1 Introduction

Orthogonal frequency division multiplexing (OFDM) has many well-known advantages such as robustness in frequency selective fading channels, high bandwidth efficiency, and so on. For coherent OFDM systems, a robust channel estimator is required to be critical in order to improve the performance of systems. By now, various OFDM channel estimation schemes have been proposed in the literature.

In general, the LS estimator is the simplest channel estimation [4]. This algorithm has lower complexity. However, it has larger mean-square error (MSE) and is easily influenced by noise and inter-carrier interference. The minimum mean-square error MMSE [17] estimator is a well-known channel estimator. However, the statistical characteristics of a channel, i.e., the autocorrelation matrix of channel frequency response and signal-to-noise ratio (SNR), must be obtained in advance. Usually, this is impossible because of wide-varying channel conditions. So, high computation complexity is a serious disadvantage. By partitioning off channel covariance matrix into some small matrices on the basis of coherent bandwidth, Noh et al. [14] proposed a low-complexity LMMSE estimation method. However, the modified LMMSE methods still have high computational complexity. An alternative is to use recursive least-square (RLS) method to track the channel [21]. Although RLS algorithm is quite effective, its computational complexity is also very high.

Another popular estimator is the DFT-based estimator. Based on the fact that a channel impulse response (CIR) is usually not longer than the guard interval in the OFDM system and most of the channel energy is contained in a small number of samples, Dowler et al. [3] showed the performance of various channel estimation methods and concluded that the DFT-based estimator can achieve significant performance benefits if the maximum channel delay is known. Minn et al. [15] improved this idea by considering only the most significant channel taps. Similarly, Athaudage et al. [1] proposed a delay spread estimation method using cyclic prefix (CP) that can be useful to improve the DFT-based estimation. You et al. [11] proposed a channel estimation method based on a time-domain threshold which is the standard deviation of noise obtained by wavelet decomposition. The threshold value becomes a crucial factor, so it is required to find an optimal threshold value to obtain the desired results. Based on discriminant analysis, Fan el al. [6] presented dynamic time-domain threshold for channel estimation. These techniques work well when multipath delays are integer multiples of the sampling time. However, this hardly happens in practical transmission environment. As a result, performance of the DFT-based algorithm may degrade considerably. In the OFDM systems, inter-carrier interference (ICI) is one of the main problems that should also be considered in OFDM system. However, identifying channel taps is the concerns of this paper, so ICI problem is beyond the scope of this paper.

By suppressing time-domain noise, conventional DFT-based channel estimations improve the performance. However, in a multipath channel with non-sample-spaced time delays, this approach will result in an error floor. If there is some non-sample-spaced path delay, the equivalent discrete CIR will be dispersive in time domain. Yang et al. [20] have proposed an inter-path interference cancellation (IPIC) delay locked loop (DLL) based method to track the channel multipath time delays. It is shown that it can track the multipath time delays quite accurately. But the complexity of ESPRIT algorithm is very high. Usually, the CIR is composed of noise and channel taps. How to distinguish those taps can be regarded as a problem of discriminant analysis. Based on the property of CIR and Mahalanobis distance [9], we design a discrimination function to detect the significant channel taps. Simulation results show that our proposed channel estimators can eliminate the error floor substantially and perform quite close to the situation with LMMSE. Hence, our channel estimators can be used in the OFDM modulation to achieve high-power and frequency-efficient transmission for high-speed wireless communications.

The rest of the paper is organized as follows. In Sect. 2, we briefly go over the OFDM system model and multipath delay channel model. In Sect. 3, we provide a review of conventional DFT-based channel estimation algorithms. In Sect. 4, we introduce simply the discriminant analysis and propose a new method. Simulation results of bit error rate (BER) are presented in Sect. 5. The final conclusions are given in Sect. 6.

