Integrating Sensing, Computing, and Communication in 6G Wireless Networks: Design and Optimization

First, we deal with the sensing signal. To improve the accuracy of target sensing, all sensors collaborate to sense the targets. Thus, a linear unbiased estimator is deployed at the BS to estimate the reflection coefficients of the targets. To obtain an accurate reflection coefficient $r_{i},i\in\Omega_{I},$ for target sensing, the BS carries out receive beamforming to enhance the desired signal and suppress the co-channel interference. Hence, the estimated reflection coefficient ${{{\hat{r}}}_{i}}$ of the $i$-th target can be expressed as

where $\mathbf{v}_{i}\in\mathbb{C}^{N\times 1}$ is the sensing receive beamforming vector adopted at the BS for the $i$-th target. Following relevant parameter estimation works, we adopt the mean squared error (MSE) as the performance metric of target sensing. It is seen from Eq. (3) that the $\hat{r}_{i}$ is not only a function of $s_{k,i}^{\text{sens}}$, but also the other sensing signals from different sensors for different targets, the computing signals, and communication signals. In this context, we can minimize the MSE between the estimated reflection coefficient ${{{\hat{r}}}_{i}}$ and the actual reflection coefficient $r_{i}$ to suppress the interference from other signals and achieve a high-accuracy target sensing. Thus, the MSE is given by

where $\sigma_{n}^{2}$ is the noise power, and ${{R}_{m}}$ is the root mean squared (RMS) value of the reflection coefficient ${{r}_{m}},\forall m\in\{\Omega_{I}\cup\Omega_{O}\}$ according to the a-priori probability of occurrence for each type [32]. To present more intuitively the impact of reflection coefficient estimation, we provide an application result on target imaging [33]. As is shown in Fig. 2, the higher the sensing accuracy (the smaller MSE between the estimated reflection coefficient and the actual reflection coefficient), the more accurate the target imaging.

Secondly, we handle the computing signal. In a time slot, a shared learning process for AirFL with a global model $\mathbf{s}^{\text{comp}}\in\mathbb{R}^{L\times 1}$ is trained cooperatively by multiple intelligent sensors [23, 24]. Each intelligent sensor has its own local dataset $\mathcal{D}_{k},\forall k\in\Omega_{K}$, and the sum of local datasets is $\mathcal{D}$, where $\left|\mathcal{D}\right|=\sum\limits_{k=1}^{K}{\left|{{\mathcal{D}}_{k}}\right|}$. Thus, the desired global model at the BS by computing local models is given by

where $s_{k,l}^{\text{comp}}$ is the $l$-th computing signal contributed by the pre-processed local model parameter at the $k$-th sensor, and each local model parameter is multiplied by a weighting factor ${{\xi}_{k}}=\frac{\left|{{\mathcal{D}}_{k}}\right|}{\left|\mathcal{D}\right|}$ before transmission. To reduce the computation error in the presence of interference and channel fading, we need to implement receive beamforming at the BS. As a result, the received signal for AirFL at the BS can be expressed as

where $\mathbf{z}_{l}\in\mathbb{C}^{N\times 1}$ is the computing receive beamforming vector at the BS for the $l$-th model parameter. In general, the distortion of the model aggregation is measured by the MSE between ${s}_{l}^{\text{comp}}$ and ${\hat{s}_{l}^{\text{comp}}}$, which can be expressed as

Herein, we also provide a classical application case to vividly present the function of computing, i.e. image recognition based on AirFL[34]. It is seen from Table I that the higher computing accuracy (the smaller computing MSE between the aggregated computing signal and the desired global model), the more accurate the image recognition.

Finally, we discuss the communication signal. To promote the quality of the communication signal, the BS performs corresponding receive beamforming to mitigate the interference. Hence, the received signal at the BS for the communication signal $s_{k,j}^{\text{comm}},\forall k\in\Omega_{K},j\in\Omega_{J},$ is given by

where $\mathbf{u}_{k,j}\in\mathbb{C}^{N\times 1}$ is the communication receive beamforming vector of the $j$-th communication signal from the $k$-th sensor. Since the received SINR determines the quality of the communication signal, it is usually regarded as the performance metric for communication. The signal-to-interference-plus-noise ratio (SINR) related to $s_{k,j}^{\text{comm}}$ can be expressed as

where

It is observed from (4), (7), and (9) that the performance of sensing, computing, and communication are jointly determined by the transmit beamforming vectors $\mathbf{a}_{k,i}$, $\mathbf{b}_{k,l}$, and $\mathbf{c}_{k,j}$ at the sensors, and receive beamforming vectors $\mathbf{v}_{i}$, $\mathbf{z}_{l}$, and $\mathbf{u}_{k,j}$ at the BS. Although the three performance metrics are all desirable to the system, they are competitive for the system resources. Hence, it is desired to establish a joint beamforming design framework to enhance the overall performance of ISCC over the limited resources in 6G wireless networks.

