Open Access
Add to BibSonomy
Abstract
Photon counting is a promising solution to detecting low-power optical signals for ultraviolet (UV) communications in the forthcoming sixth-generation (6G) network. Different from the conventional additive white Gaussian noise (AWGN) model, the discrete signal-dependent Poisson shot noise poses challenges to the signal processing of photon-counting systems. In this paper, a joint design of precoder and equalizer is proposed for photon-counting multiple-input multiple-output (PhC-MIMO) UV systems. To circumvent the impasse arising from the signal-dependent shot noise, we propose an alternating optimization algorithm based on the minimum mean squared error (MMSE) criterion. The algorithm decomposes the joint design into convex subproblems solved in an alternating manner, and guarantees at least a stationary point solution. Numerical results corroborate that the proposed system exhibits robustness to turbulence fading and offers high throughput while mitigating the adverse effect of background radiation noise. Specifically, the 32 × 8 system can achieve a bit error rate (BER) of 10−5 at the signal energy of −154.0 dBJ per bit under strong Gamma-Gamma turbulence with the scintillation index (S.I.) of 3.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Ultraviolet (UV) non-line-of-sight (NLOS) communication has emerged as an attractive technology for the sixth-generation (6G) network, which offers broad coverage and security [1–3]. However, outdoor atmospheric turbulence severely degrades the performance of UV communications [4], requiring robust technologies to mitigate atmospheric scintillation fading.
Multiple-input multiple-output (MIMO) technology provides a promising solution for UV communications to combat turbulence fading [5]. Shi et al. [6] studied the diversity reception of the UV communication system by utilizing multiple receivers, which had reasonable bit error rate (BER) performance in a moving condition. Arya et al. [7] presented a diversity technique based on a switch and stay combining strategy in UV NLOS systems to mitigate channel turbulence. In addition, MIMO techniques enable optical wireless communication (OWC) systems to transmit multi-stream data simultaneously by utilizing spatial multiplexing, which can satisfy the rapidly rising demand for high data rates [8]. Gupta et al. [9] investigated a $2 \times 2$ UV MIMO system using a multi-beam approach, illustrating the spatial multiplexing advantage of the MIMO technique.
The potential of photon counting has been recognized for its unparalleled detection performance for low-power optical signals [10,11]. Considering the potential to extend the transmission distance, photon counting comprises a prospective technology in the outdoor Internet of Things (IoT) and other applications of the 6G networks [12]. It has superior power efficiency to the conventional OWC systems using positive-intrinsic-negative (PIN) diodes or avalanche photodiodes (APDs) [13]. In low-power signal scenarios, a critical difference between the photon-counting system and the conventional OWC system with the additive white Gaussian noise (AWGN) model lies in the signal-dependent shot noise [14]. Note that the shot-noise-limited photon-counting system is modeled by the Poisson counting process (PCP) [15]. These characteristics render algorithms developed for the AWGN model inapplicable to photon-counting systems, calling for novel signal processing techniques in the Poisson shot-noise-limited model. Regarding equalizing techniques, Gong et al. [16] presented the linear minimum mean squared error (LMMSE) equalizer for photon-counting single-input multiple-output (SIMO) systems using on-off keying (OOK) or pulse position modulation (PPM). In terms of equalization techniques, minimum mean squared error (MMSE) precoding methods have been considered in conventional OWC systems with the AWGN model [17–19].
To the best of our knowledge, no precoding design has been to date put forward in the literature for the Poisson shot-noise-limited photon-counting systems. To bridge this gap, this paper proposes a new photon-counting MIMO (PhC-MIMO) precoding UV system by jointly designing the precoder and the equalizer, as shown in Fig. 1. The key contributions of the paper are summarized as follows:
- • A joint design of precoder and equalizer for PhC-MIMO systems is proposed based on the MMSE criterion. Our proposed system is robust to turbulence fading by exploiting significant diversity gain, and achieves considerable throughput by efficiently utilizing spatial multiplexing gain.
- • An alternating optimization algorithm is proposed to solve the multivariate optimization problem of the PhC-MIMO design. The proposed algorithm benefits from the iterations between the convex subproblems, guaranteeing at least a stationary point solution.
- • An element-wise alternating optimization algorithm is further designed as a computationally efficient alternative to the alternating optimization algorithm, requiring considerably fewer iterations to converge.