2 System Description and Channel Model

2.1 OFDM System Model

We consider an OFDM system that consists of N subcarriers, and each subcarrier consists of data symbol X[k], where k represents the subcarrier index. The OFDM transmitter uses an inverse DFT of size N for modulation. Then the transmitted OFDM signals in discrete-time domain can be expressed as

$$ x[n] = \frac{1}{N}\sum_{k = 0}^{N - 1} X[k] \exp \biggl(j2\pi \frac{nk}{N}\biggr),\quad 0 \le n \le N - 1, $$
(1)

where n is the time-domain sample index of OFDM signals. In order to avoid ISI caused by multipath channel and consequent ICI, GI is appended to the beginning of the Nth sample of IDFT. After passing through a time-varying multipath fading channel and removing the GI, one received signal y[n] consists of channel distorted versions of the transmitted OFDM symbols, which is represented by

$$ y[n] = x[n] \otimes h[n] + w[n], \quad 0 \le n \le N - 1, $$
(2)

where ⊗ denotes the cyclic convolution operation, w[n] is independent and identically distributed additive white Gaussian noise (AWGN) sample in time domain with zero mean and variance \(\sigma_{\mathrm{wt}}^{2}\),

$$ \sigma_{\mathrm{wt}}^{2} = E\bigl[\bigl \vert w[n] \bigr \vert ^{2}\bigr], $$
(3)

and h[n] is the discrete-time CIR.

In this paper, it is assumed that the ISI is completely eliminated by inserting the CP.

2.2 Channel Model

In multipath fading channel, the channel impulse response in time domain [2], h(t,τ), can be expressed as

$$ h(t,\tau ) = \sum_{l = 0}^{L - 1} \alpha_{l}(t)\delta (\tau - \tau_{l}T_{s}), $$
(4)

where α l (t) (l=1,2,…,L) is used to control the energy of the lth path, and L is the total number of paths; τ l T s is the lth path delay, T s is the sampling interval. Usually, each path gain is a complex Gaussian random process generated using the Jake power spectrum. The channel frequency response by Fourier transform of CIR can be expressed as

$$ H(f,t) = \sum_{l = 0}^{L - 1} \alpha_{l}(t)\exp ( - j2\pi f\tau_{l}T_{s}). $$
(5)

Channel estimation can be got by the estimation of the channel response function with an IFFT. By Eq. (4), the CIR h(t,τ) is a time-limited pulse train, so the frequency response will span over the whole frequency domain. However, the OFDM signal is frequency limited. Hence, the receiver can only probe part of the frequency response. In this case, the channel impulse response can be denoted as

$$ h(n) = \frac{1}{N}\sum_{l = 0}^{L - 1} \alpha_{l}\operatorname{e}^{ - j\frac{\pi}{N}(n + (N - 1)\tau_{l})}\frac{\sin (\pi \tau_{l})}{\sin (\pi (\tau_{l} - n)/N)}. $$
(6)

From Eq. (6) we see that each delta function in the original impulse response will be convoluted with the sinc(•) function and no longer time limited. Furthermore, when we do sampling in the frequency domain, the time-domain impulse response will be shifted and added up. Thus, for sample-spaced channels, τ l is an integer, and from Fig. 1, all the energy from α l is mapped to tap \(h_{\tau_{l}}\). If the time delays between different paths are not sample interval spaced, then after these operations there will be path taps at the nonsampled instants. So, the path energy will leak to all the other taps, and if τ l is not an integer, from Fig. 2 we see that the energy of α l will leak to all taps h(n) [7, 8]. The conventional DFT method will not only eliminate the noise, but also lose the useful leaked channel impulse response. The error floor is caused by loss of leakage power.

Fig. 1
figure 1

Channel impulse response of sample-spaced channel

Full size image
Fig. 2
figure 2

Channel impulse response of non-sample-spaced channel

Full size image

3 DFT-Based Channel Estimation

DFT-based channel estimation is a pilot-aided technique for OFDM systems, which achieves excellent performance by neglecting nonsignificant taps and has relatively low complexity based on DFT/IDFT hardware and software.

