Stacked Intelligent Metasurfaces for
Task-Oriented Semantic Communications

Specifically, the transmitter consists of two components: i) a single transmitting antenna, and ii) a SIM in front of it.

The wireless channel is assumed to be block fading, denoted by $\mathbf{H}=\left[\mathbf{h}_{1},\mathbf{h}_{2},\ldots,\mathbf{h}_{M}\right]^{T}\in\mathbb{C}^{M\times N}$, where $\mathbf{h}_{m}^{T}\in\mathbb{C}^{1\times N}$, $m\in\mathcal{M}$, denotes the channel from the output layer of SIM to the $m$-th antenna of the receiver. In this letter, we adopt the Rician fading model [11], yielding

where $p$ is the path loss, $K$ is the Rician factor, $\mathbf{H}_{\mathrm{LOS}}$ is the line-of-sight (LOS) component, characterizing the direct path between SIM and receiver, which is defined by ${[\mathbf{H}_{\mathrm{LOS}}]}_{m,n}=e^{-j\frac{2\pi}{\lambda}d_{m,n}}$, where $d_{m,n}$ denotes the distance from the $n$-th meta-atom on the output layer of the SIM to the $m$-th antenna at the receiving array. $\mathbf{H}_{\mathrm{NLOS}}$ is the non-LOS (NLOS) component, following the circularly symmetric complex Gaussian distribution with a zero mean and an identity covariance matrix. In this paper, we focus on the optimization of SIM to enable semantic encoding by assuming that the wireless channel coefficients have been estimated using advanced channel estimation approaches [11].

As a result, the received signal can be modeled as

where $\mathbf{y}=\left[y_{1},y_{2},\ldots,y_{M}\right]\in\mathbb{C}^{M}$ represents the signal received at the receiving array, $\mathbf{v}\in\mathbb{C}^{M}$ is the additive white Gaussian noise (AWGN), satisfying $\textbf{v}\sim\mathcal{CN}\left(\mathbf{0},\sigma^{2}_{v}\mathbf{I_{M}}\right)$, where $\mathbf{I_{M}}\in\mathbb{C}^{M\times M}$ denotes the identity matrix and $\sigma^{2}_{v}$ represents the average noise power.

As shown in Fig. 1, $M$ antennas are arranged at the receiver in a uniform linear array with spacing $d_{r}$. Each antenna corresponds to an image class. In this letter, we aim to train the SIM-aided EMNN to steer the wireless signals $\tilde{\mathbf{x}}$ towards the corresponding antenna. As a result, the image can be recognized by comparing the magnitude of the signal at each antenna. This process can be formulated as

where $\hat{m}$ is the index of the antenna with maximum received energy, also indicating the corresponding class of the image.

The cross-entropy (CE) is used as the loss function to measure the difference between the power pattern across the receiving array and the expected probability distribution of recognizing the input image. Specifically, we use the softmax function to normalize the power at the receiving array, i.e., $\tilde{\mathbf{y}}=\text{softmax}(\mathbf{|y|}^{2})=\left[\tilde{y}_{1},\tilde{y}_{2},\ldots,\tilde{y}_{M}\right]\in\mathbb{R}^{M}$, where $\tilde{y}_{m}=\frac{e^{|{y_{m}}|^{2}}}{\sum_{i=1}^{M}e^{|{y_{i}}|^{2}}}$, $m\in\mathcal{M}$. Moreover, let $\mathbf{q}=\left[q_{1},q_{2},\ldots,q_{M}\right]\in\mathbb{R}^{M}$ denote the probability distribution, which is defined by

As a result, the CE is defined as

Note that a smaller value of CE indicates that the energy distribution at the receiving array is closer to the expected probability distribution.

