Journal of Applied Science and Engineering, Vol. 16, No. 4, pp. 385 394 (2013) DOI: 10.6180/jase.2013.16.4.06 Maximum-Likelihood vs. Least Squares Schemes for OFDM Channel Estimation Using Techniques of Repeated Training Blocks Shu-Han Liao 1, Min-Hui Ho 1, Rainfield Y. Yen 1 * and Hong-Yu Liu 2 1 Department of Electrical Engineering, Tamkang University, Tamsui, Taiwan 251, R.O.C. 2 Department of Electrical Engineering, Fu Jen Catholic University, Xinzhuang, Taiwan 242, R.O.C. Abstract We present techniques of using repeated training blocks for maximum-likelihood (ML) and least-squares (LS) based channel estimation of orthogonal frequency division multiplexing (OFDM) systems over slow quasi-static fading channels. Both frequency-selective and frequency-nonselective channels are considered. Explicit analytical expressions of channel estimators and the corresponding estimator mean square errors (MSEs) are derived for both ML and LS estimations. For the ML channel estimators, closed-form expressions of Cramer-Rao lower bounds (CRLBs) are also obtained to compare with the MSEs. Both theoretical and numerical results are presented and compared. Key Words: Orthogonal Frequency Division Multiplexing, Maximum-Likelihood Estimation, Least-Squares Estimation, Cramer-Rao Lower Bound 1. Introduction *Corresponding author. E-mail: rainfieldy@yahoo.com For OFDM communications, channel estimation must be made for accurate data detection. The maximum-likelihood (ML) estimation and least-squares (LS) estimation are two powerful estimation techniques [1 9]. To perform ML parameter estimations, one needs first to form a likelihood function (or log-likelihood function) in terms of the parameters to be estimated [10]. To perform LS estimations, one needs to formulate an overdetermined system of linear simultaneous equations in terms of the parameters to be estimated [11]. For both ML and LS parameter estimations, the parameters to be estimated must be independent of each other. The formulation of either the log-likelihood function for ML estimation or the overdetermined system for LS estimation is determined by how the statistical samples are collected. It is important to note that ML estimations possess the invariance property [10]. That is, the ML estimator for a function of a parameter is simply that function of the ML estimator for that parameter. On the other hand, LS estimations do not possess the invariance property. In a frequency-selective channel, tap channel impulse responses (CIRs) are uncorrelated or independent [12]. When an OFDM system operates in a frequencyselective channel, the subcarrier channel frequency responses (CFRs) are discrete Fourier transforms (DFTs) of CIRs (hence are functions of CIRs). Therefore, these subcarrier CFRs are correlated. As a result, ML or LS channel estimations for OFDM systems in frequency-selective channels are usually performed on CIRs. Once the CIR estimators are obtained, the CFR estimators can be directly determined via DFT. In the case of ML estimations, the CFR estimators thus obtained are still ML estimators, while in the case of LS estimations, such CFR estimators can no more be called LS estimators. In the literature, most channel estimations for OFDM
386 Shu-Han Liao et al. systems use one OFDM block of pilot or training data to formulate ML or LS algorithms [1 9], [15 17]. In this paper, rather than using a single training block, we employ repeated or multiple identical training blocks to perform channel estimations in OFDM systems. As will be seen, the estimation accuracy will be increased by the use of multiple training blocks. However, this performance gain is not without price. Not only it is more costly and more complex to use multiple training blocks, it apparently also reduces data throughput. Therefore, in practice depending on applications, a trade-off must be made between cost/complexity and system performance. We note that, for mobile wireless communications, use of repeated OFDM training blocks requires that the channel fading is slow. More specifically, channel fading remains essentially unchanged at least over the period of the repeated blocks. In substance, much of the theory presented here seems well established. Our contribution is to analytically derive expressions for the channel estimators and the corresponding mean square errors (MSEs) or variances that have not specifically given in the literature. For ML estimations, we will also derive expressions for Cramer-Rao lower bounds (CRLBs). We show how closely the ML and LS estimators are related. In addition, we clarify some misconceptions that have been overlooked. We further provide a proof of optimal training sequence for our channel estimation. Section 2 presents ML channel estimation. Section 3 evaluates the estimation performance for the ML channel estimator. In addition, analysis on Cramer-Rao lower bound (CRLB) is given. Section 4 presents LS channel estimation. Section 5 evaluates the estimation performance for the LS channel estimator. Then, section 6 gives numerical results. Finally, section 7 draws the conclusion. 