Channel Estimation for RIS-Assisted MIMO Systems in Millimeter Wave Communications

The large number of estimated parameters in a reconfigurable intelligent surface (RIS) makes it difficult to achieve accurate channel estimation accuracy in 6G. Therefore, we suggest a novel two-phase channel estimation framework for uplink multiuser communication. In this context, we propose an orthogonal matching pursuit (OMP)-based linear minimum mean square error (LMMSE) channel estimation approach. The OMP algorithm is used in the proposed algorithm to update the support set and pick the columns of the sensing matrix that are most correlated with the residual signal, which successfully reduces pilot overhead by removing redundancy. Here, we use LMMSE’s advantages for handling noise to address the problem of inadequate channel estimation accuracy when the signal-to-noise ratio (SNR) is low. Simulation findings demonstrate that the proposed approach outperforms least-squares (LS), traditional OMP, and other OMP-based algorithms in terms of estimate accuracy.


Introduction
One of the important 6G technologies is reconfigurable intelligent surface (RIS), which has received a lot of research attention. It consists of a controller and a sizable number of inexpensive passive reflecting components without RF chains. The wireless communication environment can be intelligently manipulated by RIS by adjusting the coefficients of its constituents [1][2][3]. Wireless communication coverage, throughput, and energy efficiency can all be considerably increased by coherently combining and guiding wireless signals in the desired direction [4,5]. Due to passive reflection, RIS also consumes less energy and has lower hardware costs than conventional active relays or beamforming methods [6,7]. However, precise channel state information (CSI) is needed for RIS to operate to its full potential. Therefore, obtaining precise CSI is essential for RIS-assisted communication systems.
To get accurate CSI, there are two key obstacles to overcome. First, many passive elements are incapable of actively transmitting or receiving signals because they lack the ability to analyze signals. Only active antennas on the base station (BS) and user equipment (UE) can be used to determine CSI [8,9]. As a result, it is difficult to estimate BS-RIS and RIS-UE channels using RIS-assisted channel estimation. Second, a large number of antennas for multiple input multiple output (MIMO) systems leads to decreased channel estimate accuracy [10]. Therefore, channel estimation is difficult in RIS-aided MIMO communication systems.
Many different approaches have recently been put forth to investigate channel estimation in RIS-assisted communication systems [11][12][13][14][15][16][17][18]. The work [11] specifically proposed an ON/OFF-based channel-estimating approach to directly estimate cascaded channels. In accordance with this technique, N time slots are sufficient to accurately estimate all reflective channels of the user in the scenario that there is no received noise at the BS, thus decreasing the pilot overhead necessary for channel estimation. However, channel estimation accuracy may be decreased since only one RIS element can reflect the pilot signal

•
We present a linear minimal mean square error (LMMSE) channel estimation approach based on OMP to estimate the BS-UE direct channel without taking RIS into account in the model we develop in mmWave MIMO communication system. We employ the OMP approach to obtain the support set and the LMMSE algorithm to estimate the channel after converting the channel estimation problem into a sparse signal recovery problem. The proposed approach can increase estimation accuracy while requiring less pilot overhead. • By using the double-structured sparsity of the angle cascaded channel in mmWave, we present an LMMSE channel estimation technique based on DS-OMP to estimate the BS-RIS-UE cascaded channel. The DS-OMP approach is used to get the support sets for the angle cascaded channel, and the LMMSE algorithm is used to estimate the channel. The proposed technique successfully manages noise to produce a more accurate estimation.
The rest of this article is arranged as follows. The system model is introduced in Section 2, which also discusses the direct channel and cascade channel estimation problems. We present the proposed channel estimate algorithms for direct and cascaded channels, respectively, in Section 3. The simulation results in Section 4 demonstrate the viability of the proposed approach. Section 5 presents the conclusion of the article.

System Model and Problem Formulation
In this section, we introduce the system models of the two stages of channel estimation and describe their channel estimation issues, respectively.

