature through the most common approach using the power spectral density (PSD) [8, 9, 13, 14]. This model is based on the outofband radiation which is determined by the PSD model of multicarrier signals. However, this model does not always give accurate results. For example, in multiuser CPOFDM when the timing offset does not exceed the cyclic prefix duration, the interference comes only from the same subchannel and the other subchannels do not contribute to this interference. Unfortunately, in this case, the PSD modeling still shows that the other carriers contribute in the resulting interference.
It is worth mentioning that there are already several works in the literature showing the FBMC robustness to time asynchronism. In [15–17], the authors demonstrate that in an asynchronous multiuser scenario, FBMC systems are more robust than OFDM systems to time and frequency misalignments among the users. Moreover, the sensitivity of the different FBMC waveforms is investigated in [18]. However, we want to indicate that the interference modeling proposed in this paper is general and can be used for any multicarrier scheme. Moreover, this model is a more efficient alternative to overcome the limitations of the PSDbased modeling which is commonly used in the analysis of interference in coexistence contexts.
In this paper, the impact of timing asynchronism on the performance of OFDM and FBMC systems is addressed. As a matter of fact, we would like to:

Properly estimate the interference part introduced by the timing asynchronism for OFDM systems.

Propose an extension of this model to the FBMC case, as it has, to our knowledge, never been considered in the literature.

Extend the interference analysis to the case of frequency selective environments.

Evaluate the accuracy of the proposed model through different simulation results.
The rest of this paper is organized as follows. Section 2 presents the general notion of the socalled interference tables, where we give the PSDbased interference tables of CPOFDM and FBMC considering PHYDYAS and IOTA prototype filters. In Section 3, we derive the OFDM/FBMC instantaneous interference tables taking into account the timing asynchronism in addition to the spectral distance between the interfering user and the victim one. Next, we extend the interference analysis to the case of frequency selective environments in Section 4. The accuracy of the proposed interference modeling is then investigated in Section 5. We finally conclude the paper in Section 6.
1.1 Definitions and notations
In this paper, we are calculating interference weights that can be used in the estimation of interference in coexistence contexts. Since these weights are computed as a function of two parameters, the timing offset and the spectral distance between the coexisting systems, we then obtain a 2D table of these weights for each multicarrier scheme. Thus, we find it appropriate to call them ‘interference tables’. For simplicity sake, we also use this definition in the PSDbased approach.
2 The general concept of interference tables
Let us consider two asynchronous systems (A) and (B) that coexist in the same geographical area. We assume that both systems share a given frequency band where ${\mathcal{F}}_{A}$ and ${\mathcal{F}}_{B}$ are the frequency subbands occupied by systems (A) and (B), respectively.
Due to the nonorthogonality between their respective transmit signals, some amount of the power is spilled from a system to the other. In order to analyze this interaction, the PSD is generally used to evaluate the mutual interference between both systems [9, 13]. According to [13], the normalized mutual interference between the colocated systems is defined as
where:

l is the spectral distance between the two interacting subcarriers.

Δ f is the subcarrier spacing.

