# Joint Channel Estimation Algorithm Based on DFT and DWT
## 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 6 th generation IEEE WLAN standard--IEEE 802.11 ax (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 802.11 ax 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.