Extensive simulations indicate that our proposed algorithms outperform the pseudo-inverse solution in terms of BER performance. The low-complexity element-wise algorithm can converge substantially faster than the alternating optimization algorithm. In particular, it can cut the convergence iterations by about 30% to 80% at a loss of 1 to 4 dB in BER performance.
The rest of this paper is organized as follows. In Section 2, the system framework is introduced. In Section 3, the joint precoder and equalizer optimization problem is formulated, and the proposed algorithms are delineated. Section 4 presents simulation results, followed by conclusions in Section 5.
2. System model
We consider a PhC-MIMO precoding system with $N_{\mathrm {t}}$ number of UV light-emitting diode (LED) transmitters and $N_{\mathrm {r}}$ number of photon-detector (PD) receivers over atmospheric turbulence channels. The schematic diagram of the proposed system is illustrated in Fig. 1.
2.1 PAM precoding transmitter
This subsection describes the pre-processing of the data signal on the transmitter side. Pulse amplitude modulation (PAM) is adopted in our system for its high spectral efficiency and flexible implementation [20]. For $Q$-order PAM, $\mathbf {s} = [s_{1} \; s_{2} \; \cdots \; s_{N_{\mathrm {s}}}]^{\mathrm {T}}$ represents the $N_{\mathrm {s}} \times 1$ PAM signal with its elements taking values independently, where $(\cdot )^{\mathrm {T}}$ indicates the transpose operation. The PAM symbols are uniformly distributed in the range of $[-1,1]$ with discrete values as $(2m-1-Q)/(Q-1), 1 \leq m \leq Q$ [21]. We assume that the probabilities of the PAM symbols are equal. Note that a time slot consists of $N_{\mathrm {s}} \mathrm {log}{_{2}} Q$ coded bits for $Q$-order PAM. The precoded signal is represented by $\mathbf {x} = [x_{1} \; x_{2} \; \cdots \; x_{N_{\mathrm {t}}}]^{\mathrm {T}}$ after precoding and adding the direct current bias (DC-bias), as given by
where $\mathbf {W}$ is the $N_{\mathrm {t}} \times N_{\mathrm {s}}$ precoding matrix, the design of which is articulated in Section 3, and $\mathbf {B} = [B \; B \; \cdots \; B]^{\mathrm {T}}$ denotes the $N_{\mathrm {t}} \times 1$ DC-bias vector. Note that $x_{i}$ indicates the number of transmitted photons at the $i$-th LED. The relationship between $x_{i}$ and the transmitted optical power $P_{\textrm {t},i}$ of the $i$-th LED yields $x_{i}=P_{\textrm {t},i} \tau / (h \nu )$, where $\tau$, $h$, and $\nu$ denote the time duration of a slot, Planck’s constant, and the optical frequency, respectively.2.2 Atmospheric turbulence channel model
Considering the large divergence angle and less coherent beam, LEDs are preferred in UV communications. The fading induced by atmospheric turbulence is the key factor of the channel model considered in our system [16,3,22–23]. The proposed scheme is readily applicable to systems with no turbulence. By taking the possible turbulence into account, the proposed system offers generic applicability, especially in bad weather conditions. The Gamma-Gamma distribution is a widely accepted model to describe turbulence fading [24]. The probability distribution function (PDF) of the optical intensity fading $I_{i,j}$ between the $i$-th LED and the $j$-th PD is given by [24]
We assume the perfect channel state information (CSI) in this paper. In practice, the OWC channel is typically slow-fading and the coherence time of the channel is much larger than a symbol duration [25,26]. The CSI can be estimated at the receivers based on the pilot signals using least-squares (LS) or LMMSE estimation, and then fed back to the transmitter [27].
2.3 Photon-counting receiver
In this subsection, we describe the post-processing of the received data signal. After propagating through the atmospheric turbulence channel, the optical signal is received by the photon counters. The PCP model is adopted on the receiver side. The outputs of the photon counters are digital counts, denoted by the $N_{\mathrm {r}} \times 1$ vector $\mathbf {y} = [y_{1} \; y_{2} \; \cdots \; y_{N_{\mathrm {r}}}]^{\mathrm {T}}$. At each slot, the received count $y_{j}$ follows the Poisson distribution, as given by [13]
where $\mathrm {Pr}(\cdot )$ indicates the probability, $r$ is a possible value of the received count, and $\lambda _{j}$ denotes the average number of received signal photons at the $j$-th receiver, as given byHere, $\eta$ is the quantum efficiency of the photon counter, $\mathbf {I}_{j}^{\mathrm {T}} = [I_{1,j} \; I_{2,j} \; \cdots \; I_{N_{\mathrm {t}},j}]$ represents the $j$-th row of the channel matrix $\mathbf {I}$, i.e., $\mathbf {I} = [\mathbf {I}_{1} \; \mathbf {I}_{2} \; \cdots \; \mathbf {I}_{N_{\mathrm {r}}}]^{\mathrm {T}}$, and $n_{\mathrm {b}}=\eta P_{\mathrm {b}} \tau / (h \nu )$ is the background radiation per slot with $P_{\mathrm {b}}$ being the incident background power. As observed in (5) and (6), the received count depends on both the transmitted signal and the background radiation.
