A Double-Threshold Channel Estimation Method Based on Adaptive Frame Statistics

Abstract

Channel estimation is an important module to enhance the performance of orthogonal frequency division multiplexing (OFDM) systems. However, the presence of a large amount of noise in time-varying multipath fading channels significantly affects the channel estimation accuracy and thus the recovery quality of the received signals. Therefore, this paper proposes a double-threshold (DT) channel estimation method based on adaptive frame statistics (AFS). The method first adaptively determines the number of statistical frames based on the temporal correlation of the received signals, and preliminarily detects the channel structure by analyzing the distribution characteristics of multipath sampling points and noise sampling points during adjacent frames. Subsequently, a multi-frame averaging technique is used to expand the distinction between multipath and noise sampling points. Finally, the DT is designed to better recover the channel based on the preliminary detection results. Simulation results show that the proposed adaptive frame statistics-double-threshold (AFS-DT) channel estimation method is effective and has better performance compared with many existing channel estimation methods.

1. Introduction

Orthogonal frequency division multiplexing (OFDM) technology is extensively employed in modern communication systems due to its exceptional performance and high spectral efficiency [1,2]. By inserting a cyclic prefix (CP) which is greater than the maximum delay spread of the multipath channel among adjacent OFDM symbols as the guard interval (GI), it not only eliminates inter-symbol interference (ISI) but also greatly simplifies the design of the frequency domain equalizer [3]. To accurately recover the signals at the receiver, channel estimation is essential. The current channel estimation methods for OFDM systems can be divided into three categories: blind channel estimation [4], semi-blind channel estimation [5], and pilot-based channel estimation [6]. Although blind and semi-blind channel estimation methods have higher spectral efficiency, their high computational complexity is not ideal in practical applications. The pilot-based channel estimation methods are often preferred due to their reliability and simplicity. Usually, the pilots are inserted in block-type [7], comb-type [8], or scatter-type arrangements [9]. In the block-type arrangement, the pilots appear in a few OFDM symbols of all subcarriers. In the comb-type arrangement, the pilots appear in a few subcarriers of all OFDM symbols. In the scatter-type arrangement, the pilots appear in a few subcarriers of a few OFDM symbols. Therefore, the number of pilots in the scatter-type is less than that in the block-type and comb-type. However, the comb-type pilots can better recover the channel state information (CSI) in time-varying channels [10], which obtain a good trade-off between the number of pilots and the CSI recovery performance. Therefore, this paper focuses on the channel estimation techniques based on comb-type pilots. Figure 1 gives the mainstream channel estimation methods based on comb-type pilots in OFDM systems, which are error square minimization, multi-frame averaging, basis expansion model (BEM), compressed sensing (CS), and threshold denoising. The threshold denoising methods in the red-dashed box and the multi-frame averaging methods in the blue-dashed box in the figure are the theoretical sources of the proposed method in this paper.

At the receiver, the channel is estimated using the known and received pilots. There are many conventional channel estimation techniques such as the least square (LS) method [11], minimum mean square error (MMSE) method [12], and linear MMSE (LMMSE) method [13]. The LS method is the simplest channel estimation method and has been widely used for many years. However, it ignores the effect of noise, which greatly reduces its channel estimation accuracy. The MMSE method uses a priori knowledge in the form of second-order statistics of the channel and has better estimation performance. However, it involves matrix inverse operations and has high computational complexity. The LMMSE method is a simplification of the MMSE method. Although it reduces the computational complexity, it still requires prior channel knowledge to calculate the channel autocorrelation matrix. Tang et al. [14] employed singular value decomposition (SVD) to decompose the channel autocorrelation matrix, which avoids inverse operations and further reduces the complexity of the LMMSE method. The LS-based channel estimator is simple to implement and does not require prior knowledge of the channel. To enhance its performance, numerous denoising strategies have been proposed in the existing literatures. Wu et al. [15] proposed a weighted averaging channel estimation method according to the temporal correlation of wireless channels. Based on the LS estimation, the channel coefficients of two adjacent OFDM symbols are weight-averaged. The noise is better suppressed under this method, but it is not effective in dynamic channels. Zettas et al. [16] adaptively adjusted the buffer according to the Doppler shift, and thus the average number of OFDM frames is determined to adapt to dynamic channels. Based on this, Zhang et al. [17] considered the effect of the signal-to-noise ratio (SNR) on the average number of OFDM frames and proposed a more reasonable multi-frame weighted averaging scheme. However, the multi-frame averaging technique causes Doppler distortion in dynamic channels. In fast time-varying channel environments, the performance loss of multi-frame averaging could be higher than the performance gain of noise suppression. Therefore, the methods proposed by Zettas et al. [16] and Zhang et al. [17] still have some limitations.

To improve the accuracy of channel estimation in the time-varying channels, channel estimation methods based on BEM have been widely studied [18]. The BEM methods use a linear combination of basis functions to fit the channel impulse response (CIR), which transform the estimation of channel parameters into the estimation of BEM coefficients, and reduce the number of parameters to be estimated in the time-varying channels. According to the different basis functions used, commonly used BEMs include complex exponential BEM (CE-BEM) [19], polynomials BEM (P-BEM) [20], and Karhunen–Loeve BEM (KL-BEM) [21]. BEM-based channel estimation methods can effectively track the time-varying channels. However, they require knowledge of the maximum Doppler shift and involve complex matrix operations. Based on the strong performance of deep learning, deep learning-based channel estimation methods are promising [22]. With the help of deep neural networks (DNNs), they are able to learn complex nonlinear channel mapping relationships, thereby achieving more accurate and robust channel estimation. However, deep learning-based channel estimation methods require a large amount of training samples, which are often difficult to obtain in practice [23].

For some broadband wireless channels, the CIR often exhibits a sparse structure due to delay differences and relatively high sampling rates. CS algorithms have been successfully applied to the recovery of sparse channel support sets. They are mainly classified into convex optimization [24] and greedy optimization [25]. Convex optimization mainly uses the basis pursuit (BP) [26] to solve the parametric minimization problem. Its reconstruction accuracy for sparse channel is high. However, it is not applicable to real-time systems and has high complexity. Unlike convex optimization, which minimizes the objective function, greedy optimization determines the location of the non-zero sampling points of the sparse channel by multiple iterations. Orthogonal matching pursuit (OMP) [27], block OMP (BOMP) [28], and compressive sampling matching pursuit (CoSaMP) [29] are the most commonly used CS algorithms. Jiang et al. [30] proposed a separable CoSaMP (SCoSaMP) algorithm based on the introduction of backtracking idea. It effectively improves the channel estimation accuracy while reducing the complexity. For some scenarios, where the channels satisfy the joint sparse model (JSM), the simultaneous OMP (SOMP) [31] algorithm effectively utilizes the joint sparse property to obtain a better sparse channel reconstruction performance. Based upon this, Wang et al. [32] proposed an improved SOMP algorithm to achieve ordinary channel taps detection and dynamic channel taps pursuit. Its channel recovery performance is significantly improved under the sparse channel model with the same partial support sets. However, the above CS algorithms [27,28,29,30,31,32] usually require a large number of iterations to reduce the approximation error, which brings high complexity. Moreover, channel estimation methods based on CS algorithms require a known number of channel common support sets to achieve optimal performance, which limits their scope of application.

