Time-Varying Phase Noise and Channel Estimation in MIMO Systems

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1 This is an author-produced peer-reviewed version of this artice. The fina definitive version of this document can be found onine at 2012 IEEE 13th Internationa Worshop on Signa Processing Advances in Wireess Communications pubished b IEEE. Copright restrictions ma app. doi: Time-Varing Phase Noise and Channe Estimation in MIMO Sstems Hani Mehrpouan Ai A. Nasir Thomas Erisson Steven D. Bostein George K. Karagiannidis Tomm Svensson Department of S2 Chamers Universit of Technoog Sweden. Research Schoo of Engineering the Austraian Nationa Universit Austraia. Department of ECE Queen s Universit Canada. Department of ECE Aristote Universit of Thessaonii Greece. emai: hani.mehr@ieee.org ai.nasir@anu.edu.au thomase tomm.svensson@chamers.se geoarag@auth.gr. Abstract Performance of high speed communication sstems is negative affected b osciator phase noise PN). In this paper joint estimation of channe gains and Wiener PN in muti-input muti-output MIMO) sstems is anazed. The signa mode for the estimation probem is outined in detai. In order to reduce overhead a ow compexit data-aided eastsquares LS) estimator for joint obtaining the channe gains and PN parameters is derived. In order to trac PN processes over a frame a new decision-directed extended Kaman fiter EKF) is proposed. Numerica resuts show that the proposed LS and EKF based PN estimator performances are cose to the CRLB and simuation resuts indicate that b empoing the proposed estimators the bit-error rate BER) performance of a MIMO sstem can be significant improved in the presence of PN. I. INTRODUCTION The throughput demands on point-to-point wireess ins are expected to grow in the foreseeabe future. For exampe microwave bachau ins that interconnect base stations to base station controers are expected to support data rates of mutipe Giga bits per second 1]. However traditiona schemes of increasing bandwidth efficienc e.g. using higher order moduations are not capabe of meeting this demand. One approach is to use Muti-input muti-output MIMO) sstems to significant enhance the bandwidth efficienc of high speed wireess sstems 2]. However osciator imperfections that resut in phase noise PN) are great detrimenta to the performance of wireess sstems 3] 6] with a more pronounced impact at higher carrier frequencies e.g. E-band GHz) 7]. Moreover in the case of MIMO sstems each transmit and receive antenna ma be equipped with an independent osciator. For exampe in the case of ine-of-sight LoS) MIMO sstems a singe osciator cannot be used for a the transmit or receive antennas since the antennas need to be paced far apart from one another 8] 1. Simiar in muti-user MIMO or space division mutipe access SDMA) sstems mutipe This research is in parts supported b the Swedish Agenc for Innovation Sstems VINNOVA) within the project P MAGIC Ericsson AB Qamcom AB and NSERC Discover Grant For a 4 4 LoS MIMO sstem operating at 10 GHz and with a transmitter and receiver distance of 2 m the optima antenna spacing is 3.8 m 9]. users with independent osciators transmit their signas to a common receiver 10]. Thus schemes that can joint estimate mutipe PN processes at the receiver are of particuar interest 8] 10]. In singe carrier communication sstems PN is mutipicative 6]. Cramér-Rao ower bounds CRLB) and agorithms for estimation of PN in singe-input singe-output SISO) sstems are thorough anazed in 6] 11] 12]. However these resuts are not appicabe to MIMO sstems where a received signa ma be affected b mutipe PNs 8]. The effect of PN on the capacit and performance of MIMO sstems has been anazed in 13] where it is demonstrated that PN can great imit the performance of muti-antenna sstems. In 8] piot-aided estimation of PN parameters in a MIMO sstem is investigated. However the scheme in 8] uses piot smbos that are transmitted based on time division mutipexing TDM) to trac the MIMO PN parameters which is bandwidth inefficient. In addition in 8] the MIMO channe is assumed to be nown and the estimation performance imits have not been anazed. In orthogona frequenc division mutipexing OFDM) sstems PN is convoved with data smbos 14]. Therefore PN resuts in inter-carrier interference ICI) and significant performance oss 14]. Considerabe research has been carried out to eiminate or reduce the effect of ICI causing PN in MIMO OFDM sstems 4] 14]. However 4] 14] are based on the assumption that the MIMO sstem is affected b a singe PN parameter and are not appicabe to LoS and or muti-user MIMO sstems. In this paper joint estimation of mutipe PN and channe gains in a MIMO sstem is anazed. A data-aided east squares LS) estimator for joint obtaining the channe gains and PN is proposed. Next the piot and estimated data smbos in combination with a decision-directed extended Kaman fiter EKF) are used for tracing the PN processes over a frame. The contributions and organization of this paper can be summarized as foows: In Sections II and III joint estimation of channe gains and PN in a MIMO sstem is parameterized and new CRLBs for the muti-parameter estimation probem is presented. The CRLBs are then used to benchmar the