Direct Channel
As shown in Figure 1, we take into account a wireless system with K single-antenna users interacting with a base station (BS) in the uplink while the BS has M antennas and uses a uniform planar array (UPA) [19]. All RIS components remain closed when there is communication between the UE and the BS. h d,k , k = 1, 2, · · · , K, denotes the direct channel from the kth user to the BS. In this paper, we simulate the direct channel using the Saleh-Valenzuela channel [20,21] as where L d,k is the number of paths between the BS and UE, α d,k denotes the complex gain and the azimuth (elevation) angle at the BS for the l 1 path, respectively.
represents the normalized array steering vector at the BS. b ϑ d,k where d is the antenna spacing, λ is the signal wavelength, and ⊗ represents the Kronecker product.
We define the array response matrix as which constitutes the dictionary of our CS formulation, and the rows of the matrix are orthogonal [22]. Then, we can rewrite (1) utilizing the array response matrix, A R , as follows: where α = α d,k 1 , α d,k 2 , · · · , α d,k L d,k . All users transmit the known pilot symbols to the BS over Q time slots in accordance with the commonly utilized orthogonal pilot transmission scheme [23] for uplink channel estimation. At this point, all RIS components are turned off; i.e., φ n,q = 0, n = 1, 2, · · · , N, q = 1, 2, · · · , Q. Specifically, in the th ( = 1, 2, , ) time slot, the effective received signal for the th user at the BS is denoted by the following equation:  Specifically, in the qth (q = 1, 2, ···, Q) time slot, the effective received signal for the kth user at the BS is denoted by the following equation: where x k,q is the pilot symbol sent by the kth user, and n k,q ∼ CN 0, σ 2 I M is the M × 1 received additive white Gaussian noise (AWGN), with σ 2 representing the noise power.
Vectorizing the received signal, y k,q , is fundamental to formulate the channel estimation issue as

