Deep Residual Learning for Channel Estimation in Intelligent Reflecting Surface-Assisted Multi-User Communications

Recently, intelligent reflecting surface (IRS), which has the capability of shaping the wireless channels between the users and the base station (BS) to enhance the system performance, has been proposed as a promising technology for the future smart radio environment [2, 3, 4, 5, 6]. In particular, thanks to the development of advanced materials, a reconfigurable and passive metasurface made of electromagnetic materials has been introduced to design the IRS to make it deployable and sustainable in practice [7, 8]. Generally, an IRS is composed of a large number of passive reflecting elements while each element is reconfigurable and can be independently controlled to adapt its phase shift to the actual environment to alter the reflection of the incident signals. As such, by jointly adjusting the phase shifts of all the passive elements, a desirable reflection pattern can be obtained which establishes a favourable wireless channel to improve the transmission quality with a low system power consumption [9, 10, 11]. Based on this, an IRS can be adopted in communication systems to enhance the energy efficiency or the spectral efficiency of communication networks through passive beamforming techniques, i.e., designing an efficient configuration of phase shifts to improve the received signal-to-noise ratio (SNR) at the desired receivers [12]. Therefore, IRS-assisted communication systems and the related studies such as the network capacity or received SNR maximization [13, 14], the energy efficiency or spectral efficiency maximization [15, 16], and the IRS-assisted physical layer security [17, 18], etc., have drawn vast attention from both the academia and the industry.

In practice, the promised performance gain brought by an IRS relies on accurate channel state information (CSI) for beamforming. Yet, the aforementioned studies were conducted under the assumption of perfect knowledge of CSI, which is usually not available. In fact, an indispensable task for realizing IRS-assisted communication systems is to perform accurate channel estimation. In contrast to the channel estimation in traditional systems, the IRS is passive and cannot perform training sequence transmission/reception or signal processing, i.e., the channel of IRS to user/BS is generally not available separately and only a cascaded channel of user-to-IRS-to-BS can be estimated. More importantly, the cascaded channel brings two main challenges to IRS-assisted communication systems which are listed as follows: (i) Limited channel estimation accuracy: Note that the cascaded user-to-IRS-to-BS channel does not follow the conventional Rayleigh fading model. In this case, the optimal minimum mean square error (MMSE) estimator involves a multidimensional integration which is overly computationally intensive for practical implementation. Meanwhile, the performance of the available linear MMSE (LMMSE) and least squares (LS) estimators still has a large gap compared with that of the optimal MMSE estimator. Thus, the channel estimation accuracy is unsatisfactory for practical IRS-assisted communication systems. (ii) Large channel estimation training overhead: The IRS generally consists of a large number of elements. Thus, the cascaded channel is with a high dimension and the corresponding channel estimation via conventional methods, e.g., the LS method and the LMMSE method, is computationally costly.

To overcome the challenge of limited channel estimation performance in IRS-assisted systems (i.e., challenge (i)), a variety of effective algorithms and efficient schemes have been proposed recently. For example, in [19], a binary reflection controlled least squares (LS) channel estimation scheme was developed for single-user systems by switching on only one reflecting element of the IRS and switching off the rest reflecting elements for each time slot. In this case, the BS only receives interference from the direct link and does not receive any interference from the other reflection elements [20]. Thus, the BS can estimate the cascaded channel successively, which paves the way for the channel estimation in IRS-assisted systems. However, since the binary reflection scheme only active one element each time, only a small received SNR can be obtained at the BS for estimation. Besides, an exceeding long delay may introduce to the system if there is a large number of IRS elements. To further improve the received SNR and shorten the required time for channel estimation, [21] proposed to switch on all the reflecting elements of the IRS for each time slot. In particular, the authors proposed a discrete Fourier transform (DFT) training sequence-based minimum variance unbiased estimator which can achieve satisfactory estimation accuracy. Moreover, the authors in [22] developed two parallel factor (PARAFAC)-based methods where the LS estimator was first adopted to obtain coarse channel coefficients and then more accurate estimation results can be obtained through exploiting the least squares Khatri-Rao factorization and the bilinear estimation techniques. On the other hand, some initial attempts have been devoted to the design of efficient schemes to reduce the required training overhead of channel estimation (i.e., challenge (ii)). For instance, [23] and [24] developed a subsurface-based channel estimation scheme where the IRS is divided into several independent subsurfaces and each subsurface is composed of multiple adjacent reflecting elements applying one common phase shift. Hence, by adopting the sharing strategy, the training overhead is dramatically reduced. Besides, [25] introduced a cascaded channel estimation framework and proposed the related algorithm based on the sparse matrix factorization and the matrix completion. Moreover, [26] studied the channel estimation in IRS-aided multiple-input multiple-output (MIMO) systems. Specifically, the authors first formulated the channel estimation as a problem of recovering a sparse channel matrix and then proposed a compressed sensing-based scheme to explore the sparsity of the cascaded channel. Also, in [27] and [28], the authors investigated the application of an IRS to a multi-user system and developed a three-phase channel estimation framework to shorten the required time duration for channel estimation by exploiting the redundancy in the reflecting channels of different users. Different from the previous works, [29] developed a novel IRS architecture equipped with a single active radio frequency (RF) chain for channel estimation at the IRS side, which further shortens the training overhead. However, despite the significant research efforts devoted in the aforementioned literature, the problem of limited channel estimation performance in (i) has not been well addressed, thus an effective and practical algorithm which can further improve the estimation accuracy is expected.