Considering that the high computational complexity of the MMSE detection limits the real-time efficiency of the system, we adopt a linear receiver architecture for the PhC-MIMO system to estimate the raw PAM signals. The expression of the MMSE estimation is provided in Appendix B. The LMMSE estimation of the PAM symbol $s_{j}$ is $\hat {s}_{j}=\mathbf {d}_{j}^{\mathrm {T}}[1 \; \mathbf {y}^{\mathrm {T}} ]^{\mathrm {T}}$. The equalizing vector $\mathbf {d}_{j}$ can be obtained as [16]
where $\mathbf {R}_{yy}$ and $\mathbf {r}_{ys_{j}}$ are the correlation matrix and correlation vector, respectively, as will be elaborated on in Section 3.3. Joint precoder and equalizer design for the PhC-MIMO system
In this section, the precoder and the equalizer are jointly designed for the new PhC-MIMO system. The mean squared error (MSE) is regarded as the performance metric to evaluate the design. Since the Poisson shot noise is signal-dependent, the received count of the photon counter cannot be directly formulated as an equation of the transmitted signal. To tackle this issue, we adopt the MMSE criterion in the proposed PhC-MIMO signal processing algorithms, where the MSE is established statistically (based on the probability and mean) without the need for the exact expression of the received signal. We decompose the joint problem into two subproblems and optimize them in an alternating manner. Then, we introduce an element-wise alternating optimization algorithm to reduce the complexity.
3.1 Problem formulation
In the new PhC-MIMO system, the precoding matrix $\mathbf {W}$ and the equalizing matrix $\mathbf {D}$ are jointly optimized based on the MMSE criterion. We assume that the number of data streams is equal to that of receivers, i.e., $N_{\mathrm {s}}=N_{\mathrm {r}}$. The MSE of the $j$-th receiver is given by $\mathrm {E}[(\hat {s}_{j}-s_{j})^{2}]$. The joint optimization problem can be formulated as
The $(\xi, \delta )$-th element of the matrix $\mathrm {E}(\mathbf {y} \mathbf {y}^{\mathrm {T}})$, denoted by $R_{\xi,\delta }$, is given by
3.2 Proposed alternating optimization algorithm
In order to solve problem (8), we propose an alternating optimization algorithm, which decouples the original problem into two more tractable subproblems to separately solve the precoder $\mathbf {W}$ and the equalizer $\mathbf {D}$. The precoder and the equalizer are solved and updated in an alternating manner.
1) Subproblem for solving $\mathbf {W}$:
For this subproblem, we aim to optimize the precoding matrix $\mathbf {W}$ for a given fixed feasible point $\mathbf {D}$. In other words, the objective function is viewed as an optimization problem concerning the precoder. In (9), we omit the constant terms and only focus on the terms related to $\mathbf {W}$, which are part of $f_{\mathrm {MSE}}{ }^{\prime }$, as given in the following
It can be seen from (14) that $f_{\mathrm {MSE,eq}}$ is a convex function. As the constraints $\mathrm {C}_{1}$ – $\mathrm {C}_{3}$ are clearly convex, subproblem (15) is a convex optimization problem, which can be solved by using the off-the-shelf CVX toolbox in MATLAB.
2) Subproblem for solving $\mathbf {D}$:
For this subproblem, we aim to optimize the equalizing matrix $\mathbf {D}$ while fixing the precoding matrix $\mathbf {W}$. In this case, the LMMSE equalizing matrix $\mathbf {D}$ can be readily calculated by
Algorithm 1 summarizes the proposed alternating optimization algorithm.
