Joint Channel Estimation Algorithm Based on DFT and DWT

Abstract

Channel estimation is an important component of orthogonal frequency division multiplexing (OFDM) systems. The existence of virtual subcarriers leads to energy spreading in the time-domain when using Inverse Fast Fourier Transform (IFFT), resulting in poor noise reduction by the conventional Discrete Fourier Transform (DFT)-based channel estimation algorithm. To tackle this problem, this paper first proposes a segmental threshold-assisted DFT-based channel estimation algorithm. The key idea is that, by utilizing the distribution characteristics of the channel and the noise components of the channel impulse response in the time-domain, different thresholds for channel estimation under different SNR conditions are set. Compared with the traditional single-threshold DFT-based algorithm, the performance of the proposed algorithm is improved. However, it still has an estimation performance floor under high SNR. Motivated by the fact that the discrete wavelet transform (DWT)-based channel estimation algorithm can achieve better estimation performance under high SNR, we propose a joint channel estimation algorithm based on DFT and DWT, which can achieve dynamic optimal selection of the two estimation methods without any prior information. Simulation results of the Wi-Fi 6 system show that the mean square error (MSE) simulation performance of the joint channel estimation algorithm is close to its theoretical approximation. It achieves the optimal estimation of MSE and BER performance across the entire SNR range compared with the separated DFT-based or DWT-based channel estimation algorithms.

1. Introduction

With the exponential growth in the number of electronic devices available, traditional wireless local area network (WLAN) transmission standards are no longer a good solution to problems of mutual interference and limited transmission capacity during transmission between multiple terminals in areas where there is a high density of electronic devices, such as stadiums and train stations [1]. IEEE 802.11 officially released the 6th generation IEEE WLAN standard—IEEE 802.11ax (Wi-Fi 6) in 2021. It is designed to be deployed in ultra-dense environments, both indoors and outdoors, on the 2.4 GHz and 5 GHz bands, achieving an increase in the average throughput per user of at least fourfold as well as greater robustness of transmission [2,3,4].

To ensure a high bandwidth efficiency and low bit error, the receiver needs to obtain more accurate Channel State Information (CSI). Therefore, channel estimation is one of the key technologies used in MIMO-OFDM communication systems. IEEE 802.11ax standard specifies that the High Efficiency-Long Training Field (HE-LTF) in the data frame preamble is used for channel estimation, where all valid subcarriers of each OFDM symbol are used as pilots, that is, to form a block pilot. The channel can be approximated to be unchanged during the transmission period when the coherence time is greater than the frame transmission time. In this case, the complete transmission channel can be reconstructed simply by using the block pilot [5].

Among the traditional channel estimation algorithms, the Least Square (LS) algorithm has the lowest implementation complexity; however, it ignores noise interference in the signal, and the estimation effect is poor under low SNR [6]. The Linear Minimum Mean Square Error (LMMSE) algorithm considers the influence of noise. Although the estimation performance of the LMMSE is better than that of the LS algorithm, it needs to know the channel’s prior information and involves some complex matrix calculation [7]. The DFT-based channel estimation algorithm exploits the feature whereby the channel power in the time-domain is concentrated on only a few samples (taps) [8]. The basic idea can be explained as follows: First, the channel frequency response (CFR) estimated by the LS algorithm is transformed into the channel impulse response (CIR). Then, samples outside the maximum delay spread are treated as noise and set to zero. Finally, the CIR after noise reduction is transformed back to the frequency domain by DFT, thus obtaining the denoised CFR estimates. It is worth pointing out that the receiver usually cannot obtain the maximum delay spread of the real-time channel, so the cyclic prefix (CP) length is used instead. To further suppress noise in the range of the maximum delay spread by utilizing the sparseness of the channel multipath distribution, papers [9,10] propose that the most powerful part of the CIR should be retained. Paper [11] proposes to set twice the average noise power as the threshold and set the samples whose power is less than the threshold in the CP length in the estimated CIR to zero. However, these algorithms are only for OFDM systems without virtual subcarriers.

In practical OFDM systems, the Fast Fourier Transform (FFT) is often used to reduce the computational complexity of the DFT-based channel estimation algorithm. Therefore, when the number of pilot subcarriers is not an integer power of two, it is necessary to implement FFT by padding with zeros. The appearance of virtual subcarriers makes CFR generate power dispersion in the process of transforming to CIR; that is, the power of the channel expands in all the time-domain samples. The time-domain noise reduction channel estimation algorithm based on DFT will remove the channel components on all zero-set samples together, resulting in a rapid estimation performance floor with the increase of SNR. In order to solve this problem, in papers [5,12,13], paths with a channel power component smaller than its noise power component are judged as valid paths, and these valid paths are retained as much as possible by a set threshold value. The ideal threshold selection principle should be to classify the samples where the noise component is larger than the channel power component as the noise taps and set them to zero, meanwhile the remaining samples are determined as the valid taps and reserved. However, since the exact percentage of the channel power component and noise component at each sample is not available, this method is still subject to misclassification. Although the above threshold-based DFT channel estimation algorithms can alleviate this by choosing the appropriate wavelet basis and wavelet decomposition layers, the wavelet-denoising algorithm can significantly improve the estimation performance floor problem of the DFT algorithm under high SNR. Under high SNR conditions, it still needs to be further improved. In addition, if the threshold setting is too conservative, fewer noise components will be removed, and the algorithm estimation performance will be degraded under low SNR.

Some studies have applied wavelet-denoising to channel estimation algorithms [14,15,16,17,18]. The basic principle of the wavelet-denoising algorithm is to transform the CFR estimated by the LS algorithm to the wavelet domain, process the wavelet coefficients using the threshold method, and then reconstruct the signal to achieve noise cancellation. By choosing the appropriate wavelet basis and wavelet decomposition layers, the wavelet-denoising algorithm can significantly improve the estimation performance floor problem of the DFT algorithm under high SNR conditions.

The main contributions of our work can be summarized as follows: For OFDM systems with virtual subcarriers, this paper first proposes the Segmented Threshold DFT-based (ST-DFT) channel estimation algorithm. By using the different distribution characteristics of the channel and noise components on the time-domain samples, a new CIR effective taps decision threshold is proposed to improve the estimation performance under the condition of channel energy dispersion. Then, motivated by the fact that the DWT algorithm and the ST-DFT algorithm have better detection performances under high and low SNR conditions, respectively, we further propose a joint channel estimation algorithm based on ST-DFT and DWT. The MSE estimation performance of the two algorithms under different channel conditions shows that the joint channel estimation algorithm can dynamically select the algorithm with the best estimation performance under different SNR conditions. The simulation results also show that, compared with the single estimation algorithm, the DFT channel estimation algorithm with an improved threshold produces better overall estimation performances. Furthermore, the proposed joint channel estimation algorithm can determine the optimal channel estimation solution under the full SNR range without any prior conditions.