In this subsection, we aim to train the transmission coefficients, including the amplitude vectors $\{\mathbf{a}^{1},\mathbf{a}^{2},\ldots,\mathbf{a}^{L}\}$ and phase vectors $\{\bm{\phi}^{1},\bm{\phi}^{2},\ldots,\bm{\phi}^{L}\}$, of the SIM for minimizing the total loss function of all image samples in the dataset³³3For discrete phase shifts, one can optimize the SIM-aided EMNN using an alternating optimization approach [23].. The training process mainly includes six steps:

1. 1.

   Initialize the transmission coefficients of the SIM, where $a^{l}_{n}$ and $\phi^{l}_{n}$, $n\in\mathcal{N}$, $l\in\mathcal{L}/\{0\}$ obey the uniform distribution of $\mathcal{U}\left[0,1\right)$ and $\mathcal{U}\left[0,2\pi\right)$ respectively.

2. 2.

   Extract $B$ samples from the image dataset to create a mini-batch and preprocess each image to match the size of the source encoding layer.

3. 3.

   For each image in the mini-batch, generate a set of received signal samples according to (5).

4. 4.

   Calculate the loss function $\mathcal{L}_{\mathrm{CE}}$ according to (8) for each image in the mini-batch and add them up to obtain the total loss function over the mini-batch, denoted by $\mathcal{L}_{\text{total}}$.

5. 5.

   Use the Adam optimization algorithm [24] to iteratively update the transmission coefficients of the SIM. It is worth mentioning that both parameters $\mathbf{a}^{l}$ and $\bm{\phi}^{l}$ are updated simultaneously.

6. 6.

   At each iteration, project the SIM transmission coefficients to the feasible set by applying the MinMaxScaler normalization method [25].

After applying the mini-batch gradient descent method in several iterations, the loss function value converges. The detailed steps for training the SIM-aided EMNN to enable image recognition tasks in the wave domain are given in Algorithm 1, where $\nabla$ in step 7 represents the gradient operator. Additionally, $\alpha>0$ in step $7$ is the learning rate, while $\mathbf{m}_{t}^{\{\mathbf{a}_{t}^{l},\bm{\phi}_{t}^{l}\}}$ and $\mathbf{v}_{t}^{\{\mathbf{a}_{t}^{l},\bm{\phi}_{t}^{l}\}}$ are respectively the first-order and second-order moment vectors. Additionally, $\beta_{1}$ and $\beta_{2}$ represent the decay coefficients, while $\epsilon$ is a small positive constant used for enhancing numerical stability. Appropriately adjusting these hyperparameters can strike tradeoffs between convergence speed and image recognition accuracy.

In this section, we use the MNIST dataset to evaluate the performance of the proposed SIM-based EMNN for implementing image recognition tasks in the wave domain. It consists of approximately 70,000 images of handwritten digits from 0 to 9 with $28\times 28$ grayscale pixels. The cardinality ratio of the training set to the testing set is 6:1, and 20% of the training set is allocated for validation purposes. Note that the noise is added to the received signal during the training process to improve the system’s robustness. Moreover, the SemCom system operates at $28$ GHz, corresponding to a wavelength of $\lambda=10.7$ mm. The thickness of the SIM is set to $D=10\lambda$, while the spacing between adjacent meta-atoms is a wavelength, yielding $S_{a}=\lambda^{2}$. Additionally, the horizontal distance from the transmitting antenna to the input layer of the SIM is set to $d_{t,s}=d_{s}$. The distance between adjacent receiving antennas is set to $d_{r}=\lambda/2$. For the wireless link, the Rician factor is set to $K=3$ dB, and the path loss from the SIM to the receiver is modeled as $p=C_{0}d_{s,r}^{-\gamma}$, where $C_{0}=-35$ dB denotes the path loss at a reference distance of 1 m. $d_{s,r}$ denotes the distance between the SIM and the receiving array, and $\gamma=2.8$ is the path loss exponent in our simulations [26]. Additionally, the transmit power and the noise power are set to $p_{t}=40$ dBm and $\sigma^{2}_{v}=-104$ dBm, respectively.

