Abstract
Intelligent reflecting surface (IRS) is a key enabling technology to reshape the electromagnetic propagation environment and enhance the communication performance. Current single IRSaided or multiple distributed IRSsaided wireless communication systems leave interIRSs collaboration out of consideration, and as a result, the system performance may be severely restricted. For cooperative double IRSsaided wireless communication systems, dyadic backscatter channel model is widely used in the performance analysis and optimization. However, the impact of factors such as the size and gain of IRS elements is omitted. As a result, the performance quantification and evaluation are inaccurate. In order to avoid the above limitations, spatial scattering channel model is leveraged to quantify the path loss of the double reflection link in typical application scenarios of double IRSsaided wireless communication systems. When the nearfield condition is satisfied, the electromagnetic wave signal transmitted between IRSs is a spherical wave, which leads to highrank channel and a lower signal to noise ratio. This paper considers the rank1 interIRSs equivalent channel and derives the closedform received signal power which reveals its relationship with the deployment of IRSs and the physical and electromagnetic properties of IRSs. Taking the impact of near/farfield effects of IRS on signal propagation further into consideration, the network configurations under which double cooperative IRSs can enhance the system performance are recognized. Simulation results show that whether double IRSs should be selected to assist in the communication between the transmitter and the receiver depends on practical network configurations, and the same number of elements should be assigned to the two IRSs to maximize the system performance if they are adopted.
Introduction
As a key enabling technology for the sixth generation (6G) wireless communication systems, intelligent reflecting surface (IRS) is capable of reshaping the electromagnetic propagation environment and dramatically enhancing the communication performance by smartly tuning the amplitude and/or the phase shift of the incident electromagnetic wave via a large number of lowcost elements integrated on it^{1,2}. Different from conventional active relays which require bulky transmit radio frequency chains, IRS acts as a passive array and directly reflects the incident signal, and its low power property conforms to the development tendency of future communication technologies^{3}.
However, most of the existing research focuses on single IRSaided wireless communication systems and multiple distributed IRSsaided wireless communication systems without considering the impact of interIRSs collaboration, which may lead to inferior system performance. Although IRS can enhance the endtoend communication by creating virtual connections, the equivalent path loss of the cascaded base station (BS)IRSuser link is the product (instead of the sum) of the path losses of the BSIRS and IRSuser sublinks in single IRSaided wireless communication systems. In order to improve the IRS array gain to a reasonable level, a large number of elements are required to compensate for the path loss caused by the multiplicative fading effects^{4}. Therefore, the academia and industry explore to utilize double IRSs to assist the communication between the BS and the user to further enhance the system performance^{5,6}. In this case, the equivalent path loss of the double reflection link (BSIRS1IRS2user) is the product of the path losses of the BSIRS, interIRSs and IRSuser sublinks, and the system performance is also constrained by multiplicative fading effects. Reasonable deployment of double IRSs and taking advantage of the rich scattering environment created by IRSs are helpful in reducing the path loss of double IRSsaided wireless communication systems. However, whether cooperative double IRSsaided wireless communication system is superior to its single counterpart requires indepth investigation.
Accurate channel characterization and modeling is an important basis for the performance analysis and optimization of IRSaided wireless communication systems. In general, there are mainly two techniques to model the cascaded channel via IRS, i.e., dyadic backscatter channel model and spatial scattering channel model^{7}. Dyadic backscatter channel model is widely used in the performance analysis and optimization of IRSaided wireless communication systems, and the impact of IRS on signal propagation is modeled as a diagonal matrix. The linear independence between signals reflected by neighboring IRS elements will lead to inaccurate system performance evaluation^{8}. By taking the direction of arrival (DoA) and angle of arrival (AoA) of the incident signal, the size and gain of IRS elements into account, spatial scattering channel model can help quantify the cascaded path loss of IRSaided wireless communication systems accurately^{9}. However, current spatial scattering channel modelbased path loss modeling developed for single IRSaided wireless communication systems^{9} cannot effectively quantify the path loss of the double reflection link. To the best of the authors’ knowledge, no work has been done to quantify the path loss of double IRSsaided wireless communication systems by applying spatial scattering channel model, even for simple freespace propagation. Therefore, specialized spatial scattering channel modelbased path loss modeling for double IRSsaided wireless communication systems is urgently needed to lay necessary foundations for the performance analysis and optimization of such systems. This motivates us to develop a closedform path loss of the double reflection link based on spatial scattering channel modeling for the typical application scenario of double IRSsaided wireless communication systems shown in Fig. 1, that is, the BS can only communicate with the user through the double reflection link, and other links are severely blocked by obstacles. The research results provide theoretical basis for the performance analysis of double IRSsaided wireless communication systems. The innovations of this paper are summarized as follows:

To avoid the limitations of dyadic backscatter channel model, spatial scattering channel model is firstly leveraged to quantify the cascaded path loss in cooperative double IRSsaided wireless communication system. The relationship between the received power at the user and various system parameters such as the transmit power of the BS, the gains of the transmit antenna and the receiving antenna, the number of IRS elements, the size and gain of each IRS element, the carrier wavelength and the deployment of IRSs is revealed.

Taking the near/farfield effects of IRS further into account, the above cascaded path loss model is adopted to evaluate the performance of cooperative double IRSsaided wireless communication system. The network configurations under which double cooperative IRSs can effectively enhance the system performance are recognized. In addition, the same number of elements should be assigned to the two IRSs to maximize the received signal to noise ratio (SNR) at the user, the channel capacity and minimize bit error rate (BER) if they are adopted. The optimal positions of double IRSs in farfield case are also identified to achieve the maximum received SNR at the user.
