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Channel estimation in OFDM systems operating under high mobility using Wiener filter combined basis expansion model
EURASIP Journal on Wireless Communications and Networking volume 2012, Article number: 186 (2012)
Abstract
In this article, we first thoroughly analyze Wiener filter combined least squares based channel estimation (WFLS) and then illustrate its limitation in highspeed mobile environments. Based on the analysis, we propose to combine WF with basis expansion model (BEM) based channel estimation to deal with channel estimation in various mobile environments, especially in highspeed cases. The expression for Wiener filter combined BEM based channel estimation (WFBEM) is derived and the result explicitly considers the effect of intercarrier interference (ICI) that occurs in orthogonal frequency division multiplexing (OFDM) systems when operating under high mobility. The simulation results demonstrate that our proposed WFBEM is better than WFLS in timevarying channels, and the performance improvement is significant especially in fast timevarying channels.
1 Introduction
Due to its high data rate transmission capability and its robustness to multipath delay spread, orthogonal frequency division multiplexing (OFDM) has been adopted in most parts of modern wireless communication sytstems, such as wireless local area networks (WLAN) [1], digital audio and video broadcasting [2], and so as OFDM being standardized for the future wireless communication systems, such as wireless metropolitan area networks (WiMAX) [3] and 3GPP long term evolution (LTE) [4]. On the other hand, with highspeed railway construction worldwide, the moving speed of highspeed train has been reported to be able to operate as high as more than 400 km/h, or, even higher onboard highspeed vehicles such as aircraft. Consequently, mobility support is widely regarded as one of the key features in the abovementioned OFDM systems. Meanwhile, for coherent detection in a wireless communication system, channel state information is indispensable. Therefore, to ensure effective communication of OFDM systems operating in high speed scenarios or alternatively very fast timevarying wireless channels, channel estimation method applicable to such highspeed environments has remained largely an open issue.
Although blind channel estimation methods can save bandwidth by avoiding the transmission of any pilot or training symbols, they must exploit the statistic of the received signals, either explicitly or implicitly, which requires the wireless channel is time invariant during several OFDM symbols (e.g., cyclic prefix (CP) [5] or virtual carriers [6] based blind channel estimation method, etc.). Therefore, blind channel estimation methods are only capable of tracking slow channel variations. In fast fading channels, they will suffer severe performance degradation. For channel estimation in wireless communication systems operating under high mobility, pilot symbols (i.e., symbols with known pilot tones) must be inserted periodically into the transmission frame to track fast channel variations (also called pilotassisted transmission). Channel estimation in fading channels with very high mobilities usually consists of two steps. As shown in Figure 1, channel estimation at pilot symbols is the first step in completing channel estimation in communication systems. The next step, we need to perform interpolation between the pilot symbols to obtain the channel estimate of data symbols (i.e., symbols only with unknown data subcarriers), which are transmitted between these pilot symbols. On one hand, traditional training symbols or pilots aided channel estimation methods adopted to obtain the channel estimate of pilot symbols assume that the wireless channel is static during an OFDM symbol period, which means that the channel frequency response matrix is a diagonal matrix [7]. This assumption holds when the users are in indoor environments or is approximately valid in slow speed moving environments. Channel estimation problems in this situation have been well studied and solved in the literature. For example, least squares (LS) or minimum mean square error (MMSE) based channel estimation methods [8] can be used to estimate the diagonal of the channel frequency response matrix, or we can further proceed to eliminate noise in transformdomain [9]. On the other hand, Considering that Wiener filter (WF) is the best interpolation method at the time dimension in terms of minimizing MSE [10], Dong et al. [11] and Zheng and Xiao [12] derive the expression for Wiener filter combined LS based channel estimation (hereafter called WFLS), combining LS based channel estimation with WF. The result is based on the assumption that the channel coefficients within an OFDM symbol period are constants and thus the effect of intercarrier interference (ICI) is completely neglected. Although the proposed WFLS method can deal with slow to moderate fading channels, however, in timevarying channels, especially in fast timevarying cases, the latter assumption may be often violated in practice and the channel coefficients can vary in an OFDM symbol period, and thus ICI occurs, where the offdiagonal elements in the channel frequency response matrix represent ICI [7]. Therefore, the performance of WFLS will degrade significantly or even lose its efficiency in this situation.
Basis expansion model (BEM) based channel estimation method can accurately estimate both slow and fast timevarying coefficients of the wireless channel in an OFDM symbol period using the coefficients of BEM that are far less than the length of wireless channel [13, 14]. Combining BEM based channel estimation with WF, in this article we derive the expression for Wiener filter combined BEM based channel estimation (hereafter called WFBEM), which explicitly considers the effect of ICI. The simulation results show that WFBEM is better than WFLS in timevarying channels, especially in fast timevarying cases.
The organization of this article is as follows. Section 2 describes the OFDM system model. Section 3 gives further analysis of WFLS and then illustrates its limitation in timevarying channels. Section 4 derives the expression for WFBEM based channel estimation. Section 5 gives some simulation results that demonstrate the effectiveness of the proposed algorithm. Finally, conclusions are drawn in the final Section 6.
