1112

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 3, MARCH 2012

A Novel Filter-Bank Multicarrier Scheme to Mitigate the Intrinsic Interference: Application to MIMO Systems Rostom Zakaria and Didier Le Ruyet, Senior Member, IEEE

Abstract—Filter-bank multicarrier (FBMC) transmission system was proposed as an alternative approach to orthogonal frequency division multiplexing (OFDM) system since it has a higher spectral efficiency. One of the characteristics of FBMC is that the demodulated transmitted symbols are accompanied by interference terms caused by the neighboring transmitted data in time-frequency domain. The presence of this interference is an issue for some multiple-input multiple-output (MIMO) schemes and until today their combination with FBMC remains an open problem. We can cite, among these techniques, the Alamouti scheme and the maximum likelihood detection (MLD) with spatial multiplexing (SM). In this paper, we shall propose a new FBMC scheme and transmission strategy in order to avoid this interference term. This proposed scheme (called FFTFBMC) transforms the FBMC system into an equivalent system formulated as OFDM regardless of some residual interference. Thus, any OFDM transmission technique can be performed straightforwardly to the proposed FBMC scheme with a corresponding complexity growth compared to the classical FBMC. First, we will develop the FFT-FBMC in the case of singleinput single-output (SISO) configuration. Then, we extend its application to SM-MIMO configuration with MLD and Alamouti coding scheme. Simulation results show that FFT-FBMC can almost reach the OFDM performance, but it remains slightly outperformed by OFDM. Index Terms—Filter-bank multicarrier, FBMC, MIMO, maximum likelihood detection, MLD, OFDM/OQAM, spatial multiplexing, space-time block coding.

I. I NTRODUCTION RTHOGONAL frequency division multiplexing (OFDM) with the cyclic prefix (CP) insertion 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. Nevertheless, this technique causes a loss of spectral efficiency due to the CP. Furthermore, the 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, which means a loss of spectral efficiency too. To avoid these drawbacks, filter-bank multicarrier (FBMC) was

O

Manuscript received April 4, 2011; revised August 8, 2011 and December 6, 2011; accepted January 9, 2012. The associate editor coordinating the review of this paper and approving it for publication was Y.-C. Ko. R. Zakaria and D. Le Ruyet are with the CEDRIC/LAETITIA Laboratory, CNAM, Paris 75141, France (e-mail: {rostom.zakaria, didier.le ruyet}@cnam.fr). This work has been carried out within the FP7 research project No. 211887, PHYDYAS. Digital Object Identifier 10.1109/TWC.2012.012412.110607

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]. In the literature we find several FBMC systems based on different structures, which are listed most of them in [4]. In particular, we have focused on the Saltzberg’s scheme [7] namely OFDM/OQAM. Saltzberg showed, in [7], that by introducing a shift of half the symbol period between the in-phase and quadrature components of QAM symbols, it is possible to achieve a baud-rate spacing between adjacent subcarrier channels and still recover the information symbol free of intersymbol interference (ISI) and intercarrier interference (ICI). Thus, each subcarrier is modulated with an offset QAM (OQAM) and the orthogonality conditions are considered only 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. An efficient discrete Fourier transform (DFT) implementation of this modulation method has been proposed by Hirosaki [8]. In recent years, FBMC has attracted a lot of interest and many equalization and synchronization methods have been developed for this modulation. However, most of these works are related to single-input single-output (SISO) systems. It has been shown in [9] that spatial multiplexing (SM) could be directly applied to FBMC with the minimum mean square error (MMSE) equalizer. SM with maximum likelihood detection (SM-MLD) and the Alamouti space time block coding (STBC) are some of the classical techniques which constitute a central ingredient in advanced wireless communication. Unfortunately, their combination with FBMC does not work well and has remained an open problem. Despite the fact that solutions were recently proposed, they are not completely satisfying either because of their spectral efficiency loss or because of their performance degradation. We have proposed in [12] a solution for MLD based on interference estimation and cancelation, but the obtained performance was still far from the optimum. Regarding the Alamouti scheme, its application in a straightforward manner to the FBMC makes an inherent interference appear that cannot be easily removed [14]. Many works have been carried out on this topic such as [14] where the authors show that Alamouti coding can be performed but only when it is combined with code division multiple access (CDMA). The pseudo-Alamouti scheme was introduced in [16] at the expense of the spectral efficiency since it requires the

c 2012 IEEE 1536-1276/12$31.00 

ZAKARIA and LE RUYET: A NOVEL FILTER-BANK MULTICARRIER SCHEME TO MITIGATE THE INTRINSIC INTERFERENCE: APPLICATION TO MIMO . . . 1113

appending of a CP to the FBMC signal. Another solution was proposed by Renfors et al. in [17] where the Alamouti coding is performed in a block-wise manner inserting gaps (zero-symbols and pilots) in order to isolate the blocks. In this paper, we propose a novel FBMC scheme, that we call FFT-FBMC, and a transmission strategy in order to get rid of the interference. We will show that this scheme provides an equivalent system model formulated as the OFDM one. Thus, the strong point of this scheme will be the fact that all the transmission techniques applied in OFDM can be straightforwardly performed in FFT-FBMC with nearly similar performance. The remainder of this paper is organized as follows. A description of FBMC modulation is presented in section II, where we give an overview of FBMC transmission over a multipath channel. Then, in section III, we present our scheme called FFT-FBMC where we perform additional IFFT and FFT operations on each subcarrier, respectively, at the TX and RX sides. We develop this scheme and provide the system model in the SISO context. After that, we extend its application to the multiple-input multiple-output (MIMO) configuration in section V where we show that Alamouti coding and the SM-MLD can be easily applied. Simulation results are provided in section VI, where the performance comparisons between the proposed scheme and OFDM are carried out. Finally, discussion and concluding remarks are given in section VII. II. T HE FBMC (OFDM/OQAM) M ODULATION We write, at the transmitter side, the discrete time FBMC signal as follows [2] s[m] =

M−1  

ak,n g[m − nM/2]ej

2π D M k(m− 2 )

ejφk,n

(1)

k=0 n∈Z

s[m] =

M−1 

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. In the case of no channel, the demodulated symbol over the k  th subcarrier and the n th instant is determined using the inner product of s[m] and gk ,n [m] r

= s, g

k ,n

=

+∞ 

=

m=−∞ +∞ 

gk0 ,n0 [m]gk∗ ,n [m] 2π

g[m]g[m − ΔnM/2]ej M Δk( 2 −m)

s[m]gk∗ ,n [m]

+∞ 

M−1 

ak,n gk,n [m]gk∗ ,n [m].

m=−∞ π

m=−∞ k=0 n∈Z

The transmultiplexer impulse response can be derived assuming null data except at one time-frequency position (k0 , n0 )

(4)

where Δn = n − n0 and Δk = k  − k0 . We notice that the impulse response of the transmultiplexer depends on k0 . Indeed, the sign of some impulse response coefficients depends on the parity of k0 . In this work, we essentially exploit this property. Several pulse shaping prototype filters g[m] can be used according to their properties. In this paper, we consider two different pulse shapes. The first one is the pulse shape based on the so-called Isotropic Orthogonal Transform Algorithm (IOTA) function introduced by Alard [5]. The second one is referred as the PHYDYAS prototype filter proposed by Bellanger in [3]. The overlapping factor of both pulse shapes is K = 4. All the prototype filters g[m] are designed to satisfy the real orthogonality condition given by [2]    ∞ ∗ gk ,n [m]gk,n [m] = δk,k δn,n (5) Re m=−∞