2. Maximum-Likelihood Channel Estimation for OFDM In this section, we present ML channel estimations for OFDM systems using repeated OFDM training blocks. The use of repeated OFDM blocks requires that channel fading remains unchanged at least over the period of the repeated blocks (slow quasi-static fading). This implies that the maximum Doppler frequency must satisfy f M = f c /c <1/Tcorresponding to a mobile speed < c/(f c T), where c is the speed of light, T is one OFDM block time length in seconds, and f c is the carrier frequency in Hz. Using a 802.11a standard with f c = 5 GHz and f =1/T = 312.5 khz, this requires < 67,500 km/hr. Apparently, this requirement for slow quasi-static fading is easily met in practice. For example, if = 60 km/hr, then f M 10-3 /T, it is thus reasonable to assume fading to remain unchanged over several hundred OFDM blocks. We shall adopt the following notations for baseband signals and systems: X =[X 0, X 1,,X N 1 ] T, T denoting transpose, is the transmitted data vector representing one OFDM block of N data samples. R m =[R m,0, R m,1,, R m,n 1 ] T and W m =[W m,0, W m,1,,w m,n 1 ] T, m =1,2,,M, represent respectively the received noisy signal vector and the noise vector of the mth OFDM block at the N-point DFT output, where W m is a Gaussian random vector having mean 0 N 1 and covariance 2 W I N N with 0 N 1 being an N 1 all zero vector and I N N an N N identity matrix. Using the familiar abbreviation for normal or Gaussian random variables, we thus write W m ~N(0 N 1, 2 W I N N ). For an OFDM system operated in a frequency-selective channel with dispersion length L, we denote the CIR vector and the subcarrier CFR vector respectively by h =[h 0, h 1,,h L ] T and H=[H 0, H 1,,H N 1 ] T. His related to h by the relation where (1) (2) We note here that, conventionally a vector is denoted by a boldfaced small letter and a matrix by a boldfaced capital letter. Then, time-domain quantities are expressed by small letters and frequency-domain quantities by capital letters. Here to avoid confusion, we use an arrow over a boldfaced capital letter for a frequency-domain vector.
Maximum-Likelihood vs. Least Squares Schemes for OFDM Channel Estimation Using Techniques of Repeated Training Blocks 387 Assuming the channel length L is known and using M repeated OFDM sample blocks, the receiver DFT output can be expressed as (8a) (8b) (3) where X = diag{x 0, X 1,,X N 1 } is a diagonal matrix. Then, with the knowledge of the channel dispersion length, we use (1) to get from which, the log-likelihood function is given by (4) (5) where denotes vector norm. By setting / h =0, we can obtain the ML estimator for the CIR vector h as (6) where H denotes Hermitian transpose. By the invariance property of the ML estimation, the ML estimator for the CFR vector H can be obtained as (7) Note that because the elements {H k, k =0,1,,N 1} in H are correlated with each other as according to (1), we could not directly obtain H ML from the log-likelihood function. Therefore, we had to transform Hinto h as given by (4) and then obtained h ML through (5), and finally used the ML invariance property to find H ML by DFT. Several special cases can be derived. First, when the channel is frequency-nonselective or frequency-flat, then L = 1 and H k = h 0 for all k. Equations (4) through (7) would reduce to (8c) Now, if only one OFDM sample block is used, we can simply take M = 1 in (3) through (8) to obtain all the corresponding results for frequency-selective and -nonselective channels. Next, if the channel length is not known a priori,we can assume full channel length. That is, we let L = N. It then can be readily shown that, for the frequency-selective channel, (6) and (7) respectively become (9a) (9b) 1 H where we have used the fact that FN N FN N/ N.Al- though simpler in expressions, we certainly expect that the estimators of (9) will be outperformed by (6) and (7). Note that we can also replace M =1in(9)toobtain the ML channel estimators using a single OFDM sample block. Another possible arrangement is to use the same single symbol sample X k for each kth subcarrier over repeated blocks. Then (3) can be decomposed into N sets of M equations, while each set of the M equations refers to only one subcarrier. For the kth subcarrier, we get (10) We have thus decoupled all subcarriers. We can now independently estimate the kth CFR H k basedonthem noisy symbol samples of R m,k without the correlation