Notation: Matrices and vectors are denoted by boldface letters. A(m, n) is the (m, n)th entry of the matrix A. x(m) denotes the mth element of the vector x. A hat over a variable indicates an estimate of the variable (e.g.,$\widehat{H}$). E{·} denotes the expectedvalue operator. Superscripts [·]^{T} , [·]^{1}, [·]^{H} , and [·]* denote the transpose, the matrix inversion, the Hermitian and the complex conjugate operations, respectively. I is the identity matrix.
2 System description
Let ${\widehat{\mathbf{P}}}_{k}={\left[{\widehat{\mathbf{H}}}_{{p}_{0}}\left(k\right){\widehat{\mathbf{H}}}_{{p}_{1}}\left(k\right)\phantom{\rule{2.77695pt}{0ex}}.\phantom{\rule{2.77695pt}{0ex}}.\phantom{\rule{2.77695pt}{0ex}}.\phantom{\rule{2.77695pt}{0ex}}{\widehat{\mathbf{H}}}_{{p}_{M1}}\left(k\right)\right]}^{T}$ be a column vector that represents the estimate of channel frequency response at the kth subcarrier of pilot symbols in an OFDM system, k = 0, 1, ..., N_{ p }  1 represents the pilot subcarrier index, which is evenly located in pilot symbols, N_{ p } is the number of pilot subcarriers in one pilot symbol, p_{ i }, i = 0, 1, ..., M  1 represents the position of the p_{ i }th pilot symbol at the time dimension, M represents the pilot symbol numbers (see Figure 1 for illustration), ${\widehat{\mathbf{H}}}_{{p}_{i}}={\left[{\mathit{\u0124}}_{{p}_{i}}^{\left(0\right)}{\mathit{\u0124}}_{{p}_{i}}^{\left(1\right)}\phantom{\rule{2.77695pt}{0ex}}.\phantom{\rule{2.77695pt}{0ex}}.\phantom{\rule{2.77695pt}{0ex}}.\phantom{\rule{2.77695pt}{0ex}}{\mathit{\u0124}}_{{p}_{i}}^{\left({N}_{p}1\right)}\right]}^{T}$, and the pilot symbol insertion rate r at the time dimension satisfies r ≥ 2f_{ D }T, f_{ D }T is the normalized Doppler frequency. ZadoffChu sequence [15] is adopted at the p_{ i }th pilot symbol ${\mathbf{X}}_{{p}_{i}}={\left[{X}_{{p}_{i}}^{\left(0\right)}{X}_{{p}_{i}}^{\left(1\right)}\phantom{\rule{2.77695pt}{0ex}}.\phantom{\rule{2.77695pt}{0ex}}.\phantom{\rule{2.77695pt}{0ex}}.\phantom{\rule{2.77695pt}{0ex}}{X}_{{p}_{i}}^{\left({N}_{p}1\right)}\right]}^{T}$ in this article, so ${\mathbf{X}}_{{p}_{i}}\left(k\right){}^{2}=1$. The channel estimate ${\widehat{\mathbf{H}}}_{d}\left(k\right)$ at the data symbol position d can be estimated by interpolating ${\widehat{\mathbf{P}}}_{k}$. Let the interpolation row vector be C_{ d,k }, and ${\widehat{\mathbf{H}}}_{d}\left(k\right)$ can be obtained by ${\widehat{\mathbf{H}}}_{d}\left(k\right)={{\mathbf{C}}_{d}}_{,k}{\widehat{\mathbf{P}}}_{k}$. By minimizing the estimation MSE ${\epsilon}_{d}=E\left\{{\mathbf{H}}_{d}\left(k\right){\widehat{\mathbf{H}}}_{d}\left(k\right){}^{2}\right\}$, namely,
the optimal interpolation vector, i.e., the WF can be obtained as [10]
Dong et al. [11] and Zheng and Xiao [12] uses LS based channel estimation to obtain ${\widehat{\mathbf{P}}}_{k}^{\mathsf{\text{LS}}}$ and then derive the interpolation vector of WFLS ${\widehat{\mathbf{C}}}_{d,k}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}$, which completely neglects the effect of ICI. Therefore, the expression for WFLS is expressed as ${\widehat{\mathbf{H}}}_{d}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(k\right)={\widehat{\mathbf{C}}}_{d,k}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}{\widehat{\mathbf{P}}}_{k}^{\mathsf{\text{LS}}}$.
In this article, Channel and noise are assumed to be independent from each other and noise is complex Gaussian distribution with zero mean and σ^{2} variance.
