Downlink Precoder and Equalizer Designs for Multi-User MIMO FBMC/OQAM

Transcription

1 WSA 016 March 9-11, 016, Munich, Germany Downin Precoder and Equaizer Designs for Muti-User MIMO FBMC/OQAM Oier De Candido, Sher Ai Cheema, Leonardo G Batar, Martin Haardt, Josef A Nosse Institute for Circuit Theory and Signa Processing, Technische Uniersität München 8090 Munich, Germany, Emai: {oierde-candido, eobatar}@tumde Communications Research Laboratory, Imenau Uniersity of Technoogy PO Box , D Imenau, Germany, Emai: {sher-aicheema, martinhaardt}@tu-imenaude Next Generation and Standards, Inte Deutschand GmbH Am Campeon 10-1, Neubiberg, Germany, Emai: eobatar@intecom Abstract In this contribution we propose two new iteratie precoder and equaizer designs for the Downin DL scenario of Muti-User MU-MIMO systems which empoy Fiter Ban based Muti-Carrier FBMC with Offset-Quadrature Ampitude Moduation O-QAM In a MU-MIMO DL scenario, we must design our per-subcarrier fiters to compensate the inter-symbo and intercarrier interference ISI and ICI present in an FBMC/OQAM system in addition to muti-user interference MUI The first method presented taes adantage of the Mean Squared Error MSE-duaity to design Minimum MSE MMSE-based precoders and equaizers The second method oos at maximizing the Signa-to-Leaage Ratio SLR in the transmitter and the Signato-Interference pus Noise Ratio SINR in the receier Through numerica simuations we wi eauate the performance of these methods and compare them to recent approaches found in the iterature I INTRODUCTION In recent years, FBMC systems hae receied attention as a promising aternatie to Orthogona Frequency Diision Mutipexing with Cycic-Prefix CP-OFDM for the physica ayer of the new 5-th generation mobie communication systems 5G CP-OFDM is aready a widey empoyed muticarrier soution due to the simpe equaization gien the CP and an efficient impementation using the Fast Fourier Transform FFT Howeer, this comes at the price of a oss in spectra efficiency due to the CP, which is extremey ong in the presence of highy frequenty seectie channes Furthermore, CP-OFDM comes with difficut synchronization requirements in the Base Station BS and the User Equipment UE Due to the spectray we designed Synthesis and Anaysis Fiter Bans, SFB and AFB, at the transmitters and the receiers 1], FBMC systems hae a much ower out-of-band radiation compared with CP-OFDM ] By introducing the O-QAM, FBMC/OQAM systems do not require a CP and thus hae an improed spectra efficiency Using an appropriate design of the puse shaping fiters imits the ICI whist contributing to more ISI within each sub-carrier Furthermore, FBMC/OQAM systems are more efficient in the presence of highy frequency seectie channes compared with CP-OFDM This comes at the price of sighty higher computationa compexity 3], 4] To tae adantage of the MU-MIMO DL scenario with Space Diision Mutipe Access SDMA, we must introduce a muti-tap, fractionay spaced, compex aued, finite impuse response fiters in the transmitters and/or receiers These shoud be designed to mitigate the ISI, ICI and MUI In 5], a non-inear spatia Tominson Harashima precoder STHP design was introduced which showed promising resuts compared with CP-OFDM Howeer, this design was imited to a MU-MISO with a fat channe frequency response The authors additionay ooed at a boc diagonaization design in 6] to mitigate the MUI and use a zero forcing based design to remoe the remaining interference In 7], the non-inear STHP design from 5] was generaized and a further iteratie precoder and equaizer design was introduced to accommodate a mutistream MU-MIMO scenario Howeer, this design was again imited to a fat channe frequency response Furthermore, in 8] the authors oo into spitting the computationa compexity between the transmitter and receier They used two inear designs based on a maximization of the SLNR and SINR in the transmitter and receier, respectiey In 9], an iteratie design for a quasi MMSE-based precoder fiters and MMSE-based equaizer fiters was introduced for the MU-MISO DL scenario This design was extended in 10] to the MU-MIMO DL scenario and compared with an SLR-based precoder design Howeer, in both designs