Algorithm 1. The alternating optimization algorithm to solve the joint problem
The initial point of Algorithm 1 can be obtained by designing the precoder $\mathbf {W}_{0}$ and the equalizer $\mathbf {D}_{0}$ separately. By reasonably assuming $N_{\mathrm {t}} \ge N_{\mathrm {r}}$, $\mathbf {W}_{0}$ can be calculated by
where the detailed derivation is provided in Appendix. The expression for $\mathbf {W}_{0}$ contains the pseudo-inverse matrix of the channel matrix $\mathbf {I}$. Then, $\mathbf {D}_{0}$ can be obtained, according to (16). A so-called pseudo-inverse solution to the considered problem can take $\mathbf {W}_{0}$ and $\mathbf {D}_{0}$ as the precoder and the equalizer, respectively. The precoding matrix of the pseudo-inverse solution, which does not take the distribution characteristics of the transmitted signal into account, aims to make the form of the equivalent channel a diagonal matrix. In other words, the precoding matrix of the pseudo-inverse solution only depends on the channel matrix.The proposed alternating optimization algorithm is convergent because the MSE has a lower bound, and the objective function $f_{\mathrm {MSE}}$ decreases monotonically throughout iterations. Moreover, $f_{\mathrm {MSE}}$ is continuously differentiable, the constraints only depend on $\mathbf {W}$, and each subproblem attains a globally optimal solution. According to [29], the limit point of the proposed alternating optimization algorithm is at least a stationary point solution to problem (8).
The complexity of the CVX interior point method is $\mathcal {O}(\max \{N_{\mathrm {c}}, N_{\mathrm {v}}\}^{4} \sqrt {N_{\mathrm {v}}})$, where $N_{\mathrm {c}}$ and $N_{\mathrm {v}}$ are the numbers of constraints and scalar variables, respectively [30]. For the subproblem solving $\mathbf {W}$, $N_{\mathrm {c}}=N_{\mathrm {t}}$ and $N_{\mathrm {v}}=N_{\mathrm {t}}N_{\mathrm {r}}$, the complexity of solving (15) is $\mathcal {O}((N_{\mathrm {t}}N_{\mathrm {r}})^{4.5})$. For the subproblem solving $\mathbf {D}$, the complexity is $\mathcal {O}(N_{\mathrm {r}}^{3})$. As a result, the overall complexity of Algorithm 1 is $\mathcal {O}(t_{\mathrm {iter}}(N_{\mathrm {t}}N_{\mathrm {r}})^{4.5})$, where $t_{\mathrm {iter}}$ denotes the number of iterations needed to converge.
3.3 Proposed element-wise alternating optimization algorithm
To reduce the complexity, we propose an element-wise alternating optimization algorithm to solve the joint precoder and equalizer design problem. In Algorithm 1, problem (15) can be divided into several subproblems to solve $w_{k,l}$ with closed-form solutions. Each of the subproblems is formulated as
In (14), if the element $w_{k,l}$ in $\mathbf {W}$ is the only variable, the objective function is a quadratic function of $w_{k,l}$. As a result, the minimum of $f_{\mathrm {MSE}}(w_{k,l})$ in a closed interval can only be taken at the vertex or endpoint. The vertex value can be calculated by setting the partial derivative to zero, as given by
Algorithm 2 summarizes the proposed element-wise alternating optimization algorithm.
Algorithm 2. The element-wise alternating optimization algorithm to solve the joint problem
The complexity of solving the subproblem for $w_{k,l}$ is $\mathcal {O}(N_{\mathrm {t}} N_{\mathrm {r}}^{3})$. As a result, the overall computational complexity of Algorithm 2 is $\mathcal {O}(t_{\mathrm {iter}} N_{\mathrm {t}}^{2} N_{\mathrm {r}}^{4})$, which is much lower than that of Algorithm 1. The reason is that each subproblem of Algorithm 2 has a closed-form optimal solution. To this end, the element-wise algorithm offers a low-complexity and effective solution to the considered joint precoder and equalizer optimization problem.
4. Numerical results
In this section, we carry out extensive simulations to demonstrate the attainable performance of the proposed PhC-MIMO precoding system in strong atmospheric turbulence channels. The pseudo-inverse solution, in which the form of the precoding matrix is the same as [17], serves as the benchmark. We adopt the following previously reported simulation parameters [16,31]: The light wavelength is set to 280 nm, which is in the deep UV band [32]; the channel satisfies a strong Gamma-Gamma distribution with $\alpha =1$ and $\beta =1$ [16]; the quantum efficiency $\eta$ is set to 0.06 [16]; and the background radiation energy per bit is $-188$ dBJ [31]. The Monte Carlo method is employed to simulate the turbulence. The simulation parameters considered are listed in Table 1.
