LS FFT-based Channel Estimators Using Pilot-Embedded Data ...

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LS FFT-based Channel Estimators Using Pilot-Embedded Data ...

in the case that K < L t L and L ≤ K, the use of L tOFDM blocks for training can guarantee the existence of theLS channel estimator and other better channel estimators, suchas the LMMSE channel estimator [3].B. Pilot-Embedded Data-Bearing LS Channel EstimationWe first extract the pilot part from the received signal matrixY(m). By using the null-space and orthogonality properties in(6), respectively, we are able to extract the pilot part by simplypost multiplying Y(m) in (5) by C T , and then dividing by α,to arrive atY(m)C TαLet Y 1 (m) = Y(m)CTα= H(m)+ Z(m)CTα. (9)and Z 1 (m) = Z(m)CTα,wehaveY 1 (m) =H(m)+Z 1 (m). (10)The PEDB-LS channel estimator can be obtained by minimizingthe following mean-squared-error objective function,Ĥ LS (m) = minH(m) ‖Y 1(m) − H(m)‖ 2 . (11)It is straightforward to show thatĤ LS (m) =Y 1 (m) = Y(m)CTα. (12)C. Pilot-Embedded Data-Bearing ML Data DetectionWe now describe the procedure of PEDB-ML data detection.First, we extract the data part from the received signal matrixY(m). Using the null-space and orthogonality properties in (7),respectively, we are able to extract the data part by simply postmultiplying Y(m) in (5) by B T , and then dividing by β,Y(m)B TβLet Y 2 (m) = Y(m)BTβ= H(m)D(m)+ Z(m)BTβ. (13)and Z 2 (m) = Z(m)BTβ,wehaveY 2 (m) =H(m)D(m)+Z 2 (m). (14)From the orthogonality of B in (7), we note that∑ KMj ′ =1 |B i ′ ,j ′|2 = β, ∀i ′ . Therefore, the data-bearerprojectednoise Z 2 (m) is AWGN with zero-mean and varianceσ 22β I (KL rN×KL rN) per real dimension. Due to the i.i.d whiteGaussian distribution of Z 2 (m), the PEDB-ML receiver jointlydecides the codewords for the d th OFDM block in the m thSF-coded data block by solving the following minimizationproblem,ˆD i,j (m) = min Di,j ‖Y 2s,j (m) − Ĥs,i(m)D i,j (m)‖ 2 ,i =1:KL t , j =(d − 1)K +1:dK,s =1:L r , and d =1,...,N, (15)where Ĥ(m) is the estimated channel matrix, e.g. Ĥ(m) =Ĥ LS (m). The codeword transmitted from the b th transmitter isrepresented by ˆD ib ,j(m), with i b =(b − 1)K +1:bK.IV. THE LS FFT-BASED CHANNEL ESTIMATION ANDPERFORMANCE ANALYSISAs shown in (9), the PEDB-LS channel estimate contains thechannel frequency response that is contaminated by AWGN.In this section, we improve the performance of the PEDB-LSchannel estimator by employing the basic concepts of the FFTbasedapproach proposed in [4].A. LS FFT-Based Channel Estimation ApproachAs suggested in [4], the FFT-based channel estimation approachfirst calculates the temporal LS channel estimate byusing L significant taps, i.e. L’s largest ∑ L tb=1 |ĤLS ab(m, k)| 2 .The resulting temporal LS channel estimate is then FFT transformedto obtain the K-subcarrier channel frequency response.Now let us propose the LS FFT-based channel estimatorin details. From (12), we have Ĥ LSab (m) =([ĤLS(m)] a,(b−1)K+1:bK ) T . From the channel model in (2),Ĥ LSab (m) can be expressed asĤ LSab (m) =Fh ab (m)+Z 1ab (m), (16)where F is the K × L matrix whose element [F] xy is definedby exp[(−j2π/K)(x − 1)τ y ],x=1,...,K, y =0,...,L− 1;h ab (m) =[h ab (m, 0),...,h ab (m, L − 1)] T ; and Z 1ab (m) =([Z 1 (m)] a,(b−1)K+1:bK ) T .Transforming the PEDB-LS channel estimate in (16) tothe temporal PEDB-LS channel estimate by using the K ×K IFFT matrix G, whose element [G] xy is defined by1Kexp(j2π/K)(x − 1)(y − 1), x,y =1,...,K,wehaveĥ LSab (m) =GĤLS