Φ(f) is the PSD which depends on the considered multicarrier technique.
In the case of CPOFDM system, the normalized PSD is given in [13] by
where T_{OFDM} is the OFDM symbol duration given by the sum of the useful symbol duration T and the CP duration Δ.
The red curve of Figure 1 depicts the normalized OFDM PSD. The mutual interference power when l=1 corresponds to the graycolored area.
The PSD of FBMC systems has been computed in [9], for two waveforms: IOTA and PHYDYAS. The respective expressions are as follows:
Here, ${g}_{1,\sqrt{2}/2,\sqrt{2}/2}\left(t\right)$ is the impulse response of the IOTA filter. The derivation of the latter is detailed in [10]. G(f) is the square root of the PHYDYAS filter frequency response which is given by
where the coefficients G_{ k } are given by [12, 19]
Based on Equation 1, we can construct a table of mutual interference as a function of the spectral distance l. The PSDbased interference tables for CPOFDM are given in Table 1 for different values of CP duration: Δ=0,T/8 and T/4. Moreover, we give in Table 2 the PSDbased interference tables for FBMC considering PHYDYAS and IOTA waveforms [9].
It is worth to point out that these tables are symmetrical as shown in Figure 2.
In Figure 3, we illustrate how to compute the interference caused by the different subcarriers of ${\mathcal{F}}_{A}$ to a given subcarrier m of system (B). The total interference is the sum of the contribution of each interfering subcarrier m^{′}, and can thus be written as
where P(m^{′}) is the transmitted power on the m^{′}th interfering subcarrier and I(m^{′}−m) is the PSDbased interference weight computed in Tables 1 and 2.
Various analysis have been developed based on this interference estimation, e.g., SINR, spectral efficiency analysis in [9, 20], and resource allocation algorithms [14, 20].
According to (1), one can see that the interference remains the same for any timing misalignment between the transmitted signals of both systems since the signals are considered to be nonorthogonal. However, in CPOFDM systems, the orthogonality between the different transmit signals is maintained as long as the timing misalignment does not exceed the cyclic prefix duration. This example highlights the overestimation of the asynchronous interference term. In fact, the real asynchronous interference is always a function of the timing offset between the considered asynchronous systems that is not taken into account in the PSDbased interference tables generation.
In the next section, we propose new interference tables that model the correlation between the interfering subcarrier and the victim one considering the timing offset between them in addition to the different parameters already considered by the PSDbased interference tables.
3 Instantaneous OFDM/FBMC interference tables
To take into account the detrimental effects of interference caused by the imperfect synchronization in multicarrier techniques, we consider the system model depicted in Figure 4. We refer, here, to a receiver which suffers from the interference coming from an asynchronous transmitter. This receiver is assumed to be perfectly synchronized with its corresponding transmitter. Moreover, the timing offset τ and the phase offset φ are assumed to be uniform random variables that are distributed on τ∈ [ 0,T] and φ∈ [ 0,2π], respectively. In the following analysis, we will be interested in the impact of the interfering signal s(t−τ) on the reference receiver.
3.1 CPOFDM case
Consider the following asynchronous signal coming from the interferer on the mth subcarrier
where

x_{m,n} are the complex data symbols transmitted by the interferer.

T and Δ are the useful OFDM symbol duration and the CP duration, respectively.