Cascaded Channel
As shown in Figure 2, we keep all the RIS elements turned on. There are M antennas and N elements in BS and RIS, respectively, and they are both UPA to serve K single-antenna users simultaneously. F ∈ C N×M indicates the RIS-BS channel; h r,k ∈ C N×1 indicates the channel from the kth user to the RIS (k = 1, 2, · · · , K). To obtain the channel between RIS and BS, we employ the Saleh-Valenzuela channel model, as shown below: where L F is the quantity of paths connecting the RIS and BS, and is the complex gain, the azimuth (elevation) angle at the BS and RIS for the l 2 path, respectively. Likewise, the channel between UE and RIS can be depicted by where L r,k is the quantity of paths between the kth user and RIS, α r,k l 3 , ϑ r,k l 3 ψ r,k l 3 , is the complex gain, the azimuth (elevation) angle at the RIS for the l 3 path, respectively. a(ϑ, ψ) ∈ C N×1 represents the steering vector for the normalized array at the RIS. For a typical N 1 × N 2 (N = N 1 × N 2 ) UPA, a(ϑ, ψ) can be defined by [24] a(ϑ, ψ) = 1 √ N e −j2πdsin(ϑ)cos(ψ)n 1 /λ e −j2πdsin(ψ)n 2 /λ , where n 1 = [0, 1, · · · , N 1 − 1], n 2 = [0, 1, · · · , N 2 − 1], d is the spacing between the antennas, λ is the wavelength, and represents the Kronecker product. We define the kth user?s N × M cascaded channel as H k = Fdiag h r,k , and we convert it to angular domain representation as where ∼ H k is the N × M angle cascaded channel, and U M and U N are the BS?s and RIS?s respective M × M and N × N dictionary unitary matrices [20]. There are a few non-zero elements in the angle cascaded channel, which exhibits sparsity, as a result of the minimal scattering near BS and RIS.
All users transmit known pilot symbols to BS through RIS in Q time periods using an orthogonal pilot transmission approach to estimate the uplink channel. The effective received signal for the kth user at the BS in the qth (q = 1, 2, ···, Q) time slot can be expressed as y k,q ∈ C M×1 after the direct channel effect between BS and UE has been removed as y k,q = Fdiag h r,k θ q x k,q + n k,q , (11) where x k,q is the pilot symbol that the kth user sends, θ q = θ q,1 , θ q,2 , · · · , θ q,N T is the reflecting vector of RIS, the reflecting coefficient at the nth RIS element (n = 1, 2, · · · , N) in the qth time slot is given by θ q,n , and n k,q ∼ CN 0, σ 2 I M is the M × 1 received noise with σ 2 representing the noise power. According to H k = Fdiag h r,k , (11) can be written as Sensors 2023, 23, 5516 where is the × angle cascaded channel, and and are the BS's and RIS's respective × and × dictionary unitary matrices [20]. There are a few non-zero elements in the angle cascaded channel, which exhibits sparsity, as a result of the minimal scattering near BS and RIS. All users transmit known pilot symbols to BS through RIS in Q time periods using an orthogonal pilot transmission approach to estimate the uplink channel. The effective received signal for the kth user at the BS in the th ( = 1, 2, , ) time slot can be expressed as , ∈ ℂ × after the direct channel effect between BS and UE has been removed as where , is the pilot symbol that the th user sends, = [ , , , , ⋯ , , ] is the reflecting vector of RIS, the reflecting coefficient at the th RIS element ( = 1,2, ⋯ , ) in the th time slot is given by , , and , ~( , ) is the × 1 received noise with representing the noise power. According to = ( , ), (11) can be written as , = , + , .
For Q time slots after pilot transmission, let , = 1, and we obtain the overall measurement matrix of = [ , , , , ⋯ , , ] as where = [ , , ⋯ , ], and = , , , , ⋯ , , . We can obtain the transformation of by substituting (10) into (13) as = ( ) is the effective measurement matrix, and = ( ) is the effective noise matrix. Based on the above formula, we can determine a CS model as follows where = ( ) denotes the sensing matrix. By making full use of the double structured sparsity of the angle cascaded channel, we can estimate the channel. However, For Q time slots after pilot transmission, let x k,q = 1, and we obtain the overall measurement matrix of Y k = y k,1 , y k,2 , · · · , y k,Q as where Θ = θ 1 , θ 2 , · · · , θ Q , and N k = n k,1 , n k,2 , · · · , n k,Q . We can obtain the transformation of Y k by substituting (10) into (13) as matrix. Based on the above formula, we can determine a CS model as follows where ∼ Θ = U T N Θ H denotes the sensing matrix. By making full use of the double structured sparsity of the angle cascaded channel, we can estimate the channel. However, under the premise of low SNR, the estimation accuracy of channel estimation algorithms based on DS-OMP still needs to be improved.

Direct Channel Estimation
Based on (5), the channel between BS and UE for the kth user can be estimated separately. Traditional channel estimation algorithms, such as the LS algorithm, have obvious advantages due to their simple implementation and low computational complexity. However, due to the lack of noise processing, the estimation accuracy cannot meet the requirements of practical applications. In addition, its estimation performance is very limited in certain specific scenarios. Based on this, we propose an OMP-based LMMSE algorithm, which can effectively solve the challenges above. Finally, we evaluate the proposed algorithm?s computational complexity.

Least Square (LS) Algorithm
The goal of the LS algorithm is to reduce the distance between signals that are received from their optimal distance; i.e.,Ĥ To minimize the sum of squares of errors, we make (16)?s first order partial derivative in relation toĤ equal to 0; i.e.,Ĥ At this point, the sum of squares of the obtained estimates is the minimum, which is the solution of the LS channel estimation. The LS algorithm is frequently used in practice, has a low computing complexity, and is reasonably easy to implement. However, it has two significant issues that must Sensors 2023, 23, 5516 6 of 14 be resolved. First, because the LS algorithm ignores the impact of noise, estimation accuracy is significantly impacted in low SNR situations. Second, the number of antennas at the BS is typically enormous, which could lead to overly large dimensionality of channel estimation problems, making it challenging to employ the LS algorithm for mmWave communication [24].