The remainder of this paper is organized as follows: Section 2 outlines the system model. Section 3 introduces the existing DFT algorithm and proposes the ST-DFT algorithm. Section 4 briefly describes the conventional DWT algorithm. Section 5 describes the proposed joint channel estimation algorithm based on ST-DFT and DWT in detail. In addition, Section 6 presents simulations and the discussion, and Section 7 gives the conclusions of this paper.

2. System Model

Assume that there are N subcarriers in an OFDM system; in order to avoid inter-symbol interference (ISI), a cyclic prefix of length $N_{\mathrm{CP}}$ is usually inserted before each OFDM symbol, i.e., the cyclic prefix length is greater than channel pulse length. $X\left[k\right]$ is the frequency domain symbol modulated on the kth subcarrier. The time-domain signal $x\left[n\right]$ formed by the IDFT of $X\left[k\right]$ is denoted as [13]:

$$x\left[n\right]=\frac{1}{N}\sum _{k=0}^{N-1}X\left[k\right]·e^{\frac{j2\pi kn}{N}},-N_{\mathrm{CP}}\le n\le N-1.$$

(1)

The time-domain channel impulse response of a multipath fading channel can be written as [13]:

$$h\left[n\right]=\sum _{l=0}^{L-1}{h}_l\delta \left[n-{\tau }_l\right],$$

(2)

where L denotes the number of paths of the multipath channel, and $h_l$ and ${\tau }_l$ denote the complex channel gain and channel delay of the lth path, respectively. Correspondingly, the channel frequency response is [11]:

$$H\left[k\right]=\sum _{l=0}^{L-1}{h}_l{e}^{-j\frac{2\pi }{N}k{\tau }_l},0\le k\le N-1.$$

(3)

Assuming ideal synchronization of the receiver, the received signal in the time-domain can be expressed as [11]:

$$y\left[n\right]=x\left[n\right]\ast h\left[n\right]+w\left[n\right],$$

(4)

where $w\left[n\right]$ denotes the additive Gaussian white noise, and “∗” represents linear convolution. After removing the cyclic prefix from Equation (4), the DFT can be used to obtain the received frequency domain signal of the kth subcarrier, i.e., [11]:

$$Y\left[k\right]=X\left[k\right]H\left[k\right]+W\left[k\right],0\le k\le N-1,$$

(5)

where $W\left[k\right]$ denotes the noise component of the kth subcarrier with a variance of ${\sigma }_{wf}^2$.

If $X\left[k\right]$ is used as the pilot symbol in the frequency domain, the LS estimation of the channel frequency response of the kth subcarrier can be given by [11]:

$${\widehat{H}}_{\mathrm{LS}}\left[k\right]=H\left[k\right]+\frac{W\left[k\right]}{X\left[k\right]}.$$

(6)

The MSE of the kth subcarrier of the LS channel estimate can be expressed as [11]:

$$\mathrm{MS}{\mathrm{E}}_{\mathrm{LS}}\left[k\right]=\frac{\beta }{\mathrm{SNR}},$$

(7)

where $\beta =E\left[{\left|X\left[k\right]\right|}^2\right]E\left[{\left|X\left[k\right]\right|}^{-2}\right]$. Without a loss of generality, let $E\left[{\left|X\left[k\right]\right|}^2\right]=1$, so $\beta =1$. Since $\mathrm{SNR}=E\left[{\left|X\left[k\right]\right|}^2\right]/{\sigma }_{wf}^2$, we have [11]

$$\mathrm{MS}{\mathrm{E}}_{\mathrm{LS}}\left[k\right]=\frac{1}{\mathrm{SNR}}={\sigma }_{wf}^2.$$

(8)

That is, the MSE of the LS algorithm is the same as the average power of the frequency domain noise.

3. DFT-Based Channel Estimation Algorithm

3.1. Traditional DFT-Based Channel Estimation Algorithm

The DFT-based channel estimation algorithm takes advantage of the concentration of channel power in a few samples in the CIR for noise reduction to further improve the estimation accuracy. The conventional DFT-based channel estimation algorithm first converts the CFR estimated by the LS algorithm to the time-domain via IDFT to obtain the estimated time-domain channel impulse response, i.e., [11]:

$${\widehat{h}}_{\mathrm{LS}}\left[n\right]=\frac{1}{N}\sum _{k=0}^{N-1}{\widehat{H}}_{\mathrm{LS}}\left[k\right]exp\left(\frac{j2\pi kn}{N}\right)=h_{\mathrm{LS}}\left[n\right]+w\left[n\right],$$

(9)

where $h_{\mathrm{LS}}\left[n\right]$ denotes the channel component on the CIR, and $w\left[n\right]$ is the time-domain complex Gaussian white noise.

In the absence of virtual subcarriers, the channel components exist only within the CP length of ${\widehat{h}}_{\mathrm{LS}}\left[n\right]$, so all samples outside the CP length can be considered noise and are removed by setting their values to zero. This process can be regarded as adding a window to the whole ${\widehat{h}}_{\mathrm{LS}}\left[n\right]$. The ${\widehat{h}}_{\mathrm{LS}}\left[n\right]$ after windowing can be expressed as

$${\widehat{h}}_{\mathrm{W}\text{-}\mathrm{DFT}}\left[n\right]={\widehat{h}}_{\mathrm{LS}}\left[n\right]R_{\mathrm{DFT}}\left[n\right].$$

(10)

$$R_{\mathrm{DFT}}\left[n\right]=\left\{\begin{array}{cc}1,& n=0,1,\cdots ,N_{\mathrm{CP}}-1,\\ 0,& otherwise.\end{array}\right.$$

(11)

Finally, the DFT of ${\widehat{h}}_{\mathrm{W}\text{-}\mathrm{DFT}}\left[n\right]$ is performed to obtain the denoised CFR. The MSE of kth subcarrier of the plus-window DFT algorithm is [11]

$$\mathrm{MS}{\mathrm{E}}_{\mathrm{W}\text{-}\mathrm{DFT}}\left[k\right]=\frac{N-K}{N}\frac{\beta }{\mathrm{SNR}},$$

(12)

where K is the number of zero-set samples.