Related works
Based on different assumptions and application scenarios, a substantial number of studies leverage different channel models to evaluate the performance gains achieved by single IRS or double IRSsaided wireless communication systems, and the relevant works are briefly summarized in Table 1. Dyadic backscatter channel model is widely used in the performance analysis and optimization of IRSaided wireless communication systems, and its general model is summarized as below:
where H_{r} and H_{t} denote the equivalent channel matrices from the IRS to the receiver and from the transmitter to the IRS, respectively. The impact of IRS on signal propagation is modeled as a diagonal matrix. The linear independence between signals reflected by neighboring IRS elements will lead to inaccurate system performance evaluation of IRSaided wireless communication systems^{8}. In addition, the existing works usually assume that each item in the channel matrix obeys certain statistical distribution, such as the Rician distribution^{10}. In general, the difference from the actual channel distribution is known to have a negative impact on system performance^{7}.
On the basis of dyadic backscatter channel model, alternating optimization (AO) and successive convex approximation (SCA) algorithms are used by^{11} to solve the confidentiality maximization problem with hardware constraints in IRSaided millimeter wave (mmWave) communication system. The received signal at the user is shown in Eq. (2):
where h^{H} denotes the IRSuser channel matrix; \({\boldsymbol{\Theta}}\) denotes the IRS reflection coefficient matrix; G denotes the APIRS channel matrix; F_{RF} denotes the analog beamforming codebook; \({\boldsymbol{\omega}}\) denotes the digital beamforming vector; s denotes the transmit signal; \(\mathcal{Q}\)(⋅) denotes the 1bit quantizer, and n denotes the additive Gaussian white noise. Fractional programming and AO algorithms are used by^{12} to solve the problem of maximizing the energy efficiency of IRSaided multicast communication system, and the k^{th} (k = 1,…, K) received signal of the mobile user is:
where H denotes the channel matrix from the BS to the IRS; \({\boldsymbol{\Phi}}\) is a diagonal matrix which denotes the effective phase shifts adopted by all the reflecting elements on the IRS; \({\mathbf{t}}_{k}^{{\text{H}}}\) denotes the channel vector from the IRS to the k^{th} mobile user; \({\mathbf{g}}_{k}^{{\text{H}}}\) denotes the BS to the user channel vector, and z_{k} denotes the cyclic symmetric complex Gaussian noise with zero mean and unit variance at the k^{th} mobile user. In^{13}, maximizing the achievable rate of the IRSaided mmWave nonorthogonal multiple access (NOMA) system while satisfying the user’s minimum rate and transmit power constraints is divided into three suboptimization problems, i.e., power allocation, joint phase shifts and analog beamforming optimization, and digital beamforming design, then solved by using alternating manifold optimization and SCA. The received signal at the k^{th} user in the n^{th} (n = 1,…, N) group is:
where P denotes the power allocation matrix; W denotes the digital beamforming matrix; F denotes the analog beamforming matrix; h_{n,k} denotes the channel vector from IRS to the k^{th} user in the n^{th} group; μ_{n,k} denotes the noise at the k^{th} user in the n^{th} group. Block coordinate descent algorithm is used by^{14} to optimize the AP transmit beamforming vector and the IRS passive beamforming in IRSaided broadcast network with power splitting to ensure the users’ quality of service and selfsustainability of the IRS. The received signal at the k^{th} user can be expressed as:
where \({\mathbf{h}}_{d,k}^{{\text{H}}}\) denotes the channel vector from AP to the k^{th} user; \({\mathbf{h}}_{r,k}^{{\text{H}}}\) denotes the channel vector from IRS to the k^{th} user; \({{\varvec{\upomega}}}\) denotes the transmit beamforming vector; x denotes the transmitted signal, and n_{k} denotes the Gaussian white noise at the k^{th} user.
Spatial scattering channel model can avoid the limitations of dyadic backscatter channel model and better demonstrate the propagation mechanism through an IRS. To be specific, each IRS element is regarded as a reflector in the environment creating a distinct propagation path. Therefore, the cascaded channel via IRS (with Q elements) is the superposition of all paths, as shown in Eq. (6).
where α_{q} is the channel gain excluding the effects of element q. Γ_{q} is the control effect of element q. \({{\varvec{\upalpha}}}_{{\mathbf{R}}}\) and \({{\varvec{\upalpha}}}_{{\mathbf{T}}}\) are the array steering vectors at the receiver and the transmitter, respectively. θ_{R,q} and φ_{R,q} are the elevation and azimuth angles of element q with respect to the receiver. Similarly, θ_{T,q} and φ_{T,q} are defined for element q with respect to the transmitter. Assuming that an IRS with Q reflecting elements is deployed on the ground plane^{15}, and the IRS is regarded as a specular reflector. Therefore, the total received power from the direct link and cascaded reflection link via IRS is:
where P_{t} is the transmission power; λ denotes the carrier wavelength; d_{1} and d_{2} are the distance from the IRS to the transmitter and the receiver, respectively. From Eq. (7), it can be seen that the received power is inversely proportional to the square of the length of the cascaded reflection link via IRS, i.e., (d_{1} + d_{2})^{2}. However, this conjecture is disproven by^{16} and it might hold for an infinitely large IRS or the nearfield case. Based on physical optics techniques, the following path loss model is proposed in^{16}:
where G_{t} and G_{r} are the gains of the transmit antenna and the receiving antenna, respectively; X × Y is the physical size of IRS; θ_{i}, θ_{s} and θ_{r} are the incident angle from the transmitter to the IRS, the observation angle and the desired reflection angle, respectively. (a) follows when θ_{s} = θ_{r}. Equation (8) indicates that the path loss is positively proportional to (d_{1}d_{2})^{2} instead of (d_{1} + d_{2})^{2}. It also explains why the surface consists of many elements that individually act as diffuse scatters can jointly beamform the signal in a desired direction with a certain beamwidth. The radiation density based on the scattered electric field intensity in the near field of IRS is calculated by^{17}. The path loss is described as a function of the Euclidean distance from the transmit antenna to each element on the IRS, the wave numbers, the element impedance, the input antenna current, and the radiation vector generated by the current. However, the properties of the IRS elements are not involved. Based on the farfield received power model of a metal reflector, the optimal received power of a passive reflectorenhanced nonlineofsight (NLOS) link in the mmWave band is derived by^{18}, as shown in Eq. (9).