ony a singe tap, rea aued equaizer with a Maxima-Ratio Combining MRC design was used at the receiers In this contribution we propose two new iteratie designs for the singe stream MU-MIMO DL scenario The first taes adantage of the MSE-duaity, 11], 1], between UL and DL scenarios, such that we ony need to design MMSE-based MIMO equaizers and transform them into precoders In the second method an iteratie design wi extend the SLR design in 10] to accommodate compex aued muti-tap equaizers at the receier that maximize the SINR This paper is organized as foows; in Section II we briefy describe the MU-MIMO FBMC/OQAM mode we inestigated In Section III and Section IV the two proposed precoders and equaizers designs wi be discussed Finay, in Section V and Section VI we wi discuss the simuation resuts and draw our concusions 454 VDE VERLAG GMBH Berin Offenbach, Germany

2 WSA 016 March 9-11, 016, Munich, Germany II FBMC SYSTEM MODEL In a MIMO FBMC/OQAM system, the SFB in each transmitter antenna combines the M, compex aued QAM input signas d s m] generated at a rate of 1/T s, into a singe, compex aued signa generated at a higher rate of M/T s The signa is transmitted across a highy frequency seectie additie white Gaussian noise channe to the receier In our system, M corresponds to the tota number of sub-channes and M u to the number of sub-carriers we transmit across corresponds to the sub-carrier index and s to the user index The AFB separates the receied signa bac into its M u components at a ow rate per sub-carrier The first operation in the SFB is the O-QAM staggering O of the input symbos d s m] The input symbo ds m] is spit into its rea and imaginary parts, up-samped by a factor of, then depending on which sub-carrier we obsere, either the R{d s m]} or j I{ds m]} symbo is deayed by T s/ and finay these components are added together When the subcarrier index is een, the R{d s m]} symbo is deayed and when the sub-carrier index is odd, the j I{d s m]} symbo is deayed Therefore, the symbo x s n] at the output of our O operation has an O-QAM structure, ie, each symbo is either purey rea or purey imaginary at a symbo rate n], which is doube the symbo rate of the input signas d s m] Due to this characteristic of the O-QAM symbos, there is a phase change of π/ between immediatey adjacent subcarriers, ensuring orthogonaity between sub-carriers At the receier, the AFB appies O-QAM de-staggering to reconstruct the compex QAM ˆd s m] symbos at the origina symbo rate from the equaized ˆx s n] symbos After the O-QAM staggering the signas x s n] are fitered by the muti-tap precoders, upsamped by M/ and puse-shaped by narrowband fiters that aow a good spectra containment of each sub-carrier At the AFB, simiar fiters are appied, a downsamping by M/ and fitering by the equaizers are performed to the signas before the O-QAM de-staggering Efficient reaization of the FBMC system can be achieed by taing adantage of exponentiay moduated fiters on both SFB and AFB gien by h r] h pr] exp j πm r Lp 1,r0,,L p 1, where h p r] is a owpass narrowband prototype fiter, here a Root Raised Cosine RRC, with ength L p KM+1, with K representing the oerapping factor of the symbos in the time domain K shoud be ept as sma as possibe not ony to imit the compexity, but aso to reduce the time-domain spreading of the symbos and the transmission atency Furthermore, by taing adantage of the poyphase decomposition of h p r] a the fitering can be performed at a rate of ony /T s The compex moduation is perfomed by a FFT To minimize the compexity in the cacuations of the equaizer and precoder fiters, we set K 4 and the rooff factor of our RRC fiter equa to one Thus, the frequency response of the fiter h ony significanty oeraps with the two adjacent fiters In our MU-MIMO FBMC/OQAM DL system, we hae assumed the BS to hae a tota of N t transmitter antennas, each with an SFB and each UE to hae a tota of N rs receier antennas In the MU-MIMO UL system we assume that the BS has N r N t receier antennas and each UE has N ts N rs transmitter antennas The tota number of users is U To simpify the system mode we define the foowing notation, h s,r,j n] h h s ch,r,j h t] tnm/ This represents