METHOD FOR JOINT ACTIVE AND PASSIVE BEAMFORMING AND RECEIVED SIGNAL OPTIMIZATION IN ISAC SYSTEM ASSISTED BY DUAL IRSS

Abstract

A method for the joint active and passive beamforming and received signal optimization in an ISAC system assisted by dual IRSs is provided. The method jointly optimizes active beamforming at Base Station (BS), reception of sensing signals at the Base Station (BS), and passive beamforming at IRSs, so as to maximize communication sum-rate of users while ensuring that SNR of sensing signals meets a minimum requirement. To address the complex non-convex optimization problem, the method first applies fractional programming to decouple problem, then adopts successive convex approximation algorithm and alternating direction method of multipliers to transform intractable non-convex problem into multiple tractable subproblems, and finally employs an alternating optimization method to efficiently acquire the high-quality suboptimal solutions. The simulation results demonstrate that disclosed scheme exhibits satisfactory convergence and effectiveness, and can significantly improve the performance of IRS-assisted ISAC systems.

Claims

1. A method for joint active and passive beamforming and received signal optimization in an integrated sensing and communication (ISAC) system assisted by dual Intelligent Reflecting Surfaces (IRSs), comprising: S1, model establishing: establishing an ISAC system model assisted by the dual IRSs, wherein the ISAC system model comprises a multi-antenna Base Station (BS) configured to simultaneously provide communication services for a plurality of users and perform sensing for targets, wherein the dual IRSs assist the multi-antenna BS in operation, and reliable channels between the multi-antenna BS and communication users and sensing targets are established by dispersing the dual IRSs on a surface of a building; and S2, alternating optimization: S21: deriving a signal-to-interference-plus-noise ratio (SINR) for the communication users and a signal-to-noise ratio (SNR) lower bound for the sensing targets based on the reliable channels for communication and sensing, and formulating an optimization problem based on the SINR and the SNR lower bound, the optimization problem jointly designing active beamforming w at a BS end, received signal processing u at the multi-antenna BS for a sensing signal, and passive beamforming .sub.1,.sub.2 at the dual IRSs, under an objective of maximizing a communication sum-rate of the communication users, while ensuring that a sensing SNR satisfies a minimum requirement, and satisfying a power constraint at the BS end and a constant modulus constraint at IRSs; S22: introducing auxiliary variables and , and decoupling the optimization problem using a fractional programming method; S23: updating the auxiliary variables and by obtaining partial derivatives, and updating the received signal processing u using Rayleigh quotient under given premise conditions w,.sub.1,.sub.2,,; S24: representing an original optimization problem as a second-order cone programming (SOCP) problem under given conditions u,,,.sub.1,.sub.2, and solving and updating the active beamforming w using a convex optimization (CVX) toolbox; S25: introducing an auxiliary variable as well as a dual variable , successively updating first reflection coefficient vectors .sub.1,, using an alternating direction method of multipliers (ADMM) under given conditions u,,,w,.sub.2, and updating a second reflection coefficient vector .sub.2 using a same method; S26: repeating S25 until values of .sub.1,.sub.2 converges; and S27: repeating S23, S24, S25, and S26 until an algorithm converges, thereby obtaining an optimal solution w.sup.opt,u.sup.opt, ${}_1^{\mathrm{opt}},{}_2^{\mathrm{opt}}.$

2. The method for the joint active and passive beamforming and the received signal optimization in the ISAC system assisted by the dual IRSs according to claim 1, wherein the ISAC system model assisted by the dual IRSs in the S1 is defined as: in the ISAC system model assisted by the dual IRSs, a multi-antenna BS simultaneously performs multi-user communication and target detection with assistance of two N-element IRSs, specifically: a multi-antenna BS end is equipped with M transmit antennas as well as M receive antennas, arranged in a uniform linear array (ULA) with a half-wavelength interval, and simultaneously transmits data to K single-antenna users while detecting T sensing targets, wherein a transmit signal of the multi-antenna Base Station (BS) is represented as: $x=Ws=W_c{s}_c+W_r{s}_r$ wherein W.sub.e.sup.MK as well as W.sub.r.sup.MM are, respectively, a communication beamforming matrix and a sensing beamforming matrix, s.sub.c.sup.K is a vector representing communication signals satisfying $E\left\{s_c{s}_c^H\right\}=I_K,$ s.sub.r.sup.M is a vector representing sensing signals satisfying $E\left\{s_r{s}_r^H\right\}=I_M,$ assuming they are statistically independent of each other and satisfying $E\left\{s_c{s}_r^H\right\}=0,$ W[W.sub.c W.sub.r].sup.M(K+M) is defined as an entire beamforming matrix, and $s{\left[s_c^T{s}_r^T\right]}^T{K+M}^{}$ is defined as a transmit symbol vector.

3. The method for the joint active and passive beamforming and the received signal optimization in the ISAC system assisted by the dual IRSs according to claim 1, wherein the reliable channels in the S1 specifically comprise: considering the ISAC system model assisted by the dual IRSs is in a crowded environment, a channel between a BS and the IRSs is modeled as a Rician fading channel, represented as: $G=\sqrt{\frac{{}_R}{1+{}_R}}{G}_{\mathrm{LoS}})+\sqrt{\frac{1}{1+{}_R}}{G}_{\mathrm{NLoS}},$ wherein .sub.R is a Rician factor of a BS-IRS link, which degenerates to a Line-of-Sight (LoS) scenario when .sub.R approaches infinity, and becomes a Rayleigh channel when .sub.R approaches zero, G.sub.NLoS.sup.NM is a Rayleigh fading component, each of a term satisfying CN(0,1) distribution, G.sub.Los.sup.NM is an LoS channel component, because the multi-antenna BS and the IRSs are modeled as, respectively, a uniform linear array (ULA) and a uniform planar array (UPA), and a channel matrix G.sub.LoS is represented as: $G_{LoS}=\sqrt{}{e}^j{a}_R\left({}_R\right)a_T^H\left({}_T\right),$ wherein is a large-scale channel gain, is a random phase uniformly distributed in [0,2], a.sub.T is a transmit steering vector of the BS, and a.sub.R is a receive steering vector of the IRSs, wherein a steering vector is represented as a()=[e.sup.j2d sin()/, . . . , e.sup.j2d(M1)sin()/].sup.T wherein d is a distance between array elements, which is typically set to a half-wavelength, and is a signal wavelength; a channel between the multi-antenna BS and each user is composed of two parts, which are a direct link between the BS and a user and a cascaded link between the BS, the IRSs, and the user, therefore, a received signal at a k.sup.th user from the multi-antenna BS is represented as: $y_k=\left(H_k^H+F_{1,k}^H{}_1^H{G}_1+F_{2,k}^H{}_2^H{G}_2\right)x+n_k,$ wherein H.sub.k.sup.M, F.sub.1,k.sup.N, F.sub.2,k.sup.N, G.sub.1.sup.NM, G.sub.2.sup.NM are defined as effective channels between the BS and the k.sup.th user, between IRS1 and the k.sup.th user, between IRS2 and the k.sup.th user, between the BS and the IRS1, and between the BS and the IRS2, respectively, and a reflection matrix of an IRS is defined as .sub.1diag{.sub.i}, wherein .sub.i=[.sub.i1, . . . , .sub.iN].sup.T is a reflection coefficient vector satisfying |.sub.in|=1, i{1,2}, n.