Threshold-based channel estimation methods have a lower complexity than those based on CS algorithms and mostly do not require a known number of channel common support sets. They typically use the LS method to obtain the initial CIR. Then, the amplitude of each sample point is compared with a given threshold to determine the multipath sample points. The performance of channel CIR support sets recovery heavily depends on the setting of detection thresholds. Most conventional spectrum sensing methods use a fixed threshold to distinguish the multipath and noise sampling points. For example, Kang et al. [33] proposed a double-noise variance threshold (DNT) by calculating the noise variance from sampling points outside the CP. Tripta et al. [34] implemented the estimation of noise standard deviation by wavelet decomposition and set it directly as the threshold. However, it is difficult to guarantee the noise removal rate using a fixed threshold. Its performance degrades faster, especially when the noise power fluctuates. Xu et al. [35] proposed a piecewise suboptimal threshold (PSOT) for selecting the most significant samples (MSSs) in the estimated CIR. This threshold is set by setting the first-order derivative of the mean square error (MSE) to zero to filter out the possible noise samples in the MSSs as much as possible. Zhang et al. [36] proposed a channel estimation method based on the combination of adaptive multi-frame averaging and improved MSE optimal threshold (IMOT). In this method, most of the noise is suppressed without significantly increasing the computational complexity. To obtain a more desirable performance, both PSOT and IMOT require a priori channel sparsity for assistance, which is often difficult to obtain in practice. In a completely unknown channel environment, Sure et al. [37] proposed a weighted noise threshold (WNT) by introducing a modified interpretation of the hypothesis testing problem. To some extent, the MSE degradation problem caused by the estimation of priori channel information is overcome. Bahonar et al. [38] proposed a sparse recovery method based on sparse domain smoothing. It is mainly divided into three parts: time domain residue computation, sparsity domain smoothing, and adaptive thresholding sparsifying. Its performance of channel recovery is considerably improved at the expense of certain complexity. However, the performance of existing threshold denoising methods is often unsatisfactory in the low SNR range. The reason is that, regardless of how the threshold is selected in the low SNR range, there is the problem of the misclassification of multipath sampling points with lower energy and noise sampling points with higher energy.

To improve the accuracy of channel estimation in fast time-varying channels and reduce computational complexity, this paper proposes a double-threshold (DT) estimation channel method based on adaptive frame statistics (AFS) by utilizing the sparsity and temporal correlation of wireless channels. Its performance in terms of normalized mean square error (NMSE), channel structure correctness detection rate (SCDR), and bit error rate (BER) is better than many existing threshold denoising methods in the low SNR range. The main novelties and contributions are as follows:

(1)

The temporal correlation of the received signals is used to analyze the time-varying characteristics of the channel, so that the number of OFDM statistical frames can be adaptively determined.

(2)

The channel estimation accuracy is further improved by designing the DT based on the preliminary detection results combined with the distribution characteristics of the sampling points.

(3)

To fully utilize the cache resources and improve the performance of DT, a multi-frame averaging technique is used to expand the distinction between multipath and noise sampling points after the preliminary statistics.

The rest of this paper is organized as follows. Section 2 presents the system model. Section 3 derives the proposed AFS-DT channel estimation method in detail. The experimental results of the proposed method and other conventional methods are given in Section 4, and the computational complexity of the different methods is compared and analyzed. Section 5 summarizes the research work.

2. System Model

It is supposed that one OFDM symbol is transmitted in one frame in this paper. Figure 2 shows the CP-OFDM system model using the proposed AFS-DT channel estimation method. The main research work of this paper is shown in the red-dashed box in the figure. At the transmitter, the input binary bits are grouped and mapped by 4-quadrature amplitude modulation (4QAM) or 16-QAM (16QAM). After inserting the comb-type pilots with uniform interval and length $N_{\mathrm{P}}$, a serial-to-parallel (S/P) transformation is performed. The inverse fast Fourier transform (IFFT) block converts the data with N rows into time-domain signals, where N is the number of subcarriers. Then, the CP of length $N_{\mathrm{G}}$ is added to the time-domain OFDM symbols, which should not be smaller than the maximum channel delay spread L (in terms of samples) [39]. The transmitted signals will pass through the multipath fading channel with additive white Gaussian noise (AWGN) after parallel-to-serial (P/S) transformation.

At the receiver, it is assumed to be perfectly synchronized with the transmitter. The CP is removed after the S/P transformation. The fast Fourier transform (FFT) output of the pilot symbols is expressed as:

$$\mathit{Y}(m,k)=\mathit{H}(m,k)\mathit{X}(m,k)+\mathit{N}(m,k),$$

(1)

where $\mathit{Y}(m,k)$ and $\mathit{X}(m,k)$ represent the received and transmitted pilot of the mth $(m=1,2,\cdots ,N_{\mathrm{P}})$ subcarrier in the kth $(k=1,2,\cdots ,K)$ OFDM symbol, respectively. $\mathit{H}(m,k)$ is the true channel frequency response (CFR) and $\mathit{N}(m,k)$ represents AWGN.

