On the Impact of the Prototype Filter on FBMC Sensitivity to Time Asynchronism Y. Medjahdi, D. Le Ruyet, D. Roviras, H. Shaiek, R. Zakaria CEDRIC/LAETITIA Laboratory, CNAM, 292 rue Saint Martin, 75141, Paris, France {yahia.medjahdi, didier.le ruyet, daniel.roviras, hmaied.shaiek, rostom.zakaria}@cnam.fr

Abstract—This paper investigates the impact of the prototype filter on the sensitivity of the FBMC system to timing synchronization errors. A global evaluation of the average error rate has been made for OFDM and two FBMC systems using PHYDYAS and IOTA prototype filters. This evaluation has been performed taking into account different parameters: the timing error range, the guard band length and the load factor of the network. The validity of the simulation results will be verified by comparison with those obtained with the analytical expressions. Index Terms—Time asynchronism, OFDM, FBMC, PHYDYAS, IOTA

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I. I NTRODUCTION Today, we witness a metamorphosis of applications for wireless communications. In fact, image, music and video sharing between users are reshaping consumption models and imposing higher and higher data rate constraints on today’s communication systems. Multicarrier techniques are promising and potential candidates to achieve these high required data rates. Indeed, thanks to its efficient FFT-based implementation and its robustness to multi-path effects, OFDM has been considered as the solution for systems such as WiFi (IEEE 802.11), WiMax (IEEE 802.16), LTE, LTE-advanced, ... etc Recently, a number of papers [1], [2], [3], [4], [5] have focused on a new alternative called Filter Bank based MultiCarrier system (FBMC) which can offer a number of advantages over CP-OFDM system such as the improved spectral efficiency by not using a redundant CP and by having much better control of out-of-band emission due to the timefrequency localized shaping pulses [6]. In the literature, we find two typical waveforms that are used in filter bank systems: the isotropic orthogonal transform algorithm (IOTA) [7] and the reference PHYDYAS prototype filter [8]. However due to many factors e.g. the propagation delays and the spatial distribution of users, asynchronism inherently exists in several communication systems such as cognitive radio, non cooperative base stations,... As one of the most challenging issue in design of communication systems, the asynchronism can harmfully affect the system performance by causing the so-called asynchronous interference. Consequently, it is relevant to evaluate the impact of this asynchronism on the system performance. In this paper, the sensitivity of IOTA and PHYDYAS waveforms to the timing synchronization errors is addressed. The

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u

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4 Reference BS Useful signal

Interfering BSs

Interfering signal

Reference mobile user

Fig. 1.

Asynchronous OFDM/FBMC multicellular network

rest of the paper is organized as follows. The system model is described in Section II. The different analytical expressions of the signal to interference plus noise ratio (SINR) and the average bit error rate (BER) are given in Section III. Simulation results are presented and discussed in Section IV. Finally, conclusions are summarized in Section V. II. S YSTEM M ODEL We consider the downlink transmission in OFDM/FBMC based multi-cellular networks depicted in Fig. 1. The reference mobile user is located at (u, v). We assume that the reference base station is located at the origin (u0 , v0 ) = (0, 0). In this analysis, we consider all signals transmitted by the different cells that are in the neighborhood of the reference √ mobile. R stands for the cell radius in Fig. 1. Let dk = (uk − u)2 + (vk − v)2 be the distance between the k-th base station and the reference mobile user. Concerning the frequency reuse scheme, the subcarriers are allocated

hk (t) =

L−1 ∑

hk,i δ(t −

i=0

nk,i T) N

(1)

0 OFDM,∆ = T /8

−10 −20 Interference level [dB]

according to the so-called block subcarrier assignment scheme. We assume in this scheme that a given BS k occupies a block Fk of Lk contiguous subcarriers. Also, two adjacent blocks are separated by a guard band of δ adjacent subcarriers. All signals propagate through different multipath channels using a similar propagation model, where the impulse response of the multipath channel between the k-th base station and the reference mobile user is given by

