Quentin Bodinier

Faouzi Bader

Jacques Palicot

Coexistence in 5G: Analysis of Cross-Interference between OFDM/OQAM and Legacy Users

To optimize the use of the spectrum, it is expected that the next generation of wireless networks (5G) will enable coexistence of newly introduced services with legacy cellular networks. These new services, like Device-To-Device (D2D) communication, should require limited synchronization with the legacy cell to limit the amount of signaling overhead in the network. However, it is known that Cyclic Prefix-Orthogonal Frequency Division Multiplexing (CP-OFDM) used in Long Term Evolution-Advanced (LTE-A) is not fit for asynchronous environments. This has motivated the search for a new waveform, able to enhance coexistence with CP-OFDM. Namely, it has been widely suggested that new devices could use OFDM/Offset-Quadrature Amplitude Modulation (OFDM/OQAM) to reduce the interference they inject to legacy cellular users. However, values of interference are usually measured at the input antenna of the receiver, based on the PSD of the interfering signal. We showed in previous works that this measurement is not representative of the actual interference that is seen after the demodulation operations. Building on this finding, we provide in this paper the first exact closed forms of cross-interference between OFDM/OQAM and CP-OFDM users. Our results prove that using OFDM/OQAM only marginally reduces interference to legacy users, in contradiction with many results in the literature.

I. INTRODUCTION

The advent of the 5th Generation of wireless communication systems (5G) is envisioned to bring flexibility to cellular networks. New services as Device-To-Device (D2D) or Machine-To-Machine (M2M) communications are expected to be massively deployed in the near future. Such new communication devices have to coexist with incumbent legacy systems in the cell, i.e. Long-Term-Evolution Advanced (LTE-A) users. In such heterogeneous environments, perfect synchronization between the different types of systems is not feasible. This loss of synchronization will cause harmful interference between active users, which will in turn degrade the overall system performance. This hurdle can be overcome through the design of new waveforms that are robust against asynchronism, and well localized in frequency. As a matter of fact, as soon as the orthogonality between CP-OFDM users is destroyed, for example because of the coexistence of unsynchronized incumbent and secondary systems, their performance shrinks dramatically [START_REF] Medjahdi | Performance Analysis in the Downlink of Asynchronous OFDM/FBMC Based Multi-Cellular Networks[END_REF]. This is mainly due to the fact that CP-OFDM systems filter symbols with a time-rectangular window, which causes poor frequency localization [START_REF] Farhang-Boroujeny | OFDM versus Filter Bank Multicarrier[END_REF]- [START_REF] Mahmoud | OFDM for Cognitive Radio: Merits and Challenges[END_REF] and high asynchronism sensitivity in the multi-user context [START_REF] Mahmoud | OFDM for Cognitive Radio: Merits and Challenges[END_REF]- [START_REF] Aminjavaheri | Impact of Timing and Frequency Offsets on Multicarrier Waveform Candidates for 5G[END_REF].

OFDM with Offset Quadrature Amplitude Modulation (OFDM/OQAM) [START_REF] Farhang-Boroujeny | OFDM versus Filter Bank Multicarrier[END_REF], [START_REF] Bellanger | FBMC Physical Layer: a Primer[END_REF], is one of the main new wave-form schemes explored by the research community. Indeed, it overcomes the cited CP-OFDM limitations and enables both higher flexibility and reduction of interference leakage for multistandard systems coexistence [START_REF] Farhang-Boroujeny | OFDM versus Filter Bank Multicarrier[END_REF], [START_REF] Bellanger | FBMC Physical Layer: a Primer[END_REF]. The main selling point of OFDM/OQAM lies in its improved spectral containment that is obtained through the filtering of each subcarrier with a highly selective prototype filter. In this paper, we will study the case of OFDM/OQAM systems using the PHYDYAS filter [START_REF] Bellanger | FBMC Physical Layer: a Primer[END_REF]. Thanks to the enhanced spectral localization of the latter, OFDM/OQAM boasts hardly measurable out-of-band (OOB) emissions, as its Power Spectral Density (PSD) rapidly decreases below the ambient noise level.

Building on this, a number of papers, for example [START_REF] Shaat | Computationally Efficient Power Allocation Algorithm in Multicarrier-based Cognitive Radio Networks: OFDM and FBMC Systems[END_REF]- [START_REF] Lin | Flexible Configured OFDM for 5G Air Interface[END_REF], have suggested to use OFDM/OQAM to coexist efficiently with CP-OFDM based networks. All the studies on that matter use the PSD-based model, originally proposed in [START_REF] Weiss | Mutual Interference in OFDM-Basedased Spectrum Pooling Systems[END_REF], to rate the cross-interference between the OFDM/OQAM and CP-OFDM systems. This model consists in integrating the PSD of the interfering signal on each subcarrier of the user that suffers from interference. Because of the advantageous PSD properties of OFDM/OQAM, studies using the PSD-based model predicted that using this waveform instead of CP-OFDM for coexistence with LTE-A users would be highly beneficial.

However, the expected gains were not confirmed through simulations [START_REF] Bodinier | 5G Waveforms for Overlay D2D Communications: Effects of Time-Frequency Misalignment[END_REF], [START_REF]Modeling Interference Between OFDM / OQAM and CP-OFDM : Limitations of the PSD-Based Model[END_REF]. Moreover, in [START_REF]Modeling Interference Between OFDM / OQAM and CP-OFDM : Limitations of the PSD-Based Model[END_REF], we explained in a qualitative manner why the PSD-based model was not fit to properly estimate the cross-interference that is seen after the demodulation operations at both the receivers of OFDM/OQAM and CP-OFDM systems. Furthermore, we showed through numerical simulations that the gains expected by following the PSD-based model were highly overestimated. However, a thorough mathematical analysis of the postdemodulation cross-interference arising between coexisting OFDM/OQAM and CP-OFDM systems is still lacking. This paper aims at resolving this issue by providing mathematical closed forms of cross-interference injected by OFDM/OQAM onto CP-OFDM and vice-versa. The provided closed forms prove that OFDM/OQAM fails to protect incumbent legacy CP-OFDM users.

The remainder of this paper is organized as follows: Section II presents the system model and a short overview on CP-OFDM and OFDM/OQAM waveforms. In Section III, the analysis of cross-interference is led. In Section IV, the validity of the derived closed forms is asserted by comparison with Notations: scalars are noted x,vectors are bold-faced as x, and ensembles are represented by a calligraphy letter X . n is the discrete symbol index, m indexes subcarriers and t is the continuous time. * is the convolution operation, x * is the complex conjugate of x, E α {} is the mathematical expectation with regards to the random variable α and R{x} is the real part of x.

II. BACKGROUND AND SYSTEM MODEL

A. Analyzed Coexistence Scenario

We consider a coexistence scenario where an incumbent CP-OFDM based system and an OFDM/OQAM secondary system share the same spectral band. The incumbent is assigned a set of active subcarriers M i and the secondary a set M s . We assume that both systems use the same subcarrier spacing ∆F and time-symbol T = 1 ∆F . To focus on cross-interference between users, we consider a transmission on an additive white gaussian noise (AWGN) channel and do not take into account any pathloss or shadowing effect. The corresponding system model is represented in Fig. 1 and an example of subcarriers distribution is given in Fig. 2.

Note that, strictly speaking, each interfering signal arrives at the receiver it interferes on with a certain delay. For example, naming δ t,s the propagation delay between the secondary transmitter and the incumbent receiver, the interfering signal received at the incumbent is not s s (t) but s s (tδ t,s ). However, we showed through simulation in [START_REF] Bodinier | 5G Waveforms for Overlay D2D Communications: Effects of Time-Frequency Misalignment[END_REF] that this propagation delay has only little impact on the interference between OFDM/OQAM and CP-OFDM users. Without loss of generality, we will therefore neglect the propagation delays in our analysis. Now that the system model is laid out, we present in the following Section a short background on CP-OFDM and OFDM/OQAM signal models.

B. CP-OFDM Incumbent System

We consider a CP-OFDM incumbent system with M subcarriers, time-symbol T and a CP duration of T CP . As previously mentioned, it has a set of active subcarriers M i . The CP-OFDM time-domain signal transmitted on each active subcarrier m i ∈ M i is expressed as

s m i (t) = n i ∈Z d m i [n i ]f T,i (t -n i (T + T CP )) e j2πm i t-n i T CP T , (1)
with d m i the data vector of quadrature amplitude modulated (QAM) symbols transmitted on subcarrier m i and f T,i the CP-OFDM transmit filter defined as:

f T,i (t) = 1 √ T , t ∈ [-T CP , T ] 0,
elsewhere.

(

The total transmitted signal is expressed as

s i (t) = m i ∈M i s m i (t), ∀t ∈ R. (3)
In this study, we consider an AWGN interference channel. Therefore, the signal at the input antenna of the CP-OFDM incumbent receiver is expressed as

y i (t) = s i (t) + s s (t) + w i (t), ∀t ∈ R (4)
where s s (t) is the interfering OFDM/OQAM secondary signal whose expression will be detailed in the next section and w i (t) is the AWGN seen at the incumbent receiver.