The timing offset and the phase offset between the reference receiver and the interferer are respectively denoted by τ and φ.
Here, f_{ T }(t) and f_{ R }(t) are, respectively, the transmit and the receiver pulse shapes,
The m_{0}th output of the receiver filter on the n_{0}th signaling interval coming from s_{ m }(t−τ,φ), will be
where, 〈.,.〉 stands for the inner product.
In the general case, we see that the product f_{ T }(t−n(T+Δ)−τ)f_{ R }(t−n_{0}(T+Δ)) and the choice of τ determine the limits of the integral appearing in (9), we have then two cases to analyze
3.1.1 Case 1: (0<τ<Δ)
In this case, the positions of the receiver window and the interferer window are depicted in Figure 5 and the signal ${y}_{{m}_{0},{n}_{0}}\left(\tau \right)$ becomes
Here, the timing offset τ is absorbed by the cyclic prefix Δ. The interference will only occur on the same subcarrier m=m_{0}, the other subcarriers are free of interference due to the orthogonality between them.
3.1.2 Case 2: (Δ<τ<T+Δ)
As illustrated in Figure 6, the product f_{ T }(t−n(T+Δ)−τ)f_{ R }(t−n_{0}(T+Δ)) is nonzero when n=n_{0}−1 and n=n_{0}, simultaneously. Consequently, the signal ${y}_{{m}_{0},{n}_{0}}\left(\tau \right)$ can be written as follows:
When m≠m_{0} equation (11) is reduced, upon the change of variables t=t^{′}−n_{0}(T+Δ), to
Using some trigonometric transformations, the signal ${y}_{{m}_{0},{n}_{0}}\left(\tau \right)$ can be written in the following form:
when m=m_{0}, the signal ${y}_{{m}_{0},{n}_{0}}\left(\tau \right)$ is given by
Accordingly, when the timing offset τ is larger than the cyclic prefix duration Δ, the orthogonality between subcarriers is damaged. Thus, the interference is caused by all subcarriers.
As the communication symbols x_{m,n} are zero mean uncorrelated variables, the corresponding interference power is the sum of the interference power coming respectively from two successive data symbols (x_{m,n−1}, x_{m,n}). Without loss of generality, we assume that $\mathbb{E}\left[\right{x}_{m,n}{}^{2}]=1$, we can define the instantaneous interference tables by the following expression:
where δ(l) is the Kronecker delta and l=m−m_{0} is the spectral distance between the interfering subcarrier and the victim one.
According to (15), the OFDM interference power tables do not depend on the phase offset φ. In Table 3, we give some examples of the instantaneous interference tables for τ=T/4,T/3 and T/2. These examples are also depicted in Figure 7 where we see that the interference level vary with respect to the timing offset τ and the spectral distance l.
3.1.3 OFDM mean interference table
In order to calculate the mean interference table $\overline{I}\left(l\right)$, we assume a timing offset τ uniformly distributed in [ 0,T+Δ],
where p(τ) is the probability density function of the random variable τ,
Substituting (15) and (17) into (16), we obtain case 1 (l=0)
case 2 (l≠0)
The OFDM mean interference table is compared to the PSDbased OFDM interference table in Figure 8 and Table 4. We can see that the two interference levels computed by the PSD and the proposed interference tables are quite different. In fact, the bigger the spectral distance is, the greater the gap in interference level becomes. It must be noted that, except for l=0, the interference levels presented by the proposed model is higher than the PSDbased one.
3.2 FBMC case
In this scheme, the idea is to transmit offset quadrature amplitude modulation (OQAM) data symbols instead of conventional QAM ones, where the inphase and the quadrature components are time staggered by half a symbol period, T/2 [21]. The second specificity of this scheme is that considering two successive subcarriers, the time delay T/2 is introduced into the imaginary part of the QAM symbols on one of the subcarriers, whereas it is introduced into the real part of the symbols on the other one [22, 23]. It is worth noticing that the spacing between two successive subcarriers is 1/T.
According to [22], the continuoustime baseband FBMC transmit signal can be written
where N is the number of subcarriers, a_{m,n} are the realvalued transmitted data symbols and γ_{m,n} is defined by
and with
According to (20), we can define the asynchronous signal coming from the interferer on the mth subcarrier s_{ m }(t−τ,φ) as follows:
The m_{0}th output of the receiver filter on the n_{0}th signalling interval (i.e., t=n_{0}T/2) coming from s_{ m }(t−τ,φ), will be
Substituting (23) and (21) in (24), the signal ${y}_{{m}_{0},{n}_{0}}\left(\tau \right)$ becomes
In order to get simplified, easiertomanipulate expressions, let us define the following integral
If we consider the PHYDYAS prototype filter, the explicit form expressions of this integral are given in (27) and (28), respectively, for l=0 and l≠0. The details of calculus are given in Appendix A.
where A is the normalization factor
Without loss of generality, let us assume that the prototype filter g(t) can be non zero only when t∈ [ 0,K T] where K represents its overlapping factor. Accordingly, the product g(t)g(t−τ) can be nonzero only when the timing offset τ∈ [−K T,+K T]. Therefore in order to compute ${y}_{{m}_{0},{n}_{0}}\left(\tau \right)$, we have to consider two cases:
3.2.1 Case 1 : ($({n}_{0}n)\frac{T}{2}<\tau $)
In this case, (25) and (26) yield
where ⌊α⌋ denotes the floor function (the largest integer less than or equal to α).
3.2.2 Case 2 : ($\tau <({n}_{0}n)\frac{T}{2}$)
According also to (25) and (26), we obtain
where ⌈α⌉ is the ceil function (the smallest integer greater than or equal to α).
After the OQAM decision, we can write the total complex symbol y_{ t o t }(τ,φ) as follows:
and the corresponding interference power table I(τ,l) can thus be given by the following expression:
In Figures 9 and 10, PHYDYAS and IOTA interference tables I(τ,l) are plotted for different values of the timing offset τ=T/4,T/3,T/2. We also see that the interference varies with respect to the timing offset τ and the spectral distance l between the interfering subcarrier and the victim one. Furthermore, PHYDYAS and IOTA mean interference tables are compared to the respective PSDbased interference tables in Figures 11 and 12, respectively. Looking at these figures, we find that the two models lead to different results. However, this difference is negligible when the interference level is high.
4 Asynchronous interference in frequency selective channels
In this section, we refer to the system model shown in Figure 13. The asynchronous OFDM/FBMC transmit signal s(t−τ,φ) propagates through a frequency selective multipath channel, where its equivalent samplespaced impulse response [24] is given by
where n_{0}<n_{1}<...<n_{L−1}<C and C is the maximum delay spread of the channel normalized by the sampling period (T/N), and h_{ i } are the complex channel path gains, which are assumed mutually independent, where $\mathbb{E}\left[{h}_{i}{h}_{i}^{\ast}\right]={\gamma}_{i}$, and $\mathbb{E}\left[{h}_{i}{h}_{j}^{\ast}\right]=0$ when i≠j. We further assume that the power is normalized such that $\sum _{i=0}^{L1}{\gamma}_{i}=1$.
In this case, the interference signal received at the input of the multicarrier demodulator r(t−τ,φ) is given by
where ⋆ stands for the convolution product.
In the following analysis, we investigate the effects of the propagation channel on the asynchronous interference signal coming from the mth subcarrier s_{ m }(t−τ,φ). In the following analysis, two cases will be investigated : CPOFDM case and FBMC one.
4.1 CPOFDM case
According to (8) and (35), the interference signal at the input of the OFDM receiver can be written as follows:
In weakly and mildly frequency selective channels, $\frac{{n}_{i}}{N}T$ is small enough. We can thus consider that ${f}_{T}(tn(T+\Delta )\tau \frac{{n}_{i}}{N}T)={f}_{T}(tn(T+\Delta )\tau )$. Hence, the signal r(t−τ,φ) becomes
where $H\left(m\right)=\sum _{i=0}^{L1}{h}_{i}{e}^{j\frac{2\pi {n}_{i}}{N}m}$ representing the complex channel gain at the mth subcarrier.
Based on (37), (10) and (14), the m_{0}th output of the receiver filter on the n_{0}th signalling interval resulting from the received interference signal r(t−τ,φ) is expressed for both cases 0<τ<Δ and Δ<τ<T+Δ as follows:
4.1.1 Case 1: (0<τ<Δ)
4.1.2 Case 2: (Δ<τ<T+Δ)
When m≠m_{0}
When m=m_{0}, the signal ${y}_{{m}_{0},{n}_{0}}\left(\tau \right)$ is given by
Consequently, the resulting interference power P_{interf} will be the product of the channel power gain of the interfering subchannel and the corresponding interference weight given in (3). Then, we write the interference power in m_{0}th subchannel as
where,

