Single-tap equalizer for MIMO FBMC systems under doubly selective channels

Offset-QAM-based filterbank multicarrier (FBMC-OQAM) modulations are known to progressively loose their orthogonality as the channel gets more selective in time and frequency. The effect of channel frequency selectivity on FBMC-OQAM systems has been extensively studied in the literature. Many compensations methods have been proposed to combat it. However, most of them have a significant implementation complexity and do not take into account the time selective nature of the channel. In this paper, we propose a MIMO equalizing structure for doubly selective channel based on a simple single-tap per-subcarrier decoding matrix. The decoding matrices are designed to minimize the mean squared error of the symbol estimate. This decoder exploits the degrees of freedom offered by the extra antennas at the receiver to compensate for the distortion induced by time and frequency selectivity. Simulation results demonstrate the performance gain of the proposed design with respect to classical designs.


INTRODUCTION
Offset-QAM-based filterbank multicarrier (FBMC-OQAM) modulations are seen as an interesting alternative to orthogonal frequency division multiplexing, thanks to its increased spectral efficiency and its good time-frequency localization [1].One of the drawbacks of FBMC-OQAM occurs when the channel gets strongly selective in time and/or in frequency, which progressively destroys the FBMC-OQAM orthogonality due to inter-carrier interference (ICI) and inter-symbol interference (ISI).
The techniques to deal with channel frequency selectivity have been extensively studied in numerous works, in the SISO case [2][3][4][5] and in the MIMO case [6][7][8].Most of those approaches are based on the design of multi-tap fractionally spaced equalizers.The work originally devised for the SISO case in [9] and later extended for the MIMO case in [10] proposes instead a parallel multi-stage processing architecture at both sides of the communication link.
Conversely, equalization for time selective channels has only been studied in very few works.In [11] and [12], the authors propose an adaptive equalizer for doubly selective channels, that tracks the channel variations in time.In this paper, we propose a very simple ˚The research reported herein was partly funded by Fonds pour la Formation à la Recherche dans l'Industrie et dans l'Agriculture (F.R.I.A.) and by the Catalan and Spanish governments under grants 2014SGR1567 and TEC2014-59255-C3-1.
design based on single-tap per-subcarrier decoding matrix, which extends the work of [13,14] to doubly selective channels.The equalizing matrices are designed to minimize the mean squared error (MSE) of the symbol estimate at each subcarrier and each multicarrier symbol of interest, taking into account a first order approximation of the distortion caused by channel time and frequency selectivity.It is shown that as soon as the receiver has more antennas than the number of transmitted streams, it can tune those extra degrees of freedom to compensate for the channel distortion.

Notations
Vectors and matrices are denoted by bold lowercase and uppercase letters, respectively (resp.).Superscripts ˚, T and H stand for conjugate, transpose and Hermitian transpose operators.The symbols tr, E, and denote the trace, expectation, imaginary and real parts, respectively. is the imaginary unit.IN denotes the identity matrix of size N ˆN .b stands for the Kronecker product.Symbol O pxq denotes a matrix of possibly increasing dimensions whose entries remain bounded when x Ñ 0.