Let us consider the SISO FBMC transmission. When passing through the radio channel and adding noise contribution γk ,n , equation (3) becomes [10] rk ,n =hk ,n ak ,n + γk ,n +∞   + hk,n ak,n gk,n [m]gk∗ ,n [m], m=−∞







Ik ,n

(6) where hk ,n is the channel coefficient at subcarrier k  and time index n , and the term Ik ,n is defined as an intrinsic interference. The most part of the energy of the impulse response is localized in a restricted set around the considered symbol [11]. Consequently, we assume that the intrinsic interference term depends only on this restricted set (denoted by Ωk ,n ). Moreover, assuming that the channel is constant at least over this summation zone, we can write as in [10] rk ,n ≈ hk ,n (ak ,n + Iˆk ,n ) + γk ,n ,

(7)

where  (k,n)∈Ωk ,n

(3)

D

× ejπ(Δk+k0 )Δn e−j 2 (Δk+Δn) ,

Iˆk ,n =

m=−∞

=

+∞ 

rk ,n =

(k,n)=(k ,n )

where M is an even number of subcarriers, g[m] is the prototype filter taking values in real field, D 2 is the delay term which depends on the length (Lg ) of g[m]. We have D = KM − 1 and Lg = KM , where K is the overlapping factor. The transmitted symbols ak,n are real-valued symbols which are the real or the imaginary parts of QAM symbols. φk,n = π2 (n + k) − πnk is an additional phase term. We can rewrite (1) in a simple manner

k ,n

where a unit impulse is applied. Then, equation (3) becomes

ak,n

+∞ 

gk,n [m]gk∗ ,n [m].

m=−∞

According to (5) and because ak,n is real-valued, the intrinsic interference Iˆk ,n = juk ,n is pure imaginary. Thus, the demodulated signal can be given by rk ,n ≈ hk ,n (ak ,n + juk ,n ) + γk ,n .

1114

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 3, MARCH 2012

III. T HE P ROPOSED FFT-FBMC S CHEME When combining FBMC with MIMO techniques such as STBC or SM-MLD, the presence of the interference term uk ,n (which can be seen as a 2D-ISI) causes problems and makes the detection process from rk,n very hard if not impossible. In this section, we propose a modified FBMC system scheme in order to get rid of the intrinsic interference term. A. System model

Fig. 1.

Let us denote the transmultiplexer impulse response ob(k ) tained in equation (4) by fΔk0 (Δn). The prototype filters are designed in a way that they provide a well-localized spectrum and spread over only a few adjacent subcarriers. The most important interference comes form the same considered subcarrier and immediate neighboring ones. Without loss of generality and for the sake of simplicity, we consider only the interference coming from adjacent subcarriers. Hence, we can split (3) into three terms and write rk,n

∼ =

Δ 

(k) f0 (i)ak,n−i

i=−Δ





Δ 

+

+



(k+1)







(i)ak+1,n−i ,



(8)



tk+1

where ak,n are the inputs of the FBMC modulator on the kth (k) subcarrier and the nth time instant, fj (i) are the coefficients of the transmultiplexer impulse response when the impulse is applied on the kth subcarrier, Δ = 2K − 1 is the one side maximum response spread in time domain. Hence, i ∈ / (k) [−Δ, Δ] ⇒ fj (i) = 0. Now, we consider each term ti in (8) as a point-to-point multipath channel. So, as in conventional OFDM, we can overcome ISI just by performing an IDFT on blocks of length N ≥ 2Δ at the TX side, and inserting a CP of sufficient length L ≥ 2Δ. At the RX side, the first L samples of each symbol block are discarded. Taking the DFT of the remaining block, one can write (k) (k−1) (k+1) Yk,n ∼ = F0,n dk,n + F1,n dk−1,n + F−1,n dk+1,n .          Tk

Tk−1

Tk+1

where N −1 −j2πnn 1  Yk,n = √ rk,n e N , N n =0

dk,n

N −1 −j2πnn 1  = √ ak,n e N , N n =0

and (k)

Fj,n =

N 2

−1



n =− N 2

+∞ 

g[m]g[m − ΔnM/2]e−j 2 Δn ejπkΔn . π

m=−∞

tk−1

f−1

i=−Δ

f0 (Δn) =

(k−1) f1 (i)ak−1,n−i

tk Δ 

(k)

The sum for Fj,n is going from −N/2 to N/2 − 1 because, (k) according to (8), the function fj (i) is considered as an acausal FIR filter. Fig. 1 depicts the proposed FFT-FBMC scheme. Unlike in the conventional FBMC, the symbols ak,n are no longer real since they are the IDFT output samples. According to (4), we have (k)

i=−Δ



The proposed FFT-FBMC scheme.

(9)

(10) As a consequence of the orthogonality condition (equation (k) (5)), the coefficients f0 (Δn) are zeros when Δn is a nonzero even integer. Moreover, one can notice from (10) that (k) (k) f0 (−Δn) = f0 (Δn)∗ . Hence, we can easily show that (k) F0,n can be written, for n = 0, ..., N − 1, as (k) F0,n

−1 

N 4

= 1−2j

(k) f0 (2n +1) sin

n =0



2πn  (2n + 1) . (11) N

Therefore, we can derive that (k)

(k)

F0,n − 1 = 1 − F0,|n+N/2| , N

n = 0, ..., N − 1,

where |.|N stands for modulo operation by N . That is, one half of the symbols in each subcarrier are amplified (when (k) (k) |F0,n | > 1), but the rest are deeply faded (when |F0,n | < 1); and this yields a severe performance degradation. The solution that we propose is to transmit no data on the faded positions and double the modulation order (by transmitting complex symbols instead of real ones) on the remaining positions to keep the same throughput. (k) As aforementioned, f0 (Δn) depends on the parity of the subcarrier index k. Let us consider two adjacent subcarriers k and k + 1, we have  π (k+1) (Δn) = g[m]g[m − ΔnM/2]e−j 2 Δn ejπkΔn ejπΔn f0 m (k)

= f0 (Δn)ejπΔn .

(13)

Taking the DFT of both members of this equation, we find that (k+1)

F0,n

(k)

= F0,|n+N/2| . N

(k+1) F0,n

(k)

fj (n )e

−j2πnn N

.

(12)

(k)

Therefore, is the circularly shifted version of F0,n by N samples. Substituting the latter into (12), we obtain 2 (k)

(k+1)

F0,n − 1 = 1 − F0,n

.

ZAKARIA and LE RUYET: A NOVEL FILTER-BANK MULTICARRIER SCHEME TO MITIGATE THE INTRINSIC INTERFERENCE: APPLICATION TO MIMO . . . 1115

of elementary operations per a block of N FBMC symbols, is 5N M log2 ( N2 )+N M for the TX, and 5N M log2 ( M 2 )−2N M for the RX. B. Signal-to-noise ratio (SNR) evaluation According to (15), the average SNR at the output of the demodulator can be written as (k)

SN Rk,n Fig. 2.

