On Spatial Data Multiplexing over Coded Filter-Bank Multicarrier with ML Detection R. Zakaria, D. Le Ruyet Electronics and Communications Laboratory, CNAM, 292 rue Saint Martin, 75141, Paris, France [email protected], [email protected]

Abstract—Multicarrier modulation and especially CP-OFDM is widely used nowadays in several radio communications. However, Filter Bank Multicarrier (FBMC) is a potential alternative to CP-OFDM since it does not require cyclic prefix and thus it has a higher spectral efficiency. One of the characteristics of FBMC is that it induces intrinsic interference among the transmitted data. The presence of this interference complicates the application of the MLD (Maximum Likelihood Detection) in the MIMO Spatial Data Multiplexing configuration which is known to improve the capacity. In this paper, we are interested in the encoded data case and we propose two receiver schemes. Both schemes are based on interference estimation and cancellation. The first one uses MMSE equalization in order to get an estimation of the intrinsic interference. The second one improves the interference estimation by exploiting the decided symbols around the considered timefrequency position. The performance of both approaches are assessed by simulation, and we show that the second approach exhibits either the same performance or better than OFDM.

I. I NTRODUCTION Orthogonal frequency division multiplexing with the cyclic prefix insertion (CP-OFDM) is the most widespread modulation among all the multicarrier modulations. This is because of its simplicity and especially its robustness against multipath fading thanks to the cyclic prefix (CP). Nevertheless, this technique causes a loss of spectral efficiency due to the CP. Furthermore, CP-OFDM spectrum is not compact due to the large sidelobe levels resulting from the rectangular pulse, what leads us to insert null subcarriers at frequency boundaries in order to avoid overlappings with neighboring systems. Hence it means a loss of spectral efficiency too. To avoid these drawbacks, filter bank multicarrier (FBMC) was proposed as an alternative approach to multicarrier OFDM [1]. In FBMC, there is no need to insert any guard interval and also it uses a time-frequency well-localized pulse shaping and hence it provides a higher spectral efficiency [2]. Each subcarrier is modulated with an Offset Quadrature Amplitude Modulation (OQAM) and the orthogonality conditions are considered in the real field [2]. Indeed, the data at the receiver side is carried only by the real (or imaginary) components of the signal, and the imaginary (or real) parts appear as interference terms. Although the data is always orthogonal to the interference term, this term of interference becomes a source of problems when combining FBMC with MIMO systems [4]. Some works were carried out to deal with this issue such as in [5, 6] for the Alamouti coding, and in [7] for the MMSE equalization in Spatial Data Multiplexing (SDM)

context. In [9], we have considered the SDM configuration with Maximum Likelihood (ML) detection in the uncoded data case. The obtained performance was far from the optimum due to the interference. In this work, we consider coded SDM and propose two schemes with ML detection based on interference cancellation, and the second one improves the interference estimation and cancellation by exploiting the channel code. The organization of the paper is as follows. In section II, we give a brief description of the FBMC modulation and we introduce the system model. In section III, we extend the system model to the multiple-antenna context, and then, the different proposed receiver schemes are detailed. Simulation results are presented and discussed in section IV. Finally, we finish by a conclusion. 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

2π

D

ak,n g[m − nM/2]ej 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

ak,n gk,n [m],

(2)

k=0 n∈Z

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

(3)

ak,n gk,n [m]gk∗′ ,n′ [m].

m=−∞ k=0 n∈Z

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

  (11) (1) h rk,n  .   k,n  .  =  ..  .   . (N ) (N 1) rk,nr hk,nr | {z } | 

rk,n

  (1)   (1)  (1N ) (1) n hk,n t ak,n + juk,n    k,n  .. ..   +  ..  . . .    .  (N N ) (N ) (N ) (N ) ak,nt + juk,nt nk,nr · · · hk,nr t {z }| {z } | {z } ··· .. .

ak,n +juk,n

Hk,n

Then, we can rewrite equation (3) as: rk,n = ak,n +

X X

ak′ ,n′

gk,n [m]gk∗′ ,n′ [m] . (6)

m=−∞

k′ 6=k n′ 6=n

|

+∞ X

{z

Ik,n :intrinsic interference

}

TABLE I T RANSMULTIPLEXER IMPULSE RESPONSE ( MAIN PART ) k0 − 1 k0 k0 + 1

n0 − 3 n0 − 2 0.043j −0.125j −0.067j 0 −0.043j −0.125j

n0 − 1 n0 n0 + 1 n0 + 2 −0.206j 0.239j 0.206j −0.125j 0.564j 1 0.564j 0 0.206j 0.239j −0.206j −0.125j

n0 + 3 −0.043j −0.067j 0.043j

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 ak′ ,n′ Γδk,δn . (8) Ik,n = (k′ ,n′ )∈Ω∗ k,n

