In underwater acoustic communications in shallow seas, signal propagation is significantly affected by reflections, diffraction, and scattering from both the sea surface and seafloor. This results in a complex multipath structure of the underwater acoustic channel. The underwater acoustic channel exhibits a significant sparse characteristic due to signal energy absorption by seawater during most of the multipath propagation. The mathematical expression for the channel impact response of the underwater acoustic time-varying channel is given by (Cheng and Wang, 2022).

where h ( t ) is the channel impulse response at time t, and L is the multipath number, the h i ( t ) and τ i ( t ) are denoted as the gain and delay of the ith path at time t, respectively.

Consider an OFDM system with N subcarriers, L pilots, and channel coherence time significantly exceeding the OFDM symbol period. If the impulse response of the channel remains time-invariant within one OFDM symbol period, (Equation 1) can be expressed as follows:

Assuming the cyclic prefix of OFDM symbols exceeds the maximum multipath delay of the channel, the frequency domain expression for the OFDM communication system is:

where y ∈ ℂ N × 1 is the received signal, X ∈ ℂ N × N is the diagonalization matrix with diagonal elements representing the transmitted signals. F ∈ ℂ N × M is the DFT matrix, and h ∈ ℂ M × 1 is the time domain channel impulse response. The v follows C N ( 0 , λ Ι N ) of Gaussian white noise. From the received signal y out of the pilot signal, the received model of the pilot signal is:

where y p ∈ ℂ p × 1 is the received pilot signal, X p ∈ ℂ p × p is the diagonalization matrix with diagonal elements representing the known pilot signals. F p ∈ ℂ p × M is the corresponding DFT matrix at the pilot position. The system model described in (Equation 4) is a single-measurement model. A multi-measurement model is considered: several different OFDM symbols are modeled with the following expressions:

Among them. Y p = [ y p , 1 , y p , 2 , … , y p , L ] ∈ ℂ p × L represents the received pilot matrix of L OFDM symbols, and Φ p ∈ ℂ p × M is the perception matrix, H = [ h 1 , h 2 , … , h L ] ∈ ℂ M × L .

The article uses the TMSBL algorithm (Qiao et al., 2018), using temporal correlation to jointly estimate (Equation 5) of H the estimation reconstruction problem. Firstly, the priori probability of each H _i is modeled as:

where H _i is the ith row of H , i.e., the number of channels tapping coefficients for different OFDM symbols at the same moment; γ_i is the nonnegative hyperparameter matrix that controls the sparsity of each row in H . When γ i = 0 , H i = 0 ; B _i is a positive definite matrix, which represents the temporal correlation structure among the elements within H _i and can be estimated using the TMSBL algorithm for the positive definite matrix B . (Equation 6) can be written as:

where Γ is the hyperparameter matrix Γ = diag ( γ ) = diag ( [ γ 1 , γ 2 , … , γ M ] T ) .

H _i , Obeying the mean Gaussian probability distribution, its posterior probability can be written as:

where the mean and covariance can be expressed as:

Of these, μ _l and ℳ are respectively the estimated values of H _i and H . Γ ( r ) is the estimated value of the Γ update matrix for the first r iteration of the Expectation Maximization (EM) algorithm. The hyperparameters are estimated using the EM algorithm. The E-step update rule of the EM algorithm is given in (Equations 9, 10). The M-step update rule is given in:

where ‖ ⋅ ‖ F 2 denotes the quadratic of the F-parameter of the vector T r ( ⋅ ) is the trace of the matrix. The joint estimation of the channel impulse response after the iteration of the EM algorithm is completed H ^ = ℳ .

(Equation 13) is the updated formula for the noise variance, which is calculated using the null subcarriers of the OFDM system:

where Y _n is the frequency domain null subcarrier. Using (Equation 14) to obtain a more accurate noise variance, a more accurate channel estimation can be obtained. This reduces the influence of the noise variance by the TMSBL input parameters such as the number of iterations, the threshold, and the received frequency-conducting matrix. At the same time, it can reduce the TMSBL algorithm for the σ ² update step in the TMSBL algorithm, reducing the complexity of the algorithm.

