High-Resolution Channel Estimation for Intelligent Reflecting Surface-Assisted MmWave Communications

Millimeter wave (mmWave), as an essential technology for 5G mobile communications, has attracted considerable attentions from both academia and industry thanks to its sufficient unoccupied spectrum resource [1]. However, mmWave transmission inherently suffers from severe path-loss due to its high operating frequency in the range of 30-300 GHz, which poses a critical challenge for practical implementation [2]. Aiming to compensate the link-budget gap, large-scale antenna arrays are usually equipped at transceivers to achieve directional beamforming with focused energy. However, the high directionality coupled with poor penetrability makes mmWave communications vulnerable to blockage events, which significantly deteriorates the coverage capability and limits its applications in urban cellular systems.

Recently, intelligent reflecting surface (IRS) is emerging as a promising technology to achieve high spectral and energy efficiency, and has been integrated into various communication systems on sub-6G band [3, 4, 5]. Specifically, IRS, acting as a planar array, is composed of a large number of low-cost passive reflective elements, which can smartly steer the incident signals towards the dedicated directions via a software-controlled manner [6]. Most recently, IRS is also considered as an energy-efficient solution to combat the blockage of mmWave communications by creating the virtual light-of-sight (LOS) path [7, 8, 9]. Specifically, the authors in [7, 8, 9] investigated the joint beamforming issue and verified that the presence of IRS can improve the mmWave coverage obviously. However, most of these works are based on the assumption of perfect channel state information (CSI), while CSI acquirement is actually a very challenging task, especially for IRS-assisted mmWave communications. Specifically, on the one hand, the introduce of IRS complicates the communication topology, and the user (UE)-IRS and IRS-base station (BS) channels need to be estimated at the same time. On the other hand, the passive IRS does not possess any signal transmitting/receiving/processsing capabilities due to the lack of active radio frequency (RF) chains, which results in that the cascaded channel can be only estimated at receivers.

There are already some channel estimation schemes for IRS-assisted wireless communications on sub-6GHz band [10, 11, 12, 13, 14]. However, these schemes are not applicable to the IRS-assisted mmWave multiple-input-multiple-output (MIMO) communications. The reasons are listed as follows: 1) the channel matrix has a large size due to the large-scale antenna arrays deployed at both transceivers and IRS for mmWave communications, which makes the entry-wise channel estimation schemes suffer from prohibitive computation complexity and pilot overhead; 2) the random phase of IRS is not reasonable for high-directional mmWave links since beam misalignment can significantly degrade the channel estimation performance. Despite these difficulties, the research on channel estimation for IRS-assisted mmWave communications is still very limited [15, 16, 17, 18]. Specifically, by exploiting the sparsity of mmWave channels, the authors in [15, 16] studied the compressive sensing (CS) based cascaded channel estimation for IRS-assisted multiple-input-single-output (MISO) systems with single-antenna transmitter/receiver. Moreover, the authors in [17] divided the cascaded channel estimation into two stages, i.e., estimating the BS-IRS channel and UE-IRS channel by equipping IRS with active RF chains, which makes channel estimation a high energy-consuming operation. Most recently, the authors in [18] proposed a beam training based channel estimation scheme. However, due to the imperfect hardware manufacturing technique, it is impractical to fabricate IRS with infinite phase-resolution, which severely degrades the accuracy of channel estimation. We can observe that there is little discussion on high-resolution cascaded channel estimation for IRS-assisted mmWave MIMO communications, while it is imperative for further system optimization and data transmission.

In this paper, by exploiting the sparsity of mmWave channels, we formulate the cascaded channel estimation into a sparse recovery problem. Taking the hardware-constraint into consideration, we propose a two-step cascaded channel estimation protocol to acquire high-resolution CSI. In the first step, the beam training procedure is utilized to estimate coarse angular domain information (ADI), which is regarded as an important prior information for uplink establishment and further high-resolution channel estimation. In the second step, enhanced by the prior ADI acquired by the first step, a low-complexity adaptive grid matching pursuit (AGMP) algorithm is proposed to estimate the high-resolution cascaded CSI. In particular, the beam training based channel estimation scheme [18] is adopted as the benchmark. The simulation results show that the proposed two-step channel estimation protocol provides a more accurate CSI as compared to the benchmark and reaps better system performance in terms of spectral efficiency (SE).

Notation: $j=\sqrt{-1}$. $\mathbf{A}$, $\mathbf{a}$ and $\mathcal{A}$ are matrix, vector and set, respectively. $\mathbf{A}^{H}$, $\mathbf{A}^{*}$ and $\mathbf{A}^{T}$ denote conjugate transpose, conjugate and transpose of $\mathbf{A}$, respectively. $\mathbf{A}(i)$ denotes the $i$th row of $\mathbf{A}$. $\mathbb{E}\left(\cdot\right)$ denotes the expectation operation. $\left\|\mathbf{A}\right\|_{F}$ is the Frobenius norm of $\mathbf{A}$. $\textrm{vec}\left(\mathbf{A}\right)$ represents the column-wise vectorization of $\mathbf{A}$, and its inverse operation is $\textrm{unvec}\left(\mathbf{a}\right)$. $\mathbf{A}\otimes\mathbf{B}$ denotes the Kronecker product of $\mathbf{A}$ and $\mathbf{B}$.

We consider an IRS-assisted mmWave MIMO system containing a single BS, a single IRS and a single user, which is shown in Fig. 1. More specifically, the IRS is deployed to provide a virtual LOS link for the user that is blocked by the obstacles. Similar to [19], we consider that both BS and user are equipped with an analog antenna architecture consisting of a full-connection uniform linear array, where the numbers of antennas are $N_{BS}$ and $N_{UE}$, respectively. The IRS is equipped with a planar array consisting of $N_{I}$ passive reflective elements whose amplitude and phase can be adjusted dynamically via the IRS-controller.

The geometric channel model is adopted to characterize the channels for the considered IRS-assisted mmWave MIMO communications, and the channel matrix $\mathbf{H}\in\mathbb{C}^{N_{R}\times N_{T}}$ is expressed as [20]

where $N_{T}$ and $N_{R}$ denotes the number of antennas at transmitter and receiver, respectively. $L$ is the number of path, and $\alpha_{l}$ is the channel complex gain of the $l$th path. $\theta$ and $\varphi$ denote the angle-of-arrival (AoA) and angle-of-departure (AoD). For convenience, we have $\varPhi=\text{cos}\left(\theta\right)$ and $\varOmega=\text{cos}\left(\varphi\right)$, which are called AoA and AoD in the rest of this paper. $\mathbf{a}_{R}\left(\cdot\right)$ and $\mathbf{a}_{T}\left(\cdot\right)$ are the steering vectors at receiver and transmitter, respectively, which are given by

where $\lambda$ is the wavelength of the carrier, and $d$ is the antenna spacing which is equal to the half-wavelength. The channel model in (1) can be written in a more compact form as

where $\mathbf{H}_{a}=\sqrt{\frac{N_{T}N_{R}}{L}}\text{diag}\left(\alpha_{1},...,\alpha_{L}\right)$ represents the angular domain channel, $\mathbf{A}_{R}=\left[\mathbf{a}_{R}\left(N_{R},\varPhi_{1}\right),...,\mathbf{a}_{R}\left(N_{R},\varPhi_{L}\right)\right]\in\mathbb{C}^{N_{R}\times L}$, $\mathbf{A}_{T}=\left[\mathbf{a}_{T}\left(N_{T},\varOmega_{1}\right),...,\mathbf{a}_{T}\left(N_{T},\varOmega_{L}\right)\right]\in\mathbb{C}^{N_{T}\times L}$.

For the considered IRS-assisted mmWave system, let $\mathbf{H}^{B}\in\mathbb{C}^{N_{I}\times N_{BS}}$ denotes the channel from BS to IRS, $\mathbf{H}^{I}\in\mathbb{C}^{N_{UE}\times N_{I}}$ denotes the channel from IRS to UE. Assuming that the direct link from BS to UE is blocked, we mainly focus on the cascaded channel acquirement. The BS-IRS-UE cascaded channel $\mathbf{H}^{S}\in\mathbb{C}^{N_{UE}\times N_{BS}}$ can be expressed as

where $\mathbf{\Theta}=\text{diag}\left(\beta_{1}e^{j\phi_{1}},...,\beta_{N_{I}}e^{j\phi_{N_{I}}}\right)$ denotes the phase-shift matrix of IRS with $\beta_{n}\in\left[0,1\right]$ and $\phi_{n}\in\left[0,2\pi\right)$ being the reflection amplitude and phase-shift parameters of the $n$th passive reflective element, respectively. Without loss of generality, we assume $\beta_{n}=1$ for all elements to maximize beamforming gain brought by IRS [6].

In this section, we propose a novel two-step cascaded channel estimation protocol to acquire high-resolution CSI, and the specific frame structure is shown in Fig. 2. First, a downlink beam training procedure is performed to estimate the coarse ADI, and further the rubost UE-IRS-BS communication link is established. In the second step, the coarse ADI is utilized to facilitate the uplink cascaded channel estimation, and meanwhile, the AGMP algorithm is proposed to estimate the high-resolution CSI, which is vitally important to enable further high-quality data transmission. More details will be described in the sequel.

In this section, the performance of the proposed two-step cascaded channel estimation protocol is evaluated via computer simulation. In our simulation, the corresponding system parameters are set as following: the number of antennas at BS $N_{BS}=64$, the number of antennas at IRS $N_{I}=64$, the number of antennas at user $N_{UE}=16$, the carrier frequency $f_{c}=28$ GHz, the number of paths from IRS to user $L=3$.

We evaluate the normalized mean square error (NMSE) performance of our proposed two-step concatenated channel estimation protocol against signal-to-noise ratio (SNR). As shown in Fig. 4, assuming the dictionary resolution $\widetilde{G}=5$, the NMSE performance against SNR is compared under different iterations of the AMGP algorithm. The NMSE is defined as

where $\mathbf{H}$ is the perfect CSI, and $\widehat{\mathbf{H}}$ is the estimated CSI. Moreover, we assume that there are a single LOS path and two NLOS paths between the IRS and user, and the Rician K-factor is set as 20dB. Obviously, even with few iterations, the proposed scheme shows great superiority over the benchmark scheme [18], in which the CSI is obtained directly by beam training. Moreover, we can observe that increasing the iteration will improve the NMSE performance. In particular, the beam training based channel estimation scheme suffers from poor NMSE performance even at high-SNR, and thus we can conclude that the ADI estimation plays a dominant role for the accuracy of channel estimation, especially for high-directional mmWave channel.

Fig. 5 shows the comparison of SE performance against SNR. In particular, the SE performance achieved by perfect CSI is adopted as the upper bound, and the random beamforming is shown as the lower bound. We can observe that our proposed two-step channel estimation protocol outperforms the benchmark scheme even with limited dictionary resolution ($\widetilde{G}=5$) and iteration $\zeta=4$, and meanwhile can achieve the near-optimal SE performance obtained by perfect CSI with the increase of iteration. Moreover, we also present the SE performance for the case without IRS, which verifies that the presence of IRS indeed improves the SE performance of mmWave communications due to the additional beamforming gain brought by IRS. Comparing Fig. 4 with Fig. 5, we can conclude that the beam training based channel estimation scheme can only estimate coarse ADI due to the limited phase-resolution of IRS, while the estimated channel gain is not accurate drastically due to the effects of the angular estimation error, multi-path and noise. When the coarse CSI is adopted for data detection or system optimization, poor performance may be generated. In this regard, we propose a high-resolution channel estimation scheme to mitigate the influence of these factors by exploiting an AGMP algorithm. Since the IRS can reinforce the signal power through directional beamforming, which is equivalent to degrade the influence of the noise, the proposed protocol can still reap a great NMSE performance even at low-SNR for IRS-assisted mmWave MIMO communications.

Fig. 6 and Fig. 7 compare the NMSE performance and SE performance against the dictionary resolution $\widetilde{G}$ under different SNR cases, respectively. With the increase of the dictionary resolution $\widetilde{G}$, the NMSE and SE performance can be significantly improved for different SNR cases. In particular, the SE performance converges to the upper bound obtained by perfect CSI quickly even when $\widetilde{G}=3$, which suggests that $\widetilde{G}=3$ is the optimal selection for the given system settings.

In this paper, we have studied the high-resolution cascaded channel estimation issue for IRS-assisted mmWave MIMO communications. Specifically, by exploiting the sparsity of mmWave channels, we formulate the cascaded channel estimation problem into a sparse recovery problem, and propose a two-step cascaded channel estimation protocol to achieve high-resolution channel estimation. The simulation results verify the advantages of our proposed protocol in terms of SE and NMSE performance as compared to the beam training based cascaded channel estimation scheme. Moreover, our proposed AGMP algorithm can approach the optimal performance achieved by perfect CSI with low computation complexity.