Data transmission strategy within a data block.

(k)

(k+1)

This means that when |F0,n | > 1, we have |F0,n | < 1 and (k−1) (k) |F0,n | < 1. Hence, when |F0,n | > 1, we transmit a complex data dk,n on the position (k, n) and we set the positions (k ± (k±1) 1, n) to zero because |F0,n | < 1, and vice versa. Thus, thanks to this transmission strategy, illustrated in Fig. 2, we also remove the interference terms Tk−1 and Tk+1 in (9) since dk±1,n = 0 when dk,n is nonzero. Then, we can write  (k) F0,n dk,n n ∈ Ω(k) ∼ Yk,n = (14) (k1 ) (k+1) F1,n dk−1,n + F−1,n dk+1,n n∈ / Ω(k) where Ω(k) denotes the set of the time indices where we transmit useful data. Hence, Ω(k) = {0, 1, ... N2 − 1} when k is odd, and Ω(k) = { N2 , ...N − 1} when k is even. Therefore, at the RX side we consider only the positions where n ∈ Ω(k) to recover the transmitted symbols dk,n . Thus, the equalization becomes easier since the received symbols are now free of ISI. If we activate only one subcarrier k0 , then the transmitted signal is s[m] = n ak0 ,n gk0 ,n [m]. Hence, the signal PSD is S(f ) ∼ |G(f − k0 /M )|2 Sa (f ), where Sa (f ) is the PSD of ak0 ,n and G(f ) is the Fourier transform of g[m]. Then, the transmitted spectrum is still confined in the prototype filter spectrum. In the presence of the radio channel, we assume that the channel is invariant during N multicarrier symbols, and we can rewrite (14), for n ∈ Ω(k) , as (k) Yk,n ∼ = hk,n F0,n dk,n + Γk,n

(15)

where Γk,n is the noise term at the output of the demodulator, and hk,n is the channel coefficient. Clearly, the added complexity compared to FBMC lies in the extra M IFFTs and FFTs of size N in the TX and the RX, respectively. Since N/2 samples in each subcarrier are zeros at the TX, and also only N/2 samples are needed in each subcarrier at the RX, we can use the pruned IFFT/FFT algorithms to reduce the added complexity. According to Skinner’s algorithm [19], a pruning input sample with length N/2 of an N-point IFFT requires 2N log2 ( N2 ) real multiplications and 3N log2 ( N2 )+N real additions. While Markel’s algorithm [20] shows that a pruning output samples with length N/2 of an N-point FFT requires 2N log2 ( N4 ) real multiplications and 3N log2 ( N2 ) real additions. Therefore, the added complexity in FFT-FBMC compared to FBMC system, in terms of number

E{|hk,n |2 }|F0,n |2 σd2 . = σs2 E{|Γk,n |2 }

(16)

where σd2 denotes the data power, E{.} is the expected value, hk,n is the channel gain and is assumed to be a complex Gaussian random variable with zero mean and unit variance. σs2 is the average signal power at the modulator output used to normalize the transmitted power to unity. Indeed, the FFTFBMC modulator affects the transmitted power and σs2 = σd2 . First of all, let us evaluate σs2 . From (1), we have σs2 [m] = E{s[m]s∗ [m]} =

 M−1 

E{ak,n a∗k ,n }g[m − n

n,n k,k =0

M M ]g[m − n ]ejΦ 2 2 (17)

D π   with Φ = 2πδk M (m− 2 )+ 2 (δk+δn)−π(kn−k n ) is a phase   term, where δk = k − k and δn = n − n . It is obvious that if k  = k, then ak,n and ak ,n are statically independent. That is, E{ak,n a∗k ,n } = 0 for k  = k. Hence, we can simplify (17) as

σs2 [m] =

M−1 

E{ak,n a∗k,n }g[m − n

k=0 n,n

M M ]g[m − n ] 2 2

1

× ejπ( 2 −k)δn .

(18)

Now, let us determine the analytic expression of E{ak,n a∗k,n }. According to the aforementioned transmission strategy, we have ⎧ ⎨ √1 N2 −1 d j2π nl N for k odd k,l,p e N l=0 ∀n ∈ Bp , ak,n = nl N −1 1 j2π ⎩√ N for k even N dk,l,p e l= N 2

where dk,l,p denotes the data dk,l transmitted in the pth block of length N , and Bp is the indices set of the symbols ak,n belonging to the pth block of length N + L, that is, Bp = {(p − 1)(N + L), ..., p(N + L) − 1} . In a closed form and without loss of generality, we rewrite the latter expression as 2 −1 nl 1  √ = dk,l,p ej2π N ejπn(k+1) , N l=0 N

ak,n

∀n ∈ Bp .

Hence, we obtain 2 −1  2 −1 2π(nl−n l ) 1  N = E{dk,l,p d∗k,l ,p }ej N  N

E{ak,n a∗k,n }

N

l=0 l =0

× ejπ(k+1)δn ,

(n, n ) ∈ Bp × Bp .

1116

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 3, MARCH 2012

TABLE I T HE VALUES OF THE AVERAGE OUTPUT SIGNAL POWER σs2 DEPENDING ON THE BLOCK LENGTH N AND THE GUARD INTERVAL L FOR BOTH USED PROTOTYPE FILTERS IOTA AND PHYDYAS.

hhhh hhh Values of L hhhh Values of N h 16 32 64 128

PHYDYAS

IOTA

0

2

4

0

2

4

1.6856 1.7194 1.7342 1.7411

1.6911 1.7208 1.7346 1.7412

1.6955 1.7221 1.7350 1.7413

1.5253 1.5492 1.5598 1.5647

1.5293 1.5503 1.5601 1.5648

1.5325 1.5512 1.5603 1.5649

2 −1 3π 2πlδn M  M  M ej 2 δn g[m − n ]g[m − n ] ej N . N p 2 2 l=0 (n,n )∈Bp2    N

σs2 [m]

=

σd2

(19)

A(m,N,L)

Since the data dk,l,p are assumed to be statically independent, we have E{dk,l,p d∗k,l ,p } = σd2 δl,l δp,p . Therefore, the equation above becomes E{ak,n a∗k,n } =  2 N σd N

2πlδn 2 −1 j N ejπ(k+1)δn , l=0 e

0,

(n, n ) ∈ Bp2 elsewhere

Substituting this last expression into (18), one can obtain the equation (19). We show in Appendix A that σs2 [m] is a periodic function with a period of M 2 (N + L). Then, we consider the average output signal power σs2 as the mean of σs2 [m] over one period and write σs2 = A(N, L)σd2 ,

(20)

where A(N, L) is the mean of A(m, N, L) over one period. In Appendix B, we show that σs2 does not depend on the subcarrier number M . Table I depicts the values of σs2 as a function of N and L for both prototype filters IOTA and PHYDYAS, and setting σd2 = 1. We remind the reader that the goal in this section is to evaluate the SNR at the FFT-FBMC demodulator given by (k) (16). Now, let us evaluate |F0,n |2 and E{|Γk,n |2 }. According to (10), we have (k)

F0,n =

+∞ 

−1 

N 2

g[m]

n =− N 2

m=−∞



  1 2π M g[m − n ]ejπn (k− 2 ) e−j N nn 2   (k)

Gm,n

=

+∞ 

g[m]G(k) m,n .