According to the theory of sparse decomposition, Y _p can be decomposed into Y p = X p F p H + V p = A ( X s + X n ) . Where A ∈ ℂ p × J is the redundant dictionary matrix, X_s and X_n are the sparse coefficient matrices corresponding to X p F p H and V _p , respectively. As the signal is sparse, whereas the noise is not sparse, and the coefficient values are generally very small, existing only in a finite number of non-zero coefficients, the approximated signal obtained by the linear combination of these sparse counterparts of the atoms contains the vast majority of the information about the signal, while the vast majority of the noise is discarded, thus realizing the purpose of noise cancellation. Let X = X s + X n , then Y p = A X . Where X ∈ ℂ J × L is the sparse coefficient matrix. To achieve the sparse decomposition of the signal, a suitable dictionary needs to be constructed. The dictionary learning algorithm is employed to construct a suitable redundant dictionary to enhance signal reconstruction.

The K-SVD dictionary learning algorithm is a new dictionary learning algorithm proposed by Aharon and Elad et al (Aharon et al., 2006). The primary concept behind the K-SVD algorithm involves updating a set of atoms in the dictionary along with their sparse coefficients simultaneously. Through iterations of updating a set of atoms within the dictionary, if, when updating any atom, the remaining atoms remain unchanged, the dictionary is then updated with the sparse coefficients. a _i with no change in the remaining atoms, the a _i new sparse coefficients will be obtained after the update X _i . When the error reaches the threshold, the whole dictionary post-sparse matrix is updated. Its solution model is:

where ‖ Y p − A X ‖ F 2 = ‖ Y p − ∑ i = 1 J a i x T i ‖ F 2 = ‖ ( Y p − ∑ i ≠ K a i x T i ) − a K x T K ‖ F 2 . a _i denotes the ith atom in the redundant dictionary A , and x T i denotes the ith row vector of the sparse coefficient matrix X . a _K is the updated atom, and x T K is the sparse solution corresponding to the updated atom. Then the error matrix of the signal is:

where E _K is the error matrix of the signal. At this point the solution model can be described as:

To avoid the loss of sparsity in the sparse solution, the E _K in the corresponding x T K non-zero positions is extracted to obtain a new E K ' , the corresponding sparse coefficient vector is x T K ' , then (17) can be converted to:

Then the singular value decomposition algorithm is used to E K ' solving:

where U is the left singular matrices, take its first column as the update atom, i.e., a K = U ( ⋅ , 1 ) . V is the right singular matrix, take its first row with the first singular value as the x T K " = Σ ( 1 , 1 ) V T ( 1 , ⋅ ) . Then the corresponding update is obtained as x T K , i.e., by updating each atom of the redundant dictionary in turn, the optimal sparse solution corresponding to each atom can be obtained. When all the atoms are updated, the updated dictionary and the optimal sparse coefficient matrix are obtained.

The GOMP algorithm is employed to achieve a sparse representation of the received pilot matrix, yielding the sparse coefficient matrix. Following this, the K-SVD dictionary learning algorithm is applied to mitigate the noise present in the received signal, resulting in the acquisition of the noise-reduced received pilot matrix Y p ' . The specific noise reduction process is shown in Figure 2 .

Obtaining the time-domain shock response of the underwater acoustic channel using GOMP channel estimation algorithm h _GOMP. To obtain the a priori knowledge of the TMSBL, the initial parameters of the EM algorithm are chosen based on the characteristics of the underwater acoustic channel, effectively reducing the algorithm’s complexity.

Time-domain impact response of underwater acoustic channel obtained using GOMP channel estimation algorithm.

where Φ p , Λ i is the perceptual matrix corresponding to the set of indexes after the ith update of atoms, and h _i is the channel estimate after the ith iteration.

Set the average energy superposition function Q of the channel to be (Hong et al., 2022):

Among them. h _GOMP i is the ith column of h GOMP , i.e., the channel time-domain impulse response of the ith OFDM symbol.

Set the threshold as T Q = α max ( Q ) where α is the energy coefficient, determined based on the characteristics of the underwater acoustic channel and considering the computational complexity. Compare the channel average energy superposition function Q and the threshold T_Q of the channel:

where Ω GOMP is the hyperparameter vector of TMSBL. When the channel average energy superposition function Q is greater than the threshold, indicating that, at this time, the hyperparameter control channel impulse response is the channel tapping coefficient, Ω GOMP = 1 ; on the contrary, it is considered that the hyperparameter is too small and its control channel impulse response probability is the noise, the Ω GOMP = 0 .

Define the initial hyperparameter matrix as Γ = diag ( Ω GOMP ) , whose diagonal elements are hyperparameter vectors Ω GOMP . Take the positions Γ whose diagonal elements are equal to zero as the indexed set s and eliminate the atoms corresponding to s in the dictionary matrix Φ p to obtain the initialized dictionary matrix Φ p ' of the improved TMSBL algorithm.

