Journal of Systems Engineering and Electronics

Multiple-input multiple-output (MIMO) is advocated as one of the 5th generation (5G) core solutions in realizing high spectral efficiency. The merge of MIMO and high-frequency technologies has become an inevitable trend. Intelligent reflecting surface (IRS) has been identified as an appealing complementary technique to improve spectral and energy efficiency for MIMO systems [1-3].

The incorporation of IRSs into MIMO systems is not straightforward due to the adverse effect of practically imperfect channel state information (CSI). The perfect and instantaneous CSI (I-CSI) is vital for reaping the promising gains of IRSs. Although it is possible to acquire nearly perfect CSI by virtue of widely studied IRS-related cascaded channel estimation [4,5], the estimated CSI cannot be used immediately to optimize the beamforming at the base station (BS) due to a non-negligible feedback delay [6]. On the other hand, non-negligible IRS configuration overhead can severely degrade the spectrum utilization. To reduce the training overhead, a twin-timescale beamforming method was presented in [7] and [8], where the expectation of the sum-rate was maximized over all random channel samples during a time frame. The IRS passive beamforming (PBF) was designed based on the statistical CSI (S-CSI) at a large timescale, and the BS precoding was produced according to the I-CSI. However, the estimated CSI in each slot can be quickly outdated as a result of signaling and feedback overhead [9].

This paper studies a twin-timescale beamforming (TTSBF) strategy in an IRS-assisted MIMO system, where the BS employs the singular value decomposition (SVD) precoder. By further taking the zero-forcing (ZF) criteria, the inter-stream interference (ISI) resulting from the outdated CSI (O-CSI) is suppressed. Our contributions are summarized as follows:

(i) This paper considers a more practical channel model, which is fast-changing and follows an first-order autoregressive (AR) process. To overcome the O-CSI effect, we introduce an “SVD-ZF” transceiver architecture for IRS-aided MIMO systems. Then, we develop a twin-timescale PBF and power allocation protocol to reduce the high feedback overhead.

(ii) The formulated TTSBF optimization problem is a nonconvex stochastic optimization problem. For long-timescale PBF, we transform the original problem into an average achievable rate (AAR) maximization problem. For the short-timescale power allocation, we derive a closed-form solution via the water-filling method.

(iii) The long-timescale PBF problem is essentially a trace-of-inverse-covariance minimization (TicMin) problem, which is different from the quadratically constrained quadratic programming (QCQP) form in [7] and [8]. Since the new problem cannot be transformed to a familiar semidefinite relaxation (SDR), this paper proposes a mini-batch sampling (mbs)-based particle swarm optimization (PSO) algorithm.

2.1    System model and channel model

We consider an IRS-assisted MIMO link from an N_b-antenna BS to an N_u-antenna user equipment (UE). The IRS is an N-element uniform planar array. ${{\boldsymbol{H}}_{\text{d}}}$ denotes the direct BS-UE channel. ${\boldsymbol{G}}$ and ${{\boldsymbol{H}}_{\text{r}}}$ denote the BS-IRS and IRS-UE channels, respectively. We define $\mathcal{H} = $$ \left\{ {{\boldsymbol{G}},{{\boldsymbol{H}}_{\text{d}}},{{\boldsymbol{H}}_{\text{r}}}} \right\}$ as the full channel ensemble.

In general, the IRS is typically deployed to achieve the line-of-sight (LoS) propagation between the BS and the IRS. Thus, there are few scatters around the BS and the IRS, but rich scatters around the UE. We assume the rank-one BS-IRS channel and the multi-path UE-IRS channel. Due to the stationarity of the BS and IRS, the BS-IRS channel is time-invariant, as given by

where ${L_{{\text{br}}}}$ accounts for the large-scale fading, ${\boldsymbol{a}}{\text{(}}\phi {\text{)}}$ is the array steering vector of the BS, and ${\boldsymbol{b}}{\text{(}}{\phi _{\text{e}}},{\phi _{\text{a}}}{\text{)}}$ is the array steering vector of the IRS.