Recently, in contrast to the traditional model-driven approaches, data-driven deep learning (DL) techniques [30, 31, 32] have proved their effectiveness in IRS-assisted communication systems, such as the DL-based passive beamforming design [33, 34, 35, 36], the deep reinforcement learning (DRL)-based phase shift optimization [37], and the DRL-based secure wireless communications for IRS-MUC systems [38]. Although these DL-based methods are promising, they still require the availability of perfect CSI for implementation in IRS-assisted systems. Note that the channel estimation problem is essentially a denoising problem. In particular, the DL technique, especially the deep residual learning (DReL) which adopts a deep residual network (DRN) to improve the network performance and accelerate the network training speed, has been recognized by its powerful capability in denoising in various research areas [39, 40, 41]. Motivated by these facts, in this paper, we focus on IRS-assisted multi-user communication (IRS-MUC) systems and adopt a DReL approach to intelligently exploit the channel features to further improve the channel estimation accuracy. Note that different from the existing work in [33] where the IRS requires the installation of additional active channel sensors, our work focuses on a general IRS without deploying any active sensors to address the challenging channel estimation problem in IRS-assisted communication systems. Besides, in contrast to the MMSE channel estimation [42, 43, 44], our proposed method adopts the LS-based channel estimation value as the initial coarse estimated value, models the channel estimation as a denoising problem, and exploits the neural network to design a denoiser. Thus, our proposed method is model-free and does not require any prior statistical information. The main contributions of this work are listed as follows¹¹1 Note that this paper mainly focuses on improving the estimation accuracy of CSI, but not on reducing the training overhead. As such, the proposed method in this paper adopts the LS-based channel estimation value as the network input and exploits a neural network to further improve the estimation accuracy. In this case, the training overhead of our proposed method comes from the LS method which can be addressed by adopting the low overhead grouping scheme for IRS elements [23, 24].:

1. (1)

   In contrast to existing channel estimation methods, e.g., [19, 21, 22, 23, 24, 25, 26, 27, 28, 42, 43, 44], we model the channel estimation problem in IRS-MUC systems as a denoising problem and develop a DReL-based channel estimation framework which adopts a DRN to implicitly learn the residual noise for recovering the channel coefficients from the noisy pilot-based observations. Specifically, according to the MMSE criterion, a DRN-based MMSE (DRN-MMSE) estimator is derived in terms of Bayesian philosophy which enables the design of an efficient estimator.

2. (2)

   To realize the developed framework, we adopt a convolutional neural network (CNN) to facilitate the DReL and propose a CNN-based DRN (CDRN) for channel estimation, in which a CNN-based denoising block with an element-wise subtraction structure is specifically designed to exploit both the spatial features of the noisy channel matrices and the additive nature of the noise simultaneously. Inheriting from the superiorities of CNN and DReL in feature extraction and denoising, the proposed CDRN method could further improve the estimation accuracy.

3. (3)

   Although it is generally intractable to analyze the performance of a neural network, we formulate the proposed CDRN as a mathematical function and derive an explicit expression of the proposed CDRN estimator to explain the mechanism of the proposed algorithm theoretically, which proves that the proposed CDRN estimator can achieve the same performance as that of the LMMSE estimator.

