A Low-Complexity KL Expansion-Based Channel Estimator for OFDM Systems

This paper ﬁrst proposes a computationally e ﬃ cient, pilot-aided linear minimum mean square error (MMSE) batch channel estimation algorithm for OFDM systems in unknown wireless fading channels. The proposed approach employs a convenient representation of the discrete multipath fading channel based on the Karhunen-Loeve (KL) orthogonal expansion and ﬁnds MMSE estimates of the uncorrelated KL series expansion coe ﬃ cients. Based on such an expansion, no matrix inversion is required in the proposed MMSE estimator. Moreover, optimal rank reduction is achieved by exploiting the optimal truncation property of the KL expansion resulting in a smaller computational load on the estimation algorithm. The performance of the proposed approach is studied through analytical and experimental results. We then consider the stochastic Cram´er-Rao bound and derive the closed-form expression for the random KL coe ﬃ cients and consequently exploit the performance of the MMSE channel estimator based on the evaluation of minimum Bayesian MSE. We also analyze the e ﬀ ect of a modelling mismatch on the estimator performance. To further reduce the complexity, we extend the batch linear MMSE to the sequential linear MMSE estimator. With the fast convergence property and the simple structure, the sequential linear MMSE estimator provides an attractive alternative to the implementation of channel estimator.


INTRODUCTION
With unprecedented demands on bandwidth due to the explosive growth of broadband wireless services usage, there is an acute need for a high-rate and bandwidth-efficient digital transmission.In response to this need, the research community has been extensively investigating efficient schemes that make efficient utilization of the limited bandwidth and cope with the adverse access environments [1].These access environments may create different channel impairments and dictate unique sets of advanced signal processing algorithms to combat specific impairments.
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Multicarrier (MC) transmission scheme, especially orthogonal frequency-division multiplexing (OFDM), has recently attracted considerable attention since it has been shown to be an effective technique to combat delay spread or frequency-selective fading of wireless or wireline channels, thereby improving the capacity and enhancing the performance of transmission.This approach has been adopted as the standard in several outdoor and indoor high-speed wireless and wireline data applications, including terrestrial digital broadcasting (DAB and DVB) in Europe, and high-speed modems over digital subscriber lines in the US.It has also been implemented for broadband indoor wireless systems including IEEE802.11a,MMAC, and HIPERLAN/2.
An OFDM system operating over a frequency-selective wireless communication channel effectively forms a number of parallel frequency-nonselective fading channels, thereby reducing intersymbol interference (ISI) and obviating the need for complex equalization, thus greatly simplifying channel estimation/equalization task.Moreover, OFDM is bandwidth efficient since the spectra of the neighboring subchannels overlap, yet channels can still be separated through the use of orthogonality of the carriers.Furthermore, its structure also allows efficient hardware implementations using fast Fourier transform (FFT) and polyphase filtering [2].
Although the structure of OFDM signalling avoids ISI arising due to channel memory, fading multipath channel still introduces random attenuations on each tone.Furthermore, simple frequency-domain equalization, which divides the FFT output by the corresponding channel frequency response, does not assure symbol recovery if the channel has nulls on some subcarriers.Hence, advanced signal processing algorithms have to be used for accurate channel estimation to improve the performance of the OFDM systems.Numerous pilot-aided channel estimation methods for OFDM have been developed [3,4,5,6].In particular, a low-rank approximation is applied to linear MMSE estimator for the estimation of subcarrier channel attenuations by using the frequency correlation of the channel [3].Two pilot-aided MLE and MMSE schemes are revisited and compared in terms of computational complexity in [4].In [5], an MMSE channel estimator, which makes full use of the time and frequency correlation of the time-varying dispersive channel, was proposed.Moreover, low-complexity MMSE doubly channel estimation approaches were presented in [6] based on embedding Kronecker-delta pilot sequences.
Multipath fading channels have been studied extensively, and several models have been developed to describe their variations [7].In many cases, the channel taps are modelled as general lowpass stochastic processes (e.g., [8]), the statistics depend on mobility parameters.A different approach explicitly models the multipath channel taps by the Karhunen-Loeve (KL) series representation [9].KL expansion models have also been used previously in modelling the multipath channel within a CDMA scenario [10].In the case of KL series representation of stochastic process, a convenient choice of orthogonal basis set is one that makes the expansion coefficient random variables uncorrelated.When these orthogonal bases are employed to expand the channel taps of the multipath channel, uncorrelated coefficients indeed represent the multipath channel.Therefore, KL representation allows one to tackle the estimation of correlated multipath parameters as a parameter estimation problem of the uncorrelated coefficients.Exploiting KL expansion, the main contribution of this paper is to propose a computationally efficient, pilot-aided MMSE channel estimation algorithm.Based on such representation, no matrix inversion is required in the proposed batch approach.Moreover, optimal rank reduction is achieved by exploiting the optimal truncation property of the KL expansion resulting in a smaller computational load on the estimation algorithm.The performance of the proposed batch approach is explored based on the evaluation of the stochastic Cramér-Rao bound for the random KL coefficients.We also analyze the effect of a modelling mismatch on the estimator performance.In contrast to [3], the proposed batch approach employs KL expansion of multipath channel parameters and reduces the complexity of the singular value decomposition (SVD) used in eigendecomposition by estimating multipath channel parameters instead of channel attenuations on each tone.In addition, we propose the simple sequential MMSE implementation for the estimation of the KL expansion coefficients, which does not require to perform matrix inversion as well.
The rest of the paper is organized as follows.Section 2 describes a general model for OFDM systems and briefly introduces the channel estimation task.Section 3 derives a basic and simplified approach to MMSE batch channel estimation for OFDM systems.To show its efficiency, the performance bounds are analyzed and the performance degradation due to a mismatch of the estimator to the channel statistics as well as the SNR is demonstrated.The sequential MMSE estimator is introduced in Section 4 and its convergence behavior is also analyzed.Some simulation examples are provided in Section 5. Finally, conclusions are drawn in Section 6.