4. The method for the joint active and passive beamforming and the received signal optimization in the ISAC system assisted by the dual IRSs according to claim 1, wherein the SINR in the S21 is defined as: a scalar $n_k\mathrm{CN}\left(0,{}_k^2\right)$ is an additive white gaussian noise (AWGN) at a k.sup.th user, therefore, the SINR of the k.sup.th user is calculated as: ${}_k=\frac{{.Math.h_k^H{w}_k.Math.}^2}{{.Math.}_{jk}^{K+M}{.Math.h_k^H{w}_j.Math.}^2+{}_k^2},$ wherein an effective channel h.sub.k.sup.M is defined as $h_k^H=H_k^H+F_{1,k}^H{}_1^H{G}_1+F_{2,k}^H{}_2^H{G}_2,$ wherein w.sub.j denotes a j.sup.th column of a beamforming matrix W, i.e., W[w.sub.1, . . . , w.sub.K+M]; considering a link between the multi-antenna BS and a t.sup.h target is blocked, an echo signal received through a path assisted by the IRSs is represented as: $y_{r,t}={}_t\left(G_1^H{}_1{z}_{1,t}+G_2^H{}_2{z}_{2,t}\right)\left(z_{1,t}^H{}_1^H{G}_1+z_{2,t}^H{}_2^H{G}_2\right)Ws+n_r,$ further considering target detection is performed under far-field conditions, wherein .sub.t represents radar cross section (RCS) of the t.sup.th target, i.e., .sub.t=4P.sub./S, in which P.sub. denotes a radiated power density of a target's scattered wave, and S denotes a power density of an incident wave; calculating a sensing target SNR in terms of expectation; for simplifying a formula expression, utilizing ${}_t^2$ to represent an expected value .sub.t, i.e., $E\left\{{.Math.{}_t.Math.}^2\right\}={}_t^2,$ wherein z.sub.1,t.sup.N and z.sub.2,t.sup.N denote baseband channels between IRS1, IRS2 and the t.sup.th target, respectively, and a vector $n_{r}CN\left(0,{}_r^2{I}_M\right)$ represents the AWGN; assuming the path between the IRSs and a target is LoS, and required angles of arrival/departure (AoA/AoD) are known, an equivalent channel matrix of the echo signal H, is redefined as: $H_t\left(G_1^H{}_1{z}_{1,t}+G_2^H{}_2{z}_{2,t}\right)\left(z_{1,t}^H{}_1^H{G}_1+z_{2,t}^H{}_2^H{G}_2\right);$ defining wvec{W} as vectorizing a matrix W, and .Math. as a Kronecker product, then y.sub.r,t is re-expressed as: $y_{r,t}={}_t\left(S{S}^H.Math.H_t\right)w+n_r.$

5. The method for the joint active and passive beamforming and the received signal optimization in the ISAC system assisted by the dual IRSs according to claim 1, wherein the SNR lower bound in the S21 is defined as: configuring the multi-antenna BS to process a received sensing signal y.sub.r,t as: $u^H{y}_{r,t}={}_t{u}^H\left(S{S}^H.Math.H_t\right)w+u^H{n}_r,$ therefore, an SNR of a t.sup.th sensing target is calculated as: ${}_{r,t}=\frac{{}_t^{2}E\left\{{.Math.u^H\left(S{S}^H.Math.H_t\right)w.Math.}^2\right\}}{{}_r^2{u}^{H}u};$ for simplifying a formula expression, redefining E{SS.sup.H}=I.sub.K+M; utilizing a Jensen's inequality, i.e., E{(x)}(E{x}), a resulting SNR lower bound is obtained as: $r_{r,t}\frac{{}_t^2{.Math.u^H\left(I_{K+M}.Math.H_t\right)w.Math.}^2}{{}_r^2{u}^{H}u}.$

6. The method for the joint active and passive beamforming and the received signal optimization in the ISAC system assisted by the dual IRSs according to claim 1, wherein the optimization problem in the S21 is defined as: multi-antenna BS received signal processing configuration u and a dual IRSs passive beamforming .sub.1 as well as .sub.2 are optimized to maximize the communication sum-rate of a multi-user, while simultaneously satisfying a worst-case sensing SNR .sub.t, a transmit power budget P and a unit modulus constraint of reflection coefficients; therefore, the optimization problem is formulated as: $\underset{,W}{\max }{R}_{sum}\left(,W\right)=\underset{k=1}{\overset{K}{.Math.}}{\log }_2\left(1+{}_k\right)$ $s.t.\begin{array}{c}{C}_1:{}_{r,t}{}_t,t\\ C_2:{.Math.W.Math.}_F^{2}P\\ C_3:.Math.{}_{i,n}.Math.=1,i\left\{1,2\right\},nN\end{array}.$

