Issue 
TST
Volume 13, Number 2, June 2020



Page(s)  51  60  
DOI  https://doi.org/10.1051/tst/2020132051  
Published online  29 January 2021 
Invited Paper
Channel estimation for intelligent reflecting surface aided multiuser MISO terahertz system
University of Electronic Science and Technology of China, Chengdu, 611731, China
^{*} Email: chenzhi@uestc.edu.cn
Received:
17
June
2020
Intelligent reflecting surface (IRS) is considered as a promising application in terahertz (THz) communications since it is able to enhance the THz communication with no additional power consumptions. In this letter, we consider the channel estimation problem for an IRSaided THz multiuser multiinput singleoutput (MISO) system with lens antenna array. The main challenge of the problem is that we need to estimate multiple channels and some of the channels are cascaded. To deal with the problem, we propose a twostage channel estimation scheme, where we set different IRS modes to estimate different channels for each stage. In stage 1, we set the IRS to an absorbing mode and estimate the channel without IRS. Removing the influence of the prior estimated channel, in stage 2, we estimate the channel with IRS by setting the IRS to a perfect reflecting mode. And we decompose the total channel estimation problem into a series of independent problems, where we estimate each independent channel component with a least square method.
Key words: Intelligent reflecting surface (IRS) / Terahertz (THz) communication / channel estimation
© The Author(s) 2020
This is an Open Access article distributed under the terms of the Creative Commons Attribution License CCBY (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, except for commercial purposes, provided the original work is properly cited.
1. Introduction
Intelligent reflecting surface (IRS), as a promising technique for future wireless systems, such as terahertz (THz) communications, has attracted growing research interest in both academia and industry over recent years [1, 2]. An IRS is a physical metasurface consisting of a large number of reflecting elements, where each element is equipped with a simple lowcost sensor [3]. And each element is able to reflect incident electromagnetic waves independently by adjusting its phaseshift. Compared to traditional relay schemes that enhance sourcedestination transmission by generating new signals, IRS does not buffer or process any incoming signals but only reflects the wireless signal as a passive planar array, which incurs no additional power consumptions [4, 5].
Previous works about the IRS are mainly focused on optimizing secrecyrate and datarate by designing the phaseshifts of the IRS while assuming perfect channel state information (CSI) is obtained by both the base station (BS) and the IRS [6–10]. However, it is difficult to obtain the perfect CSI since the IRS cannot process any induced signals or emit any pilot signals. Therefore, the BS needs to estimate all the channels between the BS and the user, which includes the channel between the (BS, IRS), (IRS, user), and (BS, user). To the best of our knowledge, there are limited literatures considering the channel estimation problem for the IRSaided system.
Thus, in this letter, we investigate the channel estimation problem for the IRSaided THz multiuser multiinput single output (MISO) system with lens antenna array. To solve the problem, we propose a twostage channel estimation scheme, where we set different IRS modes for the channel estimation in different stages. In stage 1, we estimate the channel between the BS and the user by setting the IRS to an absorbing mode which is able to absorb all induced signals by the IRS. Removing the influence of the prior estimated channel, in stage 2, we set the IRS to a perfect reflecting mode, which can reflect all induced signals by the IRS with few losses. And we find that the channel with the IRS a cascaded channel. To estimate it, we decompose the total channel estimation problem into a series of independent problems, where we estimate each channel component with a least square method.
