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1、1430IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 9, SEPTEMBER 2008Walsh Coded Training Signal Aided Time DomainChannel Estimation for MIMO-OFDM SystemsHyoung-Goo Jeon, Hyoung-Kyu Song, and Erchin SerpedinAbstractThis letter proposes a novel Walsh coded training signal design and decoding method

2、 to estimate the channel response in MIMO-OFDM systems. The Walsh coded training signals, designed to be orthogonal in the time domain, facilitate the separation of the desired training signal from the received mixed signal and the estimation of the channel response. The proposed channel estimation

3、method is directly applicable to practical MIMO-OFDM systems with null subcarriers and ex-hibits nearly the same performance as Lis original channel estimator 5 at a much reduced computational complexity.Index TermsChannel, estimation, MIMO, OFDM, training sequence.I. INTRODUCTIONREcently, MIMO-OFDM

4、 has attracted a lot of attention for achieving high speed data rates 12. In MIMO-OFDM receivers, the estimated channel frequency response is needed to separate the mixed signals received by multiple antennas. The performance of MIMO-OFDM receivers highly depends on the accuracy of the channel estim

5、ation. Thus far, there have been numerous studies on channel estimation for MIMO-OFDM systems (see e.g., 3-10). In 5, Li reported a highly accurate least-mean squares error (LMSE) based channel estimation method by using an optimal training sequence. However, this method is computationally complex d

6、ue to not only the inverse matrix calculation but also the matrix size, which is proportional to the number of transmit (Tx) antennas. In 6, the matrix size was reduced by half by decoupling the training signals, assuming that the channel responses of the adjacent sub-carriers were the same. However

7、, this assumption may cause channel estimation errors in large delay spread environments. In 7 and 8, the interfering effect of the other Tx antennas was decoupled by using training signals with different delays, resulting in no inverse matrix calculation. However, the methods 78 are not applicable

8、to practical MIMO-OFDM systems with null sub-carriers, since the unit Kronecker impulse sequence can not be obtained by taking the inverse fast Fourier transform (IFFT) of a constant modulation signal in the presence of null sub-carriers. As a possible solution to these issues, this letter proposes

9、a novel Walsh coded training signal design and decoding method to estimate the channel response in MIMO-OFDM systems withPaper approved by S. K. Wilson, the Editor for Multicarrier Modulation of the IEEE Communications Society. Manuscript received October 13, 2006; revised June 20, 2007 and October

10、21, 2007.H.-G. Jeon is with the Dept. of Information Commun. Engineering, Dongeui University, Korea (e-mail: hgjeondeu.ac.kr).H.-K.SongiswithSejongUniversity,Korea(e-mail:songhksejong.ac.kr).E. Serpedin is with the CE Dept., Texas A&M University, College Station, TX 77843-3128, USA (e-mail: serpedin

11、).Digital Object Identifier 10.1109/TCOMM.2008.060402.Fig. 1. The conceptual block diagram of the proposed channel estimation scheme.null sub-carriers. Herein, the Walsh coded training signals are designed to be orthogonal in the time domain. Using the orthogonality property, the Walsh d

12、ecoding enables the sepa-ration of the desired training signal from the received signal and the estimation of the corresponding channel. Unlike the conventional methods 5 6, the proposed method estimates the channel response without calculating FFT or IFFT. To reduce further the computational comple

13、xity of the proposed channel estimator, an additional simplified estimation method is proposed.II. PROPOSED CHANNEL ESTIMATION METHODA. Walsh Coded Training Signal DesignIn this sub-section, we will focus only on a Walsh coded training signal design for 2 2 MIMO-OFDM systems with null sub-carriers.