P_{trans}(m) is the transmitted power on the subchannel m.

I(τ,m−m_{0}) represents the interference weight for the timing offset τ and the spectral distance m−m_{0}.

H(m)^{2} is the channel power gain for subchannel m.
4.2 FBMC case
Similarly, we express the interference signal received at the input of the FBMC receiver r(t−τ,φ). According to (23) and (35), we write
Substituting (21) in (42), we obtain
We can notice that $g(t\mathit{\text{nT}}/2\tau \frac{{n}_{i}}{N}T)$ may have relatively slow variations when $\frac{{n}_{i}}{N}T\in \phantom{\rule{0.3em}{0ex}}[\phantom{\rule{0.3em}{0ex}}0,{\tau}_{\text{ds}}]$ (τ_{ds} is the maximum delay spread of the channel) [25, 26]. Indeed, compared to the coherence bandwidth B_{ c }, the filter bandwidth is very small, which also means that the time variations of the prototype filter g(t) are necessarily limited.
Consequently, the signal r(t−τ,φ) becomes
Now, let φ_{H(m)} be the phase angle of the complex channel gain H(m), i.e.,
Hence, we can write r(t−τ,φ) as follows:
According (32) and (45), the output signal after the OQAM decision is given by
where, y_{tot}(τ,φ) is given in (32).
Now, let φ and φ^{′} be two uniform random variables defined in the following intervals φ∈ [ 0,2π] and φ^{′}∈ [α,α+2π], respectively. We notice that
Therefore, the corresponding interference power P_{interf} is also the product of the channel power gain of the interfering subcarrier H(m)^{2} and the corresponding interference weight given in (33). Thus, the interference power at the m_{0}th subchannel is given by
In general, the asynchronous interference power arriving through a frequency selective channel can be calculated using the following expression:
where,

d is the distance between the interferer and the victim user.

β is the path loss exponent.

P_{trans}(m) is the transmitted power on the interfering subchannel m.

I(τ,m−m_{0}) is the interference weight for the timing offset τ and the spectral distance m−m_{0}.