The channel estimation module uses the LS method to obtain the CFR ${\widehat{\mathit{H}}}_{\mathrm{LS}}(m,k)$, which can be expressed as:

$${\widehat{\mathit{H}}}_{\mathrm{LS}}(m,k)=\frac{\mathit{Y}(m,k)}{\mathit{X}(m,k)}.$$

(2)

The CFR ${\widehat{\mathit{H}}}_{\mathrm{LS}}(m,k)$ is transformed into the time domain by the $N_{\mathrm{P}}$-point IFFT. Considering the sparsity of the wireless channels, the CIR ${\widehat{\mathit{h}}}_{\mathrm{LS}}(m,k)$ can be specifically expressed as:

$${\widehat{\mathit{h}}}_{\mathrm{LS}}(m,k)=\left\{\begin{array}{c}\mathit{h}(m,k)+{\mathit{n}}_{\mathrm{LS}}(m,k),m\in \mathit{C}\hfill \\ {\mathit{n}}_{\mathrm{LS}}(m,k),m\notin \mathit{C},\hfill \end{array}\right.$$

(3)

where $\mathit{C}$ is the true CIR support sets, which can be considered as the position of the multipath sampling points in interval sampling channels. $\mathit{h}(m,k)$ is the true CIR at the multipath sampling points, which can be modeled as a complex Gaussian random variable with a mean of 0 and a variance of ${\sigma }_m^2$ in Rayleigh fading channels. ${\mathit{n}}_{\mathrm{LS}}(m,k)$ is the complex AWGN with a mean of 0 and a variance of ${\sigma }_k^2$. Since $\mathit{h}(m,k)$ and ${\mathit{n}}_{\mathrm{LS}}(m,k)$ are independent of one another, $\mathit{h}(m,k)+{\mathit{n}}_{\mathrm{LS}}(m,k)$ also conforms to the complex Gaussian distribution. Therefore, the random variable ${\widehat{\mathit{h}}}_{\mathrm{LS}}(m,k)$ is distributed as:

$${\widehat{\mathit{h}}}_{\mathrm{LS}}(m,k)\sim \left\{\begin{array}{c}\mathcal{C}\mathcal{N}(0,{\sigma }_m^2+{\sigma }_k^2),m\in \mathit{C}\hfill \\ \mathcal{C}\mathcal{N}(0,{\sigma }_k^2),m\notin \mathit{C}.\hfill \end{array}\right.$$

(4)

Since AWGN is ignored in the LS method, the channel estimation accuracy will be further improved in the AFS-DT denoising module, which will be described in detail in Section 3. Finally, the transmitted binary bits are recovered by the 4QAM or 16QAM demodulation methods.

3. The Proposed Method

The existing threshold denoising methods have limitations in the low SNR range. Therefore, an AFS-DT-based channel estimation method is proposed in this paper. First, the channel structure is preliminarily determined by multi-frame statistics based on the distribution characteristics of multipath sampling points and noise sampling points. The number of statistical frames P is adaptively determined according to the temporal correlation of the received signals, and the derivation process will be described in detail in Section 3.3. A multi-frame averaging technique is then used to expand the distinction between multipath and noise sampling points. Finally, a cost factor is introduced to design a denoising threshold that minimizes the overall error cost, and another threshold is introduced to supplement the multipath.

3.1. Channel Structure Detection and Optimization

Perform P-frame statistics on the CIR ${\widehat{\mathit{h}}}_{\mathrm{LS}}(m,k)$ obtained by LS estimation. Mark the number of times that the real part of the mth sampling point appears in the statistical intervals $(-\infty ,0)$ and $(0,+\infty )$ as $N_{{}_{\mathrm{Re},\mathrm{Ne}}}^{m,k}$ and $N_{{}_{\mathrm{Re},\mathrm{Po}}}^{m,k}$, respectively. Similarly, mark the number of times that the imaginary part of the mth sampling point appears in the statistical intervals $(-\infty ,0)$ and $(0,+\infty )$ as $N_{{}_{\mathrm{I}\mathrm{m},\mathrm{Ne}}}^{m,k}$ and $N_{{}_{\mathrm{Im},\mathrm{P}\mathrm{o}}}^{m,k}$, respectively. The counting process can be specifically expressed as [40]:

$$N_{{}_{\mathrm{Re},\mathrm{Ne}}}^{m,k}=\mathrm{count}\left[\mathrm{Re}\right({\widehat{\mathit{h}}}_{\mathrm{LS}}(m,k),{\widehat{\mathit{h}}}_{\mathrm{LS}}(m,k+1),\cdots ,{\widehat{\mathit{h}}}_{\mathrm{LS}}(m,k+P-1))\in (-\infty ,0)],$$

(5)

$$N_{{}_{\mathrm{Re},\mathrm{Po}}}^{m,k}=\mathrm{count}\left[\mathrm{Re}\right({\widehat{\mathit{h}}}_{\mathrm{LS}}(m,k),{\widehat{\mathit{h}}}_{\mathrm{LS}}(m,k+1),\cdots ,{\widehat{\mathit{h}}}_{\mathrm{LS}}(m,k+P-1))\in (0,+\infty )],$$

(6)

$$N_{{}_{\mathrm{Im},\mathrm{Ne}}}^{m,k}=\mathrm{count}\left[\mathrm{Im}\right({\widehat{\mathit{h}}}_{\mathrm{LS}}(m,k),{\widehat{\mathit{h}}}_{\mathrm{LS}}(m,k+1),\cdots ,{\widehat{\mathit{h}}}_{\mathrm{LS}}(m,k+P-1))\in (-\infty ,0)],$$

(7)

$$N_{{}_{\mathrm{Im},\mathrm{Po}}}^{m,k}=\mathrm{count}\left[\mathrm{Im}\right({\widehat{\mathit{h}}}_{\mathrm{LS}}(m,k),{\widehat{\mathit{h}}}_{\mathrm{LS}}(m,k+1),\cdots ,{\widehat{\mathit{h}}}_{\mathrm{LS}}(m,k+P-1))\in (0,+\infty )],$$

(8)

where $\mathrm{count}\left(·\right)$ denotes the operation of counting. $\mathrm{Re}\left(·\right)$ and $\mathrm{Im}\left(·\right)$ denote the operations of taking the real and imaginary parts, respectively.

Next, define the variable $N_{max}^{m,k}$:

$$N_{max}^{m,k}=max(N_{\mathrm{Re},\mathrm{Ne}}^{m,k},N_{\mathrm{Re},\mathrm{Po}}^{m,k},N_{\mathrm{Im},\mathrm{Ne}}^{m,k},N_{\mathrm{Im},\mathrm{Po}}^{m,k}),$$

(9)

where $\max \left(·\right)$ denotes the operation of taking the maximum value.

In the time-varying channels, there exists a certain degree of correlation among adjacent OFDM symbols. Since the probability density function (PDF) of noise follows the zero-mean complex Gaussian distribution, it is known from the statistical properties of noise that if the mth sampling point is a pure noise sampling point, the number of times that its real or imaginary part appears in the intervals $(-\infty ,0)$ and $(0,+\infty )$ is similar. It can be considered that $N_{max}^{m.k}<P$. In practical communication environments, the power of the multipath in the channel is much greater than the power of noise. Considering the correlation among adjacent OFDM frames, if the mth point is a multipath sampling point, the number of times that its real or imaginary part appears in the statistical intervals $(-\infty ,0)$ and $(0,+\infty )$ approximates the number of statistical frames, i.e., $N_{max}^{m.k}=P$. Therefore, the preliminary channel structure detection matrix ${\widehat{\mathit{S}}}_{\mathrm{I}}(m,k)$ can be expressed as:

$${\widehat{\mathit{S}}}_{\mathrm{I}}(m,k)=\left\{\begin{array}{c}1,N_{max}^{m,k}=P\hfill \\ 0,N_{max}^{m,k}<P,\hfill \end{array}\right.$$

(10)

where the value of 1 represents a multipath sampling point, and the value of 0 represents a noise sampling point.