In a practical OFDM system, virtual subcarriers generally suppress out-of-band leakage. Assuming that only $N_{\mathrm{P}}(N_{\mathrm{P}}<N)$ pilot subcarriers in all N subcarriers participate in channel estimation, and the set of the pilot subcarriers is ${\mathrm{\Omega }}_{\mathrm{P}}$, then the estimated CIR under virtual subcarrier conditions can be expressed as:

$$\begin{array}{c}{\widehat{h}}_{\mathrm{V}}\left[n\right]=\frac{1}{N}{\displaystyle \sum _{k=0}^{N-1}}{\widehat{H}}_{\mathrm{LS}}\left[k\right]R_{\mathrm{P}}\left[k\right]e^{j\frac{2\pi }{N}kn}\\ ={\displaystyle \sum _{m=0}^{N-1}}h\left[m\right]r_{\mathrm{P}}{\left[\left[n-m\right]\right]}_N+\frac{1}{N}{\displaystyle \sum _{k=0}^{N-1}}\frac{W\left[k\right]R_{\mathrm{P}}\left[k\right]}{X\left[k\right]}{e}^{j\frac{2\pi }{N}kn}\\ =h_{\mathrm{V}}\left[n\right]+\tilde{w}\left[n\right],\end{array}$$

(13)

where ${\left[\left[·\right]\right]}_N$ is the modulo N operation, $h_{\mathrm{V}}\left[n\right]$ is the channel component of the CIR, $\tilde{w}\left[n\right]$ is the noise component of the CIR with a variance of ${\tilde{\sigma }}_{\mathrm{wt}}^2=\frac{N_P}{N^2}{\sigma }_{wf}^2$, and $R_{\mathrm{P}}\left[k\right]$ and $r_{\mathrm{P}}\left[n\right]$ are the frequency domain window of the pilot and its time-domain response, respectively, and

$$R_{\mathrm{P}}\left[k\right]=\left\{\begin{array}{cc}1,& k\in {\mathrm{\Omega }}_{\mathrm{P}},\\ 0,& otherwise.\end{array}\right.$$

(14)

$$r_{\mathrm{P}}\left[n\right]=\frac{1}{N}\sum _{k=0}^{N-1}{R}_{\mathrm{P}}\left[k\right]e^{j\frac{2\pi }{N}kn}.$$

(15)

Appendix A shows that the MSE of kth subcarrier of the plus-window DFT algorithm in the presence of virtual subcarriers can be approximated as:

$${\widetilde{\mathrm{MSE}}}_{\mathrm{V}\text{-}\mathrm{W}\text{-}\mathrm{DFT}}\left[k\right]\approx \sum _{n=0,n\in \varnothing }^{N-1}E\left[{\left|h_{\mathrm{V}}\left[n\right]\right|}^2\right]+\sum _{n=0,n\notin \varnothing }^{N-1}E\left[{\left|\tilde{w}\left[n\right]\right|}^2\right],$$

(16)

where ∅ is the set of samples which are set to zero in ${\widehat{h}}_{\mathrm{V}}\left[n\right]$.

In practical systems, it is obvious that the ideal multipath channel component $h_{\mathrm{V}}\left[n\right]$ cannot be obtained directly from ${\widehat{h}}_{\mathrm{V}}\left[n\right]$. Fortunately, since $E\left[{\left|{\widehat{h}}_{\mathrm{V}}\left[n\right]\right|}^2\right]=E\left[{\left|h_{\mathrm{V}}\left[n\right]\right|}^2\right]+E\left[{\left|\tilde{w}\left[n\right]\right|}^2\right]$, Equation (16) can be rewritten as:

$$\begin{array}{c}{\widetilde{\mathrm{MSE}}}_{\mathrm{V}\text{-}\mathrm{W}\text{-}\mathrm{DFT}}\left[k\right]\approx {\displaystyle \sum _{n=0,n\in \varnothing }^{N-1}}E\left[{\left|h_{\mathrm{V}}\left[n\right]\right|}^2\right]+{\displaystyle \sum _{n=0}^{N-1}}E\left[{\left|\tilde{w}\left[n\right]\right|}^2\right]-{\displaystyle \sum _{n=0,n\in \varnothing }^{N-1}}E\left[{\left|\tilde{w}\left[n\right]\right|}^2\right]\\ ={\displaystyle \sum _{n=0,n\in \varnothing }^{N-1}}E\left[{\left|{\widehat{h}}_{\mathrm{V}}\left[n\right]\right|}^2\right]+{\displaystyle \sum _{n=0}^{N-1}}E\left[{\left|\tilde{w}\left[n\right]\right|}^2\right]-2{\displaystyle \sum _{n=0,n\in \varnothing }^{N-1}}E\left[{\left|\tilde{w}\left[n\right]\right|}^2\right].\end{array}$$

(17)

It can be seen from Equation (13) that the introduction of virtual sub-carriers leads to the CIR obtained by IFFT, resulting in channel multipath power spread. Consequently, if all samples beyond the CP length range are still forced to be set to zero, the MSE performance of channel estimation will deteriorate. In addition, it is not mandatory to set a certain area beyond the CP length range to zero, but it is necessary to use the uniform threshold for the entire SNR range given by [13] to judge all the samples, which can alleviate the estimation performance floor under high SNR conditions. However, the estimation performance under medium and low SNR conditions is difficult to guarantee. This is because, at medium and low SNR, the power of the channel component in the samples beyond the CP length is generally less than the power of the noise component. If the instantaneous large noise component causes the overall power of the sample to exceed the threshold, it will inevitably lead to misjudgment of the effective channel samples, thereby reducing the channel estimation performance.

3.2. Channel Estimation Algorithm Based on Segmented Threshold DFT

To deal with the problem that the performance of the existing unified threshold DFT algorithm needs to improve under medium and low SNR conditions, this paper proposes a channel estimation algorithm based on segmented threshold DFT. By setting different thresholds in different SNR segments, the channel multipath can be estimated effectively.

According to the multipath distribution characteristics in the estimated CIR, under medium and low SNR conditions, the power of the channel component in and around the channel delay spread region is relatively high, while the power of the noise component in the rest of the region is relatively high.

When virtual subcarriers are present, the channel delay spread depends on the time-domain response of the pilot window function. Taking the Wi-Fi6 channel estimation scenario as an example, let $N=256$, $N_{\mathrm{P}}=242$ in the HE-LFT field. The time-domain impulse response of the pilot window function expressed by Equation (15) is shown in Figure 1.

As seen in Figure 1, for a single-tap channel, when there is power spreading, the power is mainly concentrated in the range with a small offset. For example, for a Wi-Fi6 system with a CP length of 1/16 (i.e., $N_{\mathrm{CP}}=16$), from Equation (15) we know that 99% of the channel power is included in $\left[0:N_{\mathrm{CP}}-1,N-N_{\mathrm{CP}}:N-1\right]$, which can be regarded as the channel observation area. For multipath channels, since the power of each path generally decreases with an increase in its time delay, the channel power that spreads outside the channel observation area is very limited, so the main power of the channel is still concentrated in the channel observation area. Correspondingly, the area outside the channel observation area can be regarded as the noise observation area ${\mathrm{\Omega }}_w$, and its range is $\left[N_{\mathrm{CP}}:N-N_{\mathrm{CP}}-1\right]$.

