1. Introduction
Orthogonal frequency division multiplexing (OFDM) system has the high frequency utilization efficiency and the strong multipath resistance. Meanwhile, the multiple-input-multiple-output (MIMO) can provide higher data rates by using parallel channels established in spatial domain over the same time and frequency without extra bandwidth. Thus, the MIMO-OFDM system that combines the advantages of the OFDM and MIMO has been considered as a key technology for the fourth generation wireless communication systems, and it has been employed in many practical applications [1].
Since the channel state information is required for the diversity combining, detection, precoding, and the spectrum sharing [2] etc., the channel estimation is essential for the MIMO-OFDM system to bring out the advantages of the MIMO-OFDM. Currently, the various channel estimation schemes have been presented for the MIMO-OFDM system [3]-[5], the most of which are based on the assumption of the quasi-static or slow time-varying fading channel within a MIMO-OFDM block. However, the assumption is invalid in the high speed mobile scenario, such as high speed railway, and the estimation errors by those techniques will be too high to be used in the equalization. In the high speed mobile environment, the variation of the channel introduces the intercarrier interference (ICI) during one OFDM symbol period, which is consequently becoming a serious challenge for the MIMO-OFDM system [6]-[7]. Therefore, the time-variation of channel within a block must be considered to support the high speed mobile channels.
In the high speed mobile scenario, many time-varying channel estimation techniques have been presented for the mobile MIMO-OFDM system [8]-[12]. Among these schemes in [8]-[10], the Kalman filter theory based the MIMO-OFDM channel estimation algorithms have been discussed and are effective methods because they can track the change of the channel gain in the high speed scenario. In [11], a dynamic autoregressive (AR)-model based channel prediction is used to the interference alignment. However, in these algorithms [8]-[11], the parameters of the AR system need to be estimated, and it is difficult to accurately decide the parameters of the AR model due to the model depending on the mobile speed, which affects the accuracy of the channel estimation or prediction. Literature [12] gives a practical MIMO-OFDM channel estimation method only using the Kalman filter without the parameter estimation of AR model. However, the scheme in [12] is only appropriate for the pilots in the frequency domain, and it cannot obtain the channel estimation of the data subcarriers with the high accuracy because the ICI caused by the time-varying channel is not considered.
Another line of research to improve the performance of the channel estimation is to consider the pilot design and channel estimation jointly [13]-[14]. Literature [13] gives a compressed sensing (CS) based channel estimation method, which improves the estimation accuracy by designing the pilot pattern to reduce the system coherence. In [14], a new data frame structure is presented to conveniently estimate the time-varying channel for the two-way relay networks. However, for the fixed frame structure, such as LTE system, these schemes will be unsuitable. Therefore, a time-varying channel estimation algorithm without changing the frame structure is desired in the high speed mobile MIMO-OFDM system.
Recently, some iterative algorithms joint channel estimation and data detection have been discussed in [15]-[17]. It is the iterative interaction between the channel estimation and the data detection to provide significant improvement compared with the non-iterative schemes. However, the detection error in these iterative algorithms is inevitable, which will result in an error floor due to the dominance of the error propagation. Therefore, it is necessary to take into account these errors in the interaction to improve the estimation accuracy of the iterative schemes. Although the detection error is considered in [18], it has a slow convergence speed due to the effect of the estimated parameters of the AR model and the covariance of the modeling error.
In our previous works, we also give some time-varying channel estimation methods for the high speed mobile scenario [19]-[20]. These algorithms are presented for the TDD-LTE SC-FDMA system with the specific subframe pattern, so they are not appropriate for the other systems due to the different frame structures. Since the considered channel model is the line-of-sight (LOS) component dominant Rician channel in our previous works, the previous schemes ignores the intercarrier interference (ICI) caused by the time-varying channel during one SC-FDMA symbol period. Therefore, these algorithms in [19]-[20] have some limitations in terms of applicability.
