Conference Paper. Ramiro Robles CISTER-TR /05/21

Transcription

1 Conference Paper Joint Beamforming, Termina Scheduing, and Adaptive Moduation with Imperfect CSIT in Rayeigh Fading Correated Channes with Cochanne Interference Ramiro Robes CISTER-TR /05/21

2 Conference Paper CISTER-TR Joint Beamforming, Termina Scheduing, and Adaptive... Joint Beamforming, Termina Scheduing, and Adaptive Moduation with Imperfect CSIT in Rayeigh Fading Correated Channes with Co-channe Interference Ramiro Robes *CISTER Research Centre Poytechnic Institute of Porto ISEP-IPP) Rua Dr. António Bernardino de Ameida, Porto Portuga Te.: , Fax: E-mai: Abstract This paper presents a joint scheduing, beamforming, and resource aocation agorithm for muti-user wireess networks affected by co-channe interference. The anaysis considers a network with one base station BS) that uses a mutipe antenna transmitter beamformer) to schedue in a time-division manner) transmissions towards a set of $J$ one-antenna terminas in the presence of $K$ persistent interferers. The transmitter is assumed to empoy maximum-ratio combining MRC) beamforming in the presence of spatiay-correated branches with channe enveopes modeed as Rayeigh-distributed processes. The BS has access to an imperfect outdated) copy of the instantaneous channe state information CSI) of each termina. Based on this CSI at the transmitter side CSIT), the BS proceeds to seect at each time interva or time-sot) the termina with the highest channe strength for purposes of transmission. This imperfect CSIT is aso used to cacuate the coefficients of the beamformer that wi be used to transmit information towards the schedued termina, as we as for seecting the most appropriate moduation format threshod-based decision). In addition, the transmission towards each schedued termina is assumed to experience persistent co-channe interference that wi degrade the quaity of the information reception process. The main merits of this work are: 1) the joint anaysis of MRC-based beamforming, termina scheduing based on maximum channe strength, and moduation assignment, and 2) joint modeing of the effects of spatia correation, co-channe interference and imperfect CSIT. CISTER Research Center 1

3 Joint Beamforming, Termina Scheduing, and Adaptive Moduation with Imperfect CSIT in Rayeigh Fading Correated Channes with Co-channe Interference Ramiro Sámano-Robes Research Centre in Rea-time and Embedded Computing Systems Intituto Poitécnico do Porto, Porto, Portuga Emai: Abstract This paper presents a resource aocation agorithm for muti-user wireess networks affected by co-channe interference. The anaysis considers a network with one base station BS) that uses a mutipe antenna transmitter beamformer) to schedue in a time-division manner) transmissions towards a set of J one-antenna terminas in the presence of K persistent interferers. The transmitter is assumed to empoy Maximum- Ratio Combining MRC) beamforming with spatiay-correated branches and channe enveopes modeed as Rayeigh-distributed processes. The BS has access to an imperfect outdated) copy of the instantaneous Channe State Information CSI) of each termina. Based on this CSI at the transmitter side CSIT), the BS proceeds to seect at each time interva or time-sot) the termina with the highest channe strength for purposes of transmission. This imperfect CSIT is aso used to cacuate the coefficients of the beamformer that wi be used to transmit information towards the schedued termina, as we as for seecting the most appropriate moduation format threshod-based decision). In addition, the transmission towards each schedued termina is assumed to experience persistent co-channe interference that wi degrade the quaity of the information reception process. The main merits of this work are the foowing: 1) joint anaysis of MRC-based beamforming, termina scheduing based on maximum channe strength, and moduation assignment, and 2) joint modeing of the effects of spatia correation, co-channe interference and imperfect CSIT. Resuts suggest that scheduing heps in rejecting co-channe interference and the degrading effects of imperfect CSIT. Spatia correation coud some times ead to better performance than the uncorreated case, particuary in the ow SNR Signa-to-Noise Ratio) regime. Conversey, uncorreated branches