When passing through the radio channel and assuming that the channel is constant at least over the summation zone Ωk,n = Ω∗k,n ∪ {(k, n)}, we can write the received signal as [12]: (9) rk,n = hk,n (ak,n + juk,n ) + nk,n , where hk,n and nk,n are, respectively, the channel coefficient and the noise term at subcarrier ”k” and time index ”n”. For the rest of the study, we consider equation (9) as the expression of the signal at the output of the demodulator. III. MIMO

WITH

FBMC

(j)

rk,n =

AND THE PROPOSED RECEIVERS

At the transmitter side, after channel coding, interleaving and mapping, data are demultiplexed onto Nt branches (corresponding to Nt antennas). Over each branch the data are

Nt X

(ji)

(i)

(i)

(j)

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

(10)

i=1

(7)

where uk,n is a real-valued P+∞interference term. ∗ Since the quantity m=−∞ gk,n [m]gk′ ,n′ [m] depends on ′ ′ the distances δk = k −k and δn = n −n [13], 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. For example, the table below depicts the main coefficients Γδk,δn of the PHYDYAS prototype filter designed in [3].

nk,n

sent to the FBMC modulator and then transmitted through the radio channel. At the receiver side, Nr antennas are used to collect the transmitted signals. At the j th receive antenna, the FBMC demodulated signal at a given time-frequency position (k, n) is expressed by:

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 ,

(5)

(ji)

where hk,n is the channel coefficient between the ith transmit and the j th receive antenna. Finally, the matrix formulation of the system can be expressed by the equation (5), and we write: (11) rk,n = Hk,n (ak,n + juk,n ) + nk,n , where Hk,n is an (Nr × Nt ) channel matrix. A. MMSE equalization The implementation of the MMSE equalization in the considered FBMC-MIMO context has been described in [7] where a virtually transmitted vector ck,n is considered instead of the effective one and defined as: ck,n = ak,n + juk,n .

(12)

The vector rk,n represents the input of the MMSE equalizer having as output the equalized virtually transmitted vector ˜k,n : c ˜k,n = GH c (13) k,n rk,n , where Gk,n is the equalization matrix based on the MMSE criterion given by [8]: 2 −1 Hk,n , Gk,n = (Hk,n HH k,n + σ0 INr )

(14)

with σ02 is the variance of the noise term, and INr is the ˜k,n (Nr × Nr ) identity matrix. Then, a real part retrieval of c ˜k,n . These equalized yields the real equalized data vector a symbols are, then, multiplexed one by one, soft demapped and deinterleaved before being decoded to recover the transmitted data bits. This scheme is the non-iterative MMSE receiver as referred in [7]. B. MMSE-ML receiver In this section, we propose a receiver proceeding by interference estimation and cancellation. This receiver is depicted on Fig. 1 (only in the dashed box). At the MMSE equalizer output (described above), an evaluation of these interference ˜k,n . terms are available by taking only the imaginary part of c This estimation is used to cancel the interference contribution

Fig. 1.

MMSE-ML scheme (in the dashed box) and Recursive ML scheme for FBMC in Nt × Nr Spatial Data Multiplexing

from the received vector rk,n . Then, we obtain a vector yk,n expressed as: ˜ k,n , yk,n = rk,n − jHk,n u ˜ k,n )) + nk,n , = Hk,n (ak,n + j(uk,n − u = Hk,n (ak,n + jǫk,n ) + nk,n ,

(15)

˜ k,n is the estimated interference vector given by where u ˜ k,n = Im{˜ ck,n }. The interference estimation error ǫk,n is u considered as an additional noise term. Its statistical parameters depend on the channel matrix Hk,n , since the reliability ˜ k,n depends also on the MMSE equalization matrix. of u Assuming almost perfect interference estimation (i.e. ǫk,n ≈ 0), we consider that the vector yk,n at the output of the interference canceler is free of interference. Thus, ignoring the presence of the term ǫk,n , we perform a simple conventional soft ML detector giving Log-Likelihood ratio (LLR) of the a posteriori probability (APP) of the encoded bits dl being +1 or -1. The LLR for the ML detector is defined for l = 1, ..., bNt as: µ ¶ P (dl = +1|yk,n ) , (16) LAP P (dl |yk,n ) = log P (dl = −1|yk,n ) where b is the number of bits that constitute the real symbol ak,n . Hence, in each subcarrier and half period Ts /2, we have bNt soft bits at the soft MLD output. By employing Bayes’ theorem and assuming statistical independence and equiprobability among the bits dl , the LLR can be written as [10, 11]: ! ÃP p(yk,n |d) d∈D+ k . (17) LAP P (dl |yk,n ) = log P d∈D− p(yk,n |d) k