Assuming perfect synchronization between the CP-OFDM transmitter and receiver, the n i -th demodulated symbol on the m i -th subcarrier of the CP-OFDM receiver is expressed, ∀n i ∈ Z, as

dm i [n i ] = d m i [n i ] + ms∈Ms η ms→m i [n i ] + w i [n i ], (5)
where w i is the filtered white gaussian noise component expressed as

w i [n i ] = ∞ -∞ f R,i (t-n i (T +T CP ))e -j2πm i t-n i T CP T w i (t)dt, ∀n i ∈ Z (6)
and η ms→m i is the interference injected by the m s -th subcarrier of the secondary onto the m i -th subcarrier of the incumbent, which is expressed ∀n i ∈ Z as

η ms→m i [n i ] = ∞ -∞ f R,i (t-n i (T +T CP ))e -j2πm i t-n i T CP T s ms (t)dt.
(7) In ( 6) and [START_REF] Bellanger | FBMC Physical Layer: a Primer[END_REF], f R,i is the receive filter of the incumbent and is expressed as

f R,i (t) = 1 √ T , t ∈ [0, T ] 0, elsewhere. ( 8
) f ∆F M 2 -1 -M 2 0 M i M s
Fig. 2: Spectral representation of the considered scenario. The incumbent and secondary systems coexist in the same spectral band, and each one is assigned a different subset of subcarriers.

C. Secondary OFDM/OQAM System

We consider an OFDM/OQAM secondary system with M subcarriers and time-symbol T , and name M s its set of active subcarriers. The time-domain signal transmitted on each active subcarrier m s ∈ M s of the secondary OFDM/OQAM system is expressed as [START_REF] Caus | Capacity Analysis of WCC-FBMC/OQAM Systems[END_REF]

s ms (t) = ns∈Z (-1) msns d ms [n s ]θ ms [n s ]f T,s t -n s T 2 e j2πms t T (9
) where d ms is the data vector of pulse amplitude modulated (PAM) real symbols and θ ms [n s ] is a phase factor added to the symbols and defined as [START_REF] Caus | Capacity Analysis of WCC-FBMC/OQAM Systems[END_REF]

θ ms [n s ] = e j π 2 ⌊ ns+ms 2 ⌋ . (10)
Besides, f T,s is the transmit filter of the secondary system expressed as

f T,s (t) = g(t) t ∈ [-KT 2 , KT 2 ] 0, elsewhere (11)
where K is called the overlapping factor, and g is the used prototype filter. In the remainder of this study, we take g as the PHYDYAS prototype filter with overlapping factor K = 4, [START_REF] Bellanger | Specification and Design of a Prototype Filter for Filter Bank Based Multicarrier Transmission[END_REF], defined as

g(t) = K-1 k=-K+1 G |k| K e j2π kt KT , t ∈ [- KT 2 , KT 2 ], (12)
with

G 0 = 1, G 1 = 0.971960, G 2 = 1 √ 2 , G 3 = 0.
235147. Note that g is a real and symmetric filter, such that g * (-t) = g(t), ∀t ∈ R.

The signal received at the input antenna of the OFDM/OQAM receiver is expressed in a similar manner as in (4) and is given by

y s (t) = s s (t) + s i (t) + w s (t), ∀t ∈ R (13)
At the OFDM/OQAM receiver, the received signal is passed through the receive filter f R,i expressed as

f R,s (t) = g * (-t) = g(t) t ∈ [-KT 2 , KT 2 ] 0, elsewhere. ( 14
) t 0 n i = -1 n i = -2 n i = 0 n i = 1 TCP T ns = 0 ns = -7 KT T 2
Fig. 3: Time axis view of the interference caused by the OFDM/OQAM transmission on CP-OFDM receiving windows n i = 0 and n i = 1. Both CP-OFDM symbols suffer the same amount of cross-interference.

Then, the real part of the filtered signal is taken to remove purely imaginary intrinsic interference [START_REF] Caus | Capacity Analysis of WCC-FBMC/OQAM Systems[END_REF]. Therefore, the n s -th demodulated symbol on the m s -th subcarrier of the OFDM/OQAM secondary receiver is expressed as

dms [n s ] = d ms [n s ] + m i ∈M i η m i →ms [n s ] + w s [n s ], (15)
with

w s [n s ] = ∞ -∞ R{f R,s (t -n s T 2 )e -j2πms t T (-1) msns w s (t)}dt (16)
η m i →ms [n s ] = ∞ -∞ R{f R,s (t -n s T 2 ) × e -j2πms t T (-1) msns s m i (t)}dt. (17)

III. CROSS-INTERFERENCE ANALYSIS A. Cross-Interference at the Incumbent CP-OFDM Receiver

Here, we derive the closed-form expression of interference seen at the incumbent CP-OFDM receiver as expressed in [START_REF] Bellanger | FBMC Physical Layer: a Primer[END_REF]. Substituting the expression of f T,s given by [START_REF] Lin | Flexible Configured OFDM for 5G Air Interface[END_REF] in [START_REF] Skrzypczak | OFDM/OQAM Modulation for Efficient Dynamic Spectrum Access[END_REF], and then putting both the resulting form of (9) and the expression of the incumbent receive filter f R,i given by ( 8) in [START_REF] Bellanger | FBMC Physical Layer: a Primer[END_REF], we obtain the expression (18) of η ms→m i [n i ]. In the following, we focus on the mean interference power caused by subcarrier m s of the OFDM/OQAM secondary onto subcarrier m i of the CP-OFDM incumbent, which we define as

I ms→m i [n i ] = E ds {|η ms→m i [n i ]| 2 } (19) = σ 2 ds T ns∈Z n i (T +TCP)+T n i (T +TCP) g t -n s T 2 × e j2π(ms-m i ) t T dt 2 , ( 20
)
η ms→m i [n i ] = 1 √ T ns∈Z (-1) msns d ms [n s ]θ ms [n s ] n i (T +TCP)+T n i (T +TCP) g t -n s T 2 e j2π(ms-m i ) t T dt, ∀n i ∈ Z. ( 18
)
with σ 2 ds the variance of the symbols modulated by the secondary OFDM/OQAM system. Note that this last expression is obtained by considering that the symbols d s are independent and identically distributed (i.i.d.). Besides, note that the sum in (20) can be reduced to the values of n s such that g(tn s T 2 ) is not null everywhere on the reception window corresponding to the n i -th symbol of the receiver. This is shown in Fig. 3, where we also point out that the value of n i does not affect the mean interference power seen at the receiver. Note also that only the difference m sm i plays a role in (20). Naming l = m sm i the spectral distance in terms of subcarriers, we therefore have

I ms→m i [n i ] = I s→i (l), ∀n i ∈ Z.
After operating the change of variable t → tn i (T + T CP ) in (20), we obtain, ∀n i ∈ Z,

I ms→m i [n i ] = I s→i(l) σ 2 ds T T 2 τ =-KT + T 2 T 0 g(t -τ )e j2π lt T dt 2 I s→i (l,τ ) (21)
By including in (21) the expression of g given in [START_REF] Weiss | Mutual Interference in OFDM-Basedased Spectrum Pooling Systems[END_REF], we obtain, ∀l ∈ Z,

I s→i (l, τ ) = 1 K 2 K-1 k=-K+1 G |k| T 0 e j2π( k(t+τ ) KT + lt T ) dt 2 (22) = T K 2 K-1 k=-K+1 G |k| e jπ k K τ sinc(π( k K + l)) 2 , (23)
and the interference power injected by an OFDM/OQAM subcarrier to a CP-OFDM subcarrier at spectral distance l is finally given by

I s→i (l) = σ 2 ds K 2 T 2 τ =-KT + T 2 K-1 k=-K+1 G |k| e jπ k K τ × sinc(π( k K + l)) 2 (24)

B. Cross-Interference at the Secondary OFDM/OQAM Receiver

We now focus on the cross-interference that is injected by the CP-OFDM incumbent transmitter onto the OFDM/OQAM receiver, as expressed in [START_REF] Brown | Synchronization Concepts[END_REF]. Substituting the expression of the CP-OFDM transmit filter (2) into (1), and then the resulting form of (1) and the expression of f R,s given by ( 14) into [START_REF] Brown | Synchronization Concepts[END_REF], we obtain the expression of (25), which is very close to (18), the only differences being the real part operator and some phase factors due to the OFDM/OQAM demodulation. In a similar way as (19), we define the mean interference power injected by subcarrier m i of the incumbent onto the m s -th subcarrier of the secondary as

I m i →ms [n s ] = E d i {|η m i →ms [n s ]| 2 } (26) = σ 2 d i 2T n i ∈Z n i (T +TCP)+T n i (T +TCP)-TCP g t -n s T 2 × e j2π(m i -ms) t T dt 2 , ( 27
)
with σ 2 d i the variance of the symbols d i . Note that the factor 1 2 comes from the fact that only the real part of the signal is taken at the OFDM/OQAM receiver. The obtained expression is similar to (19) and the different observations made in the previous section still hold true here. Therefore, following developments similar to ( 19)-( 24), the power of interference injected by a CP-OFDM subcarrier to an OFDM/OQAM subcarrier at spectral distance l is expressed as (28). Note that, if both systems transmit with the same energy per symbol E s ,

σ 2 d i 2 = σ 2 ds = Es
2 because each OQAM symbol transmits half the energy of a QAM symbol. Therefore, in the case where T CP = 0, which corresponds to a situation where the incumbent system does not use any CP, I i→s (l) = I s→i (l), ∀l ∈ Z and the incumbent and secondary systems interfere equally onto each other.