2. System model and problem formulation
2.1 System model
As shown in Fig. 1, we consider an uplink THz multiuser MISO system, where a BS, which consists of a one dimensional lens antenna array with N _{t} elements, simultaneously receives signals from K singleantenna users. To enhance the THz communication, an IRS equipped with N passive elements is installed on a surrounding wall to overcome unfavorable propagation conditions and enrich the channel with more paths. For each path, due to the severe propagation loss in the THz communication, we only consider a single reflection signal by the IRS and ignore other signals reflected by the IRS more than one time. And we assume only one data stream needs to be transmitted by each user. In tth instant, each user sends a pilot signal, denoted by , to the BS over two ways. One way is achieved by the directly channel between the BS and the user. Another way is achieved by the IRS. The IRS can reflect THz signals to the BS by a diagonal phaseshift matrix which will be discussed later. Therefore, the received signal at the BS can be expressed as(1)where s(t) = [s _{1}(t), s _{2}(t), …, s _{K}(t)]^{ T }∈ is the pilot vector for the channel estimation process, H_{ r } = [h_{r,1,} h_{r,2}, … h_{r,K}] ∈ C ^{ N×K } (resp. H_{ t } ∈ ) is the channel between the IRS and the user (resp. between the BS and the IRS), H_{ d } = [h_{ d,1,} h_{ d,2}, … h_{ d,K}] ∈ is the channel between the BS and the user, and is zeromean additive white Gaussian noise where we denote δ ^{2} as the noise power. In addition, the phaseshift matrix can be represented as where represents the phase shift for the nth reflecting element, and β ∈ [0, 1] is an amplitude reflection coefficient on the incident signals. It is worth noting that when we set β = 0, the IRS mode turns to absorbing mode, which means that all signals get to the IRS will be absorbed. And when we set β = 1, the IRS mode turns to perfect reflecting mode, which means that all signals get to the IRS can be reflected with few losses. To estimate the channels H_{t}, , and , we use total T instants for the channel estimation. And we divide T into M blocks, where each block consists of K instants. Thus, we have T = MK. For the mth block (m = 1, 2, …, M), the received signal at the BS can be written as(2)where is the mth pilot matrix. To normalize the power of the pilot signal to unit, s_{m} satisfies . And is the noise matrix.
Fig. 1 IRSaided THz multiuser MISO system with lens antenna array 
In terms of H_{t}, , and , motivated by [11], we use a modified SalehValenzuela model to capture the characteristics of the THz channel, which is comprised of several paths by reflection and directly transmission. Specifically, the channel response of H_{t}, , and can be respectively given by(3)where L _{t} (resp. L _{r} and L _{d}) is the number of paths for channel H_{t} (resp. and ), (resp. ) is the spatial direction, which can be defined as (resp. ), where (resp. ) is the physical direction, λ is the wavelength of carrier, and d is the antenna spacing or reflectingelement spacing. In addition, is the complex gain for path i, which is mainly contributed by transmission losses and molecular absorbing losses in THz communication. And is the array steering vector. For a typical uniform linear array with antennas, can be represented as .
Furthermore, the conventional channel (3) in the spatial domain can be transformed to the beamspace channel by employing the lens antenna array with a set of bases, which can be expressed as , where denotes the spatial direction. And with the transformed signals in the beamspace domain, the BS can employ a combiner to combine the above signals. Then, for the mth block, the combined signal can be obtained as(4)
2.2 Problem formulation
During the channel estimation, the estimated channel can be denoted as , , and . And the estimated combined signal is able to be represented as . Our interest lies in minimizing the Euclidean distance between and R_{m} by estimating the channel , , and , which is written as(5)
3. Channel estimation scheme
In this section, we seek to solve problem (5) with estimating the channel H_{t}, , and . And we propose a twostage channel estimation scheme, where we first estimate by turning the IRS mode to the absorbing mode, and then we estimate by removing the influence of . For the secondstage channel estimation, we decompose the total channel estimation problem into a series of independent problems, where we estimate each channel component with a least square method.
Specifically, we first multiply the know pilot matrix on the right side of (4). Since we have , the measurement matrix can be obtained by(6)
And each column of Z_{m}, denoted by , is the measurement vector for the subchannel of user k. After M block’s measurement, we can obtain a T × 1 measurement vector for user k, which can be written as(7)where vector , , .
Note that in (7), there are three channels, H_{t}, h_{r,k}, and h_{d,k} need to be estimated. And we also notice that the channel h_{d,k} is independent with the other two channels. Thus, we propose a twostage channel estimation scheme, where we first estimate the channel h_{d,k}, and then estimate the channel H_{t} and h_{r,k} by removing the influence of h_{d,k}. With setting the IRS to the absorbing mode, where Θ = 0_{N × N}, the measurement vector for the channel h_{d,k} can be obtained as(8)
Since (8) is a traditional channel estimation scheme, we can estimate the channel h_{d,k} with traditional solutions, such as [12]. After that, removing the influence from , the residual measurement vector for the channel H_{t} and h_{r,k} can be given as(9)
To estimate the channel H_{t} and h_{r,k} in (9), we set the IRS to the perfect reflecting mode, where Θ = I_{N × N}. Then, (9) can be rewritten as(10)
Although it is able to estimate the cascaded channel H_{t} h_{r,k} in (10), it is hard to separate H_{t} and h_{r,k} into H_{t} and h_{r,k}. Fortunately, for the MISO system, we do not need to separately estimate H_{t} and h_{r,k}, since we have(11)where is a N × 1 vector consisting of the diagonal elements of Θ. Therefore, for any data rate optimization problems in the MISO system, the IRS can optimize the phaseshifts by only knowing the channel H_{t} diag(h_{r,k}). Thus, we have(12)
To estimate the channel H_{t}diag(h_{r,k}) in (12), we denote the estimated as , and the estimated as . Then problem (5) can be reformulated as(13)
To solve problem (13), we first propose a proposition to prove a special property of the IRS beamspace channel, which is the base of our proposed channel estimation scheme.