Finally, the noise-canceled pilot receiver matrix Y p ' , the initial hyperparameter matrix Γ , and the initialized dictionary matrix Φ p ' are substituted into the TMSBL channel estimation method for underwater acoustic channel estimation.

The proposed algorithm is described as KSVD-GOMP-TMSBL algorithm for simplicity of expression. The main comparison algorithms for the simulation are LS, GOMP, TMSBL, and LS-TMSBL methods, setting the maximum number of iterations r max = 5000 and the error threshold is thresh = 1 × 10 − 6 .

Figure 5 displays the normalized mean square error plots for channel estimation using LS, GOMP, TMSBL, LS-TMSBL, and KSVD-GOMP-TMSBL algorithms. The received pilot matrix was denoised using the K-SVD dictionary learning method. The energy coefficient α is set to 0.05, and GOMP sparsity is fixed at 30. It can be observed from Figure 5 that the LS algorithm, devoid of channel sparsity utilization, exhibits poor estimation performance. The GOMP algorithm slightly outperforms the LS algorithm. The TMSBL algorithm, capitalizing on both the sparse nature of the channel and the temporal correlation between different symbols, demonstrates superior performance in channel estimation. The LS-TMSBL algorithm, incorporating the LS algorithm to acquire a priori knowledge of the TMSBL algorithm and influenced by the energy coefficient, exhibits performance slightly lower than the TMSBL algorithm. The KSVD-GOMP-TMSBL algorithm, leveraging the K-SVD dictionary learning algorithm for noise reduction, GOMP to obtain a priori knowledge of the TMSBL, and null subcarriers to determine noise variance, demonstrates improved performance compared to the comparison algorithms.

Figure 6 displays the normalized mean square error plots for channel estimation using LS, GOMP, TMSBL, LS-TMSBL, and KSVD-GOMP-TMSBL algorithms. No denoising was applied to the received pilot matrix. The energy coefficient α is set to 0.05, and GOMP sparsity is fixed at 30. It is evident from Figure 6 that, without noise reduction, the LS algorithm is more susceptible to noise, leading to a degradation in the performance of the LS-TMSBL algorithm. In contrast, the TMSBL algorithm and the KSVD-GOMP-TMSBL algorithm are relatively less affected by noise. The KSVD-GOMP-TMSBL algorithm improve its performance by obtaining the null subcarrier through noise variance.

Overall, the KSVD-GOMP-TMSBL algorithm exhibits improved channel estimation performance. The use of the K-SVD dictionary learning algorithm for noise reduction on the received pilot matrix enables the algorithm to achieve accurate estimation results even at low SNR. The algorithm’s performance is further enhanced by obtaining precise noise variance through null subcarriers instead of iteratively updating the noise variance. Additionally, the algorithm demonstrates some improvement in estimation performance even without noise reduction processing, showcasing its ability to mitigate certain noise interferences. The factors influencing the KSVD-GOMP-TMSBL algorithm are subsequently analyzed from various perspectives.

First, discuss the impact of the dictionary learning noise reduction method on the algorithm’s performance. Figure 7 shows the lofar plot (left) of the received pilot matrix at an SNR of -10dB and the lofar plot (right) of the received pilot matrix after noise reduction using dictionary learning. It is clear that the lofar plots are significantly clearer after noise reduction. The SNR of the received pilot matrix before noise reduction is -10.28dB, and after noise reduction, it improves to 2.36dB. This demonstrates the superior noise reduction effect of the dictionary learning algorithm on the received matrix.

In Figure 8 , compare the effectiveness of noise reduction between the dictionary learning algorithm and the singular value decomposition (SVD) noise reduction method. The figure demonstrates the impact of two noise reduction methods on the performance of both the TMSBL and GOMP-TMSBL algorithms. From Figure 8 , the dictionary learning noise reduction method significantly improves channel estimation in both algorithms compared to the singular value decomposition method. For the GOMP-TMSBL algorithm, the NMSE of channel estimation under the dictionary learning noise reduction method is 0.0566, representing an almost tenfold decrease compared to the singular value decomposition noise reduction method. This demonstrates that the dictionary learning noise reduction method is highly effective in noise reduction.

