2011 8th International Symposium on Wireless Communication Systems, Aachen

Partial ISI Cancellation with Viterbi Detection in MIMO Filter-Bank Multicarrier Modulation R. Zakaria, D. Le Ruyet CEDRIC/LAETITIA Laboratory, CNAM. 292 rue Saint Martin, 75141, Paris, France. [email protected], [email protected]

Abstract—In this paper we deal with the presence of the intrinsic intersymbol interference (ISI) in the filter-bank multicarrier systems. Indeed, the received data symbols are accompanied by 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 a receiver based on a partial ISI cancellation followed by the Viterbi detector. A simple tentative detector is first used to cancel partially the ISI. We show that under some conditions the ISI cancellation is effective and the performance converges to the optimum one. 1 Index Terms—Filter bank, FBMC, MIMO, Spatial multiplexing, ISI cancellation, Viterbi detection, M-Algorithm.

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 technique causes a loss of spectral efficiency due to the cyclic prefix. Furthermore, CP-OFDM spectrum is not compact due to the large sidelobe levels resulting from the rectangular pulse. This leads us to insert null subcarriers at frequency boundaries in order to avoid overlappings with neighboring systems. So it means a loss of spectral efficiency too. To avoid these drawbacks, filter-bank multicarrier (FBMC) was proposed as an alternative approach to OFDM [1]. In FBMC, there is no need to insert any guard interval. Furthermore, it uses a frequency well-localized pulse shaping, hence, it provides a higher spectral efficiency [2; 3]. 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 considered only 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 intrinsic 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, we consider the spatial data multiplexing (SDM) case, where the information is transmitted and received 1 This

work has been carried out within the FP7 research project N211887, PHYDYAS.

978-1-61284-402-2/11/$26.00 ©2011 IEEE

simultaneously over Nt transmit antennas and Nr receive antennas in order to increase the data rate. Linear equalizations such as ZF (Zero Forcing) or MMSE (Minimum Mean Square Error) can be applied to FBMC as shown in [4]. However, maximum likelihood (ML) detection , which is supposed to offer a diversity order equal to the number of the receive antennas [5], cannot be applied straightforwardly with FBMC due to the presence of the inherent intersymbol interference. In [6], we have proposed 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, we were inspired by the reference [7] where the authors have established the conditions under which the cancellation scheme is effective. We will show, in this paper, that satisfying these conditions allows our proposed iterative scheme to converge to the optimum performance. The remainder of this paper is organized as follows. In section II, we give a brief description of the FBMC modulation and the system model. Then, we start in section III by giving a background on the use of tentative decisions to cancel the ISI, and we show how we apply it in the FBMC context. Simulation results are presented and discussed in section IV. 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

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

322

”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

ak,n gk,n [m]gk∗′ ,n′ [m].

m=−∞

Then, we can rewrite equation (3) as:

k′ 6=k

n′ 6=n

|

ak′ ,n′

+∞ X

gk,n [m]gk∗′ ,n′ [m] . (5)

m=−∞

{z

Ik,n :intrinsic interference

rk,n = ak,n + juk,n ,

E{bk′ ,n′ b∗k,n } = σ 2

(6)

TABLE I T RANSMULTIPLEXER IMPULSE RESPONSE ( MAIN PART ) 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

gk,n [m]gk∗′ ,n′ [m]. (10)

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: rk,n =

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 , (N ) (1) (1) (N ) ak,n = [ak,n , ..., ak,nt ]T , uk,n = [uk,n , ..., uk,nt ]T , and (N ) (1) bk,n = [bk,n , ..., bk,nr ]T . III. PARTIAL INTERFERENCE CANCELLATION WITH V ITERBI DETECTION In reference [7], the authors considered the channel model depicted in Fig. 1, where f0 (ak , ak−1 , ..., ak−δ+1 ) is a function of δ data symbols that represents the target response expected by the receiver, and f1 (ak+γ , ak+γ−1 , ..., ak−λ+1 ) is a function of γ + λ data symbols that represents a small channel perturbation.

Fig. 1.

(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)}, the received signal can be, finally, written as [9]:

+∞ X

m=−∞

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 . (7) Ik,n =

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

(9)

= σ 2 Γδk,δn .

(l)

where uk,n is a real-valued interference term. P+∞ ∗ Since the quantity m=−∞ gk,n [m]gk′ ,n′ [m] depends on ′ ′ the distances δk = k − k and δn = n − n [8], 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. The table below depicts the main coefficients Γδk,δn of the PHYDYAS prototype filter designed in [3].

