Channel Estimation for Intelligent Reecting Surface Assisted mmWave Communications

Intelligent reflecting surface (IRS) consists of a large number of low-cost passive reflective elements, which can assist millimeter wave communications to solve the problems of weak penetration and short propagation distance. However, it is challenging to obtain channel state information (CSI) in IRS-aided millimeter wave communication systems. To solve this challenge, this paper proposes a regular alternating least squares (RALS) algorithm based on the canonical/parallel factor (CP) decomposition. Compared with the traditional alternate least squares (ALS) algorithm, the proposed RALS algorithm has better convergence performance, thus solving the problem of divergence or slow convergence of the conventional ALS algorithm. Besides, in order to improve the accuracy of the channel estimation, the convex optimization theory is invoked to devise the regularization parameters, and a regularization parameter selection scheme is developed to ensure that the proposed algorithm obtains the optimal solution. The simulation results verify the theoretical analysis and prove the superiority of the proposed RALS algorithm in terms of estimation error performance. This paper proposes a channel estimation scheme using regular alternating least squares algorithm for IRS assisted millimeter wave communications. Due to the introduction of regularization, the proposed scheme can avoid divergence and slow convergence in the channel estimation process, and also avoid numerical instability caused by singular value decomposition in the operation of pseudo-inverse. In order to obtain the optimal solution, the regularization parameters are devised by invoking the convex optimization theory. In simulations, the performance of the proposed scheme is compared with those of the existing algorithms. It is shown that the proposed algorithm is indeed superior to other existing algorithms in terms of estimation error performance.

2 complicated and brings certain difficulties to channel estimation. Specifically, due to the passive nature of IRS, it is quite difficult to estimate the channel from base station (BS) to IRS and the channel from IRS to mobile station (MS) separately.
In order to overcome the above difficulties, we can use separate estimation of the BS-IRS channel and the IRS-MS channel scheme or cascaded channel estimation scheme. In the following, we will discuss these two types of methods, respectively.
First, we discuss the existing work about estimating the BS-IRS and IRS-MS channels separately. In [11], the researchers proposed a new IRS hardware structure, in which a few active reflection elements where randomly deployed in the IRS.
According to this structure, a channel estimation scheme based on compressed sensing (CS) and deep learning (DL) was proposed, in which the full channels between the IRS and the BS/MS were estimated from the available channel state information (CSI) sensed at the active elements. Although this scheme can easily estimate the BS-IRS channel and the IRS-MS channel separately, in the presence of the active elements, it will increase the cost and energy consumption. Besides, in [12], a tensor model was established by using its algebraic structure, so that the two channels can be estimated separately.
Compared with the scheme in [11], this scheme reduces the cost and energy consumption at the expense of the channel estimation accuracy.
Second, we review the existing work about cascaded channel estimation schemes. In [7]- [8], in order to reduce training overhead and simplify the design of passive beamforming, a new elements grouping strategy is proposed, which only needs to estimate the cascaded channels associated with each subsurface; In [13], the channel estimation problem was transformed into a compressed sensing (CS) problem, and a new low-complexity algorithm based on CS was proposed; In [14], the investigators elaborated on the estimation of cascaded channels, and proposed a channel estimation algorithm based on sparse matrix decomposition and complete matrix. The above estimation schemes can easily estimate the cascaded channel, but they have the common limitation, i.e., the specific CSI of the BS-IRS channel or the IRS-MS channel can not be obtained. 3 In this paper, IRS is applied to the millimeter wave communication system. In order to obtain the specific SCI of the BS-IRS channel and the IRS-MS channel, this paper proposes a regular alternating least squares (RALS) algorithm. This algorithm first establishes the relationship between the received signal and the tensor decomposition to satisfy the CP decomposition model, then obtains two forms of the horizontally expanded CP decomposition. Finally, estimate the BS-IRS channel and the IRS-MS channel according to the expanded forms. In order to avoid the problem that the separate estimation will reduce the accuracy of the estimation, the algorithm adds regularization parameters, then uses convex optimization theory to analyze them to get the most suitable regularization parameters, finally obtains the optimal solution. The advantage of this algorithm is that it has a faster convergence rate than the traditional alternating least squares algorithm, avoids divergence, and also solves the problem of data instability caused by singular value decomposition when solving pseudo-inverses. The simulation results also verify the effectiveness of the proposed scheme. In summary, the contributions of this paper are listed as follows: (1) We propose a channel estimation scheme called RALS. By using CP decomposition, our proposed RALS algorithm can realize the separate estimation of the BS-IRS channel and the IRS-MS channel. Due to the addition of regularization parameters, it can speed up the convergence speed of the result and avoid divergence.
(2) In order to improve the accuracy of estimation, we use convex optimization theory to analyze the regularization parameters to obtain the optimal parameters. Ensure that the RALS algorithm is a convex problem, and the optimal solution can be obtained.