ab(m) =GFh ab (m)+GZ 1ab (m). (17)Let Z G 1 ab(m) =GZ 1ab (m), it can be shown thatĥ LSab (m) =[g(1),...,g(K)] T , (18)where g(x) = 1 ∑ L−1K l=0 h ab(m, l)f(x − 1 − τ l )e jξ(x−1−τl) +Z1 G ab(m, x) with Z1 G ab(m, x) being the x th element of Z G 1 ab(m),f(q) =sin(πq)(K−1)πsin(πq/K)is the leakage function, and ξ =K .Obviously, from (18), if τ l is an integer number, then thel th element of the L largest elements of ĥLS ab(m) is equal toh ab (m, l)+Z1 G ab(m, l), and the rest elements, which are not amember of the L largest elements, are equal to Z1 G ab(m, e), e≠l, e ∈{0,...,K−1}\W 1 with W 1 being the set of the L largestelements. As a result, by choosing L largest taps and replacingthe (K − L) remaining taps by zero is sufficient and optimal,resulting in the LS FFT-based estimate of the temporal channelimpulse response ĥFFT ab(m), since we completely capture thechannel impulse response h ab (m), and remove the excessivenoise in the (K − L) remaining taps. However, in reality, thel th multipath delay τ l is often a non-integer number, hence,the L-multipath channel impulse response dissipates to all Ktaps of ĥLS ab(m) and thus results in the model mismatch error,which increases the channel estimation error, primarily causedby the AWGN Z G 1 ab(m). This additional channel estimationerror causes the severe error floor in the MSE of the channelestimation, and the detection error probability. Once the L (orP ) largest taps are chosen and the rest taps are replaced byzero, the LS FFT-based estimated channel frequency responseis determined byĤ FFTa,b (m) =KG H ĥ FFTab (m). (19)B. Channel Estimation Error Performance AnalysisWe now analyze the performance of the PEDB-LS and LSFFT-based channel estimators by using the MSE of channelestimation as the performance measure.1521This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2006 proceedings.


1) PEDB-LS Channel Estimator: For arbitrary multipathdelay profiles, the temporal channel impulse response betweenthe a th receiver and b th transmitter can be described by,h G ab(m) =GFh ab (m). (20)The channel estimation error can be readily described by˜h LSab (m) =h G ab(m) − ĥLS ab= −Z G 1 ab(m), (21)by referring to ĥLS ab(m) in (17), and h G ab(m) in (20). Using(21), the MSE LS (a, b) of the channel estimation is expressedasE[‖˜h LSab (m)‖ 2 ]=E[‖−Z G 1 ab(m)‖ 2 ]= σ2α , (22)by using E[|Z1 G ab(m, x)| 2 ]= σ2Kα, x =1,...,K, as referringto section III.B. It is worth noticing that (22) is also the MSEof the K-tap FFT-based channel estimation.For L r -receiver L t -transmitter MIMO systems, the overallMSE LS in (22) can be expressed as follows,∑L t∑L rMSE T LS =a=1 b=1MSE LS (a, b) = σ2 L t L r. (23)α2) LS FFT-Based Channel Estimator: As we mentionedearlier, the LS FFT-based channel estimator first simply choosesthe L largest taps, and then replaces the (K −L) remaining tapsby zero. This operation can be equivalently described by usingthe K × K tap-selection matrix T given byT = diag{1, 1,...,0, 1,...,0}, (24)where 0’s and 1’s represent non-selected and selected taps,respectively. There are (K − L) 0’s and L 1’s elements inthe diagonal elements of T. By using the tap-selection matrixT, the temporal LS FFT-based-estimate of the channel impulseresponse between the a th receiver and b th transmitter can bedescribed byĥ FFTab (m) = TGĤLS ab(m)= TGFh ab (m)+TGZ 1ab (m), (25)by plugging in ĤLS ab(m) in (16).Similarly to (21), the channel estimation error can be describedby, using h G ab (m) in (20) and ĥFFT ab(m) in (25),˜h FFTab (m) = h G ab(m) − ĥFFT ab(m)= (GF − TGF)h ab (m) − TGZ 1ab (m). (26)Now let us define W 2 ∈{0,...,K−1}\W 1 to be a set of thenon-selected (K − L) less significant taps, and w 2 ∈W 2 andw 1 ∈W 1 are row indices indicating the 0’s and 1’s elementsof T, respectively. It can be shown that[GF − TGF] w2,1:L = 1 K [f(w 2 − 1)e jξ(w2−1) ,...,f(w 2 − 1 − τ L−1 )e jξ(w2−1−τL−1) ], (27)where τ 0 =0. Therefore, by substituting (27) into (26), we canexpress each part of the channel estimation error as follows,[(GF − TGF)h ab (m)] w1∈W 1=[TGZ 1ab (m)] w2∈W 2=0,= 1 K[(GF − TGF)h ab (m)] w2∈W 2∑ L−1l=0 h ab(m, l)f(w 2 − 1 − τ l )e jξ(w2−1−τl) , and[TGZ 1ab (m)] w1∈W 1=[Z G 1 ab(m)] w1∈W 1. (28)From (28), it is readily shown that the channel estimationerror of the LS FFT-based channel estimator are due totwo error sources: the model mismatch error, i.e. [(GF −TGF)h ab (m)] w2∈W 2, and the corresponding noise effect, i.e.[TGZ 1ab (m)] w1∈W 1. By substituting (28) into (26), we havethe MSE FFT (a, b) of the LS FFT-based channel estimator asE[‖˜h FFTab (m)‖ 2 ]= 1 ∑ ∑ L−1K 2 w 2∈W 2 l=0 E[|h ab(m, l)·f(w 2 − 1 − τ l )| 2 ]+ Lσ2Kα , (29)in which the assumption that the channel and noise, andchannels from different paths are mutually uncorrelated, isexploited. For L r -receiver L t -transmitter MIMO systems, theoverall MSE FFT in (29) can be expressed as follows,MSE T FFT =χ K 2 + Lσ2 L t L r, (30)KαχwhereK= ∑ L r∑ Lt∑ K 2 a=1 b=1 j=1 E[‖[ĥLS ab(m)] j ‖ 2 ] −η 2 − ∑ L r∑ Lt∑a=1 b=1 w 1∈W 1E[‖[ĥLS ab(m)] w1 ‖ 2 ] with η 2 =(K−L)σ 2 L tL rKα.First, we consider the case of the multipath delay profileswith integer delays. In this case, the model mismatch error χ isequal to zero. Since L ≤ K and by referring to (23), we can seethat the channel estimation performance of the LS FFT-basedchannel estimator is superior to that of the PEDB-LS channelestimator in this case, i.e. MSE T FFT ≤ MSE T LS. Usingχ =0,we have∑= ∑ L ra=1∑ Lr∑ Lta=1 b=1∑ Ltb=1w 1∈W 1E[‖[ĥLS ab(m)] w1 ‖ 2 ]∑ Kj=1 E[‖[ĥLS ab(m)] j ‖ 2 ] − (K−L)σ2 L tL rKα(31) .The observation in (31) indicates that in order to achieve theminimum MSE of the LS FFT-based channel estimator, theL largest taps must be capable of capturing the average totalenergy of channels in the presence of AWGN.For the case of the multipath delay profiles with non-integerdelays, a non-zero model mismatch error χ exists, as shown in(30), due to the leakage phenomenon. Hence, it is important tostudy the joint effects of the tap length L and the noise levelσ 2 on the MSE measure.V. THE PROPOSED ADAPTIVE LS FFT-BASED CHANNELESTIMATORIn this section, we propose an adaptive LS FFT-based channelestimation approach in which the number of significant tapsused in channel estimation can be adjustable in order to minimizethe model mismatch error and the corresponding noiseeffect. The model mismatch error in the LS FFT-based channelestimation stems from the fact that a fixed number of L (orP ) largest taps is used in the channel estimation process forall signal-to-noise ratio (SNR) values. In the proposed adaptiveLS FFT-based approach, the number of significant taps P ischosen to achieve the intuitive goal that the average total energyof the channels dissipating in each tap is completely capturedin order to compensate the model mismatch error. Specifically,the number of significant taps P opt used to capture the CSIin ĥLS ab(m) in (18) is obtained by solving the followingoptimization problem:1522This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2006 proceedings.