aways outperform the correated case in the high SNR regime. The use of higher numbers of antennas aso improve performance of the system. However, spatia correation tends to accumuate over the antenna array thus eading to a more noticeabe performance degradation and more aocation errors due to the outdated CSIT assumption. Keywords Beamforming; Scheduing; Resource aocation, Imperfect CSIT, Maximum Ratio Combining MRC) I. INTRODUCTION Mutipe antenna systems aso known as MIMO or Mutipe-Input Mutipe-Output systems) are expected to proiferate in the coming years, particuary in the context of 5G or fifth generation of mobie systems [1]. The growing demand for wireess connectivity, the imited transmission resources, and the outdated spectrum aocation paradigm have created the need for more efficient and higher capacity transmission systems. MIMO technoogy offers considerabe capacity growth that escaates with the number of transmitreceive antenna pairs. In addition to this, MIMO aso offers improved energetic efficiency and reduced interference with minimum spectrum expenditure [2]. From the many different types of mutipe antenna systems, perhaps beamforming technoogy represents the option with higher potentia for commercia soutions, mainy due to its maturity, fexibe impementation, and ow computationa costs. Beamforming refers to the abiity to dynamicay steer the phases of an antenna array and change the directionaity properties of the resuting radiation beams. This enabes a wide set of appications in muti-user settings, such as: interference rejection/management [3], spatia mutipexing [4], and more recenty with a few modifications) 3D beamforming with massive MIMO in 5G [5], beam-division mutipe access [6], and interference aignment [7]. In future networks, beamforming wi be key for efficienty organizing spectrum resources in dense sma ces, as we as minimizing energy expenditure, reducing eakage and/or interference to adjacent ces or terminas, and aso for improving security against potentia attacks of signa jamming or eavesdropping in the network. A these recent advances in the physica ayer of mutipe antenna systems need to be integrated with upper ayer agorithms. This has opened severa issues regarding the crossayer design and optimization of beamforming and in genera mutipe-antenna systems. One particuary important topic in this fied is the modeing of the underying mutipe antenna signa processing toos to be used in resource aocation and system-eve evauation frameworks. In arge network set ups with tens or hundreds of BSs and hundreds or thousands of terminas, a the detais of the PHYsica PHY) ayer cannot be usuay incuded in fu detai in the anaysis or simuation oop. Therefore, a trade-off must be found between the accuracy of the mode that represents the underying PHYayer and its fexibiity for purposes of resource aocation and optimisation at the system-eve. This paper attempts to partiay fi these gaps by addressing the ink-ayer interface modeing in Rayeigh fading correated channes of an adaptive wireess muti-user network using Maximum-Ratio Combining MRC) beamforming and termina scheduing based on imited outdated) feedback. The transmitter seects the most adequate Moduation and Coding Schemes MCSs) and beamforming vectors based on an estimated Channe State Information CSI). This imperfect CSI at the transmitter side i.e., CSIT) is assumed to have been initiay coected by the receiver perfect estimation), and subsequenty reported back to the transmitter via a feedback channe affected by deay. This paper presents the anaysis of the statistics of correct reception process conditiona on the decision made by the transmitter moduation format seection, beamforming and scheduing) based on the inaccurate CSIT. Link-ayer throughput is evauated by means of an interface