where a × b denotes the size of passive reflector; r_{1} and r_{2} denote the Euclidean distances from the transmit antenna to the passive reflector and from the passive reflector to the receiving antenna, respectively. However, since passive reflector is different from the IRS, the receive power model is not applicable to IRSaided wireless communication systems. By studying the physical and electromagnetic properties of IRS, a general path loss model is developed for single IRSaided wireless communication systems, and the received signal power is shown in Eq. (10)^{9}.
where G and d_{x} × d_{y} are the gain and physical size of each IRS element, respectively. N and M are the number of rows and columns of elements which are regularly arranged on IRS. \(F_{n,m}^{combine}\) accounts for the impact of the normalized power radiation patterns on the received signal power. Γ_{n,m} is the reflection coefficient of the IRS element in row n and column m, i.e., U_{n,m}. \(r_{n,m}^{t}\) and \(r_{n,m}^{r}\) are the Euclidean distances from the transmitter and receiver to U_{n,m}, respectively. To extend the application scenarios of IRS, angledependent loss factor is formulated to quantify the impact of antenna’s direction of the transmitter, receiver and IRS elements on the path loss, and the path loss model proposed in^{9} is refined for IRSaided wireless communication systems operating in the mmWave band^{19}. In addition, the relationship between the scattering gain of an IRS element and its physical size is derived, as shown in Eq. (11).
The path loss model proposed by^{20} based on the IRS elements mutual impedance theory in farfield case is not applicable to planar IRS^{19}.
Current research results show that rationally designed double IRSsaided wireless communication systems outperform single IRSaided wireless communication systems. To be specific, double IRSsaided wireless communication systems are considered for the first time in^{21}. Under the assumptions that other links are severely blocked and the interIRSs channel is of rank 1, the passive beamforming design problem is solved. The geometric relationship between the two IRSs is exploited to obtain the power gain of the user, as shown in Eq. (12).
where H is the channel gain of the cascaded reflection link. \(\alpha/d_{r}^{2}\), \(\alpha/d_{s}^{2}\) and \(\alpha/d_{t}^{2}\) represent the approximate path losses between the BS and the elements on IRS 1, between the elements on IRS 1 and IRS 2 and between the elements on IRS 2 and the user, respectively. K_{1} and K_{2} are the number of elements on IRS 1 and IRS 2, respectively. Given the total number of IRS elements K, reasonable element assignment and reflection coefficient matrix design can achieve a power gain of order \(\mathcal{O}\)(K^{4}). However, sufficient IRS elements are required to compensate for the multiplicative fading effects of the cascaded reflection link and guarantee their superior performance. The active beamforming at the BS and passive reflection beamforming at the two IRSs are jointly optimized for double IRSsassisted multiuser multiinput multioutput (MIMO) system to maximize the minimum uplink signaltointerferenceplusnoise ratio of all users^{22}. The channel model shown in Eq. (13) is applied, i.e., apart from the double reflection link, two single reflection links BSIRS 1user and BSIRS 2user are further taken into consideration to enhance the spatial multiplexing gain of double IRSsassisted wireless communication systems.
where H_{q} is the superimposed uplink channel for user q. u_{1,q} and u_{2,q} are the baseband equivalent channels for the user qIRS 1 and user qIRS 2 links, respectively. Φ_{1} and Φ_{2} are the diagonal reflection matrices which model the impact of IRS 1 and IRS 2 on signal propagation, respectively. G_{1} and G_{2} are the baseband equivalent channels for the IRS 1BS and IRS 2BS links, respectively. D is the baseband equivalent channel for the IRS 1IRS 2 link. Based on the same channel model, the impact of array response between the transmit antenna/receiving antenna and IRS is further considered in^{23}. The transmit covariance matrix and the passive beamforming matrices of the two cooperative IRSs are jointly optimized to maximize the channel capacity of double IRSsaided single user MIMO system. By further analyzing the correlation between the array responses of the BSIRS 1, BSIRS 2, IRS 1user and IRS 2user channels, the closedform channel capacity is derived for double IRSsaided single user MIMO system with rank1 and rank2 channels. Simulation results show that double IRSsaided MIMO system can achieve a channel capacity of order \(\mathcal{O}\)(M^{4}) with an asymptotically large M (the total number of IRS elements). The extended SalehValenzuela channel model in Eq. (14) is adopted by^{24}, and with the objective of maximizing the weighted sum rate of downlink transmissions, the digital precoding matrix at the BS and the analog phase shifters at the two IRSs are alternately optimized for double IRSsaided multiuser MIMO system operating in the mmWave band.
where H_{1} is the equivalent channel from the BS to IRS 1. N_{path} denotes the number of physical propagation paths between the BS and IRS 1. α_{q} is the channel gain of path q. a_{t} \(\left( {\psi_{q}^{t} ,\beta_{q}^{t} } \right)\) and a_{r} \(\left( {\psi_{q}^{r} ,\beta_{q}^{r} } \right)\) are the array response vectors of the transmit antenna and IRS associated with path q. \(\psi_{q}^{t}\) and \(\beta_{q}^{t}\) are the azimuth and elevation angles of departure of path q, respectively. \(\psi_{q}^{r}\) and \(\beta_{q}^{r}\) are the azimuth and elevation AoAs of path q, respectively. Based on the same channel model, the transmit beamforming matrix of the BS and the reflection coefficient matrices of the two IRSs are alternately optimized to maximize the weighted sum rate of a multiIRSaided multiuser MIMO system^{25}. However, the above systems leave interIRSs collaboration out of consideration, and each IRS only serves the users in its half reflection space. Offline beam training solution is proposed in^{26}, and the channel model shown in Eq. (15) is utilized to maximize the endtoend channel gain of multiIRSaided wireless networks.
where h_{0,J+1} is the equivalent multihop BSuser channel. Ω represents the multihop reflection path between the BS and the user, and Q is the total number of IRSs on the path. w_{B} is the precoding vector of the BS. \({\varvec{H}}_{{0,a_{1} }}\) is the equivalent channel between the BS and its nexthop IRS. \({{\varvec{\Phi}}}_{{a_{q} }}\) is the reflection coefficient matrix of IRS q. \({\varvec{S}}_{{a_{q} ,a_{q + 1} }}\) represents the equivalent channel matrix between IRS q and its nexthop IRS. \({{\varvec{\Phi}}}_{{a_{Q} }}\) denotes the reflection coefficient matrix of the last IRS, and \({\varvec{g}}_{{a_{Q} ,J + 1}}^{{\text{H}}}\) is the equivalent channel from the last IRS to the user.