Let the average power of the samples in ${\mathrm{\Omega }}_w$ be:

$$\overline{P}=\frac{\sum {\left|{\widehat{h}}_{\mathrm{V}}\left[n\right]\right|}^2}{N-2N_{\mathrm{CP}}},n\in {\mathrm{\Omega }}_w.$$

(18)

In addition, through Equation (13), we know that the average power of the channel component and noise component in ${\mathrm{\Omega }}_w$ can be expressed as:

$${\overline{P}}_u=\frac{\sum {\left|h_{\mathrm{V}}\left[n\right]\right|}^2}{N-2N_{\mathrm{CP}}},{\overline{P}}_w=\frac{\sum {\left|\tilde{w}\left[n\right]\right|}^2}{N-2N_{\mathrm{CP}}},n\in {\mathrm{\Omega }}_w.$$

(19)

Since the channel component and the noise component are independent of each other, the expected values of the sample power, the channel component power, and the mean value of the noise component power satisfy the relationship $E\left[\overline{P}\right]=E\left[{\overline{P}}_u\right]+E\left[{\overline{P}}_w\right]$, where $E\left[{\overline{P}}_w\right]={\tilde{\sigma }}_{wt}^2$. The results can be divided into two cases for discussion.

The first case is when $E\left[{\overline{P}}_u\right]<E\left[{\overline{P}}_w\right]$, i.e., $E\left[\overline{P}\right]<2E\left[{\overline{P}}_w\right]$, and the SNR is relatively low. The power of the noise component for all samples in the ${\mathrm{\Omega }}_w$ region can be considered to be generally greater than the power of the channel component. Under this condition, the decision threshold of the CIR effective path can be set to the maximum value of the power of the samples in the ${\mathrm{\Omega }}_w$ region, and all the samples in the ${\mathrm{\Omega }}_w$ region will be set to zero. The maximum power can be expressed as:

$$P_{\max }=max\left\{{\left|\widehat{h}\left[n\right]\right|}^2\right\},n\in {\mathrm{\Omega }}_w.$$

(20)

The other case is when $E\left[{\overline{P}}_u\right]\ge E\left[{\overline{P}}_w\right]$, i.e., $E\left[\overline{P}\right]\ge 2E\left[{\overline{P}}_w\right]$. The SNR is relatively high. The power of the noise component for all samples in the ${\mathrm{\Omega }}_w$ region can be considered to be less than that of the channel component. Under this condition, the decision threshold of the CIR effective path can be set to $2E\left[{\overline{P}}_w\right]$, in order to retain the samples with greater channel power in the ${\mathrm{\Omega }}_w$ region.

Therefore, the decision threshold of the effective taps in the proposed channel estimation algorithm based on segment threshold DFT can be expressed as:

$$T=\left\{\begin{array}{cc}{P}_{\max },& \overline{P}<2{\tilde{\sigma }}_{wt}^2,\\ 2{\tilde{\sigma }}_{wt}^2,& otherwise,\end{array}\right.$$

(21)

and the specific calculation of ${\tilde{\sigma }}_{wt}^2$ is described in detail in Section 5.

The specific steps used in the ST-DFT channel estimation algorithm are as follows:

• Step 1: Based on the received signal and training sequence, estimate the frequency domain response of the channel ${\widehat{H}}_{\mathrm{LS}}$ estimated using the LS algorithm;
• Step 2: Perform N-point IFFT to ${\widehat{H}}_{\mathrm{LS}}$ to get the time-domain channel impulse response ${\widehat{h}}_{\mathrm{V}}$ with power dispersion;
• Step 3: Calculate the threshold T according to Equation (21);
• Step 4: Perform a threshold determination on ${\widehat{h}}_{\mathrm{V}}$, and get ${\widehat{h}}_{\mathrm{ST}\text{-}\mathrm{DFT}}.$

$${\widehat{h}}_{\mathrm{ST}\text{-}\mathrm{DFT}}\left[n\right]=\left\{\begin{array}{cc}{\widehat{h}}_{\mathrm{v}}\left[n\right],& {\left|{\widehat{h}}_{\mathrm{v}}\left[n\right]\right|}^2\ge T,\\ 0,& {\left|{\widehat{h}}_{\mathrm{v}}\left[n\right]\right|}^2<T;\end{array}\right.$$

(22)

• Step 5: Perform N-point FFT to ${\widehat{h}}_{\mathrm{ST}\text{-}\mathrm{DFT}}$ to obtain the channel frequency response estimate after denoising ${\widehat{H}}_{\mathrm{ST}\text{-}\mathrm{DFT}}$.

It is assumed that the set of samples set to zero in ${\widehat{h}}_{\mathrm{v}}\left[n\right]$ is $\mathsf{\Theta }$ and the number of samples is K. The ST-DFT algorithm yields different $\mathsf{\Theta }$ for each estimation of different ${\widehat{h}}_{\mathrm{v}}\left[n\right]$. For the convenience of the engineering implementation, the expected values $E\left[{\left|{\widehat{h}}_{\mathrm{V}}\left[n\right]\right|}^2\right]$ and $E\left[{\left|\tilde{w}\left[n\right]\right|}^2\right]$ in Equation (17) are replaced by the instantaneous value in the single estimated CIR. The MSE of kth subcarrier of the ST-DFT channel estimation algorithm can be approximated as:

$${\widetilde{\mathrm{MSE}}}_{\mathrm{ST}\text{-}\mathrm{DFT}}\left[k\right]\approx \sum _{n=0,n\in \mathsf{\Theta }}^{N-1}{\left|{\widehat{h}}_{\mathrm{V}}\left[n\right]\right|}^2+\sum _{n=0}^{N-1}{\left|\tilde{w}\left[n\right]\right|}^2-2\sum _{n=0,n\in \mathsf{\Theta }}^{N-1}{\left|\tilde{w}\left[n\right]\right|}^2.$$

(23)