3 Further analysis of WFLS based channel estimation
Least squares based channel estimate is given by
where ${\mathbf{W}}_{{p}_{i}}={\left[{W}_{{p}_{i}}^{\left(0\right)}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{W}_{{p}_{i}}^{\left(1\right)}\phantom{\rule{2.77695pt}{0ex}}...\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{W}_{{p}_{i}}^{\left({N}_{p}1\right)}\right]}^{T}$ denotes additive complex Gaussian noise in the frequency domain. Substituting (3) into (2) and letting $M=2,{\hat{\mathbf{C}}}_{d,k\left(2O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}$ (with the subscript m  O representing WFLS with orders m) can be expressed as
where D^{2O}= B^{2}(0)B(p_{0}  p_{1})B(p_{1} p_{0}), D^{mO}represents the corresponding determinant of WFLS with orders m, and
is the autocorrelation function of the channel frequency response at the kth subcarrier of different OFDM symbols [16], where J_{0}(·) is the zeroth order Bessel function of the first kind and Γ(·) is the gamma function. Meanwhile, the coefficients of linear interpolation is given by
In the following, we will first show that linear interpolation is in fact a special case of WFLS with orders 2, or put it another way, WFLS with orders 2 will "degrade" into linear interpolation asymptotically. Then, we will analyze the asymptotical relationships among WFLS with different orders. Finally we will point out the limitation of WFLS. Let l = 1 in (5) and substitute the resulting (5) into (4a) and (4b), after some manipulations we can obtain the expanded expression for ${\widehat{\mathbf{C}}}_{d,k\left(2O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(i\right)$, i = 0, 1 as
Notice that (7a) and (7b) hold true when the wireless channel is static or in very slow moving environments. When SNR → ∞, that is, when σ^{2} → 0 (i.e., in the absence of noise), and inserting (6a) and (6b) into (7a) and (7b), respectively, then the relationship between the coefficients of WFLS with orders 2 and that of linear interpolation can be obtained as
From (8) it is observed that as f_{ D }T → 0, the desired result
can be established, as shown in Figure 2. It is worth noting that a totally different looking Figure 2 will be obtained if we choose another set of p_{0}, p_{1} and p, but the asymptotical behavior as f_{ D }T → 0 will definitely be the same, as proved and explained in (8). Therefore, it can be inferred that WFLS with orders 2 implicitly exists certain linearity which results in its failure in high speed communication environments, in which case the approximate linearity of the channel frequency response ${\widehat{\mathbf{H}}}_{{p}_{0}}\left(k\right)$ and ${\widehat{\mathbf{H}}}_{{p}_{1}}\left(k\right)$ between p_{0} and p_{1} disappears because of fast fading channels. On the other hand, because the channel estimation at pilot symbols is fixed to LS based channel estimation, if we want to improve the performance of WFLS, one has to resort to WFLS with higher orders (i.e., choosing M > 2 in (2)) and hopefully that by adopting WFLS with higher orders we can improve the estimation accuracy through using more pilot symbols, i.e., the MSE of channel estimation may be decreased by collecting more pilot symbols [17]. However, without properly adopting WFLS with higher orders, there will be no performance improvement. In what follows we will investigate the asymptotical behavior of WFLS with higher orders and show that WFLS with higher orders can indeed be "degraded" into WFLS with orders 2 if not carefully designed.
Substituting (3) and (5) into (2) and letting M = 3 we can obtain ${\widehat{\mathbf{C}}}_{d,k\left(3O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(0\right)$,${\widehat{\mathbf{C}}}_{d,k\left(3O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(1\right)$ and ${\widehat{\mathbf{C}}}_{d,k\left(3O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(2\right)$. Then, substituting (4a) and (4b) into the resulting ${\widehat{\mathbf{C}}}_{d,k\left(3O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(0\right)$ and ${\widehat{\mathbf{C}}}_{d,k\left(3O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(1\right)$, respectively, after a few tedious, but otherwise straightforward, algebraic manipulations, we can obtain the relationships between D^{3O}and D^{2O}, ${\widehat{\mathbf{C}}}_{d,k\left(3O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(0\right)$ and ${\widehat{\mathbf{C}}}_{d,k\left(2O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(0\right)$, ${\widehat{\mathbf{C}}}_{d,k\left(3O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(1\right)$ and ${\widehat{\mathbf{C}}}_{d,k\left(2O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(1\right)$, and finally ${\widehat{\mathbf{C}}}_{d,k\left(3O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(2\right)$ as
It is observed from (10) that as σ^{2} → 0 and by fixing p, p_{0} and p_{1} (it is assumed here that we already have or know p, p_{0} and p_{1} for WFLS with orders 2), what really matters are f_{ D }T and the distance between p_{1} and p_{2}. By using (10) and putting different values of f_{ D }T and the distance between p_{1} and p_{2}, we will obtain some interesting results as shown in Figure 3. It can be seen from Figure 3 that as p_{2} increases, ${\widehat{\mathbf{C}}}_{d,k\left(3O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(0\right)$ and ${\widehat{\mathbf{C}}}_{d,k\left(2O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(0\right)$ will converge, and so will be ${\widehat{\mathbf{C}}}_{d,k\left(3O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(1\right)$ and ${\widehat{\mathbf{C}}}_{d,k\left(2O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(1\right)$. Meanwhile, ${\widehat{\mathbf{C}}}_{d,k\left(3O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(2\right)$ will converge to zero as p_{2} increases. On the other hand, this convergence will change with f_{ D }T , as can also be observed from Figure 3. In this case, WFLS with orders 3 will "degrade" into WFLS with orders 2, which means that even more pilot symbols are adopted to estimate ${\widehat{\mathbf{H}}}_{d}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(k\right)$, there will be no performance gains at all. Also note here that as in Figure 2, if we choose another set of p_{0}, p_{1}, p_{2} and p we will obtain a different looking Figure 3, but the asymptotical convergence behavior will be the same. This asymptotical convergence behavior can be explained by investigating (10), which shows that B(Δ_{ p }) will become small and close to zero as f_{ D }T (p_{2}  p_{ i }), i = 0, 1 grows large and tends to infinity. In conclusion we have the following results in the absence of noise