We focus on a time frame, within which the S-CSI of all links remains unchanged. The time frame is divided evenly into T time slots. The channel coefficients of the links, i.e., I-SCI, do not change during a time slot. The non-LoS (NLoS) channel matrices are correlated across different time slots. At time slot t, the UE-IRS channel is given by

where $ {L_{{\text{ur}}}} $ accounts for the large-scale fading; $\kappa $ is the Rician fading-related factor; ${\bar {\boldsymbol{H}} _{\text{r}}}$ and ${\tilde {\boldsymbol{H}}_{\text{r},t}}\sim {{\rm{CN}}}{\text{(}}0,{\boldsymbol{I}}{\text{)}}$ account for the LoS and NLoS components, respectively. Likewise, the BS-UE channel is given by

where ${L_{{\text{bu}}}}$, ${\bar {\boldsymbol{H}} _{\text{d}}}$, and ${\tilde {\boldsymbol{H}}_{\text{d},t}}$ are defined in the same way as ${L_{{\text{ur}}}}$, ${\bar {\boldsymbol{H}} _{\text{r}}}$, and ${\tilde {\boldsymbol{H}}_{\text{r},t}}$, respectively.

The temporal evolution of the NLoS Rayleigh fading channel is modeled by an first-order AR process [10], as given by

where ${{\boldsymbol{E}}_t}\sim {{\rm{CN}}}{\text{(}}0,{\text{(}}1 - {\rho ^2}{\text{)}}{\boldsymbol{I}}{\text{)}}$ is a random matrix. Following the Jakes ’ model [11], the time correlation coefficient is $\rho = {J_0}{\text{(}}2{\text{π}}{f_d}\tau {\text{)}}$. Here, ${J_0}{\text{(}} \cdot {\text{)}}$ is the zeroth order Bessel function of the first kind; ${f_d}$ is the maximum Doppler frequency; $\tau $ is the delay between the time when the UE estimates the CSI and the time when the BS produces the precoder.

2.2    SVD-ZF architecture

The BS transmits M data streams to the UE with the aid of the IRS. The received signal at the UE is written as

where ${\boldsymbol{x}} \in {{{\bf{C}}}^{M \times 1}}$ is the transmit signal, ${\boldsymbol{n}}\sim {{\rm{CN}}}{\text{(}}0,\sigma _n^2{{\boldsymbol{I}}_{{N_{\text{r}}}}}{\text{)}}$ is the Gaussian noise, ${{\boldsymbol{W}}_{\text{t}}}$ is the pre-processing matrix at the BS, ${{\boldsymbol{W}}_{\text{r}}}$ is the post-processing matrix at the UE, ${\boldsymbol{\varTheta}} = {\text{diag(}}{{\text{e}}^{{\rm{j}}{\boldsymbol{\theta}} }}{\text{)}}$ is the IRS reflection matrix, and ${\overset{\smile} {\boldsymbol{H}}} \overset{\Delta }{\mathop{=}}\, {{\boldsymbol{H}}_{\text{d}}} + {\boldsymbol{G\varTheta}} {{\boldsymbol{H}}_{\text{r}}}$ defines the effective channel for notational brevity.

The BS can produce the pre-processing matrix ${{\boldsymbol{W}}_{\text{t}}}$, by using the truncated SVD of ${\overset{\smile} {\boldsymbol{H}^{\rm{H}}}} = {{\boldsymbol{UDV}}^{\rm{H}}}{\text{,}}$ where ${\boldsymbol{U}} \in {{{\bf{C}}}^{{N_{\text{u}}} \times M}}$ with ${{\boldsymbol{U}}^{\rm{H}}}{\boldsymbol{U}} = {{\boldsymbol{I}}_M}{\text{,}}$${\boldsymbol{V}} \in {{{\bf{C}}}^{{N_{\text{b}}} \times M}}$ with ${{\boldsymbol{V}}^{\rm{H}}}{\boldsymbol{V}} = {{\boldsymbol{I}}_M}$, and ${\boldsymbol{D}} = {\text{diag([}}{D_1},{D_2}, \cdots ,{D_M}{\text{]) }}$ with ${D_1} \geqslant {D_2} \geqslant \cdots \geqslant {D_M}$ is the singular value matrix. According to SVD, ${{\boldsymbol{W}}_{\text{r}}} = {\boldsymbol{U}}{\text{,}}$ and ${{\boldsymbol{W}}_{\text{t}}} = {\boldsymbol{V}}{{\boldsymbol{\varLambda}} ^{1/2}}$ with ${\boldsymbol{\varLambda }}= {\text{diag([}}{P_1},{P_2}, \cdots ,{P_M}{\text{])}}$ collecting the allocated powers.

