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

7/27/2019 Performance analysis of general pilot-aided linear channel estimation in LTE OFDMA systems with application to si

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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 == F L , G == F p and R == OL

showing that the LS estimator, a t least theoretically, is unbiased. Thus, compared to the linear interpolation estimatorgiven by (2), the LS est imator is considered as the perfectinterpolator as it sets to zero the bias term of expression(3) with A == F L (F{[Fp)- l F{[. Consequently, the errorcovariance matrix can be shown to be:

C 2 (H )-1 HHLS == G'Hp FL F p F p FLB. General approach to linear channel estimation

We remand to the simulation results section of this paperfor a comparison of their respective performances.

Given the LTE system parameters and the pilot structure,in practice, bFi[Fp is far from being a multiple of anidentity matrix: the approximation would be an equality whenK == N ,N/M > Land N /M being an integer, Le. thesystem should be dimensioned without guard-bands and thepilot should be disposed with a spacing which is dividingexactly the FFf order N, namely a power of two. Therefore,according to (8), the estimator HIFFT is biased as for the linearinterpolation case if the channel is deterministic and unbiasedfrom the Bayesian point of view.

Compared to the simple approaches presented in the previous section, more elaborated linear estimators derived fromboth deterministic and statistical viewpoint proposed in [6], [7]and [9], namely LS, Regularized LS, MMSE and MismatchedMMSE in addition to the novel estimators presented in thefollowing sections, can all be expressed under the generalformulation:

Where B, G and R are matrices that vary according to eachestimator as detailed in the following. With (1) and (11), weobtain the error expression:

-. ( H - 1 H - 'H gen == B G G +R) G Hp (11)To counter this problem, the robust regularized LS estimator

was used to yield a bet ter condi tioning of the matrix to beinverted by using the same Band G as for the LS estimator butintroducing the regularization matrix R == aIL with a beinga constant (off-line) chosen to optimize the performance ofthe estimator in a given Signal-to-Noise Ratio (SNR) workingrange.

Hge n = (FL - B (GHG + R)-l GHFp) h+- B (GHG + R)-l GHHp (12)

Hence, we can write the estimator as follows:-. ( H -1 H - 'Hreg,LS == F L F p F p+ aIL) Fp H p (18)


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The expressions for the error and the error covariance matrixof this estimator can be deduced directly from (12) and (13) bysubstitutingB, G and R with their corresponding expressions.

would consist in dividing the SNR working range into subranges and storing different versions of the matrix invertedoff-line for each sub-range.

and the error covariance matrix:

3) MMSE estimator: Using equations (11), (12) and (13),we can formulate the MMSE estimator [6] by denoting:

B = F L , G = F p and R = Ch -1 (19)lJ

(24)

5) Exponential mismatched MMSE estimator: Realisticchannel PDP are likely exponentially decaying rather thanuniform as assumed by the lnismatched-MMSE discussedabove. We therefore propose an exponential mismatchedMMSE estimator that approximates Ch by a diagonal matrixwhose entries are decaying exponentially. This is done byusing (19) and taking:

up - 1 . (-n In (2L)R = ~ C L , e x p and CL,exp = 'Y . dlag e Lh

. L-l In(2L) WIth 0 ::s n ::s L - 1 and'Y = 1/ Ln=o e - n - L -. Hence, It ISrepresented by:.-... (H aH p - 1 ) -1 HHexp-MMSE = FL Fp Fp + CL,exp Fp Hp (25)

Again, the error and the error covariance matrix can bededucted from (12) and (13) by substituting B, G and Rwith their corresponding expressions.Compared to the uniform channel distribution assumption ofMismatched-MMSE, the estimator reveals to be less sensitive

to the channel length mis-estimation due to the exponentialdecaying nature and thus less versions of the inverse ofthe matrix (F{fFp+ * C L ' ~ X P ) can be precomputed andstored.

(20)

(21)

thus giving.-... (H 2 -1) -1 H'-'"HMMSE = F L F p F p + aMp Ch F p H pAgain, applying (1) and (19) in (12) and (13), the error of

the MMSE estimator is:HMMSE =FL (IL - (F:;Fp + ali p Ch-1) -1 F:;Fp) h+

- FL (F:;Fp+ ali p Ch -1) -1 F:!Hp

4) Mismatched MMSE estimator: To avoid the estimationof the second order channel statistics Ch and of the consequenton-line inversion of a L x L matrix required in the straightforward application of the MMSE of (20), the channel PDP canbe assumed uniform [7]. Hence, in this Mismatched-MMSEformulation, Ch is imposed to have the structure of an identitymatrix.With reference to the general formulation in (11), thisscheme consists in taking the same Band G of (19) but

defining R = aH / . I L to give the expressionp

6) Simplified MMSE estimator: As already mentioned, thedirect implementation of the MMSE estimator in (20) requiresthe solution of two problems:1) The estimation of the variance of the noise and channelsecond order statistics;2) The on-line inversion of the large L x L matrix

SMMSE = F{ :F p + a ~ Ch - 1 (26)pwhenever the channel and noise statistics change.