2 This is an author-produced peer-reviewed version of this artice. The fina definitive version of this document can be found onine at 2012 IEEE 13th Internationa Worshop on Signa Processing Advances in Wireess Communications pubished b IEEE. Copright restrictions ma app. doi: performance of the proposed estimators and to quantif the effect of unnown PN on channe estimation accurac and vice versa. In Section IV nove agorithms for estimating and tracing the unnown channe gains and time-varing PN respective throughout a frame are proposed. A data-aided LS estimator for joint obtaining the MIMO channe and PN parameters is proposed. Next a new EKF based estimator is proposed that is shown to accurate trac the PN processes over a frame. In Section V simuations are carried out that investigate the performance of a MIMO sstem using the proposed estimators. Notation: Superscripts ) ) H and ) T denote conjugate conjugate transpose and transpose operators respective. Bod face sma etters e.g. x are used for vectors bod face capita aphabets e.g. X are used for matrices and X] x represents the entr in row x and coumn of X. I X X 0 X X and 1 X X denote the X X identit a zero and a 1 matrices respective. stands for Schur eement-wise) product is the absoute vaue operator x returns the phase of the compex variabe x x denotes the eement-wise absoute vaue of a vector x diagx) is used to denote a diagona matrix where the diagona eements are given b vector x. diagx) is used to denote the diagona eements of matrix X. E ] denotes the expected vaue of the argument and R and I are the rea and imaginar parts of a compex quantit respective. Fina N μ σ 2) and CN μ σ 2) denote rea and compex Gaussian distributions with mean μ and variance σ 2 respective. II. SYSTEM MODEL A MIMO sstem with N t and N r transmit and receive antennas respective is considered. Each frame of ength L f smbos is assumed to consist of a training sequence TS) of ength L t smbos data smbos and piot smbos that are transmitted ever L p smbos. In this paper the foowing set of assumptions is adopted: A1. The TSs broadcast from the N t transmit antennas are assumed to be mutua orthogona and of ength L t = N t. A2. To ensure generait each transmit and receiver antenna is assumed to be equipped with an independent osciator as depicted in Fig. 1. This ensures that the sstem mode is in ine with previous wor in the iterature 8] and is aso appicabe to various MIMO sstems e.g. LoS MIMO and SDMA MIMO sstems 10]. A3. Quasi-static and frequenc fat-fading channes are considered where the channe gains are assumed to remain constant over the ength of one frame. A4. Perfect timing snchronization is assumed which can be achieved b standard snchronization agorithms 11]. Note that Assumption A3 is reasonabe for actua wireess ins incuding in LoS microwave bachau sstems where the antennas are ocated on stationar towers with amost no obstaces between them the channes var much more sow than the phase noise process. In addition Assumptions A3 and A4 are in ine with previous PN estimation agorithms in SISO and MIMO sstems in 6] 8] 11]. More important unie the resuts in 6] 8] 11] which assume that the channe gains are estimated and equaized before PN estimation in this paper we joint estimate the channe gains and PN in a MIMO sstem. The discrete-time baseband received signa at the MIMO receiver n) 1 n) Nr n)] T isgivenb 2 n) =Θ r] n)hθ t] n)sn)+wn) = A e jbn)) sn)+wn) 1) sn) s 1 n) s Nt n)] T s n) is the nth transmitted smbo that corresponds to the th transmit antenna and consists of both piots and data smbos H is the N r N t channe matrix with h denoting the quasi-static unnown channe gain from the th transmit to the th receive antenna which is assumed to be distributed as h CNμ h σh 2 ) from frame to frame Θ r] n) diag diag e jθr] 1 n) e jθr] Nr n)) and Θ t] n) e jθt] 1 n) e jθt] N t n) ) are N r N r and N t N t matrices respective θ t] n) and θr] n) correspond to the nth sampe of the PN at the th transmit and th receive antenna respective A α 1 α Nr ] T is the N r N t channe magnitude matrix with α α 1 α Nt ] T α h Bn) β 1 n) β Nr n)] T is an N r N t matrix with β n) β 1 n) β Nt n)] T β n) θ t] n)+θr] n)+ h denotes the overa phase contributions from the osciators and channe corresponding to the th transmit and th receive antenna and wn) w 1 n) w Nr n)] T is the vector of additive white Gaussian noise AWGN) where w n) CN0σ 2 w). For free-running osciators it is found that the PN process can be modeed as a Wiener process 3] 4]. Therefore θ t] is modeed as 3] 4] θ t] n) =θt] n 1) + Δt] n) 2). In 2) the phase where a simiar mode is used for θ r] innovations for the th transmit antenna Δ t] th n) receive ) antenna Δ r] n) is assumed to be Δ t] n) N ) 0σ 2 Δ t] 2 Throughout this paper indices =1 N t =1 N r and n =1 L f are used to denote transmit antennas receive antennas and smbos respective.