(21)

m=−∞

Now, let b[m] be the noise term at the RX input. Hence, according to (3) we have γk,n =

+∞ 

2π

b[m]g[m − nM/2]ej M k( 2 −m) e−jφk,n . D

m=−∞

The noise term Γk,n obtained after performing the FFT

operation at the RX is given by N −1  1  2π γk,n e−j N nn , Γk,n = √ N n =0

which yields +∞ 

πkD

ej M Γk,n = √ N

´ (k) , b[m]e−j M km G m,n 2π

m=−∞

N −1   M −jφk,n −j 2π ´ (k) e N nn . After where G m,n = n =0 g[m − n 2 ]e processing, we can easily find that ´ (k) = G(k) ej π2 (N k−2n−k− N2 ) . G m,n m,n Now, let us evaluate the variance of the noise term Γk,n , one can write 2  +∞    1 −j 2π ´ (k) M km  } b[m]G e E{|Γk,n |2 } = E{ m,n  N m=−∞ = =

1 N σ02 N

+∞  m=−∞ +∞ 

´ (k) |2 E{|b[m]|2 } |G m,n 2 |G(k) m,n | ,

(22)

m=−∞

where σ02 denotes the noise power at the demodulator input. Finally, according to (20), (21) and (22), one can rewrite (16) as  +∞ (k) 2  N m=−∞ g[m]Gm,n × +∞ SN R0 , (23) SN Rk,n = (k) 2 A(N, L) m=−∞ |Gm,n | with SN R0 = σd2 /σ02 is the average SNR at the RX input before the power normalization. Since the SNR depends on the time index n, then the symbol-error rate (SER) also depends on the time index n. Let us denote by Pn , the SER at the time index n, with n ∈ Ω(k) . Hence, the overall symbol-error rate Pe is Pe =

2  Pn . N (k) n∈Ω

(24)

ZAKARIA and LE RUYET: A NOVEL FILTER-BANK MULTICARRIER SCHEME TO MITIGATE THE INTRINSIC INTERFERENCE: APPLICATION TO MIMO . . . 1117

TABLE II

T HE VALUES OF THE AVERAGE SNR GAINS

SNReq DEPENDING ON THE BLOCK LENGTH SNR0

FILTERS

hhhh hhh Values of L hhhh Values of N h 16 32 64 128

N AND THE GUARD INTERVAL L FOR BOTH USED PROTOTYPE IOTA AND PHYDYAS. PHYDYAS

0

2

4

0

2

4

1.0040 0.9880 0.9770 0.9709

1.0008 0.9871 0.9768 0.9708

0.9982 0.9864 0.9766 0.9708

1.0051 0.9891 0.9796 0.9745

1.0024 0.9884 0.9794 0.9745

1.0003 0.9878 0.9793 0.9745

In Rayleigh fading channel, the relationship between Pn and SN Rk,n depends on the symbol modulation type and is approximated in the limit of high SNR by [13] α Pn ≈ , (25) 2β × SN Rk,n where α and β are constants that depend on the modulation type. Then, we define an equivalent SNR SN Req , which provides, by (25), the same SER as the overall Pe . Therefore, according to equations (25) and (24), one can easily conclude that

−1  1 N . (26) SN Req ≈ 2 SN Rk,n (k) n∈Ω

Substituting SN Rk,n by its expression given in (23), the equation above becomes SN Req ≈ N2 2A(N, L)

 n∈Ω(k)

IOTA

+∞

−1 (k) 2 m=−∞ |Gm,n | SN R0 .  +∞ (k) 2  g[m]G m,n m=−∞ (27)

Numerical calculations lead us to draw up Table II, where are SN R depicted the values of SN Req0 as a function of N and L for both IOTA and PHYDYAS prototype filters. According to this table, we notice that, practically, there is no SNR loss (at worst 0.13 dB). Then, we can expect that FFT-FBMC exhibits the same performance as OFDM.

where dk = [dk,0 , ..., dk,N −1 ]T , and WH is the N-point IFFT matrix. According to the transmission strategy the data vector dk contains zero elements in either its first or second half depending on the parity of k. When the CP length is shorter than the maximum spread (2Δ), interference terms are added to the term Tk . Let us consider a received block, after CP removal, at the kth subcarrier yk = [yk,0 , ..., yk,N −1 ]T , we can write [15] yk = F0,k ak + r1 + r2 + r3 ,

(28)

− + where r1 = −Aak , r2 = B1 a+ k and r3 = B2 ak , where ak − and ak are respectively the blocks transmitted previously and subsequently at the same kth subcarrier. F0,k is an N × N circulant matrix with entries given by  

N  N (k)  F0,k (p, q) = f0 p − q + 2  − 2 , N

for (p, q) ∈ {0, ..., N − 1}2 . B1 = Tu PL and B2 = Tl P−L , where the upper triangular matrix Tu is given by (k)

Tu (p, q) = f0 (p − q + N + L), with 0 ≤ p ≤ N − 1, p ≤ q ≤ N − 1, and the lower triangular matrix Tl is given by (k)

Tl (p, q) = f0 (p − q − N − L), with 0 ≤ q ≤ N − 1, q ≤ p ≤ N − 1. Matrix A is given by

IV. G UARD I NTERVAL R EDUCTION

A = Tu P−L + Tl PL ,

The main drawback of the proposed scheme is the insertion of the CP in each subcarrier. Despite that the FBMC impulse response is spread over 2Δ = 4K − 2 time periods, we can reduce the CP in order to decrease the spectral efficiency loss but at the expense of performance degradation. In fact, reducing the CP causes an ISI and interblock interference. Clearly, the performance depends on the ratio between the signal and the interference power (SIR). (k) Since the values of f0 (Δn) are known (they depend only on the prototype filter), we can evaluate the SIR for each value of n ∈ Ω(k) and for the different values of L. Equation (9) shows that the received signal can be considered as the sum of three terms. First, let us consider only the first term Tk , and after, the same developments can be applied to the other terms. Let us denote by ak = [ak,0 , ..., ak,N −1 ]T the N-point IFFT output at the kth subcarrier which is expressed by

where PL is a permutation matrix that circularly shifts the columns to the left by L positions. All of B1 , B2 and A are sparse matrices, and when L ≥ 2Δ, these matrices are zero. Demodulating yk by taking the N-point FFT, we obtain the output vector

ak = W dk ,

PL = WH EL W,

H

Yk = Wyk . Replacing B1 , B2 and A by their expressions, we obtain Yk =WF0,k WH dk − WTu P−L WH dk − WTl PL WH dk H − + WTu PL WH d+ k + WTl P−L W dk .