Proposed LMMSE Algorithm Based on OMP
Firstly, to address the issue of noise processing in the LS algorithm, a weighting matrix, W, is added to the LS algorithm: This is the minimum mean square error (MMSE) algorithm [25,26]. However, the matrix inversion requires a large amount of computation and continuous recalculation, which greatly occupies computational resources and has poor real-time performance. Consequently, the MMSE algorithm has limitations in practical applications. The LMMSE algorithm performs linear smoothing on the basis of the MMSE algorithm, considering the MMSE algorithm? need to compute R HH + as the noise changes and the input signal changes, the inversion operation becomes very complex, so, the LMMSE algorithm uses expectation to displace the , the estimated channel is obtained from the LMMSE algorithm as [27] where R h d,kĤLS is the cross-correlation matrix ofĤ LS ; h d,k , R h d,k h d,k denotes the autocorrelation matrix of h d,k ; and β is a channel-modulation-type parameter that in this case, we set as β = 1. We assume that the number of propagation paths between the channels BS and UE is L d,k = 8 and that the formula additionally contains a matrix inversion operation. In other words, we must do an 8 by 8 matrix inversion.
Then, as the number of entries to be estimated in the CS method is proportional to the sparsity, and its sparsity level is considerably lower than MK, we utilize the CS algorithm to overcome the difficulties of implementing the LS algorithm in mmWave communication. For direct channel estimation, we take into account a conventional OMP approach and combine it with the LMMSE algorithm, which has additional advantages in noise processing. Algorithm 1 provides a summary of our scheme.
5. Find the index of the maximum value in the term. 6. Find the column and row indices of the selected index. 7. for j = 1, 2, · · · , length(column) do 8. Find the corresponding rows in ∼ Θ i and Y K 9. Compute the LS of the channel for the selected column 10. end for 11. end for 12. R HH = h d,k (:, k)h d,k (:, k)?; R HH = h d,k (:, k)ĥ OMP ?. 13. Calculate the estimated channel matrix,ĥ d,k , according to Equation (19). 14. end for Output: Estimated channel matrix,ĥ d,k . The following is the explanation of Algorithm 1?s primary procedure. First, for each user, k, find the position of the largest element in the term in step 5. For each group column, perform steps 6-8 and save the column vector of the corresponding position in the sensing matrix to ∼ Θ i . We use Moore Penrose pseudo inverse to calculateĥ OMP of the column j in step 9. Finally,ĥ d,k is obtained using the LMMSE algorithm based on (19) in step 13.

Computational Complexity Analysis
Here, we examine the computational complexity of the three-layer loop that corresponds to the appropriate section of our OMP-based LMMSE algorithm?s major computational work. Specifically, for each user, k, L cycles are executed, and the complexity of each cycle is O(KM). To calculate ∼ Θ'R, we need to find the maximum value and calculate the estimated channel,ĥ OMP . Therefore, the complexity from steps 1 to 12 is O KML 2 . In step 13, we calculate matrix multiplication according to (19), where R HS andĥ OMP are M × K matrix, and R HH is M × M matrix. Consequently, it is necessary to calculate O KM 2 times multiplication and addition. At the same time, using the pinv function to calculate the inverse matrix requires calculating O M 3 times multiplication and addition. Therefore, the complexity of step 13 is O KM 3 . In summary, the computational complexity of our proposed algorithm is O KML 2 + KM 3 .

Cascaded Channel Estimation
Based on (15), for each user, k, we are able to estimate the angle cascaded channel. The traditional CS algorithm generates high pilot overhead while ensuring great estimation accuracy. Although the DS-OMP algorithm proposed in [15] reduces the pilot overhead of cascaded channel estimation to some extent, the estimation accuracy still needs to be improved in low SNR. Therefore, we propose an improved DS-OMP algorithm to achieve higher estimation accuracy in low SNR scenarios. Finally, the computational complexity of this algorithm is analyzed.