To perform the DFT-based channel estimation, the Least Square (LS) channel estimates on the pilots are first obtained. At the receiver, the received pilot signals are extracted from Y[k]. For the pilot subcarriers, the transmitted information X[k] is known. Then the channel frequency response in pilot subchannels can be estimated by

$$ \hat{H}_{\mathrm{LS}}[k] = \frac{Y[k]}{X[k]} = H[k] + \frac{W[k]}{X[k]}\quad k \in K_{P}, $$
(7)

where K P is the ensemble of all pilot indexes. This channel estimation algorithm is called the LS estimator. The greatest advantage of the LS estimation algorithm is its simple structure and low complexity since LS does not make use of any channel information. As a result, the accuracy of this algorithm is limited; the LS algorithm is useful when channel noise is small.

Since a large number of time-domain channel power concentrates on a few first samples, the DFT-based estimation reduces the noise power that exists only outside of the CIR part. An N-point IDFT is applied to the LS channel estimate \(\hat{H}_{\mathrm{LS}}[k]\), to obtain an estimate of the CIR:

$$ \hat{h}_{\mathrm{LS}}[n] = \mathrm{IDFT}\bigl\{ \hat{H}_{\mathrm{LS}}[k]\bigr\} = h[n] + \tilde{w}[n]\quad 0 \le n \le N - 1, $$
(8)

were \(\tilde{w}[n]\) is an IDFT sample of noise, which is also AWGN because IDFT is a linear transform. The CIR is typically limited to the length of CIR L, which is less than the GI and much smaller compared with the number of subcarriers N. Conventionally, the CIR is described as

$$ h[n] = \left \{ \begin{array}{@{}l@{\quad}l@{}} \mathrm{IDFT}_{N}\{ H[k]\},& 0 \le n \le L - 1 ,\\ 0,& L \le n \le N - 1. \end{array} \right . $$
(9)

By using (9), (8) can be divided into two parts: CIR part and noise-only existing part. Then we have

$$ \hat{h}_{\mathrm{LS}}[n] = \left \{ \begin{array}{@{}l@{\quad}l@{}} h[n] + \tilde{w}[n],& 0 \le n \le L - 1, \\ \tilde{w}[n],& L \le n \le N - 1. \end{array} \right . $$
(10)

As shown in (9), all CIRs existed in the first L samples, and other samples are only noise. Because the number of paths L is unknown, by zero-padding for pure noise period, the conventional DFT-based estimator can be expressed as

$$ \hat{h}_{\mathrm{DFT}}[n] = \left \{ \begin{array}{@{}l@{\quad}l@{}} h[n] + \tilde{w}[n],& 0 \le n \le N_{\mathrm{CP}} - 1 ,\\ 0,& N_{\mathrm{CP}} \le n \le N - 1, \end{array} \right . $$
(11)

where N CP is the length of CP. From (11), the DFT-based channel estimator is denoted as

$$ \hat{H}_{\mathrm{DFT}}[k] = \mathrm{DFT}_{N}\bigl\{ \hat{h}_{\mathrm{DFT}}[n]\bigr\}\quad 0 \le k \le N - 1. $$
(12)

Conventional DFT-based channel estimations improve the performance by suppressing time-domain noise. However, the traditional DFT-based channel estimator removes only the noise in the channel impulse response beyond the length of the cyclic prefix (CP), while the noise within the length of CP is not suppressed,indicating that there is certain space to improve the performance. So, DFT-based channel estimation can suppress the interference and noise by adding a window to the estimated CIR [19] or by selecting the most significant channel taps [10]. However, this approach will cause power leakage and result in an error floor in a multipath channel with non-sample-spaced time delays.

4 Proposed Channel Estimation

As mentioned before, in the conventional DFT method, the noise is eliminated based on the length of the guard interval or the position of the significant multipath components. However, the energy of the corresponding path will leak to all the taps in the time domain when the channel path delay is usually non-sample-spaced. Then this method will not only eliminate the noise, but also lose the useful leaked channel impulse response. So the error floor is caused by loss of leakage power. In [18], a novel symmetric extension DFT method is used to decrease the loss of channel information. In [13], another approach reduces leakage power by calculating energy increasing rate.