4. (4)

   Extensive simulations have been conducted to verify the efficiency of the proposed method in terms of the impacts of SNR and channel size on normalized MSE (NMSE), respectively. Additionally, visualizations of the proposed CDRN are also provided to illustrate the denoising process. Our results show that the performance of the proposed method approaches that of the optimal MMSE estimator requiring the computation of a prior probability density function (PDF) of cascaded channel.

The reminder of our paper is organized as follows. Section \@slowromancapii@ introduces the system model of the IRS-MUC system and derives the optimal MMSE estimator, the LMMSE estimator, and the LS estimator as benchmarks. In Section \@slowromancapiii@, we model the channel estimation as a denoising problem and develop a versatile DReL-based channel estimation framework. As a realization of the developed framework, a CDRN architecture and a CDRN-based channel estimation algorithm are designed in Section \@slowromancapiv@. Extensive simulation results are provided in Section \@slowromancapv@ to verify the effectiveness of the proposed scheme, and finally Section \@slowromancapvi@ concludes the work of this paper.

The notations used in our paper are listed as follows. The superscripts $T$ and $H$ are used to represent the transpose and conjugate transpose, respectively. Terms $\mathbb{R}$ and $\mathbb{C}$ denote the sets of real numbers and complex numbers, respectively. Term $\mathbb{Z}$ denotes the set of integers. ${\mathcal{CN}}(\bm{\mu},\mathbf{\Sigma})$ is the circularly symmetric complex Gaussian (CSCG) distribution where $\bm{\mu}$ and $\mathbf{\Sigma}$ are the mean vector and the covariance matrix, respectively. The matrix ${\mathbf{I}}_{p}$ represents the $p$-by-$p$ identity matrix and the vector ${\mathbf{0}}$ represents a zero vector. $(\cdot)^{-1}$ denotes the operation of the matrix inverse. $\mathrm{Re}\{\cdot\}$ and $\mathrm{Im}\{\cdot\}$ are used to represent the operations of extracting the real part and the imaginary part of a complex-valued matrix, respectively. $\mathrm{tr}[\cdot]$ is the trace of a matrix and $\mathrm{diag}(\cdot)$ represents the construction of a diagonal matrix. $\|\cdot\|_{F}$ are used to denote the Frobenius norm of a matrix. In addition, $\exp(\cdot)$ indicates the exponential function, $E(\cdot)$ represents the statistical expectation operation, and $\mathbbm{1}_{\Omega}(\cdot)$ denotes the indicator function of an event $\Omega$.

In this paper, we consider an IRS-MUC system adopting the time division duplex (TDD) protocol, which consists of one base station (BS), one IRS, and $K$ users, as shown in Fig. 1. The BS is equipped with an $M$-element antenna array and $N$ passive reflecting elements are installed at the IRS to assist the BS to serve the $K$ single-antenna users. By joint beamforming design at both the BS and the IRS, the throughput of the IRS-MUC system could be further enhanced. However, the downlink beamforming design critically depends on the availability of accurate CSI. Due to the property of channel reciprocity in TDD systems, the downlink CSI could be acquired from the uplink channel estimation in the IRS-MUC system. The uplink system model is illustrated in Fig. 1, where $U_{k}$, $k\in\{1,2,\cdots,K\}$, denotes the $k$-th user. The channels of the $U_{k}$-BS link, the $U_{k}$-IRS link, and the IRS-BS link are represented by $\mathbf{d}_{k}\in\mathbb{C}^{M\times 1}$, $\mathbf{f}_{k}\in\mathbb{C}^{N\times 1}$, and $\mathbf{G}\in\mathbb{C}^{M\times N}$, respectively. The reflecting link $U_{k}$-IRS-BS can be regarded as a dyadic backscatter channel [2, 7], where each element at IRS combines all the arriving signals and re-scatters them to the BS behaving as a single point source. Denote $\mathbf{R}=\mathrm{diag}(\mathbf{r})\in\mathbb{C}^{N\times N}$ with $\mathbf{r}=[\beta e^{j\varphi_{1}},\beta e^{j\varphi_{2}},\cdots,\beta e^{j\varphi_{N}}]^{T}$ as the phase-shift matrix, where $0\leq\beta\leq 1$ and $0\leq\varphi_{n}\leq 2\pi$ are the amplitude and the phase shift at the $n$-th, $n\in\{1,2,\cdots,N\}$, element of IRS, respectively. The channel response from $U_{k}$ to the BS via the IRS can be expressed as $\mathbf{G}\mathbf{R}\mathbf{f}_{k}\in\mathbb{C}^{M\times 1}$. In this case, the channel response of the reflecting link $U_{k}$-IRS-BS can be altered through adjusting the phase-shift matrix $\mathbf{R}$ at the IRS. Since it is obvious that $\mathrm{diag}(\mathbf{r})\mathbf{f}_{k}=\mathrm{diag}(\mathbf{f}_{k})\mathbf{r}$, the channel of reflecting link $U_{k}$-IRS-BS can be expressed as $\mathbf{G}\mathrm{diag}(\mathbf{r})\mathbf{f}_{k}=\mathbf{G}\mathrm{diag}(\mathbf{f}_{k})\mathbf{r}$. Therefore, the objective of the uplink channel estimation is to estimate