0.38 taps, SNR = 2 dBMSE of channel estimation vs. the number of taps10 0 Mean−Squared Error (MSE) of Channel Estimation vs. SNRPilotEmbedded Data−Bearing LS CM−based matricesAdaptive LS FFTbased CM−based matricesLMMSE CM−based matrices10−tap LS FFTbased CM−based matricesMSE of channel estimation0.250.20.1516 taps, SNR = 10 dBModel mismatch error, SNR = 2 dBNoise error, SNR = 2 dBMSE, SNR = 2 dBModel mismatch error, SNR = 20 dBNoise error, SNR = 20 dBMSE, SNR = 20 dBModel mismatch error, SNR = 10 dBNoise error, SNR = 10 dBMSE, SNR = 10 dBMSE10 −110 −20.154 taps, SNR = 20 dB10 −30.0500 20 40 60 80 100 12010 −4The number of taps0 5 10 15 20SNR (dB)Fig. 1: Theoretical examples of the model mismatch error, the noiseeffect, and the overall MSE of the LS FFT-based channel estimator asa function of the number of significant taps P .Fig. 3: The graph of MSEs of the channel estimation in quasi-staticfading channels.Magnitude of l.h.s and r.h.s of (35) and (36) vs. the number of taps6Bit Error Rate (BER) vs. SNR10 0 PilotEmbedded Data−Bearing LS CM−based matricesAdaptive LS FFTbased CM−based matrices5.5LMMSE CM−based matrices10−tap LS FFTbased CM−based matrices10 −1Ideal Channel5l.h.s of (35)10 −2r.h.s. of (36)4.5r.h.s of (35)l.h.s of (36)SNR = 2 dB10 −34SNR = 20 dB3.510 −4Equilibrium points of the constraint in (32) and (35)0 20 40 60 80 100 120The number of taps0 2 4 6 8 10 12 14 16 18SNR (dB)Fig. 2: The relationship between l.h.s and r.h.s of (35) and (36) basedon numerical calculations.Fig. 4: The graph of BERs of the pilot-embedded SF-coded MIMO-OFDM system in quasi-static fading channels.P opt = min(P ) s.t.∑ Lr∑ ]Lta=1 b=1[max E increases as SNR increases. In addition it is noted that P opt canWp,|W p|=P∑i∈W p‖[ĥLS ab(m)] i ‖ 2 ≥ be approximately determined by locating the intersection point∑ Lr∑ Lt∑ Ka=1 b=1 j=1 E[‖[ĥLS ab(m)] j ‖ 2 (K−P )LtLrσ2 of the curve of the model mismatch error and the curve of the] −Kα. (32)noise error. Therefore, the problem in (33) can be formulatedIt is clear that,∑for a given P , the solution of achieving asmax Wp,|W p|=P i∈W p‖[ĥLS ab(m)] i ‖ 2 is to choose W p as the P opt = min(P ) s.t. { χ K 2 ≤ Pσ2 L t L r}. (34)Kαindices of the P largest taps (P = L in the previous sections).Or equivalently, by substituting χ, wehaveNow let us intuitively explain why (32) in details. If weP opt = min(P ) s.t.assume a perfect situation, then the most desired criteria used ∑to determine the number of significant taps P is the MSE T Lr∑ Lt∑]FFTa=1 b=1 i∈W pE[‖[ĥLS ab(m)] i ‖ 2 + Pσ2 L tL rKα≥in (30), such that the optimization solution is expressed as ( ∑Lr ∑ Lt∑ KP opt = minP {MSET a=1 b=1 j=1FFT(P )}. (33)E[‖[ĥLS ab(m)] j ‖ 2 (K−P )LtLrσ2] −KαFirst, instead of minimizing MSE T However, in practice, since WFFT(P ) directly, we want top is an unknownset, it is not feasible to compute the termtake advantage of specific observations revealed in the two terms ∑ Lr∑ Lt∑]of (30). In Fig. 1, we plot the corresponding χ/K 2 term and a=1 b=1 i∈W pE[‖[ĥLS ab(m)] i ‖ 2 directly. In stead,the noise error term Lσ2 L tL rKαas a function of P under severalfor each transmission, depending on the real observations,SNR levels with the setting parameters described in Section VI.for different P ,∑we instantaneously compute the termFrom this figure, we note that χ/K 2 converges to zero as Pmax {Wp,|W p|=P } i∈W p‖[ĥLS ab(m)] i ‖ 2 . Then empiricalincreases, meaning that more taps can be used to compensateexpectation is calculated. Overall, we compute the following ]the model mismatch error. Also, it is seen that the resulting P opt∑i∈W p‖[ĥLS ab(m)] i ‖ 2 .MagnitudeBERterm ∑ L ra=1∑ Ltb=1 E [max Wp,|W p|=P)(35) .1523This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2006 proceedings.