H(m)^{2} is the channel power gain between the interfering transmitter and the victim receiver on subchannel m.
In the next section, we investigate the accuracy of the proposed interference modeling expressed by Equation 48. Various applications and scenarios can be studied using this interference model.
5 Simulation results
In this section, we consider the uplink transmission in OFDM/FBMCbased network depicted in Figure 14a. The reference mobile user MU_{0} and the interfering one MU_{1} communicating respectively with BS_{0} and BS_{1}. Moreover, MU_{0} and MU_{1} are respectively located at distances d_{0} and d from the reference base station BS_{0}. It is assumed that, the transmitted power of each user must guarantee a target signal to noise ration S N R_{ t }=20 d B at its base station (BS_{0} for MU_{0} and BS_{1} for MU_{1}).
Concerning the frequency scheme, the subcarriers are allocated according to the scheme described in Figure 14b where, the size of each subcarrier block is set at 18 subcarriers. Here, we have chosen the practical size of subcarrier block in WiMax 802.16 [27].
All signals propagate through different multipath channels using a similar propagation model, where the impulse responses of the multipath channel between MU_{0}, MU_{1} and the reference base station BS_{0} are denoted by h_{0} and h_{1}, respectively. The considered model is the PedestrianA model whose parameters are given in Table 5. The choice of this model is based on the assumption that the subcarriers of interest experience flat fading. Therefore, the interference caused by the multipath effects are negligible compared to the one caused by time asynchronism, in the FBMC case.
Furthermore, the underlying channel model includes path loss effects which takes into account the position of the mobile user with respect to the reference base station BS_{0}. The path loss of a received signal at distance d is governed by the following expression [28] corresponding to a path loss exponent β=3.76 and a carrier frequency of 2 GHZ
On the other hand, we consider a system with N=1,024 subcarriers and a sampling frequency of 10 MHz. The noise term is considered as a thermal noise with spectral density N_{0}=−174 dBm/Hz.
Each mobile user is assumed to be perfectly synchronized with its corresponding base station but it is not synchronized with the other base station. Because of the timing misalignment between MU_{1} and BS_{0}, the signal arriving from MU_{1} at BS_{0} appears nonorthogonal to the desired signal arriving from MU_{0}. This nonorthogonality generates interference and degrades the signaltointerferenceplusnoise ratio (SINR). Using the interference tables introduced previously, the instantaneous SINR on a given subcarrier $m\in {\mathcal{F}}_{0}$ can be expressed as
where Δ f is the subcarrier spacing. Actually, the SINR expression given in (49) is established assuming the absence of ISI and ICI terms. Such an assumption is valid in the PedestrianA channel model.
Here, it is worth mentioning that the interference weight I is computed by two methods : PSDbased interference tables and our proposed interference tables.
The objective of this section is to evaluate the average SINR and the spectral efficiency expressed, respectively by
where $\mathbb{E}\left[.\right]$ stands for the statistical expectation which is computed over all channel realizations (H_{0}(m), $\left\{{H}_{1}\left({m}^{\prime}\right),{m}^{\prime}\in {\mathcal{F}}_{1}\right\}$) and all values of the timing offset τ which is uniformly distributed over [ 0,T].
Two different contexts will be analyzed as depicted in Figure 14:

The classical multicellular context: when d varies from R to 2R, i.e., MU_{1} can move from the edge to the center of cell 1