3 This is an author-produced peer-reviewed version of this artice. The fina definitive version of this document can be found onine at 2012 IEEE 13th Internationa Worshop on Signa Processing Advances in Wireess Communications pubished b IEEE. Copright restrictions ma app. doi: Δ r] n) N ) ) 0σ 2 and aso mutua independent Δ r] ). As shown in 5] for most practica osciators the PN innovation variance is sma e.g. σδ 2 =10 3 ) rad 2. III. CRAMÉR-RAO LOWER BOUNDS In this section new expressions for the Fisher s information matrices FIMs) for data-aided estimation DAE) of PN and channe gains in an N r N t MIMO sstem is derived. Corresponding to 1) the vector of parameters of interest λ is given b λ ] λ T 1 λ T T N r where λ α T β T n)] T is the parameter vector for the th receive antenna. Proposition 1: The FIM for joint DAE of channe gains and PN ] parameters over the observation T 1 N T T r with n L t +1) n)] T is a 2N t N r 2N t N r matrix where its 2N t 2N t submatrices FIM for =1 N r are given b 3) at the bottom of this page. In 3) ] T U 0 T 1)L t N t E S) T 0 T N r )L t N t is an N r L t N t matrix E diage jβ1n) e jβ N n) t ) is an N t N t matrix S s 1 s Nt ] T is an N t L t matrix with s s n L t +1) s n)] T Υ diag α 1 α Nt ) is an N t N t matrix and the th row th coumn for =1 N t eements of the N t N t matrices Π 11 Π 12 Π 21 Π 22 are given Σ 1 Σ α Σ 1 Σ b Π 11 ] ) Tr α Π 12 ] ) Tr Σ 1 Σ α Σ 1 Σ β n) Π 21 ] ) Tr Σ 1 Σ β n) Σ 1 Σ α Π 22 ] ) Tr Σ 1 Σ β n) Σ 1 Σ β n) respective and the L t L t submatrices Σ ofthen r L t N r L t covariance matrix Σ are given b 4) at the bottom Σ of this page. Note that Σ α and β n) can be easi determined and are given in 15]. Proof : See 15]. Remar 1: The CRLB for estimation of channe gains and PN parameters is given b the diagona eements of the inverse of the FIM matrix. Given the structure of the FIM it is difficut to find a cosed-form expression for the CRLB for an N r N t MIMO sstem. Remar 2: The FIM is not boc diagona. Therefore the estimation of channe magnitudes and PN parameters in a MIMO sstem are couped with one another i.e. channe estimation accurac is affected b the presence of PN and vice versa. This resut indicates that channe and PN estimation need to be carried out joint in a MIMO sstem. IV. CHANNEL AND PHASE NOISE ESTIMATION In this section the data-aided LS and decision-directed EKF channe and PN estimators are derived in detai. A. Data-Aided Estimation DAE) Since the maximum-ieihood estimator for joint estimation of channe and phase noise has a ver high computationa compexit in this subsection an LS estimator for joint estimation of A and Bn)  and ˆB DAE] respective is determined. Using the received signa mode in 1) the LS estimator is given b ˆPn) =  ) e j ˆBn) = YS H SS H ) 1 = YSH 5) L t where Y 1 Nr ] T Pn) A e jbn) and S is defined in 3). The third equait in 5) foows from Assumption A1 i.e. SS H = L t I Nt N t. Using 5) estimates of the channe and PN matrices are determined b  = YS H /L t ˆBDAE] n) = YS H /L t 6) respective. Simuation resuts in Section V show that the performance of the proposed LS estimator in 6) is cose to the CRLB over a wide range of SNR vaues. B. Decision-Directed Estimation This section presents an EKF to trac N r N t PN parameters β n) in decision directed mode. Using 2) the FIM = 2R U H Σ 1 U + Π11 ) 2I U H Σ 1 U Υ + Π12 ) 2I Υ U H Σ 1 U + Π21 ) 2R Υ U H Σ 1 U Υ + Π22 ) ] 3) Σ = =1 α2 s s H σ 2 Δ r] Ψ + σ 2 Ψ)+ N t Δ t] =1 σ 2 Ψ)+σ Δ wi 2 r] Lt L t =1 α α e jβ n) e jβ n) s s H σ 2 Δ t] =1 Ψ) α α e jβ n) e jβ n) s s H = 4)