Due to the delay between the channel estimation and data transmission, the pre-processing matrix is set to ${{\boldsymbol{W}}_{\text{t}}} = {{\boldsymbol{V}}_{t - \tau }}$ based on the O-CSI ${\overset{\smile} {\boldsymbol{H}}}$ at the BS. The UE can estimate the channel ${\overset{\smile} {\boldsymbol{H}}}_t$ by using its received pilot symbols, and subsequently detect the signals with the post-processing matrix ${{\boldsymbol{U}}_t}$. Then, the received signal at the UE can be written as

with ${\boldsymbol{Q}} \overset{\Delta }{\mathop{=}} {\boldsymbol{V}}_t^H{{\boldsymbol{V}}_{t - \tau }}.$ We see ${\boldsymbol{Q}} \ne {{\boldsymbol{I}}_M}$ results in the ISI due to ${{\boldsymbol{V}}_t} \ne {{\boldsymbol{V}}_{t - \tau }}$.

We propose a new “SVD-ZF” architecture to suppress the ISI resulting from the O-CSI, as depicted in Fig. 1.

Figure 1.  Proposed SVD-ZF scheme over a time-varying channel

Specifically, given the IRS configuration at a frame, the UE transmits the pilots to the BS at each time slot. Then the BS estimates the I-CSI ${{\overset{\smile} {\boldsymbol{H}}}_{t - \tau }}$ of the (t−τ)th slot to produce the precoder ${{\boldsymbol{V}}_{t - \tau }}$, which are adopted in the next tth slot. According to the SVD-ZF architecture, the BS sends the pilots and data symbols through the combined channel of precoder and downlink channel, i.e., ${\underline {\boldsymbol{H}} _t} \overset{\Delta }{\mathop{=}} {\boldsymbol{V}}_{t - \tau }^{\rm{H}}{{\overset{\smile} {\boldsymbol{H}}}_t}$. The UE can estimate ${\underline {\boldsymbol{H}} _t}$ by using the pilots, and then detects the data symbols using the ZF post-processing matrix

Therefore, the decoded signal is given by

The signal-to-interference-plus-noise ratio (SINR) of the mth data stream is given by

2.3    Beamforming strategy and problem formulation

The IRS-related cascaded channel needs to be estimated, every time the IRS is reconfigured. Consequently, the channel estimation would require large training and feedback overhead, if the IRS phase shifts are updated slot by slot.

We propose a TTSBF strategy, which configures the IRS once per frame based on the S-CSI and updates the beamformers frequently every slot according to the O-CSI. As illustrated in Fig. 2, within each frame, the large-scale fading ${\text{\{ }}{L_{{\text{ur}}}},{L_{{\text{bu}}}}{\text{\} }}$ and the S-CSI ensemble ${\text{\{}}{{{\bar {\boldsymbol{H}}}}_{\text{r}}},{{{\bar {\boldsymbol{H}}}}_{\text{d}}}{\text{\}}}$ can be acquired by the S-CSI estimation approach [7] at the beginning of each frame. The BS optimizes the long-term IRS configuration for the entire frame, by utilizing the S-CSI. In the rest of the frame, the IRS configures its PBF persistently, according to the IRS configuration feedback.

Figure 2.  Graphical illustration of TTSBF strategy

Under the SVD-ZF scheme, a short-timescale power allocation method is designed to accommodate the fixed PBF of the IRS as follows:

where the constraint in (10) guarantees that the total transmit power of the BS does not exceed its power budget ${P_{{\text{tot}}}},$ and the constraint in (11) specifies the unit-modulus IRS phase shifts.