The above spatial scattering channel modelingbased path loss models effectively avoid the limitations of dyadic backscatter channel modeling. However, the results derived for single IRSaided wireless communication systems cannot be directly extended to double IRSsaided systems. Although there are research results based on SalehValenzuela channel model which takes the impact of AoA and DoA into account, the physical and electromagnetic properties of IRS are still omitted. To the best of the authors’ knowledge, there is no relevant research on spatial scattering channel modelbased path loss modeling for cooperative double IRSsaided wireless communication networks, and this motivates our work in this paper. The research results in this paper lay indispensable foundations for future research on double IRSsaided wireless communication systems.
Spatial scattering channel modelbased path loss modeling for double IRSsaided wireless communication systems
As illustrated in Fig. 2, in order to minimize the path loss of the double reflection link, IRS 1 and IRS 2 are placed close to the BS and the user, respectively. They are placed in X–Y plane of Cartesian coordinate systems 1 and 2 whose origins align with the geometric centers of the two IRSs, respectively. N_{1} and M_{1} are the number of rows and columns of elements which are regularly arranged on IRS 1, and similarly, N_{2} and M_{2} are the number of rows and columns of elements on IRS 2. Without loss of generality, the above parameters are assumed to be even numbers. d_{x} × d_{y} is the size of each IRS element. U(n_{1},m_{1}) represents the element in row n_{1}(n_{1} ∈ [− N_{1}/2 + 1,N_{1}/2]) and column m_{1}(m_{1} ∈ [− M_{1}/2 + 1,M_{1}/2]) on IRS 1, and its center coordinate in Cartesian coordinate system 1 is ((m_{1} − 1/2)d_{x},(n_{1} − 1/2)d_{y},0). Its programmable reflection coefficient is \(\Gamma_{{n_{1} ,m_{1} }}\), and the gain is G_{1}. Similarly, parameters U(n_{2},m_{2}), ((m_{2} − 1/2)d_{x},(n_{2} − 1/2)d_{y},0), \(\Gamma_{{n_{2} ,m_{2} }}\) and G_{2} are defined for element U(n_{2},m_{2}) on IRS 2. F(θ,φ) is the inherent normalized power radiation pattern of IRS elements. F^{tx}(θ,φ) and F^{rx}(θ,φ) are the normalized power radiation patterns of the transmit antenna and the receiving antenna, respectively. d_{1}, d_{2} and d_{3} are the Euclidean distances from the BS to the center of IRS 1, between the centers of IRS 1 and IRS 2, from the center of IRS 2 to the user, respectively. \(r_{{n_{1} ,m_{1} }}^{t1}\), \(r_{{n_{1} ,m_{1} }}^{t2}\) and \(r_{{n_{2} ,m_{2} }}^{r}\) are the Euclidean distances from the BS to U(n_{1},m_{1}), between U(n_{1},m_{1}) and the center of IRS 2, from U(n_{2},m_{2}) to the user, respectively. \(\theta_{{n_{1} ,m_{1} }}^{t1}\) and \(\varphi_{{n_{1} ,m_{1} }}^{t1}\) are the elevation and azimuth angles from U(n_{1},m_{1}) to the BS, respectively. Similarly, parameters \(\theta_{{n_{2} ,m_{2} }}^{r}\) and \(\varphi_{{n_{2} ,m_{2} }}^{r}\) are defined for U(n_{2},m_{2}) with respect to the user. \(\theta_{{n_{12} ,m_{12} }}^{t2}\) and \(\varphi_{{n_{12} ,m_{12} }}^{t2}\) are the elevation and azimuth angles from U(n_{2},m_{2}) to U(n_{1},m_{1}), respectively. \(\theta_{{n_{12} ,m_{12} }}^{t12}\) and \(\varphi_{{n_{12} ,m_{12} }}^{t12}\) are parameters defined for the center of IRS 2 with respect to U(n_{1},m_{1}). \(\theta_{{n_{1} ,m_{1} }}^{tx1}\) and \(\varphi_{{n_{1} ,m_{1} }}^{tx1}\) are the elevation and azimuth angles from the transmit antenna of the BS to U(n_{1},m_{1}), respectively. \(\theta_{{n_{12} ,m_{12} }}^{tx2}\) and \(\varphi_{{n_{12} ,m_{12} }}^{tx2}\) are the elevation and azimuth angles from U(n_{1},m_{1}) to U(n_{2},m_{2}). \(\theta_{{n_{12} ,m_{12} }}^{tx12}\) and \(\varphi_{{n_{12} ,m_{12} }}^{tx12}\) are the elevation and azimuth angles from U(n_{1},m_{1}) to the center of IRS 2. \(\theta_{{n_{2} ,m_{2} }}^{rx}\) and \(\varphi_{{n_{2} ,m_{2} }}^{rx}\) are the elevation and azimuth angles from the receiving antenna of the user to U(n_{2},m_{2}).
The received signal power of U(n_{1},m_{1}) from the BS is:
where P_{t} is the transmit power of the BS, and G_{t} is the gain of the transmit antenna.
The reflected signal power of U(n_{1},m_{1}) is:
If IRS 1 is in the far field of IRS 2, the propagation distance of the signal reflected by U(n_{1},m_{1}) towards IRS 2 can be approximated as the Euclidean distance between U(n_{1},m_{1}) and the center of IRS 2. In this case, the power of the reflected signal received by U(n_{2},m_{2}) from U(n_{1},m_{1}) is:
where F^{tx2}(\(\theta_{{n_{12} ,m_{12} }}^{tx2}\),\(\varphi_{{n_{12} ,m_{12} }}^{tx2}\)) and F(\(\theta_{{n_{12} ,m_{12} }}^{t2}\),\(\varphi_{{n_{12} ,m_{12} }}^{t2}\)) are the normalized power radiation function of U(n_{1},m_{1}) with respect to U(n_{2},m_{2}) and the reversed normalized power radiation function, respectively.
