On ISI cancellation in MIMO-ML detection using FBMC/QAM modulation R. Zakaria, D. Le Ruyet, Y. Medjahdi CEDRIC/LAETITIA Laboratory, CNAM. 292 rue Saint Martin, 75141, Paris, France. {rostom.zakaria, didier.le_ruyet, yahia.medjahdi}@cnam.fr

Abstract—In this paper, the filter-bank multicarrier (FBMC) system is considered, and we deal with the presence of the inherent intersymbol interference (ISI). Indeed, the transmitted data symbols in FBMC are OQAM (Offset QAM) modulated, and the received data symbols are corrupted by inherent interference terms which complicate the detection in a maximum likelihood (ML) sense in the spatial multiplexing scheme. Detection schemes with ISI estimation and cancellation are not always effective due to the error propagation. We propose in this paper to modify the conventional FBMC system by transmitting QAM data symbols instead of OQAM ones in order to reduce the inherent interference. Then, we propose a receiver based on ISI estimation and cancellation. A simple tentative detector is first used to attempt to cancel the ISI before applying the ML detection. We show by simulation that with the proposed FBMC/QAM system, the ISI cancellation is effective and the performance converges to the optimum one. Index Terms—Filter bank, FBMC, MIMO, Spatial multiplexing, FBMC/QAM, maximum likelihood, ML detection, ISI cancellation, interference.

I. I NTRODUCTION Orthogonal frequency division multiplexing with the cyclic prefix insertion (CP-OFDM) is the most widespread modulation among all the multicarrier modulations, and this thanks to its simplicity and its robustness against multipath fading using the cyclic prefix (CP). Nevertheless, this cyclic prefix causes a loss of spectral efficiency. Moreover, CP-OFDM spectrum is not compact due to the large sidelobe levels resulting from the rectangular pulse. This requires us to insert null subcarriers at frequency boundaries to avoid interferences with neighboring systems. This means also a loss of spectral efficiency. Filter-bank multicarrier (FBMC) was proposed [1] as an alternative solution to overcome the OFDM shortcomings. Indeed, unlike the OFDM, there is no need to insert any guard interval in FBMC. Furthermore, FBMC provides a higher spectral efficiency [2; 3; 4] since it uses a frequency well-localized pulse shaping. In FBMC, each subcarrier is modulated with an Offset Quadrature Amplitude Modulation (OQAM) which consists in transmitting real and imaginary samples with a shift of half the symbol period between them. The FBMC orthogonality condition is only considered in the real field [2]. Consequently, the data at the receiver side is carried only by the real (or imaginary) component of the signal. The imaginary (or real) part appears as an inherent interference term. Although the data is always orthogonal to

the interference term. But, the latter becomes a source of problems when combining FBMC with some MIMO techniques. In this work, the spatial data multiplexing (SDM) is considered, where Nt symbol streams are simultaneously transmitted over Nt transmit antennas in order to increase the data rate, and they are received by Nr receive antennas. Linear equalizations such as ZF (Zero Forcing) or MMSE (Minimum Mean Square Error) can be applied to FBMC as shown in [5]. However, maximum likelihood (ML) detection, which is supposed to offer a diversity order equal to the number of the receive antennas [6], cannot be applied in a straightforward manner with FBMC due to the presence of the intrinsic intersymbol interference. We have proposed in [7] a scheme based on an iterative ISI cancellation. The obtained performance was limited and far from the optimum due to the error propagation. In order to counteract the error propagation and make the cancellation scheme effective, the authors in [8] have shown that the interference power must be small and kept under a certain threshold. We will show, in this paper, that using QAM modulated symbols instead of OQAM allows us to reduce the inherent interference power and allows our proposed iterative scheme to converge to the optimum performance. The rest of the paper is structured as following. We give in section II a general description of the FBMC modulation and its system model. In section III, we first give a brief overview on the ISI cancellation by using tentative detector, and then we highlight the interference reduction in FBMC when QAM modulation is used. In section IV, we present and discuss the simulation results. Then, we finish by a conclusion in V. II. FBMC MODULATION AND SYSTEM MODEL In baseband discrete time model, we can write at the transmitter side the FBMC signal as follows [1]: s[m] =

M −1 X X

ak,n g[m − nM/2]ej

2πk D M (m− 2 )

ejφk,n , (1)

k=0 n∈Z

with M is an even number of subcarriers, g[m] is the prototype filter, D is the filter delay term, φk,n is an additional phase term, and the transmitted symbols ak,n are real-valued symbols. We can rewrite equation (1) in a simpler manner: s[m] =

M −1 X X k=0 n∈Z

ak,n gk,n [m],

(2)

