Open AccessArticle

Joint Design of Transmitter Precoding and Optical Intelligent Reflecting Surface Configuration for Photon-Counting MIMO Systems Under Poisson Shot Noise

Jian Wang

¹,

Xiaolin Zhou

^1,*

Fanghua Li

¹,

Yongkang Chen

Chaoyi Cai

¹ and

Haoze Xu

School of Information Science and Technology, Fudan University, Shanghai 200433, China

College of Information Engineering, Shanghai Maritime University, Shanghai 201306, China

Author to whom correspondence should be addressed.

Appl. Sci. 2024, 14(24), 11994; https://doi.org/10.3390/app142411994

Submission received: 12 November 2024 / Revised: 15 December 2024 / Accepted: 19 December 2024 / Published: 21 December 2024

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Abstract

Intelligent reflecting surfaces (IRSs) have emerged as a promising technology to enhance link reliability in a cost-effective manner, especially for line-of-sight (LOS) link blocking caused by obstacles. In this paper, we investigate an IRS-assisted single-cell photon-counting communication system in the presence of building shadows, where one IRS is deployed to assist the communication between a multi-antenna base station (BS) and multiple single-antenna users. Photon counting has been widely adopted in sixth-generation (6G) optical communications due to its exceptional detection capability for low-power optical signals. However, the correlation between signal and noise complicates analyses. To this end, we first derive the channel gain of the IRS-assisted MIMO system, followed by the derivation of the mean square error (MSE) of the system using probabilistic methods. Given the constraints of the transmit power and IRS configuration, we propose an optimization problem aimed at minimizing the MSE of the system. Next, we present an alternating optimization (AO) algorithm that transforms the original problem into two convex subproblems and analyze its convergence and complexity. Finally, numerical results demonstrate that the IRS-assisted scheme significantly reduces the MSE and bit error rate (BER) of the system, outperforming other baseline schemes.

Keywords:

intelligent reflecting surface; precoding; multi-user; Poisson shot noise

1. Introduction

With the rise of new applications such as the Internet of Things (IoT) and virtual reality (VR), there is an increasing demand for higher data rates and capacities in sixth-generation (6G) communication technology. Free space optical (FSO) communication is an emerging technology that offers unique advantages in terms of high speed, anti-interference capabilities, and efficient spectrum resource utilization, making it an effective supplement for 6G technology [1,2]. Unlike fiber-optic communication, which transmits signals through optical fibers, FSO utilizes collimated beams to transmit signals through the atmosphere [3]. In scenarios where laying cables is challenging, FSO can significantly reduce costs and eliminate the difficulties associated with cable installation [4]. However, atmospheric absorption, scattering, and turbulence can adversely affect FSO transmission beams, making them vulnerable to unstable weather conditions [5].

Multiple-input, multiple-output (MIMO) technology is one of the effective ways to combat turbulence fading. By introducing spatial diversity, the performance of FSO communications in atmospheric turbulence can be improved [6,7]. In addition, as a signal processing technology used at the transmitter end, precoding technology has been widely adopted in FSO communications to combat turbulent fading [8,9]. When the transmitter is equipped with multiple antennas, data streams from multiple users can be spatially multiplexed onto independent sub-channels through precoding. By optimizing the precoder as a variable, performance requirements such as rate and energy efficiency in the communication system can be achieved [10].

Photon-counting technology is an effective method for detecting low-power optical signals in FSO communications [11,12]. This advantage becomes even more prominent in power-constrained IoT or optical communication scenarios under strong atmospheric turbulence. In such scenarios, it would be difficult to effectively detect extremely weak optical signals using traditional optical communication receivers. Photon-counting receivers, based on the photoelectric effect and high-sensitivity detectors, can effectively detect and receive low-power optical signals. However, due to quantum effects, the photon-counting process involves randomness, resulting in the generation of Poisson shot noise associated with the signal. Specifically, shot noise is modeled as a Poisson counting process (PCP) characterized by two independent parameters: the desired signal and background radiation [13]. These characteristics make it difficult to apply signal processing schemes developed based on additive white Gaussian noise (AWGN) to photon-counting systems, so it is necessary to develop new technologies to meet this challenge.

Recently, the applicability of intelligent reflective surfaces (IRSs) has been introduced to FSO scenarios to address the problem of line-of-sight (LOS) link blockage due to buildings and other obstacles [14]. Optical IRS can be implemented through either a reflector array-based design or a meta-surface-based design [15]. Considering the low cost and easy portability of IRSs, it has been applied in various scenarios in communication systems to improve communication performance.

1.1. Related Work

Currently, precoding technology in communication systems has been widely studied [16,17,18]. Liu et al. [16] proposed a symbol-level precoding method that can be applied to a dual-functional radar-communication (DFRC). This scheme not only provided additional degrees of freedom (DoFs) but also ensured that the radar sensing obtains a better instantaneous transmission beam pattern. Qi et al. [17] studied the issue of maximizing the energy efficiency (EE) of a multi-beam satellite system under total power constraints and service quality constraints, and proposed pre-programmed design schemes based on zero-forcing (ZF) and successive convex approximation (SCA), respectively. Jain et al. [18] proposed a novel precoder based on exponential singular value decomposition for massive MIMO visible light communication (VLC), which reduces the condition number of the overall channel matrix and enhances the bit error rate (BER) performance of the system.

Photon counting has been the subject of extensive research in recent years. In [11], the FSO MIMO communication problem under the Poisson photon-counting detection model was studied. Aiming at the goal of a high bit rate, Abou et al. proposed a spatial multiplexing (SMux) scheme using multi-level pulse position modulation and conducted asymptotic performance analysis. Huang et al. [12] developed a reliable symbol synchronization method based on pulse position modulation for underwater photon-counting communications, which can achieve time slot synchronization and frame synchronization, effectively reducing the BER of the system. Li et al. [19] proposed a precoder design for photon-counting ultraviolet (UV) MIMO systems that addresses the challenges of Poisson shot noise in signal processing. Simulation results show that the design can effectively combat atmospheric fading and mitigate multi-user (MU) interference. Gai et al. [20] constructed a polarization multiplexing modulation underwater wireless optical communication (UWOC) model based on photon counting and theoretically analyzed the improvements in BER and communication capacity.

As a promising emerging technology, IRSs not only provide higher link reliability and wider coverage at a lower cost but can also be potentially applied in MIMO FSO to modify channels in order to improve multiplexing gain [21,22]. Elmossallamy et al. [23] considered the downlink MU scenario and studied the problem of maximizing the dirty-paper-coding (DPC) sum rate of the IRS-assisted broadcast channel. In [24], the unique “inter-element interference (IEI) free” characteristic of the optical IRS reflection channel was revealed. Based on this analysis, Sun et al. continued to study the capacity of optical IRS-assisted MIMO VLC under different transmitter antenna power constraints and jointly optimized the optical IRS element alignment and transmitter transmit power to maximize the capacity. Additionally, Sun et al. [25] addressed the optimization problem of the IRS-assisted MIMO VLC communication system under the point source assumption, primarily focusing on the joint design of the IRS and transceiver signal processing to minimize the mean square error (MSE) of the system. In addition, Chu et al. [26] also studied the adaptive channel estimator using a general arbitrarily shaped IRS to explore time-varying millimeter wave channels. They also proposed a deep learning network framework using fixed-point iteration and cascading to reduce the complexity of the solution.