Proposed LMMSE Algorithm Based on OMP
The matrix ∼ H k in (10) can be depicted using the following equation where ∼ b(ϑ, ψ) = U H M b(ϑ, ψ) and ∼ a(ϑ, ψ) = U H  N a(ϑ, ψ). Depending on the array steering vector in the (ϑ, ψ) direction of U N and U M , each one of them has just one non-zero element. In [15], a thorough discussion of the double-structured sparsity of the angular domain cascaded channel is presented.
In this section, we consider integrating LMMSE into the classic OMP algorithm and propose an improved DS-OMP cascaded channel estimation scheme. The algorithm process is summarized in Figure 3.
The following explanation outlines the essential steps of the proposed method. First, by utilizing the sparsity of the double structure, we estimate the angle cascaded channel ∼ H k ?s completely public row support, partially public column support, and specific column support for k [15]. Once all supports have been identified, we use the LS algorithm to obtain the estimation matrix,∼ H k_LS , for the angle cascaded channel, , then useĤ k = U H M∼ H k U N to transform∼ H k_LS into the spatial cascaded channelĤ k_LS . Next, we calculate the real cascaded channel H k ?s autocorrelation matrix of and the cross-correlation matrix ofĤ k_LS and H k . Then, the estimated cascaded channel matrix, H k_LMMSE , is obtained using the following formula: where R H kĤk_LS is the cross-correlation matrix ofĤ k_LS and H k , R H k H k is the autocorrelation matrix of H k . We set β = 1 here.
where ( , ) = ( , ) and ( , ) = ( , ) . Depending on the array steering vector in the ( , ) direction of and , each one of them has just one non-zero element. In [15], a thorough discussion of the double-structured sparsity of the angular domain cascaded channel is presented.
In this section, we consider integrating LMMSE into the classic OMP algorithm and propose an improved DS-OMP cascaded channel estimation scheme. The algorithm process is summarized in Figure 3. The following explanation outlines the essential steps of the proposed method. First, by utilizing the sparsity of the double structure, we estimate the angle cascaded channel 's completely public row support, partially public column support, and specific column support for [15]. Once all supports have been identified, we use the LS algorithm to obtain the estimation matrix, _ , for the angle cascaded channel, _ , then use = to transform _ into the spatial cascaded channel _ . Next, we calculate the real cascaded channel 's autocorrelation matrix of and the cross-correlation

Computational Complexity Analysis
The computational complexity of the proposed LMMSE scheme based on DS-OMP is examined in this section. The complexity of this algorithm mainly comes from three parts: completely common row support detection, partially common column support and specific column support detection, and channel estimation matrix calculation. The complexity of the first part mainly comes from energy calculation, with a complexity of O (

Stimulation Results
To demonstrate the effectiveness of the proposed approach, we give the simulation results for the direct channel (BS-UE) and cascade channel (BS-RIS-UE) estimate phases in this section.

Direct Channel Estimation
The following simulation parameters are chosen: M = 64, K = 8, L d,k = 8, and d BU =100 m (the distance between BS and UE). A total of 500 Monte Carlo simulations are performed using MATLAB R2019a in the simulation. For performance evaluation, we employ the normalized mean square error (NMSE), which is defined as In the simulations that followed, we compared the NMSE performance of the proposed scheme with that of the LS algorithm, the conventional OMP method, and an upgraded OMP algorithm for channel estimation in order to demonstrate the proposed algorithm?s superiority.
The link between NMSE and SNR for the BS-UE channel estimation is shown in Figure 4. We contrast the proposed channel estimation algorithm?s NMSE with those of the LS, conventional, and improved OMP algorithms. The LS method is the least computationally complex of them all, but it ignores the effects of noise. The other two algorithms consider noise, but they perform poorly when the SNR is low. Our proposed channel estimation algorithm achieves higher estimation accuracy with acceptable computational complexity at low SNR. Since the LMMSE algorithm is a statistical estimation method aimed at minimizing the mean square error of channel estimation. It is based on the linear relationship between the received signal and the known transmitted signal sequence, while considering the channel noise and the correlation of the signal. In low SNR conditions, the noise in the received signal becomes the main interference factor and significantly affects the accuracy of channel estimation. The LMMSE algorithm can mitigate the impact of noise on channel estimation by optimizing the estimated mean square error. It achieves this by fully utilizing the known transmitted signal sequence and the linear relationship with the received signal. Consequently, the proposed scheme converges quickly to a performance platform. When SNR = −20 dB, the proposed algorithm can achieve an estimation accuracy of about 10 −1.3 orders of magnitude, which has significant advantages compared with the three algorithms considered. When SNR = 10 dB, about 10 −2 orders of magnitude estimation accuracy can be attained with the proposed scheme, and it is superior to the other three algorithms. Therefore, we can see the superiority of the proposed algorithm.
sors 2023, 23, x FOR PEER REVIEW 10 o advantages compared with the three algorithms considered. When = 10 dB, ab 10 orders of magnitude estimation accuracy can be attained with the proposed schem and it is superior to the other three algorithms. Therefore, we can see the superiority the proposed algorithm.  [25], conventional OMP algorithm [14], and improved OMP algorithm [26]).