14、The extension of the proposed results from 2 2 MIMO-OFDM systems to other types of MIMO-OFDM systems is possible. The conceptual block diagram of the adopted 2 2 MIMO-OFDM system is shown in Fig. 1. Since 2 Tx antennas are used, 2 orthogonal codes are needed to separate the 2 training signals. A 2 2

15、 Walsh code matrix is used to generate the orthogonal codes. The 2 2 Walsh code matrix is given byW10W1111W20W21 =11 .(1)Let T Sik denote the i-th Tx antennas Walsh coded train-ing signal in the frequency domain at the k-th sub-carrier. The training signal T Sik consists of 2 basic signals X1kand X2

16、k = X1k exp (j2kL/N ), 0 k N 1, where N and L denote the total number of sub-carriers andthe delay factor, respectively, and L = N/2. In a 4 4 MIMO-OFDM system, the delay factor should be L = N/4, and the 4 4 Walsh code and 4 basic signals Xik = X1k exp (j2k(i 1)L/N ) are used. Taking into account t

17、he presence of null sub-carriers in practical MIMO-OFDMsystems, X1k is designed as X1k = X kZ k, where X k can be selected as a binary (1) random sequence (see e.g.,0090-6778/08$25.00 c 2008 IEEEJEON et al.: WALSH CODED TRAINING SIGNAL AIDED TIME DOMAIN CHANNEL ESTIMATION FOR MIMO-OFDM SYSTEMS14319

18、for some optimal designs), and不是很明白Z k =0if k = 0, orN1g1 k N+ g2 (2)1elsewhere22where k = 0 denotes the DC (null) sub-carrier, and g1 and g2 stand for the number of null sub-carriers present within each guard band, respectively. The Walsh coded training signal T Sik is designed as followsT Sik = Wi

19、0X1k + Wi1X2k,i = 1, 2.(3)In the time domain, the Walsh coded training signal tsin is the IDFT of T Sik:tsin = Wi0x1n + Wi1x1n L, 0 n N 1 (4)where x1n stands for the IDFT of X1k, i.e., x1n =(1/N )N1 X1k exp (j2kn/N ). Since x1n is the IDFTk=0of X1k, x1n admits a period of N (= 2L) samples in the tim

20、e domain, i.e., x1n + L = x1n L. Therefore, using (4), tsin + L can be expressed astsin + L = Wi1x1n + Wi0x1n L .(5)From (4) and (5), we can see that ts1n = ts1n + L andts2n = ts2n + L. Therefore, the analog Walsh codedtraining signal exhibits the following property:ts1(t) = ts1(t Ts/2) and ts2(t) =

21、 ts2(t Ts/2),(6)where Ts is one OFDM symbol period. Now define the Walsh decoding function fl(2, tsin) by means of fl(2, tsin) :=(1/2)1 =0 Wlmtsin + mL, 0 n L 1. Therefore,mfl(2, tsin) =1(Wl0+ Wl1)ts1nif i = 1(7)21(Wl0 Wl1)ts2nif i = 22which reduces to fl(2, tsin) = tsini,l, where i,j = 1 ifi = j, a

22、nd 0 elsewhere. This means that when the Walsh coded signals ts1n and ts2n are mixed in the time domain, each of two training signals can be separated by Walsh decodingwithout calculating a FFT/IFFT.冲击响应B. Walsh Decoding and Channel Estimation at经过信道the Receiver之后的时The channel between the i-th Tx an

23、tenna and the j-th Rxantenna hij (t) can be expressed as 4, 5域延拓hij (t) =ij (u)(t ij (u),(8)uwhere (t), ij (u) and ij (u) stand for the Dirac delta function, the complex gain and time delay of the u-th path of the channel between the i-th Tx antenna and the j-th Rx antenna, respectively. Note that i

24、j (u) may be a non-sample spaced time delay in real environments. The analog baseband Rx signal is then given by2rj (t) =hij (t) tsi(t) + (t)i=12=ij (u)tsi(t ij (u) + j (t),(9)i=1uwhere and j (t) stand for convolution and the low pass filtered complex additive Gaussian noise at the j-th antennawith

25、zero mean and variance 2, respectively. The Walsh decoding function carried out after sampling of the Rx signal leads tofi(2, rj n)=112ij (u)tsi(n + mL ij (u)2Wimim=0=1u+ fi(2, j n).(10)Using (6), it follows that ts1n ij (u) = ts1n + L ij (u) and ts2n ij (u) = ts2n + L ij (u). Therefore, (10) can be

26、 expressed asfi(2, rj n)1j (u)ts1(n 1j (u)= 2 (Wi0 + Wi1) u112j (u)ts2(n 2j (u)+ 2 (Wi0 Wi1) u+ fi(2, j n)=ts1n h1j n + f1(2, j n)if i = 1 .(11)ts2n h2j n + f2(2, j n) if i = 2In matrix form, (11) can be re-written asfi(rj ) = Ti hij + fi(j ) ,T(12)withhij=(hij 0, hij 1, ., hij L0 1)T,fi(rj )=(fi(2,

