Federated Learning for Channel Estimation in Conventional and RIS-Assisted Massive MIMO

Before reception at the users, the transmitted signal is passed through the mm-Wave channel, which can be represented by a geometric model with limited scattering [11]. Let us define $\mathbf{H}_{k}[m]$ as the $N_{\mathrm{MS}}\times N_{\mathrm{BS}}$ mm-Wave channel matrix between the BS and the $k$th user. Then, $\mathbf{H}_{k}[m]$ includes the contributions of $L$ paths, each of which has the time delay $\tau_{k,l}$ with relative AoA $\bar{\phi}_{k,l}\in\Theta$ ($\Theta=[-\frac{\pi}{2},\frac{\pi}{2}]$), angle-of-departure (AoD) $\phi_{k,l}\in\Theta$, and the complex path gain $\alpha_{k,l}$ for the $k$th user and $l$th path. Let $p(\tau)$ denote a pulse shaping function for $T_{\mathrm{s}}$-spaced signaling evaluated at $\tau$ seconds. Then, the mm-Wave delay-$d$ MIMO channel matrix in time domain is given by

where $\mathbf{a}_{\mathrm{MS}}(\bar{\phi_{k,l}})$ and $\mathbf{a}_{\mathrm{BS}}(\phi_{k,l})$ are the $N_{\mathrm{MS}}\times 1$, and $N_{\mathrm{BS}}\times 1$ steering vectors representing the array responses of the antenna arrays at the users and the BS, respectively. Let $\lambda_{m}=\frac{c_{0}}{f_{m}}$ be the wavelength for the subcarrier $m$ at frequency $f_{m}$. Since the operating frequency is relatively higher than the bandwidth in mm-Wave systems and the subcarrier frequencies are close to each other (i.e., $f_{m_{1}}\approx f_{m_{2}}$, $m_{1},m_{2}\in\mathcal{M}$), we use a single operating wavelength $\lambda=\lambda_{1}=\dots=\lambda_{M}=\frac{c_{0}}{f_{c}}$, where $c_{0}$ is speed of light and $f_{c}$ is the central carrier frequency [11, 12]. This approximation also allows for a single frequency-independent analog beamformer for each subcarrier. Then, for a uniform linear array (ULA), the array response of the antenna array at the BS is

where ${d}_{\mathrm{BS}}=\lambda/2$ is the antenna spacing. The $n$th element of $\mathbf{a}_{\mathrm{MS}}(\bar{\phi})$ can be defined in a similar way as for $\mathbf{a}_{\mathrm{BS}}(\phi)$ as $\left[\mathbf{a}_{\mathrm{MS}}(\bar{\phi})\right]_{n}=e^{j\pi(n-1)\sin(\bar{\phi})}$, $n=1,\dots,N_{\mathrm{MS}}$. After performing $M$-point DFT of the delay-$d$ channel model in (1), the channel matrix of the $k$th user at subcarrier $m$ becomes

where $D\leq M$ is the CP length. The frequency domain channel in (3) is used in MIMO-OFDM systems, where the orthogonality of each subcarrier is held such that $||\mathbf{H}_{k}^{\textsf{H}}[m_{1}]\mathbf{H}_{k}[m_{2}]||_{\mathcal{F}}^{2}=0$ for $m_{1},m_{2}\in\mathcal{M}$ and $m_{1}\neq m_{2}$.

With the aforementioned block-fading channel model [11], the received signal at the $k$th user before analog processing at subcarrier $m$ is $\tilde{\mathbf{y}}_{k}[m]=\sqrt{\rho}\mathbf{H}_{k}[m]\mathbf{x}[m]$, i.e.,

where $\rho$ represents the average received power and $\mathbf{n}[m]\sim\mathcal{CN}({0},\sigma^{2}\mathbf{I}_{\mathrm{N_{\mathrm{MS}}}})$ is additive white Gaussian noise (AWGN) vector. At the $k$th user, the received signal is first processed by the analog combiner $\mathbf{w}_{\mathrm{RF},k}\in\mathbb{C}^{N_{\mathrm{MS}}}$. Then, the cyclic prefix is removed from the processed signal and $M$-point DFTs are applied to yield the signal in frequency domain. Then, the received baseband signal becomes

where the analog combiner $\mathbf{w}_{\mathrm{RF},k}$ has the constraint $\big{[}\mathbf{w}_{\mathrm{RF},k}\mathbf{w}_{\mathrm{RF},k}^{\textsf{H}}\big{]}_{i,i}=1$, similar to the RF precoder. Once the received symbols, i.e., $y_{k}[m]$ are obtained at the $k$th user, they are demodulated according to its respective modulation scheme, and the information bits are recovered for each subcarrier. To accurately recover the data streams $\mathbf{s}[m]$ in (5), the channel matrix $\mathbf{H}_{k}[m]$ should be estimated. This is usually done by using pilot signals in the preamble stage [36, 16], wherein the beamformers $\mathbf{F}_{\mathrm{RF}}$, $\mathbf{F}_{\mathrm{BB}}$ and $\mathbf{w}_{\mathrm{RF_{k}}}$ are designed accordingly (See Section III-C).