To solve the above problems, we propose a practical iterative channel estimation algorithm for the high speed mobile MIMO-OFDM system in the paper. Specifically, the time-variation of the channel is firstly approximated by the polynomial-basis expansion model (P-BEM), and then the Kalman filter is employed to estimate and track the BEM coefficients by the detected data and pilot in each iteration. To reduce the effect of the error propagation on the channel estimation, the detection error is considered as part of the noise in the Kalman recursion in each iteration. Moreover, the employed Kalman filter does not include the estimated parameters of the AR model, so the proposed method has a fast convergence speed and it is more practical for the mobile MIMO-OFDM system.
This paper is organized as follows: Section II introduces the system model. Section III presents the proposed method in detail. The simulation results and conclusions are given in Section IV and V respectively.
The adopted notations are as follows: diag{x} is a diagonal matrix with x in its main diagonal, and blckdiag{A, B} is a block diagonal matrix with the matrices A and B on its main diagonal. E{⋅} and ⊗ are the expectation and Kronecker product operations respectively. [X]i,j denotes the (i,j)th element of the matrix X. IN and 0N are the N × N identity matrix and the matrix of zeros respectively. (⋅)H, (⋅)T and (⋅)* stand respectively for Hermitian, transpose, and conjugate operations.
2. Signal Model
2.1 MIMO-OFDM System Model
Consider a MIMO-OFDM system with Nt transmit antennas and Nr receive antennas, and the duration of a MIMO-OFDM block is T=NsTs, where Ts is the sampling time and Ns=N+Nc, N and Nc are the number of the subcarriers and the length of the cyclic prefix (CP) respectively. Assume xm is the mth transmitted MIMO-OFDM block, and
where is the mth transmitted OFDM symbol by the tth transmit antenna, and is the power-normalized information symbol sent by the tth transmit antenna at the kth subcarrier. After a multi-path Releigh channel, the mth received MIMO-OFDM block can be written as
where , and is the mth received OFDM symbol by the rth receive antenna. is the NrN×1 white complex Gaussian noise vector of covariance matrix σw2INNr, and Hm is a NrN × NtN MIMO channel matrix, i.e.,
where is the channel matrix between the tth transmit antenna and rth receive antenna. The entries of channel matrix can be expressed as
where L is the number of the channel taps. (n) is the nth sampling of the lth channel tap, which follows Jakes’ power spectrumof the maximum Doppler frequency fd, and (n) is a zero mean complex Gaussian processes of variances .
Define , the correlation matrix of for the time-lag τ' is , and its element is given by
where J0(⋅) is the zero-order Bessel function of the first kind.
2.2 BEM channel model
Since the number of the samples to estimate NtNrLN is greater than the number of observation equations NrN, it is not efficient to estimate the time-varying channel in (2), and it needs to reduce the number of the estimated parameters. Therefore, the channel coefficient (n) is approximated by the BEM with the finite-parameters in the paper, and it is given as
where bn,q is the basis function, and is the corresponding BEM modeling error, Q is the order of the BEM. To simplify notation, we rewrite the BEM in vector as
where [B]n,q = bn,q is a matrix of size N × Q, and
The P-BEM based method is a simple channel modeling method without the accurate knowledge of the Doppler shift, and it is employed in the paper to model the time variation of the channel.
Using BEM in (7) and neglecting the BEM model error, the received OFDM symbol can be rewritten as
where
where fl is the lth column of the N × L Fourier matrix F and Mq is a N × N matrix, which are given by
Using the BEM coefficients, one can easily obtain as
where .
3. Proposed Method
In the proposed method, the iterative algorithm for joint Kalman filter and data detection scheme is given to estimate the time-varying channel. Specifically, the Kalman filter is from the canonical state space model, which does not include the parameters of the AR model, and the detection error is considered as part of the noise in the Kalman filter to improve the estimation accuracy. In the following, the method is derived in detail.
3.1 Kalman Filter from Canonical State Space Model
Since the available schemes need to estimate the parameters of the AR model, it will take a large computation complexity. Moreover, the accuracy of the estimated parameters of the AR model will affect the convergence speed of the iterative scheme. Therefore, the proposed method employs a canonical state space model without the parameters of the AR model in[12], which includes two parts, i.e., (a) the state equation consisting of the BEM coefficients and (b) the observation equation consisting of the pilot/ detected signal and the noise.