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

∗ w[m]gk,n [m],

m=−∞

}

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

k0 − 1 k0 k0 + 1

+∞ X

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

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

X X

bk,n =

(3)

m=−∞ k=0 n∈Z

rk,n = ak,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:

Channel model

It should be noted that, in general, both f0 and f1 may be nonlinear functions. The samples of the signal at the input of the receiver are:

(8)

323

rk =f0 (ak , ak−1 , ..., ak−δ+1 ) + f1 (ak+γ , ak+γ−1 , ..., ak−λ+1 ) + wk ,

(13)

where wk is the noise contribution. The receiver is composed by a tentative detector producing tentative decisions, and a main viterbi detector which assumes that the channel is described only by f0 . Before performing the main Viterbi detector, the tentative decisions are used to cancel the remaining ISI (RISI) represented by f1 . The receiver scheme is depicted in Fig. 2.

Fig. 2.

Receiver scheme with ISI cancellation using tentative decisions (ǫ)

Given the correct data sequence ak , and a sequence ak for a certain error event ǫ, let us define: (ǫ)

(ǫ)

(ǫ)

Now that the conditions for effective RISI cancellation are summarized, we will attempt to apply them to FBMC. Hence, the problem is, essentially, how to select the functions f0 and f1 such as these conditions are fulfilled. The intrinsic interference in FBMC can be seen as a two-dimensional intersymbol interference (2D-ISI). An extension of the works of Agazzi and Seshadri [7] to 2D-ISI channels was treated in [10] assuming that the noise is uncorrelated (which is not the case in FBMC). So in general, the target response f0 also represents a 2D-ISI channel. Hence, a 2D-Viterbi detector is required to match with f0 . Designing a 2D-Viterbi is quite challenging. Many works were carried out on this problem (2D-Viterbi) leading to somewhat complicated algorithms [11; 12]. Therefore, for simplicity reasons, we opted to set the constraint that the target response f0 is one-dimensional and that f1 covers the rest of 2D-ISI. We know that the receiver complexity depends directly on that of the Viterbi detector. Therefore, we have to choose a configuration with the least complex Viterbi detector that meets the conditions for effective RISI cancellation. We will select three configurations with different sizes of the target response f0 . According to the second condition, f0 must contains the largest coefficients (Γδk,δn ) in each configuration. Hence, from table I, the selected target responses are:

(ǫ)

(1)

∆k =f0 (ak , ak−1 , ..., ak−δ+1 ) − f0 (ak , ak−1 , ..., ak−δ+1 ), (ǫ)

(ǫ)

(ǫ)

Φ(ǫ) = [∆0 , ∆1 , ..., ∆K−1 ]T ,

f0 (ak,n ) = ak,n , (14) (15)

(2) f0 (ak,n , ak,n−1 )

(17) = ak,n + Γ0,−1 × ak,n−1 ,

(18)

and (3)

where K is assumed to be the total number of transmitted symbols. The authors in [7] classified the error events in terms of their distance d0 (ǫ) in the absence of RISI (f1 = 0), which is given, in the presence of correlated noise, by: kΦ(ǫ)k2 d0 (ǫ) = p , ΦH (ǫ)RΦ(ǫ)

(16)

where R is the normalized autocovariance matrix of the noise. The events whose distance d0 (ǫ) is minimum are called ”firstorder” error events. Similarly, events whose distance is the second smallest are called ”second-order” error events, and so on [7]. The conditions for which RISI cancellation is satisfying are summarized as follows [7]: 1) Errors affecting the main (Viterbi detector) and the tentative detector must be statically independent. 2) The RISI (described by f1 ) must be small enough to guarantee that the main Viterbi detector can make relatively reliable decisions even when the tentative detector makes a decision error, and such that the tentative detector also makes relatively reliable decisions in spite of the ISI. 3) The distance of second-order and higher-order error events that could cause error propagation must be significantly larger than that of first-order error events.

f0 (ak,n+1 , ak,n , ak,n−1 ) = Γ0,1 ak,n+1 + ak,n + Γ0,−1 ak,n−1 . (1)

(19)