From both (28) and ( 24), it can be noticed that the rectangular time window of the CP-OFDM system incurs a sum of sine-cardinal in frequency, which slowly decrease and will therefore cause high interference to both systems, despite the well shaped prototype filter used by the OFDM/OQAM system.

C. Taking into Account Frequency Misalignments

The closed forms we derived in the previous section were obtained in the case where the secondary and the incumbent systems agree on the exact same frequency basis. Therefore, the spectral distance l between each subcarrier of the secondary and each subcarrier of the incumbent is an integer, i.e. l ∈ Z. This is correct if the local oscillators (LO) of all users are perfectly synchronized. However, in a real setup, this is not the case. Indeed, LOs of mobile terminals have a typical accuracy of ±1 ppm with respect to their nominal frequency [START_REF] Brown | Synchronization Concepts[END_REF]. At a carrier frequency of 2 GHz, this can yield a misalignment between users of around 10 4 Hz, which can become significant as it is close to the LTE subcarrier width of 15 kHz.

Here, we consider that the transmitter and receiver in each system achieve perfect frequency synchronization. However,

η m i →ms [n s ] = 1 √ T n i ∈Z R d m i [n i ]e -j2πm i n i T CP T (-1) msns n i (T +TCP)+TCP n i (T +TCP)-TCP g t -n s T 2 e j2π(m i -ms) t T dt , ∀n s ∈ Z (25)
I i→s (l) = σ 2 d i 2K 2 (1 + T CP T ) T 2 τ =-KT + T 2 K-1 k=-K+1 G |k| e jπ k K τ (1+ T CP T ) sinc π 1 + T CP T k K + l 2 (28) f ∆F M 2 -1 -M 2 0 M i M s δ f
Fig. 4: Spectral representation of the considered scenario with frequency misalignment δ f between coexisting systems. the incumbent and secondary systems are not supposed to cooperate, and it is very likely that they will not be perfectly aligned in frequency. This means that, taking the incumbent system as a reference, the m s -th subcarrier of the secondary will not be modulated at base-band frequency m s ∆F but at base-band frequency (m s + δ f )∆F, where δ f ∈] -0.5, 0.5] and is the frequency misalignment value between the secondary and incumbent systems. In other words, Fig. 2 is transformed into Fig. 4 where we see that the secondary transmission is misaligned with the band it should actually transmit in.

This simply leads to rewriting ( 4) and ( 13) as

y i (t) = s i (t) + s s (t)e j2πδft + w i (t), (29)
y s (t) = s s (t) + s i (t)e -j2πδft + w s (t). (30)
Then, the analysis is led exactly as ( 7) -(28). Redefining l as l = m s + δ fm i , mathematical derivations yield the exact same expression as ( 24) and (28). Therefore, the closed forms we derived are also applicable in the presence of frequency misalignment between the incumbent and secondary systems i.e. ∈ R.

IV. NUMERICAL RESULTS

In Fig. 5, we represent the values of I s→i (l) and I i→s (l) according to the expressions of (28) and (24) for T CP = T 8 . We observe that both systems interfere almost equally onto each other. Note that, if we had set T CP = 0, the two curves would perfectly overlap.

In Fig. 6, the derived theoretical expressions are compared to results obtained through Monte-Carlo simulations. We represent the interference power seen at each subcarrier of index l when the interferer transmits at subcarrier of index 0. Parameters are set as follows: both the CP-OFDM incumbent and the OFDM/OQAM secondary systems have M = 512 subcarriers. Both systems transmit symbols with unitary energy σ 2

d i = 2σ 2 ds = E s = 1.
Besides, we consider a CP of relative duration T CP = T 8 and δ f = 0. Figures show that the expressions (24) and (28) perfectly match the simulation results.

Moreover, we show the results predicted by the PSD-based model used for example in [START_REF] Shaat | Computationally Efficient Power Allocation Algorithm in Multicarrier-based Cognitive Radio Networks: OFDM and FBMC Systems[END_REF]- [START_REF] Ihalainen | On Spectrally Efficient Multiplexing in Cognitive Radio Systems[END_REF]. It is worth noticing that the PSD-based model gives a good approximation of the interference injected by the CP-OFDM incumbent onto the OFDM/OQAM secondary. However, it completely fails in estimating the interference injected by the OFDM/OQAM secondary onto the CP-OFDM incumbent. As pointed out in [START_REF]Modeling Interference Between OFDM / OQAM and CP-OFDM : Limitations of the PSD-Based Model[END_REF], this is due to the fact that the PSD-based model does not take into account the rectangular receive window of the CP-OFDM receiver. Because the receive window of the OFDM/OQAM receiver is larger than the transmit filter of the CP-OFDM, the PSD appears to be a good measure of I i→s . On the contrary, as shown in Fig. 3, the CP-OFDM receive window truncates each OFDM/OQAM symbol and the good PSD properties of the latter are lost in the process. Finally, we compare the values of cross-interference in the considered scenario with those observed in a homogeneous scenario where both users would use CP-OFDM. To do so, in Fig. 7, we compare the expressions of ( 24) and (28) with the interference tables derived by Medjahdi et. al. in [START_REF] Medjahdi | Performance Analysis in the Downlink of Asynchronous OFDM/FBMC Based Multi-Cellular Networks[END_REF] in the case where both the secondary and the incumbent users use CP-OFDM. Note that the interference observed by the incumbent and secondary users when they both use CP-OFDM is, at maximum, only 3 dB higher to that observed when the secondary uses OFDM/OQAM. In real-world transmission, when the hardware impairments and the high power amplifier non-linearities come into play, this 3 dB gap will fade out and the benefits of using OFDM/OQAM for coexistence with CP-OFDM systems are likely to disappear.

V. CONCLUSION

In this paper, we were able to develop exact closed forms of the cross-interference between coexisting OFDM/OQAM and CP-OFDM systems. As OFDM/OQAM is seen as a potential contender for certain 5G applications, the analysis led in this paper can be useful to dimension networks in scenarios where secondary 5G devices coexist with incumbent legacy users. Besides, the presented analysis can be generalized in the case where the secondary system would use other waveforms studied for 5G, such as filtered OFDM (f-OFDM) or Universal Filtered OFDM (UF-OFDM). Indeed, our analysis showed that the cross-interference is mainly caused by the rectangular window of the incumbent system. In other words, results presented in this paper show that coexistence with legacy CP-OFDM systems cannot be drastically improved by designing enhanced waveforms only, but that it is necessary to modify the CP-OFDM receiver itself.

Hence, the derived closed forms show that the PSD-based model, widely used in the literature, is not fit to study coexistence scenarios, as it does not take into account the This observation has a wide array of consequences: indeed, it shows that all the studies which rely on OOB emissions or PSD to rate cross-interference between users are inherently flawed, especially in the context of coexistence between two systems with different waveforms. However, a large number of studies recently released on that matter rely on these two measurements to rate the performance of various waveforms. Moreover, standards like LTE-A usually define spectral masks that systems should respect to coexist smoothly. However, as these masks are based on PSD measurements as well, our results show that they are not adapted to the management and dimensioning of 5G systems. Therefore, the results we derived in a particular scenario in this paper have broader applications and consequences. Especially, it is utterly important to rethink the models used to ensure coexistence between different physical layers. Future work will therefore focus on this particular aspect.

Fig. 5 :

 5 Fig. 5: Values of cross interference injected at spectral distance l, according to (28) and (24).

Fig. 6 :Fig. 7 :

 67 Fig. 6: Cross-interference values for δ f = 0.

ACKNOWLEDGEMENT

This work was partially funded through French National Research Agency (ANR) project ACCENT5 with grant agreement code: ANR-14-CE28-0026-02.


Quentin Bodinier

Faouzi Bader

Jacques Palicot

Coexistence of Filter Banks and CP-OFDM: What are the Real Gains? (Invited Paper)

A coexistence scenario between filter bank (FB) based waveforms and legacy Cyclic Prefix-Orthogonal Frequency Division Multiplexing (CP-OFDM) is studied. It is shown that studies available on that matter use a wrong metric to rate the interference between the coexisting systems. Specifically, it is shown that even well spectrally localized FB waveforms interfere at a high level with incumbent CP-OFDM receivers. The presented results invalidate a number of studies in the literature, which stated that FB based waveforms could be very efficiently used to insert communications in parts of spectrum left free by incumbent legacy CP-OFDM users.

I. INTRODUCTION

In the course of the development of the 5th generation of wireless networks (5G), a number of new waveforms have been proposed to replace Cyclic-Prefix Orthogonal Frequency Division Multiplexing (CP-OFDM) used in Long Term Evolution-Advanced (LTE-A) [START_REF] Banelli | Modulation Formats and Waveforms for 5G Networks: Who Will Be the Heir of OFDM?[END_REF]. Indeed, the latter suffers from well-known limitations. Due to the rectangular shaping of symbols in the time domain, CP-OFDM has high out of band emissions (OOBE) and is very sensitive to asynchronism [START_REF] Mahmoud | OFDM for cognitive radio: merits and challenges[END_REF], [START_REF] Medjahdi | Interference tables: a useful model for interference analysis in asynchronous multicarrier transmission[END_REF].