Table 1. Simulation parameters of the system
4.1 Benefit 1: Diversity gain against turbulence fading
Figure 2 depicts the substantial diversity gain of the new PhC-MIMO precoding system under different numbers of transmitters. We see that the proposed algorithm outperforms the pseudo-inverse solution. With the growth of $N_{\mathrm {t}}$, a lower transmitted signal energy per bit $E_{\mathrm {b}}$ is required to achieve the same BER target. Specifically, in the PhC-MIMO system with $N_{\mathrm {t}}=32$ and $N_{\mathrm {r}}=8$, the alternating optimization algorithm can achieve the BER of $10^{-5}$ at $E_{\mathrm {b}}=-154.0$ dBJ, providing a performance gain of about 4.6 dB over the $16 \times 8$ system. This indicates that the proposed PhC-MIMO precoding system can enhance the robustness of the system to turbulence fading by exploiting significant diversity gain.
Fig. 2. BER performance of the proposed PhC-MIMO precoding system with 8 receivers under different numbers of transmitters. The PAM order is 2. The proposed alternating optimization algorithm is denoted by AO. $E_{\mathrm {b}}$ indicates the total transmitted signal energy per bit.
4.2 Benefit 2: Increased throughput thanks to multiplexing gain and PAM order
This subsection presents results for systems with different multi-stream numbers and PAM orders, contributing to the increased throughput of the system.
Figure 3 demonstrates that the simultaneous multiple streams of the raw data can enhance the system throughput. It can be observed that, as the number of data streams gets larger, the system throughput increases from 2 to 4 and 8 bits/slot, respectively. This indicates that the proposed system achieves spatial multiplexing gain.
Fig. 3. BER performance of the proposed PhC-MIMO precoding system with 16 transmitters upon different multi-stream numbers. The PAM order is 2.
Figure 4 shows the impact of the PAM order on the system. Assume that the duration of a symbol slot is 1 $\mu$s and thus the symbol rate is 1 Msps [16,33]. By increasing the PAM order to 4 and 8, the system throughput grows to 4 and 6 Mb/s, respectively. It is shown that increasing the PAM order is another way to enhance the system throughput, especially in scenarios where the respective numbers of transmitters and receivers are restricted.
Fig. 4. BER performance of the proposed system with 8 transmitters and 2 receivers upon different PAM orders. $P_{\mathrm {tx}}$ indicates the total transmitted power per bit.
4.3 Benefit 3: Resistance to background radiation
Figure 5 shows the impact of background radiation on the BER performance of the proposed system. With the increase of the background radiation, higher signal energy is required to achieve the same BER target. However, we also see that the proposed system still performs well at high background noise levels. When the background radiation energy per bit $E_{\mathrm {nb}}$ increases from $-165$ to $-155$ dBJ, the required signal energy $E_{\mathrm {b}}$ only increases by about 0.38 dB for $\mathrm {BER}=10^{-4}$. The system can achieve the BER of $10^{-4}$ at $E_{\mathrm {b}}=-151.8$ dBJ when $E_{\mathrm {nb}}$ is $-155$ dBJ. This demonstrates the tolerance of the proposed PhC-MIMO precoding system against background radiation. The reason is that the LMMSE equalizer, which takes noises into account, can effectively reduce the impact of the background radiation.
Fig. 5. BER performance of the proposed system with 16 transmitters and 4 receivers upon different background radiation levels. The PAM order is 2.
4.4 Benefit 4: Rapid convergence of the element-wise method
Figure 6 presents the convergence behaviors of the proposed algorithms. Compared to the alternating optimization algorithm, the element-wise method requires fewer iterations to converge, thus saving computational complexity. In particular, the element-wise method can cut the convergence iterations by about $30\%$ to $80\%$. For the $8 \times 4$ MIMO system, the element-wise method converges after fewer than 10 iterations, while the alternating optimization algorithm needs more than 30 iterations to converge. Besides, for the system with the same parameters, the number of iterations required for convergence increases with $E_{\mathrm {b}}$. Nevertheless, the value of the objective function at convergence decreases with the increase of $E_{\mathrm {b}}$. In other words, better system performance can be achieved.
Fig. 6. Convergence performance of the proposed optimization algorithms. The proposed element-wise alternating optimization algorithm is denoted as EW-AO.