Since the matrix F0,k and the permutation matrices PL and P−L are circulant matrices, we can write F0,k = WH F0,k W and

1118

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 3, MARCH 2012

such that EL and F0,k are diagonal matrices. Hence, the last equation becomes   Yk =F0,k dk − Tu E−L + Tl EL dk



− + T u E L d+ k + Tl E−L dk ,

(29)

where Tu = WTu WH and Tl = WTl WH . The diagonal elements of F0,k are given by −1 

N 2

F0,k (n, n) =

(k)

interference coming from the immediate neighboring subcarriers. Therefore, we consider all the possible terms Tk+l with |l| = 0, ..., Δ , where Δ is the maximum spectrum spread over the subcarriers. Hence, we can finally write Yk as

np

f0 (p)e−j2π N ,

Yk = (F0,k − D)dk + (l)

(l)

Tl

. (l)

Tu (m, n) = and



− Yk = (F0,k − D)dk − Tdk + Tu EL d+ k + Tl E−L dk . (30)

The elements of the matrices Tl = WTl WH and Tu = WTu WH are respectively given by N −1 N −1−q (n−m)q mp 1   (k) Tl (m, n) = f0 (p − L0 )e−j2π N ej2π N , N q=0 p=0

(31) Tu (m, n) =

N −1 N −1−q mp 1   (k+l) (m, n) = f (p − L0 )e−j2π N N q=0 p=0 l

× ej2π

Let us denote by D the diagonal matrix which contains the diagonal elements of the term Tu E−L + Tl EL in (29), and let T = Tu E−L + Tl EL − D. This last matrix represents the ISI in the same data vector dk . Thus, we can rewrite (29) as

N −1 q (n−m)q mp 1   (k) f0 (L0 − p)ej2π N ej2π N , N q=0 p=0

(32) (k)

where L0 =  ∗ N + L. According to (10), we have f0 (p) = (k) f0 (−p) , ∀p ∈ Z. Hence, we can easily show that the entries of matrix T can be expressed as    n−m n−m 2e−jπ N Re G(m, n)ejπ N , n = m T(m, n) = 0, n=m (33) where G(m, n) = N −1 q (n−m)q mp nL 1   (k) f0 (L0 − p)ej2π N ej2π N e−j2π N . N q=0 p=0

Also, we can show that the diagonal matrix D has as elements D(n, n) = 2Re {G(n, n)} ⎫ ⎧ ⎬ ⎨ L+N  np 2 (k) = Re (p − L)f0 (p)e−j2π N . (34) ⎭ ⎩ N p=L+1

Until now, we have considered only the first term Tk in (9). Regarding the other terms, we can proceed in the same way. Moreover, let us relax the assumption made in sub-section III-A which consisted in considering only the

(l)

− where Ql,k = −T(l) dk+l + Tu EL d+ k+l + Tl E−L dk+l , with

and those of EL are given by EL (n, n) = e

(35)

Ql,k ,

l=−Δ

p=− N 2

j2π nL N

Δ 

(l)

T

=

(n−m)q N

,

N −1 q (n−m)q mp 1   (k+l) fl (L0 − p)ej2π N ej2π N , N q=0 p=0

Tu

(0) (l)

E−L + Tl

(0)

(l)

EL − D,

l=0

Tu E−L + Tl EL − Fl,k , l = 0,

(36)

Fl,k is a diagonal matrix with entries given by −1 

N 2

Fl,k (n, n) =

(k+l)

fl

np

(p)e−j2π N .

p=− N 2

Finally, for uncorrelated zero-mean modulation symbols of equal variance, the SIRk,n at the kth subcarrier and time index n is given by   F0,k (n, n) − D(n, n)2 , for n ∈ Ω(k) , (37) SIRk,n = σI2 (k, n) where σI2 (k, n) = 

Δ 



(l)

(l)

|T(l) (n, r)|2 + |Tu (n, r)|2 + |Tl (n, r)|2 .

l=−Δ r∈Ω(k+l)

By the analogy with the definition of SN Req in (26), we consider the equivalent SIR (SIReq ) that provides the same SER floor as the one caused by all the considered SIRk,n . The values of SIReq for some combinations of L and N are depicted in Table III for both considered filters. Since the IOTA filter provides a spectrum spread beyond the immediate adjacent subcarriers, the values of SIReq do not exceed 27 dB according to the table. Indeed, when L = 16, we obtain SIReq = 27 dB for IOTA filter, whereas for PHYDYAS filter, we obtain SIReq −→ +∞. V. A PPLICATION TO MIMO S YSTEMS The motivation of this work is to propose an FBMC system which can be combined easily with MIMO techniques such as SM-MLD and STBC. In this section, we will show how the proposed scheme is straightforwardly applied to SM-MLD and STBC.

ZAKARIA and LE RUYET: A NOVEL FILTER-BANK MULTICARRIER SCHEME TO MITIGATE THE INTRINSIC INTERFERENCE: APPLICATION TO MIMO . . . 1119

TABLE III VALUES OF THE EQUIVALENT SIR DEPENDING ON N AND L

hhhh hhh Values of L hhhh Values of N h 16 32 64 128

⎡

Yk,n

(k) F0,n

⎤

⎢ ⎥ hk,1 ⎢ ⎥ ⎢

∗ ⎥ = h∗ ⎣ Yk,n+1 ⎦ k,2  (k) F0,n+1

PHYDYAS 0 15.57 18.68 21.73 24.76

2 dB dB dB dB

26.11 29.18 32.30 35.22

dB dB dB dB

34 dB 37.05 dB 40 dB 43.06 dB

0 16.41 19.09 21.47 23.42

2 dB dB dB dB

26.52 26.76 26.89 26.96

4 dB dB dB dB

26.96 dB 27 dB 27 dB 27 dB

(n, n + 1) ∈ Ω(k) × Ω(k) .

(38)

F0,n+1

Hk

i=1 (ji) hk,n

where is the channel coefficient between transmit antenna ”i” and receive antenna ”j”. The MIMO channels can be spatially correlated or uncorrelated. Finally, the matrix formulation of the system can be expressed as (k)

Yk,n = F0,n Hk,n dk,n + Γk,n , (N )

4

⎤ ⎡ Γk,n ⎡ ⎤ (k) (1) F0,n ! d ⎥ ⎢ hk,2 ⎢ k,n ⎥ ⎢ ⎥ , + ⎢ ⎣ ⎦

∗ ∗⎥ −hk,1 ⎦ ⎣ (2) Γk,n+1   dk,n (k)

In the case of (Nr × Nt ) spatial multiplexing, we transmit (i) complex symbols dk,n at the ith transmit antenna and at a given time-frequency position (k, n) such that n ∈ Ω(k) . The signal at the jth receive antenna, after demodulation, is given by Nt  (j) (k) (ji) (i) (j) hk,n dk,n + Γk,n , Yk,n = F0,n

(1)

IOTA

(1)

(N )

where Yk,n = [Yk,n , ..., Yk,nr ]T , dk,n = [dk,n , ..., dk,nt ]T , (1) (N ) Γk,n = [Γk,n , ..., Γk,nr ]T , and Hk,n is an (Nr × Nt ) matrix (ji) with entries Hk,n (j, i) = hk,n . Now since we got rid of the interference terms caused by the FBMC modulation, we can apply the MLD in a straightforward manner. This consists just in searching the data ˆ k,n which minimizes the Euclidean distance vector d " "2  " " (k) ˆ dk,n = arg min "F0,n Hk,n dk,n − Yk,n " . dk,n