7. The method for the joint active and passive beamforming and the received signal optimization in the ISAC system assisted by the dual IRSs according to claim 6, wherein the optimization problem is converted and optimized through the S22 to the S24, specifically comprising: firstly, introducing the auxiliary variable =[.sub.1, .sub.2, . . . , .sub.K].sup.T via Lagrangian duality transformation, and converting an objective function to the following form: $\underset{k=1}{\overset{K}{.Math.}}{\log }_2\left(1+{}_k\right)-\underset{k=1}{\overset{K}{.Math.}}{}_k+\underset{k=1}{\overset{K}{.Math.}}\frac{\left(1+{}_k\right){.Math.h_k^H{v}_k.Math.}^2}{{.Math.}_{j=1}^{K+M}{.Math.h_k^H{w}_j.Math.}^2+{}_k^2},$ by introducing the auxiliary variable =[.sub.1, .sub.2, . . . , .sub.K].sup.T, expanding the objective function to a quadratic form: $f\left(w,,,\right)=\underset{k=1}{\overset{K}{.Math.}}{\log }_2\left(1+{}_k\right)-\underset{k=1}{\overset{K}{.Math.}}{}_k-\underset{k=1}{\overset{K}{.Math.}}{.Math.{}_k.Math.}^2{}_k^2+\underset{k=1}{\overset{K}{.Math.}}2\sqrt{1+{}_k}\mathrm{.Math.}\left\{{}_k^{*}{h}_k^T{w}_k\right\}-\underset{k=1}{\overset{K}{.Math.}}{.Math.{}_k.Math.}^2\underset{j=1}{\overset{K+M}{.Math.}}{.Math.h_k^H{w}_j.Math.}^2,$ then a new objective function becomes (w,,,), which is converted into a more concise form through equivalent transformation: $f\left(w,,,\right)=\mathrm{.Math.}\left\{v^{H}w\right\}-{.Math.\mathrm{Bw}.Math.}^2+{}_1=\mathrm{.Math.}\left\{g^H\right\}-{}^H+{}_2;$ in an above-mentioned formula, $wvec\left\{W\right\}={\left[w_1^T,w_2^T,.Math.,w_{K+M}^T\right]}^T$ is defined, wherein w.sup.M(K+M)1 and w.sub.j.sup.M1, and extraction of w.sub.j from w is achieved by defining a permutation matrix Q.sub.j.sup.MM(K+M); is utilized to represent .sub.1 or .sub.2 for simplifying an expression; an equivalent expression of (w,,,) is obtained by applying the formula $h_k^H{w}_j=H_k^H{w}_j+F_{1,k}^H\mathrm{diag}\left\{G_1{w}_j\right\}{}_1^H+F_{2,k}^H\mathrm{diag}\left\{G_2{w}_j\right\}{}_2^H;$ remaining variables other than w are defined as follows: $v={\left[2\sqrt{1+{}_k}{}_k^H{h}_k^H,.Math.,2\sqrt{1+{}_K}{}_K^H{h}_K^H,0^H\right]}^H$ $B={\left[b_{1,1},b_{1,2},.Math.,b_{1,K+M},.Math.,b_{K,1},b_{K,2},.Math.,b_{K,K+M}\right]}^T,b_{k,j}.Math.{}_k.Math.Q_j^H{h}_k$ ${}_1=\underset{k=1}{\overset{K}{.Math.}}{\log }_2\left(1+{}_k\right)-\underset{k=1}{\overset{K}{.Math.}}{}_k-\underset{k=1}{\overset{K}{.Math.}}{.Math.{}_k.Math.}^2{}_k^2;$ similarly, remaining variables other than are defined as: $g2\underset{k=1}{\overset{K}{.Math.}}\sqrt{1+{}_k}{}_k\left(\mathrm{diag}\left\{w_k^H{G}_1^H\right\}{F}_{1,k}+\mathrm{diag}\left\{w_k^H{G}_2^H\right\}{F}_{2,k}\right)-2\underset{k=1}{\overset{K}{.Math.}}{.Math.{}_k.Math.}^2\underset{j=1}{\overset{K+M}{.Math.}}\left(\mathrm{diag}\left\{w_j^H{G}_1^H\right\}{F}_{1,k}{H}_k^H{w}_j+\mathrm{diag}\left\{w_j^H{G}_2^H\right\}{F}_{2,k}{H}_k^H{w}_j\right)$ $A\underset{k=1}{\overset{K}{.Math.}}{.Math.{}_k.Math.}^2\underset{j=1}{\overset{K+M}{.Math.}}\left(\mathrm{diag}\left\{w_j^H{G}_1^H\right\}{F}_{1,k}{F}_{1,k}^H\mathrm{diag}\left\{G_1{w}_j\right\}+\mathrm{diag}\left\{w_j^H{G}_2^H\right\}{F}_{2,k}{F}_{2,k}^H\mathrm{diag}\left\{G_2{w}_j\right\}\right)$ ${}_2{}_1+\underset{k=1}{\overset{K}{.Math.}}\left[2\sqrt{1+{}_k}\mathrm{.Math.}\left\{{}_k^H{H}_k^H{w}_k\right\}-{.Math.{}_k.Math.}^2\underset{j=1}{\overset{K+M}{.Math.}}{.Math.H_k^H{w}_j.Math.}^2\right].$