Cascaded Channel Estimation
The following simulation parameters are chosen: formed using MATLAB R2019a in the simulation. We assess performance using NM We examined the NMSE performance of three channel estimation schemes-the clas OMP method, the OMP algorithm based on row-structured sparsity, and the sche based on DS-OMP-to demonstrate the proposed approach's superiority. The relationship between the NMSE and the pilot overhead of the BS-RIS-UE chan under various is depicted in Figures 5-7, respectively. The work [15] discussed performance of their algorithm in four different scenarios with of 0, 4, 6, and 8. chose three of these ( = 0, 4, 6) for comparison to fully demonstrate the performa advantages of our algorithm. The proposed channel estimation algorithm's NMSE w contrasted with the conventional OMP approach, the OMP algorithm based on row-str tured sparsity, and the algorithm based on DS-OMP. Under the same pilot overhead c ditions, the proposed scheme outperforms the other three schemes in NMSE performan and as the training pilots increase, our scheme consistently maintains significant perf mance advantages. Taking Figure 7 as an example, the estimation accuracy of the al rithm proposed in this article can approach 10 orders of magnitude when the p overhead is 44. The estimated accuracy of the other three algorithms is around 10 . ders of magnitude lower. The proposed scheme is superior to the other three becaus requires fewer pilots to perform more optimal channel estimation.   [25], conventional OMP algorithm [14], and improved OMP algorithm [26]).