The vector d contains the bits corresponding to the transmitted symbols ak,n over all the antennas, and the set D+ l (or ) contains all the vectors d having d = +1 (or d = −1). D− l l l

The likelihood density p(yk,n |d) is given by: ¡ ¢ exp − 2σ1 2 krk,n − Hk,n ak,n (d)k2 p(yk,n |d) = , (2πσ 2 )Nr

(18)

where ak,n (d) is the transmitted real vector associated to the bit-vector d. Substituting equation (18) in (17) and applying the Max-Log approximation, the LLR calculation is simplified by: 1 min krk,n − Hk,n ak,n (d)k2 2σ 2 d∈D− k 1 − 2 min krk,n − Hk,n ak,n (d)k2 . (19) 2σ d∈D+ k

LAP P (dl |yk,n ) =

The obtained soft information at the ML output should be multiplexed, deinterleaved and fed into the soft-input decoder to recover the transmitted information source bits. This receiver is referred by MMSE-ML since we combine an MMSE equalizer with an ML detector. We recall that this receiver is built assuming that yk,n is free of interference. But unfortunately, the error term in equation (15) is non-zero (ǫk,n 6= 0) and consequently it is clear that we cannot reach optimal bit-error-rate performance as if there is no interference. C. Recursive ML (Rec-ML) receiver To improve the performance, we propose in this section a recursive structure where we use the MMSE-ML outputs to perform a second interference estimation. This time, the intrinsic interference is estimated through the decided data bits (available at the MMSE-ML output) which are within the neighborhood of the considered frequency-time position (k, n). That means that all the symbol estimations in the set Ωk,n must be available, contrary to the MMSE-ML where the interference estimation is obtained immediately from the MMSE output. For this, the estimated data bits are encoded

with the same convolutional code used in the transmitter, interleaved, mapped and demultiplexed repeating exactly the same transmission operations to provide an estimation of ˆk,n , which serve to improve the the transmitted symbols a interference estimation since the information bits are encoded ˆk,n within and some errors will be corrected. Once all a Ωk,n are reconstructed, we can easily estimate the intrinsic interference according to equation (8): X ˆIk,n = ˆk′ ,n′ Γδk,δn , (20) a

TABLE II S IMULATION PARAMETERS

Complex modulation FFT size CP size in OFDM Convolutional code Sampling frequency

We remark that ǫk,n depends on the reliability of the estiˆk′ ,n′ around the considered position (k, n). If mated symbols a we denote by Pa the probability of getting a wrong estimate a ˆk,n , we can evaluate the variance of ǫk,n by: σǫ2 ≈

2 Pa d2min , Nt

(22)

where dmin is the smallest Euclidean distance between two different symbols aP k,n . This relationship is obtained by taking into account that |Γp,q |2 ≈ 1 [12]. It is clear (p,q)∈Ω∗ k,n that as long as Pa is non-zero, the variance of the residual interference σǫ2 is also non-zero. Therefore, there will be still a gap between Rec-ML performance and the optimum one. We can iterate this method until the performance convergence. Simulation results show that the convergence is obtained after the second iteration. IV. S IMULATION RESULTS In this section, we compare FBMC to CP-OFDM in 2x2 Spatial Data Multiplexing scheme over two classes of channels. Our objective is to test the proposed receiver schemes over a low and high frequency selective channels. For that purpose, we have chosen the Pedestrian-A and the Vehicular-A channel [14]. We should note that in both chosen channels, we have not considered the time selectivity. Moreover, we assume perfect channel knowledge at the receiver side. The system performance is assessed in terms of bit-error rate (BER) as a function of the signal-to-noise ratio per bit (Eb /N0 ). In the 2x2 SDM configuration, the four sub-channels are spatially non-correlated. The simulation parameters for FBMC and CP-OFDM are summarized in the table shown below. In CP-OFDM, we

Eb /N0 = Nr

SN R T +∆ , T Rs Rc log2 (N )