Due to the extreme randomness of statistics and the time-varying nature of the channel, there are still misclassifications of noise as multipath and multipath as noise. Therefore, further optimization is needed. The P-frame averaging of the CIR can be expressed as:

$${\widehat{\mathit{h}}}_{\mathrm{A}}(m,k)=\frac{1}{P}\sum _{i=k}^{k+P-1}{\widehat{\mathit{h}}}_{\mathrm{LS}}(m,i)=\left\{\begin{array}{c}{\mathit{h}}_{\mathrm{A}}(m,k)+{\mathit{n}}_{\mathrm{A}}(m,k),m\in \mathit{C}\hfill \\ {\mathit{n}}_{\mathrm{A}}(m,k),m\notin \mathit{C}.\hfill \end{array}\right.$$

(11)

The channel information of adjacent OFDM frames is still similar due to the temporal correlation of the channel. It can be considered that ${\mathit{h}}_{\mathrm{A}}(m,k)\approx \mathit{h}(m,k)$. That is the true multipath power ${\sigma }_{m,\mathrm{A}}^2$ after averaging is approximately equal to the unaveraged true multipath power ${\sigma }_m^2$. The multipath power after superimposed noise becomes ${\sigma }_{\mathrm{R}}^2\approx {\sigma }_m^2+{\sigma }_k^2/P$, and the noise power is ${\sigma }_{\mathrm{N}}^2={\sigma }_k^2/P$. Therefore, the CIR ${\widehat{\mathit{h}}}_{\mathrm{A}}(m,k)$ after averaging conforms to the following complex Gaussian distribution:

$${\widehat{\mathit{h}}}_{\mathrm{A}}(m,k)\sim \left\{\begin{array}{c}\mathcal{C}\mathcal{N}(0,{\sigma }_m^2+{\sigma }_k^2/P),m\in \mathit{C}\hfill \\ \mathcal{C}\mathcal{N}(0,{\sigma }_k^2/P),m\notin \mathit{C}.\hfill \end{array}\right.$$

(12)

Combining (4) and (12), the change in the distinction between multipath sampling points and noise sampling points can be specifically expressed as:

$$(\frac{{\sigma }_m^2+{\sigma }_k^2/P}{{\sigma }_k^2/P})/(\frac{{\sigma }_m^2+{\sigma }_k^2}{{\sigma }_k^2})\approx P.$$

(13)

It can be seen that, by averaging the adjacent P-frame OFDM symbols, the distinction in energy between multipath sampling points and noise sampling points is approximately expanded by a factor of P. Unlike the purpose of conventional multi-frame averaging, it is not directly used to average the CIR for noise suppression. This is because multi-frame averaging in dynamic channels distorts the channel coefficients at the multipath sampling points, which means that ${\mathit{h}}_{\mathrm{A}}(m,k)\approx \mathit{h}(m,k)$ is not rigorous and even causes errors.

The square of the envelope of the CIR ${\widehat{\mathit{h}}}_{\mathrm{A}}(m,k)$ after averaging gives $|{\widehat{\mathit{h}}}_{\mathrm{A}}(m,k){|}^2$, which represents the power and follows an exponential distribution. The cumulative distribution functions (CDFs) $F_{\mathrm{R}}\left(x\right)$ and $F_{\mathrm{N}}\left(x\right)$ of the multipath power and noise power can be specifically expressed as:

$$F_{\mathrm{R}}\left(x\right)=\left\{\begin{array}{c}1-{\mathrm{e}}^{-\frac{1}{{\sigma }_{\mathrm{R}}^2}x},x>0\hfill \\ 0,x\le 0,\hfill \end{array}\right.$$

(14)

$$F_{\mathrm{N}}\left(x\right)=\left\{\begin{array}{c}1-{\mathrm{e}}^{-\frac{1}{{\sigma }_{\mathrm{N}}^2}x},x>0\hfill \\ 0,x\le 0.\hfill \end{array}\right.$$

(15)

When the power at the mth subcarrier in the kth OFDM frame is less than the denoising threshold $T_k$, the subcarrier is considered a noise sampling point, and vice versa as a multipath sampling point. According to the nature of the CDF, when $x=T_k$, the multipath error removal probability is $F_{\mathrm{R}}\left(x\right)$ and the noise correct removal probability is $F_{\mathrm{N}}\left(x\right)$. Therefore, the probabilities of missed alarm (MA) and false alarm (FA) are $F_{\mathrm{R}}\left(x\right)$ and $1-F_{\mathrm{N}}\left(T_k\right)$, respectively. The overall error cost $W\left(T_k\right)$ can be expressed as:

$$\begin{array}{c}W\left(T_k\right)=(1-\alpha )F_{\mathrm{R}}\left(T_k\right)+\alpha [1-F_{\mathrm{N}}\left(T_k\right)]=(1-\alpha )(1-{\mathrm{e}}^{-\frac{1}{{\sigma }_{\mathrm{R}}^2}{T}_k})+\alpha {\mathrm{e}}^{-\frac{1}{{\sigma }_{\mathrm{N}}^2}{T}_k},\end{array}$$

(16)

where $\alpha$ is the introduced FA cost factor, the specific value of which will be given in Section 4.2. To obtain the optimal denoising threshold, the first-order derivative of $T_k$ for the overall error cost $W\left(T_k\right)$ can be expressed as:

$$\frac{\partial W\left(T_k\right)}{\partial T_k}=\frac{1-\alpha }{{\sigma }_{\mathrm{R}}^2}{\mathrm{e}}^{-\frac{1}{{\sigma }_{\mathrm{R}}^2}{T}_k}-\frac{\alpha }{{\sigma }_{\mathrm{N}}^2}{\mathrm{e}}^{-\frac{1}{{\sigma }_{\mathrm{N}}^2}{T}_k}.$$

(17)