the interference from the BS antenna j in sub-carrier into the UE antenna r of user s in sub-carrier Where { 1,, +1}, {1,,M u }, s {1,,U}, j {1,,N t } and r {1,,N rs } To simpify notation we do not incude the sub-script index of the receier sub-carrier since the interference is aways reatie to Furthermore, in the foowing deriations we wi stac or sum the ectors of equiaent channes oer the antennas to further simpify the notation The resuting fiter has the ength Lp 1 + L ch Q, M/ with the prototype fiter ength and channe impuse response ength, L p and L ch, respectiey After the O-QAM staggering operation, the sequences of input symbos, x s n], hae the structure αm] s jβm] s αm s 1], is odd, x s n] ] T jβm] s αm] s jβm s 1], is een, where α sm] and βs m] represent the rea and imaginary part of compex moduated QAM input symbo In the foowing sections we wor with a purey rea notation and therefore define a purey rea input sequence as x s n] J x s n], ie, x s n] RB+Q 1 with J { diag 1 j 1 j ], is odd, diag j 1 j 1 ], is een The matrix J extracts the imaginary j s from the input signa In the foowing deriations, we wi mutipy the transposed conoution matrices of the equiaent channes with J and wor with purey rea notation It can be shown, 9] and 13], that cacuating the precoder or equaizer fiters with either the rea or imaginary part of the input symbo both resut in the same fiters III MSE-DUALITY BASED PRECODER AND EQUALIZER DESIGN In this section we discuss an iteratie agorithm to design joint MMSE-based precoder and equaizer fiter for the MU- MIMO DL scenario Howeer, it shoud be noted that we ony design receier fiters from both the BS and UE perspecties In the foowing sections we wi use the notation ˇ and ˆ to indicate the DL scenario and the UL scenario, respectiey In Agorithm 1, each step ony depends on ariabes from the same iteration, thus to simpify notation, we wi excude the iteration index in the deriations that foow Our agorithm starts by initiaizing the equaizer fiters in the DL scenario as a simpe deay, ie, the unit ector e with a 1 at the position B/ The initia DL to UL DL/UL MSEduaity transformation in Step 3 sets the scaing factor ˇγ in 455 VDE VERLAG GMBH Berin Offenbach, Germany

3 WSA 016 March 9-11, 016, Munich, Germany the 0th iteration equa to 1 which means the precoder fiter in the UL scenario is aso a deay As aready mentioned we ony need to design MMSEbased equaizers for the UL and DL scenarios as seen in Step 6 and Step 9 In Step 7, we use the UL/DL MSE-duaity transformation to cacuate the DL precoder fiter and in Step 10, we hae to transform the DL equaizer fiter into the UL precoder fiter for the next iteration Finay, our agorithm ends after a predefined number of iterations, n, hae been executed Agorithm 1 can be appied to a of the MSE-duaity transformation, in step 7 and 10 we obsere that our designs eeps the UL/DL and DL/UL MSE-duaity transformations the same for a iterations Agorithm 1 Joint MMSE-based Precoder and Equaizer Design using the MSE-duaity Transformations 1: Initiaization: : ˇw,0 e B/, 3: ˇγ 0 1 ˆb,1 e B/, 4: i 1 5: repeat ] 6: ŵ,i ˆα argmine,i n] α,i n ν] 7: ˆγ i UL/DL MSE-duaity transformation 8: ˇb,i ˆγ i ŵ,i ] 9: ˇw,i ˇα argmine,i n] α,i n ν] 10: ˇγ i s DL/UL MSE-duaity transformation 11: ˆb,i+1 ˇγ i ˇw,i 1: i i +1 13: unti i n A Base station Perspectie In this sub-section we inestigate the MU-MIMO UL scenario We define a muti-tap, fractionay spaced equaizer ŵ C Leq per user, sub-carrier and BS receier antenna and a muti-tap, fractionay spaced precoder ˆb C B per user, subcarrier and UE transmitter antenna In our system we hae U decentraized users, each with N rs transmitter antennas Each user transmits sequences x 1,, xu of independent and identicay distributed iid and Gaussian distributed input signas in the sub-carriers {1,,M u } to the N r centraized BS receier antennas Furthermore, we assume the O-QAM input symbos to be hae haf the] ariance of the QAM input symbos σd, ie, E x s n] xs,t n] σd /UI σ M IIn the UL scenario, the rea part of our receied signa for user in sub-carrier is defined as U +1 ˆα n] ŵ,t ˆP s xs n]+ ˆΞ, 1 s1 1 where ˆP s is the transposed conoution matrix of the equiaent