8. The method for the joint active and passive beamforming and the received signal optimization in the ISAC system assisted by the dual IRSs according to claim 6, wherein an alternating optimization method is adopted in the S24 to the S27 in order to iteratively solve respective optimization variables, specifically comprising: before optimizing a configuration u of the multi-antenna BS for received signals, both the auxiliary variables as well as are optimized first, and under given conditions , u, w, .sub.1 and .sub.2, optimization of the auxiliary variable is an unconstrained convex problem, and by calculating a partial derivative /.sub.k=0, an optimal solution for an auxiliary variable ${}_k^{\mathrm{opt}}$ is obtained as: ${}_k^{\mathrm{opt}}={}_k^{*}=\frac{{.Math.h_k^H{w}_k.Math.}^2}{{.Math.}_{jk}^{K+M}{.Math.h_k^H{w}_j.Math.}^2+{}_k^2},k;$ similarly, given ${}_k^{\mathrm{opt}},$ u, w, .sub.1 and .sub.2, by setting /.sub.k=0, an optimal solution ${}_k^{\mathrm{opt}}$ is obtained as: ${}_k^{\mathrm{opt}}=\sqrt{1+{}_k^{\mathrm{opt}}}{\left(\underset{j=1}{\overset{K+M}{.Math.}}{.Math.h_k^H{w}_j.Math.}^2+{}_k^2\right)}^{-1}{h}_k^H{w}_k,k;$ next, fixing other variables, a multi-antenna BS configuration u is then optimized specifically as follows: firstly, a maximization problem is defined as: $\underset{u}{\max }\frac{{}_t^2{.Math.u^H\left(I_{K+M}.Math.H_t\right)w.Math.}^2}{{}_r^2{u}^{H}u},t,$ and an optimal solution u.sup.opt is derived through knowledge of the Rayleigh quotient, and a result is: $u^{\mathrm{opt}}=\frac{\left(I_{K+M}.Math.H_t\right)w}{w^H\left(I_{K+M}.Math.H_t^H{H}_t\right)w},t;$ under the given conditions , , u, .sub.1 as well as .sub.2, optimization of transmit beamforming w is represented as: $\begin{array}{c}\underset{w}{\min }{.Math.\mathrm{Bw}.Math.}^2-\left\{v^{H}w\right\}\\ s.t.C_1:{}_{r,t}{}_t,t\\ C_2:{.Math.w.Math.}^{2}P\end{array},$ a sensing constraint C.sub.1 regarding an objective function is a non-convex function, in order to handle this non-convex constraint, C.sub.1 is represented as a form of second-order cone constraint (SOCP), which is formulated as: $\left\{u^H\left(I_{K+M}.Math.H_t\right)w\right\}\sqrt{{}_r^2{u}^{H}u{}_t/{}_t^2},$ the optimization problem is redefined as: $\begin{array}{c}\underset{w}{\min }{.Math.\mathrm{Bw}.Math.}^2-\left\{v^{H}w\right\}\\ s.t.C_1:\left\{u^H\left(I_{K+M}.Math.H_t\right)w\right\}\sqrt{{}_r^2{u}^{H}u{}_t/{}_r^2},t,\\ C_2:{.Math.w.Math.}^{2}P\end{array}$ then the optimization problem is redefined as a simple convex problem, which can be solved by the CVX toolbox; under the given conditions , , u, w and .sub.2, an optimization problem regarding a reflection coefficient .sub.1 is expressed as: $\begin{array}{c}\underset{{}_1}{\min }{}_1^H{}_1-\left\{g^H{}_1\right\}\\ s.t.C_1:\left\{u^H\left(I_{K+M}.Math.H_t\right)w\right\}\sqrt{{}_r^2{u}^{H}u{}_t/{}_t^2},t;\\ C_2:.Math.{}_{1n}.Math.=1,n\end{array}$ because both a radar constraint and a non-convex unit modulus constraint involve implicit functions of .sub.1, which cannot be solved directly, firstly, the non-convex constraint C.sub.1 is handled by rewriting an expression on a left-hand side of C.sub.1 concerning .sub.1, and then finding a surrogate function of C.sub.1 through a successive convex approximation (SCA) method, specifically comprising: firstly, an expression H.sub.t is expanded as: $H_t=G_1^H{}_1{z}_{1,t}{z}_{1,t}^H{}_1^H{G}_1+G_1^H{}_1{z}_{1,t}{z}_{2,t}^H{}_2^H{G}_2+G_2^H{}_2{z}_{2,t}{z}_{1,t}^H{}_1^H{G}_1+G_2^H{}_2{z}_{2,t}{z}_{2,t}^H{}_2^H{G}_2,$ by utilizing equivalent transformation .sub.1z.sub.1,t=diag{z.sub.1,t}.sub.1, .sub.1z.sub.2,t=diag{z.sub.2,t}.sub.1, .sub.2z.sub.2,t=diag{z.sub.2,t}.sub.2, .sub.2z.sub.1,t=diag{z.sub.1,t}.sub.2 and vectorized sandwich formula extraction vec(ABC)=(C.sup.T.Math.A)vec{B}, (I.sub.K+M.Math.H.sub.t)w is reformulated as: $\begin{array}{c}\left(I_{K+M}.Math.H_t\right)w=\underset{i=1}{\overset{2}{.Math.}}\underset{j=1}{\overset{2}{.Math.}}\left(I_{K+M}{G}_i^H\mathrm{diag}\left\{z_{i,t}\right\}{}_i{}_j^H\mathrm{diag}\left\{z_{j,t}^H\right\}{G}_j\right)w\\ =\underset{i=1}{\overset{2}{.Math.}}\underset{j=1}{\overset{2}{.Math.}}\mathrm{vec}\left\{G_i^H\mathrm{diag}\left\{z_{i,t}\right\}{}_i{}_j^H\mathrm{diag}\left\{z_{j,t}^H\right\}{G}_{j}W\right\}\\ =\underset{i=1}{\overset{2}{.Math.}}\underset{j=1}{\overset{2}{.Math.}}\left(W^T{G}_j^T\mathrm{diag}\left\{z_{j,t}^{*}\right\}.Math.G_i^H\mathrm{diag}\left\{z_{i,t}\right\}\right)\mathrm{vec}\left\{{}_i{}_j^H\right\}\end{array};$ to simplify the expression, an expression $\begin{array}{c}{L}_{1,t}=W^T{G}_1^T\mathrm{diag}\left\{z_{1,t}^{*}\right\}.Math.G_1^H\mathrm{diag}\left\{z_{1,t}\right\}\mathrm{and}\\ E_{1,t}=W^T{G}_2^T\mathrm{diag}\left\{z_{2,t}^{*}\right\}.Math.G_1^H\mathrm{diag}\left\{z_{1,t}\right\}+W^T{G}_1^T\mathrm{diag}\left\{z_{1,t}^{*}\right\}.Math.G_2^H\mathrm{diag}\left\{z_{2,t}\right\}\end{array}$ are defined, given .sub.2 is fixed, and E.sub.1,t.sup.M(K+M)N, a constraint then turns into an expression as $\{u^H{E}_{1,t}{}_1+u^H{L}_{1,t}\mathrm{vec}\left\{{}_1{}_1^H\right\}=\left\{u^H{E}_{1,t}{}_1+{}_1^H{L}_{1,t}{}_1\right\}\sqrt{{}_r^2{u}^{H}u{}_t/{}_t^2},$ wherein $L_{1,t}={\left(u^H{L}_{1,t}\right)}^H=L_{1,t}^{H}u,$ L.sub.1,t.sup.NN, since .sub.1.sup.HL.sub.1,t.sub.1 is a convex function of .sub.1, a lower bound is represented by utilizing an SCA algorithm: ${}_1^H{L}_{1,t}{}_1-{}_1^{\left(n\right)H}{L}_{1,t}{}_1^{\left(n\right)}+2\left\{{}_1^H{L}_{1,t}{}_1^{\left(n\right)}\right\},$ next, $q_{t}2L_{1,t}{}_1^{\left(n\right)}+E_{1,t}^{H}u$ is redefined, then the sensing constraint C.sub.1 is reformulated as: $\left\{{}_1^H{q}_t\right\}\sqrt{{}_r^2{u}^{H}u{}_t/{}_t^2}+{}_1^{\left(n\right)H}{L}_{1,t}{}_1^{\left(n\right)}={}_3;$ hereafter, an ADMM algorithm is utilized to solve a constant modulus constraint problem, specifically comprising: firstly, an auxiliary variable [.sub.1, .sub.2, . . . , .sub.N].sup.T is introduced to convert an optimization problem to be solved .sub.1 into: $\begin{array}{cc}\underset{{}_1}{\min }& f_2\left({}_1\right)={}_1^H{}_1-\left\{g^H{}_1\right\}\\ s.t.& C_1:\left\{{}_1^H{q}_t\right\}{}_3,t\\ & C_2:.Math.{}_{1,n}.Math.1,nN\\ & C_3:=\\ & C_4:.Math.{}_n.Math.1,nN\end{array};$ by utilizing the ADMM algorithm, the problem is further converted through an augmented Lagrangian function as: $\begin{array}{cc}\underset{,,}{\min }& {}_1^H{}_1-\left\{g^H{}_1\right\}+\frac{}{2}{.Math.{}_1-+/.Math.}^2\\ s.t.& C_1:\left\{{}_1^H{q}_t\right\}{}_3,t\\ & C_2:.Math.{}_{1,n}.Math.1,n\\ & C_4:.Math.{}_n.Math.=1,n\end{array},$ wherein .sup.N is the dual variable, >0 is a pre-set penalty parameter, and this multi-variable problem is solved by alternatingly updating each variable given the other variables: updating .sub.1: given as well as , an optimization problem regarding .sub.1 is a convex problem, which is solved by various existing efficient algorithms; updating : given .sub.1 and , .sup.opt is obtained through phase alignment as
.sup.opt=e.sup.j(.sup.1.sup.+); updating : given .sub.1 and , the dual variable is updated as $:=+\left({}_1-\right);$ under given conditions , , u, w and .sub.1, an optimization problem regarding a reflection coefficient .sub.2 is similar to .sub.1 optimization.