The cognitive radio context: when d varies from 0 to 2R, i.e., MU_{1} can be very close to BS_{0} while transmitting to BS_{1}.
It is worth mentioning that, in cognitive radio scenarios, we cannot always assume that both primary and secondary users are using OFDM or FBMC. However, the objective here is to highlight the potential gain that can be achieved when both primary and cognitive systems are using either OFDM or FBMC waveform.
In Figures 15, 16, and 17, we investigate, respectively, the accuracy of the SINR expression in the CPOFDM, FBMCPHYDYAS, and FBMCIOTA cases. The averaged SINRs over all subcarriers $m\in {\mathcal{F}}_{0}$ are plotted against the distance d, using the instantaneous tables (line), the PSDbased table (point markers), and numerical simulation (cross markers). The instantaneous table results depicted in these figures show a perfect match to the corresponding simulation results. However, the PSDbased method exhibits a strong inaccuracy especially in the cognitive radio context.
In the cognitive radio context, we observe a significant degradation of the OFDM SINR with respect to the target SNR (20 dB). Such a result can be explained by the high level of OFDM asynchronous interference caused by the timing misalignment which damages the orthogonality between the subcarriers. On the other hand, we notice a slight loss of the FBMC SINR with respect also to the target SNR of 20 dB. The better performance of PHYDYASFBMC compared to IOTAFBMC and CPOFDM can be justified by the fact that only the two subcarriers on the edge of the cluster (subcarrier block) ${\mathcal{F}}_{0}$ suffers from the interference caused by their immediate adjacent subcarriers in ${\mathcal{F}}_{1}$ as depicted in Figure 11; whereas in IOTAFBMC, two subcarriers at each edge are affected by the asynchronous interference coming from the two neighboring subcarriers at each edge as shown in Figure 12. Furthermore, the entire cluster ${\mathcal{F}}_{0}$ suffers, in the CPOFDM case, from the asynchronous interference caused by all subcarriers of ${\mathcal{F}}_{1}$ (see Figure 8).
In the cellular context, the asynchronous interferer MU_{1} is quite far from the reference base station BS_{0} and, at the same time, it is close to its base station BS_{1}. This means that its transmitted power is reduced and consequently the interference power received by BS_{0} will be much lower due to the path loss effect (d is quite large). Therefore, the impact of the asynchronous interference is less significant in the cellular context for all waveforms. Also, it is worth mentioning that SINRs of OFDM and FBMC converge to the target SNR (20 dB) when MU_{1} is very far from BS_{0} as the interference becomes negligible compared to the noise level.
The impact of the asynchronous interference on the average spectral efficiency has also been investigated. Figure 18 shows the average spectral efficiency over all subcarriers $m\in {\mathcal{F}}_{0}$ against the distance d, for CPOFDM (solid line −), FBMCPHYDYAS (dasheddotted line −.), and FBMCIOTA (dashed line −−). Also, the accuracy of instantaneous and PSDbased tables is examined. Figure 18 shows that timing synchronization errors cause a degradation of the spectral efficiency. The same remarks can be formulated, here, where the FBMC system still outperforms the OFDM system. Furthermore, simulation and instantaneous tables results shown in Figure 18 validate the accuracy of the proposed interference model while the PSDbased tables still present a strong inaccuracy with respect to simulation results especially in the cognitive radio context where the noise level becomes negligible compared to the asynchronous interference caused by MU_{1}. We see that both modulation schemes (OFDM and FBMC) lead to identical spectral efficiency floor when MU_{1} is close to its base station because of the predominance of the noise term. We have to remind that the actual bit rate is lower for CPOFDM because of the redundancy introduced by the cyclic prefix in each OFDM block.
6 Conclusion
In this paper, we have investigated the asynchronous interference modeling in OFDM and FBMC systems. First, the general concept of interference tables has been introduced where we have derived the PSDbased interference tables of CPOFDM and FBMC for two considered waveforms: PHYDYAS and IOTA. It has been noticed that the PSDbased tables do not consider the timing offset between the interferer and the victim user.
Next, we have proposed new interference tables that model the correlation between a given interfering subcarrier and the victim one, not only as a function of the spectral distance separating both subcarriers but also with respect to the timing misalignment between the subcarrier holders. Theoretical expressions of these tables have been derived for both OFDM and FBMC systems.
The interference analysis has been extended to the case of frequency selective environments where we have proposed a tablebased estimation method as a computationally simpler alternative to the numerical evaluation; as the latter requires huge computational efforts.
Furthermore, the accuracy of the proposed model has been validated through different simulation results, where the results based on the instantaneous tables method shows an excellent match with the corresponding simulation ones. In contrast to the instantaneous interference tables, we have shown through this evaluation that the PSD modeling exhibits a strong inaccuracy with respect to the numerical results.
Finally, through this evaluation, we have shown that in OFDM case, timing asynchronism between coexisting systems cause a severe degradation in the performance. This result is explained by the loss of orthogonality between all system subcarriers. In contrast to the OFDM system, the FBMC waveforms are demonstrated to be less sensitive to the timing misalignment between the cohabiting systems due to the better frequency localization of the prototype filter. The obtained results make FBMC a promising candidate for the physical layer of future cognitive radio systems.
Appendix
Proof of the expressions (27) and (28)
Let us first recall the impulse response of the PHYDYAS prototype filter which is given in [29],
where A is the normalization factor
Substituting the expression (52) in (26), we obtain
when l=0
Using some trigonometric transformations, the integral $\Psi (t,\tau ,0){t={t}_{1}}_{{t}_{2}}^{}$ can be written in the following form:
After integration, we get
When l≠0,
After integration, we obtain
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This work has been partially supported by European Commission through the Emphatic project (ICT318362).
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Medjahdi, Y., Terré, M., Ruyet, D.L. et al. Interference tables: a useful model for interference analysis in asynchronous multicarrier transmission. EURASIP J. Adv. Signal Process. 2014, 54 (2014) doi:10.1186/16876180201454
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Keywords
 PSD; Interference table; Time asynchronism; OFDM; FBMC; PHYDYAS; IOTA; Spectrum coexistence