Since∑ Lr∑ Lta=1]b=1[max E Wp,|W p|=P∑i∈W p‖[ĥLS ab(m)] i ‖ 2 ≥∑ Lr∑ Lt∑a=1 b=1 i∈W pE[‖[ĥLS ab(m)] i ‖ 2 ], (36)where W p is a set of the P largest taps. We note that theleft hand is based on order statistics. In our case, since thecomponents in ĥLS ab(m) follow non-identical distributions, dueto the complex nature of order statistics, it is infeasible to findthe theoretical close-form expression of the l.h.s of (36) in termof the r.h.s of (36). Numerical examples are plotted in Fig. 2 todemonstrate the relationship between the l.h.s of (35) and thel.h.s of (36). From Fig. 2, we can see that the curves of the l.h.sof (35) and (36) are close together when the number of taps aresmall until the intersection point between these two curves andthe r.h.s of (35) for both SNR = 2 and 20 dB. This phenomenonindicates that by replacing the l.h.s of (35) by the l.h.s of (36) fordetermining the minimum number of significant taps that yieldthe equality to the constraint of (35), the resulting number oftaps are mostly the same as solving (35) directly. It is worthnoticing that in the regimes beyond the intersection point, thesetwo curves are different; however, this phenomenon does notaffect the minimum number of significant taps since we neverexploit their relationship in these regimes.Based on the above observations, we propose to replace thel.h.s of (35) by the l.h.s of (36), and thus replace the inequalityconstraint in (35) by the inequality constraint in (32). Therefore,we have the proposed scheme in determining P opt as describedin (32). In this sense, we could regard the proposed scheme in(32) as a sub-optimal approach in determining P opt . However,in most cases, the P opt determined by solving the problem in(32) is almost identical to the optimum solution obtained byusing an exhaustive search for the minimum MSE T FFT in (30).While the later case serves as a theoretical ideal solution.VI. SIMULATION RESULTSSimulations are conducted under quasi-static frequencyselectiveRayleigh fading channels. The simulated SF blockcode is obtained from Alamouti’s structure as in [1], using aBPSK constellation for two transmit and two receive antennas.The COST207 typical urban (TU) six-ray normalized powerdelay profile [9] with delay spread of 5 µs is studied. The entirechannel bandwidth, 1 MH z , is divided into K = 128 subcarriersin which four subcarriers on each end are served as guard tones,and the rest (120 tones) are used to transmit data. To make thetone orthogonal to each other, the symbol duration is 128 µs,and additional 20 µs guard interval is used as the cyclic prefixlength in order to protect the ISI due to the multipath delayspread. The equal block-power allocation, i.e. β = α =0.5 W,is employed, the normalized SF-coded symbol block-power is1W,N =2, and M = N +L t =4. The CM-based structure in(8) is selected as the representative of three structures studied in[7]. In this experiment, the channel impulse response h ab (m, l)’sin (2) are based on Jake’s model [10], when f d ∗ T f =0.08(fast fading) with f d being the Doppler’s shift.In Fig.3, the MSEs of the PEDB-LS, 10-tap LS FFT-based,adaptive LS FFT-based, and LMMSE channel estimators [3], areshown. Notice that the PEDB-LS approach has a higher MSEin low SNR regimes than that of the 10-tap LS FFT-based andthe adaptive LS FFT-based approaches. In high SNR regimes,the PEDB-LS and adaptive LS FFT-based approaches performsbetter than the 10-tap LS FFT-based one in which the error floorcaused by the