Cascaded Channel Estimation
The following simulation parameters are chosen: M = 64, N = 256, K = 8, L F = 3, L r,k = 8, L c = 0, 4, 6 (the number of common paths for h r,k K k=1 ), d RU =10 m (the distance between RIS and UE), d BU =100 m. A total of 500 Monte Carlo simulations are performed using MATLAB R2019a in the simulation. We assess performance using NMSE. We examined the NMSE performance of three channel estimation schemes-the classic OMP method, the OMP algorithm based on row-structured sparsity, and the scheme based on DS-OMP-to demonstrate the proposed approach?s superiority.
The relationship between the NMSE and the pilot overhead of the BS-RIS-UE channel under various L c is depicted in Figures 5-7, respectively. The work [15] discussed the performance of their algorithm in four different scenarios with L c of 0, 4, 6, and 8. We chose three of these (L c = 0, 4, 6) for comparison to fully demonstrate the performance advantages of our algorithm. The proposed channel estimation algorithm?s NMSE was contrasted with the conventional OMP approach, the OMP algorithm based on row-structured sparsity, and the algorithm based on DS-OMP. Under the same pilot overhead conditions, the proposed scheme outperforms the other three schemes in NMSE performance, and as the training pilots increase, our scheme consistently maintains significant performance advantages. Taking Figure 7 as an example, the estimation accuracy of the algorithm proposed in this article can approach 10 −2 orders of magnitude when the pilot overhead is 44. The estimated accuracy of the other three algorithms is around 10 −1.5 orders of magnitude lower. The proposed scheme is superior to the other three because it requires fewer pilots to perform more optimal channel estimation.  [14], RS-OMP algorithm [13], and DS-OMP algorithm [15]). Figure 6. NMSE's response to the pilot signal ( = 4, compared with OMP algorithm [14], RS-OMP algorithm [13], and DS-OMP algorithm [15]).  [14], RS-OMP algorithm [13], and DS-OMP algorithm [15]).
The relationship between NMSE and SNR of the BS-RIS-UE channel under various is depicted in turn in Figures 8-10. With the classic OMP method, the OMP algorithm based on row-structured sparsity, and the algorithm based on DS-OMP, we compared the  [14], RS-OMP algorithm [13], and DS-OMP algorithm [15]). Figures 8-10. With the classic OMP method, the OMP algorithm based on row-structured sparsity, and the algorithm based on DS-OMP, we compared the NMSE of the proposed channel estimation scheme. Taking Figure 10 as an example, under the same SNR, the proposed method performs better in terms of NMSE than the other three schemes. It continuously retains a considerable performance advantage as the SNR rises, and our approach has an even greater advantage at low SNR. While the estimation accuracy of the other three schemes is roughly 10 0.6 orders of magnitude when SNR = −20 dB, the approach proposed in this study has an estimation accuracy of 10 0 orders of magnitude. The estimation accuracy of the algorithm proposed in this study can approach 10 −1.8 orders of magnitude when SNR = 10 dB, whereas the estimation accuracy of the other three algorithms is only about 10 −1.3 orders of magnitude, demonstrating the proposed algorithm?s superiority. Figure 7. NMSE's response to the pilot signal ( = 6, compared with OMP algorithm [14], RS-OMP algorithm [13], and DS-OMP algorithm [15]).

The relationship between NMSE and SNR of the BS-RIS-UE channel under various L c is depicted in turn in
The relationship between NMSE and SNR of the BS-RIS-UE channel under various is depicted in turn in Figures 8-10. With the classic OMP method, the OMP algorithm based on row-structured sparsity, and the algorithm based on DS-OMP, we compared the NMSE of the proposed channel estimation scheme. Taking Figure 10 as an example, under the same SNR, the proposed method performs better in terms of NMSE than the other three schemes. It continuously retains a considerable performance advantage as the SNR rises, and our approach has an even greater advantage at low SNR. While the estimation accuracy of the other three schemes is roughly 10 . orders of magnitude when SNR = −20 dB, the approach proposed in this study has an estimation accuracy of 10 orders of magnitude. The estimation accuracy of the algorithm proposed in this study can approach 10 . orders of magnitude when SNR = 10 dB, whereas the estimation accuracy of the other three algorithms is only about 10 . orders of magnitude, demonstrating the proposed algorithm's superiority.  [14], RS-OMP algorithm [13], and DS-OMP algorithm [15]).  [14], RS-OMP algorithm [13], and DS-OMP algorithm [15]).
The relationship between NMSE and the number of RIS elements (N) and the number of UE (K) are depicted in turn in Figures 11 and 12, respectively. Here, we chose one of the three above: = 4. In Figure 11, under the same N, the proposed method performs better in terms of NMSE than the other three schemes. It continuously retains a considerable performance advantage as N rises. Due to a certain pilot cost, the estimation accuracy decreases with the increase of N. Therefore, the image shows an upward trend, and the same trend applies to the increase of K in Figure 12. While the estimation accuracy of the best of the other three algorithms is roughly 10 . orders of magnitude when N = 64 in Figure 11, our proposed algorithm has an estimation accuracy of 10 . orders of magnitude. The estimation accuracy of the proposed algorithm can approach 10 . orders of magnitude when N = 484, whereas the estimation accuracy of the best of the other three algorithms is only about 10 . orders of magnitude. In Figure 12, the estimation accuracy of the best of the other three algorithms is roughly 10 . orders of magnitude when K = 4; our proposed algorithm has an estimation accuracy of 10 . orders of magnitude.  [14], RS-OMP algorithm [13], and DS-OMP algorithm [15]).
The relationship between NMSE and the number of RIS elements (N) and the number of UE (K) are depicted in turn in Figures 11 and 12, respectively. Here, we chose one of the three L c above: L c = 4. In Figure 11, under the same N, the proposed method performs better in terms of NMSE than the other three schemes. It continuously retains a considerable performance advantage as N rises. Due to a certain pilot cost, the estimation accuracy decreases with the increase of N. Therefore, the image shows an upward trend, and the same trend applies to the increase of K in Figure 12. While the estimation accuracy of the best of the other three algorithms is roughly 10 −2.2 orders of magnitude when N = 64 in Figure 11, our proposed algorithm has an estimation accuracy of 10 −2.8 orders of magnitude. The estimation accuracy of the proposed algorithm can approach 10 −1.4 orders of magnitude when N = 484, whereas the estimation accuracy of the best of the other three algorithms is only about 10 −1.0 orders of magnitude. In Figure 12, the estimation accuracy of the best of the other three algorithms is roughly 10 −2.0 orders of magnitude when K = 4; our proposed algorithm has an estimation accuracy of 10 −2.6 orders of magnitude. The estimation accuracy of the proposed algorithm can approach 10 −2.0 orders of magnitude when K = 18, whereas the estimation accuracy of the best of the other three algorithms is only about 10 −1.4 orders of magnitude, demonstrating the proposed algorithm?s superiority. The estimation accuracy of the proposed algorithm can approach 10 . orders of magnitude when K = 18, whereas the estimation accuracy of the best of the other three algorithms is only about 10 . orders of magnitude, demonstrating the proposed algorithm's superiority. Figure 11. NMSE's response to the number of RIS elements ( = 4, compared with OMP algorithm [14], RS-OMP algorithm [13], and DS-OMP algorithm [15]). Proposed DS-OMP RS-OMP Figure 11. NMSE?s response to the number of RIS elements (L c = 4, compared with OMP algorithm [14], RS-OMP algorithm [13], and DS-OMP algorithm [15]). Figure 11. NMSE's response to the number of RIS elements ( = 4, compared with OMP algorithm [14], RS-OMP algorithm [13], and DS-OMP algorithm [15]).

