On interference cancellation in Alamouti coding scheme for filter bank based multicarrier systems R. Zakaria, D. Le Ruyet CEDRIC/LAETITIA Laboratory, CNAM. 292 rue Saint Martin, 75141, Paris, France. {rostom.zakaria, didier.le_ruyet}@cnam.fr
Abstract—In this paper we consider the application issue of the Alamouti coding scheme in filter bank based multicarrier (FBMC) systems. The presence of the inherent interference prevents the use of Alamouti coding scheme with FBMC modulation. Receiver schemes with interference cancellation are not always effective due to the error propagation. In this work, we propose some arrangements in the space-time and frequency block coding (STBC, SFBC) in order to reduce the effect of the interference. Then, the Alamouti decoding is followed by an interference canceller. We test the proposed arrangements for the basic 2 × 1 Alamouti coding scheme. The performance is assessed in terms of the bit-error rate (BER) as a function of the signal-to-noise ratio (SNR). We will show that these proposed arrangements allow us to reach almost the optimal performance.1 Index Terms—Filter bank, FBMC, MIMO, Alamouti coding, FBMC/OQAM, interference cancellation, STBC, SFBC.
I.
INTRODUCTION
Filter-bank multicarrier (FBMC) was proposed [4] 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 [9; 3] since it uses a frequency welllocalized pulse shaping. In FBMC/OQAM, 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/OQAM orthogonality condition is only considered in the real field [9]. Consequently, the data at the receiver side is carried only by the real (or imaginary) component of the signal. Whereas, 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 [8; 10]. In this paper, the classical 2×1 STBC scheme [2] (Alamouti coding) is considered. This scheme, which is supposed to offer a diversity order equal to two, cannot be applied in a straightforward manner with FBMC/OQAM due to the presence of the self intersymbol interference. Some works have been carried out on these topics such as [5] where the authors show that Alamouti coding can be performed when it is combined with code division multiple access (CDMA). Another work was carried out in [7] where a pseudo-Alamouti 1 This
work is carried out within the European project EMPHATIC
scheme was proposed. But this solution requires the appending of the cyclic prefix (CP) to the FBMC signal. Renfors et al. in [8] have proposed a solution to combine the Alamouti scheme with FBMC, where the Alamouti coding scheme is performed in a block-wise manner inserting gaps (zero-symbols and pilots) in order to isolate the blocks. Iterative interference cancellation solutions are subject to error propagation. In order to counteract the error propagation and make the cancellation scheme effective, the authors in [1] have shown that the interference power must be small and kept under a certain threshold. In this paper, we propose some modification in the Alamouti block coding in order to reduce the FBMC self interference. The idea behind these schemes is to remove an important part of the interference only by performing the adequate Alamouti decoding. Then, the remaining interference is cancelled iteratively by interference estimation and cancellation