UL channe We hae added the UL precoder fiter into the tota transmission channe, ie, ˆp s,r,j ˆb s,r hs,r,j where s,, j, r represent the user index, sub-carrier index, BS receier antenna index and UE transmitter antenna index, respectiey Furthermore, ˆΞ R BNt 1 contains the staced rea and imaginary parts of Γ η j with Γ as an M/ downsamped, transposed conoution matrix of h which fiters the noise η j We assume the additie noise is Gaussian distributed with η j n] N C 0,σ η I The optimization probem we wish to minimize is expressed with respect to the UL MSE ˆɛ as ŵ argmine ˆα n] αn ν] ], ŵ argminˆɛ, ŵ where we define ν as the transmission atency in our system This optimization represents Step 6 of Agorithm 1 We soe the optimization probem in simiar to 13], arriing at an MMSE-based equaizer fiter for a receier antennas, U +1 ŵ s1 1 σ M ˆP s ˆP s,t + ˆR η 1 σ M ˆP e ν 3 Gien the MMSE-based equaizer we are eft with a simpified, cosed form expression for the UL MSE per user and per subcarrier defined as ˆɛ ŵ 1 e T,T ν ˆP ŵ, 4 where we define the matrices, staced oer the BS receier antennas, as ] w,t,1 w,t,n r R 1 BNr, 5 P s,r,j J P s,r,j C B B+Leq+Q, 6 ŵ,t { } { } P s,r,j R Ps,r,j I Ps,r,j 7 ˆP s Nts P r1 s Nts,r,1 P r1 s,r,n r, 8 ˆR η bocdiag ] Rη1 RηNr, 9 ] Rη,,1 R R η, η,, R R η,, R B B, 10 η,,1 Γ R Γ R,T + Γ I ΓI,T R B B, with R η,,1 σ η R η,, σ η Γ R Γ I,T 11 Γ I ΓR,T R B B 1 We use the notation to indicate taing the rea and imaginary part of a ector and stacing them on top of each other, ie, x R{x}, I{x} Now we moe onto the MU-MIMO DL scenario where, from the BS perspectie, we can define the rea part of our receie signa for user in sub-carrier as ˇα n] U +1 s1 1 { } ˇb s,t ˇQ xs n]+r ˇw,T Γ η n], 13 where ˇbs is defined as the dua stacing ector to ŵ s with N t N r Furthermore, ˇQ Nrs Q r1 N rs Q,r,1 r1,r,n t is the equiaent DL channe to moe the DL equaizer into the tota transmission 456 VDE VERLAG GMBH Berin Offenbach, Germany

4 WSA 016 March 9-11, 016, Munich, Germany chain, ie, ˇq,r,j ˇw,r h,r,j Again, we assume that the input signas x s n] and the noise are iid and Gaussian distributed with an equiaent distribution to that defined in Section III-A The optimization probem in the DL scenario we woud require to minimize is defined as ˇb argmine ˇα n] αn ν] ], ˇb U M u argminˇɛ s t ˇb M uu ˇb 1 1 By pugging 13 into the argument of our optimization probem, we arrie at a cose formed expression for the DL MSE from the BS perspectie as ˇɛ σ M U +1 s1 1 + ˇw,T Ř η ˇw ˇb s,t ˇQ,T ˇQ ˇb s B BS-Side MSE-duaity Transformations e T ν,t ˇQ ˇb In this sub-section we inestigate the four different methods, from the BS perspectie, of transforming our UL MIMO system into an equiaent DL MIMO system using the MSEduaity principe as introduced in 1] and 11] In our iteratie Agorithm 1, we are now at the UL to DL MSE-duaity transformation in Step 7 In a the MSE-duaity transformations, the Mu 1 ˆb tota power is presered 11], 1], ie, U 1 M u U A more detaied description of the MSE-duaity transformations for a MU-MISO FBMC/OQAM system can be found in 14] 1 UL/DL System-Wide Sum-MSE: First, we define a reation between the DL and UL fiters with a singe scaing factor for a users and sub-carriers such that ˇb s ˆγŵ s and ˇw ˆγ 1ˆb, ˆγ R + 15 In the next step we set the system-wide sum-mse equa between the UL and the DL scenarios, ie, U Mu 1 1 ˆɛ! U Mu 1 1 ˇɛ, where the reation! impies both sides of the equation must be equa By soing this equation we can cacuate a singe scaing factor ˆγ U Mu 1 1 σ M e T ν U 1 Mu 1,T ˆP ŵ U s1,t ˆb Ř ˆb η +1 1ŵs,T ˆP,T ˆP ŵ s 16 UL/DL User-Wise Sum-MSE: Next, we define a reation between the DL and UL fiters with a scaing factor per user such that ˇb s ˆγ s ŵ s and ˇw ˆγ 1 ˆb, ˆγ s R + 17 Foowing this, we set the user-wise sum-mse equa between the UL and the DL system, ie, M u 1 ˆɛ! M u 1 ˇɛ, {1,,U} We end up with the foowing system of inear equations, ˆγ 1 Mu 1,T 1 ˆb Ř 1 ˆb η 1 A s U ˆγ, 18 Mu U,T 1 ˆb Ř U ˆb η U }{{} y s where the matrix A s R U U has stricty positie main diagona eements, defined as M u +1 e T,T ν ˆP ŵ A s ],y 1 1 M u ŵ,t ˆP ŵ y,t,t ˆP ŵ ˆP, if y,, if y,t ˆP ŵ y 19 3 UL/DL Sub-Carrier-Wise Sum-MSE: Next, we define a reation between the DL and UL fiters with a scaing factor per sub-carrier such that ˇb s ˆγ ŵ s and ˇw ˆγ 1ˆb, ˆγ R + 0 Next we set the sub-carrier-wise sum-mse equa, ie, U 1 ˆɛ! U 1 ˇɛ, {1,,,M u} We end up with the foowing system of inear equations A ˆγ 1 ˆγ M u U,T 1 ˆb 1 U 1,T ˆb M u Ř ˆb η 1 Ř η ˆb M u, 1 where the tri-diagona matrix A R Mu Mu has stricty positie eements on the main diagona and stricty negatie off-diagona eements, defined as U e T,T ν ˆP ŵ A ],s1,m ŵ s,t ˆP,T ˆP ŵ s, if m, U ŵ s,t ˆP m,t m ˆP m ŵs m, if m 1,,s1 0 ese 4 UL/DL User and Sub-Carrier-Wise MSE: Finay, we define a reation between the DL and UL fiters with a scaing factor per user and per sub-carrier such that ˇb s ˆγ s ŵs and ˇw ˆγ 1 ˆb, ˆγ s R + 3 We then set the user and sub-carrier-wise MSE equa between the UL and the DL system, ie, ˇɛ! ˆɛ, {1,,,U} and {1,,,M u } We end up with the foowing system of inear equations A 1,1 A 1, 0 U 0 U A,1 A, A,3 0 U 0U AMu 1,M u 0 U 0 U A Mu,Mu 1 A Mu,Mu ˆγ 1 ˆγ ˆγ 3 ˆγ Mu κ 1 κ κ 3 κ Mu, VDE VERLAG GMBH Berin Offenbach, Germany

5 WSA 016 March 9-11, 016, Munich, Germany where A,m R U U and ˇγ R U +, {1,,M u } The eements on the Right-Hand-Side RHS are defined as ˆb1,T κ Ř 1 ˆb η 1,, ˆbU,T Ř U ˆb η U 5 We define the matrices A, and A,m for m as foows e T,T ν ˆP ŵ A, ],s ŵ,t ˆP,T ˆP ŵ, if s, 6 σ Mŵs,T ˆP,T ˆP ŵ s, if s, A,m ],s { ŵ m s,t ˆP,T m ˆP m ŵm, s if m 1 7 C User Equipment Perspectie In this sub-section we moe on to oo at the design of the DL MIMO MMSE-based equaizer fiter from the UE s perspectie The foowing deriations represent Steps 9 and 10 in our iteratie Agorithm 1 From the UE perspectie, the rea part of the receied signa for user in sub-carrier can be defined as U +1 ˇα n] ˇw,T ˇP s x s n]+r{γ η }, 8 s1 1 where we define ˇw as the muti-tap, fractionay spaced DL equaizer Again, we assume iid input symbos x s n] and AWGN noise as defined in Section III-A The transposed s conoution matrix ˇP is the equiaent DL channe to moe the DL precoder into the tota transmission chain, ie, ˇp s,r,j ˇb s,j h,r,j The optimization probem we wish to minimize on the UEside, is expressed with respect to the DL MSE ˇɛ as ˇw argmine ˇα n] αn ν] ], ˇw 9 argminˇɛ, ˇw To this end, the MMSE-based equaizer fiter per user, and sub-carrier in the DL system is cacuated as U +1 1 ˇw s ˇP ˇP s,t + Ř η ˇP e ν 30 s1 1 Gien the MMSE-based DL equaizer we are eft with a simpified, cosed form expression for the UE-side DL MSE, ˇɛ ˇw 1 e T,T ν ˇP ˇw 31 Now we moe on to the MU-MIMO UL scenario from the UE perspectie The rea part of our receied signa for user in sub-carrier is defined as ˆα n] U +1 s1 1 } ˆb s,t ˆQ s xs n]+ŵ,t R {ˆΓ ˆη, 3 where we define ˆb as a muti-tap, fractionay spaced precoder and ŵ is the UL MMSE-based equaizer designed in 3 s The transposed conoution matrix ˆQ is the equiaent UL channe to moe the UL equaizer into the tota transmission chain, ie, ˆq,r,j s ŵ,r hs,r,j Using 3 we can cacuate the UL MSE as U +1 ˆɛ ˆb s,t ˆQ s s,t ˆQ ˆb s ˆb,T ˆQ e ν +1 + ŵ,t s1 1 ˆR η ŵ 33 D User Equipment-Side MSE-duaity Transformations In this sub-section we inestigate the two possibe methods, from the UE perspectie, of transforming our DL equaizer fiters into equiaent UL precoder fiters These are simiar to the transformations introduced in Sub-Section III-B, howeer, since we hae decentraized users, we concuded that spreading the transmit power oer the users was not meaningfu Therefore, we end up with ony two forms of DL/UL MSE-duaity transformations, ie, the DL/UL User-Wise Sum-MSE and the DL/UL User and Sub-carrier-Wise MSE transformation For consistency we matched the DL/UL MSE-transformation to the MSE-transformation used in the UL/DL scenario, ie, where we summed oer the sub-carriers we used the DL/UL User- Wise Sum-MSE transformation and where we summed oer users or set the indiidua MSEs equa we used the DL/UL User and Sub-carrier-Wise MSE transformation 1 DL/UL User-Wise Sum-MSE: Again, we define a reation between the UL and DL fiters with a scaing factor per user such that 1 ˆb s ˇγ i s ˇws and ŵ ˇγ i ˇb, ˇγ s R + 34 By summing oer a sub-carriers as in Section III-B and setting this up for a users, we end up