Remark: It is worthy pointing out that our proposed integrated system is inherently different from multi-user communication systems in terms of signal waveform, transceiver design and function realization. Firstly, multi-user communication systems only consider one kind of signal for communication purpose, while in our proposed integrated system, sensing, computing and communication signals have different types and serve different purposes. Secondly, signals of the same type from different users interfere with each other at the BS in multi-user communication systems, and thus transceiver is designed to reduce the inter-user interference. In contrary, for the proposed integrated system, signals of different types interfere with each other, but signals of the same type may help each other. In this case, the designed transceiver not only needs to decrease the interference among signals of different types, but also needs to increase the cooperative gain of signals of the same type in terms of sensing and computing. Finally, although different performance metrics such as SINR, data rate and MSE, are chosen for optimization in multi-user communication systems, they are all used to improve the performance of the communication function. While for the proposed integrated system, we select the appropriate performance metrics based on the functions of sensing, computing and communication. By solving the formulated multi-objective optimization problem (MOOP), the desired performance of three different functions can be effectively achieved.

Now, we first study three individual single-objective optimization problem (SOOP) for sensing, computing, and communication in 6G wireless networks, respectively, which severs as building blocks for the formulation of WOPM design. The first SOOP aims at minimizing the weighted sum-MSE for sensing subject to the maximum transmit power budget, which is formulated as S-1: Weighted Sum-MSE of Sensing Minimization:

where constant $\theta_{i}^{\text{sens}}$ is the weighted coefficient related to $\text{MSE}_{i}^{\text{sens}}$, and $P_{\max,k}$ is the maximum transmit power budget at the $k$-th sensor. The objective in S-1 is to minimize the weighted summation of MSEs between estimated reflection coefficients and actual ones for all $I$ targets depicted in (4), and constraint C1 is the transmit power limitation at the $k$-th sensor. Note that each SOOP only concerns one performance metric of the proposed integrated sensing, computing and communication without considering others. For example, it is seen from S-1 that the communication beam ${{\mathbf{c}}_{k,j}}$ and the computing beam ${{\mathbf{b}}_{k,l}}$ will naturally be zero to minimize the objective, while the sensing beam ${{\mathbf{a}}_{k,i}}$ will not be zero due to the term ${{\left|\sum\limits_{k=1}^{K}{\mathbf{v}_{i}^{H}{{\mathbf{G}}_{k,i}}{{\mathbf{a}}_{k,i}}}-1\right|}^{2}}R_{\text{i}}^{2}$ in (4). Hence, no additional conditions related to communication and computing beams are required. The second SOOP is to minimize the weighted sum-MSE for computing under the limitation of transmit power consumption, which can be mathematically expressed as

S-2: Weighted Sum-MSE of Computing Minimization:

where constant $\theta_{l}^{\text{comp}}$ is the weighted coefficient relevant to $\text{MSE}_{l}^{\text{comp}}$, and the objective is minimizing the weighted summation of the computation distortions for $L$ model parameters, which was defined in (7). The third SOOP focuses on the weighted sum-rate maximization for communication subject to the transmit power limitation, which is formulated as

S-3’: Weighted Sum-Rate of Communication Maximization:

where ${{r}_{k,j}}={{\log}_{2}}\left(1+{{\Gamma}_{k,j}}\right)$ is the achievable rate of the $j$-th communication signal from the $k$-th sensor and $\theta_{k,j}^{\text{comm}}$ is the weighted coefficient associated with ${{r}_{k,j}}$. S-3’ is a classical weighted sum-rate maximization problem which is non-convex due to the complicated objective function involving log functions of fractions [35]. Moreover, the transmit beamforming vectors $\{{{\mathbf{a}}_{k,i}},{{\mathbf{b}}_{k,l}},{{\mathbf{c}}_{k,j}}\}$ and communication receive beamforming vector $\mathbf{u}_{k,j}$ are coupled in the constraints and objective function. To handle this issue as well as facilitate the design, we transform S-3’ into its equivalent log-MSE minimization problem via the following theorem.