procedure. In this work, we are interested in the PHYDYAS prototype filter proposed by Bellanger in [3]. The rest of the paper is structured as follows. We give in section II a general description of the FBMC modulation and the problem statement. In section III, we describe the proposed coding schemes and evaluate their impacts on the inherent interference. In section IV, we present and discuss the simulation results. Then, we finish by a conclusion. II. P ROBLEM STATEMENT In baseband discrete time model, we can write at the transmitter side the FBMC signal as follows [4]: 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
”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)
∗ ak0 ,n0 gk0 ,n0 [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 [9]: ( +∞ ) X ∗ Re gk0 ,n0 [m]gk,n [m] = δk,k0 δn,n0 . (4) m=−∞
Then, we can rewrite equation (3) as: rk,n = ak,n +
X X k0 6=k n0 6=n
|
+∞ X
ak0 ,n0
the error propagation through iterations[1]. In order to counteract the error propagation and make the cancellation scheme effective, the authors in [1] showed that a necessary condition to avoid the error propagation is to hold the interference power under a certain threshold, i.e. the interference cancellation technique is effective only when the ISI is small enough compared to the minimal distance d0 between two different symbols. Let us consider the basic 2 × 1 Alamouti coding: when the first antenna transmits at the kth subcarrier a block [ak,n ak,n+1 ], the second antenna transmits the block [ak,n+1 − ak,n ]. We note that the conjugate operation is aborted because the data symbols are real valued. Without loss of generality, let us consider a specific block with its immediate neighboring symbols. We can write: a0,−1 a0,0 a0,1 A1 = a1,−1 a1,0 a1,1 a2,−1 a2,0 a2,1 −a0,−2 a0,1 −a0,0 A2 = −a1,−2 a1,1 −a1,0 −a2,−2 a2,1 −a2,0
∗ 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 ,
(6)
where uk,n is a real-valued P+∞interference term. ∗ Since the quantity m=−∞ gk0 ,n0 [m]gk,n [m] depends on 0 the distances δk = k − k and δn = n0 − n [10], let us denote it by the coefficient Γδk,δn . These coefficients Γδk,δn represent the transmultiplexer impulse response in the timefrequency domain and depend on the used prototype filter. Table I depicts the main coefficients Γδk,δn of the PHYDYAS prototype filter. 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 Ik,n = ak0 ,n0 Γδk,δn . (7)
where A1 and A2 are the data blocks transmitted at the first and the second antennas, respectively. For simplicity reasons, let us only consider the interference from the immediate neighborhood of the symbol of interest. Hence, according to (7) and (5) we can write the received symbols r1,0 and r1,1 as r1,0 =h1
1 1 X X
Ã
− h2
rk,n = hk,n (ak,n + juk,n ) + bk,n ,
(8)
where hk,n and bk,n are, respectively, the channel coefficient and the noise term at subcarrier ”k” and time index ”n”. The received data symbols are corrupted by inherent interference terms which complicate the STBC decoding [5]. The interference estimation and cancellation procedure is repeated several times to improve the detection. Unfortunately, detection schemes with ISI estimation and cancellation are not always effective due to the error propagation [11]. Therefore, the challenge in ISI estimation and cancellation is mitigating