When noise is present, the convergence behavior can be obtained through similar analysis by investigating (10) and the result is shown in Figure 4. This idea of analyzing the relationship between WFLS with orders 3 and WFLS with orders 2 can be further extended to analyze the relationships among WFLS with different orders, which can be inferred directly from the above analysis that when p_{ m } is sufficiently large (for m > 2)
Therefore we conclude that we should carefully avoid these "convergence points" when we are trying to improve the performance of WFLS with low orders by using WFLS with higher orders, otherwise this goodwill will prove to be in vain. Of course, we can choose p_{ m } arbitrarily close to p_{m 1}, but this will cause transmitting more pilot symbols, which will significantly reduce the spectral efficiency. However, we can not choose p_{ m } arbitrarily large to increase the spectral efficiency, as have been proved that this will result in no performance improvement. Meanwhile the choose of p_{ m } is related to f_{ D }T. For example, we observe that when SNR = 30 dB and f_{D}T = 0.01, in order for ${\widehat{\mathbf{C}}}_{d,k\left(3O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(0\right)$, ${\widehat{\mathbf{C}}}_{d,k\left(3O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(1\right)$ and ${\widehat{\mathbf{C}}}_{d,k\left(3O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(2\right)$ to have a negligible difference between ${\widehat{\mathbf{C}}}_{d,k\left(2O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(0\right)$, ${\widehat{\mathbf{C}}}_{d,k\left(2O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(1\right)$ and 0, respectively, p_{2} should be bigger than about 54 and 56. When SNR = 30 dB and f_{ D }T = 0.1, for ${\widehat{\mathbf{C}}}_{d,k\left(3O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(0\right)$, ${\widehat{\mathbf{C}}}_{d,k\left(3O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(1\right)$ and ${\widehat{\mathbf{C}}}_{d,k\left(3O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(2\right)$ to have a negligible difference between ${\widehat{\mathbf{C}}}_{d,k\left(2O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(0\right)$, ${\widehat{\mathbf{C}}}_{d,k\left(2O\right)}^{\mathsf{\text{WF}}\mathsf{\text{LS}}}\left(1\right)$ and 0, respectively, p_{2} should be bigger than 9. The above observations are also approximately true for other SNR values. Therefore, from the above observations and analysis we may conclude that the combined parameter f_{ D }T (p_{ m }  p_{m 1}) should be smaller than about 0.5 to avoid the degradation of WF with higher orders. This result is coincide with the sampling theorem in the time domain [18], which is, f_{ D }T (p_{ m }  p_{m 1}) < 0.5. Till this point we can say that we have just proved the sampling theorem in the time domain, from a new perspective, i.e., from the coefficients of WFLS point of view. The optimal value of f_{ D }T (p_{ m }  p_{m 1}) that will simultaneously increase the performance while maintaining the spectral efficiency is dependent on the specific problem at hand and related to the pilot symbol and system design problems, which is beyond the scope of this article, but will be an interesting and a meaningful topic for the future work.
Although the performance of WFLS can be improved by adopting more pilot symbols provided that the above conditions are satisfied, WFLS has an intrinsic weakness that will lead to its decreased performance in high speed mobile environments. As will be seen in Section 5, the performance of WFLS is acceptable in slow to moderate mobile environments, but decreases significantly in high speed environments. This degradation of WFLS is due mainly to the fatal weakness of the underlining assumption of LS based channel estimation that the channel coefficients within one OFDM symbol period are constants. This assumption will generally not hold true in timevarying channels, especially in fast timevarying cases. In light of the above analysis and considering that BEM based channel estimation method [13, 14] can accurately estimate the timevarying channel coefficients of an OFDM symbol, we propose to combine BEM based channel estimation with WF to deal with channel estimation in timevarying channels.
4 The expression for the proposed WFBEM based channel estimation
Basis expansion model based channel impulse response of the p_{ i }th pilot symbol can be expressed as [13, 14]
where ${h}_{{p}_{i}}\left(n,l\right)$ represents the channel impulse response of the lth path at time n within the p_{ i }th pilot symbol period and is assumed to be modeled as a wide sense stationary (WSS) complex Gaussian process with a statistically independent path, n = 0, 1, ..., N  1, N is the symbol length (N_{ p } ≤ N), l = 0, 1, ..., L  1, L is the length of wireless channel, η_{q,l}(p_{ i }) is the coefficient of BEM, b_{n,q}is the base that captures channel time variations, and Q is the number of BEM bases. BEM is motivated by the observation that the temporal (n) variation of h(n, l) is usually rather smooth due to the channel's limited Doppler spread, and therefore, ${\left\{{b}_{n,q}\right\}}_{q=0}^{Q}$ can be chosen as a small set (i.e., Q ≪ N) of smooth functions.