For a single estimated CIR (${\widehat{h}}_{\mathrm{v}}\left[n\right]$) and a given effective tap decision threshold T, among all samples with the same or similar channel component power values, those samples that are always superimposed with less power noise are more likely to be classified as noisy samples and are set to zero. That is, the average power of the noise component superimposed on those zero-set samples is no greater than that of the noise component superimposed on all samples in the estimated CIR. At the same time, the average power of the superimposed noise component on taps (such as K taps) that are set to zero should not be less than the average power of the K noise component with the smallest superimposed power value on all samples in the CIR. Therefore, as shown in Appendix B, the range of values for the total noise power ${\displaystyle \sum _{n=0,n\in \mathsf{\Theta }}^{N-1}}{\left|\tilde{w}\left[n\right]\right|}^2$ contained in all zero-set samples can be expressed as:

$$\left(\frac{K}{N}+\left(1-\frac{K}{N}\right)ln\left(1-\frac{K}{N}\right)\right)\frac{N_{\mathrm{P}}}{N}{\sigma }_{s\text{-}wf}^2\le \sum _{n=0,n\in \mathsf{\Theta }}^{N-1}{\left|\tilde{w}\left[n\right]\right|}^2\le \frac{K{N}_{\mathrm{P}}}{N^2}{\sigma }_{s\text{-}wf}^2,$$

(24)

where ${\displaystyle \sum _{n=0}^{N-1}}{\left|\tilde{w}\left[n\right]\right|}^2=\frac{N_{\mathrm{P}}}{N}{\sigma }_{s\text{-}wf}^2$, ${\sigma }_{s\text{-}wf}^2$ is the frequency domain noise variance on the active subcarriers in the current symbol. Therefore, the value of ${\displaystyle \sum _{n=0,n\in \mathsf{\Theta }}^{N-1}}{\left|\tilde{w}\left[n\right]\right|}^2$ can be approximated as the middle value of its upper and lower bounds, i.e.,

$$\sum _{n=0,n\in \mathsf{\Theta }}^{N-1}{\left|\tilde{w}\left[n\right]\right|}^2\approx \left(\frac{K}{N}+\frac{1}{2}\left(1-\frac{K}{N}\right)ln\left(1-\frac{K}{N}\right)\right)\frac{N_{\mathrm{P}}}{N}{\sigma }_{s\text{-}wf}^2.$$

(25)

Finally, the MSE of the kth subcarrier of the ST-DFT channel estimation algorithm can be approximated as:

$${\widetilde{\mathrm{MSE}}}_{\mathrm{ST}\text{-}\mathrm{DFT}}\left[k\right]\approx \left\{\begin{array}{cc}{\displaystyle \sum _{n=0,n\in \mathsf{\Theta }}^{N-1}}{\left|{\widehat{h}}_{\mathrm{V}}\left[n\right]\right|}^2+\frac{N_P}{N}{\sigma }_{s\text{-}wf}^2-2\frac{K{N}_P}{N^2}{\sigma }_{s\text{-}wf}^2,& T=P_{\max },\\ {\displaystyle \sum _{n=0,n\in \mathsf{\Theta }}^{N-1}}{\left|{\widehat{h}}_{\mathrm{V}}\left[n\right]\right|}^2+\frac{N_P}{N}{\sigma }_{s\text{-}wf}^2-2\left(\frac{K}{N}+\frac{1}{2}\left(1-\frac{K}{N}\right)ln\left(1-\frac{K}{N}\right)\right)\frac{N_P}{N}{\sigma }_{s\text{-}wf}^2,& otherwise.\end{array}\right.$$

(26)

4. DWT-Based Channel Estimation Algorithm

The wavelet transform is often used to analyze signals with different frequencies and time resolution and to decompose the signal into detail coefficients and approximate coefficients, which are defined as high-frequency coefficients and low-frequency coefficients, respectively. Compared with the traditional Fourier transform, the wavelet transform has advantages in terms of accurately decomposing and reconstructing finite, aperiodic, and non-stationary signals. For the channel frequency response ${\widehat{H}}_{\mathrm{LS}}$ obtained by the LS channel estimation, the Mallat algorithm is used for discrete wavelet transform, and the output wavelet coefficients can be expressed as [16]:

$$\begin{array}{c}{C}_{j,k}={\displaystyle \sum _n}{C}_{j-1,n}{h}^{\ast }\left(n-2k\right),\\ D_{j,k}={\displaystyle \sum _n}{D}_{j-1,n}{g}^{\ast }\left(n-2k\right),\end{array}$$

(27)

where $C_{j,k}$ is the approximate wavelet coefficient, $D_{j,k}$ is the detailed wavelet coefficient, j is the order of wavelet decomposition, $h^{\ast }$ and $g^{\ast }$ are a set of orthogonal mirror filters, and ${\widehat{H}}_{\mathrm{LS}}=C_0=D_0$.

In the practical wavelet decomposition, the length of the coefficients after each layer of wavelet decomposition is not the number of coefficients in the previous layer halved because of the addition of filters. The number of wavelet coefficients after decomposition in each layer can be expressed as:

$$L_j=floor\left(\frac{L_{j-1}-1}{2}\right)+N_{\mathrm{FB}},$$

(28)

where $L_j$ represents the wavelet coefficients after the jth wavelet decomposition, $floor\left(·\right)$ denotes the downward rounding, and $N_{FB}$ is the length of the filter. The total wavelet coefficients $L_{\mathrm{total}}$ is the total number of wavelet coefficients in the multi-layer decomposition.

At present, there is no unified standard for the selection of wavelet bases, and all walks of life choose according to their actual needs [19,20]. Since this paper studies the channel estimation algorithm for the block pilot, the Db4 wavelet with better continuity is selected as the mother wavelet [16]. Having too few wavelet decomposition layers will lead to an unsatisfactory denoising effect, and having too many decomposition layers may misjudge the effective signal as noise and remove the valid signal. At present, the wavelet-denoising algorithm in the field of channel estimation is generally decomposed into 1∼3 layers [16,17,18]. This paper does not discuss the optimal number of decomposition layers for the DWT. To ensure that the effective signal is preserved as much as possible under high SNR condition, the intermediate value of two layers is selected as the wavelet decomposition layers.