Let the first order derivative be equal to zero, and the denoising threshold $T_k^{\mathrm{N}}$ for finding the minimum error cost can be specified as:

$$T_k^{\mathrm{N}}={\sigma }_{\mathrm{N}}^2\frac{{\sigma }_{\mathrm{R}}^2}{{\sigma }_{\mathrm{R}}^2-{\sigma }_{\mathrm{N}}^2}ln\frac{\alpha {\sigma }_{\mathrm{R}}^2}{(1-\alpha ){\sigma }_{\mathrm{N}}^2}.$$

(18)

Since the multipath power ${\sigma }_{\mathrm{R}}^2$ is usually much larger than the noise power ${\sigma }_{\mathrm{N}}^2$, the essence of this denoising threshold $T_k^{\mathrm{N}}$ is a multiple of the noise power corresponding to the kth frame. Based on this, the threshold $T_k^{\mathrm{R}}$ for supplementing the multipath can be expressed as:

$$T_k^{\mathrm{R}}={\sigma }_{\mathrm{R}}^2/[\frac{{\sigma }_{\mathrm{R}}^2}{{\sigma }_{\mathrm{R}}^2-{\sigma }_{\mathrm{N}}^2}ln\frac{\alpha {\sigma }_{\mathrm{R}}^2}{(1-\alpha ){\sigma }_{\mathrm{N}}^2}].$$

(19)

The preliminary channel structure detection matrix ${\widehat{\mathit{S}}}_{\mathrm{I}}(m,k)$ is further optimized with the denoising threshold $T_k^{\mathrm{N}}$ and the supplementary multipath threshold $T_k^{\mathrm{R}}$. Searching for possible noise sampling points among the originally judged multipath sampling points, the selection principle is that the sampling points power is less than the denoising threshold set in the current frame. Searching for possible multipath sampling points among the originally judged noise sampling points, the selection principle is that the sampling points power is greater than the supplementary multipath threshold set in the current frame. The final channel structure detection matrix ${\widehat{\mathit{S}}}_{\mathrm{F}}(m,k)$ can be specifically expressed as:

$${\widehat{\mathit{S}}}_{\mathrm{F}}(m,k)=\left\{\begin{array}{c}1,[N_{max}^{m,k}=P\mathrm{and}|{\widehat{\mathit{h}}}_{\mathrm{R}}(m,k){|}^2\ge T_k^{\mathrm{N}}]\mathrm{or}[N_{max}^{m,k}<P\mathrm{and}|{\widehat{\mathit{h}}}_{\mathrm{N}}(m,k){|}^2\ge T_k^{\mathrm{R}}]\hfill \\ 0,[N_{max}^{m,k}<P\mathrm{and}|{\widehat{\mathit{h}}}_{\mathrm{N}}(m,k){|}^2<T_k^{\mathrm{R}}]\mathrm{or}[N_{max}^{m,k}=P\mathrm{and}|{\widehat{\mathit{h}}}_{\mathrm{R}}(m,k){|}^2<T_k^{\mathrm{N}}],\hfill \end{array}\right.$$

(20)

where ${\widehat{\mathit{h}}}_{\mathrm{R}}(m,k)$ and ${\widehat{\mathit{h}}}_{\mathrm{N}}(m,k)$ are the CIRs of multipath and noise after preliminary detection, respectively.

Based on the final channel structure detection matrix ${\widehat{\mathit{S}}}_{\mathrm{F}}(m,k)$ for time-domain denoising, the final CIR ${\widehat{\mathit{h}}}_{\mathrm{F}}(m,k)$ can be expressed as:

$${\widehat{\mathit{h}}}_{\mathrm{F}}(m,k)={\widehat{\mathit{h}}}_{\mathrm{LS}}(m,k)·{\widehat{\mathit{S}}}_{\mathrm{F}}(m,k).$$

(21)

3.2. Estimation of Multipath Power and Noise Power

From (18) and (19), the optimal threshold requires the known multipath power ${\sigma }_{\mathrm{R}}^2$ and noise power ${\sigma }_{\mathrm{N}}^2$. Then, the ${\sigma }_{\mathrm{R}}^2$ and ${\sigma }_{\mathrm{N}}^2$ will be estimated based on the preliminary detection results.

The CIR ${\widehat{\mathit{h}}}_{\mathrm{R}}(m,k)$ at all multipath sampling points and the CIR ${\widehat{\mathit{h}}}_{\mathrm{N}}(m,k)$ at all noise sampling points can be obtained from the CIR ${\widehat{\mathit{h}}}_{\mathrm{A}}(m,k)$ and the preliminary channel structure detection matrix ${\widehat{\mathit{S}}}_{\mathrm{I}}(m,k)$, which are expressed as:

$${\widehat{\mathit{h}}}_{\mathrm{R}}(m,k)={\widehat{\mathit{h}}}_{\mathrm{A}}(m,k)·{\widehat{\mathit{S}}}_{\mathrm{I}}(m,k),$$

(22)

$${\widehat{\mathit{h}}}_{\mathrm{N}}(m,k)={\widehat{\mathit{h}}}_{\mathrm{A}}(m,k)·[\mathit{J}(m,k)-{\widehat{\mathit{S}}}_{\mathrm{I}}(m,k)],$$

(23)

where $\mathit{J}(m,k)$ denotes an all-one matrix with the same dimension as ${\widehat{\mathit{S}}}_{\mathrm{I}}(m,k)$. Based on the CIRs ${\widehat{\mathit{h}}}_{\mathrm{R}}(m,k)$ and ${\widehat{\mathit{h}}}_{\mathrm{N}}(m,k)$, the multipath power and noise power of the kth OFDM frame can be expressed as:

$${\sigma }_{\mathrm{R}}^2\left(k\right)=\frac{{\mathrm{sum}}_1\left(\right|{\widehat{\mathit{h}}}_{\mathrm{R}}(m,k){|}^2)}{{\mathrm{sum}}_1\left({\widehat{\mathit{S}}}_{\mathrm{I}}(m,k)\right)},$$

(24)

$${\sigma }_{\mathrm{N}}^2\left(k\right)=\mathrm{median}(|{\widehat{\mathit{h}}}_{\mathrm{N}}(m,k){|}^2),$$

(25)

where ${\mathrm{sum}}_1\left(·\right)$ denotes the summation by column, $\mathrm{median}\left(·\right)$ denotes the operation of taking the median by column. To minimize the estimation error due to the preliminary channel structure detection matrix ${\widehat{\mathit{S}}}_{\mathrm{I}}(m,k)$, the multipath power is estimated employing the mean operation. This is because the number of noise sampling points in some columns of ${\widehat{\mathit{h}}}_{\mathrm{R}}(m,k)$ may be more than the number of multipath sampling points. If the median operation is employed, the estimated multipath power may be much smaller than the true power. Similarly, the noise power is estimated by employing the median operation. This is because there may be multipath sampling points with high power in some columns of ${\widehat{\mathit{h}}}_{\mathrm{N}}(m,k)$. If the mean operation is employed, the estimated noise power may be much larger than the true power.

