Anchor-Assisted Intelligent Reflecting Surface Channel Estimation for Multiuser Communications

Recently, intelligent reflecting surface (IRS) has emerged as a promising technology to achieve high spectral and energy efficiency for future wireless networks [1], [2]. Specifically, IRS is a uniform planar array composed of a large number of low-cost, passive, and tunable reflecting elements. By adaptively varying the reflection coefficient of each element based on the user dynamic channels, IRS can achieve high beamforming and interference suppression gains cost-effectively [3]. As such, IRS has been studied recently in various wireless systems, including non-orthogonal multiple access (NOMA)[4], [5], simultaneous wireless information and power transfer (SWIPT) [6], [7], secrecy communications [8], [9], and so on.

To reap the performance gain of IRS, accurate channel state information (CSI) is required. However, the passive nature of IRS makes channel estimation fundamentally challenging in IRS-aided wireless networks. This is because without radio frequency (RF) chains, IRS can neither transmit nor receive pilot signals, thus the base station (BS)-IRS and IRS-user channels cannot be estimated separately. One alternative approach is to estimate the cascaded BS-IRS-user channel via element-wise on-off operation at each reflecting element [10], [11] or time-varying reflection patterns [12], [13]. However, by applying the above methods to estimate the cascaded channels of multiple users consecutively, the required pilot overhead is the product of the number of IRS reflecting elements and that of users, which is prohibitively large for the case of large IRS serving a high density of users nearby (e.g., in a hot spot scenario).

To tackle this problem, we propose in this paper a new anchor-assisted two-phase channel estimation scheme for IRS-aided multiuser communications, which can significantly reduce the channel training overhead by decoupling the estimation of cascaded channels and exploiting multiple antennas at the BS. As shown in Fig. 1, two anchor nodes, namely A1 and A2, are deployed near the IRS to assist its channel estimation. In the first phase, the cascaded BS-IRS-A1/A2 and A1-IRS-A2 channels are estimated separately, based on which the partial CSI of the BS-IRS link, in terms of the square of each IRS element’s channel with the BS (thus, with a +/- sign uncertainty), is obtained. In the second phase, with such partial CSI of the BS-IRS channel that is common to all users, the individual cascaded BS-IRS-user channels are efficiently estimated. Particularly, when the number of antennas at the BS ($M$) is no smaller than that ($N$) of the IRS reflecting elements, i.e. $M\!\geq\!N$, we show that the proposed scheme can estimate all users’ cascaded channels in the second phase using only one pilot symbol. Besides, even for the case of $\!M<N\!$, the minimum training overhead (in terms of number of pilot symbols) is shown to be $\left\lceil\frac{NK}{M}\right\rceil$, where $K$ is the number of IRS-served users and $\left\lceil\cdot\right\rceil$ denotes the ceiling operation, which is significantly lower than $NK$ for the conventional schemes [10, 11, 12, 13]. Moreover, in the case that the IRS-anchor channel is line-of-sight (LoS) by properly deploying the anchor, it is shown that the proposed scheme can be further simplified such that deploying only one anchor is sufficient.