These two characteristics are very problematic for certain applications and use cases that are expected to be enabled with the advent of 5G. Especially, communication types are expected to become ever more various and diverse, ranging from classical cellular networking to Device-To-Device (D2D) or Machine-To-Machine (M2M) communication [START_REF] Wunder | 5GNOW -5G Cellular Communications Scenarios and System Requirements[END_REF]. This new paradigm requires the physical layer (PHY) of 5G to be adaptable to various situations, and robust to asynchronous interference coming from neighboring communication devices [START_REF] Banelli | Modulation Formats and Waveforms for 5G Networks: Who Will Be the Heir of OFDM?[END_REF].

Besides, these new communication types will add a new burden to the radio spectrum, which is already saturated. To answer this challenge, two directions have mainly been explored:

(i) Exploit new parts of the spectrum at higher frequencies above 6 GHz. (ii) Exploit parts of the already licensed spectrum that are temporally left free by incumbent users. In this paper, we focus on the point (ii). The main challenge to overcome when re-exploiting some parts of the already licensed spectrum lies in the fact that secondary users should insert their communication without causing harmful interference to the incumbent legacy users. In the context of 5G, reusable spectrum mainly belongs to either LTE-A cellular networks, Wi-Fi or TV bands. In all these cases, inserted secondary users will have to coexist with CP-OFDM based incumbent communications, as it is the PHY used by these three technologies.

To protect the incumbent CP-OFDM users, the common policy states that secondary users should respect a certain spectrum mask at the transmitter side. This spectrum mask is usually specified by the standard followed by the incumbent users. CP-OFDM, suffering from poor spectral containment, can hardly fit efficiently in these spectrum masks and is therefore not efficiently applicable to secondary users [START_REF] Mahmoud | OFDM for cognitive radio: merits and challenges[END_REF], [START_REF] Medjahdi | Interference tables: a useful model for interference analysis in asynchronous multicarrier transmission[END_REF], [START_REF] Ihalainen | On spectrally efficient multiplexing in cognitive radio systems[END_REF], [START_REF] Datta | Flexible multicarrier PHY design for cognitive radio in White Space[END_REF]. This observation has motivated the research community to look for new waveforms whose OOBE are decreased compared to CP-OFDM, so that the secondary transmissions could fit into the specified spectrum masks.

New waveform schemes proposed for such applications all fall into the category of filter bank (FB) based multi-carrier waveforms [START_REF] Saltzberg | Performance of an Efficient Parallel Data Transmission System[END_REF], [START_REF] Farhang-Boroujeny | OFDM versus filter bank multicarrier[END_REF]. They all rely on the filtering of the transmit signal by highly frequency selective filters to reduce OOBE. Therefore, users based on this type of waveform easily fit the requirements set by incumbent users in terms of Power Spectral Density (PSD). Building on this, a number of papers have investigated the benefits of using FB based waveforms for the insertion of secondary users [START_REF] Noguet | Advances in opportunistic radio technologies for tvws[END_REF]- [START_REF] Huang | Heterogeneous LTE Downlink Spectrum Access Using Embedded-GFDM[END_REF]. Some real-world demonstrations of coexistence between FB waveforms and CP-OFDM have also been undertaken by both industrials and academics [START_REF] Jue | Implementing a Flexible Testbed for 5G Waveform Generation and Analysis -White Paper[END_REF]- [START_REF] Berg | A flexible radio transceiver for TVWS based on FBMC[END_REF].

All these studies rely on the observation of the PSD of the waveform used by the secondary. The advantages of using FB waveforms instead of CP-OFDM for secondary users are then rated in terms of the maximum performances that each waveform can achieve under the same spectral mask. However, PSD only accounts for the properties of the transmit signal, and totally neglects the effects of the CP-OFDM incumbent reception. To refine this measurement, it is necessary to study the Error Vector Magnitude (EVM) of the CP-OFDM incumbent receiver when it suffers interference from a neighboring FB waveform. To the best of our knowledge, this is only done experimentally by Keysight Technologies in [START_REF] Jue | Implementing a Flexible Testbed for 5G Waveform Generation and Analysis -White Paper[END_REF] but no comparison of the EVM of the CP-OFDM incumbent as a function of the secondary waveform is given. In [START_REF] Kaltenberger | Experimental analysis and simulative validation of dynamic spectrum access for coexistence of 4G and future 5G systems[END_REF], [START_REF] Berg | A flexible radio transceiver for TVWS based on FBMC[END_REF], the authors show that FB based secondary users do interfere less than CP-OFDM secondary onto CP-OFDM incumbent receivers. However, the gains shown are not in line with the much higher theoretical gains expected for example in [START_REF] Noguet | Advances in opportunistic radio technologies for tvws[END_REF]- [START_REF] Skrzypczak | OFDM/OQAM modulation for efficient dynamic spectrum access[END_REF].

In previous works, we already showed that results based on the PSD were misguiding, as they underestimate the interference that is seen after demodulation of the incoming signal by the CP-OFDM incumbent receiver [START_REF] Bodinier | Modeling Interference Between OFDM / OQAM and CP-OFDM : Limitations of the PSD-Based Model[END_REF]. We further showed that, considering EVM measurements, the interference caused by secondary users is not drastically reduced if the latter use FB waveforms instead of CP-OFDM [START_REF] Bodinier | Modeling Interference Between OFDM / OQAM and CP-OFDM : Limitations of the PSD-Based Model[END_REF]. As these previous results are contradictory with many results published in the literature, it is important to justify them clearly and revise some previously published results with regards to these new findings.

Therefore, in this paper, we further explain why coexistence of FB waveforms with CP-OFDM is not feasible. To do so, we show that FB waveforms belong to a class of waveforms that are not orthogonal with CP-OFDM. We propose a simplified model to replace the current misguiding PSD-based spectrum masks. Furthermore, we select a representative set of results from the literature and update them according to our interference measurements based on EVM. Both our justifications and the revised set of results we present confirm that coexistence with CP-OFDM is not drastically improved if the secondary users use FB waveforms.

The remainder of this paper is organized as follows: In Section II, we present a canonical model of coexistence. Besides, we explain the difference between the PSD based approach of the literature and the EVM based measurements of interference. In Section III, we show that proposed FB waveforms are not orthogonal with the CP-OFDM receiver, which causes high interference to the latter. In Section IV we select a specific set of results available in the literature on coexistence and update them according to our results. Finally, Section V concludes this paper.

II. IN-BAND COEXISTENCE SCENARIO

A. Typical Coexistence Scenario

A typical coexistence scenario is presented in Fig. 1a, where we see that the signals of both the FB based secondary s s and the CP-OFDM based incumbent s i add up to form the signal r at the input antenna of the incumbent CP-OFDM receiver. Note that, here, no noise and no channel effects are considered, in an effort to focus on the interference caused by the FB secondary tranmission. Then, the over-the-air signal r is passed through the CP-removal block which outputs the useful part of the signal, noted r. After demodulation, the vector of estimated symbols d is obtained. Under this simplified coexistence scenario, d = d + η η η, where d is the vector of quadrature amplitude modulation (QAM) symbols modulated by the CP-OFDM incumbent transmitter and η η η is the interference caused by the secondary FB based user. In Fig. 1b, we present an example spectral distribution of the incumbent and secondary users. The two users coexist in the same spectral band composed of M subcarriers of width ∆F, and each one is assigned a different set of subcarriers, M s for the secondary and M i for the incumbent. Under this model, the interference power caused by the secondary FB user onto the incumbent CP-OFDM receiver is expressed as the EVM seen after demodulation as

I tot = E{| d -d| 2 } = E{|η η η| 2 }, (1)
where E{.} is the mathematical expectation. I tot can be further decomposed in the sum of the power of interference caused by each subcarrier of the secondary onto each subcarrier of the incumbent as follows [START_REF] Bodinier | Modeling Interference Between OFDM / OQAM and CP-OFDM : Limitations of the PSD-Based Model[END_REF]:

I tot = m i ∈M i ,ms∈Ms I(l = m s -m i ), (2)
where I(l = m sm i ) is the interference caused by the subcarrier m s of the secondary to the subcarrier m i of the incumbent such that the spectral distance between these two subcarriers, noted l, is equal to m sm i . I(l) represents the interference power that is injected by a given subcarrier of the secondary FB user onto a subcarrier at spectral distance l of the incumbent CP-OFDM user. It is the key factor that will determine if it is possible to insert efficiently the secondary transmission. Indeed, the faster I(l) decreases with l, the more efficiently the secondary user will be able to utilize the free part of the band without interfering onto the subcarriers of the incumbent user. It is therefore crucial to model this value with great accuracy.

B. Modeling of I(l) in the literature

In the literature, the modeling of I(l) is based on the PSD of each subcarrier of the secondary FB transmitter. This model has been initially proposed to rate interference between various OFDM users in a spectrum pooling context in [START_REF] Weiss | Mutual Interference in OFDM-Basedased Spectrum Pooling Systems[END_REF]. Precisely, naming Φ s the PSD of each subcarrier of the FB based waveform used by the secondary, I(l) is usually modeled according to the PSD-based model as

I PSD (l) = l+ ∆F 2 l-∆F 2 Φ s (f ) df. (3)
The total interference is then obtained by putting (3) in [START_REF] Mahmoud | OFDM for cognitive radio: merits and challenges[END_REF].

The obtained expression of the total interference seen by the incumbent receiver is then equivalent to

I PSD,tot = M i S s (f )df, (4)
where S s (f ) is the total PSD of the signal s s .