The electric field of the reflected signal received by U(n_{2},m_{2}) from U(n_{1},m_{1}) is:
where \(\phi_{{n_{1} ,m_{1} }}\) is the phase shift introduced by U(n_{1},m_{1}) to the incident signal, and Z_{0} is the characteristic impedance of the air.
The total electric field of the reflected signal received by U(n_{2},m_{2}) from all elements on IRS 1 is:
The signal power received by U(n_{2},m_{2}) from IRS 1 is:
where \(F_{{n_{1} ,m_{1} }}^{combine}\) = F^{tx1}(\(\theta_{{n_{1} ,m_{1} }}^{tx1}\),\(\varphi_{{n_{1} ,m_{1} }}^{tx1}\))F(\(\theta_{{n_{1} ,m_{1} }}^{t1}\),\(\varphi_{{n_{1} ,m_{1} }}^{t1}\))F^{tx2}(\(\theta_{{n_{12} ,m_{12} }}^{tx2}\),\(\varphi_{{n_{12} ,m_{12} }}^{tx2}\))F(\(\theta_{{n_{12} ,m_{12} }}^{t2}\),\(\varphi_{{n_{12} ,m_{12} }}^{t2}\)). According to^{9}, when IRS 1 is in the far field of IRS 2 and all elements on IRS 1 adopt intelligent reflection, that is, F^{tx2}(\(\theta_{{n_{12} ,m_{12} }}^{tx2}\),\(\varphi_{{n_{12} ,m_{12} }}^{tx2}\))≈F^{tx2}(\(\theta_{{n_{12} ,m_{12} }}^{tx12}\),\(\varphi_{{n_{12} ,m_{12} }}^{tx12}\)), F(\(\theta_{{n_{12} ,m_{12} }}^{t2}\),\(\varphi_{{n_{12} ,m_{12} }}^{t2}\))≈F(\(\theta_{{n_{12} ,m_{12} }}^{t12}\),\(\varphi_{{n_{12} ,m_{12} }}^{t12}\)), \(\phi_{{n_{1} ,m_{1} }}\) = 2π(\(r_{{n_{1} ,m_{1} }}^{t1}\) + \(r_{{n_{{1}} ,m_{{1}} }}^{t2}\))/λ, the received signal power of U(n_{2},m_{2}) is maximized, and the value is:
The reflected signal power of U(n_{2},m_{2}) is:
The power of the reflected signal received by the user from U(n_{2},m_{2}) is:
where A_{r} is the effective area of the receiving antenna.
The electric field of the signal received by the user from U(n_{2},m_{2}) is:
where \(\phi_{{n_{2} ,m_{2} }}\) is the phase shift introduced by U(n_{2},m_{2}) to the incident signal.
The total electric field of the reflected signal received by the user from all elements on IRS 2 is:
The total signal power received by the user through the double reflection link is:
Assuming that the peak radiation direction of the signal reflected by U(n_{1},m_{1}) points to the center of IRS 2, F^{tx2}(\(\theta_{{n_{{{1}2}} ,m_{{{1}2}} }}^{{tx{1}2}}\),\(\varphi_{{n_{{{1}2}} ,m_{{{1}2}} }}^{{tx{1}2}}\))≈1 in farfield case. If \(\phi_{{n_{2} ,m_{2} }}\)=2π \(r_{{n_{2} ,m_{2} }}^{r}\)/λ, the received signal power of the user is maximized, as shown in Eq. (28).
Results and discussion
Simulation scenarios
As shown in Fig. 3, the original Cartesian coordinate system is established whose origin is aligned with the midpoint of the connecting line between the centers of IRS 1 and IRS 2, and the positive X axis is horizontal right along the connecting line. In order to apply the path loss model proposed in Eq. (28) conveniently, the original coordinates of the BS and the user are premultiplied by rotation matrices and converted into the coordinates in Cartesian coordinate systems 1 and 2, respectively, as shown in Eqs. (29) to (32).
where (\(x^{\prime}_{BS}\),\(y^{\prime}_{BS}\),\(z^{\prime}_{BS}\)) and (\(x^{\prime}_{User}\),\(y^{\prime}_{User}\),\(z^{\prime}_{User}\)) are the coordinates of the BS and the user in Cartesian coordinate system 1, respectively. β is the deviation angle from the positive X axis of Cartesian coordinate system 1 to the negative Z axis of the original Cartesian coordinate system.
where (\(x^{\prime\prime}_{BS}\),\(y^{\prime\prime}_{BS}\),\(z^{\prime\prime}_{BS}\)) and (\(x^{\prime\prime}_{User}\),\(y^{\prime\prime}_{User}\),\(z^{\prime\prime}_{User}\)) are the corresponding coordinates of the BS and the user in Cartesian coordinate system 2, respectively.