TABLE I T RANSMULTIPLEXER IMPULSE RESPONSE ( MAIN PART ) USING PHYDYAS

FILTER

k−1

n−3 0.043j

n−2 0.125j

n−1 0.206j

n 0.239j

n+1 0.206j

n+2 0.125j

n+3 0.043j

k

−0.067j

0

−0.564j

1

0.564j

0

0.067j

k+1

0.043j

−0.125j

0.206j

−0.239j

−0.206j

−0.125j

0.043j

where gk,n [m] are the shifted versions of g[m] in time and frequency. When the transmitter and the receiver are connected back to back, the signal at the receiver output, in subchannel ”k” and at a time instant ”n”, is determined using the inner product of s[m] and gk,n [m]: rk,n = hs, gk,n i =

+∞ X

∗ s[m]gk,n [m]

m=−∞ +∞ X

=

M −1 X X

Ωk,n = Ω∗k,n ∪ {(k, n)}, the received signal can be, finally, written as [10]: rk,n = hk,n (ak,n + juk,n ) + bk,n ,

where hk,n and bk,n are, respectively, the channel coefficient and the noise term at subcarrier ”k” and time index ”n”. The noise bk,n is colored and given by:

(3)

bk,n =

∗ ak0 ,n0 gk0 ,n0 [m]gk,n [m].

The prototype filter g[m] is designed such that it satisfies the real orthogonality condition given by [2]: ( +∞ ) X ∗ Re gk0 ,n0 [m]gk,n [m] = δk,k0 δn,n0 . (4)

(9)

E{b

k0 ,n0

b∗k,n }

=σ

2

+∞ X

gk,n [m]gk∗0 ,n0 [m].

m=−∞

= σ 2 Γδk,δn .

Then, we can rewrite equation (3) as:

k0 6=k n0 6=n

|

∗ w[m]gk,n [m],

where w[m] is a white Gaussian noise with variance equals σ 2 . We can easily show that:

m=−∞

rk,n = ak,n +

+∞ X m=−∞

m=−∞ k=0 n∈Z

X X

+∞ X

ak0 ,n0

∗ gk0 ,n0 [m]gk,n [m] . (5)

m=−∞

{z

}

Ik,n :intrinsic interference

According to the real orthogonality given by (4), the term Ik,n in the equation above is pure imaginary. Then, we can write: rk,n = ak,n + juk,n ,

(8)

In the case of spatial data multiplexing configuration with Nt transmit antennas and Nr receive antennas, we transmit (i) (li) in each antenna ”i” a real symbol ak,n . We denote by hk,n the channel gain between the ith transmit antenna and the lth receive antenna. Hence, the received signal collected by the lth receive antenna is given by: (l)

rk,n =

(6)

where uk,n is a real-valued interference term. P+∞ ∗ Since the quantity m=−∞ gk0 ,n0 [m]gk,n [m] depends on 0 0 the distances δk = k − k and δn = n − n [9], let us denote it by the coefficient Γδk,δn . These coefficients Γδk,δn represent the transmultiplexer impulse response in the time-frequency domain and depend on the used prototype filter. Table I depicts the main coefficients Γδk,δn of the PHYDYAS prototype filter designed in [3]. The intrinsic interference Ik,n depends only on symbols transmitted in a restricted set Ω∗k,n of time-frequency positions around the considered position (k, n). Outside of this set, the coefficients Γδk,δn are zeros. Therefore, the intrinsic interference can be expressed as: X ak0 ,n0 Γδk,δn . Ik,n = (7) (k0 ,n0 )∈Ω∗ k,n

When passing through the radio channel and assuming that the channel is constant at least over the summation zone

(10)

Nt X

(li)

(i)

(i)

(l)

hk,n (ak,n + juk,n ) + bk,n ,

(11)

i=1 (l)

where bk,n is the noise contribution at the antenna ”l”. In a matrix formulation, we can write the received vector rk,n = (1) (N ) [rk,n , ..., rk,nr ]T as: rk,n = Hk,n (ak,n + juk,n ) + bk,n ,

(12) (li)

where Hk,n is an Nr × Nt matrix whose entries are hk,n , (1) (N ) (1) (N ) ak,n = [ak,n , ..., ak,nt ]T , uk,n = [uk,n , ..., uk,nt ]T , and (1) (N ) bk,n = [bk,n , ..., bk,nr ]T . III. I NTERFERENCE ESTIMATION AND CANCELLATION Basically, a receiver based on interference estimation and cancellation is composed by a tentative detector producing tentative decisions, and a main detector which considers that the channel is free of interference. The basic receiver scheme is depicted in Fig. 1. Obviously, the main detector outputs can be injected into the interference estimator block, and the operation of the interference cancellation can be repeated several times.

Interference variance

ISI power threshold

Fig. 1.