SYSTEM MODEL
In order to eliminate ISI arising due to multipath channel and preserve orthogonality of the subcarrier frequencies (tones), conventional OFDM systems first take the IFFT of data symbols and then insert redundancy in the form of a cyclic prefix (CP) of length L CP larger than the channel order L. CP is discarded at the receiver and the remaining part of the OFDM symbol is FFT processed.Combination of IFFT and CP at the transmitter with the FFT at the receiver divides the frequency-selective channel into several separate flatfading subchannels.The block diagram in Figure 1 describes the conventional OFDM system.We consider an OFDM system with K subcarriers for the transmission of K parallel data symbols.Thus, the information stream X(n) is parsed into K-long blocks: X i = [X i (0), X i (1), . . ., X i (K − 1)] T , where i = 1, 2, . . . is the block index and the superscript (•) T indicates the vector transpose.The K × 1 symbol block is then mapped to a (K + L) × 1 vector by first taking the IFFT of X i and then replicating the last L CP elements as s i is serially transmitted over the channel.At the receiver, the CP of length L CP is removed first and FFT is performed on the remaining K × 1 vector.Therefore, we can write the output of the FFT unit in matrix form as where A i is the diagonal matrix A i = diag(X i ) and H i is the channel vector.The elements of H i are the values of the channel frequency response evaluated at the subcarriers.Therefore, we can write . . . . . .channel impulse during the ith OFDM block.Finally, η i is a K × 1 zero-mean, i.i.d.complex Gaussian vector that models additive noise in the K subchannels (tones).We have E[η i η † i ] = σ 2 I K where I K represents a K × K identity matrix, σ 2 is the variance of the additive noise entering the system, and the superscript (•) † indicates the Hermitian transpose.
Based on model (2), our main objective in this paper is to develop both batch and sequential pilot-aided channel estimation algorithm according to MMSE criterion and then explore the performance of the estimators.A batch approach adapted herein explicitly models the channel parameters by the KL series representation and estimates the uncorrelated expansion coefficients.Furthermore, the computational load of the proposed MMSE estimation technique is further reduced with the application of the KL expansion optimal truncation property [9].We then introduce batch channel MMSE approach first.