The aim in this work is to estimate the channel matrix $\mathbf{H}_{k}[m]$ via FL, as illustrated in Fig. 1b. To this end, the global NN for channel estimation (henceforth called ChannelNet) located at the BS is trained on the local datasets of the users. Let $\mathcal{D}_{k}$ denote the local dataset at the $k$th user, containing the input-output pairs $\mathcal{D}_{k}^{(i)}=(\mathcal{X}_{k}^{(i)},\mathcal{Y}_{k}^{(i)})$²²2The sizes of $\mathcal{X}_{k}^{(i)}$ and $\mathcal{Y}_{k}^{(i)}$ depend on the size of the channel matrix, and they are explicitly given in Sec. III-C and Sec. III-D for conventional and RIS-assisted massive MIMO scenario, respectively., $i=1,\dots,\textsf{D}_{k}$ and $\textsf{D}_{k}=|\mathcal{D}_{k}|$ is the size of the local dataset $\mathcal{D}_{k}$. Here, $\mathcal{X}_{k}^{(i)}$ represents the $i$th input data, i.e., the received pilot signals, $\mathcal{Y}_{k}^{(i)}$ denotes the $i$th output/label data, i.e., the channel matrix, for $k\in\mathcal{K}$, $\mathcal{K}=\{1,\dots,K\}$. Thus, for an input-output pair $(\mathcal{X},\mathcal{Y})$, ChannelNet constructs a non-linear relationship between the input and the output data as $f(\mathcal{X}|\boldsymbol{\theta})=\mathcal{Y}$, where $\boldsymbol{\theta}\in\mathbb{R}^{P}$ denotes the learnable parameters.

In Fig. 2, we present the communication interval at the user for two consecutive data transmission blocks. At the beginning of each transmission block, the received pilot signals are acquired and processed for channel estimation. This can be done by employing one of the analytical channel estimation techniques, which can be based on compressed sensing [37, 38], angle-domain processing [14] and coordinated pilot-assignment [15]. The analytical approach is only used in the training data collection stage, which is relatively smaller than the prediction stage [32]. Hence, the use of ML/FL in the prediction stage becomes more advantageous over the analytical techniques in the long term.

It is also worth to mention that the training data can be obtained via offline datasets which are prepared by collecting the data from the field measurements. In [39], authors present a channel estimation dataset, which is obtained by electromagnetic simulations tools. While this approach can also be followed, the offline collected data may not always reflect the channel characteristics and the imperfections in the mm-Wave channel. In this work, we evaluate the performance of the proposed approach on the datasets whose labels are selected as both true and estimated channel data. For the estimated channel, we assume that the training data are collected, as described in Fig. 2, by employing angle-domain channel estimation (ADCE) technique [14], which has close to minimum mean-square-error (MMSE) performance.

After channel estimation, the training data can be collected by storing the received pilot data $\mathbf{G}_{k}[m]$ and the estimated channel data $\hat{\mathbf{H}}_{k}[m]$ in the internal memory of the user. (We discuss how $\mathbf{G}_{k}[m]$ is determined in Sec. III-C.) Then, the user feedbacks the estimated channel data to the BS via uplink transmission. As a result, the local dataset $\mathcal{D}_{k}$ can be collected at the $k$th user after $i=1,\dots,\textsf{D}_{k}$ transmission blocks. This approach allows us to collect training data for different channel coherence times, which can be very short due to dynamic nature of the mm-Wave channel, such as indoor and vehicular communications [10].

The above process is the first stage of the proposed FL-based channel estimation framework. Once the training data is collected, the global model is trained (see, e.g., Fig. 1b). After training, each user can estimate its own channel via the trained NN by simply feeding the NN with $\mathbf{G}_{k}[m]$ and obtains $\hat{\mathbf{H}}_{k}[m]$, as illustrated in Fig. 2b.

We begin by introducing the training concept in conventional CL, then develop FL-based model training.