4 mode based on an instantaneous Signa-to-Interference-pus- Noise Ratio SINR) adaptive switching threshod scheme for moduation assignment. This mode aims to provide an accurate but fexibe representation of the underying PHYayer suitabe for upper-ayer design. In the proposed mode, a packet transmission using a given MCS is considered as correcty received with given vaues of BLock-Error Rate BLER) and spectra efficiency whenever the instantaneous SINR exceeds the reception threshod of the seected MCS. The reception parameters of each MCS are obtained from Look-Up-Tabes LUTs) previousy cacuated via off-ine PHY-ayer simuation. The main contribution of this work is the joint anaysis of spatia correation, imperfect CSIT and cochanne interference in ink adaptation and termina scheduing for MRC-based mutipe antenna beamforming systems. This paper is organized as foows. Section II describes previous works and the achievements of this paper with respect to the state of the art. Section III describes the system mode and the assumptions of the paper. Section IV presents the inkayer interface mode. Section V deas with the statistics of the estimated SNR and the instantaneous SINR. Section VI presents anaytic resuts and sketches of the statistics of packet reception using different network assumptions. Finay, Section VII presents the concusions of this paper. II. PREVIOUS WORKS The simpest mutipe antenna system is the MRC transceiver, which provides a reativey fexibe framework for statistica anaysis and interface modeing. The iterature of MRC transceivers has focused on the derivation of outage and bit error probabiity distributions see [18]-[26]). The effects of imperfect channe knowedge on the performance of MRC receivers in Rayeigh fading correated channes can be found in [18]-[19] foowing the anaysis with perfect channe estimation presented in [20]. A series expansion of the statistics of MRC systems with correated Rician channes is given in [21]. A unified approach for anaysis of two-stage MRC systems with hybrid seection in generaized Rice correated channes was proposed in [22]. Extensions to the case of cochanne interference are given in [23]-[26]. The present work considers the extension of outage probabiity anaysis of MRC transmitters beamformers) to the study of Adaptive Moduation and Coding AMC) in Rayeigh fading correated channes with imperfect/outdated CSIT and co-channe interference. To the best of our knowedge, this is the first attempt in the iterature that addresses these issues under the same framework. This work attempts to extend the anaysis of MRC systems towards incuding resource aocation aspects which are typica of upper ayer design radio resource management). In addition, network design and in particuar resource aocation for mutipe antenna systems is usuay conducted under the assumption perfect CSIT. Imperfect CSIT has been addressed in [27] for distributed systems and in [28] for energy efficient MIMO ink adaptation. In comparison with these works, which are focused on numerica evauation of imperfect CSIT, this work provides an anaytic framework for obtaining the statistics of errors in MCS assignment for correated MRC transmitters. A reated topic is the anaysis of the effects of imited feedback in adaptive moduation for beam-forming and mutipe antenna systems. The work in [29] provides a review of the state of the art of imited feedback in adaptation schemes for MIMO systems. The work in [30] presents the anaysis of adaptive moduation for two-antenna beam-formers considering mean CSI at the transmitter side. The work in [31] addressed the impact of outdated feedback on AMC and user seection diversity systems for MIMO systems in Rayeigh uncorreated channes. Other works with imited feedback for different types of system can be found in [33]-[35]. A these previous works consider uncorreated MIMO channes. This work goes beyond this assumption searching for a joint anaysis of imited feedback and spatia correation for adaptive MRC transmitters with co-channe interference. Notation: Bod ower case etters e.g., x) denote vector variabes, bod upper case etters e.g., A) denote matrices, ) T is the vector transpose operator, E[ ] is the statistica average operator, ) is the compex conjugate operator, f z, F z and F z denotes, respectivey, the Probabiity Density Function PDF), Cumuative Density Function CDF) and Compementary Cumuative Density Function CCDF) of any random variabe z, Z + denotes the set of positive integers, Re ) denotes the rea part operator, 0 N is the vector of ) N zeroes,i N is the identity matrix of ordern, and = ) = J 1)! 0, 1,... N 0! 