MSE FORMULATION FOR FBMC-OQAM UNDER
CHANNEL FREQUENCY SELECTIVITY

System Model
A FBMC-OQAM transceiver with NT transmit antennas and NR receive antennas is considered, as depicted in Fig. 1.Pure spatial multiplexing is assumed, so that the number of streams is equal to NT , with NR ě NT .The number of subcarriers is denoted by 2M and the number of real-valued multicarrier symbols is denoted by 2Ns.
The real-valued transmitted symbols, denoted by d m,l P R N T ˆ1, are FBMC-OQAM modulated using a prototype pulse prns of length Lp " 2κM , where κ is the overlapping factor.The transmitted signal srns P C N T ˆ1 can be written as for n " 0, . . ., p2Ns ´1qM `Lp ´1 and where p m,l rns " j l`m prn ´lM se j 2π 2M mpn´L p´1 2 q .We denote by Hrb, ns P  rrns is FBMC-OQAM demodulated using prototype pulse grns of length Lg " 2κM .The signal after demodulation, at subcarrier l0 and multicarrier symbol m0, denoted by z m 0 ,l 0 P C N R ˆ1, may be written as where g m,l rns " j l`m grn ´lM se j 2π 2M mpn´L g ´1 2 q .To compensate for the distortion effect of the channel, we use the low complex classical approach, based on a single-tap equalizing matrix B m 0 ,l 0 P C N T ˆNR .The transmitted symbol is then estimated by taking the real part, i.e., dm 0 ,l 0 " pB m 0 ,l 0 z m 0 ,l 0 q.In classical approaches, the channel is assumed not to be very selective and matrix B m 0 ,l 0 is designed to invert the channel matrix at the subcarrier and multicarrier symbol of interest given by 2M bm 0 , which means that we have B m 0 ,l 0 H m 0 ,l 0 " IN T .If the channel is not too selective and for a moderate signal-to-noise ratio (SNR), one can expect to have dm 0 ,l 0 « d m 0 ,l 0 .

MSE Expression Under Strong Channel Selectivity
If the channel is highly selective in time and/or in frequency, the channel variations will break the FBMC-OQAM orthogonality and distortion will be superimposed on the demodulated symbol, mainly due to ICI and ISI.We propose below an approximation of the MSE at the subcarrier m and multicarrier symbol l, that we define as We divided the MSE expression into two contributions, namely, the distortion power P d pm, lq and the noise power Pnpm, lq.The distortion term, including the ISI and ICI power, comes from channel selectivity or in other words the fact that the channel is not flat and constant at the subcarrier and multicarrier symbol of interest.The noise term is simply due to the additive noise and is given by (for the receive pulse energy normalized to one) Since the noise and symbol samples are uncorrelated, their effect can be studied separately.To propose an analytical and compact approximation of P d pm, lq, we make the following assumptions: pAs1q The transmit and receive prototype pulses prns and grns are of the perfect reconstruction type [15], with energy normalized to one.They are either symmetric or anti-symmetric.Furthermore, grns is obtained by the discretization of a smooth real-valued analog waveform gptq with bounded derivatives, so that pAs2q The channel is assumed to be perfectly known by the receiver.To characterize the time variations of the channel, we choose to specify the discrete delay-Doppler spectrum of the channel Prb, νs, which equivalently describes Hrb, ns: . Prb, νs is assumed to be compactly supported in delay and Doppler, i.e., Prb, νs is non zero only for b " 0, . . ., L b and ν " ´Lν , . . ., Lν .Given the maximal delay τmax and maximal Doppler shift f d of the physical channel, the parameters L b and Lν are defined as L b " t τmax2M T u and Lν " tf d N 1 s T u with T being the multicarrier symbol period.Then, we can define the time-variant frequency response of the channel as given in terms of normalized time and frequency, i.e., t P r0, 1s, ω P r0, 2πs.The quantities H pq,rq m,l are defined as H pq,rq m,l " d q dω q d r dt r Hpω, tq ˇˇˇω p´bq q p2πνq r Prb, νse , for l " 0, . . ., 2Ns ´1, m " 0, . . ., 2M ´1.Note that H p0,0q m,l " H m,l .
pAs3q The real-valued symbols d m,l are bounded, independent and identically distributed random variables with zero mean and variance Ps{2.
pAs4q The channel is inverted at the receiver, i.e., for l " 0, . . ., 2Ns ´1, m " 0, . . ., 2M ´1.This implicitly assumes that the channel H m,l is full rank.Under this assumptions, we propose the following approximation of P d pm, lq.Proposition 2.1.Under pAs1q-pAs4q, as τmax T ÝÑ 0, T f d ÝÑ 0, we can write where the different η's are pulse related quantities properly defined in the Appendix.
Proof.The MSE approximation is obtained by a first order Taylor approximation of the demodulated signal with respect to the time and frequency variations of the channel.The details of the proof are omitted due to space constraints.
Note that, the previous expression is an approximation, that holds true asymptotically as τmax T ÝÑ 0, T f d ÝÑ 0. In practice, T can be both considered large w.r.t.τmax and small w.r.t.f d such that the approximation is very accurate.