388 Shu-Han Liao et al. concern. The log-likelihood function for (10) is given by (11) (14) from which, the ML estimation for H k can readily be found as (12) The ML CIR estimate can be obtained by the inverse DFTof(12)as where the pseudo-inverse (XF N L ) + = (F N H XH L XF N L ) -1 H F N XH L. From (14), we find (15a) (13) Noticethatwehaveusedn =0,1,,N 1instead of n =0,1,,L 1. This is because we started first with the CFR estimation without relying on the channel length information. If the actual channel length were L < N, then we would expect h n, ML 0 for n L. It should be obvious that the ML estimations obtained here will be less accurate then (6) and (7). Closer examination will reveal that (12) and (13) correspond to the elements of the estimator vectors given by (9). Thus, the current arrangement is actually equivalent the previous special case of L = N. 3. Performance Evaluation for ML Channel Estimation For ML channel estimation, we can evaluate the performance using the estimator MSE or by the CRLB if the ML estimator is unbiased. We shall first consider MSE and then CRLB. 3.1 MSE 2 The MSE of the CIR estimator is hml h 2 ]. Using (4) on (6), we find E[ h ML (15b) where E[ ] denotes expectation and Tr[ ] denotes trace. From (15a), h ML is unbiased. Note that the variance of (15b) is a total variance as it is the sum of L variances, i.e., Using H F h ML N L ML along with (14), we get (15c) (16a) (16b) (16c) where we have used the fact that Tr[AB] =Tr[BA] and
Maximum-Likelihood vs. Least Squares Schemes for OFDM Channel Estimation Using Techniques of Repeated Training Blocks 389 H the fact that FN LFN L NIL L. When L = N, it can readily verified that (XF N N ) + = (XF N N ) -1 whence (15b) and (16c) respectively reduce to Using (21b) on (17), the minimum variance or MSE can be found as (22a) (17a) (22b) (17b) H H where we have used the fact FN NFN N FN NFN N = NI N N. We now constrain the transmitted power to unity as (18) We now apply Lagrange s method to minimize the variance HML subject to the constraint of (18). 2 Let (19) where is the Lagrangian multiplier. Taking the partial derivatives of f with respect to { X k 2 } and setting the results to zero, we get (20) This is seen to be independent of k.thus,all{x k }have the same amplitude regardless of k. Substituting (20) into (18), we find Thus, we have proven that the optimal training sequence is a sequence with unit constant amplitude. If (21b) is used in (15b) and (16c), the results can be further simplified to (23a) (23b) Note that, from either (15), (16), (17) or from (22), (23), we can clearly see that the estimation performances improve with increasing number of training blocks. The polyphase sequences given by Chu [13] are good candidates that satisfy (21b). The Chu sequences can avoid the large peak-to-average power ratio problem. They have constant magnitudes both in the time and frequency domain, and possess a desirable periodic autocorrelation of the Kronecker delta function [13]. In our simulations, we have chosen the Chu sequence given by j k N X e. k 2 / 3.2 CRLB Since the ML channel estimators given in the previous section are unbiased, we can also use CRLB for performance evaluation. One needs to find the Fisher information matrix first. For h ML given by (6), we use (5) to compute the Fisher information matrix as [14, Sec. 6.7] (21a) (21b) (24)
390 Shu-Han Liao et al. where m = XF N L h. The CRLB for h ML is then given by (25) which is exactly identical to the MSE of (15b). Thus, the estimator h ML is optimum in the sense that its variance or MSE is equal to CRLB. When a unit constant amplitude training sequence is used, (24) and (25) become (26) (27) H where the fact FN LF = NI N L L L has been used. Note that (27) is identical to the MSE of (23a) as expected. For the CFR estimator of (7), we cannot derive the CRLB directly from the log-likelihood function as all subcarrier CFRs are correlated. Instead, we use the fact 2 2 N (see (16c)) to get HML1 h ML1 (28) Similarly, when L = N, the CRLBs can readily be shown to be identical to (22). 4. Least Squares Channel Estimation for OFDM Like ML channel estimations, LS channel estimations normally use CIRs as the channel parameters to be estimated due to their uncorrelated nature. Using M repeated OFDM blocks, an overdetermined system for h can be formed by stacking the R m in (4) over m =1,2,,M as (29) from which, the least squares CIR estimator h LS can be obtained straightforwardly using the pseudo-inverse formula [11]. It can be readily shown that the result will be exactly identical to the ML estimator of (6). Thus, (30) The identical result of the LS and ML estimators stems from the fact that the noise vector W m in (4) has all entries being independent, identically distributed (i.i.d.) 2 Gaussian random variables, i.e., W m,k ~N (, 0 W ), m,k [10,14]. A CFR estimator can be found by H F h, N L LS which is, of course, identical to the H ML given by (7). Note that, since LS estimation does not possess the invariance property, this H is not an LS CFR estimator. We can also formulate an overdetermined system using a different approach. By utilizing the diagonal nature of X, we can rearrange (4) as (31) The new noise vector is X 1 W 2 H 1 m ~ N(0 N N, W ( X X) ). Apparently, the entries of this noise vector are not i.i.d. Gaussian random variables. By stacking X 1 R m of (31) over m = 1, 2,, M, we obtain the weighted LS system as (32) Then, by directly using the pseudo-inverse formula, we readily get (33) H where we have used the fact FN LFN L= NI L L.Thisisa different result. A word of importance is in order. Note that (32) is basically obtained from multiplying (29) by a matrix, viz., X 1. As a result, each linear equation of the overdetermined system is multiplied by a constant
Maximum-Likelihood vs. Least Squares Schemes for OFDM Channel Estimation Using Techniques of Repeated Training Blocks 391 (not necessarily all identical). Intuitively at first thought, a same solution should be obtained. But this is not true. The principle of the least squares method is to minimize the sum of squared errors. When the linear equations of an overdetermined system are respectively multiplied by different constants (even some constants may be identical), the original errors become weighted (weighted LS, [11]) and a different solution will result. On the other hand, if one starts from (31) to find an ML CIR estimator [15, Sec. 6.8], the result will remain the same as the h ML of (6). This is because in ML estimations, a determined system of simultaneous equations (not necessarily linear) is formed by partial differentiating the log-likelihood function with respect to the parameters to be estimated. Multiplying the equations of a determined system by constants will not change the solution outcome. It is relatively easy to find that the h LS of (33) is better than the estimator of (9a) but is inferior to that of (30). However, when L = N, (33) reduces to (9a). A CFR estimator can be derived from h LS as (34) This is identical to the ML CFR estimator H k, ML as given by (12). We wish to point out here that, in [1], the CFR estimator of (35) has been wrongly called an LS estimator as the fact has been overlooked that LS estimations do not possess the invariance property. Further, using one OFDM training block (M = 1), [8] uses (4) to directly formulate the CFR estimation. There, they overlook another fact that subcarrier CFRs are correlated. 5. Performance Evaluation for LS Channel Estimation We have seen that, except for the weighted LS result of (33), the regular LS channel estimators given in Section 4 are identical to the ML channel estimators given in Section II. Therefore, we will only derive the MSE of the weighted LS estimators given by (33) and (34). Substituting (31) into (33), we get (38) But this is neither called an LS nor an ML estimator. We further point out here that, for the special case of L = N and M = 1, (34) reduces to from which, we find (39a) (35) This is exactly the CFR estimator given in [1]. Finally, another possible arrangement is to stack the decoupled form of (10) to obtain an overdetermined system as The MSE of H given by (34) is simply (39b) from which, the pseudo-inverse formula gives (36) (37) (40) Note that, when a Chu sequence is used, (39b) becomes identical to (23a) and (27). 6. Numerical Examples Taking N = 64, we present simulation results of MSE
392 Shu-Han Liao et al. and CRLB against signal-to-noise ratio (SNR) for the ML CIR estimators. We consider the case of known channel length of L = 16 (eq. (6)) as well as the case of unknown channel length (eq. (9)). In Figure 1, single training block is employed (M = 1) while in Figures 2, 4 identical training blocks are used (M = 4). The SNR is defined as E[ X k 2 ]/ W 2. Note that the ML estimators of (6) and (9) are also the LS CIR estimators given by (30) and (37). In the figures, results obtained from a random square 16-QAM sequence and a Chu sequence are compared. For fairness, the signal energy of the QAM sequence has been normalized to unity. Therefore, SNR is simply given as 1/ 2 W. Next, Figures 3 and 4 plot the MSE vs. SNR curves of the LS estimator h LS respectively for M =1andM = 4. Again, both the random 16-QAM sequence and Chu sequence are considered. From all the above plots, it is observed that estimation performance is improved by use of the Chu sequence. However, when the channel length is known, the improvement rendered by a Chu sequence is small except for the case of weighted LS estimation as shown in Figures 3 and 4. We can also observe that, with QAM sequence, the weighted LS estimator h LS is seen to have a performance lying between those given by the estimators of (6) and (9a). Figure 1. MSE vs. SNR for the CIR estimators given by (6) and (9a) with M =1. Figure 3. MSE vs. SNR for the CIR estimator h LS given by (33) with N = 64, M = 1 and L =8. Figure 2. MSE vs. SNR for the CIR estimators given by (6) and (9a) with M =4. Figure 4. MSE vs. SNR for the CIR estimator h LS given by (33) with N = 64, M = 4 and L =8.