To achieve a robust channel estimation that is not affected by the distribution of channel paths, we propose an improved channel estimation method based on the property of Channel impulse response. The estimated CIR’s taps are composed of the significant channel taps and the added noise. How to distinguish those taps can be regarded as a problem of discriminant analysis. The significant channel taps can be detected on the basis of the discriminant function.

4.1 Discriminant Analysis for Two Groups

We already proposed this technology is [5, 6], but the energy leakage caused by the non-sample-spaced path was not considered. Discriminant analysis is used to analyze relationships between a non-metric-dependent variable and metric or dichotomous independent variables. Identifying channel taps from the channel impulse response (CIR) can be regarded as a problem of discriminant analysis for two groups. Mahalanobis defined the notion “equidistant” for two group values [9]. He proposed a covariance-adjusted measure according to the following equation:

$$ d^{2}(x) = (x - \mu )^{H}\varSigma^{ - 1}(x - \mu ), $$
(13)

where Σ −1 represents the inverse of the covariance matrix, and x is a vector. The Mahalanobis distance is a unitless measure and is used to identify and gauge similarity of an unknown sample set to a known one. It differs from the Euclidean distance in that it takes into account the correlations of the data set and is scale-invariant.

To classify CIR, the noise in channel taps are the group 1, and the significant channel taps are the group 2, which are both normally distributed with mean vectors μ 1 and μ 2 and covariance matrices Σ 1 and Σ 2. The Mahalanobis distances from any xto the two means of groups are given by

$$ d_{^{1}}^{2}(x) = (x - \mu_{1})^{t}\varSigma _{1}^{ - 1}(x - \mu_{1})\quad \mbox{and}\quad d_{^{2}}^{2}(x) = (x - \mu_{2})^{t} \varSigma_{2}^{ - 1}(x - \mu_{2}). $$
(14)

To determine which group x belongs to, we may judge from the Mahalanobis distance. The discriminant analysis classification rule is given as follows:

$$ \left \{ \begin{array}{@{}l@{\quad}l@{}} x \in \mathit{group} 1& \mathit{if}\ d_{^{1}}^{2} \le d_{^{2}}^{2}, \\ x \in \mathit{group}2&\mathit{if}\ d_{^{1}}^{2} > d_{^{2}}^{2}. \end{array} \right . $$
(15)

In most practical situations, the mean vectors and covariance matrices are unknown, so both of the parameters can be replaced by their sample counterparts.

4.2 Proposed Channel Estimation

Even though the energy of non-sample-spaced path leaks to all the taps, most of the energy is kept in the neighborhood of the original pulse locations. Moreover, as is shown in Fig. 2, we can see that the energy concentrates at the ends of the sequences after the energy leakage. For simplicity, the real and imaginary parts of CIR are independent and identically distributed normal random variables. The block diagram of the proposed channel estimation method is presented in Fig. 3.

Fig. 3
figure 3

Block diagram of the proposed algorithm

Full size image

Next, we introduce discriminant analysis to replace the predetermined threshold. It can be summarized as follows.

Step 1::

First, the same as the DFT-based estimator, the frequency response \(\hat{H}_{\mathrm{LS}}[k]\) is estimated by the LS method. After that, convert \(\hat{H}_{\mathrm{LS}}[k]\) to time domain, getting \(\hat{h}_{\mathrm{LS}}[n]\). From Eq. (8), the noise term \(\tilde{w}[n]\) is represented in the whole time domain. Because significant channel taps concentrate at the ends of \(\hat{h}_{\mathrm{LS}}[n]\), the leakage energy that concentrates in the middle of \(\hat{h}_{\mathrm{LS}}[n]\) can be ignored.