MMSE ESTIMATION OF KL COEFFICIENTS: BATCH APPROACH
A low-rank approximation to the frequency-domain linear MMSE channel estimator is provided by [3] to reduce the complexity of the estimator.Optimal rank reduction is achieved in this approach by using the SVD of the channel attenuations covariance matrix C H of dimension K × K.In contrast, we adopt the MMSE estimator for the estimation of multipath channel parameter h that uses covariance matrix of dimension L × L. The proposed approach employs KL expansion of multipath channel parameters and reduces the complexity of the SVD used in eigendecomposition since L is usually much less than K.We will now develop MMSE batch estimator for pilot-assisted OFDM system in the sequel.

MMSE channel estimation
Pilot-symbol-assisted techniques can provide information about an undersampled version of the channel that may be easier to identify.In this paper, we therefore address the problem of estimating multipath channel parameters by exploiting the distributed training symbols.Considering (2), and in order for the pilot symbols to be included in the output vector for our estimation purposes, we focus on an undersampled signal model.Assuming that K p pilot symbols are uniformly inserted at known locations of the ith OFDM block, the K p × 1 vector corresponding to the FFT output at the pilot locations becomes where ] is a diagonal matrix with pilot-symbol entries, ∆ is pilot spacing interval, F is a K p × L FFT matrix generated based on pilot indices, and similarly η is the undersampled noise vector.
For the estimation of h, the new linear signal model can be formed by premultiplying both sides of (3) by A † and assuming that pilot symbols are taken from a PSK constellation A † A = I Kp , then the new form of (3) becomes where Y and η are related to Y and η by the linear transformation, respectively.Furthermore, η is statistically equivalent to η. Equation ( 4) offers a Bayesian linear model representation.Based on this representation, the minimum variance estimator for the time-domain channel vector h for the ith OFDM block, that is, conditional mean of h given Y, can be obtained using MMSE estimator.We should clearly make the assumptions that h ∼ N (0, C h ), η ∼ N (0, C η ), and h is uncorrelated with η.Therefore, MMSE estimate of h is given by [11] Due to PSK pilot-symbol assumption together with the result C η = E[ η η † ] = σ 2 I Kp , we can therefore express (5) by Under the assumption that uniformly spaced pilot symbols are inserted with pilot spacing interval ∆ and K = ∆ × K p , correspondingly, F † F reduces to F † F = K p I L .Then according to (6) and F † F = K p I L , we arrive at the expression Since MMSE estimation still requires the inversion of C −1 h , it therefore suffers from a high computational complexity.However, it is possible to reduce the complexity of the MMSE algorithm by diagonalizing channel covariance matrix with a KL expansion.

KL expansion
Channel impulse response h is a zero-mean Gaussian process with covariance matrix C h .The KL transformation is therefore employed here to rotate the vector h so that all its components are uncorrelated.The vector h, representing the channel impulse response during ith OFDM block, can be expressed as a linear combination of the orthonormal basis vectors as follows: where , ψ i 's are the orthonormal basis vectors, g = [g 0 , g 1 , . . ., g L−1 ] T , and g l 's are the weights of the expansion.If we form the covariance matrix C h as where Λ g = E{gg † }, the KL expansion is the one in which Λ g of C h is a diagonal matrix (i.e., the coefficients are uncorrelated).If Λ g is diagonal, then the form ΨΛ g Ψ † is called an eigendecomposition of C h .The fact that only the eigenvectors diagonalize C h leads to the desirable property that the KL coefficients are uncorrelated.Furthermore, in the Gaussian case, the uncorrelatedness of the coefficients renders them independent as well, providing additional simplicity.Thus, the channel estimation problem in this application is equivalent to estimating the i.i.d.complex Gaussian vector g KL expansion coefficients.