In CL-based model training for channel estimation [16, 17, 32, 18, 34], the training of the global NN is performed by collecting the local datasets $\{\mathcal{D}_{k}\}_{k\in\mathcal{K}}$ from the users, as illustrated in Fig. 1a. Once the BS has collected the whole dataset $\mathcal{D}$, the training is performed by solving the following problem

where $\textsf{D}=|\mathcal{D}|$ is the number of training samples and $\mathcal{L}(\boldsymbol{\theta})$ denotes the loss function defined as

which is the MSE between the label data $\mathcal{Y}^{(i)}$ and the prediction of the NN, $f(\mathcal{X}^{(i)}|\boldsymbol{\theta})$.

On the other hand, in FL, the local datasets $\mathcal{D}_{k\in\mathcal{K}}$ are preserved at the users and not transmitted to the BS. Hence, FL-based model training is performed at the user side as

where $\mathcal{L}_{k}(\boldsymbol{\theta})=\frac{1}{\textsf{D}_{k}}\sum_{i=1}^{\textsf{D}_{k}}\|f(\mathcal{X}_{k}^{(i)}|\boldsymbol{\theta})-\mathcal{Y}_{k}^{(i)}\|_{\mathcal{F}}^{2}$. Notice that the FL-based model training in (III-B) is solved at the user while the CL problem in (III-B) is handled at the BS. To efficiently solve (III-B) and (III-B), gradient descent (GD) is employed and the problems are solved iteratively. In CL, the gradient is computed over the whole dataset as $\mathbf{g}(\boldsymbol{\theta}_{t})=\nabla\mathcal{L}(\boldsymbol{\theta}_{t})$ and the parameter update is performed as

where $\eta$ is the learning rate.

In FL, each user computes the gradients individually as $\mathbf{g}_{k}(\boldsymbol{\theta}_{t})=\nabla\mathcal{L}_{k}(\boldsymbol{\theta}_{t})$ to solve (III-B), then sends them to the BS, where the model parameters are updated as

The transmission of gradients to the BS provides more energy-efficiency than directly transmitting the model parameters as in the FedAvg algorithm [35]. The main reason is that gradients include only the model updates obtained from the GD algorithm, whereas model transmission includes already known data from the previous iteration. Hence, model transmission wastes a significant amount of transmit power from all the users [31, 30, 40].

The gradients $\mathbf{g}_{k\in\mathcal{K}}(\boldsymbol{\theta}_{t})$ are sent to the BS via wireless channel, which causes corruptions during transmission. Therefore, the corrupted model parameters and gradients at the $t$th iteration are given as [24, 41]

where $\tilde{\boldsymbol{\theta}}_{t}$ represents the noisy model parameters captured at the users, $\mathbf{g}_{k}(\tilde{\boldsymbol{\theta}}_{t})$ is the gradient vector computed at the user based on $\tilde{\boldsymbol{\theta}}_{t}$ and $\tilde{\mathbf{g}}_{k}(\tilde{\boldsymbol{\theta}}_{t})$ denotes the noisy gradient vector received at the BS. $\Delta\boldsymbol{\theta}_{t}$, $\Delta\mathbf{g}_{k}({\boldsymbol{\theta}}_{t})$ and $\Delta\mathbf{g}_{k}(\tilde{\boldsymbol{\theta}}_{t})$ represent the noise terms added onto $\boldsymbol{\theta}_{t}$, $\mathbf{g}_{k}({\boldsymbol{\theta}}_{t})$ and $\mathbf{g}_{k}(\tilde{\boldsymbol{\theta}}_{t})$, respectively. Then, the model update rule can be given by

which can be rewritten as

where $\Delta$ corresponds to the overall noise term added onto $\boldsymbol{\theta}_{t+1}=\boldsymbol{\theta}_{t}-\eta\frac{1}{K}\sum_{k=1}^{K}\mathbf{g}_{k}(\boldsymbol{\theta}_{t})$. Now, let us consider the statistics of $\Delta$. Without loss of generality, the noise terms due to wireless transmission in (11) and (13), i.e., $\Delta{\boldsymbol{\theta}}_{t}$ and $\Delta\mathbf{g}_{k}(\tilde{\boldsymbol{\theta}}_{t})$, can be modeled as AWGN with variances $\sigma_{\boldsymbol{\theta}}^{2}$ and $\tilde{\sigma}_{k}^{2}$, respectively [42, 41]. Furthermore, we define $\Delta\mathbf{g}_{k}({\boldsymbol{\theta}}_{t})$ in (12) as AWGN with variance ${\sigma}_{k}^{2}$ due to the linearity of gradient and the NN layers³³3Many of the NN layers, such as convolutional, fully connected, normalization and dropout layers, perform linear operations, whereas pooling and ReLU layers are non-linear [43, 41, 42].. Hence, the overall noise term $\Delta$ can be viewed as an AWGN with variance $\sigma_{\Delta}^{2}=\sigma_{\boldsymbol{\theta}}^{2}+\eta\frac{\sum_{k=1}^{K}(\tilde{\sigma}_{k}^{2}+{\sigma}_{k}^{2})}{K}$.