DECODER DESIGN TO COMPENSATE FOR THE CHANNEL SELECTIVITY
The goal now is to optimize the above MSE formula to obtain the expression of the optimal decoder in the asymptotic regime.Due to the channel inversion constraint (1), the multi-stream interference is mitigated.The optimal decoder should then compensate for time and frequency selectivity of the channel while keeping the noise level low enough.For the sake of clarity, in the following, the subcarrier and multicarrier symbol indexes will be omitted, such that all timefrequency depending quantities should be understood as evaluated at the subcarrier and multicarrier symbol of interest.The general expression of any decoder that satisfies (1) is given by B " H : `BP : where H : " `HH H ˘´1 H H , P : " IN R ´HH : and B is a matrix left to be optimized.The above expression of B shows that the optimal decoder can be written as the pseudo inverse of the channel plus a matrix lying on the left null space of H.In the case NR " NT , it is obvious to see that B is fixed to the inverse of the channel and there are no extra degrees of freedom.On the contrary, as NR increases relatively to NT , there is more and more degrees of freedom left to optimize and one can expect a better compensation of the noise and distortion.Note that the assumption that NR ą NT makes for instance particular sense in the uplink of cellular networks where the number of base stations could be drastically increased in a massive MIMO scenario [16].

Decoder Optimization
By using the expression of P d pm, lq and the approximation of Pnpm, lq derived above, the optimization problem can be formulated as follows, where matrix Ψ is given by This problem is a quadratic form in the variable B which can be easily solved by differentiation with respect to B˚a nd setting the derivative to zero.The expression of the optimized decoder is then given by

Asymptotic Behavior at Low and High SNR
It is easy to see that at low SNR, i.e., when N0 ÝÑ `8, the optimized decoder converges to the classical pseudo inverse of the channel B " H : .Indeed, in that regime, the noise power is dominant relatively to the distortion power and the best to do is to invert the channel combining the signals coming from each receive antenna to prevent noise amplification.
As explained above, the distortion will be better compensated as NR increases.One could be interested to know the number of receive antennas NR required to completely compensate the distortion caused by the channel selectivity in the high SNR regime, i.e., when N0 Ñ 0. In [13], it was shown that, when the receiver has twice as many antennas as the number of transmit antennas, it can completely remove the first order approximation of the distortion at high SNR.However, in that work, a quasi-static channel was assumed such that only the effect of frequency selectivity was regarded.Here, the channel time selectivity should also be mitigated.
When N0 Ñ 0, the MSE is given by the distortion power only, MSE " BΨB.Hence, in order to completely remove the first order distortion power, matrix B should on the one hand satisfy (1) and on the other hand lie in the null space of Ψ.Let us define the i-th decoding vector as b H i " e H i B where ei is the i-th column of the identity matrix For this problem to be feasible, NR has to be larger than the number of linearly independent constraints.We will assume that all matrices are full rank.Depending on the channel selectivity, we differentiate three cases: • No channel selectivity (H p0,1q " H p1,0q " 0): in this trivial case, the MSE is zero by assumption (NR ě NT ).
• Channel with frequency or time selectivity only (H p0,1q " 0 or H p1,0q " 0): in that case, at least 2NT receive antennas are required, as was the case in [13].
• Channel with time and frequency selectivity: in that case, at least 3NT receive antennas are required.