Estimation of KL coefficients
In contrast to (4) in which only h is to be estimated, we now assume that the KL coefficient vector g is unknown.Thus the data model ( 4) is rewritten for each OFDM block as (10) which is also recognized as a Bayesian linear model, and recall that g ∼ N (0, Λ g ).As a result, the MMSE estimator of g is where and λ g0 , λ g1 , . . ., λ gL−1 are the singular values of Λ g .
It is clear that the complexity of the MMSE estimator in (7) is reduced by the application of KL expansion.However, the complexity of g can be further reduced by exploiting the optimal truncation property of the KL expansion [9].MMSE estimator of g requires 4L 2 + 4LK p + 2L real multiplications.From the results presented in [4], ML estimator of g is obtained as follows: Note that, according to (13), the ML estimator of g requires 4L 2 + 4LK p real multiplications.

Truncated KL expansion
A truncated expansion g r can be formed by selecting r orthonormal basis vectors among all basis vectors that satisfy C h Ψ = ΨΛ g .The optimal one that yields the smallest average mean squared truncation error (1/L)E[ † r r ] is the one expanded with the orthonormal basis vectors associated with the first largest r eigenvalues as given by where r = g−g r .For the problem at hand, truncation property of the KL expansion results in a low-rank approximation as well.Thus, a rank-r approximation to Λ gr is defined as Since the trailing L − r variances {λ gl } L−1 l=r are small compared to the leading r variances {λ gl } r−1 l=0 , then the trailing L−r variances are set to zero to produce the approximation.However, typically the pattern of eigenvalues for Λ g splits the eigenvectors into dominant and subdominant sets.Then the choice of r is more or less obvious.The optimal truncated KL (rank-r) estimator of (11) now becomes where Since our ultimate goal is to obtain MMSE estimator for the channel frequency response H, from the invariance property of the MMSE estimator, it follows that if g is the estimate of g, then the corresponding estimate of H can be obtained for the ith OFDM block as Thus, from ( 16) and ( 17), the truncated MMSE estimator of g requires 4Lr + 4LK p + 2r real multiplications.

Performance analysis
We turn our attention to analytical performance results of the batch MMSE approach.We first consider the CRB and derive the closed-form expression for the random KL coefficients, and then exploit the performance of the MMSE channel estimator based on the evaluation of minimum Bayesian MSE.

Cramér-Rao bound for random KL coefficients
The mean squared estimation error for unbiased estimation of a nonrandom parameter has a lower bound, the Cramér-Rao bound (CRB), which defines the ultimate accuracy of unbiased estimation procedure.Suppose g is an unbiased estimator of a vector of unknown parameters g (i.e., E{ g} = g) then the mean squared error matrix is lower bounded by the inverse of the Fisher information matrix (FIM): Since the estimation of unknown random parameters g via MMSE approach is considered in this paper, the modified FIM needs to be taken into account in the derivation of stochastic CRB [12].Fortunately, the modified FIM can be obtained by a straightforward modification of (19) as where J P (g) represents the a priori information.Under the assumption that g and η are independent of each other and η is a zero mean, from [12] and (10), the conditional PDF is given by from which the derivatives follow as where the superscript (•) * indicates the conjugation operation.
Using C η = σ 2 I Kp , Ψ † Ψ = I L , and F † F = K p I L , and taking the expected value yield the following simple form: The second term in (20) is easily obtained as follows.Consider the prior PDF of g as The respective derivatives are found as Upon taking the negative expectations, the second term in (20) becomes Substituting ( 23) and ( 26) in (20) produces for the modified FIM the following: Inverting the matrix J M (g) yields

Bayesian MSE
For the MMSE estimator g, the error is Since the diagonal entries of the covariance matrix of the error represent the minimum Bayesian MSE, we now derive covariance matrix C of the error vector.From the performance of the MMSE estimator for the Bayesian linear model theorem [11], the error covariance matrix is obtained as and then the minimum Bayesian MSE of the full rank estimator becomes (see Appendix A) where SNR = 1/σ 2 and tr denotes trace operator on matrices.
Comparing (28) with (30), the error covariance matrix of the MMSE estimator coincides with the stochastic CRB of the random vector estimator.Thus, g achieves the stochastic CRB.
As the details are given in Appendix A, B MSE ( g) given in (31) can also be computed for the truncated (low-rank) case as follows: Notice that the second term in (32) is the sum of the powers in the KL transform coefficients not used in the truncated estimator.Thus, the truncated B MSE ( g r ) can be lower bounded by (1/L) L−1 i=r λ gi which will cause an irreducible error floor in the SER results.