Figure 9 compares the channel estimation error of the KSVD-GOMP-TMSBL algorithm with other comparison algorithms, without noise reduction processing, to investigate the impact of dictionary learning noise reduction algorithms on the performance of channel estimation methods. As shown in Figure 9 , the estimation error of the GOMP-TMSBL algorithm is 6.616, while the estimation error of the KSVD-GOMP-TMSBL algorithm is 0.5973 at a signal-to-noise ratio of -10 dB. The estimation error of the KSVD-GOMP-TMSBL algorithm is reduced by approximately 91.1% compared to the GOMP-TMSBL algorithm. This indicates that the performance of the channel estimation algorithm is improved by performing noise reduction on the received pilot matrix using the dictionary learning algorithm.

Next, discuss the impact of the energy coefficient (α) on the KSVD-GOMP-TMSBL algorithm. The KSVD-GOMP-TMSBL algorithm in the α value will affect the size of the threshold and, consequently, the hyperparameter matrix. When the α value is too small, the threshold becomes insufficient, leading to noise being misestimated as channel tapping coefficients, thereby affecting the accuracy of channel estimation. Conversely, when the α value is too large, the threshold becomes excessive, causing real but smaller channel tapping coefficients to be mistaken as noise, thus affecting the algorithm’s performance. The chosen value of α significantly impacts the performance of the KSVD-GOMP-TMSBL algorithm. Figure 10 shows the energy coefficient α channel estimation error when different values are taken, where the GOMP algorithm sparsity is taken to be 30. From the figure, it can be observed that when the 0.01 ≤ α ≤ 0.10 the estimation errors of the KSVD-GOMP-TMSBL algorithms are relatively close, both achieve a better performance. When α = 0.15 , the estimation performance of the KSVD-GOMP-TMSBL algorithm is closer to that when α takes the value in the range [ 0.01 ~ 0.10 ]. However, as the threshold is critical at a = 0.15, confusion between channel tapping coefficients and noise can arise, which affects the algorithm’s stability, exhibiting significant fluctuations when the signal-to-noise ratio is 11dB. For α > 0.15 , the estimation performance of the KSVD-GOMP-TMSBL algorithm is poorer, and it is slightly inferior to the TMSBL algorithm when the signal-to-noise ratio reaches a certain value. Therefore, to ensure optimal channel estimation performance and algorithm stability, the value of α should be in the range of [ 0.01 ~ 0.10 ].

The performance of the conventional GOMP algorithm is notably influenced by sparsity. Figure 11 illustrates the channel estimation error of the GOMP algorithm with varying sparsity values, with the energy coefficient (α) set to 0.05. It is evident from the figure that the subpar channel estimation performance when the sparsity is 5 is attributed to the multipath complexity of the shallow sea environment. In such an environment, where the number of propagated multipaths is higher, opting for a small sparsity value may erroneously set the channel’s tapping coefficient to 0 when deriving hyperparameters from the a priori knowledge of the TMSBL. This leads to a biased channel estimation. The performance of the KSVD-GOMP-TMSBL algorithm is more consistent when the sparsity is set to the other five values, all of which surpass the TMSBL algorithm, enabling superior estimation performance. Thus, the KSVD-GOMP-TMSBL algorithm only needs to adopt a larger sparsity value, as the algorithm is minimally affected by the specific value of sparsity. Consequently, the performance of the KSVD-GOMP-TMSBL algorithm is not influenced by the sparsity chosen by the GOMP algorithm.

The influence of noise variance calculation methods on the performance of the TMSBL algorithm and the KSVD-GOMP-TMSBL algorithm is discussed below. Figure 12 illustrates the channel estimation errors of the KSVD-GOMP-TMSBL algorithm employing two noise variance calculation methods. At an SNR of -10 dB, the NMSE of the channel estimation for the KSVD-GOMP-TMSBL algorithm is 1.058 when the noise variance is computed through iterative updating using (Equation 13). Conversely, the NMSE is reduced to 0.5579 when the noise variance is determined using the null subcarrier in conjunction with (Equation 14). Compared to the previous method, employing null subcarriers to acquire the noise variance diminishes the NMSE of the channel estimation of the KSVD-GOMP-TMSBL algorithm by approximately 47.27%. It is evident that utilizing the null subcarrier to determine noise variance in the OFDM system can enhance the performance of the KSVD-GOMP-TMSBL algorithm and improve the accuracy of channel estimation.