System model
This paper considers a downlink IRS assisted millimeter wave communication systems. The IRS is composed of NIRS passive reflective elements, in which the phase shifts and the amplitudes can be adjusted independently. Both the transmitter and the receiver of the system are equipped with multiple antennas. That is, the base station (BS) and the mobile station (MS) are equipped with M and N antennas respectively. In this case, the communication system has two propagation links, which are the BS directly to the MS and the BS to the MS through the IRS. There are three kinds of channels on these two links: 4 the channel from the BS directly to the MS L, the channel from the BS to the IRS H1, and the channel from the IRS to the MS H2. When the direct path is blocked by the obstacles, just the path provided by IRS is used, which can help millimeter wave transmission. This article assumes that the direct path is blocked by the obstacles. The considered IRS-assisted millimeter wave communication system model is shown in Fig. 1. Therefore, the received signal at the MS of the q-th (q∈{1,…,Q}) time slot in the t-th (t∈{1,…,T}) time frame can be written as Accordingly, the received signal in the t-th frame is expressed as From Eq. (2), we know that t Y is the received signal in the t-th time frame, and thus the received signal in T time frames can be written as Finally, we analyze the channel model. Due to the existence of scattering paths, the millimeter wave channel has rich geometric features, in which each scattering path should determine a single propagation path based on the geometric channel model.
Under this model, the channel between BS-IRS and the channel between IRS-MS can be respectively expressed as where 1 2 ( ) l l   is the path gain of the 1 2 ( ) l l -th path. 1 L and 2 L are the number of scattering paths on the channel H1 and the channel H2, respectively. Some previous papers assumed that the antennas follow a uniform linear array (ULA) structure for research simplicity, but this is not in reality situation. In this paper, the uniform planar array (UPA) structure is used. Then the response vectors of the antenna array in the channel can be respectively written as where  is the millimeter wave wavelength. d is the spacing distance between the antenna arrays or that between the 6 In order to make the structure more compact, we define three new variables, which are given as Therefore, according to Eq. (8), Eq. (9) and Eq. (10), Eq. (4) can be rewritten as Similar to the channel H1, the expression of the channel H2 can be rewritten as , , , , , ,

Proposed channel estimation algorithm
Currently, most of the previous papers focused on the design of the reflection matrix for millimeter wave communication system, in which it was assumed that either the CSI is fully known at the BS side or the MS side, or the channel information between the BS and the IRS is fully known. However, since the elements on the IRS are passive, it is challenging to obtain channel state information in practice. Therefore, this paper intends to estimate the channel H1 between BS-IRS and the channel H2 between IRS-MS at the same time based on the channel model developed in the last section.
It can be seen from Eq. (3) that the received signal is a third-order tensor, which makes the channel estimation analysis and calculation more complicated. To proceed, we need to establish the relationship between the received signal and the tensor decomposition.
, t Y is the t-th matrix slice on the third-order tensor According to Eq. (17), we know that the received signal Y is a third-order tensor. Thus, it is difficult to estimate the channel H1 and the channel H2 directly. In such case, the third-order tensor needs to be rearranged to become a matrix, which is called matricization of tensor. In this paper, we use the model-n matrix unfolding method for the matricization of tensor.
Then the two types of matrices unfolding can be written as , the third-order tensor problem is transformed to solve a problem with a second-order matrix, which reduces the complexity of analysis and calculation. At this time, the alternative least squares (ALS) method [18] can be used to estimate the channel H1 and the channel H2. But it is known that the operation of pseudo-inverse is required when the ALS method is used to solve the problem. Particularly, the divergence or slow convergence will occur when we solve the pseudo-inverse of the singular matrix. In order to avoid the above situation, this paper proposes a regular alternating least squares (RALS) scheme. The main concept of the RALS scheme is to regularize the cost function of alternating least squares. In doing so, the cost functions of regular alternating least squares can be reformulated as where    and    are the regularization parameters.