Proposition 1
Denote the effective channel as , where is the ith channel component of . When the number of transmission antennas N _{t} tends to infinity, we have(14)which means that any two channel components c _{k,i} and c _{k,j} in are independent.
Proof: Based on (3), the ith channel component c _{k,i} can be presented aswhere , , and x = 1, 2, ⋯, N_{t}. And it has been demonstrated in [13] when c _{k,i} is formulated as (a), Proposition 1 can be true.
According to Proposition 1 , we can estimate the channel UH_{t} h_{r,k} by estimating a series of independent channel components , which can decrease the complexity of the channel estimation. When is obtained, is able to be calculated by , where (⋅)^{+} is the pseudo inverse operation.
Summarizing the above analysis, the detailed steps of our proposed twostage channel estimation scheme are illustrated in Algorithm 1. Specifically, in stage 1, we estimate h_{d,k} by setting the IRS to the absorbing mode. And in stage 2, we estimate H_{t}diag(h_{r,k}) by setting the IRS mode to the perfect reflecting mode. In step 3, we remove the influence of the estimated channel , and obtain the measurement vector for H_{t} h_{r,k}. Then, we enumerate UH_{t} h_{r,k} by estimating its channel component one by one. For the ith component, in step 5, we detect the position of the strongest element of c_{k,i}. And in step 6, we construct a position vector , which represents the position of the most 2V + 1 strong elements in c_{k,i}, since for the function in Proposition 1, when n is more close to , can be more close to . Next, we use the 2V + 1 elements to extract the sub measurement vector from as , where can be approximated by since we have . With the reduceddimensional measurement vector , in step 7, we use a least square method to estimate by , which can effectively reduce the computational complexity. After all L _{t} channel components are estimated, in step 10, we can obtain as . And can be estimated in the end.
Proposed TwoStage Channel Estimation Scheme
4. Numerical results
In this section, numerical results are presented to demonstrate the performance of Algorithm 1. We consider an IRS aided THz multiuser MISO system, where the BS equips with a lens antenna array with N _{t} = 256 antennas, simultaneously serves to K = 16 singleantenna users with the aid of an IRS, which is with N = 16 reflecting elements. For the channel H_{ t } and , we assume the complex gain is , the spatial direction ψ follows a uniform distribution within [−0.5, 0.5], and L _{t} = L _{r} = 3 due to the sparsity of the THz channel. For simplicity, we assume which means that the directly transmissions between the BS and the user are broken down with some obstacles. Finally, all the results are averaged over 5000 random channel realizations.
Fig. 2 shows the normalized mean square error (NMSE) performance versus SNR in different schemes, where we define
Fig. 2 Normalized mean square error comparison versus SNR. 
As shown in Fig. 2, the NMSE performance can be improved with the increasing SNR. Therefore, when the SNR is smaller than 10 dB, the NMSE can be lower when V decreases, but when the SNR is larger than 10 dB, the result will be reversed.
5. Conclusions
In this letter, we investigate the channel estimation problem for the IRSaided multiuser MISO system with lens antenna array. Specifically, we propose a twostage channel estimation scheme, where we first have estimated the channel without IRS by setting the IRS to the absorbing mode, and then we have estimated the cascaded channel reflected by the IRS with removing the influence of the prior estimated channel. Since we demonstrate that the channel components of the cascaded channel are independent, in stage 2, we decompose the total channel estimation problem into a series of independent problems, where we have estimated each channel component by the least square method. Numerical results show the effectiveness of our proposed channel estimation scheme.
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All Tables
All Figures
Fig. 1 IRSaided THz multiuser MISO system with lens antenna array 

In the text 
Fig. 2 Normalized mean square error comparison versus SNR. 

In the text 