3.3. Determination of P

Assuming that the duration of each OFDM frame is $T_{\mathrm{sym}}$, the duration of the statistical frames is $T_P=P\times T_{\mathrm{sym}}$. To ensure the reliability of preliminary statistical results, there should be a strong correlation among the OFDM frames used for statistics. Therefore, $T_P$ should be smaller than the coherence time $T_{\mathrm{C}}$ of the channel, i.e.,

$$T_P=\frac{T_{\mathrm{C}}}{\beta },$$

(26)

where $\beta$ is the buffer factor and the value is greater than 1, and it is selected as $\beta =3$ for the proposed AFS-DT method. Meanwhile, the coherence time $T_{\mathrm{C}}$ of the channel can be defined as [41]:

$$T_{\mathrm{C}}=\sqrt{\frac{9}{16\mathsf{\pi }{f_{\mathrm{d}}}^2}}=\frac{0.423}{f_{\mathrm{d}}},$$

(27)

where $f_{\mathrm{d}}$ is the Doppler shift. Combining (26) and (27), P can be determined as:

$$P=\mathrm{floor}(\frac{0.423}{f_{\mathrm{d}}\beta T_{\mathrm{sym}}}),$$

(28)

where $\mathrm{floor}(·)$ stands for rounding down to ensure inter-frame correlation. From (28), it can be seen that the estimation of the Doppler shift $f_{\mathrm{d}}$ is essential for the adaptive determination of the statistical frame number P. Under Rayleigh fading channel models, the autocorrelation function of the time-domain-received signals can be expressed as a first-class zero-order Bessel function [42]:

$$r(\Delta k)=J_0(2\mathsf{\pi }{f}_{\mathrm{d}}\Delta k{T}_{\mathrm{sym}}),$$

(29)

where $\Delta k$ is the difference in the number of OFDM symbols. On the other hand, the autocorrelation function can be directly calculated from the time-domain-received signals as:

$$\widehat{r}(\Delta k)=\frac{1}{K-|\Delta k|}\sum _{k=1}^{K-|\Delta k|}\left[\mathit{y}\right(k){\mathit{y}}^{*}(k+|\Delta k\left|\right)],$$

(30)

where $\mathit{y}\left(k\right)$ denotes the kth frame of OFDM symbols received in the time domain. ${\left(·\right)}^{*}$ denotes the operation of conjugate transpose. Next, the first negative value of $\widehat{r}(\Delta k)$ is found according to (30) and let $\Delta k$ be $\widehat{z}$. Then, the first zero crossing point ${\widehat{z}}_0$ is determined by linear interpolation, which can be specified as:

$${\widehat{z}}_0=\frac{\widehat{r}\left(\widehat{z}\right)}{\widehat{r}(\widehat{z}-1)-\widehat{r}\left(\widehat{z}\right)}+\widehat{z}.$$

(31)

Since the first zero crossing point of the first-class zero-order Bessel function $J_0\left(x\right)$ is $x=2.4048$, there is $2\mathsf{\pi }{f}_{\mathrm{d}}{\widehat{z}}_0{T}_{\mathrm{sym}}=2.4048$ when $\widehat{r}(\Delta k)=0$. The resulting estimate of the Doppler shift can be specifically achieved as [17]:

$${\widehat{f}}_{\mathrm{d}}=\frac{2.4048}{2\mathsf{\pi }{\widehat{z}}_0{T}_{\mathrm{sym}}}=\frac{0.383}{{\widehat{z}}_0{T}_{\mathrm{sym}}}.$$

(32)

Combining (28) and (32), P can be determined by:

$$P=\mathrm{floor}(\frac{1.104{\widehat{z}}_0}{\beta }).$$

(33)

4. Simulation Results

To evaluate the effectiveness of the proposed method in this paper, this section conducts simulations on the NMSE, SCDR, and BER of the AFS method and the further optimized AFS-DT method. Several conventional methods are also simulated and compared, which include DNT [33], WNT [37], and IMOT [36]. The performance metrics and simulation environments including the channel models and system parameters are given in Section 4.1. For the proposed AFS-DT method, the specific determination process of the FA cost factor $\alpha$ is given in Section 4.2. For the WNT method, the weight factor q is taken as $q=0$. For the IMOT method, the prior probability $P_{\mathrm{R}}$ of multipath sampling points is taken as the local sparsity level (LSL) $P_{\mathrm{R}}={\sigma }_{\mathrm{R}}^2$. This section also simulates the channel estimation performance for two ideal cases, i.e., the known true CIR support and perfect CSI. In the first ideal case, the AWGN at all noise sampling points can be removed, but the AWGN superimposed on the multipath sampling points will still be retained. In the second ideal case, the CIR is obtained without superimposing the AWGN and represents the true CIR. Finally, the computational complexity of the different channel estimation methods is analyzed and compared in Section 4.5.

4.1. Simulation Environments and Performance Metrics

The channel models used are two new models from the Finnish Wing-TV test project [16]: the vehicular urban (VU) and the motorway rural (MR) models. These channels are all Rayleigh channels with the multipath number of 12, whose power delay profiles (PDPs) are shown in

Table 1. PDPs for VU and MR multipath fading channels.

Tap	VU			MR
	Delay ($\mathsf{\mu }$s)	Power (dB)		Delay ($\mathsf{\mu }$s)	Power (dB)
1	0.0	0.0		0.0	0.0
2	0.3	−0.5		0.5	−1.3
3	0.8	−1.0		1.0	−3.4
4	1.6	−4.1		1.8	−6.8
5	2.6	−8.8		2.5	−10.2
6	3.3	−12.6		3.1	−12.9
7	4.8	−18.6		3.9	−16.3
8	5.8	−21.6		4.8	−19.5
9	7.2	−24.6		5.5	−21.7
10	10.8	−20.7		6.4	−23.3
11	11.8	−18.8		7.0	−24.2
12	12.6	−19.5		9.0	−25.8

. The main simulation parameters of the OFDM system are shown in

Table 2. Simulation parameters of the OFDM system.