SIMULATION RESULTS
This section aims at demonstrating the performance of the proposed decoder design with respect to classical designs.By "classical" decoder, we refer to single-tap per-subcarrier decoding matrices that assume a constant and flat channel at the subcarrier level, i.e., the pseudo inverse of the channel, B m,l " H : m,l .We consider an FBMC-OQAM system with 2M " 128 subcarriers and a frame transmission composed of 2Ns " 100 real multicarrier symbols.The subcarrier spacing is fixed to 1{T " 15kHz as in LTE systems.The transmit and receive pulses are the Phydyas prototype pulse [17].Note that this pulse is only of the nearly-perfect-reconstruction type and does not fulfill pAs1q.However, given that it almost fulfills the PR constraints, the approximation of the distortion in Prop.2.1 will remain very accurate, as will be shown in the following.In the simulations, one channel realization is drawn and the performances are averaged over multiple frame transmission where data and noise samples are regenerated.The SNR of the system is 35dB.The delay-Doppler discrete channel coefficients Prb, νs are assumed zero-mean and independent with variance which models an exponentially decaying power delay profile and a classical Jake's Doppler spectrum.The constants α b and αν are chosen to normalize ř b p 2 b rbs and ř ν p 2 ν rνs to one.In the simulations, we fixed τmax " 5µs and f d " 400Hz.
Fig. 2 and Fig. 3 show the MSE in an NR " 4, NT " 2 scenario at one subcarrier and one multicarrier symbol respectively.As can be seen, the optimized decoder has a large gain of performance over the classical one.One can further check that the crosses that represent the simulated MSE perfectly match the theoretical one in dashed/solid lines, which validates the accuracy of the approximation of Prop.2.1.NT " 2 and NR " 6 antennas is drawn and the performance of the receiver using only two, four or all its six antennas is shown.One can check that in the NR " NT " 2, the performance of the classical and optimized decoders is the same, which is logical since they are the same (no extra degrees of freedom at the receiver).As the number of receive antennas increases, the distortion cannot be compensated for efficiently with a classical decoder while the optimized decoders can use their extra antennas to mitigate the distortion.

CONCLUSION
This work investigated the design of equalizers for MIMO doubly selective channels.The proposed design is based on a single-tap per-subcarrier decoding matrix and has a very low complexity.It is shown that, as soon as the receiver has more antennas than the number of streams, these extra degrees of freedom can be used to compensate for the distortion induced by the channel selectivity.

APPENDIX
We here define the pulse-related quantities η pq 1 ,r 1 q,pq 2 ,r 2 q appearing in Proposition 2.1.Given two generic pulses, p, q of length 2M κ and let P and Q denote two 2M ˆκ matrices obtained by arranging the samples of the respective pulses in columns from left to right.We will define Rpp, qq " P f J 2M Q Spp, qq " pJ2 b IM q P f J 2M Q where f denotes row-wise convolution, b denotes Kronecker product, IM (resp.JM ) are the identity (resp.exchange) matrices of order M .We define η pq 1 ,r 1 q,pq 2 ,r 2 q as η pq 1 ,r 1 q,pq 2 ,r 2 q " PsM 2 tr " U `R ´p, g pq 1 ,r 1 q ¯RT ´p, g pq 2 ,r 2 q Ū´S ´p, g pq 1 ,r 1 q ¯ST ´p, g pq 2 ,r 2 q ¯ı where U ˘" I2 b pIM ˘JM q and g pq,rq rns is defined as the sampling of the function g pq,rq ptq " d q dt q pt r gptqq, g pq,rq rns " g pq,rq ˆˆn ´2M κ ´1 2 ˙1 2M ˙.

Fig. 2 .
Fig. 2. SNDR at multicarrier symbol l " 35 as a function of the subcarrier.Crosses correspond to simulated SNDR and solid/dashed lines to the theoretical proposed approximation.

Fig. 3 .
Fig. 3. SNDR at subcarrier m " 35 as a function of the multicarrier symbol.Crosses correspond to simulated SNDR and solid/dashed lines to the theoretical proposed approximation.

Fig. 4
plots the cumulative density function (CDF) of the MSE of the classical and optimized decoders.A channel realization for

Fig. 4 .
Fig.4.CDF of the MSE for different number of antennas at the receiver using a classical decoder or the optimized one.