The first configuration (f0 ) corresponds P to the whole ISI cancellation. Since, in FBMC, we have p,q |Γp,q |2 = 2 [9], it is easy to calculate the power of the RISI (represented by f1 ) for each configuration. Regarding the first condition, it is easily satisfied when the tentative detector is different from the main one (Viterbi) [7]. We recall that we consider the case of spatial multiplexing system. Then, we chose the MMSE equalizer as the tentative detector. The third condition concerns the spectrum distances d0 (ǫ) (i) defined by (16). Hence, for each configuration (f0 , i ∈ {1, 2, 3}), we compare the non-minimum distances to the minimum one. Then, according to (16), we have to determine the matrix R. Since we consider that the target responses are one-dimensional and Viterbi algorithm is performed on each subcarrier ”k”, the matrix R is composed only by the coefficients Γ0,q , q ∈ Z (see equation (10)) and is given by:   1 Γ0,1 Γ0,2 · · · Γ0,−1 1 Γ0,1 · · ·   (20) R = Γ0,−2 Γ0,−1 1 · · ·   .. .. .. .. . . . .

324

K×K

TABLE II DISTANCES AND

(1)

First-order distance Second-order distance Third-order distance Power of the RISI

First Configuration (f0 ) 2 √ 2√ 2 2 3 1

RISI

POWER (2)

Second Configuration (f0 ) 1.8857 2.6668 3.4596 0.6819

In Table II, we summarize the values of the first, second, and third order distances obtained by using (16), and also the power of the RISI for the three considered configurations. We remark that the difference between the second-order and the first-order distances is almost the same for all the configurations (0.8 ± 0.03), so we consider (as considered in [7]) that the higher-order distances are sufficiently larger than the minimum distance for each configuration. Hence, condition 3) is fulfilled for the three configurations. Now, we have only to determine the configuration(s) for which the second condition is satisfied. Unfortunately, the determination of the RISI power for which the cancellation starts to be effective (or equivalently, error propagation ceases) is not trivial and depends also on the noise variance σ 2 [7]. We will show in the next section (by simulations) that only (3) the third configuration (f0 ) allows to obtain effective RISI cancellation. As for the receiver complexity, it depends strongly on that of the Viterbi detector. When we consider a spatial multiplexing system with Nt transmit antennas, the Viterbi detector has to compute q i×Nt branch metrics, where q is the number of all possible symbols ak,n (constellation size) and i ∈ {1, 2, 3} is (i) the number of the taps in f0 . In order to reduce the receiver complexity, we can replace the Viterbi detection algorithm by the M-Algorithm [13] which keeps only a fixed number (J) of inner states instead of all the inner states (q (i−1)×Nt ). Hence, the M-algorithm has to compute only J × q Nt branch metrics.

(3)

Third Configuration (f0 ) 1.9189 2.7137 3.2728 0.3638

0

10

Tentative detector (MMSE) PIC/Viterbi−1 PIC/Viterbi−2 PIC/Viterbi−3 OFDM−ML

−1

10

BER

S PECTRUM

−2

10

−3

10

−4

10

0

5

10

15

20

25

SNR (dB)

Performance of PIC/Viterbi receivers for 2 × 2 spatial multiplexing

Fig. 3.

0

10

IV. S IMULATION RESULTS

We call the proposed receivers ”PIC/Viterbi” (for Partial Interference Cancellation with Viterbi detector) followed by an index indicating the considered configuration. Fig. 3 depicts the performance of the MMSE equalizer (which is our tentative detector) and of the PIC/Viterbi for the three considered configurations. We notice clearly that PIC/Viterbi-3 exhibits almost the same performance as OFDM, and that the RISI

−1

10

BER

In this section, we provide the simulation results concerning the three configurations treated in the previous section. 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. We assume perfect channel knowledge at the receiver side, and the four Rayleigh sub-channels are spatially non-correlated. The complex data symbols are QPSK modulated (q = 2). The system performance is assessed in terms of bit-error rate (BER) as a function of the signal-to-noise ratio (SNR) and is compared to that of the conventional OFDM with ML detector.

Tentative decision (MMSE) PIC/M−Algo J=2 PIC/M−Algo J=4 PIC/Viterbi−3 (J=16)

−2

10

−3

10

−4

10

Fig. 4.