In order to solve (III-B) effectively in the presence of noisy model parameters, we define a regularized loss function $\tilde{\mathcal{L}}_{k}(\boldsymbol{\theta})$ as

which is widely used in stochastic optimization [44]. (16) can be obtained via first order Taylor expansion of the expectation-based loss $\mathbb{E}\{||\mathcal{L}_{k}(\boldsymbol{\theta}+\Delta)||^{2}\}$, which can be approximately written as

where the first term corresponds to the minimization of the loss function with perfect estimation and the second term is the additional cost due to noise [44, 41]. Using (16), the regularized version of FL-based training problem in (III-B) is given by

which can be effectively solved via GD in the presence of noisy model updates as

where $\nabla\bar{\mathcal{L}}(\tilde{\boldsymbol{\theta}})=\frac{1}{K}\sum_{k=1}^{K}\bar{\mathbf{g}}_{k}(\tilde{\boldsymbol{\theta}}_{t})$ and $\bar{\mathbf{g}}_{k}(\tilde{\boldsymbol{\theta}}_{t})=\nabla\tilde{\mathcal{L}}_{k}(\tilde{\boldsymbol{\theta}}_{t})=\nabla\big{[}\mathcal{L}_{k}(\tilde{\boldsymbol{\theta}}_{t})+\sigma_{\Delta}^{2}||\mathbf{g}_{k}(\tilde{\boldsymbol{\theta}}_{t})||^{2}\big{]}$.

Due to the effect of noisy gradient transmission, $\bar{\mathcal{L}}(\boldsymbol{\theta})$ converges slower than ${\mathcal{L}}(\boldsymbol{\theta})$. In the following theorem, we prove the convergence of $\bar{\mathcal{L}}(\boldsymbol{\theta})$. While the convergence of the regularized loss function was studied in different FL works [42, 41], they consider model transmission, whereas in this work we investigate the gradient transmission approach. The convergence analysis is also different from the previous gradient transmission-based works, e.g., [30, 24], which are based on the sparsity assumption of the gradient vector, which may not be always satisfied.

Theorem 1: Let $\boldsymbol{\theta}_{0}$ and $\boldsymbol{\theta}_{\star}$ be the initial and optimal model parameters, respectively. Then, the FL-based model training converges with the convergence rate $\mathcal{O}(1/t)$ as

with the learning rate $\eta\leq\frac{1}{(1+{\sigma}_{\Delta}^{2})\beta}$ for some $\beta\geq 0$.

Proof: See Appendix A.∎

In practice, the convergence of the learning model is subject to the wireless factors, such as the SNR of the transmitted/received model updates. In particular, the convergence becomes slower due to the packet errors during training [45]. Furthermore, the channel statistics change in each communication round, which entails CSI acquisition for each round. While some of the recent works assume that a single communication round between the server and the clients takes a single channel coherence time [31, 30, 24], in [46] FL-based training is completed in a single long-coherence time, which is approximately composed of 40 small-scale fading channel coherence intervals [46].

Here, we discuss how the input and output of ChannelNet are determined for massive MIMO scenario.

The input of ChannelNet is the set of received pilot signals at the preamble stage. Consider the downlink received signal model in (5) and assume that the BS activates only a single RF chain, one at a time. Let $\overline{\mathbf{f}}_{u}[m]\in\mathbb{C}^{N_{\mathrm{BS}}}$ be the resulting beamformer vector and pilot signals are $\overline{{s}}_{u}[m]$, where $u=1,\dots,M_{\mathrm{BS}}$ and $m\in\mathcal{M}$. At the receiver side, each user activates the RF chain for $M_{\mathrm{MS}}$ times and applies the beamformer vector $\overline{\mathbf{w}}_{v}[m]$, $v=1,\dots,M_{\mathrm{MS}}$ to process the received pilots [16]. Hence, the total channel use in the channel acquisition process is $M_{\mathrm{BS}}\lceil\frac{M_{\mathrm{MS}}}{N_{\mathrm{RF}}}\rceil$. Therefore, the received pilot signal at the $k$th user becomes