Figure 7 plots the BER performance of the proposed algorithms, compared to the pseudo-inverse solution. Since the throughputs of the different schemes with the same numbers of the PAM order and data streams are the same, the system throughputs of Fig. 7(a) – Fig. 7(d) are 2, 4, 8, and 8 bits/slot, respectively. It can be observed that the same BER is achieved by our algorithms with lower $E_{\mathrm {b}}$. Quantitatively, the performance gains of the proposed algorithms are seen to be about 6 to 11 dB over the pseudo-inverse solution. Furthermore, by comparing Fig. 7(a) and Fig. 7(d), the element-wise method performs closely to the alternating optimization algorithm when $N_{\mathrm {t}}$ and $N_{\mathrm {r}}$ are small, while the alternating optimization algorithm has more noticeable improvement over the element-wise algorithm with larger $N_{\mathrm {t}}$ and $N_{\mathrm {r}}$. Specifically, the alternating optimization algorithm outperforms the element-wise method by about 1 to 4 dB. This is because the elements in $\mathbf {W}$ are coupled, resulting in relatively worse convergence under the element-wise method, as shown in Fig. 6. On the other hand, as discussed in Section 3.3, the element-wise algorithm has lower complexity than the alternating optimization algorithm. Therefore, a trade-off needs to be acquired between the system performance and complexity in practical applications.
4.5 Benefit 5: Robustness under outdated CSI
In this subsection, channel uncertainty is considered. The imperfect CSI will lead to the erroneous generation of the precoding matrix and the inaccurate estimation [34]. We assume an independent additive stochastic error $\varepsilon _{\mathrm {CSI}}$ for outdated CSI, which conforms to the normal distribution with the mean of zero and the variance of $V_{e}$, i.e., $\varepsilon _{\mathrm {CSI}} \sim \mathcal {N}\left (0, V_{e}\right )$ [35].
Figure 8 demonstrates the impact of channel uncertainty on the system performance. The imperfect CSI affects the transmission and equalization accuracy, leading to a higher BER compared to the case with the perfect CSI. It is observed that for systems with the same $N_{\mathrm {r}}$, the robustness to imperfect CSI improves as the number of $N_{\mathrm {t}}$ increases. For example, in the $32 \times 4$ case, the proposed alternating optimization algorithm can achieve the BER of $10^{-4}$ at $E_{\mathrm {b}} = -152.4$ dBJ under outdated CSI when $V_{e}$ is 0.05. This indicates that the proposed PhC-MIMO system can enhance its stability under the outdated CSI by increasing the number of transmitters.
Figure 9 assesses the impact of the turbulence strength on the proposed algorithm. In practical communications scenarios, the system performance is subject to different turbulence conditions [32,36]. Under stronger turbulence, a higher transmitted signal energy per bit is required to maintain the same BER target, as shown in Fig. 9.
Fig. 9. BER performance of the proposed system with 16 transmitters and 4 receivers upon different turbulence strength.
5. Conclusion
In this paper, we introduced a new PhC-MIMO precoding scheme for UV communications in atmospheric turbulence. A joint precoder and equalizer optimization problem was formulated under the MMSE criterion. To solve the problem, we proposed the alternating optimization algorithm, which can guarantee at least a stationary point solution. Furthermore, an element-wise alternating optimization algorithm was developed to deliver a low-complexity solution. Simulation results showed that the system is robust to turbulence fading due to diversity gain and exhibits high throughput thanks to multiplexing gain.
Appendix
A. Derivation of the initial precoder
In order to obtain the initial optimization point $\mathbf {W}_{0}$, the defined error of the receiver is expressed as $\varepsilon _{0}=\mathrm {E}[(y-s)^{2}]$ with $y \sim \operatorname {Poisson}(\lambda )$. The optimal $\lambda$ is expected to minimize $\varepsilon _{0}$, as given by
Since $\varepsilon _{0}$ is a quadratic function, the optimal $\lambda$ can be calculated as $\lambda =s-1/2$. $\lambda$ can also be obtained from (6). Then, we have $\eta \mathbf {Ix}=\mathbf {s}-(n_{\mathrm {b}}+1/2)$, which comprises $N_{\mathrm {r}}$ linear equations and $N_{\mathrm {t}}$ variables. By assuming $N_{\mathrm {t}} \ge N_{\mathrm {r}}$, $\mathbf {x}$ is solvable. The precoding matrix is obtained as $\mathbf {W}_{0}=\mathbf {I}^{\mathrm {T}}(\mathbf {I I}^{\mathrm {T}})^{-1}/\eta$.