1 X
Γp,−1 a1−p,0 +
1 X
+ h2
p=−1
Γp,1 a1−p,−2 1 X
Γp,q a1−p,1−q − h2
p=−1 q=−1 1 X
! + b1,0
p=−1
1 1 X X
Ã
Γp,0 a1−p,1
p=−1
p=−1
r1,1 =h1
1 X
Γp,q a1−p,−q + h2
p=−1 q=−1
(k0 ,n0 )∈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 [6]:
a0,2 a1,2 a2,2 a0,3 a1,3 a2,3
Γp,−1 a1−p,3 +
Γp,0 a1−p,0
p=−1 1 X
Γp,1 a1−p,1
! + b1,1
p=−1
The Alamouti decoding is performed by calculating y1,0 and y1,1 as ( ∗ y1,0 = h∗1 r1,0 − h2 r1,1 (9) ∗ y1,1 = h∗1 r1,1 + h2 r1,0 After processing, we obtain ¡ ¢ Re {y1,0 } = |h1 |2 + |h2 |2 + 2Re {h∗1 h2 Γ0,1 } a1,0 ª © + Re {h∗1 h2 I1,0 } + Re h∗1 b1,0 − h2 b∗1,1 ¡ ¢ Re {y1,1 } = |h1 |2 + |h2 |2 + 2Re {h∗1 h2 Γ0,1 } a1,1 © ª + Re {h∗1 h2 I1,1 } + Re h∗1 b1,1 + h2 b∗1,0
TABLE I T RANSMULTIPLEXER IMPULSE RESPONSE ( MAIN PART ) USING PHYDYAS
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 the interference terms I1,0 and I1,1 are given by I1,0 =
1 X
block, the received symbols r1,0 and r1,2 are as follow
2Im {Γp,0 } a1−p,1 − Γp,1 a1−p,−2 − Γ∗p,−1 a1−p,2 r1,0 =h1
p=−1
I1,1 =
1 X
FILTER
1 1 X X p=−1 q=−1
Ã
−2Im {Γp,0 } a1−p,0 + Γp,−1 a1−p,3 − Γ∗p,1 a1−p,−1
− h2
p=−1
1 X
r1,2 =h1
Γp,−1 a1−p,3 +
1 X
1 1 X X
Ã
This remaining term of interference will cause a BER floor in high SNR regime. If this BER floor is not small enough, then iterative interference cancellation receiver will introduce error propagation and, thus, the interference cancellation cannot be effective. Moreover, one can remark the term 2Re {h∗1 h2 Γ0,1 } weighing the useful symbol. Hence, this term increases the BER when it is negative, and decrease the BER when it is positive. Nevertheless, the BER increase is more important than the BER decrease. III. P ROPOSED A LAMOUTI SCHEMES FOR FBMC/OQAM MODULATION
In this section, we propose some different Alamouti coding schemes where the overall signal-to-interference ratio (SIR) is improved. Let us consider the space time block coding (STBC) defined as a0,−1 a0,0 a0,1 a0,2 a0,3 A1 = a1,−1 a1,0 a1,1 a1,2 a1,3 a2,−1 a2,0 a2,1 a2,2 a2,3 a0,−3 a0,2 −a0,3 −a0,0 a0,1 A2 = a1,−3 a1,2 −a1,3 −a1,0 a1,1 a2,−3 a2,2 −a2,3 −a2,0 a2,1 where A1 and A2 are the data blocks transmitted at the first and the second antennas, respectively. We note that in this scheme there are two interleaved Alamouti blocks (shown in red and blue colors). Furthermore, there is an alternating rule in positions of the minus sign between the blue and the red blocks. Let us focus on the decoding of the blue Alamouti
+ h2
1 X
!
Γp,1 a1−p,−3
+ b1,0
p=−1 1 X
Γp,q a1−p,2−q − h2
p=−1 q=−1
E{|I1,0 |2 } = E{|I1,1 |2 } = 8|Γ1,0 |2 + 4|Γ1,1 |2 + 2|Γ0,1 |2 ' 1.2646 (10)
Γp,0 a1−p,2
p=−1
p=−1
According to Table I, we can show that
1 X
Γp,q a1−p,−q + h2
Γp,−1 a1−p,1 −
p=−1
Γp,0 a1−p,0
p=−1 1 X
Γp,1 a1−p,3
! + b1,2
p=−1
The Alamouti decoding for the blue block is performed by calculating y1,0 and y1,2 as ( ∗ y1,0 = h∗1 r1,0 − h2 r1,2 ∗ y1,2 = h∗1 r1,2 + h2 r1,0
(11)
Hence, after processing we obtain: ¡ ¢ Re {y1,0 } = |h1 |2 + |h2 |2 a1,0 + Re {h∗1 h2 I1,0 } © ª + Re h∗1 b1,0 − h2 b∗1,2 ¡ ¢ Re {y1,2 } = |h1 |2 + |h2 |2 a1,2 + Re {h∗1 h2 I1,2 } © ª + Re h∗1 b1,2 + h2 b∗1,0 where I1,0 =
1 X
2Im {Γp,0 } a1−p,2 − Γp,1 a1−p,−3 − Γ∗p,1 a1−p,1
p=−1
I1,2 =
1 X
−2Im {Γp,0 } a1−p,0 − Γp,1 a1−p,3 + Γ∗p,1 a1−p,−1
p=−1
As in the first case, we can show that E{|I1,0 |2 } = E{|I1,2 |2 } ' 1.2646. However, one can remark that, in this case, the only factors that weigh the useful symbol are |h1 |2 and |h2 |2 which are both positives. Therefore, we expect a BER floor level below the classical Alamouti one presented in the previous section. Based on the same idea, we also propose another Alamouti scheme. This time, the Alamouti scheme is the following space
frequency block coding (SFBC) defined by: a0,−1 a0,0 a0,1 a1,−1 a1,0 a1,1 A1 = a2,−1 a2,0 a2,1 a3,−1 a3,0 a3,1 a4,−1 a4,0 a4,1 a2,−1 −a2,0 a2,1 −a3,−1 a3,0 −a3,1 A2 = −a0,−1 a0,0 −a0,1 a1,−1 −a1,0 a1,1 a6,−1 −a6,0 a6,1
IV. SIMULATION RESULTS In this section we provide simulation results of the straightforward implemented Alamouti coding scheme, and the proposed ones. All the receivers apply an iterative interference cancellation. The performance is assessed in terms of BER as a function of the SNR. The performance of the different schemes is compared to the Genie-Aided one which exploits the perfect knowledge of the interference. We assume full channel state information (CSI) knowledge at the receiver side. Fig. 1 depicts the BER performance obtained with the classical Alamouti scheme in FBMC. The data symbols are OQPSK modulated. One can observe a high BER floor level and the interference cancellation is not effective. As for the proposed
The received symbols r1,0 and r3,0 are given by 1 1 X X p=−1 q=−1 1 X
− h2
(−1)q Γ−1,q a2,−q + h2
1 X
Γp,q a3−p,−q − h2
p=−1 q=−1
+ h2
IIC−Alamouti Iter=1 IIC−Alamouti Iter=2 IIC−Alamouti Iter=3 IIC−Alamouti Iter=4 Genie−Aided Alamouti Alamouti OFDM
−1
10
(−1)q Γ1,q a0,−q + b1,0
q=−1
1 1 X X
1 X
0
10
(−1)q Γ0,q a3,−q
q=−1
q=−1
r3,0 = h1
1 X
Γp,q a1−p,−q + h2
(−1)q Γ0,q a1,−q
−2
10
q=−1
(−1)q Γ−1,q a0,−q −
q=−1
1 X
BER
r1,0 = h1
1 X
−3
10
(−1)q Γ1,q a6,−q + b3,0
q=−1
The Alamouti decoding for the blue block is performed by calculating y1,0 and y3,0 as ( ∗ y1,0 = h∗1 r1,0 − h2 r3,0 (12) ∗ y3,0 = h∗1 r3,0 + h2 r1,0
−4
10
0
5
10 SNR (dB)
¡ ¢ Re {y1,0 } = |h1 |2 + |h2 |2 a1,0 + Re {h∗1 h2 I1,0 } © ª + Re h∗1 b1,0 − h2 b∗3,0 ¡ ¢ Re {y3,0 } = |h1 |2 + |h2 |2 a3,0 + Re {h∗1 h2 I3,0 } © ª + Re h∗1 b3,0 + h2 b∗1,0
Fig. 1. Performance of classical Alamouti scheme in FBMC/OQAM using OQPSK modulation
0
10
STBC iter=1 STBC iter=2 STBC iter=3 STBC iter=4 Genie−Aided Alamouti
where I1,0 =
−1
10
2Γ1,q a2,−q − Γ−1,q a4,−q +
20
STBC scheme, the BER performance with OQPSK modulation is shown in Fig. 2. Clearly, the BER performance of the first iteration is improved and the BER floor is less than 7 × 10−3 . Moreover, we can almost reach the Genie-Aided performance by using 3 iterations. Now, we show the performance of the
We obtain:
1 X
15
(−1)q Γ∗1,q a0,−q
I3,0 =
1 X
−2Γ1,q a0,−q +
Γ∗−1,q a2,−q
−
(−1)q Γ∗1,q a6,−q
BER
q=−1 −2
10
q=−1 −3
Hence, We obtain the same expressions as the STBC ones. However, we can show, according to Table I that E{|I1,0 |2 } = E{|I3,0 |2 } = 12|Γ1,1 |2 + 6|Γ1,0 |2 ' 0.8518
10
−4
10
0
5
10 SNR (dB)
15
20
(13)
Clearly, the variance of the interference terms I1,0 and I3,0 is significantly reduced. This fact will guarantee better performance compared to the previous STBC proposal.