27、 r 0), f(2, r 1), ., f(2, r L1),f ( )=jijij Ti j(fi(2, j0), fi(2, j1), ., fi(2, j L 1), and convolu-tion matrixtsi0tsiN 1 tsiN + 1 L0tsi1tsi0 tsiN + 2 L0Ti =. .tsL 1tsL 2tsL Lii i0()TNotationsandL0 standfortranspositionand channelresponse length, respectively. From (12), the channel is es-timated vi

28、a the least-squares (LS) methodhij = Tifi(rj ) = hij + Tifi(j ) ,(13)where the superscript denotes the pseudo-inverse ma-trix. The estimator (13) is unbiased since one can quicklycheck that =+ () =becauseE hijhijTi E fi jhijEfi(2, j n) = fl(2, Ej n) = 0. Straightforward ma-nipulations on (13) show t

29、hat the variance of the proposed channel estimator takes the expressionvar(h) =2trace(TH T )12L02, (14)2trace(TiH Ti)ij2i iwhere ()H and trace() stand for the Hermitian transposition and trace operator, respectively, and the lower bound in the righthand side of (14) is due to Cauchys inequality 11.

30、Note that since (11) is derived in non-sample spaced channels, the proposed method can be used in non-sample spaced channels. Non-sample spaced channels cause a channel response leakage 3, 4, resulting in a small performance degradation. The performance degradation can be reduced by means of a pre-a

31、dvancement () of the timing point 6. It was shown in1432IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 9, SEPTEMBER 2008TABLE IPARAMETERS OF MULTI-PATH CHANNELTx and Rx PathRelative mean power of 6-pathsRelative arrival time of 6-pathsTx Ant. 1- Rx Ant. 10, 2, 7, 10, 12, 15(dB)0, 2, 3.5, 5.5, 9.5

32、,16(samples)Tx Ant. 2- Rx Ant. 10, 3, 6, 11, 12, 17(dB)0, 1.5, 2.5, 7, 10.5, 16(samples)Tx Ant. 1- Rx Ant. 20, 4, 7, 9, 11, 15 (dB)0, 2.5, 6, 7.5, 10.5, 14(samples)Tx Ant. 2- Rx Ant. 20, 2, 5, 10, 13, 16(dB)0, 1.5, 3.5, 5, 9.5,14(samples)6 that the use of = 4 includes all the leakage down to approxi

33、mately 23 dB of the total energy of the path with 0 ij (u) 1. An alternative way to deal with the channel response leakage is the frequency domain channel estimation approach 4. However, herein we focus on the time domain approach due to its reduced complexity, and the frequency domain methods are o

34、ut of the scope of this letter. By designing X k adequately, most of the energy of tsin can be concentrated on small portions of the symbol interval, making tsin 0 almost everywhere and forcing many rows of Ti to have a small norm ( 0). The computational complexity of the channel estimator (13) can

35、be further reduced by removing from (12) the equations corresponding to the rows of Ti with the smallest norm. By downsizing the matrix Ti, the computational complexity of the estimator is reduced at the price of a potential performance loss. However, comprehen-sive computer simulations show that th

36、e performance loss is negligible. In the next section, assuming that the sequences ts1n and ts2n are defined in terms of X k = 1 for 0 k N/4 1 and X k = 1 for N/4 k N 1, and that L/2 equations are preserved in the simplified estimation method, it is shown that the simplified estimator suffers only a

37、 negligible performance loss relative to the estimator (13) that relies on all the L equations. In this case, notice that the peak to average power ratios (PARs) of ts1n and ts2n are 10.5% and 9.1% of the maximum PAR (= N ), respectively. An immediate intuition for this result is due to the fact tha

38、t by removing the rows of Ti with the smallest norms, we simply eliminate the observations that are heavily corrupted by noise. Therefore, the simplified estimator appears to be more robust. Notice also that by neglecting the rows of Ti with small norm ( 0), the lower bound in (14) is only slightly