B. MMSE estimation
The MMSE estimation of the PAM signal can be calculated by
As shown in (21), the high computational complexity to obtain the optimal MMSE estimation may limit the real-time detection efficiency of the system.
Funding
National Natural Science Foundation of China (62231010).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. W. Jiang, B. Han, M. A. Habibi, and H. D. Schotten, “The road towards 6G: A comprehensive survey,” IEEE Open J. Commun. Soc. 2, 334–366 (2021). [CrossRef]
2. H. Qi, D. Zou, C. Gong, and Z. Xu, “Two-dimensional intensity distribution and adaptive power allocation for ultraviolet ad-hoc network,” IEEE Trans. on Green Commun. Netw. 6(1), 558–570 (2022). [CrossRef]
3. V. Bhatia, S. Jain, K. Garg, and R. Mitra, “Performance analysis of RKHS based detectors for nonlinear NLOS ultraviolet communications,” IEEE Trans. Veh. Technol. 70(4), 3625–3639 (2021). [CrossRef]
4. M. Elamassie and M. Uysal, “Incremental diversity order for characterization of FSO communication systems over lognormal fading channels,” IEEE Commun. Lett. 24(4), 825–829 (2020). [CrossRef]
5. S. M. A. Hasan, S. Ahmed, and A. K. M. Nazrul Islam, “Simulation of a massive MIMO FSO system under atmospheric turbulence,” in 5th International Conference on Electrical Engineering and Information Communication Technology (2021), pp. 1–6.
6. J. Shi, M. Zhang, D. Han, M. Liu, Q. Huo, L. Lang, and P. Luo, “Experimental study of the mobility feature and selective maximal ratio combining algorithm for UV communication,” in 20th European Conference on Networks and Optical Communications (2015), pp. 1–4.
7. S. Arya and Y. H. Chung, “Non-line-of-sight ultraviolet communication with receiver diversity in atmospheric turbulence,” IEEE Photonics Technol. Lett. 30(10), 895–898 (2018). [CrossRef]
8. S. Jain, R. Mitra, and V. Bhatia, “Hybrid adaptive precoder and post-distorter for massive-MIMO VLC,” IEEE Commun. Lett. 24(1), 150–154 (2020). [CrossRef]
9. A. Gupta and M. Brandt-Pearce, “Receiver design for shot noise limited MIMO FSO/UV communication systems,” in Globecom Workshops (IEEE, 2012), pp. 1183–1187.
10. G. Wen, J. Huang, L. Zhang, C. Li, T. Wen, and J. Wang, “A high-speed and high-sensitivity photon-counting communication system based on multichannel SPAD detection,” IEEE Photonics J. 13(2), 1–10 (2021). [CrossRef]
11. C. Abou-Rjeily, “Spatial multiplexing for photon-counting MIMO-FSO communication systems,” IEEE Trans. Wireless Commun. 17(9), 5789–5803 (2018). [CrossRef]
12. Z. Xu, W. Liu, Z. Wang, and L. Hanzo, “Petahertz communication: Harmonizing optical spectra for wireless communications,” Digital Commun. Netw. 7(4), 605–614 (2021). [CrossRef]
13. Y. Li, M. Safari, R. Henderson, and H. Haas, “Optical OFDM with single-photon avalanche diode,” IEEE Photonics Technol. Lett. 27(9), 943–946 (2015). [CrossRef]
14. D. Guo, S. Shamai, and S. Verdu, “Mutual information and conditional mean estimation in Poisson channels,” IEEE Trans. Inf. Theory 54(5), 1837–1849 (2008). [CrossRef]
15. M. L. B. Riediger, R. Schober, and L. Lampe, “Multiple-symbol detection for photon-counting MIMO free-space optical communications,” IEEE Trans. Wireless Commun. 7(12), 5369–5379 (2008). [CrossRef]
16. C. Gong and Z. Xu, “LMMSE SIMO receiver for short-range non-line-of-sight scattering communication,” IEEE Trans. Wireless Commun. 14(10), 5338–5349 (2015). [CrossRef]
17. A. Wiesel, Y. C. Eldar, and S. Shamai, “Zero-forcing precoding and generalized inverses,” IEEE Trans. Signal Process. 56(9), 4409–4418 (2008). [CrossRef]
18. H. Ma, L. Lampe, and S. Hranilovic, “Coordinated broadcasting for multiuser indoor visible light communication systems,” IEEE Trans. Commun. 63(9), 3313–3324 (2015). [CrossRef]
19. B. Li, J. Wang, R. Zhang, H. Shen, C. Zhao, and L. Hanzo, “Multiuser MISO transceiver design for indoor downlink visible light communication under per-LED optical power constraints,” IEEE Photonics J. 7(4), 1–15 (2015). [CrossRef]
20. L. Wang, X. Wang, J. Kang, and C. P. Yue, “A 75-Mb/s RGB PAM-4 visible light communication transceiver system with pre- and post-equalization,” J. Lightwave Technol. 39(5), 1381–1390 (2021). [CrossRef]
21. Y. Zhai, H. Chi, J. Tong, and J. Xi, “Capacity maximized linear precoder design for spatial-multiplexing MIMO VLC systems,” IEEE Access 8, 63901–63909 (2020). [CrossRef]
22. J. Liu, Z. Pan, and S. Shimamoto, “Performance analysis of image sensor communication over free space optical channel,” in International Conference on Imaging Systems and Techniques Proceedings (IEEE, 2012), pp. 627–632.