−30

FBMC,IOTA

−40 −50 −60 −70

where nk,0 < nk,1 < ... < nk,(L−1) < C and C is the maximum delay spread of the channel normalized by the sampling period (T /N ), and hk,i are the complex gains of channel paths, which are assumed mutually independent, where E[hk,i h∗k,i ] = γk,i , and E[hk,i h∗k,j ] = 0 when i ̸= j. We further assume that the power is normalized for each channel L−1 ∑ such that γk,i = 1, ∀k. We notice also that N and T are

FBMC,PHYDYAS −80 −90 −6

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0 2 subcarrier offset (l)

4

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Fig. 2. The mean interference level in CP-OFDM (b) (∆ = T /8) and FBMC (PHYDYAS (r) and IOTA (g) prototype filters, K = 4)

i=0

the total number of subcarriers and the duration of the useful OFDM symbol, respectively. The propagation channels are assumed to be stationary over one OFDM symbol. Furthermore, the underlying channel model includes path-loss effects which take into account the positions of the base stations with respect to the reference mobile user. Moreover, it is assumed that the reference mobile user is perfectly synchronized with its base station. However, this assumption is no longer valid with the other base stations. Accordingly, the signals arriving from the cells in the neighborhood will be non-orthogonal to the useful signal, because of the timing synchronization errors between the neighboring cells and the reference one. This loss of orthogonality between these signals will generate interference and will degrade the SINR. In the next section, we examine this degradation. III. SINR

AND

BER

where, • d is the distance between the interferer and the victim user • β is the path loss exponent • Ptrans (m) is the transmitted power on the interfering subchannel m • I(τ, |m − m0 |) is the interference table coefficient for the timing offset τ and the spectral distance |m − m0 | 2 • |H(m)| is the power channel gain between the interfering transmitter and the victim receiver on the subchannel m. Accordingly, we can easily express the total interference power caused by the surrounding BSs on the m-th subchannel of the reference user [10], Pinterf (m, {τk , k = 1, ..., K}) = K ∑ ∑

ANALYSIS

In [3] and [9], a new class of interference tables so-called instantaneous tables has been derived. These tables model the correlation between the interfering subcarrier and the victim one considering the timing offset and the spectral distance between them. In Fig. 2, we plot the mean interference tables with respect to the subcarrier offset between the victim subcarrier and the interfering 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 PHYDYAS [8] and IOTA [7] prototype filters with an overlapping factor of 4. The asynchronous interference analysis has been extended to the case of frequency selective environments in [9] and [10]. It has been demonstrated that the asynchronous interference power arriving through a selective frequency channel can be calculated using the following expression Pinterf (m, τ ) = d−β Ptrans (m′ )I(τ, |m′ − m|) |H(m′ )| (2) 2

′ ′ ′ d−β k Ptrans (m )I(τk , |m − m|) |Hk (m )|

2

(3)

k=1 m′ ∈Fk

Here, Fk stands for the set of subcarriers that are assigned to 2 the k-th BS. Also, |Hk (m′ )| and τk are respectively the power channel gain and the timing offset between the interfering base station k and the reference mobile user. Since the reference mobile user is assumed to be perfectly synchronized with its base station, the power of the useful part of the received signal 2 will be d−β 0 Ptrans (m) |H0 (m)| . Consequently, the instantaneous SINR can be given by, SINR(m) = d−β 0 Ptrans (m) |H0 (m)|

2

K ∑

∑

k=1 m′ ∈Fk

′ ′ ′ d−β k Ptrans (m )I(τk , |m − m|) |Hk (m )| + N0 Bsc 2

(4) where N0 denotes the noise power spectral density and Bsc is the bandwidth of the m-th subchannel. We rewrite (4) in the

TABLE I C HANNEL PARAMETERS USED IN

following form |H0 (m)| K ∑

∑

k=1 m′ ∈Fk

where

[

Ak,m,m′

dk = d0

]−β

b=

(5)

Ak,m,m′ |Hk

2 (m′ )|

+b

Ptrans (m′ ) I(τk , |m′ − m|) Ptrans (m)

N0 Bsc −β d0 Ptrans (m)

−1

(6)

BERaverage = +∞ ∫ K −1 1 1 e−z(1+2b) ∏ √ − √ ILk + 2 Ωk DkA z dz 2 2 π z

(7)