In the following sections, the proposed algorithm is compared with other methods. Figure 13 illustrates the NMSE of various channel estimation methods. The SBL method (Wipf and Rao, 2004) addresses the multi-measurement model by sequentially processing columns to estimate the channel. The TMSBL method (Qiao et al., 2018) leverages temporal correlation between channels to address the multi-measurement model. The SVD-LS-TMSBL method (Hong et al., 2022) utilizes SVD to reduce noise, LS to acquire a priori knowledge for the TMSBL algorithm, and TMSBL to achieve joint channel estimation. The figure illustrates that the SBL method exhibits the highest NMSE, which can be attributed to its failure to leverage the correlation between the channels. In contrast, TMSBL demonstrates superior channel estimation accuracy compared to the SBL algorithm due to its ability to capitalize on the temporal correlation between the underwater acoustic channels. The SVD-LS-TMSBL algorithm achieves lower NMSE in channel estimation than the TMSBL algorithm because it integrates singular value decomposition for noise reduction and employs the LS method to acquire the priori knowledge for the TMSBL algorithm. The KSVD-GOMP-TMSBL algorithm employs K-SVD dictionary learning for noise reduction, with a priori knowledge for the TMSBL algorithm obtained through the GOMP algorithm. This approach effectively addresses the limitations of the SVD-LS-TMSBL algorithm, where the LS channel estimation method is more sensitive to noise, and the noise reduction capability of SVD is suboptimal in low SNR environments, leading to reduced channel estimation accuracy. These results demonstrate that the KSVD-GOMP-TMSBL method achieves superior channel estimation performance compared to the other algorithms. It can be observed that the proposed algorithm is capable of effectively reducing the impact of noise on the TMSBL algorithm, thereby enhancing the accuracy of channel estimation in OFDM communication systems.

The subsequent analysis centers on the influence of the KSVD-GOMP-TMSBL algorithm on the performance of the OFDM communication system. Figure 14 illustrates the bit error rate (BER) of various channel estimation methods, suggesting that the performance of the OFDM communication system is improved by the KSVD-GOMP-TMSBL algorithm. With an increase in SNR, all five channel estimation methods represented in the figure demonstrate a reduction in the BER of the OFDM communication system. At an SNR of 15 dB, the BERs for the LS, GOMP, TMSBL, LS-TMSBL, and KSVD-GOMP-TMSBL algorithms are 6.2%, 5.7%, 5.8%, 5.3%, and 4.5%, respectively. In comparison to the benchmark algorithms, the BER of the KSVD-GOMP-TMSBL algorithm decreases by at least approximately 15.1%. Based on the aforementioned analysis, the KSVD-GOMP-TMSBL algorithm contributes to the enhancement of the OFDM communication system’s performance.

Table 2 presents the estimation errors of various channel estimation methods under identical signal-to-noise ratio conditions. The results clearly indicate that the estimation accuracy of the KSVD-GOMP-TMSBL algorithm is markedly improved compared to other methods. At a signal-to-noise ratio of -10 dB, the channel estimation error of the KSVD-GOMP-TMSBL algorithm decreases by around 96.6%, 95.4%, 92.2%, and 92.1% compared to the LS, GOMP, TMSBL, and LS-TMSBL channel estimation methods, respectively. The estimation error of the KSVD-GOMP-TMSBL algorithm similarly decreases across other signal-to-noise ratio values. This confirms the significant enhancement in channel estimation accuracy achieved by the KSVD-GOMP-TMSBL algorithm.

The following section will analyze the convergence and convergence rate of the KSVD-GOMP-TMSBL algorithm. Figure 15 illustrates the NMSE of the TMSBL and KSVD-GOMP-TMSBL algorithms for varying numbers of iterations at a signal-to-noise ratio (SNR) of -10 dB. It can be observed that both algorithms demonstrate convergence as the number of iterations increases. The KSVD-GOMP-TMSBL algorithm reaches convergence at 7 iterations, while the TMSBL algorithm reaches convergence at 26 iterations. The results demonstrate that the KSVD-GOMP-TMSBL algorithm exhibits a faster convergence rate than the TMSBL algorithm. It has been demonstrated that the integration of the KSVD-GOMP-TMSBL algorithm for noise reduction based on a KSVD dictionary and the utilization of GOMP to derive the prior knowledge of the TMSBL algorithm can effectively reduce the number of iterations of the TMSBL algorithm and accelerate the convergence speed.