From the expression (4), it is clear that the PSD-based model considers interference in the channel, before the input antenna of the incumbent CP-OFDM receiver, and is not consistent with the actual expression of the interference expressed in (1).

III. OBSERVATION OF FB SIGNAL AFTER CP-OFDM DEMODULATION

In the former section, we showed that the PSD-based model is not relevant to study the interference η η η and therefore, to rate the EVM I tot as defined in [START_REF] Banelli | Modulation Formats and Waveforms for 5G Networks: Who Will Be the Heir of OFDM?[END_REF], which corresponds to the actual value of interference seen by the CP-OFDM incumbent receiver after the OFDM demodulation operations. However, one may think that the PSD-based model is still a good approximation that will give a good idea of the actual values of interference. Hereafter, we explain why it is not the case.

FB waveforms rely on the filtering of each symbol by a filter g that is longer than the time symbol T = 1 ∆F . This means that each FB symbol has a temporal support higher than the time symbol T . On the opposite, the CP-OFDM receiver of the incumbent truncates the incoming signal in windows of duration T before performing FFT operations on each of these windows. Therefore, the FB and CP-OFDM signals are not orthogonal, as they are not based on the same temporal support. An example of the way the filter g used by the FB is cut is shown in Fig. 2. It is shown that only a certain portion of each FB symbol, represented by a solid line, is taken into account by the CP-OFDM receive window. More detailed explanation on this aspect can be found in [START_REF] Bodinier | Modeling Interference Between OFDM / OQAM and CP-OFDM : Limitations of the PSD-Based Model[END_REF].

Note that the rigorous mathematical analysis of the effects of this truncation is out of the scope of this paper, but has been led for a particular FB waveform in [START_REF] Bodinier | Coexistence in 5G: Analysis of Cross-Interference between OFDM/OQAM and Legacy Users[END_REF]. However, it is readily understood that the rectangular truncation in time of the FB filter g is equivalent, in frequency, to convolving it with a function sinc(f T ). From this observation, it is clear that the selective properties of the FB filter g are lost when passing through the CP-OFDM receiver. To illustrate this, we study a particular type of FB waveform, OFDM/OQAM with the PHYDYAS filter. More information on this waveform can be found in [START_REF] Bellanger | FBMC physical layer: a primer[END_REF]. In Fig. 3, we show the PSD of one OFDM/OQAM subcarrier. It is clear from that figure that, after passing through the CP-OFDM receiver, the spectral shape of the OFDM/OQAM signal is deeply altered. Whereas the original OFDM/OQAM signal is almost perfectly localized in frequency, high frequency ripples appear after the CP-OFDM receiver, which are due to the rectangular truncation shown in Fig. 3. Actually, the spectral shape of the altered OFDM/OQAM signal is very close to that of an OFDM signal. To strengthen our point, we compare in Fig. 4 the two ways of computing the values of interference injected by the OFDM/OQAM secondary onto a CP-OFDM incumbent, either through EVM or through PSD. It is very clear from this figure that the PSD measurements are completely irrelevant to measure the interference actually seen by the CP-OFDM incumbent, and do not approximate it in a satisfying way at all.

Though the results presented in Fig. 3 and Fig. 4 concern here only a particular waveform, the same behavior will be shown by other FB waveforms, as the limiting factor is the receive window of the CP-OFDM incumbent. More generally, because of the effects shown in Fig. 2 based on EVM measurements have been obtained through simulations in [START_REF] Bodinier | 5G Waveforms for Overlay D2D Communications: Effects of Time-Frequency Misalignment[END_REF] for a number of other waveforms, such as Generalized Frequency Division Multiplexing (GFDM) [START_REF] Michailow | Generalized Frequency Division Multiplexing for 5th Generation Cellular Networks[END_REF], Filtered Multi-Tone (FMT) [START_REF] Renfors | D2.1 : FB-MC and Enhanced OFDM Schemes[END_REF] and Lapped FBMC [START_REF] Bellanger | A Filter Bank Multicarrier Scheme Running at Symbol Rate for FutureWireless Systems[END_REF]. All these waveforms were shown to interfere approximately as much as OFDM/OQAM onto CP-OFDM incumbent users in [START_REF] Bodinier | 5G Waveforms for Overlay D2D Communications: Effects of Time-Frequency Misalignment[END_REF]. These results show that, in the context of 5G, to cope with heterogeneity at the PHY level in the network, usual spectrum masks need to be replaced to efficiently protect incumbent CP-OFDM users. Indeed, FB based users could easily fit the currently used spectrum masks based on PSD, but still interfere in a critical way onto CP-OFDM incumbent users. More specifically, instead of defining the transmit masks of secondary users, future standards should include the possibility for each incumbent user to define its maximum acceptable EVM post-demodulation, and let the secondary users adapt their transmission to respect that limitation. These new types of masks will be more complex to specify than PSD based spectrum masks, as they need both the properties of the transmitter and the receiver to be taken into account.

IV. REVISING PUBLISHED RESULTS

In this section, we present two coexistence scenarios that are usually studied in the literature according to the PSDbased model, which we invalidated in the previous sections. We compare results shown in the literature with the ones we obtain when rating the interference caused onto the CP-OFDM incumbent through measurements of EVM after demodulation. Once again, we show results for OFDM/OQAM with Phydyas filter only, but the nature of the presented results is extensible to other FB based waveforms.

A. Reduction of Guard Bands

A first way of studying the impact of the waveform used by the secondary user on the amount of interference injected onto the incumbent is to consider the width of the guard band that is necessary to protect the incumbent system. This is the approach followed for example in [START_REF] Ihalainen | On spectrally efficient multiplexing in cognitive radio systems[END_REF], [START_REF] Datta | Flexible multicarrier PHY design for cognitive radio in White Space[END_REF], [START_REF] Skrzypczak | OFDM/OQAM modulation for efficient dynamic spectrum access[END_REF], [START_REF] Jue | Implementing a Flexible Testbed for 5G Waveform Generation and Analysis -White Paper[END_REF], [START_REF] Berg | A flexible radio transceiver for TVWS based on FBMC[END_REF]. Here we study this problem in the following layout: we consider that the secondary and the incumbent users both use 20 subcarriers, and we measure the number of guard subcarriers that is necessary between them to reach the amount of protection expected by the incumbent. Simulation results are reported in Fig. 5. According to the PSD based modeling of interference, a maximum of three subcarriers is needed to protect the CP-OFDM incumbent if the secondary utilizes the considered FB waveform. In reality, when the interference is rated through EVM measurements, the gains of using a FB waveform are much more limited. For example, if the CP-OFDM incumbent sets a maximum interference constraint of -50 dB, the guard band must be 407 subcarriers wide for a CP-OFDM secondary and 301 subcarriers wide for a FB secondary. This is an interesting gain of 26% but is much less than what is expected based on the PSD, which predicts a relative gain of 1 -3 216 = 99%.

B. Optimal Power Allocation for Secondary Users

A second way to study the coexistence scenarios analyzed in this paper in the literature is to follow the approach of [START_REF] Shaat | Computationally efficient power allocation algorithm in multicarrier-based cognitive radio networks: OFDM and FBMC systems[END_REF]. In that work, the secondary user aims to optimize its capacity while respecting its total power budget and the maximum interference that is allowed by the incumbent.

Here, we update the results presented in [START_REF] Shaat | Computationally efficient power allocation algorithm in multicarrier-based cognitive radio networks: OFDM and FBMC systems[END_REF]. We consider a coexistence scenario where 60 subcarriers are available, an incumbent occupies subcarriers 20 to 39 and a secondary tries to exploit subcarriers 0 to 19 and 40 to 59. The secondary uses the PI-algorithm presented in [START_REF] Shaat | Computationally efficient power allocation algorithm in multicarrier-based cognitive radio networks: OFDM and FBMC systems[END_REF] to distribute its total power of 1W on its subcarriers. We show in Fig. 6 the capacity it achieves as a function of the interference that is tolerated by the incumbent CP-OFDM user. We compare the results obtained when interference is computed through PSD or EVM measurements. In Fig. 6, the gains of using an FB waveform at the secondary seem significant when the PSDbased measurements of interference are used as in [START_REF] Shaat | Computationally efficient power allocation algorithm in multicarrier-based cognitive radio networks: OFDM and FBMC systems[END_REF], but vanish when the actual EVM measurements of interference are considered. Note that differences between EVM-based and PSD-based results in the case where both the secondary and the incumbent users utilize CP-OFDM are well explained in [START_REF] Medjahdi | Interference tables: a useful model for interference analysis in asynchronous multicarrier transmission[END_REF].

V. CONCLUSION In this paper, we explained that the metric commonly used to rate the interference between FB waveforms and CP-OFDM incumbent users does not correspond to the actual interference experienced by the CP-OFDM receiver after the demodulation of the incoming signal. We showed that because the FB signals are not orthogonal with the receiver of the CP-OFDM incumbent receiver, the spectral properties of the former are lost. We demonstrated that the actual gains of using FB based waveforms for coexistence with CP-OFDM are very limited. Based on our observations, we proposed to define new masks based on EVM instead of PSD as it is currently done in the literature in order to efficiently protect the incumbent users. Furthermore, we updated results available in the literature that were built on the PSD-based model and therefore were predicting inaccurate results.