Theorem 1: The relationship between the received SINR $\Gamma_{k,j}$ and the minimum MSE (MMSE) $e_{k,j}^{\text{comm}}$ for the communication signal $s_{k,j}^{\text{comm}}$ can be expressed as

According to Theorem 1, we can transform the objective function of S-3’ to

which is equivalent to minimizing the weighted sum-MSE of communication $\text{MSE}_{k,j}^{\text{comm}}$ with the MMSE receiver ${{\mathbf{u}}_{k,j}}=\Xi^{-1}{{\mathbf{H}}_{k}}{{\mathbf{c}}_{k,j}}$, given in Appendix A. Thus, (15) can be reformulated as

However, the transformed objective function (16) still remains non-convex caused by the structure of sum of logarithmic function. To this end, we introduce a weight variable $\omega_{k,j}$ for the $\text{MSE}_{k,j}^{\text{comm}}$ [36], and then (16) is transformed as

As for unconstrained optimization problems in terms of the weight variable $\omega_{k,j}$, by letting the first-derivative of the objective function (17) equal to zero, we can obtain the optimal solution as follows

which makes (16) and (17) are equivalent. After the above conversion, S-3’ can be transformed as

S-3: Weighted Sum-Modified MSE of Communication Minimization:

To jointly optimize the three performance of sensing, computing, and communication, we adopted the weighted sum method to formulate the corresponding MOOP [37, 38]. The WOPM is formulated as follows:

M-1: Weighted Overall Performance Maximization:

where ${{\Psi}_{p}},p=1,2,3$ is the normalized objective function⁵⁵5Since the objective value of S-1, S-2 and S-3 have different ranges, it is desired to perform the normalization for each objective function in order to coordinate the performance of sensing, computing, and communication as well as to facilitate the convergence of objective value for M-1. In this paper, we define $\mathcal{F}_{p},p=1,2,3,$ as the objective function of S-$p$ and ${{\Psi}_{p}}={\left({\mathcal{F}_{p}}-\mathcal{F}_{p}^{*}\right)}/{\left|\mathcal{F}_{p}^{*}\right|}\;$, where $\mathcal{F}_{p}^{*}$ are the corresponding performance limits of sensing, computing and communication in M-1, which can be obtained by respectively solving S-$p$ by applying the same algorithm for solving M-1. of S-1, S-2 and S-3, respectively. $\alpha_{p}\geq 0$ is the priority of the $p$-th objective function, which represents the preference of the system operator and is satisfied with $\alpha_{1}+\alpha_{2}+\alpha_{3}=1$. By varying the value of $\alpha_{p}$, M-1 can yield different solutions. Note that M-1 is equivalent to S-p when $\alpha_{p}=1$ and $\alpha_{q}=0,\forall p\neq q$, which means M-1 is a general formulation of S-1, S-2 and S-3. Hence, we aim at solving M-1 with given priorities $\alpha_{p}$ in the following.

Since multiple variables are inter-coupled in the objective function of M-1, i.e., transmit beams $\{\mathbf{a}_{k,i},{{\mathbf{b}}_{k,l}},{{\mathbf{c}}_{k,j}}\}$, receive beams $\{{{\mathbf{v}}_{i}},\mathbf{u}_{k,j},{{\mathbf{z}}_{l}}\}$, and weight variables $\{\omega_{k,j}\}$, M-1 is non-convex, which makes it impossible to obtain an optimal solution in polynomial time. In this context, we turn to find a sub-optimal solution for exploring the trade-off relationship among sensing, computing, and communication in 6G wireless networks. By examining M-1, although it is not a joint convex function of all variables, it is a convex one in terms of transmit beams, receive beams, and weight variables, respectively. Based on this observation, an alternating optimization (AO) method is applied to divide M-1 into three subproblems, i.e., optimizing receive beams by fixing transmit beams and weight variables, optimizing weight variables by fixing transmit and receive beams, and optimizing transmit beams by fixing receive beams and weight variables. In particular, enabled with the AO method, solution procedure for M-1 will stop until the objective value converges in the iterations [39]. Let us first address the subproblem of optimizing receive beams $\{{{\mathbf{v}}_{i}},\mathbf{u}_{k,j},{{\mathbf{z}}_{l}}\}$ while other variables remain fixed. By applying the Karush-Kuhn-Tucher (KKT) conditions in M-1, i.e.,