9. The method for the joint active and passive beamforming and the received signal optimization in the ISAC system assisted by the dual IRSs according to claim 3, wherein the SINR in the S21 is defined as: a scalar n.sub.kCN (0,.sub.k.sup.2) is an additive white gaussian noise (AWGN) at the k.sup.th user, therefore, the SINR of the k.sup.th user is calculated as: ${}_k=\frac{{.Math.h_k^H{w}_k.Math.}^2}{{.Math.}_{jk}^{K+M}{.Math.h_k^H{w}_j.Math.}^2+{}_k^2},$ wherein an effective channel h.sub.k.sup.M is defined as $h_k^H=H_k^H+F_{1,k}^H{}_1^H{G}_1+F_{2,k}^H{}_2^H{G}_2,$ wherein w.sub.j denotes a j.sup.th column of a beamforming matrix W, i.e., W[w.sub.1, . . . , w.sub.K+M]; considering a link between the multi-antenna BS and a t.sup.th target is blocked, an echo signal received through a path assisted by the IRSs is represented as: $y_{r,t}={}_t\left(G_1^H{}_1{z}_{1,t}+G_2^H{}_2{z}_{2,t}\right)\left(z_{1,t}^H,{}_1^H{G}_1+z_2^H,{}_2^H{G}_2\right)\mathrm{Ws}+n_r;$ further considering target detection is performed under far-field conditions, wherein .sub.t represents radar cross section (RCS) of the t.sup.th target, i.e., .sub.t=4P.sub.1/S, in which P.sub.1 denotes a radiated power density of a target's scattered wave, and S denotes a power density of an incident wave; calculating a sensing target SNR in terms of expectation; for simplifying a formula expression, utilizing .sub.t.sup.2 to represent an expected value .sub.t, i.e., $E\left\{{.Math.{}_t.Math.}^2\right\}={}_t^2,$ wherein z.sub.1,t.sup.N and z.sub.2,t.sup.N denote baseband channels between IRS1, IRS2 and the t.sup.th target, respectively, and a vector $n_r\mathrm{CN}\left(0,{}_r^2{I}_M\right)$ represents the AWGN; assuming the path between the IRSs and a target is LoS, and required angles of arrival/departure (AoA/AoD) are known, an equivalent channel matrix of the echo signal H, is redefined as: $H_t\left(G_1^H{}_1{z}_{1,t}+G_2^H{}_2{z}_{2,t}\right)\left(z_{1,t}^H{}_1^H{G}_1+z_{2,t}^H{}_2^H{G}_2\right);$ defining wvec{W} as vectorizing a matrix W, and .Math. as a Kronecker product, then y.sub.r,t is re-expressed as: $y_{r,t}={}_t\left({\mathrm{SS}}^H.Math.H_t\right)w+n_r.$

10. The method for the joint active and passive beamforming and the received signal optimization in the ISAC system assisted by the dual IRSs according to claim 3, wherein the SNR lower bound in the S21 is defined as: configuring the multi-antenna BS to process a received sensing signal y.sub.r,t as: $u^H{y}_{r,t}={}_t{u}^H\left({\mathrm{SS}}^H.Math.H_t\right)w+u^H{n}_r,$ therefore, an SNR of a t.sup.th sensing target is calculated as: ${}_{r,t}=\frac{{}_t^{2}E\left\{{.Math.u^H\left({\mathrm{SS}}^H.Math.H_t\right)w.Math.}^2\right\}}{{}_r^2{u}^{H}u};$ for simplifying a formula expression, redefining E{SS.sup.H}I.sub.K+M; utilizing a Jensen's inequality, i.e., E{(x)}(E{x}), a resulting SNR lower bound is obtained as: ${}_{r,t}\frac{{}_t^2{.Math.u^H\left(I_{K+M}.Math.H_t\right)w.Math.}^2}{{}_r^2{u}^{H}u}.$

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0073] FIG. 1 is a diagram illustrating the actual scenario of the ISAC system assisted by dual IRSs.

[0074] FIG. 2 is a diagram illustrating the simulation scenario of the ISAC system assisted by dual IRSs.

[0075] FIG. 3 is a curve graph illustrating system sum-rate versus the number of iterations.

[0076] FIG. 4 is a curve graph illustrating the system sum-rate versus maximum transmit power P of the Base Station (BS).

[0077] FIG. 5 is a curve graph illustrating the system sum-rate versus the number of IRS elements N.

[0078] FIG. 6 is a curve graph illustrating the system sum-rate versus the sensing signal SNR requirement.

[0079] FIG. 7 is a parameter setting table for the method for the joint active and passive beamforming and received signal optimization in an ISAC system assisted by dual IRSs.

DETAILED DESCRIPTION OF THE EMBODIMENTS

[0080] The following provides the further detailed description of the present invention in conjunction with the drawings and embodiments.

[0081] The present invention discloses method for joint active and passive beamforming and received signal optimization in an ISAC system assisted by dual IRSs, including: [0082] S1, model establishing: [0083] establishing the ISAC system model assisted by dual IRSs, wherein a multi-antenna Base Station (BS) simultaneously provides communication services for a plurality of users and performs sensing for a target, utilizing dual IRSs to assist multi-antenna Base Station (BS) in operation, and establishing reliable channels between multi-antenna Base Station (BS) and communication user and sensing targets by deploying dual IRSs dispersedly on a surface of a building; [0084] S2, alternating optimization: [0085] S21: deriving a SINR for communication user and a SNR lower bound for sensing targets based on reliable channels for communication as well as sensing, and formulating an optimization problem based on SINR and SNR lower bound, the optimization problem jointly designing active beamforming w at Base Station (BS) end, received signal processing u for sensing signal at multi-antenna Base Station (BS), and passive beamforming .sub.1,.sub.2 at the dual IRSs, under the objective of maximizing the communication sum-rate of communication user, while ensuring that sensing SNR satisfies a minimum requirement, and satisfying a power constraint at Base Station (BS) end and a constant modulus constraint at the IRSs; [0086] S22: introducing auxiliary variables as well as , and decoupling optimization problem using a fractional programming method; [0087] S23: updating auxiliary variables as well as by obtaining partial derivatives, and updating received signal processing u using Rayleigh quotient under the given premise conditions w,.sub.1,.sub.2,,; [0088] S24: representing the original optimization problem as SOCP problem under given conditions u,,,.sub.1,.sub.2, and solving and updating the active beamforming w using a CVX toolbox; [0089] S25: introducing auxiliary variable as well as dual variable , and successively updating the first reflection coefficient vectors .sub.1,, using ADMM under given conditions u,,,w,.sub.2, and updating a second reflection coefficient vector .sub.2 using the same method; [0090] S26: repeating S25 until the values of .sub.1,.sub.2 converges; [0091] S27: repeating S23, S24, S25, and S26 until algorithm converges, thereby obtaining optimal solutions w.sup.opt,u.sup.opt,

[00052]${}_1^{\mathrm{opt}},{}_2^{\mathrm{opt}}.$

[0092] The model establishing is specifically as follows:

[0093] As shown in FIG. 1, the present invention establishes the ISAC system model assisted by dual IRSs, wherein a multi-antenna Base Station (BS) executes sensing target detection while providing communication services for a plurality of users. Considering that obstacles may exist between the Base Station (BS) and the sensing target; the IRS is utilized to construct a reliable link between the Base Station (BS) and the sensing target to achieve the sensing function.