The sphere decoding can be also applied instead of the basic MLD in order to reduce the complexity especially when large number of antennas or high modulation are used. As for Alamouti coding [6], we consider the basic scheme with two transmit antennas and one receive antenna. In each (1) subcarrier k, a complex symbol dk,n is transmitted at the time (k) from the first antenna whereas the second instant n ∈ Ω (2) antenna transmits a second symbol dk,n . Then, at time instant (2) n+ 1 ∈ Ω(k) , the first antenna transmits (−dk,n )∗ whereas the (1) ∗ second one transmits (dk,n ) . We refer by hk,1 and hk,2 the complex and gaussian channel gains at the kth subcarrier from, respectively, the first and the second antenna to the receive (k) one. According to (15) and since the coefficient F0,n depends on the time instant n, we can write the equation (38). Note that Hk is an orthogonal matrix and 2 2 HH k Hk = (|hk,1 | + |hk,2 | )I2 ,

where I2 is the identity matrix of size 2. Hence, Alamouti (1) coding could be performed, and the data estimates dˆk,n and (2) dˆk,n are obtained by using the maximum ratio combining (MRC) equalization [6]. VI. S IMULATION R ESULTS The performance analysis is based on the bit-error rate (BER) assessment as a function of the SNR. For all the simulations, we assume a perfect channel state information (CSI) at the RX. The sampling period is Ts = 100 ns, and the carrier frequency is fc = 2.5 GHz. The number of subcarriers is M = 512. For the FFT-FBMC, we consider different configurations corresponding to all the combinations of the values of N ∈ {16, 32, 64} and L ∈ {0, 2}. As for OFDM, the CP size (Lcp ) is adapted to the channel spread length. We define the spectral efficiency loss by μ = NL+L for the FFTLcp FBMC system, whereas for OFDM μ = M+L . We assume cp pedestrian scenario where the velocity is v ≈ 3 km/h. For this application target, the coherence time is Tc ≈ c/(2fc v) ≈ 72 ms. We define Nc = Tc /( M 2 Ts ) as the coherence number of multicarrier symbols, thus, Nc ≈ 2800. Since N  Nc , we can pretend that the channel is invariant within an N-block. We test the FFT-FBMC and compare it to OFDM in both MIMO contexts presented in section V, namely: Spatial Multiplexing with ML detection and Alamouti coding scheme. Regarding SM-MLD, we considered a basic configuration with two spatially uncorrelated antennas at both TX and RX. First, we consider the Pedestrian-A channel [18] where the parameters are given by: • Delays = [0 110 190 410] ns, • Powers = [0 -9.7 -19.2 -22.8] dB. Hence, for OFDM, we set Lcp = 5 (μ ≈ 0.97%). The data are QPSK modulated. Fig. 3 and Fig. 4 show the performance of the FFT-FBMC, respectively, for PHYDYAS and IOTA filters and give a comparison with OFDM and the conventional FBMC using the method of [12] based on interference estimation and cancelation. For both IOTA and PHYDYAS filters, we observe clearly that the BER degradation depends strongly on L due to the BER floor effect caused by the insufficiency of the CP. We

1120

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 3, MARCH 2012

0

0

10

10

−1

−1

10

10

L=0

−2

−2

10

BER

BER

10

−3

10

N=16 L=0 (=0%) N=16 L=2 (=11.11%) N=32 L=0 (=0%) N=32 L=2 (=5.88%) N=64 L=0 (=0%) N=64 L=2 (=3.03%) OFDM (=0.97%) FBMC (=0%)

−4

10

−5

10

FBMC (=0%) N=16 L=2 (=11.11%) N=32 L=2 (=5.88%) N=64 L=2 (=3.03%) N=16 L=0 (=0%) N=32 L=0 (=0%) N=64 L=0 (=0%) OFDM (=4.66%)

−4

10

L=2

−5

10

−6

10

−3

10

−6

0

5

10

15 SNR (dB)

20

25

10

30

Fig. 3. Performance of FFT-FBMC using PHYDYAS prototype filter with MIMO (2×2) spatial multiplexing and QPSK modulation in Ped-A channel.

0

5

10

15 SNR(dB)

20

25

30

Fig. 5. Performance of FFT-FBMC using PHYDYAS prototype filter with MIMO (2×2) spatial multiplexing and QPSK modulation in Veh-A channel without mobility.

0

10

0

10

−1

10

−1

−3

10

BER

BER

10

L=0

−2

10

N=16 L=0 (=0%) N=16 L=2 (=11.11%) N=32 L=0 (=0%) N=32 L=2 (=5.88%) N=64 L=0 (=0%) N=64 L=2 (=3.03%) OFDM (=0.97%) FBMC (=0%)

−4

10

−5

10

−2

10

FBMC (=0%) OFDM (=0.97%) N=16 L=0 (=0%) N=32 L=0 (=0%) N=64 L=0 (=0%) N=16 L=2 (=11.11%) N=32 L=2 (=5.88%) N=64 L=2 (=3.03%)

−3

10

L=2

−6

10

0

5

10

15 SNR (dB)

20

25

30

Fig. 4. Performance of FFT-FBMC using IOTA prototype filter with MIMO (2×2) spatial multiplexing and QPSK modulation in Ped-A channel.

notice that the BER floor is much higher when there is no CP inserted (L = 0). We also notice that the BER floors lower by increasing N . This is explained by the fact that the power of the interference within a block caused by an insufficient CP remains almost the same whatever the value of N > 2Δ. However, after applying the FFT at the RX, this interference is scattered over the whole block. Hence, when we double N , the distribution of the interference power is halved, and thus, the SIR is also doubled (+3 dB). This matches well with the values in Table III; we notice, for PHYDYAS filter, that for each L, the SIR is incremented by about +3 dB when N is doubled. As for IOTA filter, the situation is different because the SIR is limited by the ICI caused by the subcarriers k ± 2. The SIR values corresponding to the BER limits match well with Table III. One can check this by projecting the BER floor values on the OFDM curve and taking the corresponding SNR values. Since the SIR values are relatively high when L = 2, the BER floor would be observed beyond 30 dB. Hence, as long as the SNR is less than 30 dB, we can consider that the FFT-

−4

10

0

5

10

15

20 SNR (dB)

25

30

35

40

Fig. 6. Performance of FFT-FBMC using PHYDYAS prototype filter with MIMO (2×2) spatial multiplexing and 16-QAM modulation in Ped-A channel.