In this section, we reformulate problem (P1) as a joint design of power allocation at the BS and PBF at the IRS, and design a new modified PSO algorithm to solve it efficiently.

We first rewrite the SINR of the mth data stream at slot t, ${\gamma _m}{\text{(}}{P_m},{\boldsymbol{\varTheta}} ,\mathcal{H}{\text{)}}$, as

Denoting ${\underline {\boldsymbol{R}} _t} \triangleq {\underline {\boldsymbol{H}} _t}\underline {\boldsymbol{H}} _t^{\rm{H}}$ by the covariance matrix of the combined channel, we define ${f_{m,t}}{\text{(}}\theta {\text{)}} = {{\text{[}}\underline {\boldsymbol{R}} _t^{ - 1}{\text{(}}{\boldsymbol{\theta}} {\text{)]}}_{{\text{(}}m,m{\text{)}}}}$ as the mth diagonal element of the inverse of the covariance matrix. Problem (P1) can be rewritten as

where ${\boldsymbol{p}} = {{\text{[}}{P_1},{P_2}, \cdots ,{P_M}{\text{]}}^{\rm{T}}}$ collects the transmit powers of M data streams.

Problem (P1 ’) is a stochastic optimization problem, because of the per-slot optimization of the power allocation. At each slot, the power allocation is based on O-CSI, and the fixed phase-shifts of the IRS configured at the long timescale. The IRS PBF optimization needs to predictively maximize the expectation of the sum-rate over random channel samples and potential per-slot power allocations throughout the frame. Although we can transforms Problem (P1 ’) into a deterministic optimization problem, it confronts serious challenges: lack of explicit relation between ${\gamma _i}$ and $\theta $ in the formulated TicMin problem, and nonconvex unit-modulus constraints on ${\boldsymbol{\varTheta }}$.

The key problem decomposition and the associated schemes are summarized in Fig. 3.

Figure 3.  An overview of the proposed twin-timescale resource alloca-tion scheme

3.1    Short-timescale power allocation scheme

Provided that the PBF of the IRS (i.e., the IRS configuration) is optimized at the beginning of the current frame, problem (P1) is written as

Problem (P2) is convex with respect to ${\boldsymbol{p}}$, and can be solved using the off-the-peg water-filling algorithm. Specifically, the Lagrangian of problem (P2) is given by

where $\mu $ is the Lagrange multiplier corresponding to constraint in (13). The optimal power solution is given by

with ${{\text{[}}x{\text{]}}^ + } = {\text{max\{}}x,0{\text{\} }}$. As a result, the corresponding per-slot sum-rate is given by

where ${r_{m,t}} = {\text{log}_2}(1 + {\gamma _{m,t}}{\text{)}}$.

3.2    Long-timescale PBF design

The per-frame PBF is a stochastic optimization problem due to the expectation operation in the objective of Problem (P1 ’). To circumvent this impasse, we first rewrite the objective in a deterministic form. Specifically, we take the average rate of ${L_B}$ number of random channel samples to approximate the expectation in the objective.

The lth channel sample set in the tth time slot is collected by $\hat{ \mathcal{H}}_{\mathrm{irs}, t}^l=\{{\hat {\boldsymbol{H}}}_{\mathrm{d}, t-\tau}^l, \hat{{{\boldsymbol{H}}}}_{\mathrm{r}, t-\tau}^l\}$, where $l = 1,2, \cdots ,{L_B}.$ The AAR of the mth data stream can be approximated by

The long-timescale PBF problem is rewritten as

Apart from the nonconvex constant-modulus constraint, the power allocation variables are closely coupled with the reflection phase-shifts of the IRS. As ${{\text{[}}\underline {\boldsymbol{R}} _t^{ - 1}{\text{(}}{\boldsymbol{\theta}} {\text{)]}}_{{\text{(}}m,m{\text{)}}}}$ cannot be written as an explicit function of ${\boldsymbol{\varTheta}} {\text{,}}$ traditional successive convex relaxation methods cannot be applied to solve problem (P3). The AAR obtained by solving (P3) can be considered as an upper bound for problem (P1).