4 This is an author-produced peer-reviewed version of this artice. The fina definitive version of this document can be found onine at 2012 IEEE 13th Internationa Worshop on Signa Processing Advances in Wireess Communications pubished b IEEE. Copright restrictions ma app. doi: 10 1 LS Est. σ 2 Δ =10 3 LS Est. σ 2 Δ =10 3 LS Est. σ 2 Δ =10 4 LS Est. σ 2 Δ =10 5 LS Est. σ 2 Δ =10 4 LS Est. σ 2 Δ =10 5 CRLB σ 2 Δ =10 3 CRLB σ 2 Δ =10 3 Channe Est. MSE.29 CRLB σ 2 Δ =10 4 CRLB σ 2 Δ =10 5 Phase Noise Est. MSE CRLB σ 2 Δ =10 4 CRLB σ 2 Δ = SNR db] Fig. 1. MSE for DAE of channepn) in 5) for a 2 2 MIMO sstem for different PN variances. unnown state vector φn) β T 1 n) β T Nr n)]t is given b φn) =φn 1) + δn). 7) The state noise vector δn) Δ 11 n) Δ 1Nt n) Δ Nr1n) Δ NrN t n)] T is distributed as δn) N0 NtN r 1 Q). The state noise covariance matrix Q = E δn)δn) T isann t N r N t N r matrix where its submatrices Q for =1 N r can be determined as Q = σ 2 Δ r] diagσ 2 Δ t] 1 1 Nt N t + diagσ 2 σ 2 Δ t] 1 Δ t] σ 2 Δ t] N t ) N t ) =. 8) The observation equation for the Kaman fiter is given b 1). Let us define zn) z 1 n) z Nr n)] T = A e jbn) ) sn). The N r N r N t Jacobian matrix for the EKF is evauated b computing the first order partia derivative of zn) with respect to the state vector φn) i.e. D φn) ) = zn)/ φ T n). Using 7) and 8) the EKF equations can be written as ˆφn n 1) =ˆφn 1 n 1) Mn n 1) =Mn 1 n 1)+Q 9a) 9b) D ) φn)) ˆφn n 1) =D φn)= 9c) ˆφn n 1) Kn) =Mn n 1)D H ) ˆφn n 1) 9d) D ) ˆφn n 1) Mn n 1)D H ) ) 1 ˆφn n 1) + W ˆφn n) =ˆφn n 1) + R Kn) Mn n) =RMn n 1) n)  e j ˆBn) ) sn) Kn)D ˆφn n 1) ) Mn n 1) 9e) ] 9f) SNR db] Fig. 2. MSE for DAE of PN for a 2 2 MIMO sstem for different PN variances. where ˆφn n 1) = ˆβT 1 n n 1) ˆβ ] T Nr T n n 1) is the predicted state vector at the nth smbo W = σwi 2 Nr N r ˆBn) = Bn) φn)= Kn) is the N ˆφn n 1) t N r N r Kaman gain matrix  is given in 6) and Mn n) is the N t N r N t N r fitering error covariance matrix. In order to ensure convergence of the EKF the state vector is initiaized with the LS estimates and ML t L t )=0.1I NrN t N rn t. V. NUMERICAL RESULTS AND DISCUSSIONS In this section L t = N t the observation ength in the decision-directed mode L o = N t and σ 2 = σ 2 = σ Δ t] Δ r] Δ 2 and σw 2 = 1/SNR. The MIMO channe matrix is generated as a sum of LoS and non-los NLoS) components using the mode in 9]. K denotes the Rician factor. The antenna spacing at the transmitters and receiver are set to their optimum vaues 9] the carrier frequenc is set to 70 GHz and K = 2 db. Wash-Hadamard codes with binar phase-shift eing BPSK) are used for the TSs. The phase unwrapping agorithm in 6] is appied here. Without oss of generait on the mean-square errors MSEs) for estimation of channe gain and phase noise for the first antenna eement are presented 3. A. Estimation Performance Figs. 1 and 2 show the CRLB and MSE for DAE of MIMO channes Pn) in 5) and time-varing PN respective versus SNR. The CRLB in Section III is numerica evauated for PN variances σδ 2 =10 3 ] rad 2. Note that the PN variance σδ 2 =10 3 rad 2 corresponds to ver strong PN 5]. The CRLB resuts in Fig. 1 show that in the presence of PN estimation of the MIMO channe 3 Note that simiar resuts are obtained for the estimation of the parameters for the remaining antennas and are not presented here to avoid repetition.