Future work will therefore be twofold. On the one hand, we will develop our analysis of the studied coexistence scenario to find new ways of protecting incumbent CP-OFDM users. The impact of FB interference on synchronization and channel estimation operations performed at the CP-OFDM incumbent receiver should also be considered. On the other hand, we will build a demonstrator to experimentally validate the observations made in this paper and convince the community that FB waveforms proposed so far fail to protect incumbent CP-OFDM users as much as is expected in the literature.

Fig. 1 :

 1 Fig. 1: Considered coexistence scenario (a) Coexistence layout : the signals of the secondary and the incumbent sum up and are passed through the demodulator of the CP-OFDM incumbent receiver. (b) Spectral representation : the incumbent and secondary systems coexist in the same spectral band, and each one is assigned a different subset of subcarriers.

Fig. 2 :

 2 Fig. 2: Truncation of the FB filter operated by the CP-OFDM receiver on the receiving window corresponding to a single CP-OFDM symbol.

Fig. 3 :

 3 Fig. 3: Comparison of the PSD of the FB signal before and after passing through the CP-OFDM demodulator. The CP-OFDM receive operations destroy the good spectral shape of the FB signal, which becomes almost as poorly spectrally localized as an OFDM signal.

0Fig. 4 :

 4 Fig.4: Comparison of values of interference caused by FB secondary onto CP-OFDM primary when using either PSD or EVM measurements.

Fig. 5 :

 5 Fig. 5: Number of guard subcarriers necessary between the secondary and the incumbent as a function of the interference constraint set by the incumbent.

Fig. 6 :

 6 Fig.6: Maximum capacity achieved by the secondary user with the PI-algorithm[START_REF] Shaat | Computationally efficient power allocation algorithm in multicarrier-based cognitive radio networks: OFDM and FBMC systems[END_REF]. Gains obtained with the use of FB waveforms is significantly decreased when actual EVM values of interference are used instead of the flawed PSDbased approach.

ACKNOWLEDGEMENT

This work was partially funded through French National Research Agency (ANR) project ACCENT5 with grant agreement code: ANR-14-CE28-0026-02. The authors also extend their sincere appreciation to Dr. Musbah Shaat, Associate Researcher at CTTC research center (Spain), for his valuable support and help with the PI-algorithm which enabled us to present comparative results in this paper.


Philippe Helluy

Conservative scheme for two-fluid compressible flows

without pressure oscillations

Conservative scheme for two-fluid compressible flows without pressure oscillations Helluy, Philippe (joint work with Jung, Jonathan) Compressible two-fluid flows are difficult to numerically simulate. Indeed, classic conservative finite volume schemes do not preserve the velocity and pressure equilibrium at the two-fluid interface. This leads to oscillations, lack of precision and even, in some liquid-gas simulations, to the crash of the computation. Several cures have been proposed to obtain better schemes (see [START_REF] Bachmann | Random sampling remap for compressible two-phase flows[END_REF] and included references). The resulting schemes are generally not conservative. Based on ideas of [START_REF] Chalons | Computing material fronts with a Lagrange-Projection approach[END_REF], we propose a new Lagrange-Projection scheme. The projection step is based on a random sampling strategy at the interface. The scheme has the following properties: it preserves constant velocity and pressure at the two-fluid interface, it preserves a perfectly sharp interface and it is fully conservative (in a statistical sense). The scheme can be extended to higher space dimensions through Strang dimensional splitting. Finally, it is very simple to implement and thus well adapted to massively parallel GPU computations.

We are interested in the numerical resolution of the following system of partial differential equations, modeling a liquid-gas compressible flow (1)

∂ t W + ∂ x F (W ) = 0,
where W = (ρ, ρu, ρv, ρE, ρϕ) T , F (W ) = (ρu, ρu 2 + p, ρuv, (ρE + p)u, ρuϕ) T .

The unknowns are the density ρ, the two components of the velocity u, v, the internal energy e and the mass fraction of gas ϕ. The unknowns depend on the space variables x and on the time variable t. The total energy E is the sum of the internal energy and the kinetic energy E = e+(u 2 +v 2 )/2.The pressure p of the twofluid medium is a function of the other thermodynamical parameters p = p(ρ, e, ϕ).

We consider a stiffened gas pressure law p(ρ, e, ϕ) = (γ(ϕ)-1)ρe-γ(ϕ)π(ϕ),where γ and π are given functions of the mass fraction ϕ, and γ(ϕ) > 1.

At the initial time, the mass fraction ϕ(x, y, 0) = 1 if the point (x, y) is in the gas region and ϕ(x, y, 0) = 0 if the point (x, y) is in the liquid region. The mass fraction is transported with the flow, which implies that for any time t > 0, ϕ(x, y, t) can take only the two values 0 or 1. However, classic numerical schemes generally produce an artificial diffusion of the mass fraction, and in the numerical approximation we may observe 1 > ϕ > 0. In classic conservative schemes, the artificial mixing zone implies a loss of the velocity and pressure equilibrium at the interface.

We construct a better numerical scheme for solving [START_REF] Bachmann | Random sampling remap for compressible two-phase flows[END_REF]. We consider a sequence of time t n , n ∈ N such that the time step τ n = t n+1 -t n > 0. We consider also a space step h. We define the cell centers by x i = ih. The cell C i is the interval ]x i-1/2 , x i+1/2 [. We look for an approximationW n i W (x i , t n ). Each time step of the scheme is made of two stages: an Arbitrary Lagrangian Eulerian (ALE) step and a Projection step. ALE stage. In the first stage, we allow the cell boundaries x i+1/2 to move at a velocity ξ n i+1/2 . At the end of the first stage, the cell boundary is

x n+1,- i+1/2 = x i+1/2 + τ n ξ n i+1/2 .

Integrating the conservation law (1) on the moving cells, we obtain the following finite volume approximation

h n+1,- i W n+1,- i -hW n i + τ n (F n i+1/2 -F n i-1/2 ) = 0. The new size of cell i is given by h n+1,- i = x n+1,- i+1/2 -x n+1,- i-1/2 = h + τ n (ξ n i+1/2 -ξ n i-1/2
). The numerical flux is of the form

F n i+1/2 = F (W n i+1/2 ) -ξ n i+1/2 W n i+1/2 . The intermediate state W n
i+1/2 is obtained by the resolution of a Riemann problem. More precisely, we consider the entropy solution of

∂ t V + ∂ x F (V ) = 0, V (x, 0) = V L if x < 0, V R if x > 0, which is denoted by R(V L , V R , x/t) = V (x, t). The intermediate state is then W n i+1/2 = R(W n i , W n i+1 , ξ n i+1/2
). In practice, R can also be an approximate Riemann solver.

Finally, the interface velocity is defined by

(2) ξ n i+1/2 = u n i+1/2 if (ϕ n i -1/2)(ϕ n i+1 -1/2) < 0, 0 else.
The numerical flux is thus a classic Godunov flux in the pure fluid. It is a Lagrangian numerical flux at the two-fluid interface. Projection step. The second stage of the time step is needed for returning to the initial mesh. We have to compute on the cells C i of the initial mesh the averages of W n+1,- i , defined on the moved cells C n+1,- i =]x n+1,- i-1/2 , x n+1,- i+1/2 [. Instead of a standard integral averaging method, we rather consider a random sampling averaging process. We consider a pseudo random sequence ω n ∈ [0, 1[ and we perform the following sampling

(3) W n+1 i =      W n+1,- i-1 , if ω n < ξ n i-1/2 τn h , W n+1,- i , if ξ n i-1/2 τn h ≤ ω n ≤ 1 + ξ n i+1/2 τn h , W n+1,- i+1 , if ω n > 1 + ξ n i+1/2 τn h .
A good choice for the pseudo-random sequence ω n is the (k 1 , k 2 ) van der Corput sequence. In practice, we consider the (5, 3) van der Corput sequence. We can extend the scheme to higher dimensions with dimensional splitting (more details in [START_REF] Helluy | OpenCL numerical simulations of two-fluid compressible flows with a 2D random choice method[END_REF]). It is remarkable that the same random number can be used for one time step in the x and y directions. We present in Figure 1 the results of a two-dimensional shock-droplet GPU simulation. We observe that we are able to capture a sharp interface and small Kelvin-Helmholtz vortices. The numerical noise is moderate, despite the random nature of the scheme.

Figure 1 .

 1 Figure 1. Shock-droplet simulation. Density plot.


Brian Trammell

David Gugelmann

Nevil Brownlee

Inline Data Integrity Signals for Passive Measurement

de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction and Related Work

Network measurement is the practice of examining traffic data in order to deduce or derive information about the properties of the measured network. This traffic can either be generated for the purposes of measurement (in active measurement), or can be generated by the network's users (in passive measurement). In all cases, the measurement infrastructure itself injects some error or bias into the measurement, such that interpretation of the results must take care to account for this error. This fact is too often ignored in the network measurement literature.

Measurement studies can use control traffic with known parameters to provide ground truth, whether part of the protocol for active measurements, or injected alongside the measured traffic for passive measurements. In all cases, the data produced are a function of the actual traffic measured, the mismatch between the assumed and actual behavior of the network, and inaccuracies injected by design or implementation flaws in measurement tools.

Even the simplest of active measurement approaches, the venerable ping utility, is not immune from this conflict between the ideal and the real. For example, Pelsser et al [START_REF] Pelsser | From Paris to Tokyo: On the suitability of ping to Measure Latency[END_REF] recently found that two-way delay measured across a link by ping can have significant dependencies on the flow identifier, an artifact of in-path load balancing.