where $\overline{\mathbf{F}}[m]=[\overline{\mathbf{f}}_{1}[m],\overline{\mathbf{f}}_{2}[m],\dots,\overline{\mathbf{f}}_{M_{\mathrm{BS}}}[m]]$ and $\overline{\mathbf{W}}[m]=[\overline{\mathbf{w}}_{1}[m],\overline{\mathbf{w}}_{2}[m],\dots,\overline{\mathbf{w}}_{M_{\mathrm{MS}}}[m]]$ are $N_{\mathrm{BS}}\times M_{\mathrm{BS}}$ and $N_{\mathrm{MS}}\times M_{\mathrm{MS}}$ beamformer matrices, respectively. $\overline{\mathbf{S}}[m]=\mathrm{diag}\{\overline{s}_{1}[m],\dots,\overline{s}_{M_{\mathrm{BS}}}[m]\}$ denotes pilot signals and $\widetilde{\mathbf{N}}_{k}[m]=\overline{\mathbf{W}}^{\textsf{H}}\overline{\mathbf{N}}_{k}[m]$ is the effective noise matrix, where $\overline{\mathbf{N}}_{k}[m]\sim\mathcal{N}(0,\sigma^{2}\mathbf{I}_{M_{\mathrm{MS}}})$. Without loss of generality, we assume that $\overline{\mathbf{F}}[m]=\overline{\mathbf{F}}$ and $\overline{\mathbf{W}}[m]=\overline{\mathbf{W}}$ and $\overline{\mathbf{S}}[m]=\mathbf{I}_{M_{\mathrm{BS}}}$ $\forall m\in\mathcal{M}$. Then, the received signal (21) becomes

Using $\mathbf{\overline{Y}}_{k}[m]$, we define the input of ChannelNet $\mathbf{G}_{k}[m]$ as

where $\mathbf{T}_{\mathrm{MS}}=\begin{dcases}\overline{\mathbf{W}},&M_{\mathrm{MS}}<N_{\mathrm{MS}}\\ (\overline{\mathbf{W}}\overline{\mathbf{W}}^{\textsf{H}})^{-1}\overline{\mathbf{W}},&M_{\mathrm{MS}}\geq N_{\mathrm{MS}},\end{dcases}$ and $\mathbf{T}_{\mathrm{BS}}=\begin{dcases}\overline{\mathbf{F}}^{\textsf{H}},&M_{\mathrm{BS}}<N_{\mathrm{BS}}\\ \overline{\mathbf{F}}^{\textsf{H}}(\overline{\mathbf{F}}\overline{\mathbf{F}}^{\textsf{H}})^{-1},&M_{\mathrm{BS}}\geq N_{\mathrm{BS}}.\end{dcases}$ Here, $\mathbf{T}_{\mathrm{BS}}$ and $\mathbf{T}_{\mathrm{MS}}$ clear the effect of unitary matrices $\overline{\mathbf{F}}$ and $\overline{\mathbf{W}}$ in (22), respectively. Since ChannelNet accepts real-valued data, we construct the final form of the input $\mathcal{X}_{k}$ as three “channel” tensors. Thus, the first and second “channel” of $\mathcal{X}_{k}$ are the real and imaginary parts of $\mathbf{G}_{k}[m]$, i.e., $[\mathcal{X}_{k}]_{1}=\operatorname{Re}\{\mathbf{G}_{k}[m]\}$ and $[\mathcal{X}_{k}]_{2}=\operatorname{Im}\{\mathbf{G}_{k}[m]\}$, respectively. Finally, the third “channel” is given by $[\mathcal{X}_{k}]_{3}=\angle\{\mathbf{G}_{k}[m]\}$. We note here that the use of three “channel” input (e.g., real, imaginary and angle information of $\mathbf{G}_{k}[m]$) provides better feature representation [19, 18, 36]. As a result, the size of the input data is $N_{\mathrm{MS}}\times N_{\mathrm{BS}}\times 3$.

The output of ChannelNet is given by a $2N_{\mathrm{BS}}N_{\mathrm{MS}}\times 1$ real-valued vector as

As a result, ChannelNet maps the received pilot signals $\mathbf{G}_{k}[m]$ to the channel matrix $\mathbf{H}_{k}[m]$.

In this part, we examine the channel estimation problem in RIS-assisted massive MIMO, which is shown in Fig. 3. First, we present the received signal model including both direct (BS-user) and cascaded (BS-RIS-user) channels⁴⁴4Channel estimation is required to design the passive beamformer weights. Although the BS-user, BS-RIS and RIS-user channels can be estimated separately [47], the estimation of the direct and the cascaded channels is sufficient for beamformer design [48, 49].. Then, we show how input-output pairs of ChannelNet are obtained for RIS-assisted scenario.