Parameters	Specifications
System model	CP-OFDM
Channel distribution	Rayleigh
Baseband bandwidth B	5 MHz
Modulation mode	4QAM/16QAM
The number of OFDM frame K	400
Subcarrier number (without CP) N	960
CP length $N_{\mathrm{G}}$	240
Pilot interval $\Delta P$	3

. There are 1200 subcarriers, among which the CP occupies 240 subcarriers. Therefore, the total number of pilot subcarriers and data subcarriers is $N=1200-240=960$. In the pilot-based channel estimation methods, the way of inserting the pilots is very important. To achieve accurate channel recovery, the pilots’ insertion interval needs to satisfy the Nyquist sampling theorem. Respecting the frequency selectivity of the channels, the pilot subcarriers spacing $\Delta f=\frac{N}{N_{\mathrm{P}}}\frac{1}{T_{\mathrm{sym}}}$ should not exceed the coherence bandwidth, i.e., $\Delta f\le \frac{N}{L}\frac{1}{T_{\mathrm{sym}}}$, which leads to $N_{\mathrm{P}}\ge L$. Since L is unknown, this paper chooses $N_{\mathrm{P}}=N_{\mathrm{G}}$, which guarantees that the above constraint is respected. Therefore, the pilot interval $\Delta P$ is equal to 3. In the dynamic multipath channel environments, the system carrier frequency $f_{\mathrm{c}}$ is set to 800 MHz. The speeds of user equipment (UE) in VU and MR are set to 40 km/h and 100 km/h, respectively. The final simulation results are obtained by averaging a total of 1000 trail runs.

In this paper, the channel estimation performance is evaluated in terms of NMSE, SCDR, and BER, which can be defined as (34), (35), and (36), respectively:

$$e_{\mathrm{NMSE}}=\sqrt{\frac{\mathrm{E}\left(\right|\mathit{h}-\widehat{\mathit{h}}{|}^2)}{\mathrm{E}\left(\right|\mathit{h}{|}^2)}},$$

(34)

$$r_{\mathrm{SCDR}}=1-\frac{\mathrm{E}[{\mathrm{sum}}_2\left(\right|\mathit{S}-\widehat{\mathit{S}}\left|\right)]}{{\mathrm{sum}}_2\left(\mathit{J}\right)},$$

(35)

$$r_{\mathrm{BER}}=\frac{N_{\mathrm{e}}}{N_{\mathrm{a}}},$$

(36)

where $\mathrm{E}\left(·\right)$ denotes the operation of taking the mean value. $\mathit{h}$ and $\widehat{\mathit{h}}$ represent the true CIR and CIR obtained by various channel estimation methods, respectively. ${\mathrm{sum}}_2\left(·\right)$ means summing all elements of the matrix. $\mathit{S}$ and $\widehat{\mathit{S}}$ represent the true channel structure detection matrix and the channel structure detection matrix obtained by various threshold channel estimation methods, respectively. $N_{\mathrm{e}}$ and $N_{\mathrm{a}}$ are the number of error bits and the total number of bits which are sent into the transmitter, respectively.

4.2. Determination of $\alpha$

Since most of the sampling points in an OFDM frame are noise sampling points, the value of $\alpha$ should theoretically be within the interval $[0.5,1)$. To verify the theoretical reasoning and find the optimal $\alpha$, the NMSE simulation curves for the VU channel with a UE speed of 40 km/h and the MR channel with a UE speed of 100 km/h when $\alpha =\left\{0.1,0.5,0.9,0.99,0.999\right\}$ are shown in Figure 3a and Figure 3b, respectively. The trends of the simulation curves are similar in both channels, but the difference in NMSE among different $\alpha$ values is more obvious in the MR channel. This is because the fast time-varying characteristics of the MR channel lead to a high probability of misclassification in the AFS method, and the DT can more significantly improve the channel estimation performance. Therefore, the result of the MR channel is described in detail as an example.

As shown in Figure 3b, the value of NMSE becomes smaller as $\alpha$ increases when $\alpha =\left\{0.1,0.5,0.9\right\}$. From (18) and (19), it can be seen that, as $\alpha$ becomes larger, the denoising threshold becomes larger, while the complementary multipath threshold becomes smaller, which makes the DT have better denoising and a supplementary multipath performance. It also further verifies the theoretical reasoning that the optimal $\alpha$ value is taken within the interval $[0.5,1)$. When $\alpha =\left\{0.9,0.99,0.999\right\}$, the value of NMSE is still smaller when $\alpha$ takes a larger value in the relatively high SNR range, but the performance improvement is not significant. Moreover, when $\alpha =0.99$ or $\alpha =0.999$, the NMSE will sharply deteriorate in the lower SNR range. This is because an excessively large $\alpha$ causes the denoising threshold to incorrectly remove some multipath sampling points, while the supplementary multipath threshold also incorrectly supplements some noise sampling points. Therefore, $\alpha =0.9$ is finally selected for the experiment in this section.

4.3. Analysis of NMSE and SCDR

Since both NMSE and SCDR reflect the degree of channel recovery to some extent, this subsection will simulate and analyze them together. The NMSE and SCDR of different channel estimation methods for the VU channel with a UE speed of 40 km/h are shown in Figure 4 and Figure 5, respectively. The NMSE and SCDR of different channel estimation methods for the MR channel with a UE speed of 100 km/h are shown in Figure 6 and Figure 7, respectively. Figure 4a, Figure 5a, Figure 6a, and Figure 7a are obtained under 4QAM modulation. Figure 4b, Figure 5b, Figure 6b, and Figure 7b are obtained under 16QAM modulation.

As shown in Figure 4a, the NMSE of all seven methods decreased with the increase in SNR. The proposed AFS-DT method has the best NMSE in the overall SNR range in addition to the ideal method of true CIR support. The AFS method without further structural optimization shows a suboptimal performance in the lower SNR range. It outperforms the true CIR support method. This is because, in the case of a very low SNR, the multipath power on some multipath sampling points is low, but the power of AWGN superimposed on them is high. In the AFS method, these multipath sampling points will be judged as noise sampling points and removed. In the true CIR support method, these multipath sampling points superimposed with a large amount of AWGN will be retained, and the error caused by AWGN will be amplified in the subsequent interpolation. Figure 4b shows that the downward trend of the simulation curves obtained with 16QAM modulation is the same as that obtained with 4QAM modulation, but the NMSE is larger. Compared with Figure 6a, the AFS-DT method in Figure 4a has a slightly lower performance gain compared with the conventional methods. Similar results can be obtained in Figure 4b and Figure 6b. The reason is that the number of statistical frames becomes smaller and the noise removal rate is lower to ensure the reliability of the preliminary channel structure detection under the MR channel.