To the best of our knowledge, there is no research on applying optical IRS technology to photon-counting communication systems and even less work on the configuration optimization of optical IRS. Therefore, it is very meaningful to combine the research on signal processing technology for MU photon-counting communication systems with IRS configuration optimization to solve the non-line-of-sight (NLOS) transmission problem. This research can not only fill the gap in this field but also provide inspiration for the further development of other subsequent work.

1.2. Contributions

In this paper, we consider a downlink optical IRS-assisted photon-counting communication system in which the signals emitted via multiple LEDs are transmitted through NLOS paths, reflected via optical IRS mirrors, and finally received via multiple photodetectors (PDs). Based on the statistical idea, we use the knowledge of expectation and conditional expectation to derive the MSE of the system under Poisson shot noise. Then, we formulate an optimization problem that aims to minimize the MSE of the system while jointly optimizing the IRS configuration and transmitter precoding. We also propose the corresponding alternating optimization algorithm to solve it. The simulation results demonstrate that our proposed scheme outperforms other baseline schemes in terms of MSE and BER performance. The main contributions of this paper are outlined as follows:

First, we consider the IRS model in the very near field and study the MSE optimization problem of a downlink optical IRS-assisted photon-counting communication system. Taking into account the power constraints in optical communication and the configuration constraints of the optical IRS, we transform it into a joint optimization problem of the optical IRS configuration matrix and the precoder.
Next, we use statistical knowledge such as conditional probability and conditional expectation to derive the MSE closed-form expression of the IRS-assisted MU photon-counting system. Then we decompose the original problem into two convex subproblems through a series of convexification methods, and we analyze the convergence and complexity of the proposed alternating optimization (AO) algorithm.
Finally, we numerically evaluate the IRS-assisted photon-counting communication system. We select several baseline schemes for comparison, including the IRS random allocation scheme and the ZF precoding scheme. The numerical results show that the proposed A0 algorithm effectively reduces the MSE and BER of the system, outperforms other schemes, and exhibits robustness under different turbulence intensities, background radiation, and imperfect CSI.

The rest of this paper is organized as follows. In Section 2, we introduce the IRS-assisted MU photon counting system and present the framework of the whole system, especially the detailed introduction of channel modeling. In Section 3, we provide a detailed derivation of the IRS-assisted photon-counting system based on the MMSE principle, and we formulate the optimization problem. In Section 4, we introduce an AO algorithm and analyze its convergence and complexity in detail. We then present our simulation results and discuss their implications in Section 5. Finally, we summarize the paper in Section 6.

Notations: Scalars are denoted by regular letters, while vectors and matrices are represented by bold letters. The symbol

R

represents the set of real numbers,

\Pr (\cdot)

denotes the probability,

\Pr (\cdot | \cdot)

denotes the conditional probability,

E (\cdot)

and

E (\cdot | \cdot)

stand for expectation and conditional expectation, respectively,

Var (\cdot)

denotes variance,

{(\cdot)}^{T}

represents the transpose, and

\mod (\cdot)

represents the modulus operator. The Hadamard product, vectorization operator, and floor operator are represented as ⊙,

vec (\cdot)

, and

⌊ \cdot ⌋

, respectively.

2. System Model

As illustrated in Figure 1, we consider a downlink IRS-assisted MU MISO photon-counting communication system. In this scenario, optical signals are absorbed or reflected by obstacles such as buildings, which weakens the signal strength or changes the signal transmission path, leading to signal interruption or the degradation of communication quality. Therefore, we introduced IRS to facilitate NLOS transmission, using the reflection and refraction principles of light to guide optical signals to the target receiver and thereby bypassing buildings. Optical IRS can be physically implemented based on the design of reflector arrays or metasurfaces, and through reasonable configuration with the transmitting LED and receiving PD, a stable and reliable signal transmission link can be formed. In our proposed system, the base station (BS) is equipped with M LEDs to transmit information to K users, each of whom has a single PD for signal reception.

2.1. Transmitter

Considering the ease of implementation and the widespread use of on-off keying (OOK) modulation in optical communications, we also adopted this modulation in our system. Let

u_{k}

represent the binary data symbol sent via the BS to the k-th user, and let the vector

u = {[u_{1}, \dots, u_{K}]}^{T}

collect the symbols sent via the BS to all K users in a symbol duration. After the transmitted symbol vector

u

is precoded using the precoding matrix and adding the direct current (DC) bias, the final transmitted signal

x \in R^{M \times 1}

is expressed as

x = Wu + D = \sum_{k = 1}^{K} w_{k} u_{k} + D,

(1)

where

W \in R^{M \times K}

represents the precoding matrix,

w_{k}

is the k-th column of

W

, and

D = {[D_{1}, D_{2}, \dots, D_{M}]}^{T}

denotes the DC bias vector. Note that

x_{i}

is the expected number of photons to be sent on the i-th LED, and its relationship with the transmitted optical power

P_{t, i}

satisfies

x_{i} = \frac{P_{t, i} τ}{h ν}

, where

τ, h, ν

are the symbol duration, the Planck’s constant, and the optical frequency, respectively.

2.2. Channel Model

As the most common transmission medium of FSO systems, the atmosphere will inevitably cause the attenuation of the transmitted optical signal. Although the introduction of IRSs realizes NLOS transmission, considering the deployment environment of IRSs themselves and the settings of related physical parameters, this fading will be more complicated.

2.2.1. Channel Gains

With the consideration of atmospheric loss, turbulence-induced fading, and geometric and misalignment losses, the FSO channel gain can be modeled as follows [27,28]:

h = h_{p} h_{a} h_{g},

(2)

where

h_{p}

represents the loss of beam energy due to absorption and scattering in the transmission link.

h_{a}

and

h_{g}

correspond to the influence of atmospheric turbulence and the matching distortion error of the IRS, respectively.

In addition to the reflection effect, IRSs can also absorb or scatter part of the beam power. Let

ζ

stand for the reflection efficiency. For the FSO system operating at

λ = 1550 nm

, the value of

ζ

typically ranges from 0.7 to 1 [29]. The absorption at the IRS could be considered part of the atmospheric loss,

h_{p}

. Specifically,

h_{p}

can be modeled as [30]

h_{p} = ζ 10^{- κ d_{e 2 e} / 10},

(3)

where

κ

represents the weather-dependent attenuation coefficient of the FSO link, and

d_{e 2 e}

represents the end-to-end distance between the LED and the PD.

The gamma–gamma distribution stands out as a key channel model for FSO links, given its capacity to effectively simulate diverse atmospheric turbulence scenarios. Its close match with empirical data at different turbulence levels further solidifies its suitability as a pivotal channel model for FSO communications [31,32]. The probability density function (PDF) of the optical intensity fading is expressed as [33]

\Pr (h_{a}) = \frac{2 {(α β)}^{\frac{α + β}{2}}}{Γ (α) Γ (β)} h_{a}^{\frac{α + β}{2} - 1} K_{α - β} (2 \sqrt{α β h_{a}}), h_{a} > 0,

(4)

where

Γ (\cdot)

is the standard Gamma function, and

K_{ν} (\cdot)

denotes the second kind of the modified Bessel function. The scintillation parameters,

α

and

β

, satisfy

\begin{matrix} α = {[exp (\frac{0.49 σ_{R}^{2}}{{(1 + 1.11 σ_{R}^{12 / 5})}^{7 / 6}}) - 1]}^{- 1}, \end{matrix}

(5)

\begin{matrix} β = {[exp (\frac{0.51 σ_{R}^{2}}{{(1 + 0.69 σ_{R}^{12 / 5})}^{5 / 6}}) - 1]}^{- 1}, \end{matrix}

(6)

where

σ_{R}^{2}

stands for the Rytov variance. The parameter

σ_{R}^{2}

serves as a pivotal descriptor, quantifying the intensity of turbulence across a spectrum of scenarios. In specific terms, turbulence is categorized as weak when

σ_{R}^{2} \leq 0.3

, moderate within the range of 0.3 to 5, and strong for values surpassing 5. In scenarios of weak turbulence disturbances, the gamma–gamma distribution shares similarities with the lognormal distribution, both adept at characterizing FSO links operating across unobstructed atmospheric conditions [34].

The matching distortion error arises from the divergence of the beam over the propagation distance, as well as the pointing error between the laser beam and the center of the PD, which is caused by building sway. Moreover, the introduction of random jitter from buildings equipped with intelligent reflective surfaces heightens the unpredictability of the optical pathway. A two-dimensional topological model of the light path based on the optical intelligent reflecting surface proposed by Najafi et al. [28] is shown in Figure 2.

For illustration convenience, we denote the angles between the laser beam, IRS plane, and PD and the horizontal direction as

θ_{L}

θ_{I R S}

, and

θ_{P}

, respectively. In addition, we denote the half-length of IRS and PD as

r_{i r s}

and

r_{p}

. The angle between the laser source and the PD is

ψ = θ_{L} + θ_{P} - 2 θ_{I R S}

. The beam width at distance d is defined as [35]

w (d) = w_{0} \sqrt{1 + (1 + \frac{2 w_{0}^{2}}{ρ^{2} (d)}) {(\frac{λ d}{π w_{0}^{2}})}^{2}},

(7)

where

w_{0}

denotes the beam waist radius.

ρ

is the coherence length, defined as

ρ (d) = {(0.55 C_{n}^{2} k^{2} d)}^{- 3 / 5}

, where

C_{n}^{2}

is the medium refraction coefficient, and k is the wave number.

According to Proposition 2 in [28], when Gaussian fluctuations for the building sway are assumed, the PDF of

h_{g}

\Pr (h_{g}) = \frac{ϖ}{2 A_{0} \sqrt{π}} {[ln (\frac{A_{0}}{h_{g}})]}^{- \frac{1}{2}} {(\frac{h_{g}}{A_{0}})}^{ϖ - 1}, 0 \leq h_{g} \leq A_{0},

(8)

where

A_{0} = \erf (v), v = \frac{\sqrt{2} sin (ψ) r_{p}}{w (d_{l 2 p})}, ϖ = \frac{t w^{2} (d_{l 2 p})}{4 σ^{2}}

σ^{2}

represents the variance in misalignment between the center of the beam footprint and the center of the PD.

2.2.2. IRS-Aided MIMO Channel

The geometric optics approximation is demonstrated to be highly accurate in this extremely near-field environment, as long as the size of the IRS unit significantly exceeds the wavelength of light. Numerical experiments’ results indicate that the power density of the reflected beam in the near-infrared is highly concentrated, with energy focused along a straight line [36]. From geometric optics, it is understood that the light beam reflected by the optical IRS is equivalent to the beam emitted from an image source that is symmetrical to the IRS unit [28]. Based on the above analysis, under the condition of an extremely near field, there is almost no interference between the reflective elements of the optical IRS reflection channel. We can assume that a given IRS unit provides NLOS channel gain only when it is aligned with the LED/PD pair [25].

For convenience, let

M ≜ \{1, \dots, M\}

K ≜ \{1, \dots, K\}

N ≜ \{1, \dots, N\}

. Define two binary matrices,

R

and

F

, to represent the alignment between the IRS unit and the transmitting LED and receiving PD, respectively, in the following form:

\begin{matrix} R ≜ & [r_{1}, \dots, r_{M}] \in {\{0, 1\}}^{N \times M}, r_{m} \in R^{N \times 1}, m \in M, \end{matrix}

(9)

\begin{matrix} F ≜ & [f_{1}, \dots, f_{K}] \in {\{0, 1\}}^{N \times K}, f_{k} \in R^{N \times 1}, k \in K . \end{matrix}

(10)

We represent the association between the n-th IRS unit and the corresponding LED/PD pair using the n-th row of the

R

and

F

matrix. Specifically, if

r_{n, m} = f_{n, k} = 1

is satisfied, it means that the LED/PD pair consisting of the m-th LED and the k-th PD should be aligned with the n-th IRS unit. Thus, the IRS-assisted NLOS MIMO channel matrix should be written as follows:

G = [\begin{matrix} {(f_{1} ⊙ r_{1})}^{T} h_{1, 1} & \dots & {(f_{1} ⊙ r_{M})}^{T} h_{1, M} \\ ⋮ & ⋱ & ⋮ \\ {(f_{K} ⊙ r_{1})}^{T} h_{K, 1} & \dots & {(f_{K} ⊙ r_{M})}^{T} h_{K, M} \end{matrix}],

(11)

where the channel gain

h_{k, m} \in R^{N \times 1}

between the m-th LED and the k-th PD is expressed as

h_{k, m} ≜ {[h_{k, 1, m}, \dots, h_{k, N, m}]}^{T} .

(12)

Therefore, the channel vector

g_{k} \in R^{M \times 1}

of user k is expressed as

g_{k} = {[{(f_{k} ⊙ r_{1})}^{T} h_{k, 1}, \dots, {(f_{k} ⊙ r_{M})}^{T} h_{k, M}]}^{T} .

(13)

Furthermore, we impose row sum constraints on

R

and

F

to ensure that each optical IRS unit can be assigned to at most one LED/PD pair [24,25]:

\sum_{m = 1}^{M} r_{n, m} \leq 1, \sum_{k = 1}^{K} f_{n, k} \leq 1, \forall n \in N .

(14)

2.3. Receiver

For the receiving end, each user is equipped with a PD, which converts the detected photons into photo-electrons through the photoelectric effect and then triggers the built-in counter to count these photo-electrons. This process is described by the PCP. Within the symbol duration

τ

, the electron counts received by user k, denoted as

y_{k}

, provide probabilistic measurements of the received photon counts [37]. The PDF of

y_{k}

is given by

\Pr (y_{k} = r | u) = \frac{{(λ_{k})}^{y_{k}}}{y_{k}!} exp (- λ_{k}),

(15)

where

r = 0, 1, \dots, \infty

is a non-negative integer representing the possible values of the received count, and

λ_{k}

denotes the expected number of signal photons received by the k-th user, as given by

λ_{k} = E (y_{k} | u) = η g_{k}^{T} (Wu + D) + n_{b} .

(16)

Here,

η

is the PD efficiency and

n_{b} = η \frac{P_{b} τ}{h ν}

represents the average number of photons generated via background radiation during each symbol duration, where

P_{b}

denotes the incident background power. The expected number of received photons can be divided into two parts:

E (y_{k} | u) = \underset{e x p e c t e d s i g n a l o f u s e r k}{\underset{︸}{η g_{k}^{T} Wu}} + \underset{e x p e c t e d e x t r a n e o u s c o u n t s f o r u s e r k}{\underset{︸}{η g_{k}^{T} D + n_{b}} .}

(17)

For convenience in subsequent representations, we define

Υ_{k} = η g_{k}^{T} D + n_{b}

as the extraneous counts.

3. Problem Formulation

The MSE between the transmitted and received signals is often used as an optimization indicator for BER performance, and MSE-based transceiver design has been widely studied in MIMO systems [38]. Based on this, we derive the MSE expression for MU photon-counting systems and construct an optimization problem to minimize the MSE under power constraints and optical IRS configuration constraints.

Based on the definition of MSE, we give the expression of user k,

{MSE}_{k} ≜ E [{(y_{k} - u_{k})}^{2}] = E (y_{k}^{2}) + E (u_{k}^{2}) - 2 E (y_{k} u_{k}),

(18)

where

y_{k}

is the received signal, and

u_{k}

is the original data. Note that the expectation is taken over variables

y_{k}

and

u_{k}

Differing from the AWGN model, which is independent of the signal, the MSE expression can be directly obtained. In the photon-counting communication system under Poisson shot noise, we need to use the distribution of the transmitted signal

u_{k}

and statistical information, such as the expectation of the received signal

y_{k}

, to indirectly derive the MSE expression.

Without a loss of generality, we assume that

u_{k} = \pm 1, i . e ., \Pr (u_{k} = - 1) = \Pr (u_{k} = 1) = 1 / 2, \forall k \in K

[39]. Obviously, we can get

E (u_{k}^{2}) = 1 \times \Pr (u_{k} = - 1) + 1 \times \Pr (u_{k} = 1) = 1 .

(19)

Regarding the solution of

E (y_{k}^{2})

, we can use the conditional expectation method to achieve it. Since

y_{k}

follows the Poisson distribution, we can use the property of the Poisson distribution,

E (X) = Var (X)

, and get

E (y_{k}^{2}) = \sum_{u \in U} E (y_{k}^{2} | u) \Pr (u) = \sum_{u \in U} \{{[E (y_{k} | u)]}^{2} + E (y_{k} | u)\} \Pr (u),

(20)

where the set

U_{0}

is used to represent all possible values of

u

, expressed as

U_{0} = {u^{(i)} \in \{- 1, 1\} | i = 1, \dots, 2^{K}}

. Combining Equation (17) and expanding the above formula, we get

\begin{matrix} E (y_{k}^{2}) = \sum_{u \in U} \{{(η g_{k}^{T} Wu + Υ_{k})}^{2} + (η g_{k}^{T} Wu + Υ_{k})\} \Pr (u) \\ = \frac{1}{2^{K}} \sum_{i = 1}^{2^{K}} {(η g_{k}^{T} {Wu}^{(j)})}^{2} + (2 Υ_{k} + 1) η g_{k}^{T} W \underset{= 0}{\underset{︸}{\sum_{i = 1}^{2^{K}} u^{(i)}}} + Υ_{k}^{2} + Υ_{k} \\ = \frac{1}{2^{K}} η^{2} g_{k}^{T} W \underset{= I_{K}}{\underset{︸}{(\sum_{i = 1}^{2^{K}} u^{(i)} {[u^{(i)}]}^{T})}} W^{T} g_{k} + η^{2} g_{k}^{T} {DD}^{T} g_{k} + η (2 n_{b} + 1) g_{k}^{T} D + n_{b}^{2} + n_{b} \\ = η^{2} g_{k}^{T} {WW}^{T} g_{k} + η^{2} g_{k}^{T} {DD}^{T} g_{k} + η (2 n_{b} + 1) g_{k}^{T} D + n_{b}^{2} + n_{b} . \end{matrix}

(21)

where

I_{K} \in R^{K \times K}

represents the K × K dimensional identity matrix.

Similarly, we can expand

E (y_{k} u_{k})

into

\begin{matrix} E (y_{k} u_{k}) & = \sum_{u_{k} \in \{\pm 1\}} E (y_{k} u_{k} | u_{k}) \Pr (u_{k}) \\ = \frac{1}{2} E (y_{k} | u_{k} = 1) - \frac{1}{2} E (y_{k} | u_{k} = - 1) \end{matrix}

(22)

According to ([19], Lemma 2), we can get the conclusion of (22), as given by

E (y_{k} u_{k}) = η g_{k}^{T} w_{k},

(23)

By combining Equations (19), (20) and (22), we obtain the MSE expression for user k. Summing the MSE of all K users yields the overall system MSE.

\begin{matrix} MSE (R, F, W) = \sum_{k = 1}^{K} {MSE}_{k} \\ = \sum_{k = 1}^{K} η^{2} g_{k}^{T} {WW}^{T} g_{k} + η g_{k}^{T} w_{k} + η^{2} g_{k}^{T} {DD}^{T} g_{k} + η (2 n_{b} + 1) g_{k}^{T} D + n_{b}^{2} + n_{b} + 1 . \end{matrix}

(24)

Considering the practical application of optical communication, we need to add some inequality constraints to ensure the working effect of optical devices. Since the transmitted signal in optical communication is non-negative, we need to add the following DC bias constraint:

\sum_{k = 1}^{K} w_{k} u_{k} + D \geq 0 .

(25)

where

0

stands for the M-dimensional vectors with all 0 elements.

In addition, we must ensure that the signal transmit power cannot exceed the maximum power:

\sum_{k = 1}^{K} w_{k} u_{k} + D \leq \frac{P_{max} τ}{h ν} 1,

(26)

where

P_{max}

is the maximum transmit power of the LEDs, and

1

stands for the M-dimensional vectors with the whole one element.

Constraints (25) and (26) ensure that the transmit power of each LED remains within physically feasible ranges.

Based on the above analysis, we can construct the optimization problem, which aims to minimize the MSE of IRS-assisted photon-counting MIMO by jointly designing the transmitter precoding

W

and IRS configurations

F

and

R

. Specifically, the problem is expressed as follows:

\begin{matrix} P 1 : min_{R, F, W} & MSE (R, F, W) \\ s . t . & C 1 : \sum_{k = 1}^{K} w_{k} u_{k} + D \geq 0, \\ C 2 : \sum_{k = 1}^{K} w_{k} u_{k} + D \leq \frac{P_{max} τ}{h ν} 1, \\ C 3 : \sum_{m = 1}^{M} r_{n, m} \leq 1, \forall n \in N, \\ C 4 : \sum_{k = 1}^{K} f_{n, k} \leq 1, \forall n \in N, \\ C 5 : f_{n, k}, r_{n, m} \in \{0, 1\}, \forall k \in K, m \in M, \end{matrix}

(27)

where constraints

C 1

–

C 2

are derived from power constraints in optical communications, and constraints

C 3

–

C 5

originate from the characteristics of IRS and the fact that one IRS can only be assigned to one LED/PD pair.

4. Proposed Alternating Optimization Algorithm to Minimize MSE

For the optimization problem

P 1

, which is a combinatorial integer programming problem, there is typically no general polynomial-time solution algorithm available. Therefore, in this section, we propose an AO algorithm to solve the relaxed form of

P 1

. Specifically, we first transform

P 1

into a two-variable problem based on the optimization variables, and then we propose corresponding solution methods for the two sub-problems, respectively.

4.1. Optimization Problem Transformation

From problem

P 1

, we can easily see the high similarity of the configuration matrices

R

and

G

. We introduce an auxiliary matrix

V

to express this fact, which is expressed as

V = [v_{1}, \dots, v_{M K}] \in {\{0, 1\}}^{N \times M K}, v_{i} \in R^{N \times 1}, i \in I,

(28)

where

I

is defined as

I ≜ \{1, 2, \dots, M K\}

. More specifically, each column of the

V

matrix can be given by

v_{k + (m - 1) K} = f_{k} ⊙ r_{m}, \forall k \in K, m \in M .

(29)

Correspondingly, constraint

C 3

–

C 5

can be transformed into

\begin{matrix} \sum_{i = 1}^{M K} v_{n, i} \leq 1, \forall n \in N, \end{matrix}

(30)

\begin{matrix} v_{n, i} \in \{0, 1\}, \forall i \in I . \end{matrix}

(31)

Similarly, for the convenience of subsequent expression, we restate the channel vector of user k as

\begin{matrix} g_{k} & = {[{(f_{k} ⊙ r_{1})}^{T} h_{k, 1}, \dots, {(f_{k} ⊙ r_{M})}^{T} h_{k, M}]}^{T} \\ = vec \{[v_{k}, v_{k + K}, \dots, v_{k + (M - 1) K}] ⊙ [h_{k, 1}, h_{k, 2}, \dots, h_{k, M}]\} \\ = vec (V_{k} ⊙ H_{k}) \end{matrix}

(32)

where

V_{k} ≜ [v_{k}, v_{k + K}, \dots, v_{k + (M - 1) K}]

represents the configuration vector in the IRS auxiliary matrix corresponding to user k, and

H_{k} ≜ [h_{k, 1}, h_{k, 2}, \dots, h_{k, M}]

represents all possible channel vectors corresponding to user k.

Substituting the above results into Equation (24), we can get the overall system MSE:

\begin{matrix} MSE (V, W) = \sum_{k = 1}^{K} {MSE}_{k} \\ = \sum_{k = 1}^{K} \{\begin{matrix} η^{2} {(vec {(V_{k} ⊙ H_{k})}^{T} W)}^{2} + η vec {(V_{k} ⊙ H_{k})}^{T} w_{k} + η^{2} {(vec {(V_{k} ⊙ H_{k})}^{T} D)}^{2} \\ + η (2 n_{b} + 1) vec {(V_{k} ⊙ H_{k})}^{T} D + n_{b}^{2} + n_{b} + 1 \end{matrix}\} \end{matrix}

(33)

Based on the above discussion, we can transform problem

P 1

into the following problem

P 2

\begin{matrix} P 2 : min_{W, V} & MSE (W, V) \\ s . t . & C 1 : \sum_{k = 1}^{K} w_{k} u_{k} + D \geq 0 . \\ C 2 : \sum_{k = 1}^{K} w_{k} u_{k} + D \leq \frac{P_{max} τ}{h ν} 1, \\ C 3 : \sum_{i = 1}^{M K} v_{n, i} \leq 1, \forall n \in N, \\ C 4 : v_{n, i} \in \{0, 1\}, \forall i \in I . \end{matrix}

(34)

Proposition 1.

The problem

P 2

is equivalent to

P 1

Proof.

See Appendix A. □

4.2. Optimizations of Precoding Matrix

When the IRS auxiliary matrix

V^{(r)}

is specified at the r-th iteration, the problem

P 2

can be simplified to the following form:

\begin{matrix} P 2 - a : min_{W} & MSE (W, V^{(r)}) \\ s . t . & C 1 : \sum_{k = 1}^{K} w_{k} u_{k} + D \geq 0, \\ C 2 : \sum_{k = 1}^{K} w_{k} u_{k} + D \leq \frac{P_{max} τ}{h ν} 1 . \end{matrix}

(35)

From (33), it is evident that

MSE (W, V^{(r)})

is a quadratic function, indicating that

MSE (W, V^{(t)})

is a convex in terms of

W

. However, the two constraints of problem

P 2 - a

do not satisfy convexity yet since it becomes uncertain, depending on the transmitted symbol vector

u

. To eliminate this uncertainty, we consider two extreme cases, where

u_{k} = - 1, \forall k \in K

and

u_{k} = 1, \forall k \in K

. Constraint

C 1

turns into

\sum_{k = 1}^{K} w_{k} + D \geq 0

and

\sum_{k = 1}^{K} (- w_{k}) + D \geq 0

in these two extreme cases, respectively. Considering these two cases together, we can arrive at a more general expression:

\sum_{k = 1}^{K} |w_{k}| \leq D .

(36)

Similarly, constraint

C 2

can be updated as

\sum_{k = 1}^{K} |w_{k}| \leq \frac{P_{max} τ}{h ν} 1 - D .

(37)

As a result, problem

P 2 - a

is transformed into

\begin{matrix} P 3 - a : min_{W} & MSE (W, V^{(r)}) \\ s . t . & (36), (37) . \end{matrix}

(38)

For the optimization problem

P 3 - a

, the objective function is convex, and constraints (36) and (37) are also convex (specifically, affine). Therefore,

P 3 - a

constitutes a standard convex problem, which can be directly solved using the CVX toolbox [40] in MATLAB (2023b).

4.3. Optimization of IRS Configuration

Similarly, if the precoding matrix

W^{(r)}

is provided at the r-th iteration, the problem

P 2

can be simplified to

\begin{matrix} P 2 - b : min_{V} & MSE (V, W^{(r)}) \\ s . t . & (30), (31) . \end{matrix}

(39)

In practice, solving for the auxiliary matrix

V

as an integer is challenging. To address this computational complexity, we adopt a more practical approach by relaxing the values of

V

. We constrain each element of

V

to lie between 0 and 1, and then we restore

V

, based on the criterion of minimizing the projection distance.

The relaxed form of problem

P 2 - b

\begin{matrix} P 3 - b : min_{V} & MSE (V, W^{(t)}) \\ s . t . & C 1 : \sum_{i = 1}^{M K} v_{n, i} \leq 1, \forall n \in N, \\ C 2 : 0 \leq v_{n, i} \leq 1, \forall i \in I . \end{matrix}

(40)

According to (33), the expression for

MSE (V, W^{(t)})

is the sum of two quadratic functions of

V

. Since the sum of convex functions remains convex,

MSE (V, W^{(t)})

is a convex function of

V

. Constraint

C 1

is also convex (affine), while constraint

C 2

ensures that the elements of

V

are constrained between 0 and 1, forming a convex set. Therefore, problem

P 3 - b

is convex and can be solved directly using CVX.

After MSE converges, we can recover the two configuration matrices,

F

and

R

, of IRS from

V

solved in problem

P 3 - b

. Let

{\bar{v}}_{n} \in R_{+}^{1 \times M K}

denote the n-th row of

V

, and let

e_{p}

be a one-hot row vector of the same dimension as

{\bar{v}}_{n}

, with the p-th element equal to 1. In addition, define

e_{0} = 0

. The index corresponding to the minimum distance projection of the n-th IRS unit is

p^{*} = \arg min_{p} {∥{\bar{v}}_{n} - e_{p}∥}_{2}^{2},

(41)

where

e_{p} \in \{e_{0}, e_{1}, \dots, e_{M K}\}

Based on the (29) and Proposition 1, the associated LED and PD can be obtained via

\begin{matrix} m = ⌊ (p^{*} - 1) / N ⌋ + 1, \end{matrix}

(42)

\begin{matrix} k = \mod (p^{*} - 1, N) + 1, \end{matrix}

(43)

where

p^{*} \in \{0\} \cup I

4.4. Algorithm Summary and Analysis

Algorithm 1 outlines the proposed AO algorithm for the precoding matrix and IRS configuration matrix. For simplicity, we assume that each IRS unit is paired with a matched LED and PD, ensuring that every unit of the IRS is utilized. Steps 1 to 2 initialize

W^{(0)}

and

V^{(0)}

. In Steps 3 to 10, the algorithm iteratively addresses the two convex subproblems,

P 3 - a

and

P 3 - b

, refining and updating

W^{(r + 1)}

and

V^{(r + 1)}

. This iterative process continues until the MSE difference between successive iterations falls below a predefined threshold, indicating convergence. Finally, we discretize the obtained matrix

V

into binary variables based on (42) and (43) to derive

F

and

R

Algorithm 1: Proposed precoding and IRS optimization algorithm

Next, we discuss the algorithmic complexity of Algorithm 1. The convergence of the proposed AO algorithm is guaranteed by the convexity of the two subproblems. Specifically, let

V^{(r)}

and

W^{(r)}

denote the relaxed configuration matrix and the precoding matrix at the r-th iteration, respectively. In the subsequent iteration,

W^{(r + 1)}

is obtained by solving

P 3 - a

, leading to the inequality

MSE (W^{(r + 1)}, V^{(r)}) \leq MSE (W^{(r)}, V^{(r)})

. Similarly, solving problem

P 3 - b

yields

V^{(r + 1)}

, which results in the inequality

MSE (W^{(r + 1)}, V^{(r + 1)}) \leq MSE (W^{(r + 1)}, V^{(r)})

. Consequently, after several iterations, the value of MSE converges to a non-negative value.

The complexity of Algorithm 1 is influenced by the number of iterations required for the two subproblems and the number of matrix operations performed in each iteration. Given a precision of

ε

, the number of iterations needed to obtain the globally optimal solution for subproblem

P 3 - a

O ({(M K)}^{2} log (1 / ε))

, where

M, K

are the dimensions of

W

. The time complexity of the interior point method for each iteration is

O (M K^{2})

[41]. Therefore, the overall complexity of

P 3 - a

O (M^{3} K^{4} log (1 / ε))

. Similarly, the overall complexity for

P 3 - b

O (N^{3} {(M K)}^{4} log (1 / ε))

. Consequently, the total complexity of

P 3

O (M^{3} K^{4} log (1 / ε) + N^{3} {(M K)}^{4} log (1 / ε))

. Due to the dominance of the subproblem

P 3 - b

, and given the low priority of addition, the overall complexity of Algorithm 1 is

O (I_{i t e r} (N^{3} {(M K)}^{4} log (1 / ε)))

, where

I_{i t e r}

denotes the number of iterations required for convergence.

4.5. Assumptions Summary

Given the maturity of channel estimation technology, we assume that the signal transmitter [42] knows perfect channel state information (CSI). This can be achieved by estimating the CSI based on the pilot signal using least-squares (LS) estimation at the receiver and then feeding it back to the transmitter. At the transmitter, we add DC bias constraints to ensure that the optical signal is non-negative and power constraints to ensure that the LED is within the feasible operating range. Without a loss of generality, we assume that the DC bias added to all transmitting LEDs is the same. In addition, the constraints on IRSs come from the uniqueness of the configuration of each IRS unit. In the algorithm initialization, in order to ensure the efficient use of IRS, we assume that each IRS unit has a paired LED/PD pair and is in operation.

5. Numerical Results

In this section, we conduct extensive simulation experiments to evaluate the communication performance of of our proposed algorithm in an IRS-assisted photon-counting MU downlink communication system. Specifically, we use the widely adopted Monte Carlo method to randomly generate transmission data for all users and evaluate the MSE and BER performance of the communication system. To reduce the error, we take the average of 100 independent tests for each data point in the figure.

In the subsequent simulation experiments, we set the wavelength to

λ = 1550 nm

and the PD efficiency to

η = 0.5

. According to [19], we set the scintillation parameters

α = 1

and

β = 1

, which indicate the situation of strong turbulence fading. In fact, the scintillation index is defined as

S . I . = 1 / α + 1 / β + 1 / (α β)

[43], and its value represents the strength of the channel turbulence fading. In addition, we set the symbol duration to

τ = 1

µs and the background radiation energy per bit is

- 170 dBJ

. Our subsequent research on the robustness of the solution can be conducted by setting different scintillation indexes and background radiation values. Table 1 shows the detailed parameter settings of the simulation system.

To evaluate the performance improvement offered via the proposed AO algorithm, we selected two comparison schemes. (1) ZF precoding: This scheme employs typical ZF precoding under the AWGN model as a reference. The specific calculation method is expressed as

W_{Z F} = G^{T} {({GG}^{T})}^{- 1}

and needs to be scaled by multiplying the coefficient to satisfy constraints (36) and (37). (2) Random IRS scheme: in this scheme, the binary matrix

V

P 3 - b

is generated randomly, needing only to satisfy constraints (30) and (31).

To begin with, we present the convergence performance curve of the proposed algorithm, illustrated in Figure 3. The simulation parameters are configured for a 12 × 4 communication system assisted by 32 optical IRS units. As shown in Figure 3, the proposed AO algorithm achieves convergence after 10 iterations, while the only precoding

W

scheme reaches convergence after 15 iterations. The convergence speed in both cases is very fast, indicating the effectiveness of the proposed AO algorithm. However, the AO algorithm has fewer iterations, indicating that it is more efficient. In addition, the simulation results indicate that compared with the only precoding scheme, the convergence value of the AO algorithm is lower and can reach close to

10^{- 2}

. This shows the necessity of adding IRS optimization, which can significantly reduce the MSE of the system and improve the system communication performance.

Next, we explore the comparison between the proposed scheme and other baseline schemes, as shown in Figure 4. Regarding the coding methods, Figure 4 clearly illustrates that the performance of the proposed MMSE precoding scheme under Poisson noise is significantly superior to that of the ZF coding scheme under the AWGN model. Specifically, when the transmitted signal energy per bit is

E_{b} = - 163 dBJ

, the MSE of the ZF scheme is less than

10^{- 1}

, while the MSE of our proposed MMSE scheme can reach

10^{- 3}

. This improvement arises because the ZF scheme fails to capture the correlation between the noise and signal in the photon-counting system. Consequently, even with our proposed IRS configuration scheme, the MSE performance of the ZF method can only achieve approximately

10^{- 2}

. On the other hand, when compared to the randomized IRS configuration scheme, our proposed IRS configuration scheme significantly reduces the MSE of the system, highlighting the effectiveness of our IRS design.

In Figure 5, we simulate the MSEs of several baseline schemes alongside our proposed method with varying numbers of IRS units. Notably, the MSE for all four schemes decreases as the number of IRS units increases. However, for the randomized IRS configuration scheme, the downward trend becomes less obvious after

N = 32

. This occurs because the randomized configuration fails to effectively leverage the additional IRS units to enhance communication performance. As the number of IRS units increases significantly, the potential benefits diminish due to the ineffective scheduling of the IRS. In contrast, the MSE for our proposed algorithm demonstrates a consistent and significant decline with the increasing number of IRS units. This trend is expected, as a greater number of IRS units expands the feasible space of the NLOS channel matrix, allowing our IRS configuration scheme to fully capitalize on the benefits provided by the additional units and thereby enhancing the overall system performance.

Additionally, Figure 6 presents the BER performance of our AO algorithm. It is evident that, as the number of emitting LEDs increases, the average BER for users decreases significantly. Specifically, when the transmission energy per bit is

E_{b} = - 163 dBJ

, the BER performance of a

16 \times 4

system can reach

10^{- 5}

. These results demonstrate the effectiveness of our precoding design scheme, which leverages diversity gain to enhance the system’s resilience against turbulent fading.

We also investigated the BER performance of the system with varying numbers of IRS units (i.e., 12, 24, 40, and 60), as depicted in Figure 7. It is evident from the figure that, when the number of IRS units is small, the NLOS channel provided by the IRS struggles to achieve satisfactory communication performance. Specifically, the BER does not exhibit a significant downward trend with increasing transmission energy per bit,

E_{b}

. For instance, with only 12 IRS units, even at a transmitted signal energy per bit of

E_{b} = - 163 dBJ

, the average BER remains around

10^{- 1}

. However, as the amount of IRS units rises, the system’s average BER exhibits a rapid decline with a rising

E_{b}

, demonstrating that our proposed method effectively configures all IRS units to achieve excellent BER performance. Specifically, when 60 IRS units are configured and the transmitted signal energy per bit,

E_{b}

, is −163 dBJ, the average BER of the system can reach

10^{- 5}

In addition, considering the influence of environmental factors and uncertainties, we also design experiments to verify the robustness of the algorithm. As shown in Figure 8, we explored the effect of turbulence intensity on the average BER performance of the proposed algorithm, where

M = 12

N = 36

, and

K = 4

. It can be seen from Figure 8 that, under different turbulence intensities, the BER performance of the proposed scheme shows a downward trend. However, under stronger turbulence intensities, a higher

E_{b}

must be used if the same BER target needs to be achieved.

Figure 9 shows the effect of background radiation intensity on the average BER performance of the system, where

M = 16

N = 40

, and

K = 4

. The experimental results show that our proposed scheme can resist the influence of strong background radiation. Specifically, even under strong background light of

E_{b} = - 164 dBJ

, the BER of the system can reach

10^{- 4}

. In addition, as the background radiation increases, higher signal energy is required to achieve the same BER. When the background radiation energy per bit

E_{n b}

increases from

- 172

dBJ to

- 168

dBJ, the required signal energy

E_{b}

for

BER = 10^{- 4}

only increases by about 1.2 dB. These results demonstrate the tolerance of the proposed scheme to background radiation.

In Figure 10, we consider the impact of channel uncertainty on system performance. Imperfect CSI may be caused by estimation errors or delayed feedback from the receiver, which can lead to the incorrect generation of the precoding matrix and thus affect the performance of the algorithm. We characterize the channel irradiance as

{\hat{h}}_{k, m} = δ (h_{k, m})

, where the estimation error

δ

follows an independent and uniformly random distribution within the interval

[- δ, δ]

, with

δ

representing the maximum error percentage [44]. In Figure 10,

δ = 0

represents perfect CSI, and it can be observed that the proposed algorithm exhibits excellent robustness. Specifically, even with an estimation error percentage as high as

δ = 0.8

, our scheme can achieve a BER performance of

10^{- 3}

E_{b} = - 164 dBJ

6. Conclusions

In this paper, we have proposed a novel IRS-assisted MU photon-counting system and jointly optimized the precoder and the IRS configuration based on the MMSE principle. Under the assumption of extreme near-field conditions, we discussed the crosstalk-free properties of the NLOS channel achieved via optical IRS reflection and simulated the IRS configuration by aligning the IRS units with various LED/PD pairs. Subsequently, we decomposed the non-convex optimization problem into two convex subproblems through variable substitution and relaxation, and we developed an AO algorithm to solve it effectively. Additionally, we analyzed the convergence and complexity of the algorithm. Finally, the simulation results demonstrated the superiority of our AO algorithm compared to other benchmark schemes in terms of MSE and BER, highlighting that the optimization of the IRS configuration significantly enhances the performance of the communication system.

It should be pointed out that the above simulation experiment did not consider the impact of dead time in the photon-counting system. Dead time refers to the minimum time delay required for the detector to detect two consecutive photons. In subsequent work, we can analyze the impact of dead time and optimization schemes on the system throughput. In addition, this system considers an MU system equipped with a single antenna, which can be expanded to an MU system equipped with multiple PDs in the future. In order to save transmission power, it is also a feasible option to optimize the DC bias as an optimization variable, which can be further explored in subsequent work.

Author Contributions

Methodology, J.W., Y.C., F.L. and X.Z.; software, J.W.; writing—original draft, J.W.; writing—review and editing, J.W., X.Z., C.C. and H.X. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported in part by the Natural Science Foundation of Shanghai under Grant 24ZR1407100 and in part by the National Natural Science Foundation of China under Grant 62231010, Grant 62071126, and Grant 61571135.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

The following abbreviations are used in this manuscript:

IRS	Intelligent reflecting surfaces
MIMO	Multiple-input, multiple-output
NLOS	Non-line-of-sight
MSE	Mean square error
AO	Alternating optimization
IoT	Internet of Things
FSO	Free-space optical
PCP	Poisson counting process
AWGN	Additive white Gaussian noise
ZF	Zero forcing
BER	Bit error rate
CSI	Channel state information
OOK	On–off keying
PD	Photo-detector
LEDs	Light-emitting diodes
DC	Direct current
LEDs	Light-emitting diodes
PDF	Probability density function

Appendix A

First, we prove sufficiency. In fact, the construction of

F

and

R

can be directly obtained from the given

V

. Specifically, given

v_{n, k + (m - 1) K} = 1

, we can directly obtain the following conclusions:

r_{n, m} = f_{n, k} = 1

and

r_{n, i} = f_{n, j} = 0, \forall i \neq m, j \neq k

As for the necessity of this proposition, we can start from the characteristics of

F

and

G

. According to the definition of the

F

matrix and the row constraints, we know that the support set of

f_{k}

does not intersect with the support set of

f_{i}, \forall i \neq k

, which shows that the column orthogonality of

F

. A similar column orthogonality property also holds for the

R

matrix, which we express as

f_{k_{1}}^{T} f_{k_{2}} = δ (k_{1} - k_{2}), r_{m_{1}}^{T} r_{m_{2}} = δ (m_{1} - m_{2}),

(A1)

where

δ (\cdot)

is the discrete Dirac function. Based on the above properties, we continue to get

{(f_{k_{1}} ⊙ r_{m_{1}})}^{T} (f_{k_{2}} ⊙ r_{m_{2}}) = δ (k_{1} - k_{2}) δ (m_{1} - m_{2}),

(A2)

which is nonzero if and only if

k_{1} - k_{2}

and

m_{1} - m_{2}

. Combining (29) and (A2), we can obtain proof that the support set of

v_{i}

is disjointed from the support set of

v_{j}

, where

\forall j \in I, i \neq j

, which guarantees the column orthogonality of

V

Based on the above discussion, the proposition is proven. For more details, please refer to ([25], Proposition 1).

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Figure 1. IRS-assisted MU downlink MISO photon-counting communication system.

Figure 2. Two-dimensional topological model of optical path based on IRS.

Figure 3. Convergence performance comparison of the proposed AO algorithm and the scheme that only optimizes the precoding matrix.

Figure 4. Comparison of normalized MSE of different schemes, where

M = 16, K = 4

and

N = 64

Figure 4. Comparison of normalized MSE of different schemes, where

M = 16, K = 4

and

N = 64

Figure 5. Comparison of normalized MSE of different schemes under different numbers of IRS units, where

M = 16, K = 4

and

E_{b} = - 170 dBJ

Figure 5. Comparison of normalized MSE of different schemes under different numbers of IRS units, where

M = 16, K = 4

and

E_{b} = - 170 dBJ

Figure 6. Average BER performance of AO algorithm under different transmitting LEDs, where

N = 36

Figure 6. Average BER performance of AO algorithm under different transmitting LEDs, where

N = 36

Figure 7. Average BER performance of AO algorithm with different numbers of IRS units, where

M = 12

and

K = 4

Figure 7. Average BER performance of AO algorithm with different numbers of IRS units, where

M = 12

and

K = 4

Figure 8. Average BER performance of AO algorithm under different turbulence channels, where

M = 12

N = 36

, and

K = 4

Figure 8. Average BER performance of AO algorithm under different turbulence channels, where

M = 12

N = 36

, and

K = 4

Figure 9. Average BER performance of AO algorithm under different background radiations, where

M = 16

N = 40

, and

K = 4

Figure 9. Average BER performance of AO algorithm under different background radiations, where

M = 16

N = 40

, and

K = 4

Figure 10. Average BER performance of AO algorithm under imperfect CSI, where

M = 16

N = 40

, and

K = 4

Figure 10. Average BER performance of AO algorithm under imperfect CSI, where

M = 16

N = 40

, and

K = 4

Table 1. Simulation parameters of the system.

Parameter	Value
Optical wavelength ( $λ$ )	1550 nm
Symbol duration ( $τ$ )	1 µs
PD efficiency ( $η$ )	0.5
IRS panel energy reflection efficiency ( $ζ$ )	1
End-to-end distance ( $d_{e 2 e}$ )	800 m
Horizontal angle of emitted beam ( $θ_{L}$ )	$π / 4$
Horizontal angle of IRS panel ( $θ_{I R S}$ )	$π / 10$
Horizontal angle of PD panel ( $θ_{P}$ )	$π / 3$
PD aperture radius ( $r_{p}$ )	2.5 cm
Area of an IRS unit	$10 \times 10 {cm}^{2}$
Maximum number of iterations ( $T_{max}$ )	30
Scintillation parameters ( $α, β$ )	1, 1
Background radiation energy per bit ( $E_{n b}$ )	−170 dBJ

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Wang, J.; Zhou, X.; Li, F.; Chen, Y.; Cai, C.; Xu, H. Joint Design of Transmitter Precoding and Optical Intelligent Reflecting Surface Configuration for Photon-Counting MIMO Systems Under Poisson Shot Noise. Appl. Sci. 2024, 14, 11994. https://doi.org/10.3390/app142411994

AMA Style

Wang J, Zhou X, Li F, Chen Y, Cai C, Xu H. Joint Design of Transmitter Precoding and Optical Intelligent Reflecting Surface Configuration for Photon-Counting MIMO Systems Under Poisson Shot Noise. Applied Sciences. 2024; 14(24):11994. https://doi.org/10.3390/app142411994

Chicago/Turabian Style

Wang, Jian, Xiaolin Zhou, Fanghua Li, Yongkang Chen, Chaoyi Cai, and Haoze Xu. 2024. "Joint Design of Transmitter Precoding and Optical Intelligent Reflecting Surface Configuration for Photon-Counting MIMO Systems Under Poisson Shot Noise" Applied Sciences 14, no. 24: 11994. https://doi.org/10.3390/app142411994

APA Style

Wang, J., Zhou, X., Li, F., Chen, Y., Cai, C., & Xu, H. (2024). Joint Design of Transmitter Precoding and Optical Intelligent Reflecting Surface Configuration for Photon-Counting MIMO Systems Under Poisson Shot Noise. Applied Sciences, 14(24), 11994. https://doi.org/10.3390/app142411994

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Joint Design of Transmitter Precoding and Optical Intelligent Reflecting Surface Configuration for Photon-Counting MIMO Systems Under Poisson Shot Noise

Abstract

1. Introduction

1.1. Related Work

1.2. Contributions

2. System Model

2.1. Transmitter

2.2. Channel Model

2.2.1. Channel Gains

2.2.2. IRS-Aided MIMO Channel

2.3. Receiver

3. Problem Formulation

4. Proposed Alternating Optimization Algorithm to Minimize MSE

4.1. Optimization Problem Transformation

4.2. Optimizations of Precoding Matrix

4.3. Optimization of IRS Configuration

4.4. Algorithm Summary and Analysis

4.5. Assumptions Summary

5. Numerical Results

6. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest

Abbreviations

Appendix A

References

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