1!,... N is the mutinomia combinatoria number of and N coefficients 0, 1,... N arranged in the vector = [ 0, 1,..., N ] T. III. SYSTEM MODEL AND ASSUMPTIONS Consider the network depicted in Figure 1 with one Base Station BS) scheduing transmissions towards J terminas in a time-division fashion), each one with one receiving antenna, and a set of K persistent singe-antenna interferers. The BS uses an N-antenna Maximum-Ratio Combining MRC) beamformer that is used to transmit information to a given termina at specific time sots. The channe vector between the BS and thejth termina is denoted byh j = [h j 1),h j 2),...h j N)] T. A instantaneous channe variabes wi be modeed as zeromean compex circuar Gaussian random variabes with variance γ: h j n) CN0 N,γI N ). The estimated channe variabe avaiabe at the transmitter side is given by ĥj = [ĥj1),ĥj2),...ĥjn)] T. This information is used by the BS for purposes of beamforming, termina scheduing and resource aocation moduation format assignment). The channe between the interferer k towards termina j is denoted by h k,j and is modeed as a zero-mean compex circuar Gaussian random variabe with variance λ: h k,j CN0,λ). The transmitter seects one of M moduation formats, which are arranged in increasing order according to their target Signa-to-Interference pus Noise Ratio SINR). The target SINR of the mth MCS wi be denoted by β m. The variabes θ m and η m wi denote, respectivey, the BLER and spectra efficiency in bps/hz) considering operation at the target SINR of the mth MCS. It is assumed that the receiver monitors the quaity of the channe and reports it back to the transmitter. Based on this coected Channe State Information CSI), the transmitter seects the most appropriate MCS using a correction for the decision threshods denoted here by ˆβ m. This paper considers perfect channe estimation at the receiver side and imperfect channe state information at the transmitter side CSIT). Imperfect CSIT is assumed to be mainy due to a

5 TABLE I. LIST OF VARIABLES. Spatia orreatio Tra s itter: S hedui g a d ea for i g Esti ated SNR I sta ta eous SINR Feed a k deay Ti e orreatio I terfere e Ter i as I sta ta eous SNR Figure 1. Wireess network with one transmitter using imperfect CSIT for scheduing, moduation assignment and beamforming information to a set of terminas in the presence of co-channe interference. Variabe N ρ ρ c P σ 2 v γ λ J M h j w j ĥ j ˆX j Γ j I j h k,j K s j š k β m Meaning Number of antennas at the transmitter side Spatia correation coefficient Tempora correation coefficient Transmit power Noise variance Channe variance Interferer channe variance Number of terminas Number of moduation formats Channe vector of termina j Beamforming vector for termina j Estimated channe vector Estimated SNR for termina j Instantaneous SINR of termina j Interference experienced by termina j Channe between interferer k and termina j Number of persistent interferers Symbo transmitted towards termina j Symbo transmitted by interferer k Reception SINR target threshod for moduation format m feedback channe affected by deay. The beamforming vector is denoted by w j = [w j 1),w j 2),...,w j N)] T, which using the MRC criterion is given by w j = h j. Therefore, the signa received by the schedued termina can be mathematicay written as foows K r j = wj T h j s j + h k,j š k +v j, 1) k=1 where s j is the information symbo transmitted towards termina j, š k is the symbo transmitted by interferer k, and v j is the additive white Gaussian noise experienced by termina j with variance σv: 2 v j CN0,σv). 2 Considering the symbo transmit power constraint E[s j s j] = P, the estimated SNR at the transmitter side from1) is given by: ĥ ˆX H j j = ĥje[s j s N j] = P ĥjn) 2. 2) σ 2 v Note that in this paper it is assumed that an estimate of interference I j = K k=1 h k,jš k in 1) is not avaiabe at the transmitter. Therefore, a decisions wi be based on an estimate of the SNR in 2). The estimated channes wi be generated using the foowing inear correation mode: σ 2 v ĥ j n) = 1 ρz j n)+ ρg j, 3) where ρ is the spatia correation coefficient and the terms Z j n) and G j are the zero-mean compex circuar Gaussian variabes with variance γ. Note that this correation mode compies with E[ĥjn) ĥ jñ)] = ργ, n ñ, and E[ĥjn) ĥ jn)] = γ. This correation mode constitutes an approximation of rea-ife settings by assuming that a eements experience the same correation with each other. In rea-ife systems, antennas farther apart from each other experience ess correation