Step 2::

As mentioned in step 1, the channel taps \(\hat{h}_{\mathrm{LS}}[n]\) can be divided into three groups: 0≤n<N cp, N cp≤n<N−N cp, N−N cp≤n<N−1. In the region N cp≤n<N−N cp, the channel taps \(\hat{h}_{\mathrm{LS}}[n]\) can be handled as the taps of noise, which are noise sample counterparts. From these taps, the mean vectors and covariance matrices of noise group are determined by

$$ \begin{aligned} \mu_{1} & = \frac{1}{N - 2N_{\mathrm{cp}}}\sum_{n = N_{\mathrm{cp}}}^{N - N_{\mathrm{cp}} - 1} \hat{h}_{\mathrm{LS}}[n]\quad \mbox{and}\quad \\ \varSigma_{1} & = \frac{1}{N - 2N_{\mathrm{cp}} - 1}\sum_{n = N_{\mathrm{cp}}}^{N - N_{\mathrm{cp}} - 1} \bigl( \hat{h}_{\mathrm{LS}}[n] - \mu_{1}\bigr) \bigl(\hat{h}_{\mathrm{LS}}[n] - \mu_{1}\bigr)'. \end{aligned} $$
(16)

In this region, we can treat is as zero. Then the CIR can be expressed as

$$ \hat{h}[n] = 0\quad N_{\mathrm{cp}} \le n \le N - N_{\mathrm{cp}} - 1. $$
(17)
Step 3::

In the region 0≤n<N cp and N−N cp≤n<N−1, according to the Parseval theorem in the DFT form [16], we design the threshold λ based on the noise power. If \(\vert \hat{h}_{\mathrm{LS}}[n] \vert ^{2} > \lambda\), \(\hat{h}[n] = \hat{h}_{\mathrm{LS}}[n]\) and \(\hat{h}_{\mathrm{LS}}[n] = 0\). From these significant channel taps, the mean vector μ 2 and covariance matrix Σ 2 of significant group are also calculated in the same way as μ 1 and Σ 1.

Step 4::

For the other nonzero channel taps \(\hat{h}_{\mathrm{LS}}[n]\), compute the Mahalanobis distance \(d_{^{1}} \) and d 2 by Eq. (14). If \(d_{^{1}} < d_{2}\), \(\hat{h}_{\mathrm{LS}}[n]=0\); otherwise, \(\hat{h}[n] = \hat{h}_{\mathrm{LS}}[n]\) and \(\hat{h}_{\mathrm{LS}}[n] = 0\), and renew μ 2 and Σ 2. This procedure is repeated until all of the nonzero channel taps \(\hat{h}_{\mathrm{LS}}[n]\) have been carried out.

Step 5::

Finally, the channel estimation can be achieved by

$$ \hat{H}[k] = \mathrm{DFT}_{N}\bigl\{ \hat{h}[n]\bigr\}. $$
(18)

We name the above improved channel estimation as time-domain discriminant analysis channel estimation (TD-DACE).

As mention above, channel estimation based on Mahalanobis distance discriminant analysis is a dynamic time-domain threshold algorithm. Performance analysis of Mahalanobis distance discriminant analysis can be found in [9].

5 Simulation Results and Analysis

5.1 Simulation Performance

In this section, we investigate the performance of the proposed algorithm on non-sample-spaced multipath channels, and the bit-error-rate (BER) performance is examined. The estimator in [10] is only suitable for an ideal channel. In addition, it cannot be used in practical environment because its statistical data is insufficient. Because of the incomplete data, it can lead to estimate the amount of bias and variance increases, but it can also reduce statistical method efficiency. Our goal is to find an applied estimation algorithm. The simulations are concentrated on comparison of the LS algorithm, conventional DFT algorithm, LMMSE algorithm, and the proposed algorithm.

Before presenting the simulation results, we will first describe the OFDM system and channel model. For simulation, we have considered an IEEE 802.16e OFDMA system. Table 1 summarizes major system parameters. In our simulation, we test different algorithms under ITU-R M.1225 vehicular A channel model, which is applied assuming the maximum mobile speed of v max=120 km/h. It is a typical non-sample-spaced multipath channel, and its parameters are summarized in Table 2. In addition, perfect synchronization is assumed since it is hard to estimate the channel frequency response beyond guard band.