where

is a cascaded channel. Therefore, the downlink channels can be obtained by the conjugate transpose of the uplink channels via exploiting the channel reciprocity in TDD systems.

Based on these, we then develop a channel estimation protocol for the considered IRS-MUC system. As shown in Fig. 2, each frame in the data stream has an identical frame structure which consists of a channel estimation phase and a data transmission phase. In this paper, we mainly focus on the channel estimation phase to estimate $\mathbf{H}_{k}$ as defined in (1). In the channel estimation phase, the IRS generates $C$, $C\geq N+1$, different reflection patterns through $C$ different phase-shift matrices, denoted by

Here, $\mathbf{R}_{c}=\mathrm{diag}(\mathbf{r}_{c})$ is the $c$-th, $c\in\{1,2,\cdots,C\}$, phase-shift matrix with $\mathbf{r}_{c}=[\beta_{c}e^{j\varphi_{c,1}},\beta_{c}e^{j\varphi_{c,2}},$ $\cdots,\beta_{c}e^{j\varphi_{c,N}}]$, where $\beta_{c}\in[0,1]$ and $0\leq\varphi_{c,n}\leq 2\pi$ are the amplitude and the phase shift at the $n$-th element of the IRS under the $c$-th phase-shift matrix, respectively. In addition, $K$ orthogonal pilot sequences are adopted to distinguish different users, i.e., for $U_{k}$, $\forall k$, a pilot sequence with a length of $L$, $L\geq K$, is adopted for each IRS phase-shift matrix, denoted by $\mathbf{u}_{k}=[u_{k,1},u_{k,2},\cdots,u_{k,L}]^{T}$ with $\mathbf{u}_{k}^{H}\mathbf{u}_{k}=\mathcal{P}L$ and $\mathbf{u}_{a}^{H}\mathbf{u}_{b}=0$, where $\mathcal{P}$ is the power of each user, $a,b\in\{1,2,\cdots,K\}$, and $a\neq b$. According to Fig. 2, the channel estimation phase consists of $C$ sub-frames and each sub-frame consists of $L$ pilot symbols. For the IRS, the phase-shift matrix keeps unchanged within one sub-frame and it switches to different phase-shift matrices for different sub-frames. As for the $k$-th user, $U_{k}$, $\forall k$, it sends its identity pilot sequence $\mathbf{u}_{k}$ in each sub-frame. Therefore, the $l$-th, $l\in\{1,2,\cdots,L\}$, received pilot signal vector at the BS in the $c$-th sub-frame can be expressed as

Here, $\mathbf{H}_{k}$ is the channel matrix needed to be estimated as defined in (1) and $\mathbf{p}_{c}=[1,\mathbf{r}_{c}]^{T}\in\mathbb{C}^{(N+1)\times 1}$. Equation (5) is due to the property of $\mathrm{diag}(\mathbf{r}_{c})\mathbf{f}_{k}=\mathrm{diag}(\mathbf{f}_{k})\mathbf{r}_{c}$. In addition, $\mathbf{v}_{c,l}\in\mathbb{C}^{M\times 1}$ is the $l$-th sampling noise vector at the BS in the $c$-th sub-frame. Generally, $\mathbf{v}_{c,l}$ is assumed to be a CSCG random vector, i.e., $\mathbf{v}_{c,l}\sim\mathcal{CN}(\mathbf{0},\sigma_{v}^{2}\mathbf{I}_{M})$, where $\sigma_{v}^{2}$ denotes the noise variance of each antenna at the BS. Stacking the $L$ received pilot signal vectors at the BS during the $c$-th sub-frame into a matrix form:

where $\mathbf{S}_{c}=[\mathbf{s}_{c,1},\mathbf{s}_{c,2},\cdots,\mathbf{s}_{c,L}]$ and $\mathbf{V}_{c}=[\mathbf{v}_{c,1},\mathbf{v}_{c,2},\cdots,\mathbf{v}_{c,L}]$. Since the pilot sequences of each two users are orthogonal, the received signal from $U_{k}$ can be separated by multiplying a sequence $\mathbf{u}_{k}$ with $\mathbf{S}_{c}$ in (6), i.e.,

Here, ${\mathbf{x}}_{c,k}=\frac{1}{\mathcal{P}L}\mathbf{S}_{c}\mathbf{u}_{k}\in\mathbb{C}^{M\times 1}$ is the received signal vector at the BS from $U_{k}$ within the $c$-th sub-frame. ${\mathbf{z}}_{c,k}=\frac{1}{\mathcal{P}L}\mathbf{V}_{c}\mathbf{u}_{k}\in\mathbb{C}^{M\times 1}$ with ${\mathbf{z}}_{c,k}\sim\mathcal{CN}(\mathbf{0},\sigma_{z}^{2}\mathbf{I}_{M})$, where $\sigma_{z}^{2}$ is the variance of each element of ${\mathbf{z}}_{c,k}$. Based on this and after $C$ subframes, we can then obtain the matrix form of (7): $\mathbf{X}_{k}=[\mathbf{x}_{1,k},\mathbf{x}_{2,k},\cdots,\mathbf{x}_{C,k}]$ which can be expressed as

where $\mathbf{P}=[\mathbf{p}_{1},\mathbf{p}_{2},\cdots,\mathbf{p}_{C}]\in\mathbb{C}^{(N+1)\times C}$ with $\mathbf{p}_{c}=[1,\mathbf{r}_{c}]^{T}$ as defined in (5) and $\mathbf{Z}_{k}=[\mathbf{z}_{1,k},\mathbf{z}_{2,k},\cdots,\mathbf{z}_{C,k}]\in\mathbb{C}^{M\times C}$. According to [21], an optimal scheme of $\mathbf{P}$ in terms of improving received signal power at the BS is to design $\mathbf{P}$ as a discrete Fourier transform (DFT), i.e.,

where $W_{C}=e^{j2\pi/C}$ and $\mathbf{P}\mathbf{P}^{H}=C\mathbf{I}_{N+1}$. The $C$ different phase-shift matrices $\mathbf{\Lambda}=[\mathbf{R}_{1},\mathbf{R}_{2},$ $\cdots,\mathbf{R}_{C}]^{T}$ defined in (3) are set based on $\mathbf{P}$ in (9).

Therefore, the task of channel estimation in IRS-MUC is to recover $\mathbf{H}_{k}$, $\forall k$, by exploiting $\mathbf{X}_{k}$ and $\mathbf{P}$. Based on the models in (8) and (9), we then first introduce three commonly adopted methods as three benchmarks, namely the LS method, the MMSE method, and the LMMSE method.

(a) LS Channel Estimator:

According to [45], the estimated $\mathbf{H}_{k}$ using LS estimator is given by

where $\mathbf{P}^{\dagger}=\mathbf{P}^{H}(\mathbf{P}\mathbf{P}^{H})^{-1}$ denotes the pseudoinverse of $\mathbf{P}$.

(b) MMSE Channel Estimator:

Different from the LS method where $\mathbf{H}_{k}$ is assumed to be an unknown but deterministic constant, the Bayesian approach assumes that $\mathbf{H}_{k}$ is a random variable with a prior PDF $p(\mathbf{H}_{k})$. Therefore, the Bayesian approach can take advantage of prior knowledge to further improve the estimation accuracy. According to [46], the optimal Bayesian estimator is the MMSE estimator which can be expressed as

where $p({\mathbf{H}_{k}}|{\mathbf{X}_{k}})=\frac{{p({\mathbf{X}_{k}}|{\mathbf{H}_{k}})p({\mathbf{H}_{k}})}}{{\int{p({\mathbf{X}_{k}}|{\mathbf{H}_{k}})p({\mathbf{H}_{k}})d{\mathbf{H}_{k}}}}}$ is the posterior PDF and $p({\mathbf{X}_{k}}|{\mathbf{H}_{k}})$ is the conditional PDF.