we can acquire the optimal receive beamforming vectors, also called MMSE receiver, as below

and

respectively, where $\bm{\Phi}=\sum\limits_{k=1}^{K}{\sum\limits_{n=1,n\neq i}^{I}{R_{n}^{2}{{\mathbf{G}}_{k,n}}{{\mathbf{a}}_{k,n}}\mathbf{a}_{k,n}^{H}\mathbf{G}_{k,n}^{H}}}+\sum\limits_{k=1}^{K}{\sum\limits_{o=1}^{O}{\sum\limits_{m=1}^{I}{R_{o}^{2}{{\mathbf{F}}_{k,o}}{{\mathbf{a}}_{k,m}}\mathbf{a}_{k,m}^{H}\mathbf{F}_{k,o}^{H}}}}+\sum\limits_{k=1}^{K}{\sum\limits_{l=1}^{L}{{{\mathbf{H}}_{k}}{{\mathbf{b}}_{k,l}}\mathbf{b}_{k,l}^{H}\mathbf{H}_{k}^{H}}}+\sum\limits_{k=1}^{K}{\sum\limits_{m=1}^{J}{{{\mathbf{H}}_{k}}{{\mathbf{c}}_{k,m}}}}\mathbf{c}_{k,m}^{H}\mathbf{H}_{k}^{H}+\sigma_{n}^{2}{{\mathbf{I}}_{N}}$, ${{\bm{\Phi}}_{i}}=R_{i}^{2}\sum\limits_{k=1}^{K}{{{\mathbf{G}}_{k,i}}{{\mathbf{a}}_{k,i}}}$, and $\bm{\Xi}=\bm{\Phi}+\sum\limits_{k=1}^{K}{\sum\limits_{n=1}^{I}{R_{n}^{2}{{\mathbf{G}}_{k,n}}{{\mathbf{a}}_{k,n}}\mathbf{a}_{k,n}^{H}\mathbf{G}_{k,n}^{H}}}$ is defined in Appendix A. Next, for the second subproblem in terms of optimizing weight variables $\{\omega_{k,j}\}$, the optimal solution is given in (18), i.e., $\omega_{k,j}^{*}={{\left(\ln 2\cdot\text{MSE}_{k,j}^{\text{comm}}\right)}^{-1}}$. Finally, with the MMSE receivers and weight variables, the last subproblem of optimizing transmit beams $\{\mathbf{a}_{k,i},{{\mathbf{b}}_{k,l}},{{\mathbf{c}}_{k,j}}\}$ for maximizing the weighted overall performance is a standard convex quadratic constrained quadratic programming (QCQP) problem, which can be solved by an inter-point method (IPM) [[42], Chapter 11]. At first, we utilize the barrier method to transform this subproblem into an unconstrained convex optimization problem by adding the logarithmic barrier function. Then, the Newton’s method can be applied to obtain the solution. Specifically, the transformed unconstrained convex optimization problem can be written as

where $f_{k}^{\text{con}}=\sum\limits_{i=1}^{I}{{{\left\|{{\mathbf{a}}_{k,i}}\right\|}^{2}}}+\sum\limits_{l=1}^{L}{{{\left\|{{\mathbf{b}}_{k,l}}\right\|}^{2}}}+\sum\limits_{j=1}^{J}{{{\left\|{{\mathbf{c}}_{k,j}}\right\|}^{2}}}-{{P}_{\max,k}}$ and $\varepsilon>0$ is a barrier parameter that sets the accuracy of the approximation. Then, we compute the variables’ gradients and Hessian for Newton’s method as follows

and

with

and

where $\nabla_{\mathbf{x}}$ and $\nabla_{\mathbf{x}}^{2}$ denote the gradient and the Hessian of the variable for the objective function (25), respectively. Moreover, ${{{\tilde{\alpha}}}_{1}}=\frac{{{\alpha}_{1}}}{\left|\mathcal{F}_{1}^{*}\right|}$, ${{{\tilde{\alpha}}}_{1}}=\frac{{{\alpha}_{2}}}{\left|\mathcal{F}_{2}^{*}\right|}$, and ${{{\tilde{\alpha}}}_{3}}=\frac{{{\alpha}_{3}}}{\left|\mathcal{F}_{3}^{*}\right|}$. With the gradients and Hessian, the transmit beams can be computed by the following updating rules: