### 2.1 System model

In this paper, we contemplate an equivalent multiple-input-single-output (MISO) downlink wireless communication system enabled by two distributed IRSs, as shown in Fig. 1, where the BS, IRS1, IRS2, and user are equipped with an M-element uniform linear array (ULA), N_{1} reflecting elements arranged in an ULA, N_{2} reflecting elements arranged in an ULA and a single antenna, respectively. Due to the lack of an LoS propagation path between the BS and the user caused by severe obstructions, IRS1 and IRS2 are judiciously and constructively deployed on Building A near the BS and Building B near the user, respectively, to reap a cascaded BS-IRS1-IRS2-user link and improve the system performance. In addition, each IRS is joined to the BS by a dedicated transmission link to regulate signal reflection and exchange information.

Although the LoS path between the BS and the user is blocked, LoS paths can still be created in the channels between the BS and IRS1, between IRS1 and IRS2, and between IRS2 and the user by tactfully deploying the two IRSs in the wireless propagation environment. The three channels, therefore, can be modeled as Rician fading channels and expressed as follows [16, 17]:

$$ {\mathbf{H}}_{1\mathrm{B}}=\sqrt{\alpha_{1\mathrm{B}}}\left(\sqrt{\frac{K_{1\mathrm{B}}}{K_{1\mathrm{B}}+1}}{\overline{\mathbf{H}}}_{1\mathrm{B}}+\sqrt{\frac{1}{K_{1\mathrm{B}}+1}}{\tilde{\mathbf{H}}}_{1\mathrm{B}}\right) $$

(1)

$$ {\mathbf{H}}_{21}=\sqrt{\alpha_{21}}\left(\sqrt{\frac{K_{21}}{K_{21}+1}}{\overline{\mathbf{H}}}_{21}+\sqrt{\frac{1}{K_{21}+1}}{\tilde{\mathbf{H}}}_{21}\right) $$

(2)

$$ {\mathbf{h}}_{\mathrm{u}2}=\sqrt{\alpha_{\mathrm{u}2}}\left(\sqrt{\frac{K_{\mathrm{u}2}}{K_{\mathrm{u}2}+1}}{\overline{\mathbf{h}}}_{\mathrm{u}2}+\sqrt{\frac{1}{K_{\mathrm{u}2}+1}}{\tilde{\mathbf{h}}}_{\mathrm{u}2}\right) $$

(3)

where \( {\mathbf{H}}_{1\mathrm{B}}\in {\mathrm{\mathbb{C}}}^{N_1\times M} \), \( {\mathbf{H}}_{21}\in {\mathrm{\mathbb{C}}}^{N_2\times {N}_1} \)_{,} and \( {\mathbf{h}}_{\mathrm{u}2}\in {\mathrm{\mathbb{C}}}^{N_2\times 1} \) and *α*_{1B}, *α*_{21,} and *α*_{u2} denote the large-scale path losses of the BS-IRS1 link, IRS1-IRS2 link, and IRS2-user link, respectively. *K*_{1B}, *K*_{21}, and *K*_{u2} refer to the Rician factors of the BS-IRS1 link, IRS1-IRS2 link and IRS2-user link, respectively. \( {\overline{\mathbf{H}}}_{1\mathrm{B}}\in {\mathrm{\mathbb{C}}}^{N_1\times M} \), \( {\overline{\mathbf{H}}}_{21}\in {\mathrm{\mathbb{C}}}^{N_2\times {N}_1} \)_{,} and \( {\overline{\mathbf{h}}}_{\mathrm{u}2}\in {\mathrm{\mathbb{C}}}^{N_2\times 1} \) represent the LoS components of their respective links. \( {\tilde{\mathbf{H}}}_{1\mathrm{B}}\in {\mathrm{\mathbb{C}}}^{N_1\times M} \), \( {\tilde{\mathbf{H}}}_{21}\in {\mathrm{\mathbb{C}}}^{N_2\times {N}_1} \)_{,} and \( {\tilde{\mathbf{h}}}_{\mathrm{u}2}\in {\mathrm{\mathbb{C}}}^{N_2\times 1} \) depict the non-line-of-sight (NLoS) components of their respective links, whose elements are independently and identically distributed complex Gaussian random variables following the distribution of *CN* (0,1).

The LoS components are delineated by the response of the ULA. The array response of an *M*-element ULA is as follows:

$$ {\mathbf{a}}_M\left(\phi \right)={\left[1,{e}^{j2\pi \frac{d}{\lambda}\sin \phi },\dots, {e}^{j2\pi \frac{d}{\lambda}\left(M-1\right)\sin \phi}\right]}^T $$

(4)

where *ϕ* is the angle of arrival (AoA) or angle of departure (AoD) of a signal. In this case, the LoS component \( {\overline{\mathbf{H}}}_{1\mathrm{B}}\in {\mathrm{\mathbb{C}}}^{N_1\times M} \) can be defined as follows:

$$ {\overline{\mathbf{H}}}_{1\mathrm{B}}={\mathbf{a}}_{N_1}\left({\phi}_{\mathrm{AoA},1\mathrm{B}}\right){\mathbf{a}}_M^H\left({\phi}_{\mathrm{AoD},1\mathrm{B}}\right) $$

(5)

where *ϕ*_{AoA, 1B} is the AoA to ULA at IRS1 and *ϕ*_{AoD, 1B} is the AoD from the ULA at the BS. Similarly, the LoS components \( {\overline{\mathbf{H}}}_{21}\in {\mathrm{\mathbb{C}}}^{N_2\times {N}_1} \) and \( {\overline{\mathbf{h}}}_{\mathrm{u}2}\in {\mathrm{\mathbb{C}}}^{N_2\times 1} \) are defined as follows:

$$ {\overline{\mathbf{H}}}_{21}={\mathbf{a}}_{N_2}\left({\phi}_{\mathrm{AoA},21}\right){\mathbf{a}}_{N_1}^H\left({\phi}_{\mathrm{AoD},21}\right) $$

(6)

$$ {\overline{\mathbf{h}}}_{\mathrm{u}2}={\mathbf{a}}_{N_2}\left({\phi}_{\mathrm{AoD},\mathrm{u}2}\right) $$

(7)

where *ϕ*_{AoA, 21}, *ϕ*_{AoD, 21,} and *ϕ*_{AoD, u2} are the AoA to ULA at IRS2, the AoD from the ULA at IRS1, and the AoD from the ULA at IRS2, respectively.

As the BS-IRS1-IRS2-user link is a cascaded channel, the received signal at the user can be stated as follows:

$$ \mathrm{y}=\left({\mathbf{h}}_{\mathrm{u}2}^H{\boldsymbol{\Theta}}_2{\mathbf{H}}_{21}{\boldsymbol{\Theta}}_1{\mathbf{H}}_{1\mathrm{B}}\right)\mathbf{w}s+n $$

(8)

where \( {\boldsymbol{\Theta}}_1=\mathit{\operatorname{diag}}\left({\eta}_1^1{e}^{j{\theta}_1^1},\dots, {\eta}_{N_1}^1{e}^{j{\theta}_{N_1}^1}\right)\in {\mathrm{\mathbb{C}}}^{N_1\times {N}_1} \) and \( {\boldsymbol{\Theta}}_2=\mathit{\operatorname{diag}}\left({\eta}_1^2{e}^{j{\theta}_1^2},\dots, {\eta}_{N_2}^2{e}^{j{\theta}_{N_2}^2}\right)\in {\mathrm{\mathbb{C}}}^{N_2\times {N}_2} \) denote the reflection matrices of IRS1 and IRS2, respectively. \( {\eta}_m^1\in \left[0,1\right] \) and \( {\eta}_l^2\in \left[0,1\right] \) are amplitude reflection coefficients of IRS1 and IRS2, while \( {\theta}_m^1\in \left[0,2\uppi \right] \) and \( {\theta}_l^2\in \left[0,2\uppi \right] \) are the phase shifts introduced by IRS1 and IRS2. For simplicity, we will set \( {\eta}_m^1={\eta}_l^2=1 \) in the sequel to this paper. **w** ∈ ℂ^{M × 1} and *s*~*CN*(0, 1) indicate the transmit beamforming vector and the transmitted symbol, respectively. *n* is additive white Gaussian noise (AWGN) with zero mean and variance *σ*^{2}. According to formula (8), we can obtain the signal power received at the user as follows:

$$ \beta ={\left|\left({\mathbf{h}}_{\mathrm{u}2}^H{\boldsymbol{\Theta}}_2{\mathbf{H}}_{21}{\boldsymbol{\Theta}}_1{\mathbf{H}}_{1\mathrm{B}}\right)\mathbf{w}\right|}^2 $$

(9)

### 2.2 Problem formulation

Assume that P is the maximum transmit power, \( {\boldsymbol{\uptheta}}^1={\left[{\theta}_1^1,\dots, {\theta}_{N_1}^1\right]}^T \)_{,} and \( {\boldsymbol{\uptheta}}^2={\left[{\theta}_1^2,\dots, {\theta}_{N_2}^2\right]}^T \). We aim to maximize the received signal power *β* by cooperatively optimizing the transmit beamforming **w** and the phase shifts **θ**^{1} and **θ**^{2}, subject to the maximum transmit power constraint at the BS. The corresponding optimization problem can be written as follows:

$$ \left(\mathrm{P}1\right)\kern1.44em \underset{\mathbf{W},{\boldsymbol{\uptheta}}^{\mathbf{1}},{\boldsymbol{\uptheta}}^{\mathbf{2}}}{\max}\kern0.7em {\left|\left({\mathbf{h}}_{\mathrm{u}2}^H{\boldsymbol{\Theta}}_2{\mathbf{H}}_{21}{\boldsymbol{\Theta}}_1{\mathbf{H}}_{1\mathrm{B}}\right)\mathbf{w}\right|}^2 $$

(10)

$$ \mathrm{s}.\mathrm{t}.\kern2.0em {\left\Vert \mathbf{w}\right\Vert}^2\le \mathrm{P}, $$

(11)

$$ 0\le {\theta}_m^1\le 2\pi, \forall m=1,\dots, {N}_1, $$

(12)

$$ 0\le {\theta}_l^2\le 2\pi, \forall l=1,\dots, {N}_2. $$

(13)

We postulate that CSI is always acquirable. Therefore, for any specific **θ**^{1} and **θ**^{2}, it is not difficult to find that the maximum-ratio transmission (MRT) is the optimal transmit beamforming method for the optimization problem (P1) [18], i.e., \( {\mathbf{w}}^{\ast }=\sqrt{\mathrm{P}}\frac{{\left({\mathbf{h}}_{\mathrm{u}2}^H{\boldsymbol{\Theta}}_2{\mathbf{H}}_{21}{\boldsymbol{\Theta}}_1{\mathbf{H}}_{1\mathrm{B}}\right)}^H}{\left\Vert \left({\mathbf{h}}_{\mathrm{u}2}^H{\boldsymbol{\Theta}}_2{\mathbf{H}}_{21}{\boldsymbol{\Theta}}_1{\mathbf{H}}_{1\mathrm{B}}\right)\right\Vert}\overset{\Delta}{=}{\mathbf{w}}_{\mathrm{MRT}} \). By substituting **w**^{∗} into formula (10), (P1) can be transformed to the following equivalent problem:

$$ \left(\mathrm{P}2\right)\kern1.68em \underset{{\boldsymbol{\uptheta}}^{\mathbf{1}},{\boldsymbol{\uptheta}}^{\mathbf{2}}}{\max}\kern0.3em {\left\Vert \left({\mathbf{h}}_{\mathrm{u}2}^H{\boldsymbol{\Theta}}_2{\mathbf{H}}_{21}{\boldsymbol{\Theta}}_1{\mathbf{H}}_{1\mathrm{B}}\right)\right\Vert}^2 $$

(14)

$$ \mathrm{s}.\mathrm{t}.\kern1.8em 0\le {\theta}_m^1\le 2\pi, \forall m=1,\dots, {N}_1, $$

(15)

$$ 0\le {\theta}_l^2\le 2\pi, \forall l=1,\dots, {N}_2. $$

(16)

Although constraints (15) and (16) are convex, problem (P2) remains a non-convex optimization problem because of the non-concave objective function with respect to **θ**^{1} and **θ**^{2}. In practice, there is no standard solution to such a non-convex optimization problem. Accordingly, we propose to resort to the PSO algorithm to optimize problem P2, which will be analyzed in the next section.