FBMC with L = 2 exhibits almost no degradation compared to OFDM and outperforms the FBMC system. We notice that the curves of the FFT-FBMC/IOTA with L = 2 are almost independent of N . This matches well with Table III which shows that FFT-FBMC/IOTA has a SIR limit at 27 dB. Now, we test the FFT-FBMC/PHYDYAS with a more frequency selective channel and QPSK modulation, we chose the same channel parameters as the Vehicular-A model (without considering the velocity), that is: • Delays = [0 300 700 1100 1700 2500] ns, • Powers = [0 -1 -9 -10 -15 -20] dB. Then, Lcp = 25 (μ ≈ 4.66%). Fig. 5 shows that we obtain, for all the configurations, almost the same BER performance as in the case of Pedestrian-A channel model. Fig. 6 and Fig. 7 depict the BER performance of FFTFBMC/PHYDYAS scheme with 16-QAM modulated data, respectively, in the Ped-A and Veh-A channel model. Clearly, the BER floor effects are more significant. Indeed, FFT-FBMC with L = 0 has a very high BER limits. However, we can

ZAKARIA and LE RUYET: A NOVEL FILTER-BANK MULTICARRIER SCHEME TO MITIGATE THE INTRINSIC INTERFERENCE: APPLICATION TO MIMO . . . 1121

0

0

10

10

N=16 L=0 (=0%) N=16 L=2 (=11.11%) N=32 L=0 (=0%) N=32 L=2 (=5.88%) N=64 L=0 (=0%) N=64 L=2 (=3.03%) OFDM (=0.97%)

−1

10 −1

10

−2

BER

BER

10

−2

10

FBMC (=0%) OFDM (=4.66%) N=16 L=0 (=0%) N=32 L=0 (=0%) N=64 L=0 (=0%) N=16 L=2 (=11.11%) N=32 L=2 (=5.88%) N=64 L=2 (=3.03%)

−3

10

−4

10

−5

10

−4

10

−6

0

5

10

15

20 SNR (dB)

25

30

35

40

Fig. 7. Performance of FFT-FBMC using PHYDYAS prototype filter with MIMO (2×2) spatial multiplexing and 16-QAM modulation in Veh-A channel without mobility.

N=16 L=0 (=0%) N=16 L=2 (=11.11%) N=32 L=0 (=0%) N=32 L=2 (=5.88%) N=64 L=0 (=0%) N=64 L=2 (=3.03%) OFDM (=0.97%)

−1

10

−2

10

BER

10

0

5

10

15 SNR (dB)

20

25

30

Fig. 9. Performance of FFT-FBMC using PHYDYAS prototype filter with (2×1) Alamouti coding scheme and QPSK modulation in Ped-A channel.

N = 16) that SN Req is slightly smaller than SN R0 .

0

10

VII. C ONCLUSION

−3

10

−4

10

−5

10

−6

10

−3

10

0

5

10

15 SNR (dB)

20

25

30

Fig. 8. Performance of FFT-FBMC using IOTA prototype filter with (2×1) Alamouti coding scheme and QPSK modulation in Ped-A channel.

obtain acceptable performance with L = 2. As for the Alamouti scheme, we provide the performance results for PHYDYAS and IOTA filters, respectively, in Fig. 8 and Fig. 9. We notice that with IOTA filter, we obtain almost the same performance as OFDM for both values of L = 0 and L = 2. A very slight degradation with respect to OFDM can be pointed out in high SNR regime. Unlike the SM case, the nonuse of the CP has not resulted in considerable degradations. One can notice, at worst, about 1 dB of SNR loss at BER = 10−4 . Regarding PHYDYAS filter, we obtain also almost no degradation compared to OFDM, except the case when N = 16 and L = 0 where we have about a 2.75 dB SNR loss w.r.t OFDM at BER = 10−4 , and less than 1 dB at BER = 10−2 . In all the BER figures, one can notice that OFDM outperforms the FFT-FBMC system in the whole SNR region considered. This is due to the ISI caused by the insufficient CP considered in the system and the neglected ISI in (8). Further, the values of the SN Req /SN R0 depicted in table II show (except when

In this paper, we have proposed a new FBMC scheme (called FFT-FBMC) in order to get rid of the intrinsic interference which is an issue when we combine the FBMC with some MIMO techniques such as SM-MLD and Alamouti coding. The FFT-FBMC scheme consists in performing an IDFT and DFT on each subcarrier, respectively, at the TX and RX sides inserting also a CP as in the conventional OFDM. This makes FFT-FBMC more computationally complex than FBMC. The transmission strategy depicted in Fig. 2 is applied in order to isolate the adjacent subcarriers. In this way, the equivalent system became formulated as OFDM, and all MIMO techniques can be applied in a straightforward manner. We have proposed to reduce the CP and evaluate the corresponding performance degradation. We tested the proposed scheme with two MIMO techniques: (2×2) SM-MLD and (2×1) Alamouti coding, and also with two values of the CP L = 2 and L = 0. Simulation results showed that we can almost obtain the same performance as OFDM in some configurations. However, FFTFBMC remains slightly outperformed by OFDM because the residual interference. A PPENDIX A: P ERIODICITY OF σs2 [m] Let m = m + M 2 (N + L). According to (19), we have the expression written in (39). Let n1 = n − N − L, and n1 = n − N − L. When n ∈ Bp = {(p − 1)(N + L), ..., p(N + L) − 1}, then n1 ∈ Bp−1 . Since the difference n1 − n1 also equals δn and substituting p by p + 1, σs2 [m ] can be written as in (40). Hence, ∀m,

σs2 [m +

M (N + L)] = σs2 [m]. 2

That is, σs2 [m] is periodic with a period of

M 2 (N

+ L).

1122

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 3, MARCH 2012

σs2 [m ]

=

M σd2 N

2 −1 2πlδn M M   g[m − (n − N − L) ]g[m − (n − N − L) ] ej N 2 2 N





e

j 3π 2 δn

p (n,n )∈Bp2

M  σs2 [m ] = σd2 N p

 2 A(N, L) = N (N + L) p

(39)

l=0

ej

3π 2

(n1 ,n1 )∈Bp2



2 −1  M 2πlδn δn  M g[m − n1 ]g[m − n1 ] ej N 2 2 N



e

j 3π 2 δn

(n,n )∈Bp2

(40)

l=0

−1 

N 2

l=0

e

j 2πlδn N

M 2

(N +L)−1



M M ]g[m − n ] 2 2  

g[m − n

m=0



(41)

In,n

A PPENDIX B: I NDEPENDENCE OF σs2 ON M Let us calculate the mean of A(m, N, L) over one period, thus we obtain (41). The term In,n is certainly zero when g[m − n

M M ]g[m − n ] = 0 2 2

between m = 0 and m = M 2 (N + L) − 1. Since g[m] = 0 for m ∈ / {1, ..., KM − 1}, then In,n = 0  when (n, n ) ∈ / Ψ = {1 − 2K, ..., N + L − 1}2 . Ψ overlaps only with B0 and B1 , and the intersection is C1 ∪ B12 , where C1 = {1 − 2K, ..., −1}2. Let C3 = {N + L − 2K + 1, ..., N + L − 1}2 be a subset of B12 , and C2 be the relative complement of C3 in B12 , that is, C2 = B12 \C3 = ({1, ..., N + L − 2K} × B1 ) ∪ (B1 × {1, ..., N + L − 2K}). When (n, n ) ∈ C2 , it means that n ∈ {1, ..., N + L − 2K} or n ∈ {1, ..., N + L − 2K}. As a consequence, the M term g[m − n M 2 ]g[m − n 2 ] is zero beyond the summation M set {0, ..., 2 (N + L) − 1}. Hence, In,n equals the FBMC (k) coefficient f0 (Δn) to within a phase rotation. Then, In,n is a constant independent on M when (n, n ) ∈ C2 . When (n, n ) ∈ C1 , we can show that the term 

c1 =

e

j 3π 2 δn

(n,n )∈C1

−1 

N 2

ej

2πlδn N

In,n

l=0

is independent on M . Let n1 = −2K −n and n1 = −2K −n. We have (n, n ) ∈ C1 ⇐⇒ (n1 , n1 ) ∈ C1 . Because of the prototype filter symmetry (g[m] = g[KM − m − 1]), In1 ,n1 is written as M 2