[0094] In ISAC system model assisted by dual IRSs, a multi-antenna Base Station (BS) simultaneously executes multi-user communication and target detection with assistance of two N-element IRSs, specifically as follows: multi-antenna Base Station (BS) end is equipped with M transmit antennas as well as M receive antennas, arranged in the ULA with half-wavelength interval. While transmitting data to K single-antenna users, the Base Station (BS) detects T sensing targets. The transmit signal of the multi-antenna Base Station (BS) is represented as: text use z,

[00053]$x=\mathrm{Ws}=W_c{s}_c+W_r{s}_r,$

wherein

[0095] W.sub.c.sup.MK and W.sub.r.sup.MM are, respectively, the communication beamforming matrix as well as sensing beamforming matrix. Vector s.sub.c.sup.K represents communication signals satisfying

[00054]$E\left\{s_c{s}_c^H\right\}=I_K,$

and vector s.sub.r.sup.M represents sensing signals satisfying

[00055]$E\left\{s_r{s}_r^H\right\}=I_M.$

Assuming that they are statistically independent of each other and satisfying

[00056]$E\left\{s_c{s}_r^H\right\}=0.$

For simplicity, beamforming matrix is defined as W[W.sub.c W.sub.r].sup.M(K+M), and the transmit symbol vector is defined as

[00057]$s{\left[s_c^T{s}_r^T\right]}^T{}^{K+M}.$

[0096] In a practical scenario, considering the ISAC system is in a crowded environment, channel between the Base Station (BS) and the IRS is modelled as a Rician fading channel, represented as follows:

[00058]$G=\sqrt{\frac{{}_R}{1+{}_R}}{G}_{\mathrm{LoS}}+\sqrt{\frac{1}{1+{}_R}}{G}_{\mathrm{NLoS}},$ wherein [0097] .sub.R is Rician factor of BS-IRS link. When infinity is approached, it degenerates to a LoS scenario. When zero is approached, it becomes a Rayleigh channel. G.sub.NLoS.sup.NM is the Rayleigh fading component, with each of terms satisfying the CN (0,1) distribution. G.sub.LoS.sup.NM is LoS channel component. Because the Base Station (BS) and the IRS are modelled as ULA and UPA, respectively, the channel matrix G.sub.LoS can be represented as:

[00059]$G_{\mathrm{LoS}}=\sqrt{}{e}^j{a}_R\left({}_R\right)a_T^H\left({}_T\right),$ wherein [0098] is the large-scale channel gain, is a random phase uniformly distributed in [0, 2], a.sub.T is transmit steering vector of Base Station (BS), and a.sub.R is receive steering vector of IRSs. Therefore, steering vector is represented as a()=[e.sup.j2d sin()/, . . . , e.sup.j2d(M1)sin()/].sup.T wherein d is the distance between the array elements, typically set as half-wavelength, and 2 is the signal wavelength.

[0099] In the communication system model of the present invention, the channel between the multi-antenna Base Station (BS) and each user consists of two parts: the BS-user direct link and the BS-IRSs-user cascade link. Therefore, the received signal from the multi-antenna Base Station (BS) to the k.sup.th user can be expressed as:

[00060]$y_k=\left(H_k^H+F_{1,k}^H{}_1^H{G}_1+F_{2,k}^H{}_2^H{G}_2\right)x+n_k,$ wherein [0100] effective channels between the Base Station (BS) and the k.sup.th user, between IRS1 and the k.sup.th user, between IRS2 and the k.sup.th user, between the Base Station (BS) and IRS1, and between the Base Station (BS) and IRS2 are respectively defined by H.sub.k.sup.M, F.sub.1,k.sup.N, F.sub.2,k.sup.N, G.sub.1.sup.NM, G.sub.2.sup.NM IRS reflection matrix is defined as .sub.idiag{.sub.i}, wherein .sub.i=[.sub.i1, . . . , .sub.iN].sup.T is reflection coefficient vector satisfying |.sub.in|=1, i{1,2}, n. Scalar

[00061]$n_k\mathrm{CN}\left(0,{}_k^2\right)$ is AWGN at the k.sup.th user. Therefore, the SINR of the k.sup.th user can be calculated as:

[00062]${}_k=\frac{{.Math.h_k^H{w}_k.Math.}^2}{{.Math.}_{jk}^{K+M}{.Math.h_k^H{w}_j.Math.}^2+{}_k^2}.$

[0101] For simplicity, h.sub.k.sup.M is defined as

[00063]$h_k^H=H_k^H+F_{1,k}^H{}_1^H{G}_1+F_{2,k}^H{}_2^H{G}_2,$

where w.sub.j denotes the j.sup.th column of beamforming matrix W, i.e., W[w.sub.1, . . . , w.sub.K+M].

[0102] In perception model of present invention, considering link between multi-antenna Base Station (BS) and t.sup.th target is blocked, an echo signal received through a path assisted by the IRSs is represented as:

[00064]$y_{r,t}={}_t\left(G_1^H{}_1{z}_{1,t}+G_2^H{}_2{z}_{2,t}\right)\left(z_{1,t}^H{}_1^H{G}_1+z_{2,t}^H{}_2^H{G}_2\right)\mathrm{Ws}+n_r;$ [0103] further considering target detection is performed under far-field conditions, wherein .sub.t represents RCS of t.sup.th target, i.e., .sub.t=4P/S, in which P denotes radiated power density of target's scattered wave, and S denotes power density of incident wave; calculating the sensing target SNR in terms of expectation; for simplifying formula expression, utilizing .sub.t.sup.2 to represent the expected value .sub.t, i.e.,

[00065]$E\left\{{.Math.{}_t.Math.}^2\right\}={}_t^2,$ where z.sub.1,t.sup.N and z.sub.2t.sup.N denote the baseband channels between IRS1, IRS2 and t.sup.th target, respectively, and vector

[00066]$n_r\mathrm{CN}\left(0,{}_r^2{I}_M\right)$

represents AWGN; assuming the path between the IRSs and the target is LoS, and required AoA/AoD are known, the equivalent channel matrix of the echo signal H.sub.t is redefined as:

[00067]$H_t\left(G_1^H{}_1{z}_{1,t}+G_2^H{}_2{z}_{2,t}\right)\left(z_{1,t}^H{}_1^H{G}_1+z_{2,t}^H{}_2^H{G}_2\right);$ [0104] defining wvec{W} as vectorizing matrix W, and .Math. as Kronecker product, then y.sub.r,t is re-expressed as:

[00068]$y_{r,t}={}_t\left(S{S}^H.Math.H_t\right)w+n_r.$

[0105] At the same time, the present invention considers optimization of multi-antenna Base Station (BS) reception of the sensing signal. Since there are clutter signals in sensing signal received by multi-antenna Base Station (BS), in order to obtain satisfactory target detection performance, the present invention further optimizes multi-antenna Base Station (BS) configuration u.sup.M(K+M) to process the received sensing signal Y.sub.r,t, namely:

[00069]$u^H{y}_{r,t}={}_t{u}^H\left(S{S}^H.Math.H_t\right)w+u^H{n}_r,$ [0106] therefore, the SNR of the t.sup.th sensing target is calculated as:

[00070]${}_{r,t}=\frac{{}_t^{2}E\left\{{.Math.u^H\left(S{S}^H.Math.H_t\right)w.Math.}^2\right\}}{{}_r^2{u}^{H}u};$ [0107] for simplifying formula expression, redefining E{SS.sup.H}=I.sub.K+M; utilizing Jensen's inequality, i.e., E{(x)}(E{x}), the resulting SNR lower bound is obtained as:

[00071]${}_{r,t}\frac{{}_t^2{.Math.u^H\left(I_{K+M}.Math.H_t\right)w.Math.}^2}{{}_r^2{u}^{H}u}.$

[0108] The alternating optimization is as follows:

[0109] The goal of the present invention is to jointly optimize the active beamforming w at multi-antenna Base Station (BS), the reception configuration of the sensing signal u at the multi-antenna Base Station (BS), and the passive beamforming .sub.1 and .sub.2 at dual IRSs to maximize the communication sum-rate of multi-user, while simultaneously satisfying the worst-case sensing SNR .sub.t, the transmit power budget P and unit modulus constraint of reflection coefficients; therefore, the optimization problem is formulated as:

[00072]$\underset{,W}{\max }{R}_{sum}\left(,W\right)=\underset{k=1}{\overset{K}{.Math.}}{\log }_2\left(1+{}_k\right)$$s.t.C_1:{}_{r,t}{}_t,t$$C_2:{.Math.W.Math.}_F^{2}P$$C_3:.Math.{}_{i,n}.Math.=1,i\left\{1,2\right\},nN.$

[0110] To address the above non-convex problem, a joint optimization design of active and passive beamforming of the ISAC system assisted by dual IRSs and the reception of sensing signals by multi-antenna Base Station (BS) is considered, maximizing the system communication performance while ensuring that the sensing signals meet minimum SNR requirements. However, due to coupled variables between the objective function and the sensing signal SNR in the constraint C.sub.1, and the unit modulus constraint in the constraint C.sub.3 imposed by IRSs, the entire optimization problem is coupled and non-convex, making it difficult to solve. The present invention discloses using the fractional programming (FP), the SCA, and the ADMM to transform the problem into multiple tractable subproblems, which are then solved iteratively using an alternating optimization approach.

[0111] Due to the presence of logarithmic and fractional terms, the objective function is complex. It can be simplified using closed-form FP approach. Firstly, introducing auxiliary variable =[.sub.1, .sub.2, . . . , .sub.K].sup.T through the Lagrangian duality transformation, converting the objective function to the following form:

[00073]$\underset{k=1}{\overset{K}{.Math.}}{\log }_2\left(1+{}_k\right)-\underset{k=1}{\overset{K}{.Math.}}{}_k+\underset{k=1}{\overset{K}{.Math.}}\frac{\left(1+{}_k\right){.Math.h_k^H{w}_k.Math.}^2}{{.Math.}_{j=1}^{K+M}{.Math.h_k^H{w}_j.Math.}^2+{}_k^2},$ [0112] by introducing the auxiliary variable =[.sub.1,.sub.2, . . . , .sub.K].sup.T, expanding the objective function to a quadratic form:

[00074]$f\left(w,,,\right)=\underset{k=1}{\overset{K}{.Math.}}{\log }_2\left(1+{}_k\right)-\underset{k=1}{\overset{K}{.Math.}}{}_k-\underset{k=1}{\overset{K}{.Math.}}{.Math.{}_k.Math.}^2{}_k^2+\underset{k=1}{\overset{K}{.Math.}}2\sqrt{1+{}_k}\left\{{}_k^{*}{h}_k^T{w}_k\right\}-\underset{k=1}{\overset{K}{.Math.}}{.Math.{}_k.Math.}^2\underset{j=1}{\overset{K+M}{.Math.}}{.Math.h_k^H{w}_j.Math.}^2,$ [0113] then new objective function becomes (w,,,), which is converted into a more concise form through equivalent transformation:

[00075]$f\left(w,,p,\right)=\left\{v^{H}w\right\}-{.Math.\mathrm{Bw}.Math.}^2+{}_1=\left\{g^H\right\}-{}^H+{}_2;$ [0114] in above-mentioned formula,

[00076]$w\mathrm{vec}\left\{W\right\}={\left[w_1^T,w_2^T,.Math.,w_{K+M}^T\right]}^T$ is defined, where w.sup.M(K+M)1 and w.sub.j.sup.M1, and the extraction of w.sub.j from w is achieved by defining a permutation matrix Q.sup.MM(K+M). is utilized to represent .sub.1 or .sub.2 for simplifying the expression; the equivalent expression of (w,,,) is obtained by applying the formula

[00077]$h_k^H{w}_j=H_k^H{w}_j+F_{1,k}^H\mathrm{diag}\left\{G_1{w}_j\right\}{}_1^H+F_{2,k}^H\mathrm{diag}\left\{G_2{w}_j\right\}{}_2^H;$ the remaining variables other than w are defined as follows:

[00078]$v={\left[2\sqrt{1+{}_k}{}_k^H{h}_k^H,.Math.,2\sqrt{1+{}_K}{}_K^H{h}_K^H,0^H\right]}^H$$B={\left[b_{1,1},b_{1,2},.Math.,b_{1,K+M},.Math.,b_{K,1},b_{K,2},.Math.,b_{K,K+M}\right]}^T,b_{k,j}.Math.{}_k.Math.Q_j^H{h}_k$${}_1=\underset{k=1}{\overset{K}{.Math.}}{\log }_2\left(1+{}_k\right)-\underset{k=1}{\overset{K}{.Math.}}{}_k-\underset{k=1}{\overset{K}{.Math.}}{.Math.{}_k.Math.}^2{}_k^2$ [0115] similarly, the remaining variables other than 6 are defined as:

[00079]$g2\underset{k=1}{\overset{K}{.Math.}}\sqrt{1+{}_k}{}_k\left(\mathrm{diag}\left\{w_k^H{G}_1^H\right\}{F}_{1,k}+\mathrm{diag}\left\{w_k^H{G}_2^H\right\}{F}_{2,k}\right)-2\underset{k=1}{\overset{K}{.Math.}}{.Math.{}_k.Math.}^2\underset{j=1}{\overset{K+M}{.Math.}}\left(\mathrm{diag}\left\{w_j^H{G}_1^H\right\}{F}_{1,k}{H}_k^H{w}_j+\mathrm{diag}\left\{w_j^H{G}_2^H\right\}{F}_{2,k}{H}_k^H{w}_j\right)$$\underset{k=1}{\overset{K}{.Math.}}{.Math.{}_k.Math.}^2\underset{j=1}{\overset{K+M}{.Math.}}\left(\mathrm{diag}\left\{w_j^H{G}_1^H\right\}{F}_{1,k}{F}_{1,k}^H\mathrm{diag}\left\{G_1{w}_j\right\}+\mathrm{diag}\left\{w_j^H{G}_2^H\right\}{F}_{2,k}{F}_{2,k}^H\mathrm{diag}\left\{G_2{w}_j\right\}\right)$${}_2{}_1+\underset{k=1}{\overset{K}{.Math.}}\left[2\sqrt{1+{}_k}\left\{{}_k^H{H}_k^H{w}_k\right\}-{.Math.{}_k.Math.}^2\underset{j=1}{\overset{K+M}{.Math.}}{.Math.H_k^H{w}_j.Math.}^2\right]$