Define the state space model for the MIMO-OFDM system as , where ρ is the size of the state vector. And then the state equation will be given as
where S1 is the ρNtNrQL × ρNtNrQL state transition matrix, and it is given as
In (14), , and its dimension is ρNtNrQL × NtNrQL. um is the NtNrQL × 1 vector, and um = cm.
The observation equation can be expressed from (10) as
where .
Using the Kalman filter for the state space model in (14) and (16), one can obtain as
where ĝm|m-1 and ĝm|m are the priori and posteriori state estimate respectively. pm|m-1 and pm|m are the priori and posteriori error estimate covariance matrix of size ρNtNrQL × ρNtNrQL respectively. In addition, ĝ0|0 = 0pLQNr and p0|0 = IρNtNrQL×ρNtNrQL are initiated. In (17), Cum is the covariance matrix of um, and
where
In (18), δ2 = σw2INNr, and Km is the Kalman gain.
3.2 Iterative algorithm for Joint Kalman Filter and Data Detection
To improve the accuracy of the channel estimation, the detected data along with pilots is employed. In one OFDM symbol, assume that Np pilots subcarriers are evenly inserted into the N subcarriers. The algorithm proceeds as follows:
3.3 Iterative Algorithm considering Data Detection Error
In the iterative schemes, the error propagation caused by the data detection cannot be ignored and it will result in an error floor. To reduce the effect of the detection error on the channel estimation, the detection error will be considered as a part of the noise in the Kalman filter of the proposed method. Therefore, the covariance of the detection error needs to be given.
By [18], we divide the received signal in (16) into three parts, i.e., detected data, detection error and white noise, which is shown as
where Ŝm is obtained by the pilots and detected data. ψm = (Sm - Ŝm)gm + wm is the equivalent noise with zero mean. According to the derivation in Appendix, the covariance matrix of ψm is given by
where
where is the variance of the detection error of the signal.
Using the state equation (14) and the observation model in (16), one can employ the Kalman filter to obtain the BEM coefficients, and δ2 = δψm in (18).
In (22), the variances , n =0,..., N - 1 are needed for the calculation of the matrix δψm. Since for the pilot, one only need to calculate the for the data symbol. To simplify the implementation of the Kalman filter, the expected value of is used to obtain the variance of the detection error of the data symbol by [18]. Then, one has
where since is the power-normalized symbol as we considered in section II, and is the detected signal. Assuming that is ergodic, one has
where M is the number of OFDM symbols in one subframe, and Nd is the number of the data subcarriers during one OFDM duration. Then, the simplified estimator will consist in using an approximate covariance matrix of the equivalent noise ψ(m) in (21)
where 1P is an N × 1 vector composed of 0 at the pilot index, and 1 elsewhere.
Note that the type of the modulation affects the performance of the simplified covariance [18]. For the constant modulation (such as QPSK), , so the first term of (24) is equal to one, and the difference between the exact expression in (22) and approximate one in (26) is only the . However, in the case of the non-constant modulation, the approximate covariance in (26) has more performance degradation due to the approximation of and .
3.4 Computation Complexity Analysis
In the section, the complexity of the proposed method and those of the conventional methods in [12][15][18] are compared. For the these algorithms, the main difference of the complexity focuses on the calculations of the matrices S1, Cum, and δ2. In [12], since the matrix S1 is a constant matrix which only contains 0 and 1 and the data detection error is not considered, the computation complexity is focused on the calculation of the matrix Cum, which is about 2Nr2Nt2Np floating-points (FLOPS). For the scheme in [15], since the data detection error is also not considered, the computation complexities caused by S1 and Cum are respectively about L[8NQ2+2N2Q+3Q3+5ο(Q3)] FLOPS and L[Q+2Q3+ο(Q3)] FLOPS, where ο(Q3) is the computation complexity of the matrix inversion for the Q×Q matrix. In [18], the computation complexities of S1 and Cum are L[8NQ2+3N2Q+4Q3+7ο(Q3)] FLOPS, and L[4NQ2+N2Q+Q3+2ο(Q3)] FLOPS respectively, and the complexity of δ2 is about 2N[Q2LN(8N-1)+3] FLOPS for the first iteration and the others is 6N FLOPS. In the proposed method, S1 is a constant matrix and composed of only 0 and 1 and it can be pre-computed and stored, the computation complexity of Cum is about L[4NQ2+N2Q+Q3+2ο(Q3)] FLOPS. Since the detection error is considered in the proposed method, the complexity of its covariance matrix in δ2 is about 2N[Q2LN(8N-1)+3] FLOPS for the first iteration and the others is 6N FLOPS.