IOTA

−2

10

theoretical simulation

−3

10

where, • Ωk is the square root of the variance-covariance matrix 2 of the RVs {|Hk (m′ )| , m′ ∈ Fk }. • ILk is a Lk × Lk identity matrix and Lk denotes the cardinal of Fk . A • Dk is a diagonal matrix of diagonal elements, i ∈ Fk

OFDM

PHYDYAS

k=1

DkA (i, i) = Ak,m,i

value [0 110 190 410] ns [0 -9.7 -19.2 -22.8] dB

10

In [10], it has been demonstrated that the average error rate can be written as follows

0

Parameter Pedestrian-A Relative Delay Pedestrian-A Average Power

BER

SINR(m) =

SIMULATIONS

2

(8)

In contrast to direct complex analytical methods, these expressions present an efficient approach to compute the average BER with a significantly reduced computational complexity. IV. S IMULATION R ESULTS In this section we will discuss the results of the simulations carried out. We consider here the system model described in Section II. The reference mobile user is located at the center of the network composed of 12 cells as shown in Fig. 1. The cells have the same area with a radius R = 1 km. For all simulations, the Pedestrian-A model as a Rayleigh fading propagation channel has been chosen [11]. Table I gives the parameters of this channel model, for which, we assume that the subcarriers of interest see flat fading channels. For this channel model, the intra-cell interference is negligible with the FBMC system. This makes it possible to focus on the impact of the inter-cell asynchronous interference. The path loss of a received signal at a distance d is governed by the following expression [12] Γloss (d) = 128.1 + 37.6 log10 (d)[dB] In this expression the path loss exponent β = 3.76 and the carrier frequency is 2 GHz. We consider a system with N = 1024 subcarriers, a QPSK modulated data at a sampling frequency of 10 M Hz. The noise term is characterized by a thermal noise density of −174 dBm/Hz. The cyclic prefix duration, for the OFDM system, is fixed at ∆ = T /8, and the length of the subcarrier block

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Optimal

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SNR [dB]

Fig. 3. The OFDM/FBMC average BER against the SNR for τ ∈ [0, T ], the guard band size δ = 0

is fixed to 18 subcarriers. For the FBMC system, we use the PHYDYAS and IOTA prototype filters with an overlapping factor of 4. In each simulation the results are compared to the so-called optimal scenario in which the orthogonality between the different subchannels is assumed to be maintained for both OFDM and FBMC systems. In the first simulation, we consider the scenario of a zero guard-band interval between the clusters of the several cells (δ = 0). In this case, we assume that the timing offset τ is a uniform random variable in the interval [0, T ]. The BER of OFDM and FBMC modulations against the SNR is shown in Fig. 3. As can be seen for Fig. 3, the average BER of both OFDM and FBMC systems is highly affected with the timing offset errors. Indeed, up to 10 dB of SNR, the degradation for OFDM and FBMC systems is very similar. However exceeding this value, we can notice a better performance of the asynchronous FBMC when compared to the asynchronous OFDM. Such a gain can be explained by the fact that only a single subcarrier (PHYDYAS case) or two subcarriers (IOTA case) on each edge suffer from the interference caused by their adjacent subcarriers in the FBMC case. In the case of the OFDM modulation, all neighboring cells contribute to the interference induced on the entire reference cluster. For the so-called optimal case, both OFDM and FBMC systems show the same performance. In Fig. 4 and Fig. 5, we plot respectively the OFDM and FBMC average BER versus the SNR with different timing offset scenarios: [0, T /7] in scenario (a), [0, T /4] in scenario

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OFDM (c)

OFDM (b)

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IOTA (a,b,c)

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BER

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BER

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PHYDYAS (a,b,c)

OFDM (a)

theoretical simulation

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theoretical simulation

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Optimal

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Optimal

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SNR [dB]

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Fig. 4. The OFDM average BER for different timing offset intervals: Optimal case (a). τ ∈[0,T/7] (b). τ ∈[0,T/4] (c). τ ∈[0,T], the guard-band size δ =0

Fig. 5. The FBMC average BER for different timing offset intervals: Optimal case (a). τ ∈[0,T/7] (b). τ ∈[0,T/4] (c). τ ∈[0,T], the guard-band size δ =0