We propose a new PSO framework to solve problem (P3). The PSO, a widely-used meta-heuristic bionic optimization algorithm [12], can find the optimal solution with the benefits of fewer parameters, simple calculation implementation, and fast convergence [13]. In the PSO method, the position of each particle stands for a potential solution. The fitness function of the particle is typically defined to be the optimization objective. We define an IRS refection angle vector ${\boldsymbol{\theta}}$ as the position of a particle, which corresponds to ${\boldsymbol{\varTheta}} = {\text{diag(}}{{\rm{e}}^{{\rm{j}}{\boldsymbol{\theta}} }}{\text{)}}{\text{.}}$ The objective (i.e., AAR) of problem (P3) is used as the fitness function, as given by

According to (15), for each particle ${\boldsymbol{\theta}} $, the optimal power solution can be uniquely found. This ensures the uniqueness of the fitness value in (18). We assume that P particles are employed to seek for the optimal solutions in the constrained search space and each is assigned with an individual velocity at every iteration. The set of particle positions, denoted by ${\mathcal{L}_P} = {\text{\{ }}{{\boldsymbol{\theta}} _1},{{\boldsymbol{\theta}} _2}, \cdots ,{{\boldsymbol{\theta}} _P}{\text{\} }}$, corresponds to the set of PBF matrices ${\text{\{ }}{{\boldsymbol{\varTheta}} _1},{{\boldsymbol{\varTheta}} _2}, \cdots ,{{\boldsymbol{\varTheta}} _P}{\text{\} }}$. In every iteration, each particle is assessed based on its fitness function to determine whether the current position implies a good solution. Each particle can record the best position or experience ever found by itself.

The global optimal position is selected from the best positions of all particles. Let $\dot {\boldsymbol{\theta}} _p^{{\text{(}}i{\text{)}}}$ and $\ddot {\boldsymbol{\theta}} _p^{{\text{(}}i{\text{)}}}$ denote the best position of the pth particle and the global optimal position of all particles at the ith iteration, respectively. The pth particle updates its position by using its velocity ${\boldsymbol{v}}_p^{{\text{(}}i{\text{)}}}$ at the ith iteration. The resulting new position is used to update the best position of this particle, if the fitness value of the previous best position is lower than that of the new position. Otherwise, the best position of this particle remains unchanged. After one round of evaluation of the fitness value, the global optimal position of all particles is accordingly updated. Both the velocities and positions of the pth particle at the ith iteration are updated by

where w is the nonnegative inertia weight of a particle; ${c_1}$ and ${c_2}$ denote cognitive and social scaling factors, respectively; and ε₁ and ε₂ are two independent random variables with a uniform distribution in (0,1).

Each particle requires a notable amount of calculations for the fitness function evaluation per iteration, as the fitness function is evaluated over large number of random channel samples. It can be computationally prohibitive to take all channel samples (which are generally high-dimensional matrices, e.g., ${N_{\text{b}}} \times {N_{\text{u}}} \times {L_B} \times P$) to obtain a fitness value in each iteration, potentially hindering the PSO convergence.

We develop an mbs-PSO algorithm which is able to substantially reduce the number of channel samples used to evaluate the fitness function while reaping satisfactory performance. Specifically, we introduce a mini-batch recursive sampling surrogate function to replace the fitness function in (17). All ${L_B}$ random channel samples are partitioned into ${N_B}$ batches with ${L_{{\text{mb}}}} = {L_B}{\text{/}}{N_B}$ samples per batch. At each iteration, the surrogate function is given by

where ${\mu ^{{\text{(}}i{\text{)}}}} = {i^{ - 0.8}}$ is an iteration-dependent constant accounting for the decay weight coefficient associated with the new sampling of the AAR.

From (12), it is seen that the main operations for each particle that contributes to the computational complexity lies on one matrix addition, four matrix multiplications and one matrix inversion using one channel sample. The overall computational complexity of the proposed mbs-PSO algorithm is scaled with a $P \times {L_{{\text{mb}}}}$-fold increase according to (21), while that of using (17) is scaled with a $P \times {L_B}$-fold increase, which significantly reduces the computational costs.

In practical, the PSO algorithm is deployed at the BS and then informs the instructions to the UE via the IRS controller. As the BS server exhibits the powerful computational capability, a real-time IRS configuration can be enabled. In addition, a moderate number of particles and iterations are generally required to achieve a desirable performance. Moreover, the calculations dominating by the fitness evaluation can be simply parallelized through pipelining in the field programmable gate array (FPGA) [14, 15], which is promising to guarantee strict real-time applications.

Remark 1　In the proposed mbs-PSO algorithm, the fitness function represents the AAR objective. The positions of all particles represent the potential PBF solutions, which are updated by evaluating the fitness value in each iteration. Note that the optimal positions are recorded by comparing with the historical fitness values. This results in a monotonically non-decreasing sequence of AAR values with corresponding PBF solutions updated. On the other hand, the AAR is upper bounded due to the finite transmit powers. Therefore, the convergence can be guaranteed.

Our proposed method for the point-to-point MIMO system can be easily extended to the multi-user MIMO system. This is because the multi-user MIMO channel can be decomposed into parallel single-user MIMO channels by the classical block diagonalization (BD) technique [16]. Therefore, the proposed algorithm can still be valid for each single-user MIMO channel.

To be specific, in the K-user MIMO model, the combined channel matrix is given by ${\boldsymbol{H}} = {{\text{[}}{\boldsymbol{H}}_1^{\rm{H}},{\boldsymbol{H}}_2^{\rm{H}}, \cdots ,{\boldsymbol{H}}_K^{\rm{H}}{\text{]}}^{\rm{H}}}$. The BD precoding matrix $ {{\boldsymbol{V}}_{{\text{BD,}}k}} $ can be constructed by the null spaces of matrix families derived from ${\boldsymbol{H}}$, similar to the SVD procedure. When the estimated CSI ${\boldsymbol{H}}$ is outdated, the combined channel matrix $ {{\boldsymbol{H}}_{{\text{c,}}k}} = {\boldsymbol{H}}{{\boldsymbol{V}}_{{\text{BD,}}k}} $ would confront the multi-user interference due to the mismatch between the null-space and actual channel. This can be treated as the ISI in the single-user MIMO system. Therefore, the ZF detector in the proposed transmission architecture can still eliminate the multi-user interference. The received signal can be written as

It is seen that this signal model is similar to that under the single-user MIMO scenario. Consequently, the following PBF and power allocation problems can also be solved by the proposed method.

Simulation results are provided to evaluate the performance of our proposed method, including the transmission architecture and optimization algorithm. In the simulations, the path loss model is set according to ${L_{{\text{out}}}} = {L_{{\text{in}}}}{{\text{(}}d{\text{/}}{d_0}{\text{)}}^{ - \alpha }}$ where ${L_{{\text{in}}}}$ is the reference path loss at ${d_0} = 1{\text{ m}}$ and $\alpha $ is the path loss exponent. We consider the point-to-point MIMO system in the 3D Cartesian coordinates. Both the locations of the BS and IRS are [0,0,5] m and [100,0,5] m. The UE is located at height 1 m right below the IRS, centered around [100,10] m with radius of 10 m. Unless otherwise stated, other simulation parameters are set as follows: ${N_{\text{b}}} = {N_{\text{u}}} = 8{\text{, }}N = 64{\text{, }}\kappa = 3{\text{,}}$ ${\alpha _{{\text{ur}}}} = 3.0{\text{,}}$ ${\alpha _{{\text{bu}}}} = 3.4{\text{,}}$ ${\alpha _{{\text{br}}}} = 2.2{\text{, }}$ ${P_{{\text{tot}}}} = 20{\text{ dBm, }}$ $\sigma _n^2 = $$ - 80{\text{ dBm,}}$ ${I_{{\text{iter}}}} = 200{\text{,}}$ ${\bar f_d} = {10^{ - 2}}{\text{, }}$ $P = 500{\text{, }}{L_B} = {10^4}{\text{, and }} $$ {c_1} = {c_2} = 1.494\;45$.