According to the above analysis, the comparison of the total computation complexity in one iteration for the different channel estimation methods is given in Table 1. From Table 1, one can see that the scheme in [12] has a lowest complexity due to its processing in frequency domain and it without considering the detection error. The computation complexity of the proposed method is lower than that of the scheme in [15] when the detection error is not considered, and the complexity will increase when the detection error is considered in the proposed method, but its performance will be greatly improved (which can be seen from the simulation results in section IV). Moreover, the increased complexity of the proposed method can be acceptably in the signal processing. Compared with the method in [18], the proposed method has a lower complexity since it does not need to estimate the parameters of the AR model and the covariance of the modeling error.
Table 1.The comparison of the computation complexity
4. Simulation Results
To evaluate the performance of the proposed method, a MIMO-OFDM system with one transmit antenna and two receiver antennas is considered. The parameters of the system are as follows: the FFT size is 128, the CP length is chosen to be one-eighth of the FFT size, and the number of the pilots is 32 and the distance between two adjacent pilots is 4. The carrier frequency is 10GHz. The modulation scheme on the data subcarriers is QPSK. The vehicle speed is 300km/h, and the corresponding fdT=0.2. ρ=2, where ρ is the size of the state vector. In the simulation, we assume Rayleigh uncorrelated fading condition from different transmitters to different receivers, which is considered to be a 6-tap channel with an exponentially decaying power delay profile specified by exp(-l), where l is the tap index. The ZF equalizer is used in the simulation. In comparison with the proposed method, the algorithms in [12], [15] and [18] are also simulated.
To describe the convergence speed of the proposed algorithm, the convergence condition is that the hard decision after the current iteration is the same as that after the previous iteration in the simulation. Fig. 1 shows the average iterative numbers curve vs. SNRs. In Fig. 1, ‘proposed accurate’ is the proposed method with the accurate covariance of the detection error obtained by Eq. (22), and ‘proposed approx’ is the proposed method with the approximate covariance of the detection error obtained by Eq. (26). From Fig. 1, one can see that the iterative number decreases as the SNR increases, and the iterative number of the proposed approx is greater than that of the proposed accurate due to the error covariance approximation. Moreover, the maximum iterative number is less than 5 when the SNR is 0dB for both the proposed accurate and proposed approx, which indicates that the convergence speed of the proposed algorithm is relatively fast.
Fig. 1.Iterative numbers curve vs. SNRs for the proposed method with QPSK, fdT=0.2 and Np =32.
To further analyze the convergence of the proposed method, Fig. 2 gives the mean square error (MSE) vs. iterative numbers in the different SNRs for the proposed method. In Fig. 2, we give the performance of the proposed method using different iterations, e.g., proposed approx-1 uses one iteration, and proposed approx-2 uses two iterations. ‘All Data-aided’ is the BEM-based Kalman filter estimation under the assumption that all the data is known at receiver. It can be seen that the MSE performance of the proposed method improves as the iterations increase, and the performance loss in the proposed approx is more obvious for the lower iterations and gradually decreases as the iterations increase. The MSE performance of the proposed approx using five iterations (proposed approx-5) is essentially coincident with that of the all data aided, and the proposed accurate using three iterations (proposed accurate-3) is essentially coincident with the all data aided. Moreover, when the iterations are greater than 5 and 3, the MSE performance of the proposed accurate and proposed approx are no longer significantly improved respectively, which can be seen proposed accurate-5 and proposed approx-7 in the figure.
Fig. 2.MSE vs. iterative numbers in the different SNRs for the proposed method with QPSK, fdT=0.2 and Np =32.