with a system of inear equations in the same form as 18 Now the RHS of the equation is defined as y s Mu 1 A s ],y 1,T ˇb ˆR 1 ˇb η 1 and the matrix A s is defined as M u +1 e T ν 1 1 M u ˇw,T Mu 1,T ˇP ˇw ˇP s ˇw y,t s,t ˇP ˇw ˇP y U,T ˇb ˆR U ˇb η U, 35 y,t ˇP ˇw y, if y,, if y 36 DL/UL User and Sub-Carrier-Wise MSE: Finay, we define a reation between the UL and DL fiters with a scaing factor per user and per sub-carrier such that ˆb s ˇγ s ˇws and ŵ ˇγ 1 ˇb, ˇγ s R + 37 We then set the UL and DL MSE equa for each user and sub-carrier simiar to Section III-B4 Again, we end up with a system of inear equations simiar to 4 The eements on the RHS of the system of equations are defined as ˇb1,T κ ˆR 1 ˇb η 1,, ˇbU,T ˆR U ˇb η U VDE VERLAG GMBH Berin Offenbach, Germany

6 WSA 016 March 9-11, 016, Munich, Germany We define the tri-diagona matrices A, and A,m for m as foows e T,T ν ˇP ˇw A, ],s ˇw,T ˇP,T ˇP ˇw, if s, 39 s,t ˇP, if s, σ M ˇws,T A,m ],s { σ M ˇw s,t m ˇP s s ˇP m ˇw s ˇP s,t m ˇws m, if m 1 40 IV SLR-BASED PRECODER AND SINR-BASED EQUALIZER DESIGN The second design method we consider is a precoder designed to maximize the SLR and the equaizer designed to maximize the SINR The SLR precoder design is simiar to 10], howeer, in this contribution we empoy compex aued, muti-tap equaizers The iteratie aternating SLR/SINR design Agorithm is simiar to Agorithm 1 used to design the MMSE-based precoder and equaizer fiters Again, we initiaize the SINR equaizer fiter as a simpe deay ector with a one at position L eq / The SLR-maximizing precoder is designed taing the equaizer fiter from the preious iteration into account and the SINR-maximizing equaizer is designed taing the precoder fiter from the current iteration into account Anaogousy to the MSE-Duaity deriations, we hae omitted the iteration index i in the foowing deriations to simpify notation Agorithm Joint SLR-based Precoder and SINR-based Equaizer Design 1: Initiaization: : ˇw,0 e L eq/, 3: i 1 4: repeat 5: ˇb,i argmaxslr,i ˇw,i 1 ˇb,i 6: ˇw,i argmax ˇw,i 7: i i +1 8: unti i n SINR,i ˇb,i A SLR-Maximizing Precoder In this section we design a compex aued, muti-tap precoder fiter based on a maximization of the SLR, ie, Step 5 of Agorithm To derie a cosed form expression for the SLR we must define the ICI, ISI, MUI and the noiseess receied symbo for user in sub-carrier Again, we assume iid input symbos x s and AWGN noise as defined in Section III-A The sum ICI eaed into the imaginary part of the two adjacent sub-carriers is defined as c +1 1, E ˇb,T ˇΦ x n] ], 41 where ˇΦ R BNt Q+B+Leq is a stacing matrix of the transposed conoution matrix of the equiaent channe channe, q,r,j ˇw,r h,r,j and we define { } { } Φ,r,j I Q,r,j R Q,r,j R B Q+B+L eq The ISI within sub-carrier of user is defined as ˇb,T s E ˇQ,ISI x n] ] 4 ˇQ,ISI with the matrix ˇQ I e νe T ν Furthermore, the MUI eaed into the other users and their adjacent sub-carriers is defined as U +1 ˇb,T u E ˇΦ s x n] + ˇb,T ˇQ s x n] ] s1 1 s 43 Additionay, we require the effectie channe of user in sub-carrier from BS transmitter antenna j to a UE receier antennas N r for the noiseess receied signa, defined as ˇq,eff N r r1 q,eff,r,j Q,r,je ν 44 Nr r1 Finay, using 41, 43, 4 and 44, we end up with a cosed form expression for the SLR of user in sub-carrier, expressed in 45 The precoder for user in sub-carrier is the maximizer of ˇb argmax ˇb SLR, where the soution is cacuated as the principa eigenector corresponding to the maximum eigenaue of the matrix {C 1 A} Since the eigenectors are aready normaized to one, no further normaization of the precoder fiters is required B SINR-Maximizing Equaizer In this section we design a compex aued, muti-tap equaizer ector based on a maximization of the SINR, ie, Step 6 of Agorithm To derie a cosed form expression for the SINR we must again define the ICI, MUI, ISI, fitered noise and the noiseess receied symbo for user in sub-carrier Furthermore, we assume iid input symbos x s n] and AWGN noise as defined in Section III-A The ICI receied from the adjacent sub-carriers is defined as +1 c E ˇw,T ˇP x n] ], 46 1 where ˇP R BNt Q+B+Leq is a stacing matrix of the transposed conoution matrix of the equiaent channe channe, p s,r,j ˇb s,j h,r,j, equiaent to the matrix defined in Section III The MUI receied from the other users and from the adjacent sub-carriers is defined as U +1 ũ E ˇw,T ˇP s x s n] ] 47 s1 1 s The ISI within sub-carrier of user is defined as s E ˇw,T ˇP,ISI x n] ] 48 with the the ISI matrix equiaent to that found in Section IV-A, ie, ˇP I Q e ν e T ν ˇP,ISI We require the effectie channe of user in sub-carrier from BS transmitter antenna j to a UE receier antennas N r for the noiseess receied signa, defined as 459 VDE VERLAG GMBH Berin Offenbach, Germany