It is harder to control for these errors in passive measurement than active measurement, and the mismatch between real and assumed behavior of the measurement tools plays a greater role. For example, Hofstede et al [START_REF] Hofstede | Measurement Artifacts in NetFlow Data[END_REF] characterized inaccuracy in timing and TCP flags in NetFlow data in three commercial flow meters. Cunha et al [START_REF] Cunha | Uncovering artifacts of flow measurement tools[END_REF] detailed timing errors in publicly available Abilene and GEANT flow traces injected by Juniper's J-Flow. An author of the present work [START_REF] Trammell | Peeling Away Timing Error in NetFlow Data[END_REF] even found timing error inherent in the design of Cisco's Netflow Version 9 protocol. Error injected by passive measurement design, implementation, and deployment issues was treated systematically by Kögel [START_REF] Kögel | One-way delay measurement based on flow data: Quantification and compensation of errors by exporter profiling[END_REF], who advocated exporter profiling to correct for these errors post-measurement.

In this paper, we consider the QoF (pronounced "quaff") TCP-aware flow meter, which leverages the flexibility of the IPFIX [START_REF] Claise | Specification of the IP Flow Information Export (IPFIX) Protocol for the Exchange of IP Flow Information[END_REF] protocol to export perflow statistics relevant to TCP performance. QoF was developed to allow per-flow observation of TCP flows, for operational troubleshooting as well as for research purposes. As such, it requires precise timing for the estimation of TCP roundtrip time (RTT) as well as very low or no observation loss -packets which were delivered, but not observed. Because QoF is a purely passive flow meter, a selfvalidation approach using injected traffic is not an option. Therefore, we sought to observe signals present in the data in order to increase our confidence about the suitability of the data for the measurement techniques employed by QoF.

We instrumented QoF's data structures for signaling detected gaps in the input data in order to report on observation loss. The literature on the effects of packet sampling treats the problem of known random observation loss on the fidelity of flow data [START_REF] Brauckhoff | Impact of Packet Sampling on Anomaly Detection Metrics[END_REF][START_REF] Zseby | Packet Sampling for Flow Accounting: Challenges and Limitations[END_REF]; a key contribution of the present work, in essence, is the export of information about the unknown, not necessarily random sampling attributable to a measurement infrastructure, and export of this information along with measured data, so that it can be used in subsequent analysis. While this information is often available in the logs of flow meters and analysis tools, QoF's approach of providing information on loss in-band greatly increases its usefulness.

In this paper, we briefly introduce QoF and the techniques it uses for passive TCP performance measurement, as background for understanding the data integrity checking approach it uses, in section 2. We then detail the methodology for observation loss and timestamp frequency variance measurement in section 3, and apply it to several data sets and network observation points in 5. We note that the tool is particularly applicable to the verification of packet and flow data quality in research applications.

Background

QoF is a fork of the CERT Network Situational Awareness Group's YAF (Yet Another Flowmeter) [START_REF] Inacio | Yaf: Yet another flowmeter[END_REF]. Our fork of the code removes all payload inspection and export facilities from YAF to increase performance and reduce end-user privacy risk, and replaces the packet capture code with WAND's libtrace [START_REF] Alcock | Libtrace: a packet capture and analysis library[END_REF] to add flexibility and accelerate capture with general-purpose network interfaces. The overall goal is to produce a flow meter operating on unsampled packet header observations which is useful for both performance research and operational purposes. Performance is an explicit design goal, so we avoid techniques which marginally improve accuracy at the cost of significant analysis or state requirements at the metering process. Additional functionality in QoF includes passive RTT and transport loss event estimation. As round-trip time (RTT) is a basic parameter of all TCP connections, passive measurement of RTT is a fairly well studied area. QoF uses an approach based on previous work in the area [START_REF] Veal | New methods for passive estimation of tcp roundtrip times[END_REF][START_REF] Mellia | Passive analysis of tcp anomalies[END_REF].

As in figure 1(a), RTT is estimated by adding the interval between seeing a sequence number and the corresponding ACK RT T f wd to the interval between seeing the sequence number in the reverse direction and the corresponding ACK RT T rev . Since many flows are mainly unidirectional, QoF also uses intervals between a TCP timestamp value on a pure ACK and its corresponding echo as well. In contrast to the method described in [START_REF] Strowes | Passively measuring tcp round-trip times[END_REF], timestamp echoes are considered only if they would decrease or keep equal the estimate for the component on which they were measured; this simple heuristic serves to minimize delay, especially that due to an application not having data to send.

QoF estimates transport loss -packets that did not arrive at the receiver -by observing the sequence of TCP sequence numbers. QoF tracks the initial sequence number as well as the next-expected sequence number in each direction of a flow. Since gaps in the sequence number space should be relatively rareoccurring transiently and refilled within an RTT in most cases -the module tracks gaps in a gap stack per flow direction.

Each unseen range in sequence number space is kept in descending order of sequence number. When a segment is observed that begins beyond the next expected sequence number, an out-of-order segment is signaled and the resulting gap is pushed onto the gap stack. Conversely, when a segment is observed that does not advance the sequence number, it is compared with the gaps in the gap stack, most recent gap first. If covering a gap, the gap is filled; otherwise, a retransmission is signaled.

If a loss occurs in the path after the observation point (Observer A in figure 1(b)), the observer will see the retransmitted segment(s) up to one RTT after the loss occurred. If the loss occurs in the path before the observation point (Observer B), the observer will see a gap in sequence number space immediately after the loss, but not observe a retransmission; the size of the gap in the sequence number space is an indication of how many segments were lost. QoF handles spurious retransmissions in part by grouping transport loss indications into RTT-sized windows.

These two features complement each other, and assume the observation of every packet in each flow in order to work properly: transport loss requires the whole sequence of sequence numbers, and RTT estimates on missing ACKs and timestamp echos would be erroneously long. We therefore sought indications in the measured data that this assumption holds.

Observation Loss Methodology

In contrast to transport loss, observation loss occurs when a packet arrives at the receiver but is not seen by the observation point. We use the gap stack to measure observation loss as well as transport loss. Gaps which are not filled before they are pushed off the stack by newer gaps, or that are still in the gap stack when a flow completes, represent portions of the sequence number space that we presume the TCP receiver saw, since progress through sequence number space continued, but which were not observed by QoF. QoF exports information about observation loss for each direction of each TCP flow.

Packet capture devices and libraries often make available information about how many packets were dropped -partially observed or inferred but not captured -on a per-interface basis. However, this is not the only type of observation loss. For example, packet capture based on packet replication at a switch will fail to observe packets the switch fails to forward down the span port; our method will count this as observation loss as well.

Though this measurement is admittedly rather obvious, the innovation of counting observation loss on a per-flow basis as opposed to a per-interface basis also allows certain analyses to proceed even in the face of unobserved packets, and to investigate any dependencies on packet properties in observation loss. Though we can only caluclate observation loss for TCP flows, we assume that most observation loss processes are transport protocol independent, and can therefore extrapolate observation loss at a given time on a given path for coincident non-TCP traffic.

We note that observation loss measurement using a gap stack is subject to overcounting in three specific conditions. First, the gap stack has a fixed size per flow, so particularly "frothy" flows -those with many reorderings -may have observation loss overcounted, if a gap falls off the end of the stack before it would have been filled. QoF's compiled-in default gapstack size is 8, chosen empirically through analysis a set of test traces. Overcounting due to froth can be reduced by increasing this, at the expense of memory efficiency.

Second, if a gap would be filled after the flow is expired by idle timeout or the observation of a FIN or RST segment, that gap will be counted as observation loss as well. We consider the performance of this structure to be worth the potential overcounting; future work involves additional tweaks to the algorithm to minimize overcounting while maintaining performance. We explore this further in section 5.2.

Additionally, QoF's decoder drops any packet it cannot succeessfully decode, and is paranoid about what it accepts, a legacy of its heritage as a security monitoring tool. This can lead to observation loss caused by QoF itself. QoF also uses a fixed capture length per packet, allowing it to disregard octets it will never use, thereby improving its performance. However, packets where the layer 2, IP, and TCP headers together are longer than the compile-time default of 96 octets, or read from capture files with a shorter snaplen, will lead to observation loss due to such decode failures. This is possible especially in cases of excessive IPv6 encapsulation and/or extension headers. As with the gapstack size, observation loss on long packets can be reduced by increasing the compile-time constant, at the expense of I/O performance.

Evaluation

To evaluate the utility of per-flow loss counting to detect impaired measurement, we added detuning functionality to QoF to model packet loss and packet delay in the measurement path 3 . Loss and delay due to residence in a bottleneck queue are modeled using a leaky bucket with a specified size and drain rate. Additionally, we model non-queue-related loss with a test of a specified loss probability against a linearly smoothed uniform random die, and non-queuerelated delay using a linearly smoothed uniform delay generator. Since QoF is built under the assumption that packet timestamps are monotonic strictly not decreasing, all delay imparted by the detune module is clamped to a minimum value in order to avoid reordering packets.