(23)

where T is the useful OFDM symbol duration, ∆ is the cyclic prefix duration, N is the modulation order, Rc is the channel coding rate, Rs is the space-time coding rate, and SN R is the Signal-to-Noise ratio. For FBMC, the expression of Eb /N0 is obtained by setting the CP size to zero (∆ = 0). Since the CP duration affects OFDM performance and in order to compare FBMC to the best OFDM performance case, we have chosen, in table II, the smallest possible CP size for each channel model. For receivers based on ML detection, we define the Genie-Aided performance as the fictional one obtained when the symbols serving to estimate the interference are identical to the transmitted ones (perfect interference estimation). In uncoded configuration, we can show that regardless the efficiency loss due to the CP, the Genie-Aided receiver outperforms CP-OFDM by about 1 dB. Hence, it holds interesting to compare its performance to the OFDM and the proposed receivers when using convolutional coding. Pedestrian−A 1024 Code(171,133)

0

10

MMSE MMSE−ML Rec−ML Genie−Aided Rec−ML CP−OFDM ∆=8

−1

10

−2

10

BER

(k ,n )∈Ωk,n

Vehicular-A QPSK 1024 32 (171, 133) 10 MHz

define Eb /N0 by:

(k′ ,n′ )∈Ω∗ k,n

Once this interference is estimated, its contribution is canceled again from the received vector rk,n , and then, we perform one time more the soft ML detection in order to improve the performance. The complete receiver scheme is depicted in Fig. 1. We should notice that the interference estimation block in this scheme induces a processing delay since the set Ωk,n contains some future positions. At the ML input, the error term ǫk,n of the interference estimation in (15) is expressed, this time, as:     X ˆk′ ,n′ ) . (21) ǫk,n = Im Γδk,δn (ak′ ,n′ − a  ′ ′ ∗ 

Pedestrian-A QPSK 1024 8 (171, 133) 10 MHz

−3

10

−4

10

−5

10

−6

10

0

5

10 Eb/N0 (dB)

15

20

Fig. 2. BER performance comparision between CP-OFDM and FBMC receivers in 2 × 2 MIMO case over Ped.-A channel

Our main objective in this work is the implementation of the ML detector in FBMC context and compare its performance to OFDM-ML. We have proposed two receivers with ML detection (MMSE-ML and Rec-ML). Fig. 2 captures the performance of CP-OFDM with ML and that of FBMC with all the proposed receivers (including MMSE) over the

Pedestrian-A channel. The curves show that the MMSE-ML scheme outperforms the MMSE equalizer, but the performance is still far from the CP-OFDM with ML. The gain obtained by MMSE-ML with respect to MMSE equalizer is about 2.5 dB at BER = 10−4 , whereas OFDM-ML provides a 5 dB SNR gain compared to MMSE. However, Rec-ML receiver exhibits almost the same performance as OFDM-ML. It is worth recalling that CP-OFDM performance is obtained with the smallest possible CP size (∆ = 8). Increasing ∆ yields a performance degradation for CP-OFDM, and thus, FBMC with Rec-ML receiver will outperform CP-OFDM. For example, as in IEEE 802.16e standard [15], if we set ∆ = T8 = 128, we obtain a degradation of about 0.48 dB. Regarding the Vehicular-A channel, Fig. 3 shows the performance of the different receivers in this propagation channel. Firstly, as in the Pedestrian-A channel case, we remark a considerable SNR gain is obtained by MMSE-ML receiver compared to MMSE equalizer, we have a gain of about 2 dB at BER = 10−4 . Secondly, we can observe clearly that the obtained performance of Rec-ML is better than that obtained with CP-OFDM from Eb /N0 = 6 dB, and tends to reach the Genie-Aided performance in high Eb /N0 regime. Vehicular−A 1024 Code(171,133)

0

10

MMSE MMSE−ML Rec−ML Genie−Aided Rec−ML CP−OFDM ∆=32

−1

10

−2

BER

10

−3

10

−4

10

−5

10

−6

10

0

2

4

6

8 Eb/N0 (dB)

10

12

14

16

Fig. 3. BER performance comparision between CP-OFDM and FBMC receivers in 2 × 2 MIMO case over Veh.-A channel