We consider the downlink channel estimation, where the BS has $N_{\mathrm{BS}}$ antennas to serve $K$ single-antenna users with the assistance of RIS, which is composed of $N_{\mathrm{RIS}}$ reflective elements, as shown in Fig. 3. The incoming signal from the BS is reflected from the RIS, where each RIS element introduces a phase shift $\varphi_{n}$, for $n=1,\dots,N_{\mathrm{RIS}}$. This phase shift can be adjusted through the PIN (positive-intrinsic-negative) diodes, which are controlled by the RIS-controller connected to the BS over the backhaul link. As a result, RIS allows the users receive the signal transmitted from the BS when they are distant from the BS or there is a blockage among them. Let $\overline{\mathbf{S}}_{\mathrm{RIS}}\in\mathbb{C}^{N_{\mathrm{BS}}\times M_{\mathrm{BS}}}$, $(N_{\mathrm{BS}}\leq M_{\mathrm{BS}})$ be the pilot signals transmitted from the BS, then the received signal at the $k$th user becomes

where $\mathbf{y}_{k}=[y_{1,k},\dots,y_{M_{\mathrm{BS}},k}]$ and $\mathbf{n}_{k}=[n_{1,k},\dots,n_{M_{\mathrm{BS}},k}]$ are $1\times M_{\mathrm{BS}}$ row vectors and $\mathbf{h}_{\mathrm{B},k}\in\mathbb{C}^{N_{\mathrm{BS}}}$ represents the channel for the communication link between the BS and the $k$th user. $\boldsymbol{\psi}=[\psi_{1},\dots,\psi_{N_{\mathrm{RIS}}}]^{\textsf{T}}\in\mathbb{C}^{N_{\mathrm{RIS}}}$ is the reflecting beamformer vector, whose $n$th entry is $\psi_{n}=a_{n}e^{j\varphi_{n}}$, where $a_{n}\in\{0,1\}$ denotes the on/off stage of the $n$th element of the RIS and $\varphi_{n}\in[0,2\pi]$ is the phase shift introduced by the RIS. In practice, the RIS elements cannot be perfectly turned on/off, hence, they can be modeled as $a_{n}=\left\{\begin{array}[]{cc}1-\epsilon_{1}&\mathrm{ON}\\ 0+\epsilon_{0}&\mathrm{OFF}\end{array}\right.,$ for small $\epsilon_{1},\epsilon_{0}\geq 0$, which represent the insertion loss of the reflecting elements [18]. In (25), $\mathbf{V}_{k}\in\mathbb{C}^{N_{\mathrm{BS}}\times N_{\mathrm{RIS}}}$ denotes the cascaded channel for the BS-RIS-user link and it can be defined in terms of the channel between BS-RIS and RIS-user as

where $\mathbf{H}_{\mathrm{B}}\in\mathbb{C}^{N_{\mathrm{BS}}\times N_{\mathrm{RIS}}}$ is the channel between the BS and the RIS and it can be defined similar to (1) as

where $L_{\mathrm{RIS}}$ and $\alpha_{l}^{\mathrm{RIS}}$ are the number of received paths and the complex gain respectively. $\mathbf{a}_{\mathrm{BS}}(\phi_{l}^{\mathrm{BS}})\in\mathbb{C}^{N_{\mathrm{BS}}}$ and $\mathbf{a}_{\mathrm{RIS}}({\phi}_{l}^{\mathrm{RIS}})\in\mathbb{C}^{N_{\mathrm{RIS}}}$ are the steering vectors corresponding to the BS and RIS with the AoA and AoD angles of $\phi_{l}^{\mathrm{BS}},\phi_{l}^{\mathrm{RIS}}$, respectively. In (26), $\boldsymbol{\Lambda}_{k}=\mathrm{diag}\{\mathbf{h}_{\mathrm{S},k}\}$ and $\mathbf{h}_{\mathrm{S},k}\in\mathbb{C}^{N_{\mathrm{RIS}}}$ represents the channel between the RIS and the $k$th user. $\mathbf{h}_{\mathrm{S},k}$ and $\mathbf{h}_{\mathrm{B},k}$ have similar structure and they can be defined as follows

where $L_{\mathrm{B}}$, $\alpha_{k,l}^{\mathrm{B}}$ and $\mathbf{a}_{\mathrm{BS}}(\phi_{k,l}^{\mathrm{B}})$ ($L_{\mathrm{S}}$, $\alpha_{k,l}^{\mathrm{S}}$, $\mathbf{a}_{\mathrm{RIS}}(\phi_{k,l}^{\mathrm{B}})$) are the number of paths, complex gain and the steering vector for the BS-user (RIS-user) communication link, respectively.