(c) LMMSE Channel Estimator:

Note that the optimal MMSE estimator in (11) is computationally intensive to be implemented in practice when the channel distribution is complicated. Thus, the LMMSE estimator can be selected as a more practical approach with a simple explicit expression²²2Note that the proposed joint optimization approach in [47] is an effective scheme for realizing the LMMSE estimation in IRS-assisted systems. However, the system model in [47] is a downlink channel estimation model requiring the design of a new training sequence while our work focuses on the uplink channel estimation model for a given training sequence. Thus, the developed approach in [47] cannot be directly adopted in our work and the extension of our work to the downlink scenario is left for our future work due to the page limitation. [46]:

where $\mathbf{R}_{\mathbf{H}_{k}}=E(\mathbf{H}_{k}^{H}\mathbf{H}_{k})$ denotes the statistical channel correlation matrix [45].

Note that the LS estimator requires no prior knowledge of channel and is widely used in practice. If the channel follows the Rayleigh fading model and the statistical channel correlation matrix is available, the optimal MMSE estimator is equivalent to the LMMSE estimator [46] and it can be adopted to capture the statistical characteristics of the channel to further improve the estimation performance. However, for IRS-MUC systems, $\mathbf{H}_{k}$ includes a cascaded channel which generally does not follow the Rayleigh fading model. In this case, the optimal MMSE estimator involves the calculation of a multidimensional integration and thus it is intractable to be implemented in practice due to its intensive computational cost. Meanwhile, the LMMSE estimator still has a performance gap compared with the optimal MMSE estimator [46].

To further improve the estimation performance, in the next section, we still retain the MSE criterion but adopt a data driven DL approach to design a practical channel estimation method, i.e., a DReL-based channel estimation scheme.

Note that the additive noise in the system model is a key obstruction for recovering the channel coefficients. In this section, we first model the channel estimation as a denoising problem and then develop a DReL-based framework to learn the residual noise from the noisy observations for paving the way in recovering the channel coefficients. In the developed DReL framework, a DRN with a subtraction architecture is designed to exploit both the features of received observations and the additive nature of the noise to further improve the estimation performance. In the following, we will introduce the derived denoising model and the developed DReL-based channel estimation framework, respectively.

In this section, we introduce a practical approach for implementing the developed DReL framework via a CNN. Note that CNN has powerful capability in extracting features from the noisy observation matrices, meanwhile the subtraction structure of DReL contributes to exploiting the additive nature of the noise [39, 40, 41]. Therefore, we adopt a CNN to facilitate the application of DReL, resulting a CDRN for channel estimation. In the following, we will introduce the CDRN architecture and the CDRN-based channel estimation algorithm, respectively.

In this section, extensive simulations are provided to verify the efficientness of the proposed algorithm. In the simulations, an IRS-MUC system consists of a BS equipped with $M$ antennas, an IRS with $N$ reflecting elements, and $K$ single-antenna users is considered, as defined in Fig. 1. Unless further specified, the IRS operates with continuous phase shifts and the default settings of the considered IRS-MUC system are $M=8$, $N=32$, $K=6$, and $C=N+1=33$. A path loss model is adopted in the simulations and the path losses of each channel can be modeled as $\alpha_{k}^{\mathrm{UB}}=\alpha_{0}(\lambda_{k}^{\mathrm{UB}}/\lambda_{0})^{-\gamma_{1}}$, $\alpha_{k}^{\mathrm{UI}}=\alpha_{0}(\lambda_{k}^{\mathrm{UI}}/\lambda_{0})^{-\gamma_{2}}$, and $\alpha^{\mathrm{IB}}=\alpha_{0}(\lambda^{\mathrm{IB}}/\lambda_{0})^{-\gamma_{3}}$, where $\gamma_{i}$, $i\in\{1,2,3\}$, is the path loss exponent, $\lambda_{0}=10~{}\mathrm{m}$ is the reference distance, $\alpha_{0}=-15~{}\mathrm{dB}$ is the path loss at the reference distance, and $\lambda_{k}^{\mathrm{UB}}$, $\lambda_{k}^{\mathrm{UI}}$, and $\lambda^{\mathrm{IB}}$ represent the distances of $U_{k}$-BS, $U_{k}$-IRS, and IRS-BS, respectively. Specifically, we set $\lambda_{k}^{\mathrm{UB}}=100$ m, $\lambda_{k}^{\mathrm{UI}}=16$ m, and $\lambda^{\mathrm{IB}}=90$ m. In addition, a Rician fading model is considered for all the channels to characterize the small-scale fading. In this case, the channel of IRS-BS link can be expressed as