In both Pedestrian-A and Vehicular-A channel models, we notice that the Genie-Aided receiver exhibits better performance than OFDM-ML. This can be explained by the fact that FBMC uses a real constellation, while OFDM uses a complex constellation. However, the presence of the inherent interference in FBMC is an obstacle to reach this potential performance. Therefore, further investigations on interference estimation are needed to improve the obtained performance. V. C ONCLUSION In this paper we considered the association of the ML detection with the FBMC/MIMO system. The presence of the intrinsic interference due to the FBMC modulation obstructs the implementation of the ML detection in a straightforward

manner. To cope with this situation, we proposed two receiver schemes based on interference estimation and cancellation. The first scheme (MMSE-ML) estimates the interference by using the MMSE equalizer and taking the imaginary part of the equalized symbols. Once the interference contribution is removed from the received symbols, a classical ML detector is used to recover the transmitted data. We have shown that this detection method is suboptimal because of the assumption of perfect interference estimation (which is not the case) when performing ML detection. In order to improve the interference estimation, we have proposed a second scheme (Rec-ML) which exploits the MMSE-ML decoded outputs to obtain an estimation of the transmitted symbols and also to estimate again the intrinsic interference. This time, the interference estimation is done by considering the estimated transmitted symbols within the neighborhood of the considered position. Simulation results showed that the performance obtained with this last scheme is either similar or better than that obtained with CP-OFDM depending on the propagation channel. However, the drawback of this scheme is without any doubt its complexity. R EFERENCES [1] B. Le Foch, M. Alard and C. Berrou, ”Coded Orthogonal Frequency Division Multiplex”, Proceeding of the IEEE, Vol. 83, No. 6, Jun. 1995 [2] P. Siohan, C. Siclet, and N. Lacaille, ”Analysis and design of OFDM/OQAM systems based on filterbank theory”, IEEE Transactions on Signal Processing, vol. 50, no. 5, pp. 1170-1183, May 2002. [3] M. G. Bellanger, ”Specification and design of a prototype filter for filter bank based multicarrier transmission,” in Proc. IEEE ICASSP, pp. 24172420, Salt Lake City, USA, May 2001. [4] M. Payar´o, A. Pascual-Iserte and M. N´ajar, ”Performance comparison between FBMC and OFDM in MIMO systems under channel uncertainty,” European Wireless Conference (EW), pp.1023-1030, April 2010 [5] M. Renfors, T. Ihalainen, and T. H. Stitz, ”A block-Alamouti scheme for filter bank based multicarrier transmission,” European Wireless Conference (EW ’10), pp. 1038-1041, Lucca, Italy, April 2010. [6] C. L´el´e, P. Siohan, and R. Legouable, ”The Alamouti scheme with CDMA-OFDM/OQAM”, EURASIP Journal on Advances in Signal Processing, Vol. 2010, ID 703513. [7] M. El Tabach, J. P. Javaudin and M. Helard, ”Spatial data multiplexing over OFDM/OQAM modulations”, Proceedings of IEEE-ICC 2007, Glasgow, 24-28 June 2007, pp.4201-4206. [8] N. Kim, Y. Lee, and H. Park, ”Performance Analysis of MIMO System with Linear MMSE Receiver”, IEEE Transaction on wireless communications, Vol. 7, No. 11, November 2008. [9] R. Zakaria, D. Le Ruyet, and M. Bellanger, ”Maximum Likelihood Detection in spatial multiplexing with FBMC”, European Wireless Conference (EW ’10), pp. 1038-1041, April 2010. [10] D. Le Ruyet, T. Bertozzi, and B. Ozbek, ”Breadth first algorithms for APP detectors over MIMO channels”, IEEE International Conference on Communications ICC, vol. 2, pp. 926-930, Jun. 2004. [11] B.M. Hochwald and S.T. Brink, ”Acheiving Near-Capacity on a Multiple-Antennas Channel”, IEEE Trans. Commun., vol 51, pp.389399, Mar. 2003. [12] C. L´el´e, J.-P. Javaudin, R. Legouable, A. Skrzypczak, and P. Siohan, ”Estimation Methods for Preamble-Based OFDM/OQAM Modulations”,European Wireless ’07, April 2007. [13] R. Zakaria, D. Le Ruyet, ”A novel FBMC scheme for spatial multiplexing with maximum likelihood detection,” IEEE Inter. Symp. on Wireless Comm. Sys. (ISWCS) Conf., pp. 461-465, Sept. 2010. [14] ITU-R M.1225, ”Guidelines for evaluations of radio transmission technologies for IMT-2000,” 1997. [15] IEEE 802.16e-2005, ”IEEE Standard for Local and Metropolitan Area Networks Part 16,” 3GPP, 2006.