For the pilot subcarriers in the p_{ i }th pilot symbol, an equation consisting of η_{q,l}(p_{ i }) can be expressed as [14]
where ${\mathbf{Y}}_{{p}_{i}}={\left[{Y}_{{p}_{i}}^{\left(0\right)}{Y}_{{p}_{i}}^{\left(1\right)}\dots {Y}_{{p}_{i}}^{\left({N}_{p}1\right)}\right]}^{T}$ is the received signal at the pilot subcarriers of the p_{ i }th pilot symbol in the frequency domain, ${\mathit{\eta}}_{{p}_{i}}={\left[{\eta}_{0,0}\left({p}_{i}\right),\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}{\eta}_{0,L1}\left({p}_{i}\right),\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}{\eta}_{Q,0}\left({p}_{i}\right),\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}{\eta}_{Q,L1}\left({p}_{i}\right)\right]}^{T}$ represents the coefficients of BEM to be estimated, P consists of the base, the pilot subcarriers and the Fourier transform matrix at the positions of according pilot subcarriers [14, Eq. (14)], and ${\mathbf{Z}}_{{p}_{i}}$ is the interference from data subcarriers and noise. In this article, the regularized LS method [19] is adopted to estimate ${\mathit{\eta}}_{{p}_{i}}$ as
where small value of a is to ensure the matrix P^{H}P + a I a matrix with full rank. Let us define a matrix A ≜ (P^{H}P + a I)^{1}P^{H} in (15) and A can be further divided into Q + 1 submatrices, each of which is of size L × N_{ p }. More specifically, A can be expressed as
Combing (15) and (16), ${\widehat{\eta}}_{q,l}\left({p}_{i}\right)$ can be expressed as
As shown in [20], in various mobile environments, the received signal ${\mathbf{Y}}_{{p}_{i}}\left(k\right)$ at the kth subcarrier is expressed as
where ${\mathbf{W}}_{{p}_{i}}\left(k\right)$ is an additive complex Gaussian noise in the frequency domain,
represents the channel frequency response corresponding to the desired subcarrier k and
is the ICI coefficient, which is the the offdiagonal element in the channel frequency response matrix. Since the ICI term consists of a large number of random interferences and is based on the assumption that ${h}_{{p}_{i}}\left(n,l\right)$ is WSS complex Gaussian process, we can model the term as additive white Gaussian noise according to the central limit theorem [21].
To derive the expression for WFBEM that considers the effect of ICI, the following steps are proposed:

Step (1) Substitute (17) into (13). We can get the estimate of BEM based channel impulse response as
$${\u0125}_{{p}_{i}}\left(n,\phantom{\rule{2.77695pt}{0ex}}l\right)=\sum _{q=0}^{Q}\sum _{k=0}^{{N}_{p}1}{\mathbf{A}}_{q}\left(l,\phantom{\rule{2.77695pt}{0ex}}k\right){\mathbf{Y}}_{{p}_{i}}\left(k\right){b}_{n,q},$$(21) 
Step (2) Substitute (21) into (19). Then we can obtain the estimate of BEM based channel estimation ${\widehat{\mathbf{P}}}_{k}^{\mathsf{\text{BEM}}}$ with each element given by
$${\widehat{\mathbf{H}}}_{{p}_{i}}^{\mathsf{\text{BEM}}}\left(k\right)=\frac{1}{N}\sum _{l=0}^{L1}\sum _{n=0}^{N1}\sum _{q=0}^{Q}\sum _{s=0}^{{N}_{p}1}{\mathbf{A}}_{q}\left(l,\phantom{\rule{2.77695pt}{0ex}}s\right){\mathbf{Y}}_{{p}_{i}}\left(s\right){b}_{n,q}{e}^{\frac{j2\pi kl}{N}}.$$(22) 
Step (3) Substitute (18) into (22). Hence, the relationships among ${\widehat{\mathbf{H}}}_{{p}_{i}}^{\mathsf{\text{BEM}}}\left(k\right)$, ${\mathbf{H}}_{{p}_{i}}\left(k\right)$ and the ICI term can be established. Therefore, $E\left\{{\mathbf{H}}_{d}\left(k\right){\left({\widehat{\mathbf{H}}}_{{p}_{i}}^{\mathsf{\text{BEM}}}\left(k\right)\right)}^{*}\right\}$ can be derived as
$$\begin{array}{c}E\left\{{\mathbf{H}}_{d}\left(k\right){\left({\widehat{\mathbf{H}}}_{{p}_{i}}^{\mathsf{\text{BEM}}}\left(k\right)\right)}^{*}\right\}={J}_{0}\left(2\pi \left(d{p}_{i}\right){f}_{D}T\right)\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\underset{{\beta}_{{p}_{i}}\left(k\right)}{\underset{\u23df}{\times \frac{1}{N}\sum _{l=0}^{L1}\sum _{n=0}^{N1}\sum _{q=0}^{Q}\sum _{s=0}^{{N}_{p}1}\sum _{u=0}^{L1}{\mathbf{A}}_{q}^{*}\left(l,s\right){\mathbf{X}}_{{p}_{i}}^{*}\left(s\right){b}_{n,q}^{*}{\sigma}_{u}^{2}{e}^{\frac{j2\pi \left(su+k\left(lu\right)\right)}{N}}}}.\hfill \end{array}$$(23) 