The discrete wavelet transform was performed on the received signal interfered with by Gaussian white noise. Since Gaussian white noise is independently distributed, the white noise after wavelet transform is also highly random in the wavelet domain. Therefore, the detailed coefficient obtained by the DWT corresponds to Gaussian white noise, and its size and variance will change greatly with the transformation of the wavelet scale. On the contrary, the approximate coefficient corresponds to the useful signal, and its size and variance change little with the transformation of the wavelet scale [21]. The noise mixed in the signal can be filtered out by thresholding the detail coefficients. In this paper, the unbiased likelihood estimation was used as the decision threshold for wavelet-denoising with the following calculation expression [22]:

$$T_{\mathrm{DWT}}=\frac{median\left(\left|D_i\right|\right)}{0.6745},$$

(29)

where $D_i$ is the ith wavelet coefficient in the detailed coefficients decomposed by Equation (27), $median\left(·\right)$ means to take the middle value. In order to give the processed wavelet coefficients have better continuity and smoothness, a soft threshold function is used for processing, which is given by [16]:

$$D_{j,k}^{\prime }=\left\{\begin{array}{cc}\mathrm{sgn}\left(D_{j,k}\right)\left(\left|D_{j,k}\right|-T_{\mathrm{DWT}}\right),& \left|D_{j,k}\right|\ge T_{\mathrm{DWT}},\\ 0,& \left|D_{j,k}\right|<T_{\mathrm{DWT}},\end{array}\right.$$

(30)

where $D_{j,k}^{\prime }$ is the detailed wavelet coefficient after threshold denoising, and $\mathrm{sgn}\left(·\right)$ is the symbolic function. The wavelet reconstruction equation can be expressed as [16]:

$$C_{j-1,k}^{\prime }=\sum _m{C}_{j,m}h\left(m-2k\right)+\sum _m{D}_{j,m}^{\prime }g\left(m-2k\right).$$

(31)

${\widehat{H}}_{\mathrm{LS}}$ is denoised by the discrete wavelet transform to obtain ${\widehat{H}}_{\mathrm{DWT}}$, which satisfies ${\widehat{H}}_{\mathrm{DWT}}=C_0^{\prime }$.

The specific steps of the DWT channel estimation algorithm are as follows:

• Step 1: Based on the received signal and the training sequence, use the LS algorithm to estimate the channel frequency domain response ${\widehat{H}}_{\mathrm{LS}}$;
• Step 2: Perform a two-level wavelet decomposition on ${\widehat{H}}_{\mathrm{LS}}$ and obtain $C_{j,k}$ and $D_{j,k}$;
• Step 3: Based on Equations (29) and (30), $D_{j,k}$ performs the threshold decision and soft threshold processing to obtain $D_{j,k}^{\prime }$, where the number of zero-set wavelet coefficients is $L_0$;
• Step 4: Use $C_{j,k}$ and $D_{j,k}^{\prime }$ for wavelet reconstruction to obtain the frequency domain response of the denoised channel ${\widehat{H}}_{\mathrm{DWT}}$.

In the wavelet domain, the useful signal power is concentrated in some larger wavelet coefficients, and the noise power is distributed across the whole wavelet domain [15]. Moreover, the noise still behaves as additive white Gaussian noise in the wavelet domain. Like the windowed DFT algorithm, the MSE expression of the DWT-based channel estimation algorithm can also be regarded as the total noise power in the system minus the noise power on the zero-set wavelet coefficients. Therefore, the wavelet-denoising MSE of kth subcarrier can be approximately expressed as:

$${\widetilde{\mathrm{MSE}}}_{\mathrm{DWT}}\left[k\right]\approx \frac{L_{\mathrm{total}}-L_0}{L_{\mathrm{total}}}\frac{1}{\mathrm{SNR}}=\frac{L_{\mathrm{total}}-L_0}{L_{\mathrm{total}}}{\sigma }_{wf}^2.$$

(32)

5. Joint Channel Estimation Algorithm Based on DFT and DWT

On the one hand, although the channel estimation algorithm based on segmented threshold DFT proposed in Section 3 can effectively reduce the channel estimation error under medium and low SNR conditions, due to the existence of virtual subcarriers, it still cannot eliminate the problem of an estimation performance floor under high SNR. On the other hand, the wavelet-denoising algorithm converts noisy signals to the wavelet domain for processing. Selecting the wavelet base that matches the signal and the appropriate decomposition times can separate the effective signal from the noise, which can effectively reduce the estimation performance floor of channel estimation under high SNR. Therefore, based on the estimation performance characteristics of the two algorithms in different SNR intervals, the channel estimation performance can be optimized across the entire SNR range by dynamically and reasonably selecting the two algorithms for channel estimation.

To achieve a dynamic selection of better denoising algorithms from wavelet-denoising algorithms and segmented threshold DFT algorithms, it is necessary to find an effective selection decision metric. Obviously, the most direct decision metric is the channel estimation MSE of the two algorithms under given CIR conditions. According to the theoretical (approximate) MSE values based on the threshold DFT algorithm and the wavelet-denoising algorithm presented in Section 3 and Section 4, the corresponding estimates can be obtained.

5.1. Estimation of Noise Variance in the Frequency Domain

Based on the correct wavelet basis and the number of wavelet decomposition layers, the wavelet-denoising algorithm can carry out denoising with almost no effective signal removed by mistake. Therefore, the noise variance of the pilot subcarriers can be estimated based on the CFR values before and after wavelet-denoising. Thus, the instantaneous estimated value of the mean square error of the denoised noise on each subcarrier CFR can be expressed as:

$${\widehat{\sigma }}_{s\text{-}d}^2=\frac{{\sum }_{k\in {\mathrm{\Omega }}_{\mathrm{P}}}\left[{\left|{\widehat{H}}_{\mathrm{LS}}\left[k\right]-{\widehat{H}}_{\mathrm{DWT}}\left[k\right]\right|}^2\right]}{N_{\mathrm{P}}},$$

(33)

where ${\widehat{H}}_{\mathrm{LS}}\left[k\right]$ and ${\widehat{H}}_{\mathrm{DWT}}\left[k\right]$ represent the frequency domain channel response estimated by the LS algorithm before wavelet-denoising and after denoising, respectively. From Equations (7), (8) and (32) the approximate value of the average power of the noise removed from the frequency response of each subcarrier channel after wavelet-denoising can be expressed as:

$${\sigma }_{s\text{-}d}^2\approx \mathrm{MS}{\mathrm{E}}_{\mathrm{LS}}\left[k\right]-{\widetilde{\mathrm{MSE}}}_{\mathrm{DWT}}\left[k\right]=\frac{L_0}{L_{\mathrm{total}}}{\sigma }_{wf}^2.$$

(34)

Therefore, the instantaneous frequency domain noise variance can be estimated as:

$${\widehat{\sigma }}_{s\text{-}wf}^2=\frac{L_{\mathrm{total}}}{L_0}{\widehat{\sigma }}_{s\text{-}d}^2.$$

(35)

5.2. MSE Estimation of DWT and ST-DFT Channel Estimation Algorithms

By substituting Equation (35) into Equations (26) and (32), the MSE estimates for the DWT algorithm and the ST-DFT algorithm separately:

$${\widehat{\mathrm{MSE}}}_{\mathrm{DWT}}\left[k\right]=\frac{L_{\mathrm{total}}-L_0}{L_{\mathrm{total}}}{\widehat{\sigma }}_{s\text{-}wf}^2.$$

