On the Accuracy of PSD-based Interference Modeling of Asynchronous OFDM/FBMC in Spectrum Coexistence Context Yahia Medjahdi1 , Michel Terr´e2 , Didier le Ruyet2 , Daniel Roviras2 1
2
ICTEAM, Universit´e Catholique de Louvain. Place du Levant, 2, 1348 Louvain-la-Neuve, Belgium Email:
[email protected] CEDRIC/LAETITIA, CNAM, Paris, France. Email: {michel.terre, didier.le ruyet, daniel.roviras}@cnam.fr
Abstract—The performances of multicarrier techniques in a spectrum coexistence context are commonly investigated using the power spectral density (PSD) of the coexisting systems’ signals. However due to many factors, the PSD approach does not lead to accurate results. Recently, a useful tool called “instantaneous interference tables” has been proposed as an alternative to the PSD-based method leading to more accurate evaluation. This paper investigates the accuracy of the performance evaluation when the interference modeling is based on the PSD of the transmitted signals. With this aim in mind, two multicarrier schemes are examined: cyclic prefix based orthogonal frequency division multiplexing (CP-OFDM) with a rectangular pulse shape and filter bank based multicarrier (FBMC) with PHYDYAS and IOTA waveforms. 1 . Index Terms—Accuracy, PSD, interference table, time asynchronism, OFDM, FBMC, PHYDYAS, IOTA, spectrum coexistence
I. I NTRODUCTION 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. 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. In the last few years, a number of papers e.g. [1], [2] have focused on an enhanced physical layer based on the filter bank processing called filter bank based multicarrier (FBMC) technique which can offer a number of advantages compared to CP-OFDM systems such as the improved spectral efficiency by not using a redundant CP and by having much better control of out-of-band emission thanks to the time-frequency localized shaping pulses [3], [4]. In the literature, we find two typical waveforms that are used in filter bank systems: the isotropic orthogonal transform algorithm (IOTA) [5] and the reference PHYDYAS prototype filter [6]. 1 This work has been partially supported by European Commission through the Emphatic project (ICT-318362)
However due to various factors e.g. the propagation delays and the spatial distribution of users, asynchronous interference modeling is an important problem, with numerous applications to the analysis and design of multiuser communication systems. This problem has been intensively investigated in the literature through the most common approach using the power spectral density (PSD) [3], [4], [7], [8]. This model is based on the out-of band 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. In this paper, the accuracy of the PSD-based approach is investigated in a spectrum coexistence context. The rest of this paper is organized as follows: Section II presents the general concept of the so-called interference tables, where we give the PSD-based interference tables of CP-OFDM and FBMC considering PHYDYAS and IOTA prototype filters. In Section III, OFDM/FBMC instantaneous interference tables addressing the limits of the PSD-based model are briefly described. Next, the utilization of both models in the presence of channel effects is illustrated in Section IV. Through different simulation results, the accuracy of the PSD interference modeling is investigated in Section V. We finally conclude the paper in Section VI. II. PSD- BASED 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 FA and FB are the frequency sub-bands occupied by systems (A) and (B), respectively. Due to the non-orthogonality 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 [4], [7]. According to [7], the normalized mutual