[0116] After converting the optimization problem through the above steps, the alternating optimization method is adopted below to iteratively solve respective optimization variables. Before optimizing configuration u of multi-antenna Base Station (BS) for received signals, both auxiliary variables and are optimized first, and under given conditions , u, w, .sub.1 and .sub.2, the optimization of the auxiliary variable is an unconstrained convex problem, and by calculating partial derivative of a /.sub.k=0, the optimal solution for auxiliary variable

[00080]${}_k^{\mathrm{opt}}$

is obtained as:

[00081]${}_k^{\mathrm{opt}}={}_k^{*}=\frac{{.Math.h_k^H{w}_k.Math.}^2}{{.Math.}_{jk}^{K+M}{.Math.h_k^H{w}_j.Math.}^2+{}_k^2},k;$

[0117] similarly, given

[00082]${}_k^{\mathrm{opt}},$

u, w, .sub.1 as well as .sub.2 by setting /.sub.k=0, optimal solution

[00083]${}_k^{\mathrm{opt}}$

is obtained as:

[00084]${}_k^{\mathrm{opt}}=\sqrt{1+{}_k^{\mathrm{opt}}}{\left(\underset{j=1}{\overset{K+M}{.Math.}}{.Math.h_k^H{w}_j.Math.}^2+{}_k^2\right)}^{-1}{h}_k^H{w}_k,k.$

[0118] Next, fixing other variables, the multi-antenna Base Station (BS) configuration u is optimized. However, in the actual optimization process, solving for a suitable value u by fixing other variables leads to the feasibility check problem lacking the clear objective. To accelerate the convergence and create more degrees of freedom in subsequent iterations to maximize the system sum-rate, it is proposed to update u by maximizing SNR lower bound. Firstly, the maximization problem is defined as:

[00085]$\underset{u}{\max }\frac{{}_t^2{.Math.u^H\left(I_{K+M}.Math.H_t\right)w.Math.}^2}{{}_r^2{u}^{H}u},t,$ [0119] and optimal solution u.sup.opt is derived through knowledge of the Rayleigh quotient, and the result is:

[00086]$u^{\mathrm{opt}}=\frac{\left(I_{K+M}.Math.H_t\right)w}{w^H\left(I_{K+M}.Math.H_t^H{H}_t\right)w},t;$ [0120] under given conditions , , u, .sub.1 as well as .sub.2, the optimization of transmit beamforming w is represented as:

[00087]$\begin{array}{c}\underset{w}{\min }{.Math.\mathrm{Bw}.Math.}^2-\left\{v^{H}w\right\}\\ s.t.C_1:{}_{r,t}{}_t,t\\ C_2:{.Math.w.Math.}^{2}P\end{array},$ wherein [0121] a sensing constraint C.sub.1 regarding the objective function is a non-convex function, in order to handle this non-convex constraint, C.sub.1 is represented as form of the second-order cone constraint, which is formulated as:

[00088]$\left\{u^H\left(I_{K+M}.Math.H_t\right)w\right\}\sqrt{{}_r^2{u}^{H}u{}_t/{}_t^2},$ [0122] the optimization problem is redefined as:

[00089]$\underset{w}{\min }{.Math.\mathrm{Bw}.Math.}^2-\left\{v^{H}w\right\}$$s.t.C_1:\left\{u^H\left(I_{K+M}.Math.H_t\right)w\right\}\sqrt{{}_r^2{u}^{H}u{}_t/{}_t^2},t$$C_2:{.Math.w.Math.}^{2}P,$ [0123] then the optimization problem is redefined as a simple convex problem, which can be solved by the CVX toolbox; [0124] under given conditions , , u, w and .sub.2, optimization problem regarding the reflection coefficient .sub.1 is expressed as:

[00090]$\underset{{}_1}{\min }{}_1^H{}_1-\left\{g^H{}_1\right\}$$s.t.C_1:\left\{u^H\left(I_{K+M}.Math.H_t\right)w\right\}\sqrt{{}_r^2{u}^{H}u{}_t/{}_t^2},t$$C_2:.Math.{}_{1n}.Math.=1,n;$ [0125] because both the radar constraint as well as the non-convex unit modulus constraint involve the implicit functions of .sub.1, which cannot be solved directly, firstly, the non-convex constraint C.sub.1 is handled by rewriting the expression on left-hand side of C.sub.1 concerning .sub.1, and then finding surrogate function of C.sub.1 through SCA method. Firstly, the expression H.sub.t is expanded as:

[00091]$H_t=G_1^H{}_1{z}_{1,t}{z}_{1,t}^H{}_1^H{G}_1+G_1^H{}_1{z}_{1,t}{z}_{2,t}^H{}_2^H{G}_2+G_2^H{}_2{z}_{1,t}^H{}_1^H{G}_1+G_2^H{}_2{z}_{2,t}{z}_{2,t}^H{}_2^H{G}_2,$ by utilizing the equivalent transformation .sub.1z.sub.1,t=diag{z.sub.1,t}.sub.1, .sub.1z.sub.2,t=diag{z.sub.2,t}.sub.1, .sub.2z.sub.2,t=diag{z.sub.2,t}.sub.2, .sub.2z.sub.1,t=diag{z.sub.1,t}.sub.2 and the vectorized sandwich formula extraction vec(ABC)=(C.sup.T.Math.A)vec{B}, (I.sub.K+M.Math.H.sub.t)w is reformulated as:

[00092]$\begin{array}{c}\left(I_{K+M}.Math.H_t\right)w=\underset{i=1}{\overset{2}{.Math.}}\underset{j=1}{\overset{2}{.Math.}}\left(I_{K+M}.Math.G_i^H\mathrm{diag}\left\{z_{i,t}\right\}{}_i{}_j^H\mathrm{diag}\left\{z_{j,t}^H\right\}{G}_j\right)w\\ =\underset{i=1}{\overset{2}{.Math.}}\underset{j=1}{\overset{2}{.Math.}}\mathrm{vec}\left\{G_i^H\mathrm{diag}\left\{z_{i,t}\right\}{}_i{}_j^H\mathrm{diag}\left\{z_{j,t}^H\right\}{G}_{j}W\right\}\\ =\underset{i=1}{\overset{2}{.Math.}}\underset{j=1}{\overset{2}{.Math.}}\left(W^T{G}_i^H\mathrm{diag}\left\{z_{j,t}^{}\right\}.Math.G_i^H\mathrm{diag}\left\{z_{i,t}\right\}\right)vec\left\{{}_i{}_j^H\right\}\end{array};$ [0126] to simplify the expression, the expression and