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 the**LS** 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-**LS**channel estimator by employing the basic concepts of the **FFT****based**approach 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 |Ĥ**LS** 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 **LS**ab (m) =([Ĥ**LS**(m)] a,(b−1)K+1:bK ) T . From the channel model in (2),Ĥ **LS**ab (m) can be expressed asĤ **LS**ab (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 I**FFT** matrix G, whose element [G] xy is defined by1Kexp(j2π/K)(x − 1)(y − 1), x,y =1,...,K,wehaveĥ **LS**ab (m) =GĤ**LS** ab(m) =GFh ab (m)+GZ 1ab (m). (17)Let Z G 1 ab(m) =GZ 1ab (m), it can be shown thatĥ **LS**ab (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 ĥ**LS** 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 ĥ**FFT** 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 ĥ**LS** 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 **FFT**a,b (m) =KG H ĥ **FFT**ab (m). (19)B. **Channel** Estimation Error Performance AnalysisWe now analyze the performance of the PEDB-**LS** and **LS****FFT**-**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 **LS**ab (m) =h G ab(m) − ĥ**LS** ab= −Z G 1 ab(m), (21)by referring to ĥ**LS** 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 **LS**ab (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 **FFT**ab (m) = TGĤ**LS** ab(m)= TGFh ab (m)+TGZ 1ab (m), (25)by plugging in Ĥ**LS** ab(m) in (16).Similarly to (21), the channel estimation error can be describedby, using h G ab (m) in (20) and ĥ**FFT** ab(m) in (25),˜h **FFT**ab (m) = h G ab(m) − ĥ**FFT** 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 **FFT**ab (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[‖[ĥ**LS** ab(m)] j ‖ 2 ] −η 2 − ∑ L r∑ Lt∑a=1 b=1 w 1∈W 1E[‖[ĥ**LS** 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**-**based**channel estimator is superior to that of the PEDB-**LS** channelestimator in this case, i.e. MSE T **FFT** ≤ MSE T **LS**. **Using**χ =0,we have∑= ∑ L ra=1∑ Lr∑ Lta=1 b=1∑ Ltb=1w 1∈W 1E[‖[ĥ**LS** ab(m)] w1 ‖ 2 ]∑ Kj=1 E[‖[ĥ**LS** 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 adaptive**LS** **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 ĥ**LS** 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. SNR**Pilot**−**Embedded** **Data**−Bearing **LS** CM−**based** matricesAdaptive **LS** **FFT**−**based** CM−**based** matricesLMMSE CM−**based** matrices10−tap **LS** **FFT**−**based** 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 **Pilot**−**Embedded** **Data**−Bearing **LS** CM−**based** matricesAdaptive **LS** **FFT**−**based** CM−**based** matrices5.5LMMSE CM−**based** matrices10−tap **LS** **FFT**−**based** CM−**based** matrices10 −1Ideal **Channel**5l.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) **based**on 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‖[ĥ**LS** ab(m)] i ‖ 2 ≥ be approximately determined by locating the intersection point∑ Lr∑ Lt∑ Ka=1 b=1 j=1 E[‖[ĥ**LS** 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‖[ĥ**LS** 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∑]**FFT**a=1 b=1 i∈W pE[‖[ĥ**LS** 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=1**FFT**(P )}. (33)E[‖[ĥ**LS** ab(m)] j ‖ 2 (K−P )LtLrσ2] −KαFirst, instead of minimizing MSE T However, in practice, since W**FFT**(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[‖[ĥ**LS** 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‖[ĥ**LS** 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‖[ĥ**LS** 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‖[ĥ**LS** ab(m)] i ‖ 2 ≥∑ Lr∑ Lt∑a=1 b=1 i∈W pE[‖[ĥ**LS** 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 ĥ**LS** 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**-**based**channel estimator for improving the performances of the **LS****FFT**-**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**-**based**estimator provides superior performance to that of the 10-tap **LS** **FFT**-**based** and PEDB-**LS** approaches. 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