Interestingly, we notice that this estimator is in practiceequivalent to regularized LS estimator in II-B2. where theonly difference lies in the fact that the ratio / can beupestimated and therefore adapted.

Assuming the required estimations available, we proposehere an original solution to overcome in particular the secondproblem. The idea behind our simplified MMSE estimator liesin separating the problem of the approximation of (26) into,first, considering a fixed initialization matrix Sini t , as detailedbelow, and then in enhancing the first approximation by inserting the contribution of a portion of the PDP correspondingto the strongest taps, denoted captured taps in the following,on the diagonal of the initialization matrix Sini t .

For a given channel length L, to avoid the on-line inversionof the matrix (F{fFp + ali p / . IL), the practical approach As for previous approximated methods, the dependencyfrom the noise variance can be maintained by quantization


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of the SNR into sub-ranges and storing a limited set of Sini tvalues.Let us define:

Comparing (30) with (11), the Simplified MMSE consistsin choosing:B = F L ,G = F p and R = (J'H C ini t -1 + D ~ S D H (31)p

where C ini t = ,8IL and (3 is a constant carefully chosen toprovide sufficiently good performance of the estimator.S. . - FHF + C - 1Inl t - p p vB: initp (27) The estimation error and the error covariance matrix expression of the proposed estimator can then be obtained bysubstituting (31) in (12) and (13).

The matrix SMMSE can be approximated by:SSMMSE = Sini t + D ~ S D H

where(28)

III. SIMULATION RESULTS

(32)NMSE-- = Ttr (CB:)H Ttr ( F LCh F !)where with Ttr{.} we denote the truncated trace operatorconsisting of the truncated covariance matrix considering onlythe K used sub-carriers.For the comparison in figure 3,we used raised-cosine pulseshaping filter with a roll-off factor of (3 = 0.2, the SCMAchannel and an LTE setup with N = 1024 correspondingto the 10 MHz transmission bandwidth case [3]. As forthe regularized LS estimator, we use the regularization term

a = 0.1. Connected lines represent the theoretical TNMSEwhile the dotted points represent the results of simulations.We can first conclude that the IFFT and linear interpolationmethods yield the lowest performances.

We compare the performances of the estimators by meanof T r u n c a t e d - N ~ r m a l i z e d - M e a n - S q u a r e d - E r r o r(TNMSE). Foreach estimatorH, the TNMSE is computed from its covariancematrix CB: and the true channel H = FLh using thefollowing:

Applying the Matrix Inversion Lemma, we can write:S- lSMMSE =

S-l S-l D (D H S- 1 D AS-1 ) - 1 H -1ini t - ini t in it + u D Sin i t(29)

1) D is a L x M selectormatrix called afterthe role it playsin the selection of the positions where the elements ofthe PDP profile (that correspond to the M captured taps)are going to be located on the diagonal of Sinit . The firstcolumn of the matrix D contains one only in the positionthat corresponds to the index of the first captured tap andzeros everywhere else and the second column containsone only in the position that corresponds to the index ofthe second captured tap and zeros everywhere else andso on.2) ~ S is a diagonal matrix containing the inverse of thepower of the captured taps after removing the effect ofinitialization, Le. ~ S r n , r n = ( J ' ~ ( C h ~ - ,8-1) wherehm is a vector contains the M c ~ p t u r e d taps.

It is worth mentioning that the number of significant taps interms of power is usually much less than the overall length ofthe CIR. Thus, the importance of the proposed estimator stemsfrom the fact that we take advantage of this property to reducethe size of the matrix to be inverted on-line from L x L toM x M with M L. Knowning that the number of operationsrequired to invert a matrix is proportional to the cube of itssize, we infer that our estimator reduces significantly (from L3to M3 ) the computational power compared to the traditionalMMSE estimator in (20).Another important aspect of the proposed estimator is that

the accuracy of the approximation is traded-off with the complexity by controlling the number of captured taps. Therefore,the more the number of the captured taps the larger the sizeof the matrix to be inverted on-line and vice versa.

Moreover the regularized LS and the mismatched MMSEprove to perform exactly equally and the TNMSE curve ofthe latter is therefore omitted in figure 3. The exponentialmismatched-MMSE and the simplified MMSE offer a performance gain over all other sub-optimal estimators but thelatter proves to be the one approaching the most the MMSEestimator performance particularly in the low SNR region.To highlight the robustness of our simplified MMSE, figure4 compares its performance to that of the mismatched-MMSEwhere the MMSE is used as a reference. It should be noted thatthe simulated mismatched-MMSE is further approximated by

exploiting a limited number of pilots around the sub-carriers tobe estimated in order to reduce complexity. It is evident thatthe performance of simplified MMSE exceeds for any SNRthat of mismatched-MMSE even though only 11 out of 50taps are captured.