7 WSA 016 March 9-11, 016, Munich, Germany SLR ˇb,T U +1 s1 1 ˇΦ s ˇb,T A { }} { ˇb ˇΦ s,t + ˇq,,eff U s1 s ˇq,,eff,T ˇQ s s,t ˇQ + ˇQ,ISI ˇQ,ISI,T } {{ } C ˇb 45 ˇp,eff N t j1 p,eff,r,j P,r,je ν 49 Nt j1 Additionay, we define the fitered noise of user in subcarrier as n E ˇw,T Γ ηn] ] ˇw,T Ř η ˇw Finay, a cosed form expression for the SINR of user in sub-carrier, which the equaizer fiter shoud maximize, is expressed in 50 The equaizer for user in sub-carrier is the maximizer of ˇw argmax SINR, whereby the soution ˇw is cacuated as the principe eigenector corresponding to the maximum eigenaue of the matrix { C 1 Ã} Since the equaizers are cacuated as eigenectors normaized to one, we must scae the equaizer fiters with the factor α ˇw,T Ã ˇw such that the symbos can be correcty decoded in the UEs V SIMULATION RESULTS In this section we discuss the simuation resuts of the MSE-duaity based design and the SLR/SINR based design for the DL MU-MIMO scenario The channe reaizations are from the Wireess Word Initiatie New Radio WINNER II project 15] We transmit data across M u 10 of the aaiabe M 56 sub-carriers per user and per transmitter antenna with a samping rate of f s 1536 MHz, giing a sub-carrier spacing of 60 Hz We randomy generate 16-QAM symbos and tae a boc ength of 1000 symbos per sub-carrier The channe impuse response is L ch 169 taps With these system configurations, especiay due to L ch 169 and the highy frequency seectie channe, a CP-OFDM system woud hae required a CP with a minimum ength of 168 taps 3], 4] This imits the data-throughput of the CP-OFDM to more than 50%, therefore we do not incude a direct comparison in the simuation resuts We tae the quantity of E b /N 0 to be a pseudo-signa-to-noise Ratio SNR per user for the MU- MIMO simuations We tae the uncoded Bit Error Rate BER and MSE as an aerage oer a users, and we aerage oer 400 randomy generated channe reaizations We hae a precoder ength of B 5taps and an equaizer ength of L eq 3taps Throughout our simuations we hae a system with N t 4BS transmitter antennas and U users, each with N r1 N r receier antennas We stopped both of our iteratie agorithms after n 4iterations In Fig 1 we see the uncoded BER ersus SNR for the two iteratie precoder and equaizer design agorithms introduced in this paper We hae compared our two iteratie designs with an SLR-based precoder design with a rea aue, singe tap equaizer from 10] We obsere that the MSEduaity based designs show a better performance oer the whoe SNR regime Furthermore, in the high SNR regime, the MSE-duaity transformations with the System-wide Sum- MSE and User-wise Sum-MSE UL/DL transformation show performance gains of more than 5dB compared with the other designs This is attributed to the fact that these methods aow the tota transmit power to be spread across a sub-carriers depending on the channe conditions, somewhat ie an inerse waterfiing power aocation scheme Fig shows the MSE ersus SNR of the two different iteratie designs We obsere that a four MSE-duaity based designs outperform the SLR/SINR based design in the ow SNR regime due to the fact, that the SLR-precoder design does not tae the noise ariance into account, eading to worse MSE aues In Fig 3 we see the conergence, in db, of the two different iteratie designs The soid cures show the MSE conergence of the MSE-duaity based designs, and the dashed cure shows the SINR conergence of the SLR/SINR design After 4 iterations, the aues does not significanty improe anymore, which is why we stopped our agorithms after n 4 VI CONCLUSIONS We hae presented two schemes for precoder and equaizer