Compressed Sensing Channel Estimation Algorithm Combined with Wavelet Denoising

In wireless multipath channels, there is sparsity, in which traditional channel estimation algorithms do not take advantage of and ignore the impact of noise. An effective estimation algorithm combining compressed sensing and wavelet de-noising is proposed. Before channel estimation, the pilot signal received by the receiver is de-noised by the wavelet de-noising method to obtain the de-noised measurement vector, and then the original signal is recovered by the compressed sensing technology for sparse channel estimation. This algorithm breaks away from the limitation of traditional algorithms that require predictive sparsity to achieve sparsity adaptive signal reconstruction. Through simulation analysis, it has been proven that the fusion wavelet denoising algorithm can improve the estimation performance, reduce estimation errors, and be more suitable for sparse channel estimation by denoising the signal during channel estimation.

, have obvious drawbacks, and neither algorithm takes into account the sparsity of wireless channels [3-4]   .By utilizing this characteristic in the channel, the channel estimation in sparse channels can be transformed into the problem of using greedy algorithms to recover the original signal [5] .Compared to traditional algorithms, they can use fewer pilot resources while ensuring the same performance and completing channel estimation with lower complexity but higher accuracy [6][7] .It uses the weak selection matching pursuit algorithm to set the threshold for starting, and not only one atom is selected in each iteration process, which greatly reduces the number of repetitions compared with the previous algorithm [8] .The sparsity adaptive matching pursuit algorithm to increase the atomic index through each iteration is used, which can complete channel estimation under the condition of unknown sparsity [9]   .
Although the recovery signal quality of the greedy algorithm mentioned above is good, regardless of which algorithm requires a known sparsity as a prerequisite.However, in actual environments, sparsity cannot be determined in advance and the impact of actual channel noise is not taken into account.Therefore, under the condition of sparse channels, this paper proposes a channel estimation algorithm that combines wavelet denoising and compressed sensing technology to ensure good channel estimation performance.

The sparse channel model
It is assumed that an OFDM system has N subcarriers, and ) (k X and ) (k y represents the th K  data of sending and receiving OFDM symbols, respectively.The transmission model is as follows: where represents the symbols of each subcarrier at the receiving end; ) represents the symbols of each subcarrier at the transmitting end; is the complex additive Gaussian white noise; is the column vector.The discrete channel impulse response application equation is expressed as: ( All P subcarriers in the N P  dimensional observation matrix are used to transmit pilot sequences: According to the above equation, it can be inferred that the channel response h can be roughly reconstructed by utilizing the received pilot sequence P y and recovery matrix A .

2.2Compressed sensing theory
It is assumed that the one-dimensional discrete signal N R x  is sparsely projected into a certain transformation domain, and it can be reconstructed back to the original signal through a small number of observations.The signal x can be represented by an orthogonal sparse basis dictionary matrix ] ,..., , [ where  is referred to as the projection of the original signal on a certain transformation basis.It is assumed that the observation matrix IOP Publishing doi:10.1088/1742-6596/2637/1/0120393 the original signal projection is sparsely distributed and conforms to the constraint isometry, there are: After knowing y and CS A , we use the 1 l -norm to solve the above problem: where  is the estimated value, which is the reconstructed signal, and 0  is the number of non-zero elements in  .We substitute the final  into Equation ( 4) to obtain the recovered signal x .

3.WAVELET THRESHOLD DENOISING
The basic idea of using threshold denoising in wavelet transform is to choose a wavelet basis that is most suitable for processing noisy signals.Then, the signal is continuously decomposed into wavelet coefficients of different layers [10][11] .In the wavelet domain, useful signals and noise exhibit two different characteristics.Therefore, based on the above properties, we can use the method of setting a threshold to save the approximate coefficients under the maximum number of layers, while setting the detail coefficients of different layers to different thresholds.If the threshold is lower, it will be set to zero, and if it is higher, it will be retained.Finally, the previous signal is restored through inverse transformation to achieve the effect of noise removal.
The signal From the above equation, it can be seen that the size of a determines the scale coefficient and wavelet coefficient.In practical applications, signals may not always be continuous, and if discrete signals are encountered, wavelet denoising cannot be directly performed.The parameters a and b need to be discretized first.After discretization, we can get: ) After discretization, the wavelet basis function becomes: Then, after calculating the use of ddencmp the function, a hard threshold function is selected for subsequent wavelet coefficient correlation processing.Finally, the two processed wavelet coefficients are reconstructed to obtain the frequency response ^r H after removing the noise.