As shown in Fig. 1, we consider an IRS-assisted multiple-input-single-output (MISO) communication system, which consists of a BS, an IRS and $K$ users. The number of antennas at the BS and that of reflecting elements at the IRS are denoted by $M$ and $N$, respectively. The channels from the BS to IRS and User $k$ are denoted by ${\bf{H}}_{bs}\in{\mathbb{C}^{M\times N}}$ and ${\bf{h}}_{bu_{k}}\in{\mathbb{C}^{M\times 1}}$, respectively, while that from the IRS to User $k$ is denoted by ${\bf{h}}_{su_{k}}\in{\mathbb{C}^{N\times 1}}$. We assume that two single-antenna anchor nodes¹¹1In practice, anchors can be idle user terminals and/or dedicated nodes such as adjacent IRS controllers. A1 and A2 are deployed near the IRS to assist in the channel estimation. The channels from the BS to A1 and A2 are denoted by ${\bf{h}}_{ba_{1}}\!\in\!{\mathbb{C}^{M\times 1}}$ and ${\bf{h}}_{ba_{2}}\!\in\!{\mathbb{C}^{M\times 1}}$, respectively, those from the IRS to A1 and A2 are denoted by ${\bf{h}}_{sa_{1}}\!\in\!{\mathbb{C}^{N\times 1}}$ and ${\bf{h}}_{sa_{2}}\!\in\!{\mathbb{C}^{N\times 1}}$, respectively, and that from A1 to A2 is denoted by ${{h}}_{a_{1}a_{2}}$. As a result, the cascaded BS-IRS-A1/A2/User $k$ channels are denoted by ${\bf H}_{bsa_{1}}\!=\!{\bf H}_{bs}\text{diag}({\bf h}_{sa_{1}})$, ${\bf H}_{bsa_{2}}\!=\!{\bf H}_{bs}\text{diag}({\bf h}_{sa_{2}})$ and ${\bf H}_{bsu_{k}}={\bf H}_{bs}\text{diag}({\bf h}_{su_{k}})$, respectively, while the cascaded A1-IRS-A2 channel is denoted by ${\bf h}_{a_{1}sa_{2}}={\bf h}_{sa_{2}}^{T}\text{diag}({\bf h}_{sa_{1}})$. Moreover, the phase-shift matrix of the IRS at time slot $i$ is denoted by ${\mathbf{\Phi}}_{i}=\text{diag}\left(v_{1,i},v_{2,i},....,v_{N,i}\right)$, where $v_{n,i}$ is the reflection coefficient of the $n$-th IRS element at time slot $i$, $n=1,...,N$. Since IRS is a passive reflecting device, we assume that the channel reciprocity holds for each link between IRS and any other node. The quasi-static flat-fading channel model is assumed for all the links involved.

The proposed channel estimation and data transmission protocol is shown in Fig. 2, where $T_{bs}$ and $T_{su}$ are the length of channel coherence block of ${\bf{H}}_{bs}$ and ${\bf{h}}_{su_{k}}$, $\forall k$, respectively, and $L=K$ for $M\geq N$ while $L=\left\lceil{\frac{KN}{{M}}}\right\rceil$ for $M<N$. In practice, $T_{bs}$ is usually much larger than $T_{su}$ due to the fixed locations of the BS and IRS once deployed, while users can move randomly near the IRS. The channel estimation consists of one off-line phase (Phase I) and multiple on-line phases (each termed Phase II). In Phase I, the cascaded BS-IRS-A1, BS-IRS-A2, A1-IRS-A2 channels are estimated separately, which requires $2(N+1)$ pilot symbols in total. In each Phase II, the BS estimates the cascaded BS-IRS-user channels with $K+L$ pilot symbols. For the estimation of ${\bf{h}}_{bsu_{k}}$ in Phase II, the efficiency is largely improved, especially when $M$ increases up to $M\geq N$. The detailed channel estimation scheme is elaborated as follows.

We assume that the IRS and/or the anchor node “A” can be properly deployed so that the IRS-A channel is LoS. In this case, ${\bf{h}}_{ra}$ is known a priori based on the knowledge of the positions of IRS and A, thus only one anchor node is sufficient for the proposed scheme and its training overhead can be further reduced.

The simulation setup is shown in Fig. 3. It is assumed that BS, IRS (the central point), A1, A2 and A are located at (50, 0, 20), (0, 100, 2), (2, 99, 0), (2, 101, 0) and (2, 100, 0) in meter (m), respectively. We assume that the system operates on a carrier frequency of 750 MHz with the wavelength $\lambda_{c}=0.4$ m and the path loss at the reference distance $d_{0}\!=\!1$ m is given by $L_{0}=30$ dB. Suppose that the IRS is equipped with a uniform planar array with 6 rows and 10 columns, and the element spacing is $\Delta d\!=\!3\lambda_{c}/8$; thus, we have $N=60$. The noise power is set as $\sigma_{0}^{2}\!=\!-$105 dBm. The channel from the BS to A1 is generated by ${{\mathbf{h}}_{ba_{1}}}\!=\!\sqrt{{L_{0}}d_{ba_{1}}^{-{c_{ba_{1}}}}}{{\bf{g}}_{ba_{1}}}$, where $d_{ba_{1}}$ denotes the distance from the BS to A1 and ${{\bf{g}}_{ba_{1}}}$ is the small-scale fading component. The same channel model is adopted for all other channels in general. Particularly, Rayleigh fading is assumed for the channels among the BS, IRS, A1, A2 and each User $k$ with the path loss exponents set as 3, whereas in the special channel case in Section IV, the channel between IRS and A is assumed to be LoS, with the path loss exponent set as 2. We also apply the scheme proposed in [13] for each of the users to estimate their channels consecutively, which serves as the benchmark.