than contiguous eements. The correation mode for imperfect CSIT is given by: h j n) = ρ c ĥ j n)+ 1 ρ 2 cy j n), 4) θ m η m T ˆβ m BLER for moduation format β m Spectra efficiency of moduation format β m Link Layer throughput Seection SNR threshod of moduation format m where ρ c is the tempora correation coefficient that describes the accuracy of the CSIT. This correation mode compies with E[h j n) h j n)] = ρ c γ. The instantaneous SINR is given by: Γ j = RePĥH j h j) I j +σ 2 v where I j = K k=1 P h k,j 2 is the interference created by K co-channe persistent interferers. Tabe I presents a ist of the main variabes used throughout this paper. IV. LINK LAYER MODEL The probabiity of seection of a moduation format m is given by the probabiity that the estimated SNR ˆX j at the transmitter side ies within the interva [ˆβ m, ˆβ m+1 ]: 5) Pr{ˆβ m ˆX j < ˆβ m+1 } 6) Link-ayer throughput denoted by T ) wi be expressed as a inear contribution of a possibe MCSs with their respective seection probabiities from 6) and conditiona reception probabiities, each one weighted by their conditiona throughput performance T m ): T = M E Γj [T m Γ j ) ˆβ m ˆX j < ˆβ m+1 ] m=1 Pr{ˆβ m ˆX j < ˆβ m+1 }Pr{j = argmax j ˆX j }, 7) where T m Γ j ) indicates of the ink-ayer throughput of termina j when using the mth MCS conditiona on a given vaue of the operationa SINR Γ j in 5) of the seected termina. In this paper, we consider a simpification of this expression, by

6 assuming that the term T m Γ) in 7) is a step function defined by a switching SINR threshod β m above which a packet transmissions are assumed to be correcty received with a given BLER θ m and spectra efficiency η m. The simpification can be expressed as foows: T = M B Wη m 1 θ m )Pr{Γ j β m ˆβ m ˆX j < ˆβ m+1 } m=1 Pr{ˆβ m ˆX j < ˆβ m+1 }Pr{j = argmax j ˆX j } 8) where BW is the operationa bandwidth in Hz, Pr{j = argmax j ˆX j )} is the probabiity of termina to experience the highest estimated SNR and therefore being schedued for transmission by the BS, and Pr{Γ j β m ˆβ m ˆX j < ˆβ m+1 } is the probabiity of the instantaneous SINR Γ j to surpass the threshod β m provided the estimated SNR ˆX j used for MCS seection and termina scheduing) ies in the range [ˆβ m, ˆβ m+1 ]. Note that this ast conditiona probabiity term captures the effects of imperfect CSIT on the performance of the beamforming, scheduing and adaptation scheme. In the case of perfect CSIT ρ c 0), correct reception occurs with probabiity one. Aso, note that the ink-ayer throughput expression in 8) represents ony an approximation compression) of the rea performance of the system. The simpified mode in 8) assumes packets are erroneous when the instantaneous SINR drops beow the reception threshod β m, when in practice there might be some cases where correct reception can sti occur. Conversey, some cases with higher instantaneous SNR than the reception threshod coud aso ead to erroneous packet transmissions. This type of compression/abstraction mode as in 8) has been proved accurate for system-eve simuation of networks with considerabe excursions of path-oss vaues, which are typica of ceuar systems where terminas ie at different distances from the access point. V. PERFORMANCE ANALYSIS The foowing subsections present the derivation of anaytic expressions of the different terms of the ink-ayer throughput mode in 8). For convenience, it is usefu to derive the statistics of the estimated SNR presented in Section V-A) and then dea with the statistics of the instantaneous SINR presented in Section V-B) conditiona on the MCS seection, termina scheduing, and beamforming processes. A. Statistics of estimated SNR Let us now substitute the correation mode described by 3) in the expression of the estimated SNR in 2), which yieds: N ˆX j = P ĥjn) 2 = σ 2 v N P 1 ρz j n)+ ρg j 2. 9) The statistics of the estimated SNR have been investigated in our previous work in [36]. The sub-index j is dropped in subsequent derivations due to the symmetrica network assumption. The probabiity density function PDF) and compementary cumuative distribution function CCDF) are given, σ 2 v respectivey, by [36]: and f ˆXy) = A e y +e ỹ γ F ˆXy) = Ae y +e ỹ γ = Ae y +e ỹ γ u=0 u=0 B n y n 1 γ n n 1)!, 10) n 1 B n y u γ u u! y u u γ u u! = B n, 11) where γ = P1 ρ)γ σ, = αγ + γ, α = PNρ v 2 σ, A = v γ) 2 1 N ) γ) n N 1 and Bn = γ 1. The effects of termina scheduing on the statistics of the estimated SNR wi be obtained via the theory of order statistics. The statistics of the random variabe with maximum vaue are given by the foowing formua [37]: f ˆX y) = Jf ˆXy)F ˆXy) J 1. 