To show the performance degradation of the proposed method with the approximate covariance matrix, we give the MSE performance of the proposed method with the accurate and approximate covariance of the detection error in Fig. 3. From Fig. 3, one can see that the propose accurate uses three iterations to approach to the performance of the all data aided, while the proposed approx needs five iterations due to the approximate value. Since the approximate value is relatively small, the difference of the iterative numbers between the proposed accurate and proposed approx is not large, which also can be seen from Fig. 1.
Fig. 3.MSE performance of the proposed method with accurate and approximate covariance of the detection error for QPSK, fdT=0.2 and Np =32.
Fig. 4 and Fig. 5 show the MSE and bit error rate (BER) performance for the different channel estimation methods respectively. It can be seen from the figures that the performance of the proposed method is better than those of the schemes in [12], [15] and [18] even when the iterative number is the same or not. The performance of the method in [12] is worst due to without considering the ICI caused by the time-varying channel and iteration processing. Since the data detection error is not considered for the iteration algorithm in [15], it needs more iterations and has a bad performance compared with the proposed method. In [18], the scheme needs to estimate the parameters of the AR model and the covariance of the modeling error, so it also need to more iterations to converge due to the effect of the estimation accuracy of these parameters. Moreover, the performance of the proposed method will be essentially coincident with that of the all data aided only by fewer iterations, such as three iterations for the proposed accurate and five iterations for the proposed approx.
Fig. 4.MSE performance of the different channel estimation methods for QPSK, fdT=0.2 and Np =32.
Fig. 5.BER performance of the system by using the different channel estimation methods for QPSK, fdT=0.2 and Np =32.
Fig. 6 gives the MSE performance of the different channel estimation methods for fdT=0.4 and Np=32. It is shown that in comparison to Fig. 4, the MSE performance of the whole methods are deteriorated due to the Doppler shift increasing. In the case of the same iterations, the performance of the proposed method is still better than those of the available schemes in [12], [15] and [18].
Fig. 6.MSE performance of the different channel estimation methods for QPSK, fdT=0.4, Np =32.
Fig. 7 shows the MSE performance of the different channel estimation methods for Np=16 and fdT=0.2. Compared with Fig. 4, the MSE performance of the whole methods are also deteriorated due to the pilots reducing. By the same iterations, the MSE performance of the proposed method is still better compared with the available schemes in [12], [15] and [18]. For the proposed approx with five iterations, there is a performance gap to the all data aided at the high SNRs, which rarely exists in the case of Np=32 and fdT =0.2 in Fig. 4. Compared with the case of fdT =0.4 and Np=32 in Fig. 6, one can find that the effect of the pilots reducing on the channel estimation is more serious than that of the Doppler shift increasing for the schemes in [12], [15] and [18] due to they without considering the detection error or the estimation accuracy of some parameters depends on the number of the pilots. Due to the detection error and the special Kalman filter employed in the proposed method, its performance in the case of the pilots reducing is better than that in case of the large Doppler shift.
Fig. 7.MSE performance of the different channel estimation methods for QPSK, Np=16, and fdT=0.2.
To show the effect of the modulation type on the approximate proposed algorithm (proposed approx), the case of the non-constant modulation of 16QAM is also considered and it is shown in Fig. 8. Compared with the case of the constant modulation (QPSK) given in Fig. 4, the performance of the proposed approx is deteriorated due to the lower accuracy of the approximation in (26) for the non-constant modulation (16QAM), and it needs more iterations to tend to the all data aided. However, the performance of the proposed approx is still better than those of the available methods in [12], [15] and [18].
Fig. 8.MSE performance of the different channel estimation methods for 16QAM, fdT=0.2, and Np=32.
5. Conclusion
According to the high-speed mobile scenario, we proposed a practical time-varying channel estimation algorithm for the MIMO-OFDM system. Since the Kalman filter from the canonical state space model without the parameter of the AR model is employed and the data detection error is considered as part of the noise in the Kalman recursion, the proposed method is more practical compared with the available schemes. By the proposed method, the fast time-varying channel can be effectively estimated, and the proposed method can be used to estimate the channel in the high speed railway scenario. Moreover, the proposed method is not limited to the MIMO-OFDM system, and it can also be used in the SISO-OFDM or OFDMA system to improve the accuracy of the channel estimation.
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