Fig. 4(a) shows the required training time (in terms of number of pilot symbols) versus the number of antennas, $M$, at the BS. It is observed that as $M$ increases, the proposed scheme significantly reduce the training overhead as compared to that of the benchmark scheme (which is independent of $M$). This is because the proposed scheme exploits the multiple antennas at the BS for joint IRS channel estimation, whereas in the benchmark scheme the BS antennas estimate their associated channels independently in parallel. Note that when $M=1$, the training overhead of the proposed scheme is even larger than that of the benchmark scheme. This is because additional pilot symbols are transmitted by anchors in Phase I, while the training efficiency in Phase II is not improved since $\text{max}(K,\left\lceil\frac{NK}{M}\right\rceil)=NK$ in the case of $M=1$. Moreover, it is observed that the proposed scheme under the special case of LoS IRS-anchor channel is more efficient as compared to the general channel case.

Fig. 4(b) shows the training time of the proposed scheme normalized by that of the benchmark scheme versus $K$, with $M=10$ and 60, respectively. One can observe that the performance gap between the two schemes becomes larger as $K$ increases. This is because to accommodate one more user, the additional pilot overhead required by the benchmark scheme is $N+1$, while that by the proposed scheme is $1+\text{max}(1,\left\lceil\frac{N}{M}\right\rceil)$. As a result, as $K$ increases, the pilot reduction by using the proposed scheme also increases. Also note that similar to Fig. 4(a), when $K$ is very small, the proposed scheme is even worse than the benchmark scheme. This reason is that additional $2(N+1)$/$N+1$ pilot symbols are required in Phase I, regardless of $K$, while in the benchmark scheme, only $N+1$ pilot symbols are sufficient when $K=1$.

Fig. 4(c) shows the normalized mean-squared error (MSE) of the estimations of ${\bf h}_{bu_{k}}$ and ${\bf H}_{bsu_{k}}$ versus the transmit power of pilot symbols in the on-line phase, with that of the off-line phase fixed as 40 dBm. It is observed that the MSE in the general channel case of the proposed scheme is highest, while that of the benchmark scheme is lowest. The reason is that although the proposed scheme significantly reduces the training overhead (see Figs. 4(a) and 4(b)), the estimation error in ${\bf\hat{H}}_{bsu_{k}}$ depends on both ${\bf\hat{H}}_{bs}$ and ${\bf\hat{h}}_{su_{k}}$. Specifically, in the general channel case, the error in ${\bf\hat{H}}_{bs}\odot{\bf\hat{H}}_{bs}$ comes from ${\bf\hat{H}}_{bsa_{1}}$, ${\bf\hat{H}}_{bsa_{2}}$ and ${\bf\hat{h}}_{a_{1}sa_{2}}$, while in the special channel case the error in ${\bf\hat{H}}_{bs}$ comes from ${\bf\hat{H}}_{bsa}$. In contrast, for the benchmark scheme, ${\bf H}_{bsu_{k}}$ is estimated directly, which is thus less susceptible to noise/error. However, one can observe that the MSE of the proposed scheme is substantially reduced by increasing $M$, and becomes even comparable when $M=100$.

In this paper, we propose a new anchor-assisted channel estimation scheme for IRS-aided multiuser communications. By exploiting the fact that all BS-IRS-user cascaded channels share the same BS-IRS channel, the proposed scheme first estimates this common channel with only sign ambiguity via the anchor-assisted training. Then we show that the estimation of each cascaded BS-IRS-user channel is simplified to estimating each IRS-user channel with the number of unknowns significantly reduced from $MN$ to $N$, and the sign ambiguity in the estimated common channel does not affect the uniqueness of the recovered cascaded channels. Moreover, by exploring multi-antennas at the BS, the training overhead in estimating all users’ cascaded channels is reduced from $NK$ to $\text{max}(K,\left\lceil\frac{NK}{M}\right\rceil)$. Numerical results validate the effectiveness of the proposed scheme, especially when $M$ and/or $K$ is large. Considering the trend towards massive antenna arrays at the BS and massive connectivity with machine-type communications, our proposed scheme has the great potential of significantly improving the channel estimation efficiency in future IRS-aided wireless systems.