325

0

5

10 SNR (dB)

15

20

Performance of PIC/M-Algo receiver for 2 × 2 spatial multiplexing

cancellation is effective. Hence, the value of the RISI power given in Table II for the third configuration is sufficiently small that the condition 2) is satisfied. However, a slight degradation of the PIC/Viterbi-3 performance compared to OFDM is observed beyond 22 dB. Indeed, as we have mentioned at the end of the previous section, the threshold of the RISI power from which the error propagation begins (ineffectiveness of the RISI cancellation) depends on the noise variance σ 2 . As shown -for a specific example- in [7], the threshold lowers with the SNR increase. As for the first and second configurations, the performance degradation compared to OFDM begins from about 12 dB. This relatively high degradation is due to the high values of the corresponding RISI powers causing error propagation. (3) Now we consider only the third configuration (f0 ) since the RISI cancellation is effective. In fig. 4, we show the obtained performance when we replace the Viterbi algorithm by the M-Algorithm with two values of J (J = 2, and J = 4). We notice that with J = 2 we have an SNR loss about of 2.5 dB compared to PIC/Viterbi-3 due to the suboptimality of the M-Algorithm. Moreover, PIC/M-Algo with J = 2 provides a performance worse than that the one provided by the tentative detector (MMSE) as long as the SNR is less than 12 dB. However, we can observe that PIC/M-Algo with J = 4 exhibits exactly the same performance as PIC/Viterbi-3 with much lower algorithm complexity (4 inner states instead of 16). V. C ONCLUSION In this paper, we have considered the ML detection in spatial multiplexing scheme with FBMC. The intrinsic interference in FBMC is seen as a 2D-ISI in the time-frequency plan. In order to avoid a full 2D-Viterbi detector, we have proposed a receiver based on ISI cancellation. However, we have shown that the ISI cancellation is effective only under some strict conditions. One of these conditions is that the ISI must be sufficiently small. Unfortunately, the intrinsic ISI, in FBMC, has the same power as the desired symbols. Hence, we have proposed a trade-off between a whole ISI cancellation and a full 2D-Viterbi detection. The proposed receiver is composed by a tentative detector giving decisions which serve to cancel partially the ISI, and a Viterbi detector matching to the non-canceled ISI. Three configurations were treated. The first one is called PIC/Viterbi-1 and correspond to the whole ISI cancellation. The second one is PIC/Viterbi-2, where

the Viterbi detector matches with the two largest coefficients. Then, the third one is PIC/Viterbi-3 and the Viterbi detector matches with the three largest coefficients. We have shown, by simulations, that only the PIC/Viterbi-3 receiver gives the same performance as OFDM. Both others configurations suffer from error propagation because their RISI are not sufficiently small. We have also proposed to replace the Viterbi detector by another based on M-Algorithm in order to reduce the receiver complexity. Indeed, we have shown that using the M-Algorithm with J = 4, we obtain exactly the same BER performance as PIC/Viterbi-3 with much lower complexity. R EFERENCES [1] B. Le Floch, 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 Int. Conf. Acoustics, Speech, and Signal Processing, Salt Lake City, USA, May 2001, pp. 2417-2420. [4] 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. [5] R.V. Nee, A.V. Zelst, and G. Awater, ”Maximum Likelihood Decoding in a Space Division Multiplexing System”, Vehicular Technology Conference Proceedings, 2000. VTC 2000-Spring Tokyo. 2000 IEEE 51st, vol. 1, pp. 6-10, May 2000 [6] 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. [7] O. E. Agazzi and N. Seshardsi, ”On the use of tentative decisions to cancel intersymbol interference and nonlinear distortion (with application to magnetic recording channels),” IEEE Trans. Inf. Theory, vol. 43, no. 2, pp. 394-408, Mar. 1997. [8] 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. [9] 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. [10] S. Van Beneden, J. Riani, J. W. M. Bergmans, and A. H. J. Immink, ”Cancellation of linear intersymbol interference for two-dimensional storage systems,” IEEE Trans. on Magnetics, vol. 42, no. 8, pp. 20962106, Aug. 2006. [11] L. Huang, G. Mathew, and T.C. Chong, ”Reduced complexity Viterbi detection for two-dimensional optical recording,” IEEE Trans. Consumer Electr., vol 51, no. 1, pp. 123129, Feb. 2005. [12] B. M. Kurkoski, ”Towards efficient detection of two-dimensional intersymbol interference channels,” IEICE Trans. on. Fundamentals, vol. E91-A, no. 10, Oct. 2008. [13] J.B. Anderson and S. Mohan, ”Sequential Coding Algorithms: A Survey and Cost Analysis”, IEEE Transactions on Communications, vol. 32, no. 2, pp. 169- 176, Feb. 1984.

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