FBMC inherent Interference

Receiver scheme with ISI cancellation using tentative decisions 0 SNR

In this case, we obtain a receiver with iterative interference cancellation. We have proposed in [7] a receiver scheme based on interference estimation and cancellation. However, the BER performance was suboptimal and the interference cancellation was not effective. The challenge in ISI estimation and cancellation is mitigating the error propagation through iterations. In reference [8], the authors showed that a necessary condition to avoid the error propagation is to hold the ISI power under a certain threshold, i.e. the interference cancellation technique is effective only when the ISI power is small enough and less than a certain amount. Unfortunately, this threshold is depending on the signal-to-noise ratio (SNR) and it is not trivial to obtain a closed form of the ISI power threshold [8]. Nevertheless, the authors showed -in a specific example- that the threshold lowers with the SNR increase. In other words, removing completely the ISI effects becomes more difficult in high SNR. On the other hand, the interference in FBMC is inherent, thus it does not depend on the noise variance or SNR. Hence, the ISI power in FBMC is constant whatever the value of SNR. Consequently, the error propagation will appear from a certain amount of SNR when the ISI power threshold falls below the inherent interference power. Therefore, the interference cancellation is effective only when the SNR is less than a certain amount SN R0 for which the ISI power threshold is equal to the FBMC inherent interference. Fig. 2 depicts in qualitative manner the curves of the ISI power threshold and the inherent FBMC interference as a function of the SNR. Hence, if we decrease the inherent FBMC interference, then the value of SN R0 increases. In this section, we propose an FBMC configuration in order to reduce the inherent interference, thus, we increase the SNR limit SN R0 from which the error propagation is triggered. Since the FBMC real orthogonality is lost in SM-MIMO configuration from the point of view of maximum likelihood detection, we can freely abort the orthogonality condition and transmit complex QAM symbols at each one period T . Consequently, the expression of the transmitted signal s[m]

SNR0

Fig. 2. Qualitative representation of the curves of the ISI power threshold and the FBMC inherent interference.

given in (1) becomes

s[m] =

M −1 X X

sk,n g[m − nM ]ej

D 2πk M (m− 2 )

ejφk,2n , (13)

k=0 n∈Z

where sk,n are now complex QAM symbols. We note that the phase term φk,2n can also be aborted, as we dropped the real orthogonality condition. The transmultiplexer impulse response of the proposed FBMC/QAM system can be derived from the impulse response given in Table I by decimation by a factor of 2 in time axis. That yields the coefficients depicted in Table II. TABLE II T RANSMULTIPLEXER IMPULSE RESPONSE OF THE FBMC/QAM PHYDYAS FILTER n−1 0.125j 0 −0.125j

k−1 k k+1

n 0.239j 1 −0.239j

USING

n+1 0.125j 0 −0.125j

in [10] that, in FBMC/OQAM, we have PIt is shown 2 |Γ | = 2 where Γp,q are the coefficients given in p,q p,q Table I. Hence, the ISI variance in conventional FBMC (FBMC/OQAM) is X

2 σ ´ISI =

|Γp,q |2

(p,q)6=(0,0)

=

X

|Γp,q |2 − |Γ0,0 |2 = 1.

(14)

p,q

Therefore, in FBMC/OQAM, the power of ISI has the same value as the transmitted data variance. Whereas, the ISI variance in the proposed FBMC/QAM system, using PHYDYAS

filter, is given by 2 σISI =

X

|Γp,q |2

(p,2q)6=(0,0)

=1−

X

|Γp,q |2 = 0.1770.

(15)

p,2q+1

Hence, thanks to the proposed scheme we have reduced the ISI power to 17.7%. When we consider the MIMO spatial multiplexing, the matrix equation (12) can be adapted for FBMC/QAM as rk,n = Hk,n (sk,n + Ik,n ) + bk,n ,

and we assume perfect channel knowledge at the receiver side. The Rayleigh spatial sub-channels are spatially noncorrelated, and we use the Veh-A channel model to generate the channels. The complex data symbols are QPSK modulated for OFDM and the proposed FBMC/QAM. However, since the conventional FBMC uses OQAM modulation, each transmitted symbol, on each T /2, is 2-PAM modulated.

0

10

(16) −1

10

−2

10 BER

where sk,n is the vector of the QAM transmitted data, and Ik,n is the vector of the complex inherent interference. We opted to use the MMSE equalizer as the tentative detector, and the main one is the maximum likelihood detector. Hence, the MMSE equalizer provides tentative estimations of the data vectors sk,n . Basing on these tentative estimates, the interference canceler calculates an estimation of the interference vector Ik,n , and then its contribution is removed from the received vector rk,n . After that, the ML detection is applied. Fig. 3 depicts the principal blocks of the proposed Rec-ML receiver.