c 978-1-4799-5863-4/14/$31.00 2014 IEEE
TABLE I OFDM AND FBMC MUTUAL INTERFERENCE TABLES BASED ON THE PSD: ∆ = 0, T /8
interference between the co-located systems is defined as I(l) =
(l+1/2)∆f Z
Φ(f )df
(1)
(l−1/2)∆f
where: • l is the spectral distance between the two interacting subcarriers • ∆f is the subcarrier spacing • Φ(f ) is the PSD which depends on the considered multicarrier technique. In the case of CP-OFDM system, the normalized PSD is given in [7] by 2 sin(πf TOFDM ) ΦOFDM (f ) = TOFDM (2) πf TOFDM where TOFDM is the OFDM symbol duration given by the sum of the useful symbol duration T and the CP duration ∆. The PSDs of FBMC systems have been computed in [4], for two waveforms: IOTA and PHYDYAS. The respective expressions are 2 (3) ΦIOTA (f ) = g1,√2/2,√2/2 (t) ΦPHYDYAS (f ) = (G(f ))2
(4)
Here, g1,√2/2,√2/2 (t) is the impulse response of the IOTA filter. The derivation of this latter is detailed in [5]. G(f ) is the square root of the PHYDYAS filter frequency response which is given by, k=(K−1) X sin π f − NkK N K G(f ) = (5) Gk N K sin π f − NkK k=−(K−1) where the coefficients Gk are given by [6],
√ G0 = 1, G1 = 0.971960, G2 = 1/ 2 q G3 = 1 − G21 , Gk = 0; 4 < k < L − 1
A
OFDM [dB]
PHYDYAS [dB]
IOTA [dB]
0
-00.87
-00.58
-00.70
1
-12.00
-11.95
-09.27
2
-19.72
-65.02
-35.56
3
-23.55
-80.30
-42.55
4
-25.98
-89.22
-65.17
5
-27.62
-95.68
-71.72
6
-28.83
-100.80
-87.31
7
-29.92
-105.08
-90.53
8
-31.05
-108.80
-91.45
where P (m′ ) is the transmitted power on the m′ -th interfering subcarrier and I(m′ −m) is the PSD-based interference weight given in Table. I. Various analysis have been developed based on this interference estimation e.g. SINR, spectral efficiency analysis in [4], [9] and resource allocation algorithms [8], [9]. According to (I), 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 non-orthogonal. However in CP-OFDM 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. Indeed, 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 PSD-based interference tables. III. I NSTANTANEOUS INTERFERENCE TABLES
(6)
where, K is the overlapping factor and L the filter impulse response length. Based on equation (1), we can construct a table of mutual interference as a function of the spectral distance l. In Table. I, the PSD-based interference tables are given with respect to the spectral distance between the interfering subcarrier and the target one. It is worth noticing that this interference has been computed considering CP-OFDM system with a CP duration ∆ = T /8, where T is the OFDM symbol duration and FBMC system using, respectively, IOTA and PHYDYAS waveforms with an overlapping factor of 4. Using theses tables, we can compute the interference caused by the different subcarriers of FA to a given subcarrier m of system (B). The total interference is the sum of the contribution of each interfering subcarrier m′ ∈ FA , and can thus be written as X P (m′ )I(|m′ − m|) (7) Itot = m′ ∈F
|l|
In [10] and [11], a new generation of interference tables so-called instantaneous tables has been established. These tables model the interference weight as a function of both the timing offset and the spectral distance between the interfering subcarrier and the victim one. In Table II, we give some examples of the OFDM instantaneous tables I(τ, l) for the following timing offsets τ = T /4, T /3 and T /2. 2 . Looking at this table, one can see that the interference vary with respect to the timing offset τ and the spectral distance n separating the interfering subcarrier and the victim one.