39、increased.Since the Walsh training signals tsin are known, the pseudo-inverse matrices required by the proposed channel es-timators can be pre-calculated and stored when the estimators are initialized. Notice further that Lis original method requires (2L0)2 + 2N multiplications and 3 FFT/IFFT operat

40、ions per Rx antenna in 22 MIMO-OFDM systems while the proposed simplified method needs only 2(N/4)L0 multiplications. As an example, when N = 128, L0 = 0.7 (N/4) and the radix-2 algorithm is used for FFT/IFFT implementation, the proposed simplified method exhibits 60% reduction in terms of computati

41、onal complexity relative to Lis original method. The pre-calculations in Lis method and the proposed method are not counted for fair comparison.III. PERFORMANCE EVALUATIONComputer simulations were carried out to evaluate the per-formance of the proposed methods. A MIMO-OFDM system with 2 Tx antennas

42、 and 2 Rx antennas was used. There are a total of 128 sub-carriers so that the FFT/IFFT size is 128.101 Proposed methodLi simplified methodLi original methodProposed simplified method interpolation methodfd = 40 HzMSE102103246810121416182022Eb/N0dBFig. 2.MSE of channel frequency response at fd = 40

43、Hz.The DC component sub-carrier is not used, and six and five sub-carriers on each end of the spectral band, respectively, are used as guard bands. The rest of 116 sub-carriers are used to transmit data. The OFDM symbol rate is 25 Ksps, and the symbol period is 40 sec, including the cyclic prefix ti

44、me of 8 sec. The channel length L0 is assumed to be 22. Modulation in sub-carriers is QPSK. The carrier frequency is 2.4 GHz, and the multi-path Rayleigh channel assumes 6 paths. Each signal path is assumed to undergo an indepen-dent Rayleigh fading. The Rayleigh fading channel simulator (Jakes sinu

45、soid sum method) openly published in reference12 was used. The parameters of multi-path channels are given in Table I. The performance of the system is measured by the estimators mean square error (MSE) and bit error rate (BER), the results of Monte-Carlo simulations being averaged over 10,000 OFDM

46、blocks. The simulation results are shown in Figs. 2-4. Figs. 2 and 3 show MSEs obtained at Doppler frequencies of 40 Hz and 200 Hz, respectively. The MSE of Lis original method using the optimal training sequence is shown as a reference MSE in all these figures. The performance of interpolation meth

47、od 10 is also shown in Figs. 2-4. In Figs. 2 and 3, the MSEs of both the proposed method and Lis original method are decaying with an increase in Eb/N0 (SNR). The proposed method exhibits the same MSE and BER performance as Lis original method. However, the MSE of Lis simplified method shows a large

48、 estimation error, regardless of Eb/N0. The estimation error results from the guard band null sub-carriers which transmit null data. As shown in Figs. 2-4, the MSE and BER performance degradations of the proposed simplified method relative to Lis original method are less than 0.4 dB and 0.2 dB in Eb

49、/N0,respectively.JEON et al.: WALSH CODED TRAINING SIGNAL AIDED TIME DOMAIN CHANNEL ESTIMATION FOR MIMO-OFDM SYSTEMS1433101 Proposed methodLi simplified methodLi original methodProposed simplified method interpolation methodfd = 200 HzMSE102103246810121416182022Eb/N0dBFig. 3. MSE of channel frequenc

50、y response at fd = 200 Hz.101Proposed methodLI simplifiedLi originalProposed SNR methodinterpolation methodperfect102BERfd = 200 Hz103104105246810121416182022Eb/N0dBFig. 4.BER performance at fd = 200 Hz.IV. CONCLUSIONThis letter proposed a novel Walsh coded training signal design and decoding method

51、 to estimate the channel responsein MIMO-OFDM systems. Since the proposed method utilizes the orthogonality of Walsh coded training signals in the time domain, each antenna training signal can be separated without performing FFT/IFFT operations. Unlike 7 and 8, the proposed method is applicable to p

52、ractical MIMO-OFDM systems with null sub-carriers. The computer simulations show that the proposed method exhibits nearly identical performance as Lis original method at a much reduced computational complexity.ACKNOWLEDGMENT“This work was supported by the Korea Research Foun-dation Grant funded by the Korean Government (MOEHRD) (KRF- 2005-214-D00324).”REFER