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theoretical simulation −2

IOTA,δ=0

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δ=0

−2

PHYDYAS δ=0

BER

BER

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δ=1 δ=5

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δ=20

IOTA,δ=1 −3

Optimal

theoretical simulation

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Optimal, PHY(δ=1), IOTA (δ=2) −4

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SNR [dB]

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Fig. 6. The OFDM average BER for different guard-band sizes δ = 0, 1, 5 and 20, the timing offset interval τ ∈[0,T]

Fig. 7. The FBMC average BER for different guard-band sizes δ = 0,1 and 2, the timing offset interval τ ∈[0,T]

(b) and [0, T ] in scenario (c). All scenarios are compared to the optimal case. In the OFDM system, the degradation is severe and increases when the timing error interval is larger. In fact, for τ ≤ ∆, the several clusters are orthogonal. Exceeding this value, a significant asynchronous interference, induced by the loss of orthogonality, affects the reference user performances. Since the timing offset is a uniform random variable, the probability to obtain the performance of the optimal case is given by the CP duration over the whole timing offset interval (∆/τmax ). The higher is τmax , the lower is the probability to maintain orthogonal clusters. Concerning the FBMC system, the main conclusion that one can make from the results given in Fig. 5, is that both waveforms are not sensitive to the timing offset interval length. This is due to the fact that the significant interference (with a

power higher than 30 dB) is roughly invariable with respect to the timing offset value. The impact of the guard-band length δ on the performance of the OFDM and FBMC systems is depicted in Fig. 6 and Fig. 7. The first Figure shows the OFDM average BER against the SNR for different guard band values δ = 0, 1, 5 and 20 subcarriers; we assume also that the timing offset τ is a uniform random variable defined on [0, T ]. Comparing the different curves, one can see a performance improvement when increasing δ. However, there is still a gap with respect to the optimal case even for a guard-band of 20 subcarriers. In contrast to the OFDM case, the FBMC waveforms show a better performance approaching that of the optimal case for a guard-band of a single subcarrier δ =1 in PHYDYAS case and 2 subcarriers δ = 2 in IOTA case. This is based on the fact that with the PHYDYAS-FBMC waveform and if the reference

typical waveforms have been considered : IOTA and PHYDYAS. An intensive evaluation has been performed taking into account different parameters: the timing error range, the guard band length and the load factor of the network. Through this evaluation, we have shown that the PHYDYAS-FBMC system presents a better performance compared to IOTA-FBMC. This result is due to the better frequency localization of the PHYDYAS waveform compared to the IOTA one. Furthermore, the FBMC system with both waveforms outperforms the OFDM system because in this latter, the orthogonality between all system subcarriers is no longer maintained. R EFERENCES

theoretical simulation

BER

OFDM, SNR=20dB

−2

IOTA, SNR=20dB

10

PHYDYAS, SNR=20dB

Optimal, SNR=20dB

0

0.2

0.4

α [× 100%]

0.6

0.8

1

Fig. 8. The average BER against the load factor α for SNR = 20 dB, the timing offset interval τ ∈[0,T]

user is two subcarriers away from the interferong one, the interference power is 60 dB (Fig. 2) less than the useful power. The same remark can be noticed in IOTA-FBMC where the interference power is negligible compared to the noise level when spectral distance between the interfering subcarrier and the victim one is equal to 2 (δ = 2). The system performance in a partially loaded scenario is depicted by Fig. 8. For this scenario the average BER is simulated with respect to the load factor α for SNR = 20 dB. The timing offset τ is assumed to be uniformly distributed on [0, T ]. From Fig. 8, we find that the average BER increases with the load factor α. We see that PHYDYAS-FBMC still outperforms IOTA-FBMC because in this latter two subcarriers at each edge of the subcarrier block of interest are affected by interference instead of a single subcarrier in the PHYDYAS case (see Fig. 2). It is worth noticing that FBMC with both waveforms presents error rates lower that than the one of OFDM system. V. C ONCLUSION This paper addresses the problem of the sensitivity of the FBMC system to the timing synchronization errors. Two

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