# Two-Timescale Optimization for Intelligent Reflecting Surface-Assisted MIMO
Transmission in Fast-Changing Channels

Yashuai Cao, , Tiejun Lv, _Senior Member, IEEE_ ,
and Wei Ni, _Senior Member, IEEE_ Manuscript received December 20, 2021;
revised April 28, 2022; accepted June 14, 2022. _(Corresponding author: Tiejun
Lv)_. Y. Cao and T. Lv are with the School of Information and Communication
Engineering, Beijing University of Posts and Telecommunications (BUPT),
Beijing 100876, China (e-mail: {yashcao, lvtiejun}@bupt.edu.cn). W. Ni is with
Data61, Commonwealth Scientific and Industrial Research Organisation, Sydney,
NSW 2122, Australia (e-mail: wei.ni@data61.csiro.au).

###### Abstract

The application of intelligent reflecting surface (IRS) depends on the
knowledge of channel state information (CSI), and has been hindered by the
heavy overhead of channel training, estimation, and feedback in fast-changing
channels. This paper presents a new two-timescale beamforming approach to
maximizing the average achievable rate (AAR) of IRS-assisted MIMO systems,
where the IRS is configured relatively infrequently based on statistical CSI
(S-CSI) and the base station precoder and power allocation are updated
frequently based on quickly outdated instantaneous CSI (I-CSI). The key idea
is that we first reveal the optimal small-timescale power allocation based on
outdated I-CSI yields a water-filling structure. Given the optimal power
allocation, a new mini-batch sampling (mbs)-based particle swarm optimization
(PSO) algorithm is developed to optimize the large-timescale IRS configuration
with reduced channel samples. Another important aspect is that we develop a
model-driven PSO algorithm to optimize the IRS configuration, which maximizes
a lower bound of the AAR by only using the S-CSI and eliminates the need of
channel samples. The model-driven PSO serves as a dependable lower bound for
the mbs-PSO. Simulations corroborate the superiority of the new two-timescale
beamforming strategy to its alternatives in terms of the AAR and efficiency,
with the benefits of the IRS demonstrated.

###### Index Terms:

Intelligent reflecting surface, passive beamforming, two-timescale
optimization, outdated channel state information, mini-batch sampling,
particle swarm optimization.

##  I Introduction

Recent years have witnessed an unprecedentedly surging demand for mobile data
traffic. The demand is expected to continuously grow in the upcoming sixth-
generation (6G) cellular systems, resulting from new applications and services
[1]. Intelligent reflecting surface (IRS) has been identified as an appealing
complementary technology to latest multiple-input multiple-output (MIMO)
techniques, to improve spectral and energy efficiency and increase data rates
[2, 3]. An IRS is a two-dimensional meta-surface array, which can induce
favorable scattering to create anomalous reflection and programmable wireless
channels via adjusting the phase-shifts of the IRS [4]. Since IRSs are
primarily made up of low-cost passive meta-atoms without active radio
frequency (RF) chains, they can be densely deployed along with existing MIMO
systems to substantially enhance the spectral efficiency in a cost-effective
manner [5, 6]. It was reported in [7] that the required transmit power of a
base station (BS) can decrease approximately quadratically with the increasing
number of IRS meta-atoms. It was also proved that IRSs can achieve better
energy efficiency than existing relay systems [8, 9].

###  I-A Motivation

One pressing challenge arising from the incorporation of IRSs into MIMO
systems is to overcome the adverse effect of practically imperfect, typically
outdated, channel state information (CSI), especially under time-division
duplexing (TDD) settings. Most existing resource allocation algorithms
designed for IRS-assisted MIMO systems require the perfect instantaneous CSI
(I-CSI), or they can hardly reap the promising gains of the IRSs. To obtain
high-precision CSI, cascaded channel estimation has been widely studied to
estimate IRS-enhanced cascaded channels. Many significant progresses, e.g.,
[10, 11, 12, 13], have been achieved, which makes it possible to obtain nearly
perfect I-CSI at MIMO receivers. The perfect I-CSI is important to effectively
design the BS precoder and configure the IRS, but hardly available to the
transmitter due to a non-negligible feedback delay, especially in TDD systems
[14, 15, 16]. To circumvent this impasse, some studies have focused on the
robust transmissions under imperfect CSI [17, 18, 19, 20]. Two canonical CSI
error models, namely, bounded error and statistical error models, were adopted
to devise robust beamforming. In [21], it was reported that configuring the
IRS reflection amplitudes is more cost-effective than the phase-shifts under
imperfect CSI. In [22], the passive beamforming of an IRS was optimized based
on the upper bound of the ergodic spectral efficiency, where only the
statistical CSI (S-CSI) was utilized.

Another challenge arising lies in non-negligible overhead in the phase-shift
configurations of IRSs. In general, the phase-shift configurations of the IRSs
are closely coupled with MIMO precoders in IRS-assisted MIMO systems. A
holistic, joint design of the two is imperative. A common approach is to
invoke the classic alternating optimization or block coordinate descent [23]
to obtain a suboptimal solution. Therein, large-scale non-convex unit modulus
constraints induced by the physical property of passive meta-atoms need to be
dealt with to design the IRS passive beamforming. Apart from convex relaxation
methods, such as semidefinite relaxation [7], alternating direction method of
multipliers [24], and manifold optimization [25], a machine learning-based
approach was proposed in [26] to solve the joint passive beamforming design
and resource allocation problem in IRS-aided wireless systems. Some recent
works have applied the particle swarm optimization (PSO) algorithm to the
design of IRS passive beamforming, due to its demonstrated effectiveness in
solving large-scale nonconvex problems. The authors of [27] designed a PSO-
based IRS configuration algorithm to reduce the transmit power with no need
for the CSI. The algorithm searches for solutions only based on the power and
rate metrics of a given channel, and is not suitable for fast-changing
channels. An IRS-aided visible light communication system was investigated in
[28], where the secrecy rate of the system was maximized using the PSO.

To reduce training overhead, two-timescale beamforming methods [29, 30, 31]
have been increasingly studied for IRS. In [29], active and passive
beamforming were optimized at two timescales for an IRS-assisted multiple-
input-single-output system. The IRS phase shifts were optimized based on the
S-CSI on a frame basis. The BS precoding was optimized based on the I-CSI on a
slot basis. Nadeem  _et al._ [30] studied IRS-aided coverage enhancement where
direct BS-user links are unavailable. The IRS phase shifts were asymptotically
optimized based on the S-CSI knowledge. The minimum mean square error precoder
was designed by using the I-CSI. By utilizing the channel hardening property,
Wang  _et al._ [31] developed a two-timescale protocol, where the large-
timescale passive beamforming was optimized to maximize the asymptotically
achievable rate of a small-timescale maximal-ratio combining precoder. Most of
these existing IRS configuration algorithms were developed under the
assumption that the I-CSI is available and the IRSs are instantaneously
reconfigurable. In practice, the acquisition of high-precision CSI is a
computationally intensive task. Let alone the aforementioned feedback delay of
the I-CSI. However, the perfect I-CSI can be hardly available in practice.
Only outdated channel samples can be acquired as a result of signaling and
feedback overhead [32].

In the face of the challenges of outdated CSI and non-negligible IRS
configuration overhead in fast-changing channels, it is of practical interest
to design an IRS-assisted MIMO system, where the IRS is configured less
frequently but predictively to cater to the fast-changing channel conditions
and subsequently frequent BS precoding. The BS precoding needs to tolerate
rapidly outdated CSI. To the best of our knowledge, no existing studies have
to date considered fast-changing channel conditions in IRS-assisted MIMO
systems. Let alone using the outdated CSI in the systems.

###  I-B Contribution

This paper maximizes the average achievable rate (AAR) of an IRS-assisted,
downlink MIMO system under fast-changing channels resulting from mobility and
yielding a first-order autoregressive process. This problem is new and has
never been addressed in the literature, because the acquired CSI can be
quickly and substantially outdated when used for precoding in the fast-
changing channel. The key contributions of the paper are summarized as
follows.

  * •

We design a new two-timescale beamforming strategy: the IRS is configured at a
large timescale on a frame basis, while the transmit and receive beamforming
(i.e., precoder and detection) is performed at a small timescale on a slot
basis. The inherent temporal correlation of the channel is exploited between
consecutive time slots to create effective parallel data streams, with the
inter-stream interference resulting from the outdated CSI suppressed at the
user equipment (UE).

  * •

We reveal that the optimal small-timescale transmit power allocation only
depends on the outdated CSI and can be readily obtained by water-filling
techniques.

  * •

With the proposed two-timescale beamforming strategy and the optimal small-
timescale power allocation in the fast-changing channel, we design a new data-
driven PSO algorithm with a mini-batch recursive sampling surrogate function
as the fitness function, called mbs-PSO, which divides the random channel
samples into batches and expedites evaluating the fitness of the IRS passive
beamforming by sampling the batches.

  * •

We derive an analytical lower bound of the AAR, and maximize the lower bound
by solving a trace-of-inverse-covariance minimization (TicMin) problem. The
new objective of TicMin depends directly on the S-CSI, and is efficiently
solved by designing a new model-driven PSO, called LBO-PSO, which evaluates
the trace-of-inverse-covariance for the fitness. With no need for channel
samples, the LBO-PSO is a computationally efficient alternative to the mbs-PSO
algorithm.

Comprehensive simulations verify the superiority of the proposed two-timescale
beamforming strategy to five benchmarks in terms of the AAR and efficiency.
The benefits of the IRS are clearly demonstrated with useful insights shed.

###  I-C Organization and Notation

The remainder of this paper is outlined as follows. In Section II, we describe
the channel model, system architecture, and formulate the new two-timescale
beamforming strategy. In Section III, we optimize the precoding and power
allocation for the new two-timescale strategy. Water-filling-based power
allocation is developed at a small timescale. PSO-based IRS configuration are
designed at a large timescale, to maximize the AAR of the system. Simulation
results are provided in Section IV, followed by conclusions in Section V.

_Notations_ : Lower- and upper-case boldface letters indicate column-vectors
and matrices, re-spectively; $(\cdot)^{\mathsf{T}}$ and $(\cdot)^{\mathsf{H}}$
denote transpose and conjugate transpose, respectively; $(\cdot)^{-1}$ and
$(\cdot)^{\dagger}$ stand for inverse and pseudo-inverse, respectively;
$\mathrm{trace}(\cdot)$ returns the trace of a matrix;
$\mathrm{angle}\\{\cdot\\}$ returns the angle of a complex value;
$\mathrm{diag}\\{\cdot\\}$ returns a diagonal matrix; $\mathbb{E}\\{\cdot\\}$
denotes expectation; $\otimes$ is the Kronecker product operator;
$[\cdot]_{(m,n)}$ denotes the $(m,n)$-th element of a matrix;
$\mathbb{C}^{m\times n}$ is the space of $m\times n$ complex matrices; and
$\mathbf{I}_{n}$ is the $n\times n$ identity matrix.

##  II System Setting and Problem Formulation

In this section, we first describe the IRS-assisted MIMO system model and the
channel model with an emphasis on the outdated CSI. The MIMO system allows the
BS to produce its precoder based on the outdated CSI. The UE generates its
detector by using a zero-forcing (ZF) criterion and I-CSI, suppressing the
inter-stream interference arising from the outdated CSI. We put forth a new
two-timescale passive beamforming and power allocation problem with the
problem statement provided at last.

###  II-A System Model and Channel Model

As illustrated in Fig. 1, we consider an IRS-assisted MIMO link from a BS to a
UE under a TDD setting. The BS is equipped with $N_{\mathrm{t}}$ transmit
antennas. The UE is equipped with $N_{\mathrm{r}}$ receive antennas. The IRS
is a uniform square planar array (USPA) consisting of $N=N_{x}N_{y}$ meta-
atoms, with $N_{x}$ meta-atoms per row and $N_{y}$ meta-atoms per column. As
shown in Fig. 1, $\mathbf{H}_{\mathrm{d}}$ denotes the direct channel from the
BS to the UE, and $\mathbf{G}$ and $\mathbf{H}_{\mathrm{r}}$ denote the BS-IRS
and IRS-UE channels, respectively. For notational brevity, we define
$\mathcal{H}=\\{\mathbf{G},\mathbf{H}_{\mathrm{d}},\mathbf{H}_{\mathrm{r}}\\}$
as the full channel ensemble.

The IRS is deployed to achieve the line-of-sight (LoS) propagation between the
BS and the IRS. Typically, there are few scatters around the BS and the IRS,
but rich scatters around the UE. We assume that the BS-IRS channel is _rank-
one_ , and the IRS-UE channel is a multi-path channel. The geometry of the IRS
is known to the BS, and so is the BS-IRS channel. The BS-IRS channel is
assumed to be time-invariant, due to the stationarity of both the BS and IRS.
The BS-IRS channel is given by

$\displaystyle\mathbf{G}=\sqrt{L_{\mathrm{br}}}\mathbf{a}(\phi)\mathbf{b}(\phi_{\mathrm{e}},\phi_{\mathrm{a}})^{\mathsf{H}}\in\mathbb{C}^{N_{\mathrm{t}}\times
N},$ (1)

where $L_{\mathrm{br}}$ is large-scale fading, $\phi$ is the angle-of-
departure (AoD) at the BS, $\phi_{\mathrm{e}}(\text{or}\ \phi_{\mathrm{a}})$
is the elevation (or azimuth) angles-of-arrival (AoA) at the IRS,
$\mathbf{a}(\phi)$ is the array steering vector of the BS, and
$\mathbf{b}(\phi_{\mathrm{e}},\phi_{\mathrm{a}})$ is the array steering vector
of the IRS. The array steering vectors are

$\displaystyle\mathbf{a}(\phi)$
$\displaystyle=[1,\cdots,e^{-j2\pi(N_{t}-1)\frac{d}{\lambda}\sin\phi}]^{\mathsf{T}}\in\mathbb{C}^{N_{t}\times
1},$ (2) $\displaystyle\mathbf{b}(\phi_{\mathrm{e}},\phi_{\mathrm{a}})$
$\displaystyle=[1,\cdots,e^{-j2\pi\frac{d\left((N_{y}-1)\sin\phi_{\mathrm{e}}\sin\phi_{\mathrm{a}}+(N_{x}-1)\cos\phi_{\mathrm{e}}\right)}{\lambda}}]^{\mathsf{T}}$
$\displaystyle=[1,\cdots,e^{-j2\pi(N_{y}-1)\frac{d}{\lambda}\sin\phi_{\mathrm{e}}\sin\phi_{\mathrm{a}}}]^{\mathsf{T}}\otimes$
$\displaystyle\quad\quad[1,\cdots,e^{-j2\pi(N_{x}-1)\frac{d}{\lambda}\cos\phi_{\mathrm{e}}}]^{\mathsf{T}}$
$\displaystyle=\mathbf{b}_{N_{y}}(\phi_{\mathrm{e}},\phi_{\mathrm{a}})\otimes\mathbf{b}_{N_{x}}(\phi_{\mathrm{e}})\in\mathbb{C}^{N\times
1},$ (3)

where $\lambda$ is the carrier wavelength, and $d$ is the spacing between
adjacent antenna elements of the BS or adjacent passive meta-atoms of the IRS.

Figure 1: An illustration of the considered IRS-assisted downlink point-to-
point MIMO system, where both the BS and UE are equipped with multiple
antennas, and the channels are fast-changing, e.g., due to the movement of the
UE.

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-CSI, do not change during a time slot. The
non-LoS (NLoS) channel coefficient matrices are correlated across different
time slots. At time slot $t$, the IRS-UE channel can be given

$\displaystyle\mathbf{H}_{\mathrm{r}}[t]=\sqrt{L_{\mathrm{ur}}}\left(\kappa\bar{\mathbf{H}}_{\mathrm{r}}+\sqrt{1-\kappa^{2}}\tilde{\mathbf{H}}_{\mathrm{r}}[t]\right)\in\mathbb{C}^{N\times
N_{\mathrm{r}}},$ (4)

where $L_{\mathrm{ur}}$ accounts for large-scale fading; $\kappa$ is the
Rician fading-related factor; $\bar{\mathbf{H}}_{\mathrm{r}}$ and
$\tilde{\mathbf{H}}_{\mathrm{r}}\sim\mathcal{CN}(0,\mathbf{I})$ are the
deterministic LoS component and the NLoS component, respectively. Likewise,
the BS-UE channel can be given by

$\displaystyle\mathbf{H}_{\mathrm{d}}[t]=\sqrt{L_{\mathrm{bu}}}\left(\kappa\bar{\mathbf{H}}_{\mathrm{d}}+\sqrt{1-\kappa^{2}}\tilde{\mathbf{H}}_{\mathrm{d}}[t]\right)\in\mathbb{C}^{N_{\mathrm{t}}\times
N_{\mathrm{r}}}.$ (5)

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

The temporal evolution of the NLoS Rayleigh fading channel is modeled by the
first-order autoregressive process [33], as given by

$\displaystyle\tilde{\mathbf{H}}_{\mathrm{r}}[t]=\rho\tilde{\mathbf{H}}_{\mathrm{r}}[t-1]+\mathbf{E}[t],\
\tilde{\mathbf{H}}_{\mathrm{d}}[t]=\rho\tilde{\mathbf{H}}_{\mathrm{d}}[t-1]+\mathbf{E}[t],$
(6)

where $\mathbf{E}[t]\sim\mathcal{CN}\left(0,(1-\rho^{2})\mathbf{I}\right)$
yields the circular complex Gaussian distribution. Following the Jakes’ model
[34], the time correlation coefficient is $\rho=J_{0}(2\pi
f_{d}\tau)=J_{0}(2\pi\bar{f}_{d})$. Here, $J_{0}(\cdot)$ is the zero-th order
Bessel function of the first kind; $f_{d}$ is the maximum Doppler frequency;
$\tau$ is the delay between the time when the CSI is measured and the time
when the BS produces the precoders based on the CSI; and
$\bar{f}_{d}=f_{d}\tau$ is the normalized Doppler frequency.

It is reasonable to assume that the channel ensemble $\mathcal{H}$ can be
accurately estimated. In a TDD system with channel reciprocity assumed, the BS
can estimate the CSI based on the pilots sent by the UE in the uplink and then
use the CSI to precode the subsequent downlink transmissions. $\tau$ can be up
to a slot. In time-varying channels, the CSI estimated in the uplink can be
outdated when used for downlink transmissions. On the other hand, the channel
reciprocity may not hold in a practical TDD system because the RF chains are
generally different/asymmetric in the uplink and downlink. The UE needs to
estimate the CSI in the downlink based on the pilots sent by the BS, and then
feeds the CSI back to the BS in the uplink, as done in FDD systems, as
specified in 3GPP standards [35]. $\tau$ is no shorter than a slot. Given the
non-negligible delay $\tau$, the channels may have changed substantially
between when they are measured and when they are used for transmissions.

When considering both channel estimation errors and channel aging, the channel
vector can be rewritten as
$\tilde{\mathbf{H}}_{\mathrm{r}}[t]=\rho^{\prime}\tilde{\mathbf{H}}_{\mathrm{r}}^{\mathrm{est}}[t-1]+\mathbf{E}[t]+\mathbf{E}_{\mathrm{est}}[t]$,
where $\tilde{\mathbf{H}}_{\mathrm{r}}^{\mathrm{est}}$ denotes the estimated
channel, $\mathbf{E}\sim\mathcal{CN}\left(\mathbf{0},(1-\rho^{\prime
2})(1-\sigma_{e}^{2})\mathbf{I}\right)$ accounts for the channel aging,
$\mathbf{E}_{\mathrm{est}}\sim\mathcal{CN}\left(\mathbf{0},(1-\rho^{\prime
2})\sigma_{e}^{2}\mathbf{I}\right)$ accounts for the channel estimation error,
$\rho^{\prime}$ is a channel correlation coefficient, and $\sigma_{e}^{2}$ is
the estimation error variance. With the statistical independence between
$\mathbf{E}_{\mathrm{est}}$ and $\mathbf{E}$, we can rewrite the channel
vector as
$\tilde{\mathbf{H}}_{\mathrm{r}}[t]=\rho^{\prime}\tilde{\mathbf{H}}_{\mathrm{r}}^{\mathrm{est}}[t-1]+\mathbf{E}^{\prime}[t]$
where
$\mathbf{E}^{\prime}[t]=\mathbf{E}[t]+\mathbf{E}_{\mathrm{est}}[t]\sim\mathcal{CN}(\mathbf{0},(1-\rho^{\prime
2})\mathbf{I})$. The channel aging and channel estimation error can be
aggregated as one estimation error that exhibits strong analogy to (6), and
can be dealt with in the same way as (6).

###  II-B System Architecture

In the considered, IRS-assisted, $N_{\mathrm{t}}\times N_{\mathrm{r}}$ MIMO
system, the BS transmit $M(M\leq N_{\mathrm{t}})$ data streams to the UE. At
the $t$-th slot, the received signal at the UE is given by

$\displaystyle\mathbf{y}[t]$
$\displaystyle=\mathbf{W}_{\mathrm{r}}^{\mathsf{H}}[t]\underbrace{\left(\mathbf{H}_{\mathrm{d}}^{\mathsf{H}}[t]+\mathbf{H}_{\mathrm{r}}^{\mathsf{H}}[t]\mathbf{\Theta}^{\mathsf{H}}[t]\mathbf{G}^{\mathsf{H}}[t]\right)}_{\text{effective
channel}}\mathbf{W}_{\mathrm{t}}[t]$
$\displaystyle\mathbf{\Lambda}^{\frac{1}{2}}[t]\mathbf{x}[t]+\mathbf{n}[t]$
$\displaystyle=\mathbf{W}_{\mathrm{r}}^{\mathsf{H}}[t]\check{\mathbf{H}}^{\mathsf{H}}[t]\mathbf{W}_{\mathrm{t}}[t]\mathbf{\Lambda}^{\frac{1}{2}}[t]\mathbf{x}[t]+\mathbf{n}[t],$
(7)

where $\mathbf{x}\in\mathbb{C}^{M\times 1}$ is the transmit signal,
$\mathbf{n}\sim\mathcal{CN}(0,\sigma_{n}^{2}\mathbf{I}_{N_{\mathrm{r}}})$ is
the additive Gaussian noise with
$\mathbb{E}({\mathbf{n}}{\mathbf{n}}^{\mathsf{H}})=\sigma_{n}^{2}\mathbf{I}_{M}$,
$\mathbf{W}_{\mathrm{t}}$ is the precoding matrix at the BS,
$\mathbf{W}_{\mathrm{r}}$ is the detection matrix at the UE,
$\mathbf{\Lambda}=\mathrm{diag}\left\\{[P_{1},\cdots,P_{M}]\right\\}\in\mathbb{C}^{M\times
M}$ collects the transmit powers allocated to each data stream,
$\mathbf{\Theta}=\mathrm{diag}\left\\{\exp(j\bm{\theta})\right\\}\in\mathbb{C}^{N\times
N}$ is the diagonal reflection matrix of the IRS, and
$\check{\mathbf{H}}\triangleq\mathbf{H}_{\mathrm{d}}+\mathbf{G}\mathbf{\Theta}\mathbf{H}_{\mathrm{r}}\in\mathbb{C}^{N_{\mathrm{t}}\times
N_{\mathrm{r}}}$ is defined as _effective channel_ for notational brevity.

In the time-varying channels, the precoding vector produced at the BS based on
the outdated CSI can lead to non-negligible inter-stream interference,
deteriorating the AAR performance. For this reason, we adopt an SVD-ZF
beamforming approach [36] and extend it under power constraints to optimize
the power allocation based on the outdated CSI, as illustrated in Fig. 2.
Specifically, the BS produces the precoder
$\mathbf{W}_{\mathrm{t}}[t]=\mathbf{V}[t-1]$ based on the outdated CSI
available at the BS, where $\mathbf{V}[t-1]$ is the right unitary matrix of
the singular value decomposition (SVD) of $\check{\mathbf{H}}[t-1]$. Both the
pilot signals and data symbols are precoded and transmitted through the
_equivalent end-to-end channel_ ,
$\underline{\mathbf{H}}[t]\triangleq\mathbf{V}[t-1]^{\mathsf{H}}\check{\mathbf{H}}[t]\in\mathbb{C}^{M\times
N_{\mathrm{r}}}$. The UE can estimate $\underline{\mathbf{H}}[t]$ based on the
precoded pilots. The estimation is straightforward, and can be readily done on
a slot basis, as typically done in conventional wireless communication systems
[37, 38, 39].

Based on the estimated equivalent channel, i.e., I-CSI, the UE detects the
data symbols using the following zero-forcing (ZF) detection matrix:

$\displaystyle\mathbf{W}_{\mathrm{r}}[t]$
$\displaystyle\triangleq\Big{(}\check{\mathbf{H}}^{\mathsf{H}}[t]\mathbf{V}[t-1]\Big{)}^{\dagger}$
$\displaystyle=\left[\Big{(}\check{\mathbf{H}}^{\mathsf{H}}[t]\mathbf{V}[t-1]\Big{)}^{\mathsf{H}}\check{\mathbf{H}}^{\mathsf{H}}[t]\mathbf{V}[t-1]\right]^{-1}$
$\displaystyle\Big{(}\check{\mathbf{H}}^{\mathsf{H}}[t]\mathbf{V}[t-1]\Big{)}^{\mathsf{H}}$
$\displaystyle=\left(\underline{\mathbf{H}}[t]\underline{\mathbf{H}}^{\mathsf{H}}[t]\right)^{-1}\underline{\mathbf{H}}[t].$
(8)

The received signal is obtained as

$\displaystyle\mathbf{y}[t]$
$\displaystyle=\mathbf{W}_{\mathrm{r}}^{\mathsf{H}}[t]\left(\check{\mathbf{H}}^{\mathsf{H}}[t]\mathbf{V}[t-1]\mathbf{\Lambda}^{\frac{1}{2}}[t]\mathbf{x}[t]+\mathbf{n}[t]\right)$
$\displaystyle=\left(\underline{\mathbf{H}}[t]\underline{\mathbf{H}}^{\mathsf{H}}[t]\right)^{-1}\underline{\mathbf{H}}[t]\underline{\mathbf{H}}^{\mathsf{H}}[t]\mathbf{\Lambda}^{\frac{1}{2}}[t]\mathbf{x}[t]+$
$\displaystyle\left(\underline{\mathbf{H}}[t]\underline{\mathbf{H}}^{\mathsf{H}}[t]\right)^{-1}\underline{\mathbf{H}}[t]\mathbf{n}[t]$
$\displaystyle=\mathbf{\Lambda}^{\frac{1}{2}}[t]\mathbf{x}[t]+\left(\underline{\mathbf{H}}[t]\underline{\mathbf{H}}^{\mathsf{H}}[t]\right)^{-1}\underline{\mathbf{H}}[t]\mathbf{n}[t].$
(9)

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

$\displaystyle\gamma_{m}[t]$
$\displaystyle=\frac{\Big{[}\mathbf{\Lambda}[t]\Big{]}_{(m,m)}}{\sigma_{n}^{2}\left[\left((\check{\mathbf{H}}^{\mathsf{H}}[t]\mathbf{V}[t-1])^{\mathsf{H}}(\check{\mathbf{H}}^{\mathsf{H}}[t]\mathbf{V}[t-1])\right)^{-1}\right]_{(m,m)}}$
$\displaystyle=\frac{P_{m}[t]}{\sigma_{n}^{2}\left[\left(\underline{\mathbf{H}}[t]\underline{\mathbf{H}}^{\mathsf{H}}[t]\right)^{-1}\right]_{(m,m)}}.$
(10)

Figure 2: The considered transmission architecture for fast-changing channels,
where the BS precodes the data symbols and pilots based on the truncated SVD
of outdated CSI, and then the UE estimates the I-CSI based on the precoded
pilots and detects data signals.

This beamforming strategy can efficiently eliminate the inter-stream
interference resulting from the SVD precoding based on the outdated CSI. The
SVD precoding with water-filling power allocation is also known to be optimal
under I-CSI [36], [40]. It is still likely to produce effective parallel
streams based on the outdated CSI, by taking advantage of the temporal
correlation of the autoregressive channels. To this end, the small-timescale
SVD-ZF beamforming strategy is increasingly closer to the optimum, as the
channel changes more slowly and undergoes stronger temporal correlation. The
strategy becomes optimal when the channel stops changing. On the other hand,
it may not be plausible to jointly optimize the precoders and detectors in the
considered system, due to the fast-changing channels and subsequently outdated
CSI at the BS. Since the BS is the one that optimizes the precoders, by no
means can it come up with the optimal detector that requires the I-CSI.

###  II-C Beamforming Strategy and Problem Formulation

It is not straightforward to estimate the effective channel
$\check{\mathbf{H}}$, because part of it can be augmented by the IRS. The
passive IRS meta-atoms are made up by PIN-diodes, and do not involve active RF
chains or any signal processing. For this reason, cascaded channel estimation
is widely adopted to estimate the effective channels [13]. The cascaded
channel estimation needs to be conducted, every time the IRS is reconfigured.
In consequence, unacceptably large training overhead and delay would be
required, if the reflection phase-shifts of the IRS are reconfigured
excessively frequently. Moreover, it is non-trivial to synchronize the BS/UE
and the IRS if the IRS is reconfigured as frequently as the BS in fast-
changing channels. Guard intervals need to be reserved for the
reconfigurations at the cost of spectrum utilization.

We propose a new two-timescale beamforming strategy, which configures the IRS
once per frame based on the statistical knowledge of the CSI and updates the
BS precoders every time slot based on available outdated CSI. As illustrated
in Fig. 3, the large-scale fading coefficients
$\\{L_{\mathrm{ur}},L_{\mathrm{bu}}\\}$ and the S-CSI ensemble
$\\{\bar{\mathbf{H}}_{\mathrm{r}},\bar{\mathbf{H}}_{\mathrm{d}}\\}$ are
evaluated at the beginning of a frame based on all CSI estimated in the
preceding frames, using the S-CSI estimation approach [41, 42], and assumed to
remain unchanged within the frame. Based on the S-CSI, the BS optimizes the
IRS passive beamforming for the entire frame. Given the large-timescale IRS
passive beamforming for a frame, the small-timescale precoding and detection
can be conducted slot by slot throughout the frame, as described in Section
II-B. Specifically, the BS performs the SVD precoding based on the outdated
CSI, and the UE conducts the ZF detector based on the I-CSI.

Figure 3: Pictorial illustration of the new two-timescale beamforming
strategy, where the S-CSI is estimated once per frame and used for the large-
timescale passive beamforming of the IRS. In contrast, the power allocation of
the BS is updated slot by slot, based on the outdated CSI available to the BS,
due to the non-negligible delay between the I-CSI estimation (e.g., at the UE)
and the power allocation.

The goal of the new two-timescale beamforming strategy is to maximize the AAR
of the considered IRS-assisted MIMO system by jointly optimizing the long-term
passive beamforming vector of the IRS and the per-slot power allocation of the
BS. This problem is formulated as

$\displaystyle\text{(P1)}:\quad$
$\displaystyle\underset{\mathbf{\Theta}}{\max}\
\mathbb{E}\left\\{\underset{\mathbf{\Lambda}}{\max}\
\sum_{m=1}^{M}\log\Big{[}1+\gamma_{m}(\mathbf{\Theta},\mathbf{\Lambda}[t])\Big{]}\right\\}$
$\displaystyle\mathrm{s.t.}\quad$
$\displaystyle\sum_{m=1}^{M}P_{m}=P_{\mathrm{tot}},\quad P_{m}\geq 0,\forall
m,$ (11a)
$\displaystyle\left|[\mathbf{\Theta}]_{n,n}\right|=1,\quad\mathrm{angle}\left([\mathbf{\Theta}]_{n,n}\right)\in[0,2\pi),\forall
n.$ (11b)

where (11a) guarantees the total transmit power of the BS does not exceed its
power budget $P_{\mathrm{tot}}$, and (11b) specifies the unit-modulus nature
and the phase-shift range of the IRS meta-atoms.

##  III Proposed Two-timescale Beamforming Strategy

In this section, we reformulate Problem (P1) as a joint design of power
allocation at the BS and passive beamforming at the IRS, and design new
modified PSO algorithms to solve the problem efficiently. We first rewrite the
SINR of the $m$-th data stream at slot $t$,
$\gamma_{m}(P_{m}[t],\mathbf{\Theta},\mathcal{H}[t-1])$, as

$\displaystyle\gamma_{m}(P_{m}[t],\mathbf{\Theta},\mathcal{H}[t-1])$
$\displaystyle=$
$\displaystyle\frac{P_{m}[t]}{\sigma_{n}^{2}\left[\left\\{\mathbf{V}^{\mathsf{H}}[t-1]\check{\mathbf{H}}[t]\check{\mathbf{H}}[t]^{\mathsf{H}}\mathbf{V}[t-1]\right\\}^{-1}\right]_{(m,m)}}$
$\displaystyle=$
$\displaystyle\frac{P_{m}[t]}{\sigma_{n}^{2}\left[\left\\{\underline{\mathbf{H}}[t]\underline{\mathbf{H}}[t]^{\mathsf{H}}\right\\}^{-1}\right]_{(m,m)}}$
$\displaystyle=$
$\displaystyle\frac{P_{m}[t]}{\sigma_{n}^{2}\left[\mathbf{\Phi}^{-1}[t]\right]_{(m,m)}}=\frac{P_{m}[t]}{\sigma_{n}^{2}f_{m}(\bm{\theta})[t]},$
(12)

where
$\mathbf{\Phi}[t]\triangleq\underline{\mathbf{H}}[t]\underline{\mathbf{H}}^{\mathsf{H}}[t]\in\mathbb{C}^{M\times
M}$ is the covariance matrix of the equivalent channel, and
$f_{m}(\bm{\theta})[t]=\left[\mathbf{\Phi}^{-1}(\bm{\theta})[t]\right]_{(m,m)}$
is the $m$-th diagonal element of the inverse of the covariance matrix.
Problem (P1) can be rewritten as

$\displaystyle\text{(P1')}:\quad$ $\displaystyle\underset{\bm{\theta}}{\max}\
\mathbb{E}\left\\{\underset{\mathbf{p}}{\max}\
\sum_{m=1}^{M}\log\left[1+\frac{P_{m}[t]}{\sigma_{n}^{2}f_{m}(\bm{\theta})[t]}\right]\right\\}$
$\displaystyle\mathrm{s.t.}\quad$ $\displaystyle\mathbf{p}\succeq
0,\quad\mathbf{1}^{\mathsf{T}}\mathbf{p}=P_{\mathrm{tot}},$ (13a)
$\displaystyle\mathbf{\Theta}=\mathrm{diag}\left\\{e^{j\bm{\theta}}\right\\},{\theta}_{n}\in[0,2\pi),\forall
n,$ (13b)

where $\mathbf{p}=[P_{1},\cdots,P_{M}]^{\mathsf{T}}$ collects the transmit
powers of the $M$ data streams.

Problem (P1’) is a stochastic optimization problem, because of the per-slot
optimization of the power allocation at the small timescale. At each time
slot, the power allocation is based on an outdated cascaded channel
measurement, and the fixed phase-shifts of the IRS configured at the beginning
of the current frame. On the other hand, the optimization of the IRS
configuration needs to predictively maximize the expectation of the AAR over
random channel samples and potential per-slot power allocations throughout the
frame. The typical approach which transforms a stochastic optimization problem
into a deterministic non-convex optimization problem, would confront serious
challenges, i.e., 1) lack of explicit relation between $\gamma_{i}$ and
$\bm{\theta}$; 2) nonconvex unit-modulus constraints on $\mathbf{\Theta}$; and
3) the need of minimizing each diagonal entry of the inverse of the covariance
matrix, i.e., $\left[\mathbf{\Phi}^{-1}\right]_{(m,m)}$, which is arduous.

###  III-A Small-timescale Power Allocation

Provided that the IRS passive beamforming is optimized at the beginning of the
frame; at the $t$-th slot of the frame, with the outdated CSI,
$\check{\mathbf{H}}[t-1]$, the original Problem (P1) reduces to:

$\displaystyle\text{(P2)}:\quad\underset{\mathbf{p}}{\max}\
\sum_{m=1}^{M}\log\left(1+\frac{P_{m}[t]f_{m}^{-1}(\bm{\theta})[t]}{\sigma_{n}^{2}}\right)\quad\mathrm{s.t.}\
\text{(\ref{eq:power1})}.$

Problem (P2) is convex with respect to the variable $\mathbf{p}$, and can be
solved using the off-the-peg water-filling algorithm [43]. With $\mu$ being
the Lagrange multiplier for constraint (13a) and $[x]^{+}=\max\\{x,0\\}$, the
optimal per-slot power allocation is given by [43]

$\displaystyle P_{m}^{\mathrm{opt}}[t]=\left[\frac{1}{\mu\ln
2}-\sigma_{n}^{2}f_{m}(\bm{\theta})[t]\right]^{+}.$ (14)

The corresponding per-slot sum-rate is given by

$\displaystyle\sum_{m=1}^{M}\log\left(1+\frac{P_{m}[t]}{\sigma_{n}^{2}f_{m}(\bm{\theta})[t]}\right)$
$\displaystyle=$
$\displaystyle\sum_{m=1}^{M}\log\left(1+\frac{1}{\sigma_{n}^{2}}\left[\frac{1}{\mu
f_{m}(\bm{\theta})[t]\ln 2}-\sigma_{n}^{2}\right]^{+}\right).$ (15)

The proposed framework is still operational even if the channel is independent
between different time slots. In this case, the SVD precoding produces a
transmit beamformer independent of the current channel, and can be viewed as
an arbitrary precoding matrix for the current slot. Given the ZF detector at
the UE, the inter-stream interference can be suppressed. Nonetheless, the use
of an arbitrary precoding receive no benefit from the transmit power
allocation relying on effective decomposition of the channel into parallel
streams.

###  III-B Data-driven Large-timescale Passive Beamforming with Mini-batch-
sampling-based PSO

The per-frame passive beamforming 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 $L_{B}$ random channel samples to
approximate the expectation in the objective. Suppose that the $l$-th sample
of the IRS-related time-variant channels in the $t$-th time slot is
$\hat{\mathcal{H}}_{\mathrm{irs}}^{l}[t]=\\{\hat{\mathbf{H}}_{\mathrm{d}}^{l}[t-1],\hat{\mathbf{H}}_{\mathrm{r}}^{l}[t-1]\\}$,
accounting for the outdated CSI at the BS. The AAR of the $m$-th data-stream
is approximated by

$\displaystyle\bar{r}_{m}\left(P_{m}^{\mathrm{opt}}(\mathbf{\Theta},\hat{\mathcal{H}}_{\mathrm{irs}}),\mathbf{\Theta}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{T}\frac{1}{L_{B}}\sum_{t=1}^{T}\sum_{l=1}^{L_{B}}\log\left[1+\gamma_{m}\left(P_{m}[t],\mathbf{\Theta},\hat{\mathcal{H}}_{\mathrm{irs}}^{l}[t-1]\right)\right].$
(16)

The large-timescale passive beamforming problem is cast as

$\displaystyle\text{(P3)}:\quad\underset{\mathbf{\Theta}}{\max}\
\sum_{m=1}^{M}\bar{r}_{m}\left(P_{m}^{\mathrm{opt}}(\mathbf{\Theta},\hat{\mathcal{H}}_{\mathrm{irs}}),\mathbf{\Theta}\right)\quad\mathrm{s.t.}\
\mathrm{(\ref{eq:theta1})}.$

Apart from the nonconvex constant-modulus constraint, the power allocation
variables are closely coupled with the reflection phase-shifts of the IRS.
Moreover, to maximize the AAR involves minimizing the diagonal entries of the
inverse of the covariance matrix, i.e.,
$\left[\mathbf{\Phi}^{-1}\right]_{(m,m)}$, which makes it difficult to
optimize $\mathbf{\Theta}$ since $\left[\mathbf{\Phi}^{-1}\right]_{(m,m)}$
cannot be written as an explicit function of $\mathbf{\Theta}$. Conventional
convex optimization methods or convexification methods, such as successive
convex relaxation, cannot be applied to solve Problem (P3).

We propose a new PSO framework to solve Problem (P3). The PSO, a widely-used
meta-heuristic bionic optimization algorithm [15], can find the optimal
solution with little prior information. It enjoys the benefits of fewer
parameters, simple calculation implementation, and fast convergence [44]. In
the PSO method, the position of a particle stands for a potential solution.
The fitness function of a particle is typically defined to be the optimization
objective. We define a passive reflection angle vector $\bm{\theta}$ as the
position of a particle, which corresponds to a passive beamforming matrix
$\mathbf{\Theta}=\mathrm{diag}\\{e^{j\bm{\theta}}\\}$. The objective of
Problem (P3), i.e., the AAR, is used as the fitness function, as given by

$\displaystyle
J(\bm{\theta})=\sum_{m=1}^{M}\bar{r}_{m}\left(P_{m}^{\mathrm{opt}}(\bm{\theta},\hat{\mathcal{H}}_{\mathrm{irs}}),\bm{\theta}\right).$
(18)

We note that, for each particle $\bm{\theta}$, the optimal power solution can
be uniquely found according to (14). This guarantees the uniqueness of the
fitness function 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 a velocity at every iteration. The set of particle positions, denoted by
$\mathcal{L}_{P}=\\{\bm{\theta}_{1},\bm{\theta}_{2},\cdots,\bm{\theta}_{P}\\}$,
corresponds to the set of passive beamforming matrices
$\\{\mathbf{\Theta}_{1},\mathbf{\Theta}_{2},\cdots,\mathbf{\Theta}_{P}\\}$. In
every iteration, a particle is assessed based on its fitness function to
determine whether the current position implies a good solution. The particle
records its best position ever found.

The global optimal position is selected from the best positions of all
particles. Let $\dot{\bm{\theta}}_{p}^{(i)}$ and $\ddot{\bm{\theta}}^{(i)}$
denote the best position of the $p$-th particle and the global optimal
position of all particles at the $i$-th iteration, respectively. The $p$-th
particle updates its position according to its velocity $\mathbf{v}_{p}^{(i)}$
at the $i$-th 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 the fitness value evaluation,
the global optimal position of all particles is accordingly updated. Both the
velocity and position of the $p$-th particle at the $i$-th iteration are
updated by

$\displaystyle\mathbf{v}_{p}^{(i)}$
$\displaystyle=w\mathbf{v}_{p}^{(i-1)}+c_{1}\varepsilon_{1}(\dot{\bm{\theta}}_{p}^{(i-1)}-\bm{\theta}_{p}^{(i-1)})+$
$\displaystyle
c_{2}\varepsilon_{2}(\ddot{\bm{\theta}}^{(i-1)}-\bm{\theta}_{p}^{(i-1)}),$
(19) $\displaystyle\bm{\theta}_{p}^{(i)}$
$\displaystyle=\bm{\theta}_{p}^{(i-1)}+\mathbf{v}_{p}^{(i)},$ (20)

where $w$ is the nonnegative inertia weight of a particle; $c_{1}$ and $c_{2}$
denote cognitive and social scaling factors, respectively; and
$\varepsilon_{1}$ and $\varepsilon_{2}$ are two independent random variables
with a uniform distribution in $(0,1)$. In addition, the control parameters,
such as the swarm size $P$ and iteration number $I_{\mathrm{iter}}$, are
critical for the convergence rate and global search performance. Once the
parameters updated in (20) are outside $(-\pi,\pi]$, their boundary values are
taken.

Each particle requires a notable amount of calculations for the fitness
function evaluation per iteration, since the fitness function is composed of a
large number of AARs over random channel samples. It can be computationally
prohibitive to take large-scale channel samples (which are four-dimensional
matrices with size of $N_{\mathrm{t}}\times N_{\mathrm{r}}\times L_{B}\times
P$) to obtain a fitness function value per iteration, potentially hindering
the convergence of the PSO model.

We develop a new mini-batch recursive sampling (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_ [32] to replace
the fitness function in (18). All $L_{B}$ random channel samples are
partitioned into $N_{B}$ batches with $L_{\mathrm{mb}}=L_{B}/N_{B}$ samples
per batch. At each iteration, the mini-batch sampling surrogate function is
given by

$\displaystyle
J^{(i)}(\bm{\theta})=(1-\mu^{(i)})J^{(i-1)}+\mu^{(i)}\sum_{m=1}^{M}\sum_{t=1}^{T}$
$\displaystyle\sum_{l=(i-1)L_{\mathrm{mb}}+1}^{iL_{\mathrm{mb}}}\frac{\log\left[1+\gamma_{m}\left(P_{m}[t],\bm{\theta},\hat{\mathcal{H}}_{\mathrm{irs}}^{l}[t-1]\right)\right]}{TL_{\mathrm{mb}}},$
(21)

where $\mu^{(i)}$ is an iteration-dependent constant accounting for the decay
weight coefficient associated with the new sampling of the AAR. Here, we set
$\mu^{(i)}=i^{-0.2}$ according to the diminishing stepsize rules [45], which
is commonly used to guarantee more accurate sampling approximation as $i$
increases in the stochastic optimization.

The proposed mbs-PSO algorithm is summarized in Algorithm 1. The algorithm is
_data-driven_ in spirit, given its use of random channel samples. Given the
S-CSI estimated at the beginning of a frame, the BS can generate the $L_{B}$
channel samples to approximate the future channels in the frame used for
downlink transmissions. Recall that there are $T$ slots in the frame. The BS
can generate the NLOS components of $\frac{L_{B}}{T}$ channel samples for the
IRS-UE channel for the $t$-th of the $T$ slots, as given by

$\displaystyle\tilde{\mathbf{H}}_{\mathrm{r}}^{l}[t]=\rho\tilde{\mathbf{H}}_{\mathrm{r}}^{l}[t-1]+\mathbf{E}[t],\
l=1,\cdots,\frac{L_{B}}{T};t=1,\cdots,T,$ (22)

where $\tilde{\mathbf{H}}_{\mathrm{r}}^{l}\sim\mathcal{CN}(0,\mathbf{I})$ is
the NLoS component of the $l$-th IRS-UE channel sample. Then, the
$\frac{L_{B}}{T}$ channel samples of the IRS-UE channel for the $t$-th slot
are given by

$\displaystyle\mathbf{H}_{\mathrm{r}}^{l}[t]$
$\displaystyle=\sqrt{L}\left(\kappa\bar{\mathbf{H}}_{\mathrm{r}}+\sqrt{1-\kappa^{2}}\tilde{\mathbf{H}}_{\mathrm{r}}^{l}[t]\right),$
$\displaystyle l=1,\cdots,\frac{L_{B}}{T};t=1,\cdots,T,$ (23)

where $L$ is the large-scale fading coefficient and
$\bar{\mathbf{H}}_{\mathrm{r}}$ is a constant LoS component during the frame.
By repeating this process from $t=1$ to $T$, we can obtain $\frac{L_{B}}{T}$
series of time-varying channel samples to approximate the IRS-UE channels of
the upcoming $T$ slots. The $L_{B}$ samples of the BS-UE channel,
$\mathbf{H}_{\mathrm{d}}^{l}[t]$, can be obtained in the same way as
$\mathbf{H}_{\mathrm{r}}^{l}[t]$.

1 Step 1: Swarm initialization

Initialize the PSO parameters :
$\\{P,w,c_{1},c_{2},\varepsilon_{1},\varepsilon_{2},I_{\mathrm{iter}}\\}$.

Initialize the position and velocity :
$\\{\bm{\theta}_{p}^{(0)}\\}_{p=1}^{P},\\{\mathbf{v}_{p}^{(0)}\\}_{p=1}^{P}$.

2 Generate $L_{B}$ outdated channel samples according to the S-CSI and first-
order AR model;

3 Separate the total channel samples into $N_{B}$ mini batches with each
$L_{\mathrm{mb}}$ samples;

4 For each particle $\bm{\theta}_{p}^{(0)}$, obtain the associated optimal
powers $\mathbf{p}^{(0)}_{p}$ by using (14);

5 Evaluate the fitness value in (21) of each particle using first mini-batch
of samples, and obtain the initial best position set
$\\{\dot{\bm{\theta}}_{p}\\}_{p=1}^{P}$;

6 Find the global optimal position
$\ddot{\bm{\theta}}=\arg\max_{\dot{\bm{\theta}}}\\{J^{(0)}(\dot{\bm{\theta}}_{1}),J^{(0)}(\dot{\bm{\theta}}_{2}),\cdots,J^{(0)}(\dot{\bm{\theta}}_{P})\\}$;

7 end

8

9 Step 2: Iterative search

10 for  _$i=1:I_{\mathrm{iter}}$_ do

11 Update particle velocity $\\{\mathbf{v}_{p}^{(i)}\\}_{p=1}^{P}$ and
position $\\{\bm{\theta}_{p}^{(i)}\\}_{p=1}^{P}$ by using (19) and (20);

12 for  _$p=1:P$_ do

13 Given particle $\bm{\theta}_{p}^{(i)}$, calculate the optimal powers
$\mathbf{p}^{(i)}_{p}$ according to (14);

14 Evaluate the fitness value of particle $\bm{\theta}_{p}^{(i)}$ over
$i^{\mathrm{th}}$ mini-batch of samples by using (21);

15 if  _$J^{(i)} >J^{(i-1)}$_ then

16 $\dot{\bm{\theta}}_{p}=\bm{\theta}_{p}^{(i)}$;

17 else

18 $\dot{\bm{\theta}}_{p}=\bm{\theta}_{p}^{(i-1)}$;

19 end if

20 Update
$\ddot{\bm{\theta}}_{p}=\arg\max_{\dot{\bm{\theta}}}\\{J^{(i)}(\dot{\bm{\theta}}_{1}),J^{(i)}(\dot{\bm{\theta}}_{2}),\cdots,J^{(i)}(\dot{\bm{\theta}}_{P})\\}$;

21

22 end for

23

24 end for

25

26 end

27

Algorithm 1 Proposed mbs-PSO Algorithm

###  III-C Model-driven Large-timescale Passive Beamforming with Lower-Bound-
Oriented PSO

We proceed to develop a new _model-driven_ PSO method for optimizing the IRS
passive beamforming, which only exploits the S-CSI and the known random
distributions of the channels. The new model-driven PSO algorithm, referred to
as Lower-Bound-Oriented PSO (LBO-PSO), can avoid the use of random channel
samples (cf. Algorithm 1), thereby reducing its reliance on wireless data
samples and the risk of approximation inaccuracy. Following are the details of
the LBO-PSO algorithm, where the slot index $t$ is dropped for illustration
convenience.

First, by using the arithmetic-geometric inequality [46], the lower bound for
the objective of Problem (P1) is given by (24), shown at the top of the next
page.

$\displaystyle\mathbb{E}\left\\{\sum_{m=1}^{M}\log\left(1+\frac{P_{m}}{\sigma_{n}^{2}\left[\mathbf{\Phi}^{-1}\right]_{(m,m)}}\right)\right\\}\geq\mathbb{E}\left\\{M\log\left(1+\frac{M}{\sigma_{n}^{2}\sum_{m=1}^{M}P_{m}^{-1}\left[\mathbf{\Phi}^{-1}\right]_{(m,m)}}\right)\right\\}$
$\displaystyle\geq$ $\displaystyle
M\log\left(1+\frac{M}{\sigma_{n}^{2}\mathbb{E}\left\\{\sum_{m=1}^{M}P_{m}^{-1}\left[\mathbf{\Phi}^{-1}\right]_{(m,m)}\right\\}}\right)=M\log\left(1+\frac{\tilde{M}}{\sum_{m=1}^{M}P_{m}^{-1}\mathbb{E}\left\\{\left[\mathbf{\Phi}^{-1}\right]_{(m,m)}\right\\}}\right)$
$\displaystyle=$ $\displaystyle
M\log\left(1+\frac{\tilde{M}}{\mathrm{trace}\left(\mathbf{\Lambda}^{-1}\mathbb{E}\left\\{\mathbf{\Phi}^{-1}\right\\}\right)}\right),$
(24)

In (24), $\tilde{M}\triangleq M/\sigma_{n}^{2}$,
$\mathbf{\Lambda}^{-1}=\mathrm{diag}\left\\{[P_{1}^{-1},\cdots,P_{M}^{-1}]\right\\}$,
and the second inequality is based on Jensen’s inequality. Instead of directly
solving Problem (P1’), we maximize the lower bound of Problem (P1’) here, as
given by

$\displaystyle\text{(P4)}:\quad$
$\displaystyle\underset{\bm{\theta},\mathbf{p}}{\max}\
\log\left(1+\frac{\tilde{M}}{\mathrm{trace}\left(\mathbf{\Lambda}^{-1}\mathbb{E}\left\\{\mathbf{\Phi}^{-1}(\bm{\theta})\right\\}\right)}\right)$
$\displaystyle\mathrm{s.t.}\
\mathrm{(\ref{eq:power1})},\mathrm{(\ref{eq:theta1})},$

which can be further equivalently rewritten as

$\displaystyle\text{(P5)}:\quad\underset{\bm{\theta},\mathbf{p}}{\min}\
\mathrm{trace}\left(\mathbf{\Lambda}^{-1}\mathbb{E}\left\\{\mathbf{\Phi}^{-1}(\bm{\theta})\right\\}\right)\quad\mathrm{s.t.}\
\mathrm{(\ref{eq:power1})},\mathrm{(\ref{eq:theta1})}.$

Next, we derive an analytical approximation of the objective in Problem (P5)
by invoking the property of the _complex inverse Wishart matrix_ [47]. We
prove that $\mathbf{\Phi}$ yields the Wishart distribution by first proving
that $\check{\mathbf{H}}$ is a complex Gaussian distributed matrix. Without
loss of generality, we assume that
$\check{\mathbf{H}}\in\mathbb{C}^{N_{\mathrm{t}}\times N_{\mathrm{r}}}$. We
expand matrix $\check{\mathbf{H}}\in\mathbb{C}^{N_{\mathrm{t}}\times
N_{\mathrm{r}}}$ as

$\displaystyle\check{\mathbf{H}}=$
$\displaystyle\mathbf{H}_{\mathrm{d}}+\mathbf{G}\mathbf{\Theta}\mathbf{H}_{\mathrm{r}}$
$\displaystyle=$
$\displaystyle\sqrt{L_{\mathrm{bu}}}\left(\kappa\bar{\mathbf{H}}_{\mathrm{d}}+\sqrt{1-\kappa^{2}}\tilde{\mathbf{H}}_{\mathrm{d}}\right)+$
$\displaystyle\mathbf{G}\mathbf{\Theta}\left(\sqrt{L_{\mathrm{ur}}}\left(\kappa\bar{\mathbf{H}}_{\mathrm{r}}+\sqrt{1-\kappa^{2}}\tilde{\mathbf{H}}_{\mathrm{r}}\right)\right)$
$\displaystyle=$
$\displaystyle\sqrt{\kappa^{2}L_{\mathrm{bu}}}\bar{\mathbf{H}}_{\mathrm{d}}+\sqrt{\kappa^{2}L_{\mathrm{ur}}}\mathbf{G}\mathbf{\Theta}\bar{\mathbf{H}}_{\mathrm{r}}+\sqrt{\kappa_{\perp}^{2}L_{\mathrm{bu}}}\tilde{\mathbf{H}}_{\mathrm{d}}$
$\displaystyle+\sqrt{\kappa_{\perp}^{2}L_{\mathrm{ur}}}\mathbf{G}\mathbf{\Theta}\tilde{\mathbf{H}}_{\mathrm{r}},$
(25)

where
$\check{\mathbf{H}}=[\check{\mathbf{h}}_{1},\check{\mathbf{h}}_{2},\cdots,\check{\mathbf{h}}_{N_{\mathrm{r}}}]$
and $\kappa_{\perp}^{2}=1-\kappa^{2}$.

For ease of exposition, we define
$\bar{\mathbf{H}}_{\mathrm{d}}=[\bar{\mathbf{d}}_{1},\bar{\mathbf{d}}_{2},\cdots,\bar{\mathbf{d}}_{N_{\mathrm{r}}}]$,
$\tilde{\mathbf{H}}_{\mathrm{d}}=[\tilde{\mathbf{d}}_{1},\tilde{\mathbf{d}}_{2},\cdots,\tilde{\mathbf{d}}_{N_{\mathrm{r}}}]$,
$\bar{\mathbf{H}}_{\mathrm{r}}=[\bar{\mathbf{r}}_{1},\bar{\mathbf{r}}_{2},\cdots,\bar{\mathbf{r}}_{N_{\mathrm{r}}}]$
and
$\tilde{\mathbf{H}}_{\mathrm{r}}=[\tilde{\mathbf{r}}_{1},\tilde{\mathbf{r}}_{2},\cdots,\tilde{\mathbf{r}}_{N_{\mathrm{r}}}]$.
Since $\tilde{\mathbf{H}}_{\mathrm{d}}$ and $\tilde{\mathbf{H}}_{\mathrm{r}}$
are mutual independent and both contain complex Gaussian variables, we have
$\tilde{\mathbf{d}}_{j}\sim\mathcal{CN}(\mathbf{0},\mathbf{I}_{N_{\mathrm{t}}})$
and $\tilde{\mathbf{r}}_{j}\sim\mathcal{CN}(\mathbf{0},\mathbf{I}_{N}),\
\forall j=1,2,\cdots,N_{\mathrm{r}}$. Since the linear transformation of a
Gaussian random vector or the sum of independent Gaussian vectors is Gaussian
[48], the $j$-th column of $\check{\mathbf{H}}$ yields

$\displaystyle\check{\mathbf{h}}_{j}$
$\displaystyle=\sqrt{\kappa^{2}L_{\mathrm{bu}}}\bar{\mathbf{d}}_{j}+\sqrt{\kappa^{2}L_{\mathrm{ur}}}\mathbf{G}\mathbf{\Theta}\bar{\mathbf{r}}_{j}+\sqrt{\kappa_{\perp}^{2}L_{\mathrm{bu}}}\tilde{\mathbf{d}}_{j}$
$\displaystyle+\sqrt{\kappa_{\perp}^{2}L_{\mathrm{ur}}}\mathbf{G}\mathbf{\Theta}\tilde{\mathbf{r}}_{j}\sim\mathcal{CN}\Big{(}\sqrt{\kappa^{2}L_{\mathrm{bu}}}\bar{\mathbf{d}}_{j}+\sqrt{\kappa^{2}L_{\mathrm{ur}}}\mathbf{G}\mathbf{\Theta}\bar{\mathbf{r}}_{j},$
$\displaystyle\kappa_{\perp}^{2}L_{\mathrm{bu}}\mathbf{I}_{N_{\mathrm{t}}}+\kappa_{\perp}^{2}L_{\mathrm{ur}}\mathbf{G}\mathbf{G}^{\mathsf{H}}\Big{)}.$
(26)

Define
$\underline{\mathbf{H}}=[\underline{\mathbf{h}}_{1},\underline{\mathbf{h}}_{2},\cdots,\underline{\mathbf{h}}_{N_{\mathrm{r}}}]$.
Likewise, the $j$-th column of $\underline{\mathbf{H}}$ yields
$\underline{\mathbf{h}}_{j}=\mathbf{V}^{\mathsf{H}}\check{\mathbf{h}}_{j}\sim\mathcal{CN}(\sqrt{\kappa^{2}L_{\mathrm{bu}}}\mathbf{V}^{\mathsf{H}}\bar{\mathbf{d}}_{j}+\sqrt{\kappa^{2}L_{\mathrm{ur}}}\mathbf{V}^{\mathsf{H}}\mathbf{G}\mathbf{\Theta}\bar{\mathbf{r}}_{j},\kappa_{\perp}^{2}L_{\mathrm{bu}}\mathbf{I}_{M}+\kappa_{\perp}^{2}L_{\mathrm{ur}}\mathbf{V}^{\mathsf{H}}\mathbf{G}\mathbf{G}^{\mathsf{H}}\mathbf{V})$.
As a result, $\underline{\mathbf{H}}$ is a complex Gaussian distributed
matrix, i.e.,
$\underline{\mathbf{H}}\sim\mathcal{CN}\left(\mathbf{S}_{\underline{\mathbf{H}}},\mathbf{\Gamma}_{\underline{\mathbf{H}}}\right)$,
where
$\mathbf{S}_{\underline{\mathbf{H}}}=\sqrt{\kappa^{2}L_{\mathrm{bu}}}\mathbf{V}^{\mathsf{H}}\bar{\mathbf{H}}_{\mathrm{d}}+\sqrt{\kappa^{2}L_{\mathrm{ur}}}\mathbf{V}^{\mathsf{H}}\mathbf{G}\mathbf{\Theta}\bar{\mathbf{H}}_{\mathrm{r}}$
and
$\mathbf{\Gamma}_{\underline{\mathbf{H}}}=\mathbf{I}_{N_{\mathrm{r}}}\otimes\left(\kappa_{\perp}^{2}L_{\mathrm{bu}}\mathbf{I}_{M}+\kappa_{\perp}^{2}L_{\mathrm{ur}}\mathbf{V}^{\mathsf{H}}\mathbf{G}\mathbf{G}^{\mathsf{H}}\mathbf{V}\right)$.

######  Definition 1.

Suppose a random matrix $\mathbf{X}\in\mathbb{C}^{q\times p}\ (q\leq p)$,
where any two columns $\mathbf{x}_{i}$ and $\mathbf{x}_{j}$ are independent
for any $i,j\ (i\neq j)$. Each column corresponds to a covariance matrix
${\mathbf{\Sigma}}={\mathbb{E}}\left[{\mathbf{x}}_{i}{\mathbf{x}}_{i}^{\mathsf{H}}\right],\forall_{i}$.
If every element of $\mathbf{X}$ follows the complex Gaussian distribution,
i.e., $\mathbf{x}_{ij}\sim\mathcal{CN}(0,1)$ and
$\mathbb{E}(\mathbf{X})=\mathbf{0}$, then the Hermitian matrix
${\mathbf{X}}{\mathbf{X}}^{\mathsf{H}}\sim{\mathcal{W}}_{q}(p,{\mathbf{\Sigma}})$
yields the central Wishart distribution. For
$\mathbb{E}(\mathbf{X})=\mathbf{M}\neq\mathbf{0}$,
${\mathbf{X}}{\mathbf{X}}^{\mathsf{H}}\sim{\mathcal{W}}_{q}(p,\mathbf{M},{\mathbf{\Sigma}})$
follows the non-central Wishart.

According to Definition 1,
$\mathbf{\Phi}=\underline{\mathbf{H}}\underline{\mathbf{H}}^{\mathsf{H}}$
follows a non-central Wishart distribution, i.e.,

$\mathbf{\Phi}\sim\mathcal{W}_{M}\left(N_{\mathrm{r}},\mathbf{M}_{\mathbf{\Phi}},\mathbf{\Sigma}_{\mathbf{\Phi}}\right).$
(27)

In (27), for notational brevity, we define

$\displaystyle\mathbf{M}_{\mathbf{\Phi}}$
$\displaystyle=\kappa_{\perp}^{2}L_{\mathrm{bu}}\mathbf{I}_{M}+\kappa_{\perp}^{2}L_{\mathrm{ur}}\mathbf{V}^{\mathsf{H}}\mathbf{G}\mathbf{G}^{\mathsf{H}}\mathbf{V},$
(28) $\displaystyle\mathbf{\Sigma}_{\mathbf{\Phi}}$
$\displaystyle=\mathbf{M}_{\mathbf{\Phi}}^{-1}\mathbf{S}_{\underline{\mathbf{H}}}\mathbf{S}_{\underline{\mathbf{H}}}^{\mathsf{H}}=\mathbf{M}_{\mathbf{\Phi}}^{-1}\mathbf{V}^{\mathsf{H}}\Big{(}\bar{\mathbf{R}}_{\mathrm{dd}}+\bar{\mathbf{R}}_{\mathrm{dr}}\mathbf{\Theta}^{\mathsf{H}}\mathbf{G}^{\mathsf{H}}+$
$\displaystyle\mathbf{G}\mathbf{\Theta}\bar{\mathbf{R}}_{\mathrm{rd}}+\mathbf{G}\mathbf{\Theta}\bar{\mathbf{R}}_{\mathrm{rr}}\mathbf{\Theta}^{\mathsf{H}}\mathbf{G}^{\mathsf{\mathsf{H}}}\Big{)}\mathbf{V},$
(29)

where
$\bar{\mathbf{R}}_{\mathrm{dd}}=L_{\mathrm{bu}}\kappa^{2}\bar{\mathbf{H}}_{\mathrm{d}}\bar{\mathbf{H}}_{\mathrm{d}}^{\mathsf{H}}\in\mathbb{C}^{N_{\mathrm{t}}\times
N_{\mathrm{t}}}$,
$\bar{\mathbf{R}}_{\mathrm{dr}}=\kappa\kappa_{\perp}\sqrt{L_{\mathrm{bu}}L_{\mathrm{ur}}}\bar{\mathbf{H}}_{\mathrm{d}}\bar{\mathbf{H}}_{\mathrm{r}}^{\mathsf{H}}\in\mathbb{C}^{N_{\mathrm{t}}\times
N}$,
$\bar{\mathbf{R}}_{\mathrm{rd}}=\bar{\mathbf{R}}_{\mathrm{dr}}^{\mathsf{H}}$,
and
$\bar{\mathbf{R}}_{\mathrm{rr}}=L_{\mathrm{ur}}\kappa^{2}\bar{\mathbf{H}}_{\mathrm{r}}\bar{\mathbf{H}}_{\mathrm{r}}^{\mathsf{H}}\in\mathbb{C}^{N\times
N}$.

######  Definition 2.

Suppose that matrix $\mathbf{A}=\mathbf{X}\mathbf{X}^{\mathsf{H}}$ follows the
Wishart distribution of
$\mathcal{W}_{M_{A}}(N_{A},\mathbf{M}_{A},\mathbf{\Sigma}_{A})$, where $N_{A}$
is the degree of freedom (with $N_{A}\geq M_{A}$), $\mathbf{M}_{A}$ is the
mean matrix of $\mathbf{H}$, and
$\mathbf{\Sigma}_{A}\in\mathbb{C}^{M_{A}\times M_{A}}$ is the covariance
matrix of $\mathbf{H}$. Then, the inverse $\mathbf{A}^{-1}$ has the mean

$\displaystyle\mathbb{E}(\mathbf{A}^{-1})=\frac{\mathbf{\Sigma}_{A}^{-1}}{N_{A}-M_{A}}.$
(30)

where $\mathbb{E}(\mathbf{A})=N_{A}\mathbf{\Sigma}_{A}$. The mean of the
inverse is larger than the inverse of the mean, since
$\mathbb{E}(\mathbf{A})^{-1}=\frac{\mathbf{\Sigma}_{A}^{-1}}{N_{A}}<\mathbb{E}(\mathbf{A}^{-1})$.

###### Proof:

The detailed proof can be found in [49]. ∎

By approximating (27) with the central Wishart distribution with the same
first-order moment [47], we obtain the central Wishart distribution of
$\mathbf{\Phi}$ as follows:

$\displaystyle\mathbf{\Phi}\sim\mathcal{W}_{M}\left(N_{\mathrm{r}},\mathbf{M}_{\mathbf{\Phi}}+\frac{1}{N_{\mathrm{r}}}\mathbf{S}_{\underline{\mathbf{H}}}\mathbf{S}_{\underline{\mathbf{H}}}^{\mathsf{H}}\right).$
(31)

According to Definition 2, the expectation of the matrix inverse is given by
[48]

$\displaystyle\mathbb{E}\left(\mathbf{\Phi}^{-1}\right)=\frac{\left(\mathbf{M}_{\mathbf{\Phi}}+\frac{1}{N_{\mathrm{r}}}\mathbf{S}_{\underline{\mathbf{H}}}\mathbf{S}_{\underline{\mathbf{H}}}^{\mathsf{H}}\right)^{-1}}{N_{\mathrm{r}}-M}.$
(32)

The objective of Problem (P5) can be equivalently rewritten as

$\displaystyle\mathrm{trace}\left(\mathbf{\Lambda}^{-1}\mathbb{E}\left\\{\mathbf{\Phi}^{-1}(\bm{\theta})\right\\}\right)$
$\displaystyle=$
$\displaystyle\mathrm{trace}\left(\mathbf{\Lambda}^{-1}\frac{\left(\mathbf{M}_{\mathbf{\Phi}}+\frac{1}{N_{\mathrm{r}}}\mathbf{S}_{\underline{\mathbf{H}}}\mathbf{S}_{\underline{\mathbf{H}}}^{\mathsf{H}}\right)^{-1}}{N_{\mathrm{r}}-M}\right)$
$\displaystyle=$
$\displaystyle\frac{\mathrm{trace}\left(\mathbf{\Lambda}^{-1}\left(\mathbf{M}_{\mathbf{\Phi}}+\frac{1}{N_{\mathrm{r}}}\mathbf{S}_{\underline{\mathbf{H}}}\mathbf{S}_{\underline{\mathbf{H}}}^{\mathsf{H}}\right)^{-1}\right)}{N_{\mathrm{r}}-M}.$
(33)

######  Lemma 1.

Let $\mathbf{A}\succ\mathbf{0}$ be an $N\times N$ symmetric positive definite
matrix and $\mathbf{B}\succeq\mathbf{0}$ be an $N\times N$ positive
semidefinite matrix. Then,

$\displaystyle\operatorname{trace}(\mathbf{A}\mathbf{B})\leq$
$\displaystyle\|\mathbf{B}\|_{2}\operatorname{trace}(\mathbf{A})=\lambda_{\max}(\mathbf{B})\operatorname{trace}(\mathbf{A})$
$\displaystyle\leq$
$\displaystyle\operatorname{trace}(\mathbf{A})\operatorname{trace}(\mathbf{B}).$
(34)

Since $\|\mathbf{B}\|_{2}<\operatorname{trace}(\mathbf{B})$,
$\lambda_{\max}(\mathbf{B})\operatorname{trace}(\mathbf{A})$ provides a
tighter upper bound than
$\operatorname{trace}(\mathbf{A})\operatorname{trace}(\mathbf{B})$ for
$\operatorname{trace}(\mathbf{AB})$.

###### Proof:

See Appendix A. ∎

By further applying _Lemma_ 1, (33) is upper bounded by

$\displaystyle\frac{\mathrm{trace}\left(\mathbf{\Lambda}^{-1}\mathbf{D}^{-2}\right)}{N_{\mathrm{t}}-M}$
$\displaystyle\leq$
$\displaystyle\frac{\lambda_{\max}(\mathbf{\Lambda}^{-1})}{N_{\mathrm{t}}-M}\mathrm{trace}\left(\left(\mathbf{M}_{\mathbf{\Phi}}+\frac{1}{N_{\mathrm{r}}}\mathbf{S}_{\underline{\mathbf{H}}}\mathbf{S}_{\underline{\mathbf{H}}}^{\mathsf{H}}\right)^{-1}\right)$
$\displaystyle=$
$\displaystyle\frac{N_{\mathrm{r}}\lambda_{\max}(\mathbf{\Lambda}^{-1})}{N_{\mathrm{t}}-M}\mathrm{trace}\left[\left(N_{\mathrm{r}}\mathbf{M}_{\mathbf{\Phi}}+\mathbf{S}_{\underline{\mathbf{H}}}\mathbf{S}_{\underline{\mathbf{H}}}^{\mathsf{H}}\right)^{-1}\right].$
(35)

As a result, the objective of Problem (P5) can be updated as

$\displaystyle\text{(P6)}:\ \underset{\bm{\theta},\mathbf{p}}{\min}\
\mathrm{trace}\left[\left(N_{\mathrm{r}}\mathbf{M}_{\mathbf{\Phi}}+\mathbf{S}_{\underline{\mathbf{H}}}\mathbf{S}_{\underline{\mathbf{H}}}^{\mathsf{H}}\right)^{-1}\right]\
\mathrm{s.t.}\ \mathrm{(\ref{eq:power1})},\mathrm{(\ref{eq:theta1})}.$

With reference to the proposed mbs-PSO method in Algorithm 1, the proposed
LBO-PSO method takes a potential passive reflection angle vector $\bm{\theta}$
as the position of each particle. Different from the mbs-PSO method, the
fitness function of the LBO-PSO algorithm is defined to be the objective of
Problem (P6), as given by

$\displaystyle
J_{\mathrm{low}}(\bm{\theta})=\mathrm{trace}\left[\left(N_{\mathrm{r}}\mathbf{M}_{\mathbf{\Phi}}+\mathbf{S}_{\underline{\mathbf{H}}}\mathbf{S}_{\underline{\mathbf{H}}}^{\mathsf{H}}\right)^{-1}\right].$
(36)

Algorithm 2 summarizes the proposed LBO-PSO algorithm, where the fitness
function evaluation using (36) at each iteration does not require channel
samples and associated fitness evaluations (as done in Algorithm 1). With only
the S-CSI and random channel distribution of the links, each particle
$\bm{\theta}$ determines the optimal power solution using (14), and then
evaluates its fitness value using (36) at each iteration. The best position of
each particle and the global optimal position are updated according to the
historical fitness values. The velocities and positions of the particles are
also updated using (19) and (20), respectively. Upon the completion of the
LBO-PSO algorithm, the passive beamforming matrix $\mathbf{\Theta}$ is
optimized and the IRS is configured accordingly for the frame. At each time
slot of the frame, the UE measures and reports the effective channel. Based on
the reported effective channels, the BS runs the water-filling algorithm to
allocate the transmit powers for the data streams in the next time slot, as
described in Sec. III-A.

1 Step 1: Swarm initialization

Initialize the PSO parameters :
$\\{P,w,c_{1},c_{2},\varepsilon_{1},\varepsilon_{2},I_{\mathrm{iter}}\\}$.

Initialize the position and velocity :
$\\{\bm{\theta}_{p}^{(0)}\\}_{p=1}^{P},\\{\mathbf{v}_{p}^{(0)}\\}_{p=1}^{P}$.

2 For each particle $\bm{\theta}_{p}^{(0)}$, evaluate the fitness value of
each particle using (36), and obtain the initial best position set
$\\{\dot{\bm{\theta}}_{p}\\}_{p=1}^{P}$;

3 Find the global optimal position
$\ddot{\bm{\theta}}=\arg\min_{\dot{\bm{\theta}}}\\{J^{(0)}(\dot{\bm{\theta}}_{1}),J^{(0)}(\dot{\bm{\theta}}_{2}),\cdots,J^{(0)}(\dot{\bm{\theta}}_{P})\\}$;

4 end

5

6 Step 2: Iterative search

7 for  _$i=1:I_{\mathrm{iter}}$_ do

8 Update particle velocity $\\{\mathbf{v}_{p}^{(i)}\\}_{p=1}^{P}$ and position
$\\{\bm{\theta}_{p}^{(i)}\\}_{p=1}^{P}$ using (19) and (20);

9 for  _$p=1:P$_ do

10 Given particle $\bm{\theta}_{p}^{(i)}$, evaluate the fitness value of
particle $\bm{\theta}_{p}^{(i)}$ using (36);

11 if  _$J^{(i)} <J^{(i-1)}$_ then

12 $\dot{\bm{\theta}}_{p}=\bm{\theta}_{p}^{(i)}$;

13 else

14 $\dot{\bm{\theta}}_{p}=\bm{\theta}_{p}^{(i-1)}$;

15 end if

16 Update
$\ddot{\bm{\theta}}_{p}=\arg\min_{\dot{\bm{\theta}}}\\{J^{(i)}(\dot{\bm{\theta}}_{1}),J^{(i)}(\dot{\bm{\theta}}_{2}),\cdots,J^{(i)}(\dot{\bm{\theta}}_{P})\\}$;

17

18 end for

19

20 end for

21

22 end

23

Algorithm 2 Proposed LBO-PSO Algorithm

It is important to note that the objective of Problem (P6) is an upper bound
for the objective of Problem (P5). In other words, the solution to Problem
(P6) provides a lower bound for the AAR achievable by Problem (P1’). The lower
bound is derived by first exploiting the arithmetic-geometric inequality [46],
followed by the Jensen’s inequality, to reveal that the lower bound only
depends on the trace of inverse channel covariance matrix. By applying the
property of complex inverse Wishart matrix, we show that the expectation of
the inverse channel covariance matrix only depends on the S-CSI and so does
the AAR lower bound. The tightness of the lower bound depends on the curvature
of the AAR regarding the antenna number, the IRS element number, and the
channel rank (due to the use of the Jensen’s inequality), and is hard to
quantify. On the other hand, the AAR lower bound and subsequently Problem (P6)
only require the S-CSI knowledge, i.e., the mean and covariance of the CSI,
and do not need to generate channel samples (as opposed to the proposed mbs-
PSO algorithm and more generic PSO-based algorithms). To this end, the LBO-PSO
algorithm developed to solve Problem (P6) is immune to the uncertainty in the
generation of channel samples and the subsequent potential inaccuracy of using
the channel samples to approximate the real channel. Moreover, the LBO-PSO
algorithm is computationally more efficient, due to the dramatically lower
complexity for evaluating the PSO fitness value, as compared to the mbs-PSO.
For these reasons, the LBO-PSO is a computationally efficient alternative to
the mbs-PSO and provides assured results for qualifying the mbs-PSO’s results.

###  III-D Complexity Analysis

In what follows, we analyze the computational complexity of the two proposed
PSO algorithms measured by floating point operations per second (FLOPS).

mbs-PSO algorithm: According to (12), the operations dominating the
computational complexity of each particle include a matrix addition, four
matrix multiplications, and a matrix inversion per channel sample per
iteration, requiring
$F_{1}=4\left(N_{\mathrm{t}}+N_{\mathrm{r}}\right)N^{2}+4MN_{\mathrm{t}}N_{\mathrm{r}}+4M^{2}N_{\mathrm{r}}+(4M^{3}+M^{2}+M)$
FLOPS [50]. The overall computational complexity of the algorithm is $P\times
L_{\mathrm{mb}}\times F_{1}$ FLOPS, according to (21). In contrast, a direct
use of all $L_{B}$ channel samples to evaluate the fitness function per
iteration using (16) incurs $P\times L_{B}\times F_{1}$ FLOPS, which is
$N_{B}$ times the complexity of the new mbs-PSO algorithm since
$L_{B}=N_{B}L_{\mathrm{mb}}$.

LBO-PSO algorithm: According to (35), the operations dominating the complexity
of each particle include three matrix additions, three matrix multiplications,
a matrix inversion, and a matrix trace per iteration, incurring
$F_{2}=4\left(N_{\mathrm{t}}+N_{\mathrm{r}}\right)N^{2}+4N_{\mathrm{t}}^{2}N_{\mathrm{r}}+(4N_{\mathrm{t}}^{3}+N_{\mathrm{t}}^{2}+N_{\mathrm{t}})+2N_{\mathrm{t}}^{2}$
FLOPS. Requiring no channel samples, the algorithm incurs $P\times F_{2}$
FLOPS.

In general, a moderate number of particles and iterations can lead to a
reasonably good performance of the proposed Algorithms 1 and 2. Moreover, the
dominating computations of the fitness evaluation can be potentially
parallelized for real-time implementation and pipelined in the field
programmable gate array (FPGA) [51, 52, 53].

Table I: SIMULATION PARAMETERS Parameter | Value | Parameter | Value
---|---|---|---
Number of transmit antennas $N_{\mathrm{t}}$ | $8$ |  Total transmit power $P_{\mathrm{tot}}$ |  $20$ dBm
Number of receive antennas $N_{\mathrm{r}}$ | $4$ |  Average noise power $\sigma_{n}^{2}$ |  $-80$ dBm
Size of IRS meta-atoms $N_{x}\times N_{y}$ | $8\times 8$ |  Number of channel samples $L_{B}$ | $5000$
Rician factor $\kappa$ | $3$ | Number of channel samples per batch | $50$
Normalized Doppler frequency $\bar{f}_{d}$ | $0.01$ |  Swarm size $P$ | $200$
PLE of the IRS-UE link $\alpha_{\mathrm{ur}}$ | $2.2$ |  Number of PSO iterations $I_{\mathrm{iter}}$ | $100$
PLE of the BS-IRS link $\alpha_{\mathrm{br}}$ | $2.2$ |  Inertia weight of a particle $w$ | $0.9$
PLE of the BS-UE link $\alpha_{\mathrm{bu}}$ | $3.6$ |  Cognitive scaling factor $c_{1}$ | $1.49445$
Location of the BS | $[0,0,5]$ |  Social scaling factor $c_{2}$ | $1.49445$
Location of the IRS | $[100,0,5]$ |  Reference path loss $L_{\mathrm{in}}$ |  $-30$ dB

##  IV Simulation Results

Simulation results are provided to evaluate the performance of our proposed
transmission method, including the transmission architecture and resource
allocation algorithms. Our testing environment is MATLAB R2018b on a desktop
computer with 3.20 GHz Intel Core i7-8700 CPU and 8 GB random access memory
(RAM). In the simulations, the distance-dependent path loss model is given by
$L_{\mathrm{out}}=L_{\mathrm{in}}(d/d_{0})^{-\alpha}$ where $L_{\mathrm{in}}$
is the reference path loss at $d_{0}=1$ m and $\alpha$ is the path loss
exponent (PLE). Considering the 3D Cartesian coordinates, the BS and IRS are
located on a three-dimensional plane with coordinates [0, 0, 5] m and [100, 0,
5] m. The UE is 1 m high, distributed horizontally within the disk area with
the center at [100, 10] m and radius of 10 m. Unless otherwise stated, all
other simulation parameters consisting of wireless propagation and PSO related
parameters, are listed in Table I.

In addition to the proposed mbs-PSO and LBO-PSO schemes, five benchmark
schemes are evaluated for comparison:

  * •

PSO with AAR fitness: The PSO method is performed by using the objective of
problem (P3), i.e., AAR, as the fitness function, where each iteration
requires all generated channel samples to evaluate the fitness and update the
particle positions. This method serves as an upper bound for the proposed mbs-
PSO method.

  * •

Sum-path-gain maximization (SPGM)-based scheme [54]: In our proposed two-
timescale beamforming framework, the large-timescale passive beamforming
optimization is replaced with the SPGM-based method, where the IRS reflection
matrix is obtained by maximizing the effective channel gains.

  * •

SVD-based scheme: We adopt the conventional SVD precoder in the considered
IRS-assisted MIMO system, where the BS optimizes its transmit powers solely
based on the outdated CSI at each slot.

  * •

Random phase shift: Based on the studied SVD-ZF system, we adopt a randomly
generated long-term reflection matrix, and then optimize the transmit powers
by using the method developed in Sec. III-A.

  * •

Without IRS: We optimize the transmit powers to maximize the overall AAR of
the studied SVD-ZF system with the IRS disabled.

###  IV-A Convergence

Fig. 4 shows the convergence behaviors of the proposed mbs-PSO and LBO-PSO
algorithms, where the fitness values of the two algorithms, defined in (21)
and (36), are separately plotted. From the fitness value results shown in Fig.
4(a), we see that the mbs-PSO algorithm exhibits good convergence. More
importantly, the mini-batch sampling surrogate function is effective, as the
corresponding AAR significantly improves during iterations. In Fig. 4(b), the
fitness function of the LBO-PSO algorithm is also shown to be effective. It is
worth noting that the fitness value plotted in Fig. 4(b) is the objective
value of (P5), not the AAR in Problem (P1). Problem (P5) is a TicMin problem.
The fitness value measures the trace-of-inverse-covariance. Since the diagonal
entries of the involved covariance matrix take relatively small values, the
inverse of the covariance matrix (or in other words, the fitness value) is
large.

Figure 4: Convergence of the proposed mbs-PSO and LBO-PSO, where the $x$-axis
is the number of iterations, and the $y$-axis provides the fitness values of
PSO and the AAR.

Fig. 5 shows the AAR of different schemes during PSO iterations in an $8\times
4$ MIMO scenario. Since the PSO with AAR fitness uses a large number of
channel samples to reasonably approximate the channel, it serves as an upper
bound for the proposed mbs-PSO algorithm. The LBO-based algorithm exploits
only the S-CSI and serves as a lower bound for the mbs-PSO. As expected, the
mbs-PSO scheme is between the lower bound and the upper bound, and
considerably closer to the upper bound.

Figure 5: AAR achieved over iterations, with $N_{\mathrm{t}}=8$,
$N_{\mathrm{r}}=4$, $N=64$ and $P_{\mathrm{tot}}=20$ dBm.

Fig. 6 shows the impact of different fitting parameters, including mini-batch
sizes and decay weight coefficients, on the convergent AAR. It is observed
that the mbs-PSO algorithm can converge within 100 iterations under different
$L_{\mathrm{mb}}$ and $\mu^{(i)}$ values. The mbs-PSO algorithm can slightly
improve the AAR over iterations under small $L_{\mathrm{mb}}$ values, while
the improvement is substantial under large $L_{\mathrm{mb}}$ values. On the
other hand, the AAR performance achieved by the mbs-PSO algorithm is better
under the decay weight coefficient of $\mu^{(i)}=i^{-0.2}$ than it is under
$\mu^{(i)}=i^{-0.5}$. By comparing with the proposed LBO-PSO algorithm, we see
that the mbs-PSO with small $L_{\mathrm{mb}}$ and small decay $\mu^{(i)}$ can
deteriorate significantly. In particular, the LBO-PSO does not require
regenerations of channel samples. It does not suffer from approximation
inaccuracy from ill fitting parameters or dirty samples, and provides
performance assurance and reliable benchmark for the proposed mbs-PSO
algorithm.

Figure 6: Impact of fitting parameters on the PSO performance. Figure 7:
Running time per iteration vs. the number of IRS elements.

Fig. 7 shows that the running time of the proposed PSO algorithms is much
shorter than that of the conventional PSO using full batch samples. Moreover,
the processing speed of LBO-PSO is at a millisecond level when there are less
than 64 IRS meta-atoms, and is considerably faster than the speed of the mbs-
PSO algorithm.

###  IV-B Impact of IRS

To demonstrate the usefulness of the IRS, Fig. 8 compares the different
schemes under different transmit powers $P_{\mathrm{tot}}$. As
$P_{\mathrm{tot}}$ increases, the AARs of the schemes improve with the SINR.
Among the schemes, the conventional SVD scheme receives the worst performance,
since it fails to suppress the inter-stream interference resulting from the
outdated CSI. The proposed mbs-PSO algorithm, together with its upper bound,
i.e., PSO with AAR fitness, and lower bound, i.e., the proposed LBS-PSO,
surpass the other schemes, including “random phase shift” and “without IRS”,
confirming the usefulness of the IRS. Furthermore, the SPGM-based scheme
performs worse than the proposed algorithms. The reason is that the scheme
optimizes the large-timescale passive beamforming by maximizing the effective
channel gains, and does not utilize the channel correlation. In contrast, our
proposed algorithms optimize the large-timescale passive beamforming by
considering the equivalent end-to-end channel (which is made up of the
wireless channel and the precoder based on the oudated CSI), under the optimal
small-timescale power allocation and ZF detection. In this sense, the channel
correlation is better utilized in the proposed algorithms.

Figure 8: AAR performance vs. total transmit power under different schemes.

Fig. 9 evaluates the impact of the reflecting element number on the system
AAR. We see that all algorithms can improve their system AARs with the
increasing number of IRS reflecting elements, except the scheme without IRS.
This validates the effectiveness of the IRS and the proposed algorithm. We
also see that the AAR gap between the proposed mbs-PSO algorithm and the upper
bound is small, demonstrating the validity of the mini-batch sampling method.
As expected, the proposed LBO-PSO algorithm is lower than the proposed mbs-PSO
algorithm in terms of the AAR. Nevertheless, their gap is consistent across a
wide spectrum of the IRS element numbers. Fig. 10 plots the AAR of each
considered scheme with the increase of transmit antennas. The AARs of all
schemes grow with transmit antennas, but their growth rates slow down.
Increasing the number of receive antennas is more effective than increasing
the transmit antennas in terms of improving the AAR. Moreover, the SPGM-based
method performs the worst among all methods for the reason revealed in Fig. 8.

Figure 9: AAR performance vs. the number of reflecting elements. Figure 10:
AAR performance vs. the number of transmit antennas.

###  IV-C Impact of Outdated CSI

Figure 11: AAR vs. time correlation coefficient $\rho$.

Fig. 11 presents the AARs achieved by different schemes against the time
correlation coefficient $\rho$. A larger temporal correlation coefficient
$\rho$ indicates a smaller time delay $\tau$, thus resulting in smaller
channel error between consecutive slots. In the extreme case where the
outdated CSI is independent of the I-CSI, the precoding matrices of the
schemes are independent of the I-CSI and equivalent to arbitrary precoding
matrices. We see that more severely outdated CSI leads to a lower AAR. This
confirms the importance of exploiting the channel correlation between
consecutive time slots to improve the AAR performance. In the case of
$\rho=1$, the CSI remains unchanged and never outdated throughout all time
slots. It is seen that the conventional SVD scheme can achieve the same AAR as
the upper bound, since SVD precoding is known to be optimal under the I-CSI.
Under the nearly independent channel conditions, i.e., $\rho\rightarrow 0$,
the SVD precoder based on the outdated CSI is in essence an arbitrary
precoder. Nevertheless, the proposed SVD-ZF-based scheme can still outperform
the SVD-based scheme based on the outdated CSI, because of its use of the ZF
detector to suppress inter-stream interference based on the I-CSI at the UE.

##  V Conclusion

This paper developed a new SVD-ZF-based transmission strategy to overcome the
undesired outdated CSI in IRS-assisted MIMO systems. Specifically, we proposed
a two-timescale beamforming protocol, where passive beamforming at the IRS is
designed at a large timescale and power allocation at the BS is performed slot
by slot based on the outdated CSI at a small timescale. The power allocation
problem was derived with the closed-form water-filling solutions. The large-
timescale passive beamforming problem was first transformed to a non-convex
deterministic AAR maximization problem. We proposed an mbs-PSO algorithm to
efficiently solve this non-convex problem. We also provided an LBO-PSO
algorithm to obtain the passive beamformers with only S-CSI, and its resulting
AAR was utilized as a lower bound for validation of the mbs-PSO algorithm.
Finally, we evaluated the proposed two-timescale MIMO transmission strategy
and PSO algorithms by comparing with multiple benchmarks, and concluded that
the proposed mbs-PSO algorithm can approach the upper bound of the system AAR,
and outperforms the existing benchmarks.

##  Appendix A Proof of Lemma 1

Let $\mathbf{v}_{i}$ be a basis vector of $\mathbf{B}$ with
$\mathbf{B}\mathbf{v}_{i}=\lambda_{i}\mathbf{v}_{i}$. Since $\mathbf{B}$ is
symmetric,
$\mathbf{v}_{i}^{\mathsf{H}}\mathbf{B}=\lambda_{i}\mathbf{v}_{i}^{\mathsf{H}}$
and

$\displaystyle\mathrm{trace}(\mathbf{B}\mathbf{A})$
$\displaystyle=\sum_{i=1}^{N}\mathbf{v}_{i}^{\mathsf{H}}\mathbf{B}\mathbf{A}\mathbf{v}_{i}=\sum_{i=1}^{N}\lambda_{i}\mathbf{v}_{i}^{\mathsf{H}}\mathbf{A}\mathbf{v}_{i}$
$\displaystyle\leq\sum_{i=1}^{N}\lambda_{\max}(\mathbf{B})\mathbf{v}_{i}^{\mathsf{H}}\mathbf{A}\mathbf{v}_{i}=\lambda_{\max}(\mathbf{B})\sum_{i=1}^{N}\mathbf{v}_{i}^{\mathsf{H}}\mathbf{A}\mathbf{v}_{i}$
$\displaystyle=\lambda_{\max}(\mathbf{B})\mathrm{trace}(\mathbf{A}).$ (37)

This proof completes. $\blacksquare$

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|  Yashuai Cao (S’18) received the B.S. degree from Chongqing University of
Posts and Telecommunications (CQUPT), Chongqing, China, in 2017. He is
currently pursuing the Ph.D. degree in communication engineering with the
School of Information and Communication Engineering, Beijing University of
Posts and Telecommunications (BUPT), Beijing, China. His current research
interests include wireless resource allocation and signal processing
technologies for massive MIMO systems and intelligent reflecting surface
assisted wireless networks.
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|  Tiejun Lv (M’08-SM’12) received the M.S. and Ph.D. degrees in electronic
engineering from the University of Electronic Science and Technology of China
(UESTC), Chengdu, China, in 1997 and 2000, respectively. From January 2001 to
January 2003, he was a Postdoctoral Fellow with Tsinghua University, Beijing,
China. In 2005, he was promoted to a Full Professor with the School of
Information and Communication Engineering, Beijing University of Posts and
Telecommunications (BUPT). From September 2008 to March 2009, he was a
Visiting Professor with the Department of Electrical Engineering, Stanford
University, Stanford, CA, USA. He is the author of three books, more than 100
published IEEE journal papers and 200 conference papers on the physical layer
of wireless mobile communications. His current research interests include
signal processing, communications theory and networking. He was the recipient
of the Program for New Century Excellent Talents in University Award from the
Ministry of Education, China, in 2006. He received the Nature Science Award in
the Ministry of Education of China for the hierarchical cooperative
communication theory and technologies in 2015.
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|  Wei Ni (M’09-SM’15) received the B.E. and Ph.D. degrees in Electronic
Engineering from Fudan University, Shanghai, China, in 2000 and 2005,
respectively. Currently, he is a Principal Research Scientist at CSIRO,
Sydney, Australia, and an Adjunct Professor at the University of Technology
Sydney and Honorary Professor at Macquarie University. He was a Postdoctoral
Research Fellow at Shanghai Jiaotong University from 2005 to 2008; Deputy
Project Manager at the Bell Labs, Alcatel/Alcatel-Lucent from 2005 to 2008;
and Senior Researcher at Devices R&D, Nokia from 2008 to 2009. He has authored
five book chapters, more than 200 journal papers, more than 80 conference
papers, 25 patents, and ten standard proposals accepted by IEEE. His research
interests include machine learning, online learning, stochastic optimization,
as well as their applications to system efficiency and integrity. Dr Ni is the
Chair of IEEE Vehicular Technology Society (VTS) New South Wales (NSW) Chapter
since 2020, an Editor of IEEE Transactions on Wireless Communications since
2018, and an Editor of IEEE Transactions on Vehicular Technology. He served
first the Secretary and then the Vice-Chair of IEEE NSW VTS Chapter from 2015
to 2019, Track Chair for VTC-Spring 2017, Track Co-chair for IEEE VTC-Spring
2016, Publication Chair for BodyNet 2015, and Student Travel Grant Chair for
WPMC 2014.
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