Mismatch analysis
Once the true frequency-domain correlation, characterizing the channel statistics and the SNR, is known, the optimal channel estimator can be designed as indicated in Section 4.
However, in mobile wireless communications, the channel statistics depend on the particular environment, for example, indoor or outdoor, urban or suburban, and change with time.Hence, it is important to analyze the performance degradation due to a mismatch of the estimator to the channel statistics as well as the SNR, and to study the choice of the channel correlation and SNR for this estimator so that it is robust to variations in the channel statistics.As a performance measure, we use uncoded symbol error rate (SER) for QPSK signaling.The SER expression for this case is given in [13] as a function of the SNR and the average B MSE ( g) as follows: where and Ω g represents the normalized variance of the channel gains (Ω g = L−1 i=0 λ gi = 1) and SNR = 1/σ 2 .In practice, the true channel correlations and SNR are not known.If the MMSE channel estimator is designed to match the correlation of a multipath channel impulse response C h and SNR, but the true channel parameter h has the correlation C h and the true SNR, then average Bayesian MSE for the designed channel estimator is obtained as (see Appendices A and B) (i) SNR mismatch: (ii) correlation mismatch: where λ gi is the ith diagonal element of Λ g = Ψ † C h Ψ, and β i is ith diagonal element of the real part of the cross-correlation matrix between g and g.

MMSE ESTIMATION OF KL COEFFICIENTS: SEQUENTIAL APPROACH
We now turn our attention to the derivation of the sequential MMSE algorithm with simple structure.The sequential MMSE approach is proposed in this paper to follow the channel variations by exploiting only channel correlations in frequency.The block diagram for this is shown in Figure 2.
To begin with the algebraic derivation, we use (10) to write mth component of Y as where u † (m) is the mth row of FΨ and η[m] is the mth element of the noise vector η.

If an MMSE estimator of Y[m +1] can be found based on Y[m], denoted by Y m+1|m , the prediction error
We can therefore project g onto each vector separately and add the results, so that where g m+1 is the (m + 1)th estimate of g, and κ m+1 is the gain factor given as It can be seen that M m = E (g − g m ) (g − g m ) † is needed in (39), hence update equation for the minimum MSE matrix should also be given.If we substitute (38) in M m+1 = E (g − g m+1 ) (g − g m+1 ) † , we obtain an update equation for M m+1 as Based on these results, the steps of the sequential MMSE estimator for g can be summarized as follows.
Initialization.Set the parameters to some initial value g 0 = 0, M 0 = Λ g .
(2) Update the estimate of g from ( 38).(ii) Since the MMSE estimator (11) requires F † F to be equal to K p I which is satisfied only when ∆ = K/K p is an integer, however, the sequential version of ( 11) works as long as ∆ ≤ K/L.
We now analyze the complexity of the sequential MMSE algorithm.It follows from (39) in step (1) that one needs 4L 2 + 5L real multiplications to compute the gain.Similarly, from (38) in step ( 2), it requires 5L real multiplications for the estimator update.Finally, in step (3), we need 8L 2 real multiplications for the MMSE matrix update.Therefore, the total sequential MMSE algorithm requires 12L 2 + 10L real multiplications for one iteration.

Performance analysis
We turn our attention now to the performance analysis of the adaptive algorithm.We will try to evaluate its convergence properties in terms of mean square error.
Define average MSE at the mth step as MSE m = (1/L) tr(M m ), then it follows from (43) that Thus, we observe that as m → ∞, MSE m → 0 which means that g m converges to g in the mean square.

SIMULATIONS
In this section, the merits of our channel estimators are illustrated through simulations.We choose average mean square error (MSE) and symbol error rate (SER) as our figure of merits.We consider the fading multipath channel with L paths given by ( 45) with an exponentially decaying power delay profile θ(τ l ) = Ce −τl/τRMS with delays τ l that are uniformly and independently distributed over the length L CP .Note that h is chosen as complex Gaussian leading to a Rayleigh fading channel with root mean square (RMS) width τ RMS and normalizing constant C. In [3], it is shown that the normalized exponential discrete channel correlation for different subcarriers is The scenario for our simulation study consists of a wireless QPSK-OFDM system employing the pulse shape as a unit-energy Nyquist-root raised-cosine shape with rolloff α = 0.2, with a symbol period (T s ) of 0.120 microsecond, corresponding to an uncoded symbol rate of 8.33 Mbps.Transmission bandwidth (5 MHz) is divided into 1024 tones.We assume that the fading multipath channel has L = 40 paths with an exponentially decaying power delay profile (45) with τ RMS = 5 sample (0.6 microsecond) long.

Batch MMSE approach
A QPSK-OFDM sequence passes through channel taps and is corrupted by AWGN (0 dB, 5 dB, 10 dB, 15 dB, 20 dB, 25 dB, and 30 dB, respectively).We use a pilot symbol for every twenty (∆ = 20) symbols.The MSE at each SNR point is averaged over 1000 realizations.We compare the experimental MSE performance and its theoretical Bayesian MSE of the proposed full-rank MMSE estimator with maximumlikelihood (ML) estimator and its corresponding Cramér-Rao bound (CRB).Figure 3 confirms that MMSE estimator performs better than ML estimator at low SNR.However, the 2 approaches have comparable performance at high SNRs.To observe the performance, we also present the MMSE and ML estimated channel SER results together with theoretical SER in Figure 4. Due to the fact that spaces between the pilot symbols are not chosen as a factor of the number of subcarriers, an error floor is observed in Figures 3 and 4. In the case of choosing the pilot space as a factor of number of subcarriers, the error floor vanishes because of the fact that the orthogonality condition between the subcarriers at pilot locations is satisfied.In other words, the curves labeled as simulation results for MMSE estimator and ML estimator fit to the theoretical curve at high SNRs.It also shows that the MMSE estimated channel SER results are better than ML estimated channel SER especially at low SNRs.

SNR design mismatch
In order to evaluate the performance of the proposed fullrank MMSE estimator to mismatch only in SNR design, the  estimator is tested when SNRs of 10 and 30 dB are used in the design.The SER curves for a design SNR of 10, 30 dB are shown in Figure 5.The performance of the MMSE estimator for high SNR (30 dB) design is better than low SNR (10 dB) design across a range of SNR values (0-30 dB).This results confirm that channel estimation error is concealed in noise for low SNR whereas it tends to dominate for high SNR.Thus, the system performance degrades especially for low SNR design.

Correlation mismatch
To further analyze full-rank MMSE estimator's performance, we need to study sensitivity of the estimator to design errors, that is, correlation mismatch.We therefore designed the estimator for a uniform channel correlation which gives the worst MSE performance among all channels [3,5] and evaluated it for an exponentially decaying power delay profile.The uniform channel correlation between the attenuations can be obtained by letting τ RMS → ∞ in (45), resulting in Figures 6 and 7 demonstrate the estimator's sensitivity to the channel statistics in terms of average MSE and SER performance measures, respectively.As it can be seen from Figures 6 and 7, only small performance loss is observed for low SNRs when the estimator is designed for mismatched channel statistics.This justifies the result that a design for worst correlation is robust to mismatch.

Performance of the truncated estimator
The truncated estimator performance is also studied as a function of the number of KL coefficients.Figure 8 presents the MSE result of the truncated MMSE estimator for SNR = 10, 20, and 30 dB.If only a few expansion coefficients are employed to reduce the complexity of the proposed estimator, then the MSE between channel parameters becomes large.However, if the number of parameters in the expansion is increased, the irreducible error floor still occurs.

Sequential MMSE approach
The MSE results of the sequential full-rank MMSE algorithm are obtained and presented as shown in Figure 9.In order to better evaluate the performance of the proposed sequential MMSE estimation algorithm, we compare it with previously developed least mean square (LMS) and recursive least squares (RLS) algorithms.It can be seen from simulations that recursive MMSE estimator yields better performance than LMS and RLS approaches and achieves Bayesian MSE especially for low SNR.
For the convergence of the proposed adaptive algorithm, MSE versus iteration is plotted for SNR = 10, 20, 30, and 40 dB in Figure 10.As expected, the proposed sequential algorithm converges faster for high SNR values.
Finally, we wish to evaluate the performance of the algorithm for different values of pilot spacing 10, 20, 30, 40, and 50 by plotting the MSEs and SERs with respect to SNR in Figures 11 and 12, respectively.For the values pilot spacing ∆ larger than K/L, the SER and MSE performances decrease as ∆ increases.

CONCLUSION
We consider the design of low-complexity MMSE channel estimators for OFDM systems in unknown wireless dispersive fading channels.We first derive the batch MMSE estimator based on the stochastic orthogonal expansion representation of the channel via KL transform.Based on such representation, we show that no matrix inversion is needed in the MMSE algorithm.Therefore, the computational cost for implementing the proposed MMSE estimator is low and computation is numerically stable.Moreover, the performance of our proposed batch method was first studied through the derivation of stochastic CRB for Bayesian approach.Since the actual channel statistics and SNR may vary within OFDM block, we have also analyzed the effect of modelling mismatch on the estimator performance and shown both analytically and through simulations that the performance degradation due to such mismatch is negligible for low SNR values.The MMSE estimator is then extended to sequential implementation which enjoys the elegance of the simple structure and fast convergence.

B. BAYESIAN MSE FOR TRUNCATED MMSE KL ESTIMATOR UNDER CORRELATION MISMATCH
In this appendix, we derive the Bayesian MSE of the truncated MMSE KL estimator under correlation mismatch.Although the real multipath channel h has the expansion correlation C h , we designed the estimator for the multipath channel h = Ψg with correlation C h .To evaluate the estimation error g − g r in the same space, we expand h onto the eigenspace of h as h = Ψ g resulting in correlated expansion coefficients.
For the real channel, data model in (10) can be rewritten as and substituting in (16), the truncated MMSE KL estimator now becomes For the truncated MMSE estimator, the error is As a result, the average Bayesian MSE is

Figure 3 :
Figure 3: Performance of proposed MMSE and MLE together with B MSE and CRB.

Figure 5 :
Figure 5: Effects of SNR design mismatch on SER.
correlation for MMSE estimation Theoretical: true correlation for ML estimation Simulation : true correlation for MMSE estimation Simulation: correlation mismatch for MMSE estimation Simulation: true correlation for ML estimation

Figure 6 :
Figure 6: Effects of correlation mismatch on MSE.
correlation mismatch Simulation results: true correlation Theoretical results: true correlation

Figure 7 :
Figure 7: Effects of correlation mismatch on SER.

Figure 8 :
Figure 8: MSE as a function of KL expansion coefficients.

Figure 12 :B MSE g r = 1 L tr C r = 1 L 2 K p λ gi + σ 4 K p λ gi + σ 2 3 )
Figure 12: Symbol error rate of the sequential MMSE for different pilot spacings.
K p SNR λ gi λ gi − 2β i 1 + K p SNR λ gi + K p SNR λ gi λ gi + λ gi − 2β i 1 + K p SNR λ gi β is the real part of E[ gg † ] and β i 's are the diagonal elements of β.With this result, we will now highlight some special cases.(i)Letting β i = λ gi = λ gi in (B.4) for the case of no mismatch in the correlation of KL expansion coefficients, the truncated Bayesian MSE is identical to that obtained in (A.4).(ii) As r → L in (B.4), Bayesian MSE under correlation mismatch is obtained to yieldB MSE ( g) + K p SNR λ gi λ gi + λ gi − 2β i 1 + K p SNR λ gi .(B.5)(iii)Under no correlation mismatch in (B.5)where β i = λ gi = λ gi , Bayesian MSE obtained from (B.5) is identical to that in (A.6).(iv) Also note that as SNR → ∞, (B.4) reduces to MSE( g − g r ).