The KSVD-GOMP-TMSBL algorithm is primarily composed of three key components: KSVD dictionary learning for noise reduction, a priori knowledge acquisition based on GOMP channel estimation, and TMSBL algorithm channel estimation. The KSVD dictionary learning technique for noise reduction incorporates GOMP sparse coding and dictionary updating. The computational complexity of GOMP sparse coding is O ( k p J L ) , and that of dictionary updating is O ( p J L ( L + 1 ) ) . Consequently, the computational complexity of KSVD dictionary learning noise reduction is O ( p J L ( k + L + 1 ) ) . The computational complexity of acquiring a priori knowledge through GOMP channel estimation is O ( k p J L + M L ) . The computational complexity of TMSBL channel estimation is O ( N ( M 2 p + L 4 + L 2 M ) ) , where N represents the number of iterations. In conclusion, the principal computational complexity of the KSVD-GOMP-TMSBL algorithm is O ( p J L ( k + L + 1 ) + M L + N ( M 2 p + L 4 + L 2 M ) ) . It can be observed that the complexity of the proposed algorithm is predominantly dictated by the TMSBL channel estimation component. The computational complexity of the TMSBL algorithm is primarily driven by the value of N, especially when the number of iterations is substantial, which may result in a significant increase in the algorithm’s overall complexity. The KSVD-GOMP-TMSBL algorithm employs GOMP to derive the initial hyperparameter matrix and the simplified dictionary matrix. This process is analogous to the iterative updating in TMSBL during the intermediate phase, leading to a reduction in the number of iterations and an acceleration of the convergence process. This is corroborated by the results shown in Figure 15 . These findings demonstrate that the KSVD-GOMP-TMSBL algorithm effectively reduces the computational complexity of the TMSBL algorithm.

The KSVD-GOMP-TMSBL algorithm utilizes the GOMP channel estimation algorithm to derive the time-domain impulse response h _GOMP of the underwater acoustic channel. It incorporates a priori knowledge from the TMSBL and leverages the characteristics of the underwater acoustic channel to determine the initial parameters of the EM algorithm. This approach can effectively reduce the complexity of the algorithm. Utilizing the null subcarrier in conjunction with (Equation 14) to calculate the noise variance, instead of iteratively updating with (Equation 13), can theoretically decrease the number of iterations and thus simplify the complexity of the system. The computational complexity of the KSVD-GOMP-TMSBL algorithm will be subsequently analyzed by examining the algorithm’s running time. The hardware configuration for the simulation comprises a Core i5 Intel processor at 2.3GHz with 12GB of RAM.

Figure 16 illustrates the comparison of the running time among various channel estimation methods. It is evident that the TMSBL algorithm exhibits the highest running time. The LS a priori TMSBL method demonstrates a lower running time compared to the TMSBL algorithm. This is attributed to the utilization of the LS algorithm to acquire a priori knowledge for the TMSBL algorithm, thereby reducing its overall complexity. The KSVD-GOMP-TMSBL Algorithm, employing K-SVD Dictionary Learning, the incorporation of a noise reduction algorithm, and utilizing the GOMP algorithm for acquiring a priori knowledge of the TMSBL algorithm, diminishes the number of iterations and decreases the running time in comparison to the TMSBL algorithm and the LS-TMSBL algorithm. At a signal-to-noise ratio of -10 dB, the running time of TMSBL is 52.71s, the LS a priori TMSBL method records a running time of 40.3s, and the KSVD-GOMP-TMSBL algorithm demonstrates a running time of 28.66s. In comparison to the preceding two methods, the running time diminishes by approximately 45.63% and 28.89%, respectively. Although the complexity of the KSVD-GOMP-TMSBL algorithm remains relatively high compared to channel estimation methods such as LS and GOMP, Figure 9 indicates that the KSVD-GOMP-TMSBL algorithm exhibits superior channel estimation accuracy. Overall, the computational complexity of the KSVD-GOMP-TMSBL algorithm is diminished in comparison to both the TMSBL algorithm and the LS-TMSBL algorithm.

Table 3 illustrates the running time of various channel estimation methods under identical SNR conditions. As evident from Table 3 , the running time of the KSVD-GOMP-TMSBL algorithm decreases in comparison to both the TMSBL and LS-TMSBL algorithms. At a signal-to-noise ratio of 0 dB, the running time of the KSVD-GOMP-TMSBL algorithm decreases by approximately 50.3% compared to TMSBL and 33.9% compared to LS-TMSBL algorithms. The running time of the KSVD-GOMP-TMSBL algorithm also decreases at different signal-to-noise ratio values. This indicates a reduction in the computational complexity of the KSVD-GOMP-TMSBL.