Fig. 2. Performance of the proposed STBC scheme in FBMC/OQAM using OQPSK modulation
proposed SFBC scheme in Fig. 3 where the data is OQPSK
mapped. In this case, we observe that with only the second iteration we can reach the Genie-Aided performance. This is thanks to the small residual interference which does not causes significant error propagation. We have also test both proposed
enough compared to minimum distance between the different 0
10
SFBC iter=1 SFBC iter=2 SFBC iter=3 SFBC iter=4 Genie−Aided Alamouti −1
10
0
10
BER
SFBC iter=1 SFBC iter=2 SFBC iter=3 Genie−Aided Alamouti −1
10
−2
BER
10
−2
10
−3
10 −3
10
0
5
10 SNR (dB)
15
10 SNR (dB)
15
20
20
Fig. 3. Performance of the proposed SFBC scheme in FBMC/OQAM using OQPSK modulation
schemes with 16-OQAM modulation. Fig 4 and Fig. 5 depict the performance of the proposed STBC and SFBC schemes, respectively. We remark that we need more iterations than in OQPSK to remove the interference. This is explained by the fact that the minimum distance between two different symbols is smaller in 16-OQAM than in OQPSK. Hence, more errors occur when the receive signal is corrupted by interference. 0
10
−1
BER
10
STBC iter=1 STBC iter=2 STBC iter=3 STBC iter=4 STBC iter=5 Genie−Aided Alamouti
−2
10
−3
10
5
Fig. 5. Performance of the proposed SFBC scheme in FBMC/OQAM using 16-OQAM modulation
−4
10
0
0
5
10 SNR (dB)
15
20
Fig. 4. Performance of the proposed STBC scheme in FBMC/OQAM using 16-OQAM modulation
V.
CONCLUSION
in this paper, we have considered the application issue of the Alamouti coding scheme in FBMC/OQAM system. Indeed, the self-interference is not removed automatically during the decoding process. We have proposed to use iterative interference cancellation receiver to remove the remaining interference. However, when the interference is not small
useful symbols, the receiver suffers from error propagation and the cancellation is not effective. Therefore, we have proposed some different arrangement in the Alamouti coding such that the signal to interference ratio (SIR) is improved. Thus, we proposed a STBC and SFBC schemes, and we tested them using OQPSK and 16-QAM modulations. We have shown that in both proposed schemes, the BER performance can almost reach the Genie-Aided one. R EFERENCES [1] O.E. Agazzi and N. Seshadri. On the use of tentative decisions to cancel intersymbol interference and nonlinear distortion (with application to magnetic recording channels). Information Theory, IEEE Transactions on, 43(2):394 –408, mar 1997. [2] S.M. Alamouti. A simple transmit diversity technique for wireless communications. Selected Areas in Communications, IEEE Journal on, 16(8):1451 –1458, oct 1998. [3] M.G. Bellanger. Specification and design of a prototype filter for filter bank based multicarrier transmission. In Acoustics, Speech, and Signal Processing, 2001. Proceedings. (ICASSP ’01). 2001 IEEE International Conference on, volume 4, pages 2417 –2420 vol.4, 2001. [4] B. Le Floch, M. Alard, and C. Berrou. Coded orthogonal frequency division multiplex. Proceedings of the IEEE, 83(6):982 –996, jun 1995. [5] C. L´el´e, P. Siohan, and R. Legouable. The Alamouti Scheme with CDMA-OFDM/OQAM. EURASIP J. Adv. Sig. Proc., 2010, 2010. [6] C. Lele, P. Siohan, R. Legouable, and J.-P. Javaudin. Preamble-based channel estimation techniques for ofdm/oqam over the powerline. In Power Line Communications and Its Applications, 2007. ISPLC ’07. IEEE International Symposium on, pages 59 –64, march 2007. [7] H. Lin, C. L´el´e, and P. Siohan. A pseudo alamouti transceiver design for OFDM/OQAM modulation with cyclic prefix. In Signal Processing Advances in Wireless Communications, 2009. SPAWC ’09. IEEE 10th Workshop on, pages 300 –304, june 2009. [8] M. Renfors, T. Ihalainen, and T.H. Stitz. A block-Alamouti scheme for filter bank based multicarrier transmission. In Wireless Conference (EW), 2010 European, pages 1031 –1037, april 2010. [9] P. Siohan, C. Siclet, and N. Lacaille. Analysis and design of OFDM/OQAM systems based on filterbank theory . Signal Processing, IEEE Transactions on, 50(5):1170 –1183, may 2002. [10] R. Zakaria and D. Le Ruyet. A novel FBMC scheme for Spatial Multiplexing with Maximum Likelihood detection. In Wireless Communication Systems (ISWCS), 2010 7th International Symposium on, pages 461 –465, sept. 2010. [11] R. Zakaria and D. Le Ruyet. Partial ISI cancellation with viterbi detection in MIMO filter-bank multicarrier modulation. In Wireless Communication Systems (ISWCS), 2011 8th International Symposium on, pages 322 –326, nov. 2011.