8
The estimation of the channel H1 and the channel H2 is to find the minimum value of the aforementioned cost functions.
For the following comparative analysis of our proposed RALS scheme and the ALS scheme in [18], we focus on analyzing the estimation performance of the channel H2. The analysis of the channel H1 can be obtained similarly. The expression for estimating the channel H2 by the ALS scheme in [13] can be written as the following In order to make the form of Eq. (28) and that of Eq. (25) be similar, Eq. (28) needs to be rewritten. Use the Moore-Penrose inverse matrix property of Khatri-Rao product and the relationship between Khatri-Rao product and Hadamard product, which can be written as [19]  From Eq. (25) and Eq. (31), we can see that the pseudo-inverse matrix can be converted into the inverse matrix by adding the regularization parameter  . The proposed RALS scheme is to add  to each diagonal element of the singular matrix, which turns solving the pseudo-inverse of the singular matrix into solving the inverse of the nonsingular matrix.

Performance analysis of schemes
To compare the performance of the RALS scheme and the ALS scheme, we consider the two aspects, i.e., the singularity and the numerical stability. Then the mean square error (MSE) is derived to evaluate the channel estimation accuracy.
First, let us analyze the singular property, which comes from the pseudo-inverse of the singular matrix.
where min  is the minimum singular value of P W  . By comparing the condition number We can see that adding regularization parameter can significantly improve matrix performance and turn the ill-conditioned matrix into a well-conditioned one. Therefore, the regularization parameter can be added to improve numerical stability.
In summary, adding regularization parameter can solve the problem of the singular matrix and the numerical stability of ill-conditioned matrix. Therefore, the proposed RALS scheme is better than the ALS scheme in terms of the MSE 11 performance and numerical stability.

Selection of the regularization parameter
By comparing the MSE performance of the RALS scheme and the ALS scheme, the superiority of the RALS scheme is demonstrated in the last subsection. In this subsection, we will analyze the selection of the regularization parameter. Here the analysis of the channel H2 is presented firstly, and the analysis of the channel H1 can also gotten in a similar way.
Here,      36, and 49 respectively. It can be seen from Fig. 2 and Fig. 3 that as the number of elements on the IRS increases, the NMSE performance of the proposed scheme is decreasing. This is because when the number of elements on the IRS increases, the channel coefficients that need to be estimated also increase, that is, the number of unknown items increases. To deal with this problem, the length of the pilot symbol sequence should be increased or the number of time frames should be increased. Fig. 4 is a performance comparison diagram between the RALS algorithm proposed in this paper, the BALS (bilinear alternating least squares) algorithm in [12], and the conventional LS (least squares) algorithm . It can be seen from the simulations that the performance of the RALS algorithm is better than other comparison algorithms. On the one hand, the reason is that the conventional LS algorithm is used here as a cascade estimation. Although the conventional LS algorithm is simple and efficient, it ignores the influence of noise, resulting in a large NMSE at low signal noise ratio (SNR). On the other hand, the reason is that although the ALS algorithm considers the influence of noise, the two cost functions obtained are not necessarily convex functions. The final result may not converge or the optimal solution may not be obtained. The RALS algorithm proposed in this paper not only considers the influence of noise, but also adds regularization parameters to ensure convergence, so that the optimal solution can be obtained.