As shown in Figure 5a, the SCDR of all five methods becomes larger with the increase in SNR. Among them, the performance of the DNT method is the worst and the change with the SNR is flat. This is because DNT is a fixed threshold and does not change adaptively as the SNR changes. When the SNR is 0 dB, the SCDR of the proposed AFS-DT method is improved by $13.42\%$, $1.99\%$, and $1.32\%$ compared with the DNT, WNT, and IMOT methods, respectively. Figure 5b shows that the upward trend of the simulation curves obtained with 16QAM modulation is the same as that obtained with 4QAM modulation, but the SCDR is smaller. As shown in Figure 5b, when the SNR is 0 dB, the SCDR of AFS-DT method is improved by $13.49\%$, $2.51\%$, and $1.56\%$ compared with the DNT, WNT, and IMOT methods, respectively. As shown in Figure 7a, when the SNR is 0 dB, the SCDR of the AFS-DT method is improved by $13.33\%$, $1.89\%$, and $1.22\%$ compared with the DNT, WNT, and IMOT methods, respectively. As shown in Figure 7b, when the SNR is 0 dB, the SCDR of the AFS-DT method is improved by $13.38\%$, $2.41\%$, and $1.44\%$ compared with the DNT, WNT, and IMOT methods, respectively.

Comparing Figure 4a with Figure 5a, comparing Figure 6a with Figure 7a, and comparing the performance of the AFS method with those of other conventional methods in terms of NMSE and SCDR does not exactly correspond in the overall SNR range. For example, at the SNR of 10 dB, the SCDR of the AFS method is better than the IMOT method, but the NMSE is lower than the IMOT method. The same phenomenon can be found in Figure 4b compared with Figure 5b and in Figure 6b compared with Figure 7b. The reason is that the AFS method has a high noise removal rate in the high SNR range, but there is still a multipath sampling point misclassification problem, while the NMSE of the IMOT method is basically caused by the unremoved noise.

4.4. Analysis of BER

The BER of the VU channel with a UE speed of 40 km/h and the MR channel with a UE speed of 100 km/h are shown in Figure 8 and Figure 9, respectively. Figure 8a and Figure 9a are obtained under 4QAM modulation. Figure 8b and Figure 9b are obtained under 16QAM modulation. To more clearly show the BER differences among the different methods, the part of each figure shown in the black-dashed box has been enlarged.

As shown in Figure 8a, the BER of all channel estimation methods gradually improves as the SNR increases. The reason is that, as the SNR increases, the interference of the AWGN in the channel to the transmitted signals diminishes and the channel estimation accuracy improves. Except for the two ideal cases, the AFS-DT method still shows the best performance in the overall SNR range compared with the conventional methods. In the lower SNR range, it also outperforms the true CIR support method. Figure 8b and Figure 9a,b also show similar experimental results. In Figure 8a, when BER is $2\times {10}^{-1}$, the SNR gains of the AFS-DT method compared with the DNT, WNT, and IMOT methods are about 1.90 dB, 0.30 dB, and 0.17 dB. In Figure 8b, when the BER is $2\times {10}^{-1}$, the SNR gains of the AFS-DT method compared with the DNT, WNT, and IMOT methods are about 2.10 dB, 0.27 dB, and 0.10 dB. In Figure 9a, when the BER is $2\times {10}^{-1}$, the SNR gains of the AFS-DT method compared with the DNT, WNT, and IMOT methods are about 1.80 dB, 0.25 dB, and 0.10 dB. In Figure 9b, when the BER is $2\times {10}^{-1}$, the SNR gains of the AFS-DT method compared with the DNT, WNT, and IMOT methods are about 2.05 dB, 0.23 dB, and 0.15 dB.

4.5. Analysis of Computational Complexity

The computational complexity of the proposed AFS and AFS-DT methods and the conventional methods is shown in

Table 3. Comparison of the computational complexity of different channel estimation methods.

Channel Estimation Methods	Computational Complexity
LS	$O(N{log}_{2}N)$
DNT [33]	$O\left(N_{\mathrm{P}})+O\right(N{log}_{2}N)$
WNT [37]	$O\left(N_{\mathrm{P}})+O\right(N{log}_{2}N)$
IMOT [36]	$O\left(L{N}_{\mathrm{P}})+O\right(N{log}_{2}N)$
AFS	$O(N{log}_{2}N)$
AFS-DT	$O\left(N_{\mathrm{P}})+O\right(N{log}_{2}N)$

. For intuitive presentation, all methods consider only complex multiplication operations in one frame of OFDM symbols.

For the AFS method, the statistical complexity can be ignored and only interpolation is required. Its computational complexity is $O(N{log}_{2}N)$. For the AFS-DT method, the $N_{\mathrm{P}}$-point IFFT is first performed on the CFR at the pilots obtained by the LS method. Its computational complexity is $O(N_{\mathrm{P}}{log}_2{N}_{\mathrm{P}})$. Then, the DT is calculated. This mainly includes the calculations of multi-frame averaging, multipath power, and noise power, corresponding to (11), (24), and (25), respectively. Their computational complexity is approximately $O(N_{\mathrm{P}})$ and belongs to the same order of magnitude. The computational complexity of (18) and (19) can be ignored since the DT is computed only once in one frame of the OFDM symbols. In addition, the determination of P in (33) needs to be computed only once for a given channel and can also be ignored. Finally, the N-point FFT of the CIR after DT optimization is performed to achieve the estimation of the CFR at the data subcarriers. Its computational complexity is $O(N{log}_{2}N)$. Since $N>N_{\mathrm{P}}$, it can be considered that $O(N_{\mathrm{P}}{log}_2{N}_{\mathrm{P}})+O(N{log}_{2}N)\approx O(N{log}_{2}N)$. In summary, the computational complexity of the AFS-DT method is approximately $O\left(N_{\mathrm{P}})+O\right(N{log}_{2}N)$.

As shown in Table 3, the computational complexity of the proposed AFS-DT method is in the same order of magnitude as the DNT and WNT methods, which is lower than the IMOT method. Since both LS and AFS methods do not require time domain operations, only interpolation is required and their complexity is minimal. Overall, the AFS-DT method greatly improves the channel estimation accuracy with reasonable computational complexity.

5. Conclusions

To achieve low-complexity and high-accuracy channel estimation under fast time-varying channels, a double-threshold channel estimation method based on adaptive frame statistics is proposed. The method is able to adapt to time-varying channels by adaptively determining the number of statistical frames based on the received signals. In addition, to further improve the accuracy of channel estimation, a multi-frame averaging technique is used to expand the difference between the multipath sampling points and the noise sampling points.

The simulation results show that the channel estimation accuracy of the proposed AFS-DT method is better than the LS, DNT, WNT, and IMOT methods. At the same time, the method has a low computational complexity and is easy to implement. It has broad application prospects in the fields of optical communication [43], digital broadcasting [44], and satellite communication [45].