12) By substituting the expressions for the PDF and CDF of ˆX in 12) and using the formua for mutinomia theorem we obtain the foowing expression: f ˆX y) = where α = J ; N t=0 t=j 1 α e µy y τ + ; N t=0 t=j 1 α e µy ) A) t=0 B n y τ,n 13) t u=1 B u t! ) t, 14) α = α A, 15) µ = τ γ + +1, 16) τ = t t, 17) t=0 µ = 1+ τ + γ, 18) τ,n = n 1+ τ, 19) B n B n = γ n+ τ, { t,t} Z +,0 t N n 1)! 20) For detais of this derivation pease see the Appendix. B. Statistics of instantaneous SINR Let us now substitute the correation mode described by 4) into the expression of the instantaneous SINR in 5): Γ j = PρcĥH j ĥj +Re[P 1 ρ 2 N c ĥjn) Y j n)] I j +σv 2. 21)

7 Since we are interested in the reception probabiity term Pr{Γ j > β m } we can use 21) to express the term Pr{Γ j > β m }as foows: Pr{Γ j > β m } = { Pρc ĥ H j Pr ĥj +Re[P 1 ρ 2 N } c ĥjn) Y j n)] I j +σv 2 > β m By rearranging the terms of the inequaity we obtain: Pr{Γ j > β m } = Pr{Pρ c ĥ H j ĥj +Re[P 1 ρ 2 c) N ĥ j n) Y j n)] β m I j > β m σ 2 v} = Pr{ψ j > σ 2 v}. The characteristic function of ψ j conditionay on a particuar vaue of ĥj is the addition of two random variabes: a Gaussian process with meanpρ c X j and variancep 1 ρ 2 c)x j /2 and a chi-square random variabe with K degrees of freedom and parameter βλ. This can be mathematicay written as foows: Ψ ψj h j iω) = ejpρcxj+ω2 P 1 ρ 2 c )Xj/2 1+iωβ m λ) K. The CF conditiona on the decision made by the transmitter can be obtained as foows: Ψ ψj β m<x j<β m+1 iω) = Ψ ψj X j iω)fx j )dx j. β m This term wi be evauated numericay and then transformed into the PDF domain to obtain the statistics of instantaneous SINR conditiona on the decion made by the transmitter. VI. RESULTS This section presents graphica resuts of the statistics of the MRC beamformer with adaptive moduation, scheduing and co-channe interference with imperfect CSIT. Figure 2 dispays the resuts of the Cumuative Distribution Function CDF) of the SNR of the scheduer conditiona on the decision made by the transmitter based on imperfect CSIT using a hypothetica MCS seection threshod equa to ˆβ = 2). The resuts in Figure 2 have been obtained using fixed transmit power settings Pγ/σv 2 = 1) assuming no interference with different numbers of antennas N = 2,N = 4) and different vaues of correation coefficients ρ = 0.2, ρ = 0.95, ρ c = 0.2 and ρ c = 0.95). Figure 3 shows the resuts for the CDF of the SNR using the same settings as in the previous exampe, except for the transmit power which is now set to Pγ/σv 2 = 5. The objective of investigating the conditiona CDF is to observe the effects of imperfect CSIT on the instantaneous SNR experienced by the schedued terminas. The resuts show the heavy infuence of imperfect CSIT on the characteristics of the CDF. Low vaues of the correation coefficient ρ 0, see a considerabe degradation on the probabiity of correct reception. Note that a curves of the CDF depart from the hypothetica decision threshod set to ˆβ = 2. This departure to the eft-hand side of the figure is a measure of the incorrect reception due to imperfect CSIT. A the curves at the top eft of the figure are indeed the curves with worse CSIT conditions. It is observed TABLE II. SINRdB) vs BLER FOR WiMAX MODULATION AND CODING SCHEMES [39]. QPSK 1/3 QPSK 1/2 QPSK 2/3 SINR BLER SINR BLER SINR BLER e e e-3 QPSK 3/4 QPSK 4/5 16 QAM 1/3 SINR BLER SINR BLER SINR BLER e e e-3 16 QAM 1/2 16 QAM 2/3 16 QAM 3/4 SINR BLER SINR BLER SINR BLER e e e-3 that spatia correation degrades performance at high vaues of SNR, but it coud be beneficia in the ow SNR regime. In some cases, spatia diversity provided by higher numbers of antennas can even compensate for the effects of imperfect CSIT, particuary at with ow vaues of spatia correation. In a cases in both figures, it is observed that the performance of the CDF is superior with higher numbers of terminas in the scheduer, but this gain is more noticeabe in channes with ow spatia correation. It can be aso observed that user scheduing reduces the effects of spatia