To achieve a fair comparison between dyadic backscatter channel model and spatial scattering channel modelbased double IRSsaided wireless communication systems, the simulation setup in^{21} is utilized, and the detailed settings are listed in Table 2. According to^{19}, when the size of an IRS element along the X axis and Y axis d_{x} = d_{y} = 0.03 m and the carrier wavelength λ = 0.06 m, the gain of the IRS element is about 4. Therefore, its normalized power radiation pattern is defined as F(θ,φ) = cosθ. Both the BS transmit antenna and the receiving antenna of the user are assumed to be omnidirectional, and their normalized power radiation patterns are defined as F^{tx}(θ,φ) = F^{rx}(θ,φ) = 1. In addition, whether deploying double IRSs will enhance the system performance is explored by comparing with single IRSaided wireless communication systems. For single IRSaided wireless communication systems, the user is served by the BS through the single reflection link via IRS 2. In this case, the BS is in the far field of IRS 2, and according to^{9}, the total received signal power at the user is:
where θ_{tx3} and φ_{tx3} are the elevation and azimuth angles from the BS transmit antenna to the center of IRS 2, respectively. Similarly, θ_{t3} and φ_{t3} are the reversed elevation and azimuth angles, respectively. \(r_{{n_{2} ,m_{2} }}^{t3}\) is the Euclidean distance between the BS and U(n_{2},m_{2}), and \(r_{{n_{2} ,m_{2} }}^{t3}\)≈d_{4} − sinθ_{t3}cosφ_{t3}(m_{2} − 1/2)d_{x} − sinθ_{t3}sinφ_{t3}(n_{2} − 1/2)d_{y}. Here, d_{4} is the Euclidean distance from the BS to the center of IRS 2. \(r_{{n_{2} ,m_{2} }}^{r}\) is the Euclidean distance between U(n_{2},m_{2}) and the user. For a fair comparison, IRS 2 is configured with 800/1600 elements in above single IRSaided wireless communication systems while IRS 1 and IRS 2 are configured with 800/1600 elements in total in double IRSsaided wireless communication systems.
Simulation results and analysis
The number of elements on IRS 1 is gradually increased while the total number of IRS elements K is kept unchanged to observe the received SNR at the user, and the results are shown in Fig. 4.
As can be observed from Fig. 4, the received SNR at the user is maximized when the same number of elements is assigned to IRS 1 and IRS 2. When K increases from 800 to 1600, the gain of single IRSaided wireless communication systems is about 6 dB, i.e., the received power is improved by 4 times. Double IRSsaided wireless communication systems can achieve about 12 dB gain, which means that the received signal power is enhanced by 16 folds. Therefore, compared with the array gain of \(\mathcal{O}\)(K^{2}) brought by single IRS, double cooperative IRSs can provide a received power gain of \(\mathcal{O}\)(K^{4}). The above conclusions are consistent with^{21}. However, different from the conclusions drawn from^{21}, that is, when K = 800, double IRSsaided wireless communication systems are inferior to their single IRSaided competitors, the simulation results in this paper show that even though K = 800, double IRSsaided wireless communication systems still gain advantages over single IRSaided wireless communication systems. To be specific, when d_{1} = 1 m, the BS is in the near field of IRS 1, and as the number of elements on IRS 1 increases, more signal power can be received by IRS 1 from the BS. Correspondingly, the received SNR at the user gradually increases, and double cooperative IRSs can bring in about 3 dB power gain. As the number of elements on IRS 1 further increases, although IRS 1 can reflect more power towards IRS 2, the decrease of the number of elements on IRS 2 results in the decline of the received signal power. Therefore, the received SNR at the user is reduced. In addition, when K = 1600, deploying double cooperative IRSs can achieve a 9 dB gain which is higher than the 6 dB gain presented in^{21}. The reasons can be explained as below: the conclusion drawn from^{21} is based on dyadic backscatter channel model which omits various factors such as the size and gain of IRS elements and the near/farfield effects of IRS, and as a result, the actual performance gain brought by double IRSs cannot be quantified accurately. In this paper, spatial scattering channel model is applied to model the path loss of the double reflection link, and more accurate performance analysis can be achieved.
In order to further explore the impact of near/far field effects of IRS on the received SNR, the distance between the BS and IRS 1 is enlarged, and it is equal to or larger than the distance between the user and IRS 2. In this case, the BS is in the far field of IRS 1. The above simulations are repeated, and the obtained results are shown in Fig. 5.
As shown in Fig. 5, similar to the results obtained in the nearfield case, the received SNR at the user is maximized when the same number of elements is assigned to IRS 1 and IRS 2. However, even though the number of elements on IRS 1 keeps increasing, their array gain still cannot compensate for the path loss of farfield signal propagation. According to Eq. (28), if other parameters are kept unchanged, the path loss between the BS and IRS 1 in farfield case is 225 times higher than that in nearfield case, and severe path loss leads to lower received SNR.
Apart from the received SNR at the receiver, BER is also an important performance indicator of digital communication systems, and it highly depends on the input SNR of the demodulator with Gaussian white noise. Assuming that the channels in the cooperative double IRSsaided wireless communication system are constantparameter channels with ideal rectangular transmission characteristics within the frequency range of the signal. The channel noise is additive Gaussian white noise, and it affects the received signal only at the receiver side. Motivated by^{27,28,29,30,31,32,33}, the BER performance of cooperative double IRSsaided wireless communication system under BPSK, QPSK, 8PSK and 16QAM modulation and coherent demodulation is investigated, and the simulation results are shown in Fig. 6.
As shown in Fig. 6, we can observe that the cooperative double IRSsaided wireless communication system with BPSK modulation achieves its optimal BER performance when the total number of IRS elements is equally assigned to IRS 1 and IRS 2. This conclusion holds for other modulation methods, and the reason is analyzed as below: the received SNR at the user increases as the number of elements on IRS 1 increases, which results in a declined BER. The BER will increase as the phase difference between adjacent carriers decreases, which explains why the BER achieved by loworder phaseshift keying modulation is lower than highorder phaseshift keying modulation. In addition, the Euclidean distance between the 16QAM constellation points falls in between that of the QPSK and 8PSK. Therefore, the BER performance of cooperative double IRSsaided wireless communication system based on 16QAM modulation method is higher than QPSK but lower than 8PSK.
Assuming that the channels in cooperative double IRSsaided wireless communication system are bandwidthconstraint and they are affected by additive and continuous Gaussian white noise. According to the Shannon's law, the channel capacity of the cooperative double IRSsaided wireless communication system is calculated when different number of IRS elements is assigned to IRS 1, and the results are shown in Fig. 7.
As shown in Fig. 7, the channel capacity of the cooperative double IRSsaided wireless communication system first increases and then decreases as the number of elements on IRS 1 increases, and it achieves the maximum value when the number of elements on IRS 1 and IRS 2 are equal to each other. Its variation tendency can be analyzed from the change of the received SNR at the user, as shown in Fig. 4. When the total number of IRS elements increases from 800 to 1600, the channel capacity of the single IRSaided wireless communication system is improved by 0.6 bit/s/Hz, and the channel capacity of the cooperative double IRSsaided wireless communication system is increased by 0.9 bit/s/Hz under the optimal configuration. When the total number of IRS elements K = 1600, the channel capacity improvement gained by the cooperative double IRSsaided wireless communication system over its single IRS counterpart is twice the performance enhancement under K = 800. The above simulation results are consistent with the above drawn conclusions, that is, the performance gain achieved by cooperative double IRSsaided wireless communication system depends on practical network configurations.
As shown in Eq. (28), when the transmit power, antenna gains, carrier wavelength and the size of IRS are fixed, the total received signal power is only related to the Euclidean distances d_{1}, d_{2} and d_{3} which are determined by the positions of double IRSs. In order to identify the optimal positions of the double IRSs, we observe the variations of the received SNR at the user versus the interIRSs Euclidean distance d_{2}. The positions of the BS and the user are fixed while IRS 1 and IRS 2 move along the X axis in the original Cartesian coordinate system. For convenience, they are symmetric about the origin, i.e., the coordinate of IRS 1 in the original Cartesian coordinate system is (− x, 0, 0) and that of IRS 2 is (x, 0, 0). In this case, d_{2} = 2x. According to^{21}, when IRS 1 is in the far field of IRS 2, if IRS 1 beams towards one element on IRS 2, the rest elements on IRS 2 can enjoy the same power gain. However, when IRS 1 is in the near field of IRS 2, the IRS coefficient adjustment adopted cannot align all the signals, which results in performance degradation. Therefore, in order to guarantee that IRS 1 is in the far field of IRS 2, d_{2} is at least 6 m. IRS 1 and IRS 2 are configured with the optimal number of reflecting elements, i.e., all elements are divided equally between IRS 1 and IRS 2. Changing x within its feasible set x ∈ (3, 50], the received SNR at the user is shown in Fig. 8.
As shown in Fig. 8, since the path loss of double IRSsaided wireless communication system is approximately d_{2} squared times higher than that of single IRSaided system, the received SNR at the user in double IRSsaided wireless communication system changes faster as d_{2} increases. Specifically, when the total number of IRS elements is K=1600, the performance gain achieved by the double IRSsaided wireless communication system is higher than its single IRS counterpart when d_{2} is larger than 90 m; When K=800, d_{2} needs to be larger than 96 m to guarantee the higher performance of double IRSsaided wireless communication system. They both achieve the highest performance gain when d_{2}=100 m. This means that the optimal positions of the double IRSs are (− 50, 0, 0) and (50, 0, 0), which are exactly the original simulation setups in Table 2.
In all, compared with single IRSaided wireless communication systems, the performance gain brought by double IRSsaided wireless communication systems is closely related to factors such as the number of IRS elements and the location of IRSs. Whether two cooperative IRSs should be adopted needs to be determined based on practical network configurations. In addition, if double IRSs are applied and other links are seriously blocked by obstacles, they should be assigned with the same number of elements to maximize the system performance.
Conclusions
Focusing on the typical application scenarios of double IRSsaided wireless communication systems, spatial scattering channel model is firstly leveraged to quantify the path loss of the double reflection link and establish the quantitative relationship between the received signal power and various system parameters. The impact of near/farfield effects of IRS on signal propagation is further taken into consideration to recognize the network configurations under which double cooperative IRSs can enhance the system performance. Simulation results show that the system performance is maximized when the two IRSs are assigned with the same number of elements. Compared with the array gain of \(\mathcal{O}\)(K^{2}) brought by single IRS, cooperative double IRSs can achieve a power gain of \(\mathcal{O}\)(K^{4}). Specifically, in the nearfield case, even with a small number of total elements, i.e., 800, the performance gain achieved by the proposed double IRSsaided wireless communication system based on the spatial scattering channel modeling is 3 dB higher than that of dyadic backscattering modelbased system under the same parameter configurations. In addition, the channel capacity gain of the double IRSsaided wireless communication system is twice the gain of single IRSaided system when the total number of elements is increased from 800 to 1600. Moreover, the path loss is jointly determined by IRS properties and transmission distances. In the farfield case, since the path loss is heavily sustained by the double reflection link, and it is always d_{2} squared times higher than that of the single IRSaided system. As a result, double IRSsaided wireless communication systems are inferior to their single IRSaided competitors. The above conclusions are drawn with the assumption that other links are seriously blocked by obstacles. Next, we will focus on the scenarios where all links between the transmitter and the receiver are unblocked to explore the full potentials of double IRSsaided wireless communication systems.
Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
References
Wu, Q. Q. & Zhang, R. Towards smart and reconfigurable environment: intelligent reflecting surface aided wireless network. IEEE Commun. Mag. 58(1), 106–112 (2020).
Cui, T. J., Liu, S. & Zhang, L. Information metamaterials and metasurfaces. J. Mater. Chem. C. 5(15), 3644–3668 (2017).
Wu, Q. Q. & Zhang, R. Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming. IEEE Trans. Wireless Commun. 18(11), 5394–5409 (2019).
Zhang, Z. et al. Active RIS versus passive RIS: Which will prevail in 6G?. IEEE Trans. Commun. https://doi.org/10.1109/TCOMM.2022.3231893 (2022).
You, C. S., Zheng, B. X., Mei, W. D. & Zhang, R. How to deploy intelligent reflecting surfaces in wireless network: BSside, userside, or both sides?. J. Commun. Inf. Netw. 7(1), 1–10 (2022).
Mei, W. D., Zheng, B. X., You, C. S. & Zhang, R. Intelligent reflecting surfaceaided wireless networks: From singlereflection to multireflection design and optimization. Proc. IEEE. 110(9), 1380–1400 (2022).
ElMossallamy, M. A. et al. Reconfigurable intelligent surfaces for wireless communication: Principles, challenges, and opportunities. IEEE Trans. Cognit. Commun. Network. 6(3), 990–1002 (2020).