−3

10

MMSE Rec−ML iter=1 Rec−ML iter=2 Rec−ML iter=3 Rec−ML iter=4 Rec−ML iter=5 Genie−Aided Rec−ML

−4

10

0


௞,௡

MMSE Equalizer

Fig. 3.

Interference Cancellation

ML Detector

̂௞,௡

Block scheme of the Rec-ML receiver

IV. S IMULATION RESULTS In this section, we provide the simulation results of the proposed FBMC/QAM scheme compared to OFDM and also to FBMC/OQAM. The number of subcarriers is set to M = 512. According to the IEEE 802.16e standard [11], the CP duration is set to ∆ = M 8 = 64. The system performance is assessed in terms of bit-error rate (BER) as a function of the signal-tonoise ratio (SNR). For FBMC/OQAM and FBMC/QAM, the SNR is defined as Nt σs2 , σ2 where σs2 is the signal variance on each transmit antenna, and σ 2 is the noise variance on each receive antenna. However, for OFDM, the expression of the SNR is defined so that it takes into account the SNR loss due to the CP: SN R =

M + ∆ Nt σs2 . M σ2 Since the motivation of this work is to address the problem of optimum detection in spatial multiplexing with FBMC, we have considered the simple 2 × 2 spatial multiplexing scheme, SN R =

5

10

15 SNR (dB)

20

25

30

Fig. 4. Performance of Rec-ML receiver with FBMC/QAM system for 2 × 2 spatial multiplexing

We call the receiver based on iterative ISI estimation and cancellation ”Rec-ML” (for Recursive maximum likelihood). The ”Genie-Aided Rec-ML” performance is defined as the performance obtained by assuming a perfect interference estimation, i.e the exact transmitted symbols are involved to estimate the interference. Fig. 4 depicts the BER performance obtained of the proposed FBMC/QAM with the MMSE equalizer, which is our tentative detector. We also show in this figure the performance obtained using the RecML for different values of iterations and compare them to the optimum performance obtained with Genie-Aided. We notice that increasing the number of iterations or Rec-ML improves the BER performance, and the performance converge after 5 iterations, i.e. there is practically no improvement beyond 5 iterations. Hence, the Rec-ML performs correctly with 5 iterations. However, we observe a slight SNR loss less than 0.5 dB compared to the Genie-Aided performance. In Fig. 5 we compare the performance of FBMC/QAM and conventional FBMC using -for both of them- the RecML receiver. We also show the BER performance of the OFDM using ML detector. For FBMC/QAM, we present only the performance of MMSE equalizer (tentative detector) and the performance of the Rec-ML after 5 iterations. As for classical FBMC, we show the performance of MMSE and the performance of Rec-ML after 2 iterations, since it converges after only 2 iterations [7]. First of all, one can notice

must be sufficiently small and less than a certain threshold. Unfortunately, the intrinsic ISI, in conventional FBMC, has the same power as the desired symbols. In order to reduce the power of the inherent interference, we have proposed to transmit QAM-modulated symbols instead of OQAM ones. Thus, we reduced the interference variance to 17%. We have shown, by simulations, that the performance of the proposed system converges to the optimum after 5 iterations and exhibits the same performance as OFDM with MLD. We have also compared the FBMC/QAM system with the classical FBMC and shown that this latter suffers from error propagation effect, whereas the interference cancellation is effective with FBMC/QAM. R EFERENCES

0

10

−1

10

−2

BER

10

−3

10

FBMC MMSE FBMC Rec−ML FBMC/QAM MMSE FBMC/QAM Rec−ML OFDM−ML

−4

10

0

5

10

15 SNR (dB)

20

25

30

Fig. 5. Performance of Rec-ML receiver with FBMC and FBMC/QAM systems in 2 × 2 SM

that MMSE equalizer for FBMC/QAM exhibits worse BER performance compared to FBMC. This is explained by the fact that the inherent ISI term in FBMC/QAM is complex as the transmitted data symbols, whereas in FBMC the interference terms are pure imaginary and the data symbols are real-valued. However, the situation is different with Rec-ML receiver, we clearly notice that Rec-ML with conventional FBMC suffers from the error propagation effect and the BER performance converges to a suboptimal one, whereas the ISI cancellation is effective when combining Rec-ML with FBMC/QAM, and we obtain almost the same performance as OFDM-ML. V. C ONCLUSION In this paper, the association of the ML detection with the FBMC/MIMO system is considered. The presence of the inherent interference due to the FBMC modulation obstructs the implementation of the ML detection in a straightforward manner. To cope with this situation, we proposed a receiver scheme based on interference estimation and cancellation. However, we have shown that the ISI cancellation is effective only under some strict condition, which is that the ISI

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