Table 1 Major system parameters
Full size table
Table 2 Parameter of M.1225Vehicular A channel
Full size table

In the simulation, the noise power \(\sigma_{\mathrm{wt}}^{2}\) can be estimated in the time domain by averaging the noise-only existing part. Expressing the noise estimation in equation, we have:

$$ \sigma_{\mathrm{wt}}^{2} = \frac{1}{N - 2N_{\mathrm{cp}}}\sum _{n = N_{\mathrm{cp}}}^{N - N_{\mathrm{cp}} - 1} \bigl \vert \hat{h}_{\mathrm{LS}}[n] \bigr \vert ^{2} = \frac{1}{N - 2N_{\mathrm{cp}}}\sum_{n = N_{\mathrm{cp}}}^{N - N_{\mathrm{cp}} - 1} \bigl \vert \tilde{w}[n] \bigr \vert ^{2}, $$
(19)

where \(\hat{h}_{\mathrm{LS}} = \vert \hat{h}_{\mathrm{LS}}[n] \vert ^{2}\), 0≤n≤N−1, is distributed according to the chi-squared distribution with 2 degrees of freedom [12]. According to the definition of F distribution and the standard F distribution quantile table, the threshold \(\lambda = 2.5\sigma_{\mathrm{wt}}^{2}\) can be obtained. BER is used to measure the signal detection, which is the number of bit errors divided by the total number of transferred bits during a studied time interval. It can be calculated by simulation. Figure 4 shows the BER performances of four different methods under non-sample-spaced multipath channels. As shown in Fig. 4, it is obvious that the performances of TD-DACE channel estimators are better than DFT-based channel estimator. The performance ranges from the worst to the best are in the following order: the LS algorithm, conventional DFT algorithm, TD-DACE algorithm, and LMMSE algorithms. At the same time, computational complexity is becoming more and more complex. At low signal-to-noise ratio (SNR), by suppressing the time-domain noise, the conventional DFT algorithm is better than the LS algorithm. At high SNR, the performance degradation is induced when low-energy CIR samples containing useful channel information are discarded. With non-sample-spaced time delays, the channel impulse power will leak to all taps, the elimination of noise will also cause the loss of useful channel power, so the conventional DFT method has a severe error floor. Based on the property of CIR, the proposed method utilizes discriminant analysis to detect the most significant taps and achieves performance close to the LMMSE channel estimation, which is more suitable for implementation. The LMMSE algorithm with complexity of cost gets the best performance.

Fig. 4
figure 4

BER performance with different channel estimation (CE)

Full size image

5.2 Complexity Analysis

Here, we discuss the computational complexity of the proposed channel estimation. The computation complexity of the LS algorithm, conventional DFT algorithm, and LMMSE algorithm are O(N), O(Nlog2(N)), and O(N 3). The computational complexity of the proposed scheme can be addressed as follows. In Step 1, the LS method requires N complex multiplications and N-point FFT operations, which require Nlog2(N) complex multiplications; Step 2 requires N−2N cp complex multiplications and complex additions; the complex multiplications of Step 3 and Step 4 are lessen by 2N cp, and the complex additions are also lessen by 2N cp; Step 5 requires Nlog2(N) complex multiplications.

6 Conclusion

In this paper, we proposed the robust channel estimation for OFDM system in non-sample-spaced multipath fading channel. With non-sample-spaced time delays, the channel impulse power will leak to all taps, but most of the energy is kept in the neighborhood of the original pulse locations. The improved method uses discriminant analysis and calculates the dynamic threshold in order to detect significant channel taps. A wider region is designed to discriminate the loss of significant channel taps. As the leaked channel power is effectively detected, the improved method channel estimation can better track the channel state information. Simulation result shows that the proposed method can perform quite well in a non-sample-spaced multipath propagation channel, and the error floor is eliminated substantially.