Step (4) Use (20) to derive the correlation function of ${\mathbf{H}}_{{p}_{i}}^{\mathsf{\text{IC}}1}\left(k,\phantom{\rule{2.77695pt}{0ex}}v\right)$ and ${\mathbf{H}}_{{p}_{i}}^{\mathsf{\text{IC}}1}\left(k,\phantom{\rule{2.77695pt}{0ex}}z\right)$ as
$$\begin{array}{c}E\left\{{\mathbf{H}}_{{p}_{i}}^{\mathsf{\text{IC}}1}\left(k,\phantom{\rule{2.77695pt}{0ex}}v\right){\left({\mathbf{H}}_{{p}_{i}}^{\mathsf{\text{IC}}1}\left(k,\phantom{\rule{2.77695pt}{0ex}}z\right)\right)}^{*}\right\}=\frac{1}{{N}^{2}}\sum _{{n}_{1}=0}^{N1}\sum _{{n}_{2}=0}^{N1}\sum _{l=0}^{L1}{\sigma}_{l}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\times {J}_{0}\left(2\pi \left({n}_{1}{n}_{2}\right){f}_{D}{T}_{s}\right){e}^{\frac{j2\pi \left(\left(vz\right)l+\left(\left(kv\right){n}_{1}\left(kz\right){n}_{2}\right)\right)}{N}}.\hfill \end{array}$$(24)
Through the proposed Steps (1)(4) and by using (22) and (24), $E\left\{{\widehat{\mathbf{H}}}_{{p}_{i}}^{\mathsf{\text{BEM}}}\left(k\right){\left({\widehat{\mathbf{H}}}_{{p}_{j}}^{\mathsf{\text{BEM}}}\left(k\right)\right)}^{*}\right\}$ can be derived as follows:
if p_{ i } ≠ p_{ j },
and if p_{ i } = p_{ j },
where we have used the fact that $E\left\{{h}_{{p}_{i}}\left({n}_{1},l\right){h}_{{p}_{j}}^{*}\left({n}_{2},l\right)\right\}={\sigma}_{l}^{2}{J}_{0}\left(2\pi {f}_{D}\left(\left({p}_{i}{p}_{j}\right)T+\left({n}_{1}{n}_{2}\right){T}_{s}\right)\right)$ and its corresponding function in the frequency domain, ${\sigma}_{l}^{2}$ is the power of the lth path and without loss of generality we assume ${\sum}_{l=0}^{L1}{\sigma}_{l}^{2}=1,$ T_{ s } is the sampling period. It is noted that based on the assumption that ${h}_{{p}_{i}}\left(n,l\right)$ within a single OFDM symbol period are constants, the effect of ICI is neglected (i.e., in (20) ${\mathbf{H}}_{{p}_{i}}^{\mathsf{\text{ICI}}}\left(k,z\right)\equiv \phantom{\rule{2.77695pt}{0ex}}0$), the authors Dong et al. [11] and Zheng and Xiao [12] only consider the noise variance σ^{2}. However, we consider the variation of ${h}_{{p}_{i}}\left(n,l\right)$ within a single OFDM symbol period in timevarying channels and our derived result explicitly considers both the noise variance and the effect of ICI, which can provide us with details about channel variations in various mobile environments.

Step (5) Substitute (23), (25) and (26) into (2). Then, the interpolation vector of WFBEM can be obtained as
$${\widehat{\mathbf{C}}}_{d,k}^{\mathsf{\text{WFBEM}}}=\left[{J}_{0}\left(2\pi \left(d{p}_{0}\right){f}_{D}T\right){\beta}_{{p}_{0}}\left(k\right){J}_{0}\left(2\pi \left(d{p}_{1}\right){f}_{D}T\right){\beta}_{{p}_{1}}\left(k\right)\dots {J}_{0}\left(2\pi \left(d{{p}_{M}}_{1}\right){f}_{D}T\right){\beta}_{{p}_{M1}}\left(k\right)\right].$$$$\left[\begin{array}{cccc}\hfill {\phi}_{{p}_{0}}\hfill & \hfill {J}_{0}\left(2\pi \left({p}_{0}{p}_{1}\right){f}_{D}T\right){\varphi}_{{p}_{0},{p}_{1}}\hfill & \hfill \dots \hfill & \hfill {J}_{0}\left(2\pi \left({p}_{0}{p}_{M1}\right){f}_{D}T\right){\varphi}_{{p}_{0},{{p}_{M}}_{1}}\hfill \\ \hfill {J}_{0}\left(2\pi \left({p}_{1}{p}_{0}\right){f}_{D}T\right){\varphi}_{{p}_{1},{p}_{0}}\hfill & \hfill {\phi}_{p1}\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {J}_{0}\left(2\pi \left({p}_{1}{p}_{M1}\right){f}_{D}T\right){\varphi}_{{p}_{1},{{p}_{M}}_{1}}\hfill \\ \hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ \hfill {J}_{0}\left(2\pi \left({{p}_{M}}_{1}{p}_{0}\right){f}_{D}T\right){\varphi}_{{p}_{M1,}{p}_{0}}\hfill & \hfill {J}_{0}\left(2\pi \left({{p}_{M}}_{1}{p}_{1}\right){f}_{D}T\right){\varphi}_{{p}_{M1},{p}_{1}}\hfill & \hfill \dots \hfill & \hfill {\phi}_{{p}_{M1}}\hfill \end{array}\right]$$(27)
Comparing (27) with the interpolation vector of WFLS ${\widehat{\mathbf{C}}}_{d,k}^{\mathsf{\text{WFLS}}}$given in [11, Eq. (2931)] and [12, Eq. (29)], it can be seen that the coefficients ${\beta}_{{p}_{i}}\left(k\right)$, ${\varphi}_{{p}_{i},{p}_{c}}$ and ${\phi}_{{p}_{i}}$ are related with channel variations (for both the base b_{n,q}and the effect of ICI) and unique to WFBEM.