correation. Spatia correation reduces the diversity gains of the combining beamformer, and it can be accumuated over the severa antennas resuting in a more noticeabe performance reduction. User scheduing provides extra diversity gains that can compensate this reduction. The resuts presented in Figure 3 and Figure 4 have been obtained using the same settings used in the previous two exampes, except for the interference assumption. The channe power settings of the K = 2 persistent interferers were a set to λ/γ = 0.1. The resuts show the CDF of the instantaneous SINR instead of the SNR. The CDF resuts show how affected the system becomes by the presence of interference. It becomes evident that the presence of interference affects aso how the spatia correation pays a roe on the performance of the system. This wi become more evident in the resuts of throughput presented in the foowing figures. To test the performance of the agorithm in a fu wireess transmission system with different moduation formats, we have used the settings of the WiMAX standard and its different moduation schemes see Tabe II). The resuts in Figure 4 and Figure 5 present the overa throughput for a network with different numbers of users incuded in the scheduer versus different vaues of transmit average SNR. Figure 4 shows the resuts with no interference, whie Figure 5 shows the resuts with K = 2 interferers using set to λ/γ = 0.1. The resuts with interference show severa changing patterns due to the compex reation between interference and the received signa by the terminas. Surprisingy at high vaues of transmit SNR some of the curves with ow spatia correation tend to perform worse that the correated cases, which can ony be expained by the increased importance of the interference term and the parameters of the moduation formats used in the simuation. VII. CONCLUSIONS This paper has presented an anaytica framework for the study of joint MRC beamforming, termina scheduing and resource aocation moduation assignment) agorithms for mutiuser networks in the presence of persistent co-channe

8 Figure 2. CDF of instantaneous SNR conditiona on the estimated SNR being above the threshod ˆβ = 2 with fixed Tx power settings Pγ/σ 2 v) = 5) without interference and different vaes of antennas and correation coefficients. Figure 4. CDF of instantaneous SINR conditiona on the estimated SNR being above the threshod ˆβ = 2 with fixed Tx power settings Pγ/σ 2 v) = 1) in the presence of cochanne interference K = 2,λ/γ = 0.1) and different vaes of antennas and correation coefficients. Figure 3. CDF of instantaneous SNR conditiona on the estimated SNR being above the threshod ˆβ = 2 with fixed Tx power settings Pγ/σ 2 v) = 5) without interference and different vaes of antennas and correation coefficients. Figure 5. CDF of instantaneous SINR conditiona on the estimated SNR being above the threshod ˆβ = 2 with fixed Tx power settings Pγ/σ 2 v) = 5) in the presence of co-channe interference K = 2,λ/γ = 0.1) and different vaes of antennas and correation coefficients. interference. The resuts show that co-channe interference can consideraby affect the performance of beamforming, being counteracted by the effects of scheduing and higher degree of accuracy of channe state information at the transmitter side. The number of antennas tends to reduce the effects of imperfect CSIT and interference. However, channe correation can affect these gains, particuary in the high SNR regime. Conversey, in the ow SNR regime it seems that channe correation can outperform the case on uncorreated channes. Spatia correation effects tend to be accumuated when the number of antennas increases and therefore its effects wi be more ceary observed in the high SNR regime. ACKNOWLEDGMENTS This work has received funding from project SCOTT within the Eectronic Component Systems for European Leadership Joint Undertaking under grant agreement No This Joint Undertaking receives support from the European Unions Horizon 2020 research and innovation programme and Austria, Spain, Finand, Ireand, Sweden, Germany, Poand, Portuga, Netherands, Begium, Norway. Funded aso by FCT/MEC Fundacão para a Ciência e a Tecnoogia), ERDF European Regiona Deveopment Fund) under PT2020, and by CISTER Research Unit CEC/04234).

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