Finally the estimated CTF is given by:- -1-HSMMSE = FLSSMMSE H p Figure 5 compares the performances of the mismatched(30) MMSE and of the simplifiedMMSE in terms of Bit Error Rate


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50

50

40

40

30

30

20SHRedS)

20SNR(dS)

10

10

Fig. 4. CTF TNMSE vs. SNR.

o

o

_ 6 0 L . - - - - - ' - - - - - - - L . - - - . . . . I . - - - - - - 1 . - - - - L . . . . . . . - - - ~-10

-4 0

L =50, K=600, N=1024alpha=0.1, DSratio =0.68, Taps captured =11 150, SCMA

-5 0

10L=50, K=600, N=1024alpha=0.1, DS ratio =0.68, Taps captured =11 150, SCMA

Fig. 3. CTF TNMSE vs. SNR.

-4 0

-5 0

La 50, N=1024, K-600, BL=4992, CodeRate- 1/3, SCMA10 . . . . , . . . . , . . . , . . . , . . . . , . . . . , - : - : - : - : - . . . , . . . . , . . . , . . ~ . . , . . , - : - : ~ - : - : - : - : . . . . , . . . . , . . . , . . , . . . . , . . . . , . . . _ _ r : _ : _ ~ _ _ : _ : _ : _ , . - : - : - - : - ~ : _ : r : _ : _ ~ ~ . . . , . . . . , . . . . . , , . . . . , . . . . , . - : - . . , . . . . .

1 0 - 7 L = = = = = = = = : : c : = = = = = = = ~ - - i . - - - - - - - - i . . . . - _ - - - --1 0 -5 0 5 10SNR (dB)

-2 0wenI -3 0to-

Q::WCD

We have presented a framework allowing the performanceanalysis of the class of the pilot-aided linear channel estimatorsin the context of LTE OFDMA systems. Together with theanalysis of the impact of LTE system parameters, we provedthat the well known mismatched MMSE estimator is nothingbut the deterministic regularized LS estimator. The analysisalso showed that there is a large performance gap betweenthe mismatched MMSE and the MMSE since the statisticalproperties of the channel, namely the frequency correlation,are not exploited. To fill this gap, we have proposed twomodified versions of the MMSE, namely the exponentialmismatched MMSE and the simplified MMSE, aiming atreducing the complexity of the MMSE without sacrificing theperformance. This is especially achieved by the latter whichshows a great flexibility in trading off the complexity and theperformance yielding the closest results to the MMSE.

REFERENCES

with 1/3 Turbo Coding with Block Length = 4992 bits withMaximum Ratio Combining receiver for QPSK modulation.The decoding performance with simplified MMSE channelestimation outperforms that of the mismatched-MMSE by 2dB for BER lower than 1 0 -2

IV. CONCL USIONS

[ I] Ye Li et a l . ~ "Robust Channel Estimation for OFDM Systems with RapidDispersive Fading Channels", IEEE transactions on communications, Vol.pp. 903-9] 2, JuI. ]998-[2] Frieder Sanzi et aI., "An Adaptive Two-Dimensional Channel Estimatorfor Wireless OFDM with Application to Mobile DVB-T", IEEE transactions on broadcasting, Vol 46, pp. 128-133, Jun. 2000.[3] 3GPP WG], "TS 36.2]]: Physical Channels and Modulation (ReI. 8)".Available: http://www.3gpp.org/ftp/Specs/html-info/36211.htm

L4j 3GPP W G 4 ~ ''TR 25.996: Spatial channel models for MIMO simulations".Available: http://www.3gpp.org/ftp/Specs/html-info/25996.htmL5j Jan J. van de Beek e t aI., "On channel estimation in Ofdm systems",Vehicular Technology Conference, Vol. pp. 8 1 5 - 8 1 9 ~ JuI. 1995.L6j M. Morelli e t aI., "A comparison of pilot-aided Channel estimationmethods for OFDM s y s t e m s " ~ Signal Processing, IEEE Transactions on,Vol. 49, pp. 3065-3073, Dec. 2001.[7j P. Hoher et aI. , "Pilot-Symbol-Aided Channel Estimation in Time andFrequency", Proc. Communication Theory Mini-Conj. (CTMC) withinIEEE Global Telecommun. Conj. (Globecom 97), pp. 90-96, 1997.L8j G. Auer e t al., "Pi lo t aided channel es timation for OFDM: a separatedapproach for smoothing and interpolation", Communications, 2005. ICC

2005. 2005 IEEE International Conference on, Vol. 4, pp. 2173-2178,May 1997.[9] A. Ancora et aI., "Down-Sampled Impulse Response Least-SquaresChannel Estimation for LTE OFDMA", Acoustics, Speech and SignalProcessing, 2007. ICASSP 2007. IEEE International Conference on, Vol.pp. 1II-293-III-296, April 2007.

Fig. 5. BER vs. SNR.

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