In order to estimate the direct channel $\mathbf{h}_{\mathrm{B},k}$, we assume that all the RIS elements are turned off, i.e., $a_{n}=0$ for $n=1,\dots,N_{\mathrm{RIS}}$. Then, the $1\times M_{\mathrm{BS}}$ received signal at the $k$th user becomes

Then, the direct channel between BS-user $\mathbf{h}_{\mathrm{B},k}$ can be estimated from the received pilot signal $\mathbf{y}_{\mathrm{B},k}$ via LS and MMSE estimators as $\mathbf{h}_{\mathrm{B},k}^{\mathrm{LS}}=\bigg{(}\mathbf{y}_{\mathrm{B},k}\overline{\mathbf{S}}_{\mathrm{RIS}}^{\dagger}\bigg{)}^{\textsf{H}},$ and $\mathbf{h}_{\mathrm{B},k}^{\mathrm{MMSE}}=\mathbf{h}_{\mathrm{B},k}^{\mathrm{LS}}\mathbf{R}_{\mathrm{B},k}\bigg{(}\mathbf{R}_{\mathrm{B},k}+\frac{1}{\sigma^{2}}\overline{\mathbf{S}}_{\mathrm{RIS}}\overline{\mathbf{S}}_{\mathrm{RIS}}^{\textsf{H}}\bigg{)}^{-1},$ where $\mathbf{R}_{\mathrm{B},k}=\mathbb{E}\{\mathbf{h}_{\mathrm{B},k}\mathbf{h}_{\mathrm{B},k}^{\textsf{H}}\}$ [16].

Next, we consider the cascaded channel estimation. We assume that each RIS element is turned on one by one while all the other elements are turned off. This is done by the BS requesting the RIS via a micro-controller device in the backhaul link so that a single RIS element is turned on at a time. Then, the reflecting beamformer vector at the $n$th frame becomes $\boldsymbol{\psi}^{(n)}=[0,\dots,0,\psi_{n},0,\dots,0]^{\textsf{T}}$, where $a_{n}=\{0:\tilde{n}=1,\dots N_{\mathrm{RIS}},\tilde{n}\neq n\}$ and the received signal is given by

where $\mathbf{v}_{k}^{(n)}\in\mathbb{C}^{N_{\mathrm{BS}}}$ is the $n$th column of $\mathbf{V}_{k}$, i.e., $\mathbf{v}_{k}^{(n)}=\mathbf{V}_{k}\boldsymbol{\psi}^{(n)}$, where $\psi_{n}=1$. Using the estimate of $\mathbf{h}_{\mathrm{B},k}$ from (30), (31) can be solved for $\mathbf{v}_{k}^{(n)}$, $n=1,\dots,N_{\mathrm{RIS}}$, and the cascaded channel $\mathbf{V}_{k}$ can be estimated. Then, the received data for $n=1\dots,N_{\mathrm{RIS}}$ can be given by $\mathbf{Y}_{\mathrm{C},k}\in\mathbb{C}^{N_{\mathrm{RIS}}\times M_{\mathrm{BS}}}$ as $\mathbf{Y}_{\mathrm{C},k}=\left[\begin{array}[]{c}\mathbf{y}_{\mathrm{C},k}^{(n)}\\ \vdots\\ \mathbf{y}_{\mathrm{C},k}^{(N_{\mathrm{RIS}})}\end{array}\right]$. In order to train ChannelNet for RIS-assisted massive MIMO scenario, we select the input-output data pair as $\{\mathbf{y}_{\mathrm{B},k},\mathbf{h}_{\mathrm{B},k}\}$ and $\{\mathbf{Y}_{\mathrm{C},k},\mathbf{V}_{k}\}$ for direct and cascaded channels respectively. To jointly learn both channels, a single input is constructed to train a single NN as $\boldsymbol{\Upsilon}_{k}=\left[\begin{array}[]{c}\mathbf{y}_{\mathrm{B},k}\\ \mathbf{Y}_{\mathrm{C},k}\end{array}\right]\in\mathbb{C}^{(N_{\mathrm{RIS}}+1)\times M_{\mathrm{BS}}}$. Following the same strategy in the previous scenario, the three “channel” of the input data can be constructed as $[{\mathcal{X}}_{k}]_{1}=\operatorname{Re}\{\boldsymbol{\Upsilon}_{k}\}$ and $[{\mathcal{X}}_{k}]_{2}=\operatorname{Im}\{\boldsymbol{\Upsilon}_{k}\}$, $[{\mathcal{X}}_{k}]_{3}=\angle\{\boldsymbol{\Upsilon}_{k}\}$, respectively. We can define the output data as $\boldsymbol{\Sigma}_{k}=\left[\mathbf{h}_{\mathrm{B},k},\mathbf{V}_{k}\right]\in\mathbb{C}^{N_{\mathrm{BS}}\times(N_{\mathrm{RIS}}+1)}$, hence, the output label can be given by a $2N_{\mathrm{BS}}(N_{\mathrm{RIS}}+1)\times 1$ real-valued vector as

Consequently, we have the sizes of ${\mathcal{X}}_{k}$ and ${\mathcal{Y}}_{k}$ are $(N_{\mathrm{RIS}}+1)\times M_{\mathrm{BS}}\times 3$ and $2N_{\mathrm{BS}}(N_{\mathrm{RIS}}+1)\times 1$ respectively.