Huang, J. et al. Reconfigurable intelligent surfaces: Channel characterization and modeling. Proc. IEEE. 110(9), 1290–1311 (2022).
Tang, W. K. et al. Wireless communications with reconfigurable intelligent surface: Path loss modeling and experimental measurement. IEEE Trans. Wireless Commun. 20(1), 421–439 (2020).
Zhou, H., Kang, X., Liang, Y. C., Sun, S. & Shen, X. Cooperative beamforming for reconfigurable intelligent surfaceassisted symbiotic radios. IEEE Trans. Veh. Tech. 71(11), 11677–11692 (2022).
Xiu, Y., Zhao, J., Sun, W. & Zhang, Z. Secrecy rate maximization for reconfigurable intelligent surface aided millimeter wave system with lowresolution DACs. IEEE Commun. Lett. 25(7), 2166–2170 (2021).
Du, L., Zhang, W., Ma, J. & Tang, Y. Reconfigurable intelligent surfaces for energy efficiency in multicast transmissions. IEEE Trans. Veh. Tech. 70(6), 6226–6271 (2021).
Xiu, Y. et al. Reconfigurable intelligent surfaces aided mmWave NOMA: Joint power allocation, phase shifts, and hybrid beamforming optimization. IEEE Trans. Wireless Commun. 20(12), 8393–8409 (2021).
Ma, H., Zhang, H., Zhang, W. & Leung, V. C. M. Beamforming optimization for reconfigurable intelligent surface with power splitting aided broadcasting networks. IEEE Trans. Veh. Tech. 72(2), 2712–2717 (2023).
Basar, E. et al. Wireless communications through reconfigurable intelligent surfaces. IEEE Access. 7, 116753–116773 (2019).
Özdogan, Ö., Björnson, E. & Larsson, E. G. Intelligent reflecting surfaces: Physics, propagation, and pathloss modeling. EEE Wireless Commun. Lett. 9(5), 581–585 (2020).
Garcia, J. C. B., Sibille, A. & Kamoun, M. Reconfigurable intelligent surfaces: Bridging the gap between scattering and reflection. IEEE J. Sel. Areas Commun. 38(11), 2538–2547 (2020).
Maeng, S. J., Anjinappa, C. K. & Güvenç, İ. Coverage probability analysis of passive reflectors in indoor environments. IEEE Commun. Lette. 26(10), 2287–2291 (2022).
Tang, W. K. et al. Path loss modeling and measurements for reconfigurable intelligent surfaces in the millimeterwave frequency band. IEEE Trans. Commun. 70(9), 6259–6276 (2022).
Graoni, G. & Renzo, M. D. Endtoend mutual coupling aware communication model for reconfigurable intelligent surfaces: An electromagneticcompliant approach based on mutual impedances. IEEE Commun. Lett. 10(5), 938–942 (2021).
Han, Y. T., Zhang, S. W., Duan, L. J. & Zhang, R. Cooperative doubleIRS aided communication: Beamforming design and power scaling. IEEE Wireless Commun. Lett. 9(8), 1206–1210 (2020).
Zheng, B. X., You, C. S. & Zhang, R. DoubleIRS assisted multiuser MIMO: Cooperative passive beamforming design. IEEE Trans. Wireless Commun. 20(7), 4513–4526 (2021).
Han, Y. T., Zhang, S. W., Duan, L. J. & Zhang, R. DoubleIRS aided MIMO communication under LoS channels: Capacity maximization and scaling. IEEE Trans. Commun. 70(4), 2820–2837 (2022).
Niu, H. H. et al. Double intelligent reflecting surfaceassisted multiuser MIMO mmWave systems with hybrid precoding. IEEE Trans. Veh. Technol. 71(2), 1575–1587 (2021).
Zhang, L. Y., Wang, Q. & Wang, H. Z. Multiple intelligent reflecting surface aided multiuser weighted sumrate maximization using manifold optimization. In 2021 IEEE/CIC International Conference on Communications in China (ICCC). 364–369 (2021).
Mei, W. D. & Zhang, R. Distributed beam training for intelligent reflecting surface enabled multihop routing. IEEE Wireless Commun. Lett. 10(11), 2489–2493 (2021).
Tang, W. K. et al. MIMO transmission through reconfigurable intelligent surface: Systems design, analysis, and implementation. IEEE J. Sel. Areas Commun. 38(11), 2683–2699 (2020).
Tang, W. K. et al. Wireless communication with programmable metasurface: Transceiver design and experimental results. China Commun. 16(5), 46–61 (2019).
Tang, W. K. et al. Programmable metasurfacebased RF chainfree 8PSK wireless transmitter. Electron. Lett. 55(7), 417–420 (2019).
Tang, W. K. et al. Design and implementation of MIMO transmission through reconfigurable intelligent surface. In Proceedings of the IEEE 21st International Workshop on Signal Processing Advances in Wireless Communications (SPAWC). 1–5(2020).
Tang, W. K. et al. Wireless communication with programmable metasurface: New paradigms, opportunities, and challenges on transceiver design. IEEE Wireless Commun. 27(2), 180–187 (2020).
Tang, W. K. et al. Realization of reconfigurable intelligent surfacebased Alamouti spacetime transmission. In Proceedings of the International Conference on Wireless Communications and Signal Processing (WCSP). 1–6(2020).
AlaaEldin, M., Alsusa, E. & Seddik, K. G. IRSassisted physical layer network coding over twoway relay fading channels. IEEE Trans. Veh. Technol. 71(8), 8424–8440 (2022).
Acknowledgements
This work was supported in part by the National Natural Science Foundation of China under Grant 61901102.
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J.W. contributed significantly to the concept and design of this work. H.N. performed the simulations and data analysis. J.W. prepared the original manuscript. Both authors reviewed the manuscript.
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Wang, J., Ni, H. Path loss modeling and performance evaluation of double IRSsaided wireless communication systems based on spatial scattering channel model. Sci Rep 13, 8400 (2023). https://doi.org/10.1038/s41598023345625
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DOI: https://doi.org/10.1038/s41598023345625
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