Performance analysis of general pilot-aided linear channel estimation in LTE OFDMA systems with application to simplified MMSE schemes

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    Performance analysis of general pilot-aided linearchannel estimation in LTE OFDMA systems withapplication to simplified MMSE schemesSamir omar, Andrea AncoraNXP SemiconductorsBusiness Line Cellular Systems505 route des Lucioles06560 Sophia Antipolis, FranceEmail: {samir.omar.andrea.ancora}@nxp.com

    Abstract-In this paper we provide a general framework forthe performance analysis of pilot-aided linear channel estimatorsclass including the general interpolation, Least Squares (LS),regularized LS, Minimum-Mean-Squared-Error (MMSE) andapproximatedMMSE estimators. The analysis is performed fromthe perspective of Long Term Evolution Orthogonal FrequencyDivision Multiple Access (LTE OFDMA) down-link systems. Wealso propose two novel modified MMSE schemes, an ExponentialMismatched MMSE and a Simplified MMSE, to overcome thehigh implementat ion complexity o f the MMSE and offeringimprovements to other known approximated methods. At theend , we veri fy the analyt ica l results by means of Monte-Carlosimulations in terms of Normalized Mean Square Error (NMSE)and coded Bit Error Rate (BER).Index Terms-OFDMA, channel estimation, interpolation,Least-Squares, Minimum-Mean-Squared-ErrorI. INTRODUCTION AND SYSTEM DEFINITION

    In December 2004, the Third Generation Partnership Program (3GPP) members started a feasibility study on the enhancement of the Universal Terrestrial Radio Access (UTRA)in the aim of continuing the long time frame competitivenessof the 3G UMTS technology beyond HSPA (High SpeedPacket Access). This project was called Evolved-UTRANor Long Term Evolution (LTE). Orthogonal Frequency Division Multiple Access (OFDMA) was chosen as a multipleaccess scheme for the Frequency Domain Duplexing DownLink (FDD DL) transmission [3]. The discrete-time OFDMAtransceiver model is schematically depicted in Figure 1.The users ' complex constellation symbols are mapped onK - 1 occupied sub-carriers spaced by ~ f s c == 15 kHz andpadded with zeros on the DC and band-edges sub-carriers(where these last ones can be considered as a guard-band)to fit the Inverse Fast Fourier Transform (IFFf) of orderN.The resulting sequence of length N is cyclic-prefixed byLcp samples. The Cyclic-Prefix (CP) length Lcp is variableand configured by the system to be longer than the channeldelay spread L. In LTE, two cyclic-prefixes are consideredallowing flexible system deployment (small and large cellradius): a short one of duration 4.7 J..lS and a long one of

    978-1-4244-2644-7/08/$25.00 2008 IEEE

    Dirk T.M. SlockEURECOM InstituteMobile Communications Dept.2229 route des Cretes, BP 19306904 Sophia Antipolis Cedex, FranceEmail: [email protected]

    16.7 J..ls. The sequence of N + Lcp samples s(k) is convolvedwith a discrete time Finite Impulse Response (FIR) channel (eventually time varying but assumed constant over oneOFDMA symbol) modeling the wireless channel at a samplingrate Ts == 1/ ( N ~ f s c ) .At the User Equipment (UE) side, the received sequencer(k) results from the channel output signal added with complex circular white Gaussian noise w(k). Assuming perfectsynchronization, the CP samples are discarded and the remaining N samples are FFT processed to retrieve the complexconstellation symbols transmitted over the orthogonal subchannels.The system bandwidth is scalable by controlling theIFFf/FFf size N of the OFDMA symbol and keeping thesub-carrier spacing constant: table 1 resumes the main systemtransmission parameters considered for LTE. With a FFfsize varying from 128 to 2048, the supported DL bandwidth ranges from 1.25MHz to 20MHz. Figure 2 represents, without loss of generality, one possible FDD LTEslot structure, namely the short-prefixed case composed of 7OFDMA symbols. The slot structure embeds pilot symbols(represented by P), also referred to as Reference Signal (RS),used to estimate the channel at the receiver side. The channelestimation is needed for channel equalization and general linkquality measurements. These pilot symbols are interleavedwith the data symbols (represented by D) in the frequencydomain and are disposed on a uniform grid occupying thefirst and the fifth OFDMA symbols of each slot and placedevery M == 6 sub-carriers.Moreover, there are (N - K) virtual sub-carriers (VC) distributed equally at both ends of each slot to allow for radix-2FFf implementation with the given sub-carrier spacing andsystem bandwidth.

    II . PILOT-AIDED LINEAR CHANNEL ESTIMATIONAlthough the general channel estimation problem in case

    of single antenna transmissions is two-dimensional [1], Le. isto be carried jointly in the frequency and time domains, i t is

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    Substituting (1) in (2), the error of the interpolated CTFestimate is:

    interpolation filter and the interpolated CTF estimate can bewritten as:normally separated into two one-dimensional estimation steps[2] for ease of implementation.In this context, we deal in particular with the channelestimation problem over one OFDMA symbol (specificallythe symbol containing the RS) to exploit the frequency domaincharacteristics and instead we do not consider its time-varyingcharacteristics due to Doppler effect in the aim of exploitingcorrelation in time.