model mismatch error occurs, whereas the formertwo do not suffer from this severe error floor. It is worth noticingthat the LMMSE estimator serves as the channel estimationperformance bound at the price of the intensive computationalcomplexity and the additional channel correlation information.In Fig.4, the BER comparisons are shown. Notice that the10-tap LS FFT-based and adaptive LS FFT-based estimationperformances are quite close in low SNR regimes, whereas thePEDB-LS approach performs worse, in which the 2-dB SNRdifference at BER of 10 −3 is observed. In high SNR regimes,the 10-tap LS FFT-based estimator suffers from the error floor,say at BER of 2 × 10 −4 , whereas the others do not. At BER of10 −4 , the SNR differences between the ideal-channel scheme,where the true channel impulse response is employed, and theadaptive LS FFT-based and PEDB-LS estimators are 2.2 dB and3.6 dB, respectively, whereas the LMMSE estimator providesthe error probability coinciding with the ideal-channel scheme.VII. CONCLUSIONIn this paper, we have proposed an adaptive LS FFT-basedchannel estimator for improving the performances of the LSFFT-based and PEDB-LS estimators, under the framework ofpilot-embedded data-bearing approach for joint channel estimationand data detection. For quasi-static TU-profile fadingchannels, simulations show that the adaptive LS FFT-basedestimator provides superior performance to that of the 10-tap LS FFT-based and PEDB-LS approaches. For instance, atBER of 10 −4 , the SNR differences are as 2.2 dB and 3.6 dB,respectively, for the adaptive LS FFT-based and the PEDB-LSestimators compared with the ideal-channel scheme, whereasthe 10-tap LS FFT-based estimator suffers from the severe errorfloor caused by the model mismatch error.REFERENCES[1] Weifeng Su, Z. Safar, M. Olfat, and K.J.R. Liu, “Obtaining full-diversityspace-frequency codes from space-time codes via mapping,” IEEE Trans.Signal Processing, Vol. 51, No. 11, pp. 2905-2916, Nov. 2003.[2] Imad Barhumi, G. Leus, and M. Moonen, “Optimal training design forMIMO-OFDM systems in mobile wireless channels,” IEEE Trans. SignalProcessing, Vol. 51, No. 6, pp. 1615-1624, June 2003.[3] Y. Gong and K.B. Lataief, “Low complexity channel estimation for spacetimecoded wideband OFDM systems,” IEEE Trans. Wireless Commu., Vol.2, No. 5, pp. 876-882, Sept. 2003.[4] Y. (Geoffrey) Li, “Channel estimation for OFDM systems with trsmitterdiversity in mobile wireless channels,” IEEE J. Select. Areas Commu., Vol.17, No. 3, pp. 461-471, Mar. 1999.[5] J. J. van de Beek, O. Edfors, M. Sandell, S. K. Wilson, and P. O. Borjesson,“On channel estimation in OFDM systems,” Proc. 45th IEEE VTC,Chicago, IL, pp. 815-819, July 1999.[6] X. Wang and K.J.R. Liu, “Model based channel estimation frameworkfor MIMO multicarrier communication systems,” IEEE Trans. WirelessCommu., Vol. 4, No. 3, pp. 1-14, May 2005.[7] C. Pirak, Z.J. Wang, K.J.R. Liu, and S. Jitapunkul, “Performance analysisfor pilot-embedded data-bearing approach in space-time coded MIMOsystems,” Proc. IEEE ICASSP, Philadelphia, PA, pp.593-596, Mar. 2005.[8] A.V. Geramita and J. Seberry, Orthogonal Designs, Marcel Dekker, INC.,NY, 1979.[9] G. Stuber, Principles of Mobile Communications, Kluwer, MA, 2001.[10] W.C. Jakes, Jr., “Multipath Interference”, Microwave Mobile CommunicationW.C. Jakes, Jr., Ed., Wiley, NY, 1974, pp. 67-68.1524This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2006 proceedings.

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