(36)

$${\widehat{\mathrm{MSE}}}_{\mathrm{ST}\text{-}\mathrm{DFT}}\left[k\right]\approx \left\{\begin{array}{cc}{\displaystyle \sum _{n=0,n\in \mathsf{\Theta }}^{N-1}}{\left|{\widehat{h}}_{\mathrm{V}}\left[n\right]\right|}^2+\frac{N_{\mathrm{P}}}{N}{\widehat{\sigma }}_{s\text{-}wf}^2-2\frac{K{N}_{\mathrm{P}}}{N^2}{\widehat{\sigma }}_{s\text{-}wf}^2,& T=P_{\max },\\ {\displaystyle \sum _{n=0,n\in \mathsf{\Theta }}^{N-1}}{\left|{\widehat{h}}_{\mathrm{V}}\left[n\right]\right|}^2+\frac{N_{\mathrm{P}}}{N}{\widehat{\sigma }}_{s\text{-}wf}^2-2\left(\frac{K}{N}+\frac{1}{2}\left(1-\frac{K}{N}\right)ln\left(1-\frac{K}{N}\right)\right)\frac{N_{\mathrm{P}}}{N}{\widehat{\sigma }}_{s\text{-}wf}^2,& otherwise,\end{array}\right.$$

(37)

when there are no prior conditions, for a given CIR, based on the MSE estimated value of each algorithm given by Equations (36) and (37), the ST-DFT estimation algorithm and the DWT-based estimation algorithm can be selected optimally.

5.3. Steps of the Joint Channel Estimation Algorithm

The specific steps used in the joint channel estimation algorithm based on DFT and DWT are as follows:

• Step 1: Based on the received signal and training sequence, estimate ${\widehat{H}}_{\mathrm{LS}}$ using the LS channel estimation algorithm;
• Step 2: Perform wavelet-denoising on ${\widehat{H}}_{\mathrm{LS}}$ to obtain ${\widehat{H}}_{\mathrm{DWT}}$ after wavelet-denoising;
• Step 3: Calculate the estimated value ${\widehat{\sigma }}_{s\text{-}wf}^2$ of the noise variance in the frequency domain according to Equation (35);
• Step 4: Conduct N-point IFFT on ${\widehat{H}}_{\mathrm{LS}}$ to get the time-domain channel impulse response ${\widehat{h}}_{\mathrm{V}}$ with energy dispersion;
• Step 5: Calculate the threshold value T according to Equation (21) and make a threshold judgment on ${\widehat{h}}_{\mathrm{V}}$ through Equation (22) to obtain ${\widehat{h}}_{\mathrm{ST}\text{-}\mathrm{DFT}}$, and count the number K of zero-set paths in ${\widehat{h}}_{\mathrm{V}}$;
• Step 6: Based on Equations (36) and (37), calculate the MSE estimates of the wavelet-denoising algorithm and the ST-DFT algorithm ${\widehat{\mathrm{MSE}}}_{\mathrm{DWT}}$ and ${\widehat{\mathrm{MSE}}}_{\mathrm{ST}\text{-}\mathrm{DFT}}$;
• Step 7: Compare ${\widehat{\mathrm{MSE}}}_{\mathrm{DWT}}$ and ${\widehat{\mathrm{MSE}}}_{\mathrm{ST}\text{-}\mathrm{DFT}}$. If ${\widehat{\mathrm{MSE}}}_{\mathrm{DWT}}>{\widehat{\mathrm{MSE}}}_{\mathrm{ST}\text{-}\mathrm{DFT}}$, conduct N-point IFFT on ${\widehat{h}}_{\mathrm{ST}\text{-}\mathrm{DFT}}$ to obtain ${\widehat{H}}_{\mathrm{ST}\text{-}\mathrm{DFT}}$ as the channel estimation value; if ${\widehat{\mathrm{MSE}}}_{\mathrm{DWT}}\le {\widehat{\mathrm{MSE}}}_{\mathrm{ST}\text{-}\mathrm{DFT}}$, directly output ${\widehat{H}}_{\mathrm{DWT}}$ in Step 2 as the channel estimation value.

5.4. Computational Complexity Analysis

We use the number of complex multiplications as a measure of the computational complexity of the algorithm. The LS channel estimation algorithm requires $N_{\mathrm{P}}$ complex multiplications. The N-point FFT and IFFT require $\frac{N}{2}{log}_{2}N$ complex multiplications, respectively [12]. The T-DFT and ST-DFT algorithms need to calculate the power on each CIR, which requires N complex multiplications. The calculation of the noise variance estimate requires $N_{\mathrm{P}}$ complex multiplications. The computational complexity of both fast wavelet decomposition and reconstruction is $N_{\mathrm{P}}$ [16]. Then the computational complexity of different algorithms is shown in

Table 1. Comparison of computational complexity of different algorithms.

	Computational Complexity	Value
LS	$N_P$	242
DFT [11]	$N_{\mathrm{P}}+N{log}_{2}N$	2290
T-DFT [13]	$N_{\mathrm{P}}+N{log}_{2}N+N$	2546
ST-DFT	$N_{\mathrm{P}}+N{log}_{2}N+N$	2546
DWT	$4N_P$	968
Joint Algorithm	$6N_{\mathrm{P}}+N{log}_{2}N+N$	3756

.

The LS algorithm has the lowest computational complexity, followed by the DWT algorithm. The computational complexity of the different DFT-based algorithms is influenced by the computational complexity of the threshold. It can be observed that T-DFT and ST-DFT have the same complexity. Since the joint channel estimation algorithm is the combination of two algorithms, it has the highest complexity among the compared algorithms, which is increased by about 48% compared with that of the T-DFT algorithm.

6. Simulation and Analysis

In this study, the proposed channel estimation algorithm was simulated and evaluated using MATLAB software, and the simulation parameters were set as shown in

Table 2. Simulation parameter settings.

Parameter	Value	Parameter	Value
Channel bandwidth	20 MHz	OFDM Symbols	12.8 $\mathsf{\mu }$s
Poilt modulation scheme	BPSK	Circular prefix	3.2 $\mathsf{\mu }$s
Number of subcarriers	256	Number of active subcarriers	242

.

In the simulation, the D model and the urban micro-room (UMi) model in the IEEE802.11ax standard were selected as the indoor and outdoor transmission simulation scenarios, respectively [23].

Table 3. IEEE 802.11 ax channel model key parameters.

	Indoor	Outdoor
Channel Model	Model D	Model UMi
Number of taps	18	19
Maximum delay spread	390 ns	730 ns
Movement speed	0.089 km/h	3 km/h

presents the key parameters of the two models.