We then tested against a publicly available trace from the WIDE MAWI Working Group Traffic Archive4 , taken from a 150Mb/s transpacific link over a three hour period on 30 March 2012, containing 386.2 million packets. The peak flow concurrency [START_REF] Trammell | On Flow Concurrency in the Internet and its Implications for Capacity Sharing[END_REF] is 307.6 thousand concurrent flows (with 63s idle and 300s active timeouts). Reflecting the fact that the trace is taken from a backbone, 72.6% of observed flows are one-way due to asymmetric routing5 . We assume, but cannot confirm, that actual observation loss in the MAWI traces is negligble.

This evaluation shows, unsurprisingly, that there is a strong relationship between the proportion of flows with reported observation loss and the induced packet drop rate. The function of this relationship is determined by the type of loss induced. For leaky bucket loss with induced packet loss rate varying from 0.00289 to 0.138 and a queue length of 100ms, this function is linear (slope = The dominance of short flows in Internet traffic [START_REF] Trammell | On Flow Concurrency in the Internet and its Implications for Capacity Sharing[END_REF] means that packet loss will lead to unobserved flows, as well. In this evaluation, for each flow with measured observation loss, about four flows went unobserved.

The queue modeled by leaky bucket loss also induces error in packet timings, which changes QoF's estimation of round trip times: the median of the minimum estimated RTT of flows in the MAWI data set is 101ms; this shifts to 110ms using an induced drop rate of 0.302 (i.e., on the order of the loss seen in the Auckland data set treated in section 5.1 below). Since loss due to queueing or buffering (e.g., at a span port or on a capture device) is the dominant type of loss we assume, we note that this indicates that observing loss in a data set also indicates that the timings should not be trusted in further analysis.

Findings

Here we apply QoF's per-flow loss reporting features to ongoing data collection efforts on various research networks which have been used in recent publications. The goal here is to verify that the data source used in these works has appropriate observation loss characteristics for the analysis done.

Auckland and iatmon

We first applied this methodology to an observation point at the University of Auckland, implemented using a span port on inbound and outbound traffic at the campus border. During the daytime traffic peak at about 11:00 local time (12:00 UTC), we routinely found between 25% and 35% of observed flows reporting observation loss. On further investigation, we found that a span port originally configured to forward approximately 100 Mb/s of traffic in each direction over a Gigabit Ethernet interface was now being offered approximately 700 Mb/s of traffic in each direction at peak. Since 1400 Mb/s of traffic will not fit on a 1000 Mb/s link, the span port did not forward about a third of the packets to the observation point.

For the last two years the University's network operations group has been working towards upgrading its Internet-facing infrastructure, including the monitoring and measurement infrastructure. Unfortunately, this requires forklift upgrades to firewalls and boundary routers, and progress is slow. Taking a more positive view, we regard this as a chance to investigate the effects of high observation loss on traffic analysis.

In figure 2, we show the evolution of the dropped flow rate over a typical day (Tuesday, November 5, 2013). Flows experiencing observation loss during the peak hour have a significantly different shape from those which escape the span port lossless. Compare the subfigures in figure 3: while simply dropping the lossy flows in an analysis would lead to retaining about two-thirds of the flows, most of the long-duration, high-rate traffic, and therefore most of the packets, would be missing.

In [START_REF] Brownlee | One-way Traffic Monitoring with iatmon[END_REF], we monitored one-way traffic using iatmon, which measures interarrival time for packets from unsolicited trafic, and classifies such flows into ten groups based on their IAT distributions. That paper used data from the UCSD Network Telescope, i.e. header-trace data for only one-way flows.

However, we also run iatmon at Auckland using the same observation point as for our QoF work. At Auckland, iatmon filters out the one-way flows from the total traffic -one-way traffic accounts for about 3.5% of the observed packets but about 35% of the observed flows. iatmon's 'flow group' classification is fairly coarse-grained, relying only on the overall features of the IAT distributions. so 00:00:00 02:00:00 04:00:00 06:00:00 08:00:00 10:00:00 12:00:00 14:00:00 16:00:00 18:00:00 20:00:00 22:00:00 time of day it should not be affected by our high observation loss rate. However, one-way flows with only a few packets -say 10 or fewer -may not be observed, thereby distorting the distribution of flows between groups.

Horizon Extender

We then applied this methodology to HTTP traces captured at the border router of an university network. Similar to the Auckland setup, packets from the 10 Gb/s inbound and 10 Gb/s outbound link of the main router are forwarded via a single 10 Gb/s span port to the capturing device. A Myricom 10G network adapter in combination with tcpdump is used for traffic capturing. Packets are filtered before being written to disk. The network data analyzed in the following cover IPv4 TCP port 80 traffic of clients in a selected subnetwork of the university that have been recorded during 5 work days and one weekend in late 2012.

The total IP-level raw size of the analyzed traces is 1.2 TB of incoming and 43 GB of outgoing traffic, which correspond to 15M TCP streams accounting for 1.2 TB6 of incoming and 23 GB of outgoing TCP payload data. The Myricom driver on the capturing device reported no packet loss due to ring-buffer overflow, but since the 10 Gb/s up-and downstream traffic are mirrored over a single 10 Gb/s span port, packets could have been lost at the router. Here we use QoF to show that this was most likely not the case, and compare results to logs from the Bro IDS [START_REF] Paxson | Bro: a system for detecting network intruders in real-time[END_REF].

The recorded traces have been used to evaluate Horizon Extender [START_REF] Gugelmann | Horizon Extender: Long-term Preservation of Data Leakage Evidence in Web Traffic[END_REF], an approach to archive HTTP data for data leakage investigations. Because it is impossible to fully reconstruct HTTP request and response streams from TCP streams with packet loss, it was important for the evaluation that packet traces of high quality were available, that is, with as little observation loss as possible.

Figure 4 shows the average number of flows and flows with detected observation loss as observed by QoF. The rate of such flows is on the order of 2 × 10 -4 overall.

Analyzing the number of distinct clients and servers for which observation loss was observed in more detail, we found on a single weekday that while we observe on average 5 ± 6 lossy flows per 15 minute time bin, only 3 ± 3 clients and 4±4 servers are affected per time bin. In other words, certain paths between clients and servers are more likely to be affected by observation loss.

We then compared this observation loss rate to the byte loss rate reported by the TCP stream reconstruction engine in Bro, which analyzes complete packet payloads, and assembles them into full streams, as opposed to analyzing just headers as QoF does. We note that Bro reported missing bytes in on the order 10 -5 of the flows it counted 7 . Upon further investigation, we found potential faults in Bro that would lead to undercounting of observation loss, and reported these to the Bro mailing list; a quantification of this undercounting and correction thereof are subjects for future work.

While QoF in this configuration required 29 s to analyze 31 GB of traffic8 , Bro required 730 s in its default configuration and 92 s in bare mode with the connection analyzer only; i.e., QoF is 25 times faster than Bro's default configuration, and still more than three times faster than bare mode, respectively.

In summary, as even a lossy flow rate on the order to 2 per 10000 is acceptable for our analysis of Horizon Extender, we consider our analysis validated by this check. Further, the lack of any evidence of large groups of lossy flows, as with the leaky bucket used in the evaluation, indicates that the span port from which we captured traffic was never significantly overloaded.

Conclusions

In this work, we have shown the value of inline data integrity reporting for passive measurement, in the context of the QoF TCP-aware flow metering tool. Presently ongoing work includes the application of QoF to other data sets, and further investigation of other signals available in passively observed TCP traffic, e.g. timestamp frequencies from many TCP stacks, which provide an external timing reference for isolating timing jitter at observation points.

Fig. 1 .

 1 Fig. 1. Effects of observer placement on TCP observations

Fig. 2 .

 2 Fig. 2. Daily seasonality of flow loss (solid line) and observed data rate (dashed) on a typical day at Auckland

Fig. 3 .

 3 Fig. 3. Packets by mean data rate and duration of flow during a peak hour at Auckland: note that observation loss affects long-duration, high-rate flows disproportionately

Fig. 4 .

 4 Fig. 4. Average daily seasonality of flows with detected observation loss (solid line) and total number of flows (dashed line) as observed by QoF.

As these features are only useful in the context of this evaluation, they are only enabled in QoF when the --enable-detune flag is given to QoF's build configuration script.

http://mawi.wide.ad.jp/mawi/

While QoF is a biflow meter, integrity checking does not require observation of both sides of the flow to work properly, so this evaluation treats each uniflow in a biflow separately.

Since HTTP is highly asymmetric, incoming traffic is dominated by large flows, so IP layer overhead is negligible.

For comparisons to Bro, we set QoF's active timeout to 1 hour (as opposed to the default, 5 minutes) to reduce timeout-based overcounting, based on our model of potential loss overcounting in QoF (see section 3).

To provide the required data rate for this experiment, the packets have been read from a pcap file that was stored on a ramfs.

The code used in writing this paper is in the albula branch, while the master branch will always contain a stable version of the most recent feature set.

Future developments in QoF can be followed in source form: QoF is available under the GNU General Public License, and is under active development. The latest version is always available from http://github.com/britram/qof 9 .

Acknowledgments

This work was materially supported by the European Commission though the Seventh Framework Grant Agreement mPlane (FP7-318627); no endorsement of the work by the Commission is implied. Many thanks to Shane Alcock, Bernhard Ager, Jeff Boote, Mirja Kühlewind, and Jinyao Yan for the fruitful discussions during QoF's development; to Emily Sarneso, Chris Inacio, and the CERT Network Situational Awareness Group for further developing YAF, on which QoF is based; and to ITS at the University of Auckland for one of the data sets used in the development of this study.