23. Y.-H. Kim, W. A. Cahyadi, and Y. H. Chung, “Experimental demonstration of LED-based vehicle to vehicle communication under atmospheric turbulence,” in International Conference on Information and Communication Technology Convergence (2015), pp. 1143–1145.
24. E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in Gamma-Gamma fading,” IEEE Trans. Commun. 57(11), 3415–3424 (2009). [CrossRef]
25. N. Gupta Sonali, “BER analysis of BCH codes in slow fading channel with L-PPM and DPPM scheme,” in International Conference on Advanced Networks and Telecommunications Systems (2018), pp. 1–6.
26. A. A. Farid and S. Hranilovic, “Diversity gain and outage probability for MIMO free-space optical links with misalignment,” IEEE Trans. Commun. 60(2), 479–487 (2012). [CrossRef]
27. C. Gong and Z. Xu, “Channel estimation and signal detection for optical wireless scattering communication with inter-symbol interference,” IEEE Trans. Wireless Commun. 14(10), 5326–5337 (2015). [CrossRef]
28. S. Fray, F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, “Photon-counting distributions of modulated laser beams,” Phys. Rev. 153(2), 357–359 (1967). [CrossRef]
29. L. Grippo and M. Sciandrone, “On the convergence of the block nonlinear Gauss-Seidel method under convex constraints,” Operations Res. Lett. 26(3), 127–136 (2000). [CrossRef]
30. C. Sun, W. Ni, Z. Bu, and X. Wang, “Energy minimization for intelligent reflecting surface-assisted mobile edge computing,” in IEEE/CIC International Conference on Communications in China (ICCC) (2021), pp. 254–259.
31. C. Gong and Z. Xu, “Temporal spectrum sensing for optical wireless scattering communications,” J. Lightwave Technol. 33(18), 3890–3900 (2015). [CrossRef]
32. L. Liao, Z. Li, T. Lang, B. M. Sadler, and G. Chen, “Turbulence channel test and analysis for NLOS UV communication,” in Laser Communication and Propagation through the Atmosphere and Oceans III, vol. 9224A. M. J. Eijk van, C. C. Davis, and S. M. Hammel, eds., International Society for Optics and Photonics (SPIE, 2014), p. 92241A.
33. Z. Jiang, C. Gong, and Z. Xu, “Achievable rates and signal detection for photon-level photomultiplier receiver based on statistical non-linear model,” IEEE Trans. Wireless Commun. 18(12), 6015–6029 (2019). [CrossRef]
34. H. Marshoud, P. C. Sofotasios, S. Muhaidat, B. S. Sharif, and G. K. Karagiannidis, “Optical adaptive precoding for visible light communications,” IEEE Access 6, 22121–22130 (2018). [CrossRef]
35. P. Aquilina and T. Ratnarajah, “Performance analysis of IA techniques in the MIMO IBC with imperfect CSIs,” IEEE Trans. Commun. 63(4), 1259–1270 (2015). [CrossRef]
36. X. Zhou, D. Zhang, R. Zhang, and L. Hanzo, “A photon-counting spatial-diversity-and-multiplexing MIMO scheme for Poisson atmospheric channels relying on Q-ary PPM,” Opt. Express 20(24), 26379–26393 (2012). [CrossRef]