6.1. Channel Estimation Algorithm Performance Based on ST-DFT

Figure 2 and Figure 3 show the performance of different channel estimation algorithms under different channel models. The performance of the DFT-based algorithm in [11] appears floor quickly. Although the MSE of the T-DFT algorithm in [13] is significantly better than that of [11], it is still inferior to that of the LS algorithm in high SNR range. The performance of the ST-DFT algorithm proposed in this paper is better than that of the T-DFT algorithm, especially under low SNR. The reason can be explained as follows. The threshold of the T-DFT algorithm is fixed at twice the average power of the samples in the noise observation area. When the SNR is low, the large noise in the CIR cannot be effectively removed. When the SNR is high, some of the samples that should be reserved will be misjudged as noise samples and removed. The ST-DFT algorithm avoids the above problems and thus has a better channel estimation performance. However, since all samples in the time-domain contain channel components, the threshold decision will always misjudge a portion of the channel components as noise and set them to zero. T-DFT and ST-DFT algorithms will inherently suffer from performance loss under high SNR, so they still have a trend of performance floor. The wavelet-denoising algorithm transforms the signal into the wavelet domain for denoising. Based on the appropriate wavelet base and decomposition layers, the useful signals and noise can be separated correctly, so the estimation performance loss is relatively small under high SNR.

In addition, the maximum delay spread and the number of taps are different for the two models. Although the outdoor channel has more severe frequency selectivity than that of the indoor channel, the proposed algorithms are still valid. The MSE of the threshold-based DFT algorithm for the UMi model is worse than that estimated for the D model because the UMi model has a larger delay spread and more taps. When conducting N-point IFFT, more channel components diffuse into the noise observation area and thus are more likely to be misjudged during threshold judgments.

6.2. Noise Power Estimation Performance in ST-DFT Algorithm

Figure 4 and Figure 5 show the value ranges for the total power of the noise removed by the ST-DFT channel estimation algorithm under different channel models. We observed that the theoretical upper and lower bounds in the figures correspond to the maximum and minimum values in Equation (24), and the total power of the noise on the actual zero-set samples is between the upper and lower bounds. At medium and low SNR, the total noise power actually removed is closer to the upper bound; at high SNR, since the threshold has a greater probability of screening out low power noises and removing them, the actual total noise power removed gradually deviates from the upper bound and approaches the lower bound.

6.3. MSE Estimation Performance of Different Algorithms

Figure 6 and Figure 7 compare the estimated and simulated values of the frequency domain noise variance MSE from the ST-DFT channel estimation algorithm and DWT channel estimation algorithm for different channel models. As can be seen from the figures, the theoretically estimated values and simulated values of the frequency domain noise variance are similar for both channels. This indicates that the frequency domain noise variance of the current channel can be effectively estimated using Equation (35). In addition, the estimated values of both channel estimation algorithms MSE are very close to the corresponding simulation values, which verifies the correctness of the analysis of Equations (36) and (37). Therefore, the estimated MSE can be used as a decision metric for switching between the two algorithms.

6.4. Joint Channel Estimation Algorithm Performance

Figure 8 and Figure 9 show the MSE performance of the joint channel estimation algorithm under different channel models. It can be seen from the figures that, in the low SNR range, the MSE of the joint channel estimation algorithm (ST-DFT+DWT) coincides with that of the ST-DFT channel estimation algorithm. In the high SNR range, the MSE of the joint channel estimation algorithm and the DWT channel estimation algorithms coincide. Another observation is that the MSE of the joint channel estimation algorithm is smaller than that of both ST-DFT and DWT channel estimation algorithms in the mid-SNR range. The above results show that the joint channel estimation algorithm can dynamically select the algorithm with the best transient channel estimation performance based on the MSE estimated in Equations (36) and (37) as the judgment condition. Thus, its overall estimation performance will be better than that of both algorithms.

6.5. BER Performance of Different Algorithms

The simulation in this subsection will be performed under two channels, and two modulation and coding schemes will be used respectively. We further demonstrate the performance of different channel estimation algorithms by comparing the Bit Error Ratio (BER) of different algorithms.The transmission systems use Low Density Parity Check Code (LDPC) as the channel coding method.

Figure 10 and Figure 11 show the BER performance of different algorithms under two channel models and use QPSK modulation and $1/2$ coding rate. The performance of the DFT algorithm quickly appears floor because portions of the channel components are removed in the time-domain denoising process. The remaining algorithms can achieve a BER of ${10}^{-4}$, and the ST-DFT algorithm has the best BER performance, followed by the DWT algorithm. The T-DFT algorithm has inferior BER performance to the first two algorithms but is better than the LS algorithm. The BER performance of the joint channel estimation algorithm is close to that of the ST-DFT algorithm since it can always select the ST-DFT algorithm to estimate the channel under low SNR condition.

Figure 12 and Figure 13 show the BER performance of different algorithms under two channel models and use 64QAM modulation and $2/3$ coding rate. The DFT algorithm also has a fast BER performance floor, while the rest of the algorithms can achieve a BER of ${10}^{-4}$. The DWT algorithm has the best BER performance for this modulation coding method. The BER performance of the ST-DFT algorithm is close to that of the T-DFT algorithm and the LS algorithm. On the one hand, the MSEs of the three algorithms are close under high SNR; on the other hand, the gain from LDPC channel coding masks part of the gain from the T-DFT and ST-DFT algorithms. Moreover, the BER performance of the joint channel estimation algorithm is close to that of the DWT algorithm since it can always select the DWT algorithm to estimate the channel under high SNR condition.

Comparing the BER performances in channel UMi with that in channel D leads to the same conclusion as the MSE performances. In addition, the BER performance of the proposed joint channel estimation algorithm is always close to the better one of the ST-DFT and DWT algorithms, regardless of the modulation and encoding methods used in the system.

7. Conclusions

In this paper, a segmentation threshold-based channel estimation algorithm (ST-DFT) was proposed to address the energy spread of the estimated channel impulse response due to the presence of virtual subcarriers. The algorithm sets the decision threshold of the CIR effective path based on the distribution law of the channel energy dispersion, thereby obtaining a performance improvement compared with the traditional single threshold-based algorithm (T-DFT). Moreover, a joint channel estimation algorithm based on ST-DFT and DWT was also proposed to tackle the estimation performance floor problem in the DFT-based channel estimation algorithm under high SNR conditions. The simulation results show that, compared with similar DFT channel estimation algorithms, the proposed ST-DFT channel estimation algorithm has a better estimation performance under low SNR and can alleviate the problem of an estimation performance floor under high SNR. Secondly, the theoretical estimate of MSE is very close to the simulated value. Finally, based on the estimated value of MSE, the proposed joint channel estimation algorithm can achieve optimal dynamic switching of channel estimation algorithms, and its channel estimation performance is better than using ST-DFT or DWT alone in the simulated SNR range.