In terms of the training process of SIM-based EMNN, we consider 100 epochs, and the mini-batch size $B$ is set to $64$. Moreover, the learning rate is set to $\alpha=0.001$, which will be gradually decreased by multiplying $\alpha$ with a decay factor $\iota_{f}=0.8$, if the loss function value does not vary in five consecutive epochs. The decay coefficients of the first-order and the second-order moment estimates are $\beta_{1}=0.9$ and $\beta_{2}=0.999$, respectively. The first-order moment vector $\mathbf{m}_{t}$ and the second-order moment vector $\mathbf{v}_{t}$ are initialized to $\mathbf{0}$.

In Fig. 2(a), we evaluate the effect of the number $L$ of layers on the recognition accuracy, where we consider $N=441$, $d_{s,r}=10$ m. For each input image, we retain only the $21\times 21$ pixels in the center. For simplicity, $M=8$ antennas are deployed at the receiver for recognizing digits $0\sim 7$. As observed in Fig. 2(a), the recognition accuracy using the SIM-based EMNN increases with the number of metasurface layers, thanks to the enhanced inference capability of the multi-layer architecture for achieving more accurate beam steering.

Fig. 2(b) shows the recognition accuracy under different numbers $N$ of meta-atoms on each layer, where we set $L=7$ and maintain the other parameters the same as Fig. 2(a). For each input image, we preprocess it to match the size of the source encoding layer by using the bilinear interpolation method [22]. Note that the recognition accuracy improves as $N$ increases, since a larger number of learnable neurons (meta-atoms) would enhance the representational capability of the SIM. Additionally, increasing the number of meta-atoms also increases the aperture of the output metasurface layer, thus having a higher Rayleigh resolution to distinguish the signals at different receiving antennas. Specifically, the seven-layer SIM with $N=25$ meta-atoms on each layer achieves a low recognition accuracy of only $32\%$ on the testing set. As $N$ increases to $121$ and $441$, the recognition accuracy on the testing set becomes $77\%$ and $84\%$, respectively.

According to the simulation results in Fig. 2(a)-(b), increasing the value of $L$ and $N$ enhances recognition accuracy. However, the improvement of recognition accuracy is relatively small when $L$ exceeds $7$, as shown in Fig. 2(a). Similarly, Fig. 2(b) demonstrates the performance plateaus when $N$ exceeds 441. These diminishing returns suggest choosing proper SIM parameters for achieving the desired performance with minimum hardware cost.

Fig. 2(c) illustrates the recognition accuracy versus the training epoch as the distance $d_{s,r}$ between the SIM and receiver increases, with $N=441$ and other parameters unchanged. It is observed from Fig. 2(c) that a shorter propagation distance would improve the recognition accuracy. This is due to a smaller value of $d_{s,r}$ would cause stronger channel gain compared to that with a longer propagation distance. Hence, the noise effects would become increasingly negligible. Secondly, as the receiving array moves closer to the transmitter, the channels from the SIM to different antennas would become more distinguishable. As a result, decreasing the propagation distance from $d_{s,r}=50$ m to $d_{s,r}=10$ m enhances the recognition accuracy from $72\%$ to $84\%$.

Fig. 2(d) illustrates the recognition accuracy under different numbers of image classes where we set $d_{s,r}=10$ m. Specifically, we examine nine scenarios by gradually increasing both the number of image classes as well as the number of receiving antennas. Note that recognizing more handwritten digits requires the SIM to accurately focus multiple beams on an increased number of receiving antennas. As observed from Fig. 2(d), the recognition accuracy becomes worse as the number of image classes increases. Additionally, a faster convergence trend is observed as $M$ decreases.

Finally, to better illustrate the effectiveness of the proposed SIM-based EMNN, Fig. 3 depicts the input image and the corresponding energy distribution at the receiving array, while the confusion matrix of utilizing the SIM-based EMNN to recognize ten handwritten digits is shown in Fig. 4. It is demonstrated that the EMNN is capable of focusing the information-bearing signals passing through the SIM onto the corresponding antenna.