4.ALGORITHM IMPLEMENTATION
The influence of Gaussian white noise in the channel is well eliminated by the wavelet de-noising method.The wavelet de-noising theory is applied to channel estimation [12][13] .The received signal is de-noised first, and then the channel is estimated.This process can be shown in Figure 1.(2) We perform wavelet threshold denoising on 0 r to obtain the denoised observation vector 1 y ;

ICAITA-2023
(3) We calculate the inner product , select values in u which the atomic correlation is greater than the threshold  , and construct the set 0 J found in this iteration by corresponding to the column number j of A ; (4) We make 0 1 . If no new atoms are selected, we proceed to Step (7); (5) We find the least squares solution of , it proceeds directly to the last step; (8) We reconstruct the estimated coefficients of the obtained signal t  .

5.SIMULATION ANALYSIS
The simulation environment is a sparse channel, and the performance of different algorithms is compared to prove the effectiveness of the proposed algorithm.The specific simulation configuration parameters are shown in the table.The number of system subcarriers is 1, 024, the number of pilot subcarriers is 64, the modulation mode is QPSK, the channel length is set to 60, the noise is set to Gaussian white noise, and the cyclic prefix length is set to 120.
As shown in Figure 2, the sparsity K is set to 20 by default, and the impact of different threshold parameters  on the reconstruction results is compared.comparison, it can be seen that the overall effect is best when  =2.5 or  =2.8.However, when the measurement number M is between 70 and 80, the reconstruction probability of  =2.5 is higher, so it is generally set to 2.5 by default.
Figure 2. Influence of the differen  ce in the reconstruction probability graph Figure 3 shows the normalized mean square error (NMSE) simulation comparison of OMP, StOMP, and SAMP, and the algorithm proposed in this paper after wavelet denoising in the SNR ranges from 0 dB to 20 dB.From the figure, it can also be seen that the OMP algorithm has the smallest error and the SAMP algorithm has the largest error due to the known sparsity of the signal.The algorithm proposed in this article associates the noise level with the selection threshold through wavelet threshold denoising, and its performance is better than that of SAMP and the original StOMP algorithm.Considering various factors, the denoised StOMP algorithm is more suitable for sparse channel estimation.

Figure 3. Comparison diagram of NMSE of different algorithms
Figure 4 tests the bit error rate BER when the signal-to-noise ratio SNR is between 0 dB and 20 dB.In communication systems, channel estimation refers to estimating the state or parameters of a channel at the receiving end for effective signal detection and decoding, reducing the probability of information transmission errors.From the figure, it can be seen that the proposed StOMP algorithm after wavelet denoising has a performance range of 0 dB to 10 dB, which is not significantly different from other algorithms.However, as the signal-to-noise ratio (SNR) continues to increase, the performance is better than the OMP algorithm and the original StOMP algorithm, and the advantages will be more obvious.

6.CONCLUSIONS
Considering the sparsity of wireless channels, this paper introduces the compressed sensing theory into channel estimation to avoid the waste of pilot resources and reduce the complexity.In the reconstruction algorithm, the threshold with the best reconstruction effect is selected, and the estimated sparse representation coefficient of the channel is solved by using the least squares method.The maximum number of iterations or residual value is set to 0 as the stopping condition, effectively overcoming the shortcomings of traditional greedy algorithms that require signal sparsity as a prerequisite.Considering the impact of noise in the actual environment, the wavelet threshold denoising theory is integrated into channel estimation to denoise the signal received by the receiver, improving the accuracy of estimation.
This article compares and verifies the effectiveness of the proposed method from three aspects: reconstruction probability, mean square error, and bit error rate.Compared with other OMP, StOMP, and SAMP channel estimation algorithms that do not consider the impact of noise, denoising the signal can reduce the impact of noise on estimation performance.Although the algorithm complexity is slightly increased, the estimation accuracy is greatly improved.The simulation results show that the StOMP channel estimation algorithm proposed in this paper, which combines wavelet denoising with smaller estimation errors than the OMP algorithm and is more suitable for sparse channel estimation.
multiplying the original signal x by the observation matrix  in the the equation.The CS A dimension is N M  and N M  , so the problem is underdetermined and cannot obtain a unique and accurate solution.But if ICAITA-2023 Journal of Physics: Conference Series 2637 (2023) 012039

Figure 1 .
Figure 1.Global algorithm flow chartThe specific algorithm steps are as follows: We input the sensing matrix    A , the observation vector y , the number of iterations S (default to 10), and the threshold parameter  .We output the signal sparse coefficient estimation value  :(1) We initialize the residual y r  0

Figure 4 .
Figure 4. Comparison diagram of NMSE of different algorithms