design for MU-MIMO DL FBMC/OQAM systems Both are iteratie and the first one is an MMSE design based on the MSE-duaity and the second one maximizes the SLR and SINR in an aternating fashion We can see that both methods present a simiar uncoded BER performance where the MMSE based designs outperform the SLR/SINR based design and another current precoder design oer the whoe SNR regime Moreoer, we can see that a the iteratie agorithms conerge REFERENCES 1] P Siohan, C Sicet, and N Lacaie, Anaysis and design of OFDM/OQAM systems based on fiterban theory, IEEE Trans Signa Process, o 50, no 5, pp , 00 ] L G Batar, F Schaich, M Renfors, and J A Nosse, Computationa compexity anaysis of adanced physica ayers based on muticarrier moduation, in Future Networ Mobie Summit FutureNetw, 011, June 011, pp 1 8 3] L G Batar and J A Nosse, Muticarrier systems: a comparison between Fiter Ban based and Cycic Prefix based OFDM, in Proceedings of 17th Internationa OFDM Worshop 01 InOWo 1, 01, pp 1 5 4] L G Batar, D Wadhauser, and J A Nosse, Out-of-band radiation in muticarrier systems: A comparison, in 6th Internationa Worshop on Muti-Carrier Spread-Spectrum MC-SS, May 007, Herrsching, Germany 5] M Caus and A Perez-Neira, SDMA for fiterban with Tominson Harashima precoding, in IEEE Internationa Conference on Communications ICC, 013, June 013, pp VDE VERLAG GMBH Berin Offenbach, 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8 WSA 016 March 9-11, 016, Munich, Germany SINR ˇw,T U +1 s1 1 ˇw,T ˇP s ˇP s,t à { }} { ˇp,eff ˇP,T ˇP ˇp,eff,T + ˇw,ISI ˇP ˇP,ISI,T + σ M Ř η } {{ } C ˇw 50 Uncoded BER MSE-Duaity III-B1 and III-D1 MSE-Duaity III-B and III-D1 MSE-Duaity III-B3 and III-D MSE-Duaity III-B4 and III-D SLR-Precoder, SINR-Equaizer SLR-Precoder 10] E b /N 0 db Figure 1 BER of the two iteratie designs for N t 4, N r and U MSE db MSE-Duaity III-B1 and III-D1 MSE-Duaity III-B and III-D1 MSE-Duaity III-B3 and III-D MSE-Duaity III-B4 and III-D SLR/SINR Design MSE SLR/SINR Design SINR Iterations Figure 3 Agorithm conergence at E b /N 0 5dB of the iteratie designs for N t 4, N r and U SINR 1 db MSE MSE-Duaity III-B1 and III-D1 MSE-Duaity III-B and III-D1 MSE-Duaity III-B3 and III-D MSE-Duaity III-B4 and III-D SLR-Precoder, SINR-Equaizer E b /N 0 db Figure MSE of the two iteratie designs for N t 4, N r and U 6] M Caus, A Perez-Neira, and M Moretti, SDMA for FBMC with boc diagonaization, in IEEE 14th Worshop on Signa Processing Adances in Wireess Communications SPAWC, 013, June 013, pp ] Y Cheng, V Ramireddy, and M Haardt, Non-inear precoding for the downin of FBMC/OQAM based muti-user MIMO systems, in 19th Internationa ITG Worshop on Smart Antennas, 015, 015 8] M Caus and A Perez-Neira, Transmitter-Receier Designs for Highy Frequency Seectie Channes in MIMO FBMC Systems, IEEE Trans Signa Process, o 60, no 1, pp , Dec 01 9] M Newinger, L G Batar, A Swindehurst, and J A Nosse, MISO Broadcasting FBMC System for Highy Frequency Seectie Channes, in 18th Internationa ITG Worshop on Smart Antennas WSA, 014, 014, pp ] Y Cheng, L Batar, M Haardt, and J Nosse, Precoder and equaizer design for muti-user MIMO FBMC/OQAM with highy frequency seectie channes, in IEEE Internationa Conference on Acoustics, Speech and Signa Processing ICASSP, 015, Apri 015, pp ] R Hunger, M Joham, and W Utschic, On the MSE-duaity of the broadcast channe and the mutipe access channe, Internationa ITG Worshop on Smart Antennas, 009, o 57, no, pp , 009 1] A Mezghani, M Joham, R Hunger, and W Utschic, Transceier design for muti-user MIMO systems, in Internationa ITG Worshop on Smart Antennas, 006, 006, Um, Germany 13] D S Wadhauser, L G Batar, and J A Nosse, MMSE subcarrier equaization for fiter ban based muticarrier systems, in IEEE 9th Worshop on Signa Processing Adances in Wireess Communications, 008 SPAWC 008, 008, pp ] O De Candido, L G Batar, A Mezghani, and J A Nosse, SIMO/MISO MSE-Duaity for Muti-User FBMC with Highy Frequency Seectie Channes, in 19th Internationa ITG Worshop on Smart Antennas; WSA 015, March 015, pp ] L Hentiä, P Kyösti, M Käse, M Narandzic, and M Aatossaa 007, December MATLAB impementation of the WINNER Phase II Channe Mode er VDE VERLAG GMBH Berin Offenbach, Germany