Maximum-Likelihood vs. Least Squares Schemes for OFDM Channel Estimation Using Techniques of Repeated Training Blocks 393 A few words are in order. First, the M = 1 cases of Figures 1 and 3 in fact give the results of the conventional or existing single training block schemes. Specifically, with M = 1, (9b) is just the FIR estimator given in [1]. Next, we note here that, it is more costly to use multiple training blocks than a single training block. It also reduces data throughput. These shortcomings are rewarded by improved system performance with more accurate estimations as can be readily seen from the simulation results. From (15), (17), (22), (23), (25), (27), (28), (39), and (40), the estimator MSE s are all inversely proportional to the number of training blocks. Certainly, one cannot indefinitely increase the number of training blocks. In real practice, depending on the application, a trade-off should certainly be made between system cost/complexity and performance improvement. 7. Conclusion We present generalized ML and LS channel estimation schemes for OFDM systems using repeated OFDM training blocks. Expressions for ML and LS channel estimators and their MSE are derived. For ML estimations, the CRLB expressions are also derived. We also show that a unit constant amplitude sequence is the optimum training sequence for OFDM channel estimations. Although estimation performance is improved by increasing the number of training blocks, in real practice, a trade-off should be made between cost and the number of training blocks to be used. References [1] Borjesson, P. O., Edfors, O., Sandell, M., Van de Beek and Wilson, S. K., On Channel Estimation in OFDM Systems, Proceeding IEEE 45 th Vehicular Technology Conference, Chicago, Il., pp. 815 819 (1995). [2] Borjesson, P. O., Edfors, O., Sandell, M., Van de Beek and Wilson, S. K., OFDM Channel Estimation by Singular Value Decomposition, IEEE Transactions on Communications, Vol. 46, pp. 931 938 (1998). [3] Ariyavisitakul, S., Li, Y. and Seshadri, N., Channel Estimation for OFDM Systems with Transmitter Diversity in Mobile Wireless Channels, IEEE Journal on Selected Areas in Communications, Vol. 17, pp. 461 471 (1999). [4] Li, Y., Pilot-Symbol-Aided Channel Estimation for OFDM in Wireless Systems, IEEE Transactions on Vehicular Technology, Vol. 49, pp. 1207 1215 (2000). [5] Hudson, R. E., Tung, T. L. and Yao, K., Channel Estimation and Adaptive Power Allocation for Performance and Capacity Improvement of Multiple- Antenna OFDM Systems, 3rd IEEE Signal Processing Workshop on Signal Processing Advances in Wireless Communications, pp. 82 85 (2001). [6] Li, Y., Simplified Channel Estimation for OFDM Systems with Multiple Transmit Antennas, IEEE Transactions on Wireless Communications, Vol. 1, pp. 67 75 (2002). [7] Cheng, Z. and Dahlhaus, D., Time Versus Frequency Domain Channel Estimation for OFDM Systems with Antenna Arrays, ICSP 02 Proceedings, pp. 1340 1343 (2002). [8] Chouinard, J.-Y., Wang, X. and Wu, Y., Modified Channel Estimation Algorithms for OFDM Systems with Reduced Complexity, ICSP 04 Proceedings, Vol. 2, pp. 1747 1751 (2004). [9] Khan, M. Z. A., Low-Complexity ML Channel Estimation Schemes for OFDM, 2005 13 th IEEICC, Vol. 2, pp. 607 612 (2005). [10] Meyer, P. L., Introductory Probability and Statistical Applications, 2 nd ed., Addison-Wesley, Reading, MA. (1970). [11] Haykin, S., Adaptive Filter Theory, 4 th ed., Prentice- Hall, New Jersey (2002). [12] Proakis, J. G., Digital Communications, 4 th ed., Mc- Graw-Hill, New York (2001). [13] Chu, D. C., Polyphase Codes with Good Periodic Correlation Properties, IEEE Transactions on Information Theory, pp. 531 532 (1972). [14] Scharf, L. L., Statistical Signal Processing, Addison- Wesley, Reading, MA (1991). [15] Fang, K., Rugini, L. and Leus, G., Low-Complexity Block Transmission Over Doubly Selective Channels: Iterative Channel Estimation and Turbo Equalization, EURASIP Journal on Advances in Signal Processing (2010).
394 Shu-Han Liao et al. [16] Hrycak, T., Saptarshi, D., Matz, G. and Feichtinger, H. G., Practical Estimation of Rapidly Varying Channels for OFDM Systems, IEEE Transactions on Communications, Vol. 59, pp. 3040 3048 (2011). [17] Li, W., Zhang, Y., Huang, L.-K., Maple, C. and Cosma, J., Implementation and Co-Simulation of Hybrid Pilot-Aided Channel Estimation with Decision Feedback Equalizer for OFDM Systems, IEEE Transactions on Broadcasting, Vol. 58, pp. 590 602 (2012). Manuscript Received: Oct. 3, 2012 Accepted: Jun. 1, 2013