Step (6) Multiply the estimate of BEM based channel estimation ${\widehat{\mathbf{P}}}_{k}^{\mathsf{\text{BEM}}}$ obtained in Step (2) by the interpolation vector of WFBEM ${\widehat{\mathbf{C}}}_{d,k}^{\mathsf{\text{WFBEM}}}$ obtained in Step (5). Finally, the expression for WFBEM is given by
$${\widehat{\mathbf{H}}}_{d}^{\mathsf{\text{WF}}\mathsf{\text{BEM}}}\left(k\right)={\widehat{\mathbf{C}}}_{d,k}^{\mathsf{\text{WF}}\mathsf{\text{BEM}}}{\widehat{\mathbf{P}}}_{k}^{\mathsf{\text{BEM}}}.$$(28)
The proposed algorithm is summarized as follows. First, we use expression (22) to obtain the channel estimate of pilot symbols (corresponding to the Step 1 in Figure 1); then, we use expression (28) to obtain the channel estimate of data symbols (corresponding to the Step 2 in Figure 1).
In this section the interpolation vector of WFBEM at the frequency dimension is not considered for the reason that by using BEM based channel estimation, the channel impulse response at a pilot symbol can be obtained by expression (21). Then, through transferring the channel impulse response into frequency domain by expression (22), the estimate of channel frequency response at required subcarriers of the pilot symbol can be obtained.
5 Simulation results and discussions
To evaluate the performance of the proposed WFBEM algorithm, extensive computer simulations are carried out. LTE uplink system [15, 22] is considered in the simulation. The simulation parameters are listed in Table 1 and the generalized complex exponential BEM (GCEBEM) [14] is adopted.
5.1 MSE comparison
The MSE curve of BEM based channel estimation with LS based channel estimation only at pilot symbols is shown in Figure 5. From Figure 5 it can be seen that in timevarying channels, BEM based channel estimation performs consistently better than LS based channel estimation. The estimation accuracy of BEM based channel estimation is much better than that of LS based channel estimation especially in highspeed mobile environment, as can be seen from Figure 5. This result proves our analysis in Section 3 that in timevarying channels, as the assumption that the channel coefficients within one OFDM symbol period are constants no longer holds true, BEM based channel estimation will outperform LS based channel estimation. After interpolation along the time dimension by using WF, the MSE curve of WFBEM with WFLS is shown in Figure 6. It is seen from Figure 6 that due to the precondition that BEM based channel estimation is better than LS based channel estimation in timevarying channels, the estimation accuracy of the proposed WFBEM is better than WFLS.
As stated in Section 1, the channel estimation accuracy at data symbols is determined by both the channel estimation accuracy at pilot symbols and the accuracy of the interpolation method. WF is the best interpolation method at the time dimension to estimate channel at data symbols in terms of minimizing MSE. The optimality of WF is independent of the channel estimation method adopted at pilot symbols. Therefore we can infer that the reason that WFLS underperforms is because the channel estimation accuracy of LS based channel estimation is worse than that of BEM based channel estimation at pilot symbols in timevarying channels, which have been proved by Figure 5. This is also the main reason for our motivation to combine BEM based channel estimation with WF and derive the expression for WFBEM based channel estimation to deal with channel estimation in timevarying channels.
Besides BEM based channel estimation can more accurately estimate fast timevarying channels than LS based channel estimation does, another reason that a channel estimator with higher accuracy results in improved performance in timevarying channels is illustrated as follows. In OFDM systems, Doppler effects and instabilities of the transmitter and receiver carrier frequency oscillators will cause a loss of orthogonality between the subcarriers, resulting in ICI. The carrier frequency offset caused by Doppler effects in timevarying channels is a fraction of the subcarrier spacing (i.e., in most practical cases the normalized Doppler frequency is less than 0.2). If we want to compensate this effect, we are dealing with the problem of finefrequency adjustment. One effect of carrier frequency offset is the detrimental effect of a rotation of the subcarriers [23, 24]. This effect will be recognized by a channel estimator, which does not distinguish between phase offsets caused by the channel and those caused by a frequency offset. Thus, a channel equalizer appears also to have fine frequency synchronization capabilities. Therefore it can be inferred that a channel estimator with higher accuracy can more accurately compensate this rotation caused by Doppler effects and thus will perform better in timevarying channels.