where $\beta_{\mathrm{IB}}$ represents the Rician factor of IRS-BS channel, and $\mathbf{G}_{\mathrm{LOS}}$ and $\mathbf{G}_{\mathrm{NLOS}}$ are the line-of-sight (LOS) component and the Rayleigh component, respectively. Note that the above model expression is a general model which involves both the LOS channel model (when $\beta_{\mathrm{IB}}\to\infty$) and the Rayleigh channel model (when $\beta_{\mathrm{IB}}=0$). The channels of BS-$U_{k}$ and IRS-$U_{k}$ are generated similarly as defined in (38). Similarly, we define the Rician factors of channels of $U_{k}$-BS and $U_{k}$-IRS as $\beta_{\mathrm{UB}}$ and $\beta_{\mathrm{UI}}$, respectively. To consider the impacts of path loss factors, the elements in $\mathbf{G}$ should then be multiplied by $\sqrt{\alpha^{\mathrm{IB}}}$. The detailed settings of the simulations are summarized in Table \@slowromancapii@, where the settings of $U_{k}$-BS link, i.e., $\beta_{\mathrm{UB}}=0$ and $\gamma_{1}=3.6$, are due to the rich scattering environment and the relatively large distance between $U_{k}$ and the BS. Specifically, a normalized MSE (NMSE) is adopted as the estimation performance metric: $\mathrm{NMSE}=\frac{E(\|\tilde{\mathbf{H}}-\mathbf{H}\|_{F}^{2})}{E(\|\mathbf{H}\|_{F}^{2})}$ where $\tilde{\mathbf{H}}$ and $\mathbf{H}$ denote the estimated channel and the ground truth of the channel, respectively. In addition, the signal-to-noise ratio (SNR) used in the simulation results refers to the transmit SNR, i.e., $\mathrm{SNR}=\mathcal{P}/\sigma_{z}^{2}$. To evaluate the estimation performance, we also provide the simulation of the following algorithms for comparisons: the optimal MMSE method [46], the LMMSE method [46], the LS method [45], the bilinear alternating least squares (BALS) method [22], and the binary reflection controlled LMMSE (B-LMMSE) method. For the B-LMMSE method, we exploit the binary reflection controlled strategy proposed in [19] and then adopt the LMMSE estimator for channel estimation. In addition, a CDRN equipped with $D=3$ denoising blocks is adopted for our proposed method and the hyperparameters of CDRN are set according to Table \@slowromancapi@. Each simulation point of the presented results is obtained by averaging the performance over $100,000$ Monte Carlo realizations⁴⁴4The source code is available at https://github.com/XML124/CDRN-channel-estimation-IRS..

Next, we will verify the channel estimation performance of the proposed CDRN method in terms of NMSE and unveil its relationship with SNR and channel dimensionality, respectively. Besides, visualizations of our proposed CDRN are provided to illustrate the denoising process. Finally, the complexity analysis is also investigated to evaluate the computational complexity.

This paper proposed a data-driven DReL approach to address the channel estimation problem for IRS-MUC systems operating with TDD protocol. Firstly, we formulated the channel estimation problem in IRS-MUC systems as a denoising problem and developed a versatile DReL-based channel estimation framework. Specifically, according to the Bayesian MMSE criterion, a DRN-based MMSE was derived and analyzed in terms of Bayesian philosophy. Under the developed DReL-based framework, we then designed a CDRN to denoise the noisy channel matrix for channel recovery. For the proposed CDRN, a CNN-based denoising block with an element-wise subtraction structure was specifically designed to exploit both the spatial features of the noisy channel matrices and the additive nature of the noise simultaneously. Taking advantages of the superiorities of CNN and DReL in feature extraction and denoising, the proposed CDRN estimation algorithm can further improve the estimation accuracy. Simulation results demonstrated that the proposed method can achieve almost the same estimation accuracy as that of the optimal MMSE estimator based on a prior PDF of channel.