    (2)

    (5)where P is the matrix of the equivalent pulse-shaping filter,u is the discrete-time uncorrelated multipath fading channelvector and

    where H = FLh and FL is the N x L matrix obtained takingthe first L columns of the Fourier transform matrix.The error covariance matrix is:CH; = (FL - AFp) Ch (FL - AFp)H + AAH (4)

    being Ch = EhhH the channel covariance matrix, {.}H andE{ .} denoting, respectively, the Hermitian and the expectationoperators.Although pulse-shaping is not mandated in LTE, receiverfront-end consists of an anti-aliasing low-pass filtering.Therefore the channel and its covariance matrix can effec

    tively be modeled as:

    Y1 Yo

    rCP1

    ro

    :!"!ca!:ca - . -- t:i : 0en

    NFrequencyDomain

    LTE OFDMA transceiver.

    . ,.. ...............1 I .--', r(k)S ( k ) . ~ ~ \ ' - 6 )W(k)

    o 0 0o 0 0 0

    D D D Do 0 0 0DOD 0D D D 0

    o 0 0 0

    Fig. 1.

    Scp+N.1

    SCP.1

    Fig. 2. LTE OFDMA slot structure. H . (2 2 2 )u=Euu =dlag aUO,aUl, , a U LM P _ l

    2) IFFT estimator: The second natural approach to retrievethe whole CTF estimate is by IFFf interpolation. The IFFTCTFestimate interpolated over all sub-carriers can be obtainedby using in (2):

    is its diagonal covariance matrix normally assimilated to thechannel Power Delay Profile (PDP).Recalling equation (2), the first intuitive move is to use linear interpolation. Although the straightforward filter structureA is not described, the investigation of (2) reveals that the

    linear interpolation estimator is biased from the deterministicviewpoint while it is unbiased from the Bayesian viewpointregardless of the structure of A.

    (6)

    (7)Hence, the IFFT estimator is given by:

    "" 1 H ""HIFFT = pFLF p H p

    (1)where P = fK/Ml is the number of available pilots where K isthe number of occupied sub-carriers (including DC). h is the L x I Channel Impulse Response (CIR) vector.The effective channel length L :::; Lcp is assumed to beknown. F p is the P x L matrix obtained by selecting the rowscorresponding to the pilot positions and the first Lcolumns of the N x N Discrete Fourier Transform (OFf)matrix whose elements are (F)n,k = e-ij;-(nk) witho :::; n :::; N - 1 and 0 :::; k :::; N - 1; Hp is the P x 1 zero-mean complex circular white noisevector whose L x L covariance matrix is Cn = I L ;p p

    In the OFDMA LTE context, as for any comb-distributedpilot OFDM system [5], the Channel Transfer Function (CTF)is ML estimated in the frequency domain at the pilot positionsby de-correlating the constant modulus Reference Signal pilotsequence. Using matrix notation, it can be modeled as:

    A. Channel Estimation by interpolationJ) Linear interpolation estimator: The natural approach toestimate the whole CTF is to interpolate the CTF estimate onpilot positions Hp In the general case, let A be a generic

    The IFFT interpolated CTF estimate error and its covariancematrix, applying (1) and (6) into (2), becomes:HIFFT = FL (IL - ~ F : : F p ) h- ~ F L F : : H p (8)

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    Slot duration [ms] 0.5Sub-carrier spacing ~ f s c [kHz] 15Transmission BW [MHz] 1.25 2.5 5 10 15 20Sampling frequency [MHz] 1.92 3.84 7.68 15.36 23.04 30.72FFf size N 128 256 512 1024 1536 2048Occupied sub-carriers (including DC) K 76 151 301 601 901 1201

    TABLE ILTE OFDMA PARAMETERS.

    (9)

    (10)

    In the approximation of IL ~ bFi[F p, the estimator wouldbe unbiased and its error covariance matrix would reduce to:1 2 HCH ~ -pG'H- FLFLFFT p

    and its covariance matrix:CHge n = (FL-B (GHG+R)- lGHFp) Ch(FL -B(GHG+R) - lGHFp)H ++ ( T ~ p B (GHG + R) -1 GHG (GHG + RH) -1 B H

    (13)1) LS estimator: The LS estimator discussed in [6] can be

    inferred by choosing:

    Substituting (1) and (14) in (12) and (13), the error reducesto:

    with OL being the L x L matrix containing zeros. And theestimator appears as:

    2) Regularized LS estimator: As evidenced in [9], the LTEsystem parameters make the LS estimator inapplicable: the expression (FpFi[) -1 is ill-conditioned due to the large unusedportion of the spectrum corresponding to the unmodulated subcarriers.

    (16)

    (17)

    (15)

    (14)

    -. (H -1 H-'HLS == FL F p F p) F p H p

    .- (H -1 H'-HLs == -F L F p F p) F p H p

    B =

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