In1 ,n1 =

(N +L)−1



g[m + n

m=0 M 2

= =

M M + KM ]g[m + n + KM ] 2 2

(N +L)−1



g[−m − n

m=0 −1  m=− M 2 (N +L)

M M − 1]g[−m − n − 1] 2 2

g[m − n

M M ]g[m − n ]. 2 2

Hence, M 2

In,n + In1 ,n1 =

(N +L)−1



g[m − n

m=− M 2 (N +L)

M M ]g[m − n ], 2 2

M and it is independent on M because g[m−n M 2 ]g[m−n 2 ] = M  0 for all |m| ≥ 2 (N + L) when (n, n ) ∈ C1 . Now, since δn1 = n1 − n1 = n − n = δn, we can write



c1 =

e

j 3π 2 δn

−1 

N 2

(n,n )∈C1

1 = 2

ej

2πlδn N

In,n

l=0



e

j 3π 2 δn

(n,n )∈C1

−1 

N 2

ej

2πlδn N

(In,n + In1 ,n1 ).

l=0

Therefore, c1 is independent on M . As for the last term c3 =



e

(n,n )∈C3

j 3π 2 δn

−1 

N 2

ej

2πlδn N

In,n ,

l=0

we proceed in the same manner by considering n2 = 2(N + L − K) − n and n2 = 2(N + L − K) − n. We will also find that In,n + In2 ,n2 is independent on M when (n, n ) ∈ C3 . Finally, we have shown that all the three terms corresponding to the summation sets C1 , C2 , and C3 are independent on M . Then, A(N, L) is also independent on M . R EFERENCES [1] B. Le Floch, M. Alard, and C. Berrou, “Coded orthogonal frequency division multiplex,” Proc. IEEE, vol. 83, no. 6, pp. 982–996, June 1995. [2] P. Siohan, C. Siclet, and N. Lacaille, “Analysis and design of OFDM/OQAM systems based on filterbank theory,” IEEE Trans. Signal Process., 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. 2001 IEEE Int. Conf. Acoustics, Speech, and Signal Process., pp. 2417-2420. [4] B. Farhang-Boroujeny and C. H. Yuen, “Cosine modulated and offset QAM filter bank multicarrier techniques: a continuous-time prospect,” EURASIP J. Advances in Signal Process., vol. 2010, Jan. 2010. [5] M. Alard, “Construction of a multicarrier signal,” Patent WO96/35 278, 1996. [6] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Sel. Areas Commun., vol. 16, no. 18, pp. 1451–1458, 1998.

ZAKARIA and LE RUYET: A NOVEL FILTER-BANK MULTICARRIER SCHEME TO MITIGATE THE INTRINSIC INTERFERENCE: APPLICATION TO MIMO . . . 1123

[7] B. R. Saltzberg, “Performance of an efficient parallel data transmission system,” IEEE Trans. Commun. Technol., vol. 15, no. 6, pp. 805–811, Dec. 1967. [8] B. Hirosaki, “An orthogonally multiplexed QAM system using the discrete Fourier transform,” IEEE Trans. Commun., vol. 29, no. 7, pp. 982–989, July 1981. [9] M. El Tabach, J. P. Javaudin, and M. H´elard, “Spatial data multiplexing over OFDM/OQAM modulations,” in Proc. 2007 IEEE Int. Conf. Commun., pp.4201–4206. [10] D. Katselis, E. Kofidis, A. Rontogiannis, and S. Theodoridis, “Preamblebased channel estimation for CP-OFDM and OFDM/OQAM systems: a comparative study,” IEEE Trans. Signal Process., vol. 58, no. 5, pp. 2911–2916, May 2010. [11] C. L´el´e, J. P. Javaudin, R. Legouable, A. Skrzypczak, and P. Siohan, “Channel estimation methods for preamble-based OFDM/OQAM modulations,” in Proc. 2007 European Wireless Conference. [12] R. Zakaria, D. Le Ruyet, and M. Bellanger, “Maximum likelihood detection in spatial multiplexing with FBMC,” in Proc. 2010 European Wireless Conf., pp. 1038–1041. [13] A. Goldsmith, “Performance of digital modulation over wireless channels,” in Wireless Communications, Cambridge University Press, 2005. [14] C. L´el´e, P. Siohan, and R. Legouable, “The Alamouti scheme with CDMA-OFDM/OQAM,” EURASIP J. Advances Signal Process., vol. 2010, ID 703513. [15] S. Chen and C. Zhu, “ICI and ISI analysis and mitigation for OFDM systems with insufficient cyclic prefix in time-varying channels,” IEEE Trans. Consum. Electron., vol. 50, no. 1, pp. 78–83, Feb. 2004. [16] H. Lin, C. L´el´e, and P. Siohan, “A pseudo Alamouti tranceiver design for OFDM/OQAM modulation with cyclic prefix,” in Proc. 2009 IEEE Workshop Signal Processing Advances Wireless Communications, pp. 300–304. [17] M. Renfors, T. Ihalainen, and T. H. Stitz, “A block-Alamouti scheme for filter bank based multicarrier transmission,” in Proc. 2010 European Wireless Conf., pp. 1038–1041. [18] ITU-R M.1225, “Guidelines for evaluations of radio transmission technologies for IMT-2000,” 1997.

[19] D. P. Skinner, “Pruning the decimation-in-time FFT algorithm,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-24, pp. 193–194, Apr. 1976. [20] J. D. Markel, “FFT pruning,” IEEE Trans. Audio Electroacoust, vol. AU-19, pp. 305–311, Dec. 1971. Rostom Zakaria was born in 1983 in Algeria. He received the Eng. Degree in electronic from the University of Science and Technology Houari Boumediene (USTHB), Algiers, Algeria, in 2007. He then received the M.Sc. degree from Universit´e Paris-est de Marne-La-Vall´ee, France, in 2008. Currently, he is working toward the Ph.D. degree since 2008 in the CEDRIC/LAETITIA research laboratory at Conservatoire National des Arts et M´etiers (CNAM), Paris. His current research interests are the multi-antenna techniques and the filter bank based multicarrier modulations. Didier Le Ruyet received the Eng. Degree and the Ph. D. Degree from Conservatoire National des Arts et M´etiers (CNAM) in 1994 and 2001 respectively. From 1988 to 1996 he was a research engineer in the image processing and telecommunication Departments of SAGEM, France. He joined Signal and Systems Laboratory, CNAM Paris as a research assistant in 1996. From 2002 to 2009, he was an assistant professor with the Electronic and Communication Laboratory, CNAM Paris. Since 2010 he is full professor at CNAM in the CEDRIC research laboratory. He has published about 60 papers in referred journals and conferences. His main research interests lie in the areas of digital communications and signal processing including advanced channel coding, detection and estimation algorithms and multi-antenna transmission techniques for multiuser systems.