5.2 BLER comparison
Figures 7, 8, 9 and 10 report the block error rate (BLER) curve of WFBEM with WFLS, at the mobile speed of 30 km/h (corresponding to the downtown environment), 120 km/h (corresponding to the highway environment), 350 and 480 km/h (corresponding to the highspeed train environments) respectively. Figure 7 illustrates that since the wireless channel can be regarded as nearly invariant during one OFDM symbol period at 30 km/h and also because the Doppler frequency is small in this situation, the performance of WFLS is close to that of WFBEM, there is less than 1 dB SNR gain for WFBEM in this situation. The actual values of these estimates (i.e., ideal channel) are also given to facilitate comparison with the estimates. We can see that the proposed WFBEM is very close to the ideal channel case, there is only less than 0.5 dB loss compared to the ideal one.
Figure 8 illustrates that since the wireless channel changes not that fast enough during an OFDM symbol period at 120 km/h compared with that at 350 and 480 km/h, the performance gap between WFBEM and WFLS is still not that obvious, there is only about 1 dB SNR gain for WFBEM compared with WFLS in this situation (for both 16 and 64QAM). And still, there is only less than 1 dB gap between the proposed WFBEM and the ideal channel case.
However, in highspeed environments (350 and 480 km/h), the wireless channel changes quickly and drastically. Hence, in these cases the wireless channel can not be regarded as nearly static and even slow changing during one OFDM symbol period anymore. Therefore, at mobile speed of 350 km/h, compared with WFLS, about 2.5 and 3.5 dB SNR gain can be obtained in 16 and 64QAM modulation mode respectively through WFBEM, as can be seen from Figure 9. It is observed from Figure 9 that even at 350 km/h, compared to the ideal channel case, there is only about 1.5 dB loss of the proposed WFBEM in 16QAM modulation mode and about less than 3 dB loss in 64QAM mode. From Figure 9 we can infer that the use of efficient multilevel modulations, which make use of multiple signal phase and amplitude levels to carry multiple bits per symbol, may demand more precise channel estimation to demodulate the received signal, especially in highspeed environments.
Finally, the performance of WFBEM with WFLS at extremely high speed 480 km/h is presented in Figure 10. As can be observed from Figure 10, the performance of both WFBEM and WFLS rapidly degrade for an increasing speed. However, our proposed WFBEM is still much better than WFLS in 16QAM modulation mode, there is more than 3 dB SNR gain. There are two reasons for this degradation. Firstly, the wireless channel is changing so fast that by merely interpolating to obtain the channel estimate of data symbols becomes difficult and inaccurate. In this case we may need to insert pilot tones in data symbols (e.g., combtype pilot pattern) to actually estimate the channel at all OFDM symbols. Secondly, the ICI is so evident that in this case it can not be just "equalized out" as explained above. In this case, we may have to turn to ICIcancelation schemes (e.g., self ICI cancelation [25], the use of windowing [26] or the use of pilot tones [27, 28]) to mitigate or suppress the ICI. Nevertheless, to change the pilot pattern at the transmitter or to implement ICIcancelation scheme at the receiver will necessitate altering the transceiver's structure, increasing the computational complexity significantly, and moreover, the use of more pilot tones will inevitably decrease the capacity and/or throughput of the overall system. Hence, considering that the sensitivity of MQAMOFDM signals to Doppler frequency increases significantly with the alphabet size M, we recommend that in order to strike a compromise between complexity and performance as well as data transmission rate, low level modulation such as 8PSK, QPSK or BPSK modulation schemes should be adopted at 480 km/h. When operating under (including) 350 km/h the use of high level modulation 16 or 64QAM can be adopted to increase data transmission rate while at the same time maintain good performance.
5.3 ICI analysis
In Figure 11, we show the effect of ICI that is included in the derived result (see expression (26)). It is observed from Figure 11 that when the Normalized Doppler frequency is small, the effect of ICI is also small, which may indeed be ignored just as the authors of [11, 12] do. However, as the moving speed increases, the Normalized Doppler frequency will increase as well, and thus the effect of ICI can not be ignored anymore in this situation. In conclusion, the effect of ICI and the noise variance will contribute together to influence the performance of OFDM systems in timevarying channels, especially in highspeed cases.
6 Conclusions
In this article, we have further analyzed WFLS, including the asymptotical relationship between WFLS with orders 2 and linear interpolation, the asymptotical relationships among WFLS with different orders. Considering the limitation of WFLS is that LS based channel estimation assumes the channel coefficients within one OFDM symbol period are constants, we propose to combine BEM based channel estimation, which can accurately estimate the timevarying channel coefficients during one OFDM symbol period, with WF to deal with the challenging problem of channel estimation in OFDM systems operating under various mobile environments, especially in highspeed cases. The expression for WFBEM based channel estimation has been derived, and under the hypothesis of approximating the ICI term as additive white Gaussian noise, the derived result explicitly considers the effect of ICI. The simulation results showed that the proposed WFBEM provides a substantial gain, in terms of estimation accuracy in timevarying channels, with respect to WFLS, especially in fast timevarying channels.
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Acknowledgements
This work was supported in part by the Chinese Important National Science & Technology Specific Projects under Grant 2011ZX0300100701 and the National Science Foundation of China under Grant nos. 61032002, 60972029 and 60902026.
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Keywords
 Orthogonal Frequency Division Multiplex
 Channel Estimation
 Orthogonal Frequency Division Multiplex System
 Orthogonal Frequency Division Multiplex Symbol
 Pilot Symbol