We design a single CNN, i.e., ChannelNet trained on two different datasets for both conventional and RIS-assisted massive MIMO applications. The proposed network architecture is a CNN with $10$ layers. The first layer is the input layer, which accepts the input data of size $N_{\mathrm{MS}}\times N_{\mathrm{BS}}\times 3$ and $(N_{\mathrm{RIS}}+1)\times M_{\mathrm{BS}}\times 3$ for conventional and RIS-assisted massive MIMO scenario respectively. The $\{2,4,6\}$th layers are the convolutional layers with $N_{\mathrm{SF}}=128$ filters, each of which employs a $3\times 3$ kernel for 2-D spatial feature extraction. The $\{3,5,7\}$th layers are the normalization layers. The eighth layer is a fully connected layer with $N_{\mathrm{FCL}}=1024$ units, whose main purpose is to provide feature mapping. The ninth layer is a dropout layer with $\kappa=1/2$ probability. The dropout layer applies an $N_{\mathrm{FCL}}\times 1$ mask on the weights of the fully connected layer, whose elements are uniform randomly selected from $\{0,1\}$. As a result, at each iteration of FL training, randomly selected different set of weights in the fully connected layer is updated. Thus, the use of dropout layer reduces the size of $\boldsymbol{\theta}_{t}$ and $\mathbf{g}_{k}(\boldsymbol{\theta}_{t})$, thereby, reducing model transmission overhead. Finally, the last layer is output regression layer, yielding the output channel estimate of size $2N_{\mathrm{MS}}N_{\mathrm{BS}}\times 1$ and $2N_{\mathrm{BS}}(N_{\mathrm{RIS}}+1)\times 1$ for conventional and RIS-assisted massive MIMO applications respectively.

During FL-based training, the collected datasets at the users are used to compute the model updates as in Section III-B and transmitted to the BS. The collected model parameters at the BS are then aggregated as in (10) and broadcast to the users for the next iteration. This process is conducted for $t=1,\dots,T$ communication rounds until convergence.

Communication overhead can be defined as the size of the transmitted data during model training. Let $\mathcal{T}_{\mathrm{FL}}$ and $\mathcal{T}_{\mathrm{CL}}$ denote the communication overhead of FL and CL, respectively. Then, we can define $\mathcal{T}_{\mathrm{CL}}$ for both conventional and RIS-assisted scenario as

which includes the number of symbols in the uplink transmission of the training dataset from the users to the BS. In contrast, the communication overhead of FL includes the transmission of $\mathbf{g}_{k}(\boldsymbol{\theta}_{t})$ and $\boldsymbol{\theta}_{t}$ in uplink and downlink communication for $t=1,\dots,T$, respectively. Hence, $\mathcal{T}_{\mathrm{FL}}$ is given by

We can see that the dominant terms in (35) and (38) are D and $P$, which are the number of training data pairs and the number of NN parameters respectively. While D can be adjusted according to the amount of available data at the users, $P$ is usually unchanged during model training. Here, $P$ is computed as $P=\underbrace{N_{\mathrm{CL}}(CN_{\mathrm{SF}}W_{x}W_{y})}_{\mathrm{Convolutional\hskip 1.0ptLayers}}+\underbrace{\kappa N_{\mathrm{SF}}W_{x}W_{y}N_{\mathrm{FCL}}\footnotesize}_{\mathrm{Fully\hskip 1.0ptConnected\hskip 1.0ptLayers}},$ where $N_{\mathrm{CL}}=3$ is the number of convolutional layers and $C=3$ is the number of spatial “channels”. $W_{x}=W_{y}=3$ are the 2-D kernel sizes. As a result, we have $P=600,192$. Since the number of samples in the training dataset is usually larger than the number of model parameters, it is expected to have $\mathcal{T}_{\mathrm{FL}}<\mathcal{T}_{\mathrm{CL}}$ [35, 30, 31] (see Fig. 11).

We further examine the computational complexity of the proposed CNN architecture. The time complexity of the convolutional layers can be written as [16, 36]