5 This is an author-produced peer-reviewed version of this artice. The fina definitive version of this document can be found onine at 2012 IEEE 13th Internationa Worshop on Signa Processing Advances in Wireess Communications pubished b IEEE. Copright restrictions ma app. doi: BER No tracing σ 2 Δ =10 3 EKF imperf. dec. feedbac σ 2 Δ =10 3 EKF perf. dec. feedbac σ 2 Δ =10 3 EKF imperf. dec. feedbac σ 2 Δ =10 4 EKF perf. dec. feedbac σ 2 Δ =10 4 EKF imperf. dec. feedbac σ 2 Δ =10 5 Perf. Channe & phase noise nowdege SNR db] Fig. 3. Perfect and Imperfect decision feedbac BER of a 2 2 MIMO sstem for PN variances σδ 2 =10 3 ] rad 2 with BPSK and L p =10 suffers from an error foor which is direct reated to the variance of PN innovations. This resut can be anticipated since even at ver high SNR where the effect of the AWGN noise w is negigibe the noise corresponding to the PN imits estimation accurac. The same noise aso imits the estimation accurac of the nth smbo s PN parameters in the data-aided case and resuts in an error foor in Fig. 2. B. Sstem Performance The proposed LS estimator and training smbos at the start of each frame are used to estimate the MIMO channe gains and phases. Next the proposed EKF estimator is appied in both perfect and imperfect decision feedbac modes to trac the phase parameters over a frame. The frame ength is set to L f = 1000 BPSK smbos and new channes are generated for each frame. Piot spacing is set to L p =10 which corresponds to a snchronization overhead of 10%. An MMSE inear receiver given b ŝn) =ˆP H n)ˆpn)+σ 2 wi Nt N t ) 1 ˆP H n)n) 10) is used to equaize the effect of phase noise and channe gains. Fig. 3 depicts the bit error rate BER) of a 2 2 MIMO sstem using the proposed LS estimator without phase tracing and with the proposed EKF phase tracing in the presence of various PN variances. The scenario with perfect channe nowedge and snchronization is aso potted as a benchmar. The resuts in Fig. 3 demonstrate that without phase tracing the MIMO sstem performance deteriorates significant. However b combining the proposed LS and EKF estimators the BER performance of a MIMO sstem is shown to improve immense even in the presence of ver strong PN e.g. σ 2 Δ =10 3 rad 2. More important it is demonstrated that the BER of a MIMO sstem using the combination of the proposed channe and PN estimators is cose to the idea case of perfect channe and PN nowedge a performance gap of 3dB with SNR=20dB). Fina Fig. 3 shows that the overa MIMO sstem s BER suffers from an error foor at high SNRs. This resut is anticipated since at high SNR the performance of the MIMO sstem is dominated b PN instead of AWGN. VI. CONCLUSION In this paper estimation of channe and PN in MIMO sstems is anazed. The MSE of the LS estimator is shown to be cose to the derived CRLB over a wide range of SNR vaues. In order to trac the time-varing PN over a frame using the piot and estimated data smbos a new decisiondirected EKF based estimator is proposed. B empoing both the nown piot smbos and the received data smbos the resuting decision-directed EKF does not resut in error propagation. Next simuations show that the performance of a MIMO sstem using the proposed LS and EKF estimator is cose to the ideaistic setting of perfect channe and PN estimation e.g. at an SNR of 20 db for a 2 2 MIMO sstem there is a performance gap of 3dB between the two sstems. REFERENCES 1] J. Wes Muti-Gigabit Microwave and Miimeter-Wave Wireess Communications. First Edition. Artech House ] I. E. Teatar Capacit of muti-antenna Gaussian channes Europ. Trans. Commun. vo. 10 no. 6 pp Nov.-Dec ] M. R. Khanzadi et. a On modes bounds and estimation agorithms for time-varing phase noise in Proc. Int. Conf. Signa Process. and Commun. Sstems ICSPCS) Dec ] L. Tomba On the effect of Wiener phase noise in OFDM sstems IEEE Trans. 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