IV. I NTERFERENCE ANALYSIS IN FREQUENCY SELECTIVE CHANNELS
The asynchronous interference analysis has been extended to the case of frequency selective environments in [11] and [13]. It has been demonstrated that the asynchronous interference power generated through a selective frequency channel 2 More details about the analytical expressions of the instantaneous interference tables can be found in [12]
TABLE II OFDM INSTANTANEOUS INTERFERENCE TABLES FOR τ = T /4, T /3 AND T /2 BS0
|l|
τ = T /4 [dB]
τ = T /3 [dB]
τ = T /2 [dB]
0
-01.06
-01.72
-02.73
1
-15.29
-11.27
-07.63
2
-15.98
-13.28
-15.98
3
-17.18
-17.17
-24.83
4
-19.00
-24.95
-19.00
5
-21.62
-38.96
-29.27
6
-25.52
-25.58
-25.52
BS1 MU1
MU0
0
CR context
can be calculated using the following expression Pinterf (m, τ ) = d
−β
′
′
R Cellular context 2.R Useful signal
′
2
Ptrans (m )I(τ, |m − m|) |H(m )| (8)
2
Interference (a)
where, |H(m′ )| is the power channel gain between the interfering transmitter and the victim receiver on the m′ -th subchannel. In the next section, we investigate the accuracy of the proposed interference modeling expressed by Equation (8). Various applications and scenarios can be studied using this interference model. V. S IMULATION RESULTS In this section, we consider the uplink transmission in OFDM/FBMC based network depicted in Fig. 1.a. The reference mobile user MU0 and the interfering one MU1 communicate respectively with BS0 and BS1 . Moreover, MU0 and MU1 are respectively located at distances d0 and d from the reference base station BS0 . It is assumed that, the transmitted power of each user must guarantee a target signal to noise ration SNRt = 20dB at its base station (BS0 for MU0 and BS1 for MU1 ). The subcarriers are assigned according to the scheme described in Fig. 1.b. 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 [14]. In this scenario, the different multipath channels are generated using a similar propagation model, where the impulse responses of the multipath channel between MU0 , MU1 and the reference base station BS0 are denoted by h0 and h1 , respectively. The considered model is the Pedestrian-A model whose parameters are given in Table. III. It is worth to note that, 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, 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 BS0 . The path loss of a received signal at distance d is governed by the following expression [15] corresponding to a path loss
(b)
Fig. 1. Interference model: the reference user coexists with an asynchronous interferer.
Parameter Pedestrian-A Relative Delay Pedestrian-A Average Power
value [0 110 190 410] ns [0 -9.7 -19.2 -22.8] dB
TABLE III C HANNEL PARAMETERS USED IN SIMULATIONS
exponent β = 3.76 and a carrier frequency of 2 GHZ
Γloss (d) = 128.1 + 37.6 log10 (d[km])[dB]
On the other hand, we consider a system with N = 1024 subcarriers and a sampling frequency of 10 MHz. The noise term is considered as a thermal noise with spectral density N0 = −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 MU1 and BS0 , the signal arriving from MU1 at BS0 appears non-orthogonal to the desired signal arriving from MU0 . This non-orthogonality generates interference and degrades the signal to interference plus noise ratio SINR. Using the interference tables introduced previously, the instantaneous
CP−OFDM case, ∆=T/8
FBMC case
20
20
15
19.5
inst. Tables simulation PSD−Table
10
19 FBMC, IOTA SINR [dB]
5 SINR [dB]
FBMC, PHYDYAS
0 −5
18.5 18 17.5
−10
inst. Tables simulation PSD−Table
17
−15
16.5 CR context
CR context
Cellular context
−20
Cellular context
16 0.2
Fig. 2. [0, T ]
0.4
0.6
0.8
1 d/R
1.2
1.4
1.6
1.8
0.2
CP-OFDM average SINR vs. distance between MU1 and BS0 , τ ∈
SINR on a given subcarrier m ∈ F0 can be expressed as
m′ ∈F1
d−β 0 Ptrans (m) |H0 (m)|
2 2
d−β Ptrans (m′ )I(τ, |m′ − m|) |H1 (m′ )| + N0 ∆f
•
where ∆f is the subcarrier spacing. Here, it is worth mentioning that the interference table coefficient I is computed by two methods : PSD-based 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, (10) •
Caverage (m) = E [log2 (1 + SINR(m))]
0.8
1 d/R
1.2
1.4
1.6
1.8
Fig. 3. IOTA and PHYDYAS FBMC average SINRs vs. distance between MU1 and BS0 , τ ∈ [0, T ]
(9)
SINRaverage (m) = E [SINR(m)]
0.6
Comparing the SINR computed using the PSD-based table (SINRpsd ) to the actual one which is obtained by the numerical method (SINRnum ), we can distinguish two cases:
SINR(m) = P
0.4
(11)
where E[.] stands for the statistical expectation which is computed over all channel realizations (H0 (m), {H1 (m′ ), m′ ∈ F1 }) and all values of the timing offset τ which is uniformly distributed over [0, T ]. Two different contexts will be analyzed as depicted in Fig. 1: • The classical multi-cellular context: when d varies from R to 2R, i.e. MU1 can move from the edge to the center of cell 1 • The cognitive radio context: when d varies from 0 to 2R, i.e. MU1 can be very close to BS0 while transmitting to BS1 . The accuracy of the SINR expression in the CP-OFDM case is investigated in Fig. 2 where the averaged SINRs over all subcarriers m ∈ F0 are plotted against the distance d, using the instantaneous tables (line), the PSD-based table (point markers) and numerical simulation (cross markers).