Conclusions
In this paper, we proposed an innovative two-phase channel estimation framework for an RIS-assisted multi-user uplink mmWave MIMO communication system. Within this framework, we proposed two channel estimation schemes based on the OMP algorithm to estimate the direct channels (BS-UE) and the cascaded channels (BS-RIS-UE), respectively. Specifically, the direct channels are estimated with an OMP-based LMMSE channel estimation algorithm that has higher estimation accuracy when considering low SNR. Then, an improved LMMSE algorithm based on DS-OMP is proposed for cascaded channels estimation using the double structured sparsity of angular domain cascaded channels in mmWave. The algorithm we propose is an iterative algorithm that can  [14], RS-OMP algorithm [13], and DS-OMP algorithm [15]).

Conclusions
In this paper, we proposed an innovative two-phase channel estimation framework for an RIS-assisted multi-user uplink mmWave MIMO communication system. Within this framework, we proposed two channel estimation schemes based on the OMP algorithm to estimate the direct channels (BS-UE) and the cascaded channels (BS-RIS-UE), respectively. Specifically, the direct channels are estimated with an OMP-based LMMSE channel estimation algorithm that has higher estimation accuracy when considering low SNR. Then, an improved LMMSE algorithm based on DS-OMP is proposed for cascaded channels estimation using the double structured sparsity of angular domain cascaded channels in mmWave. The algorithm we propose is an iterative algorithm that can gradually improve the accuracy of channel estimation. In low SNR, using only the measurement matrix may not be able to accurately estimate the channel due to high measurement noise. However, by using the OMP algorithm to select paths with the maximum inner product, channel estimation can be performed on a smaller subset, thereby reducing the impact of noise. Moreover, LMMSE is utilized to process noise to further improve the quality of channel estimation. The simulation results show that compared with existing algorithms, our proposed algorithm has higher estimation accuracy under the same pilot overhead.