Five baselines are used for comparison, which is given as follows: (i) Upper bound: the PSO method is performed by using the AAR object in (P3) as the fitness function where each iteration requires large-scale channel samples. (ii) Sum-path-gain maximization (SPGM)-based scheme [17]: the long-term reflection matrix is obtained via the SPGM-based optimization method. (iii) SVD-based scheme: we adopt the conventional SVD-based MIMO processing architecture, where the transmit powers are optimized by considering the O-CSI as the ideal CSI in each slot. (iv) Random phase: based on the SVD-ZF, we adopt the randomly generated long-term reflection matrix to optimize the transmit powers by using (15). (v) Without IRS: based on the SVD-ZF, we just optimize the transmit powers to optimize the overall sum rate in one frame.

Fig. 4 shows the convergence behaviors of the proposed mbs-PSO algorithm. Note that the original AAR is not set as its fitness objective. From the fitness value results, we see that the mbs-PSO algorithm exhibits the good convergence. More importantly, the sampling surrogate function is effective, as the corresponding AAR is significantly improved during iterations.

Figure 4.  Convergence behaviors of proposed algorithm

Fig. 5 shows the impact of different mini-batch sizes on the convergent AAR. It is observed that all proposed algorithms with different L_mb values can converge within 200 iterations. As can be seen in the top subfigure, the proposed algorithm with small L_mb values exhibits a slight improvement of fitness gap, while the algorithm can gain a sharp improvement with large L_mb values. Comparing the two subfigures, however, we see that the subfigures do not exhibit the same upward trend. In particular, the bottom subfigure does not show a monotonic curve. This is reasonable because of the stochastic nature of the considered mini-batch sampling surrogate function.

Figure 5.  AAR versus mini-batch size

In Fig. 6, we compare different schemes under varying total powers. As ${P_{{\text{tot}}}}$ increases, their AAR performance improves due to the increasing SINR. Among these schemes, only SVD-based scheme reaps the worst capacity because it fails to deal with the ISI incurred by the O-CSI. The proposed mbs-PSO algorithm surpasses the schemes of random phase and without IRS, which verifies that the proper IRS reflection can effectively enhance the communication performance. Finally, by comparing the SPGM-based method with the scheme without IRS, we conclude that maximizing the sum-gain of all sub-channels is not equal to maximizing the AAR in our considered system.

Figure 6.  AAR versus total transmit power under different schemes

In Fig. 7, the AAR is plotted against the number of BS antennas, where the number of receive antennas is fixed as N_u = 4. It is seen that the AAR curves for all schemes show an approximately logarithmic shape. Although the number of receive antennas keeps constant, more transmit antennas can still provide a higher spatial diversity gain in the downlink transmission. Hence, the sum rate is improved with the increasing transmit antennas because the downlink SINR is increased. Also, the SPGM-based method performs the worse among all methods, which is consistent with the results in Fig. 6.

Figure 7.  AAR versus the number of BS antennas

Fig. 8 investigates the impact of O-CSI effect on the system AAR. From (4), as the time delay $\tau $ increases, the resulting time correlation coefficient $\rho $ decreases and thus the channel error increases. We see that the SVD-based scheme degrades as $\tau $ increases and it performs worst among all schemes because of the ISI. Conversely, all other schemes show a slight decline in terms of AAR. The proposed SVD-ZF strategy guarantees the transmission robustness due to the depression of the O-CSI effect. More importantly, the new SVD-ZF-based IRS assisted MIMO scheme is promising to be applied to fast-changing channels.

Figure 8.  AAR versus time delay

This paper develops an SVD-ZF-based transmission strategy to overcome the O-CSI in the considered IRS-assisted MIMO system. Since frequent IRS reconfigurations would incur large feedback overhead, we propose a TTSBF protocol, where the PBF at the IRS is designed in the long-timescale, while the BS power allocation is performed based on the O-CSI in each short-timescale slot. Particularly, the power allocation is derived with the closed-form water-filling solutions. Then, an mbs-PSO algorithm is proposed to efficiently solve the long-timescale PBF. Finally, we evaluate the proposed transmission strategy and PSO algorithms by comparing multiple baselines.

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