In the cognitive radio context, the PSD-based approach exhibits a strong inaccuracy (The difference between SINRpsd and SINRnum reaches 30 dB when the distance d = 0.1km). Such a behavior can be explained by the fact that the distance separating the interfering user MU1 from the reference base station BS0 is very small thus leading to a very low pathloss level. Consequently, the impact of interference becomes significant. It is worth mentioning that in the OFDM case, all subcarriers are suffering from interference due to high side-lobes of the modulated OFDM spectrum. By increasing the distance d, we observe a convergence of both SINR curves. Indeed, the asynchronous interferer MU1 is quite far from BS0 and, at the same time, it is close to its base station BS1 . This means that its transmit power is reduced and consequently the interference power received at BS0 becomes much lower due to the pathloss effect (d is quite large). Therefore, the impact of the asynchronous interference becomes less significant in the cellular context.
In contrast to the OFDM case, the gap between both SINRpsd and SINRnum is quite small in FBMC with both waveforms IOTA and PHYDYAS even when the distance d is small. Such a result can be explained by the fact that in FBMC systems, each subcarrier interacts only with its first or second order neighborhood (one or two subcarriers on each side) depending on the utilized prototype filter. Since we are computing the average SINR over the entire subcarrier block (18 subcarriers), the gap is reduced by a factor of 1/9 and 2/9 in PHYDYAS and IOTA cases, respectively. The accuracy of the PSD-based interference modeling has also been investigated from the average spectral efficiency
6
weights, not only as a function of the spectral distance separating both subcarriers but also with respect to the timing misalignment between the subcarrier holders. Through the obtained results, we have highlighted the weak accuracy exhibited by the PSD model. The latter leads to a significant gap with respect to direct numerical evaluation in OFDM interference-limited scenarios. This gap is almost completely reduced in the FBMC case thanks to the weak impact of asynchronous interference on filter bank systems.
FBMC, PHYDYAS
Spectral efficiency [bit/s/Hz]
5 FBMC, IOTA 4
3 CP-OFDM, ∆ = T /8 2
R EFERENCES inst. Tables simulation PSD−Table
1
0
Fig. 4.
0.2
0.4
0.6
0.8
1 d/R
1.2
1.4
1.6
1.8
Spectral efficiency vs. distance between MU1 and BS0 , τ ∈ [0, T ]
perspective. Fig. 4, shows the average spectral efficiency over all subcarriers m ∈ F0 against the distance d, for CP-OFDM (solid line −), FBMC-PHYDYAS (dashed-dotted line −.) and FBMC-IOTA (dashed line −−). Comparing the curves in this figure, we can see that the inaccuracy of the PSD approach is reduced compared to the SINR case. Indeed, the maximum gap (smaller than 1bit/s/Hz) is observed in the CP-OFDM case. Such a result is due to the reduced variation of the function y = log2 (1 + x) with respect to the function y = x when x ≥ 0. Nevertheless, it is worth to point out that relative error is still large when the distance d ∈ [0.1, 0.4]km. Finally, through this evaluation, we have shown that in the OFDM case, timing asynchronism between coexisting systems causes 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 filters. The obtained results confirm that FBMC could be a promising candidate for the physical layer of flexible access systems. VI. C ONCLUSION In this paper, we have investigated the accuracy of the PSDbased interference modeling in spectrum coexistence context. First, we have presented the PSD-based interference tables of CP-OFDM and FBMC for two considered waveforms : PHYDYAS and IOTA. After highlighting the fact that the PSD approach does not consider the timing offset between the interferer and the victim user, we have briefly described the socalled instantaneous interference tables giving the interference
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