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We consider a two–tile reconfigurable intelligent surface (RIS) assisted wireless network with a two-antenna transmitter and receiver over Rayleigh fading. We show that the average received signal-to-noise-ratio (SNR) optimal combining and transmission vectors are given by the left and right singular spaces of the RIS-receiver and transmit-RIS channel matrices, respectively. Moreover, the optimal phases at the two tiles of the RIS are determined by the phases of the elements of the latter spaces. To further study the effect of phase compensation, we statistically characterize the average SNR of all possible combinations of transmission and combining directions pertaining to the latter singular spaces by deriving novel expressions for the outage probability and throughput of each of those modes. Furthermore, for comparison, we derive the corresponding expressions in the absence of RIS. Our results show an approximate SNR improvement of 2 dB due to the phase compensation at the RIS.
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... Analogous to , the outage probability of RIS-aided multiple-input single-output (MISO) systems was approximated with a Gamma distribution in . Furthermore, the outage performance of RIS assisted 2×2 MIMO system was scrutinized by considering perfect CSI at the RIS in . Nonetheless, the above-mentioned literature assumed perfect CSI at the RIS. ...
... By combining (5), (6) and the Mellin transform φ(x) in (20) or (21), the outage probability can thus be expressed as ...
We thoroughly investigate the outage performance of reconfigurable intelligent surface (RIS) aided multi-input multi-output (MIMO) communications by exploiting only statistical channel state information (CSI). Kornecker channel model is adopted to characterize the impact of spatial correlations among MIMO antennas and reconfigurable reflectors. Mellin transform and random matrix theory are then utilized to derive the exact outage probability in closed-form. With the result, we further conduct the asymptotic outage analysis to obtain insightful findings. In particular, the asymptotic analysis reveals that the number of reflecting elements at the RIS should not be smaller than the total number of MIMO transmit and receive antennas to get rid of the rank deficiency of the cascaded MIMO channels. Moreover, we prove that the transmission rate is a monotonically increasing and convex function of the asymptotic outage probability. The numerical outcomes not only corroborate our analytical results, but also show the negative impact of the spatial correlation and the benefit of increasing the number of Zheng Shi and Guanghua Yang are with the). 2 reconfigurable reflectors. Finally, we apply the asymptotic results to optimally devise the phase shifts with a low computational complexity. Index Terms Diversity order, Kronecker model, MIMO, outage probability, reconfigurable intelligent surface (RIS), spatial correlation.
... Doing so, the corresponding reflected signals can be added either constructively or destructively with other signals to enhance the signal-to-noise ratio (SNR) and/or to suppress the co-channel interference at the receiver. Most of the current art is focused on the incident signals' phase shift optimization and/or knowledge acquisition of the channel state information (CSI) at RIS . Nevertheless, such a condition reflects on a considerably high computational complexity and power consumption. ...
A multiuser multiple-input multiple-output wireless communication system is analytically studied, which operates with the aid of a reconfigurable intelligent surface (RIS). The intermediate RIS is equipped with multiple elements and operates via random phase rotations to simultaneously serve multiple users. Independent Rayleigh fading conditions are assumed among the included channels. The system performance is analytically studied when the linear yet efficient zero-forcing detection is implemented at the receiver. In particular, the outage performance is derived in closed-form expression for different system configuration setups with regards to the available channel state information at the receiver. Further, a joint coherent/noncoherent linear detection is analytically presented. Finally, some new engineering insights are provided, such as how the channel state information and/or the volume of antenna/RIS arrays impact on the overall system performance as well as the arising efficiency on the performance/complexity tradeoff by utilizing the joint coherent/noncoherent scheme.
... Most previous work on performance analysis considers different wireless networks with single and fixed RIS setup, e.g., -  and references therein. For a given set of locations of multiple feedback location information from a part of superior RISs and then selects a RIS among these RISs feeding back. ...
The reconfigurable intelligent surface (RIS) is a promising technology that is anticipated to enable high spectrum and energy efficiencies in future wireless communication networks. This paper investigates optimum location-based RIS selection policies in RIS-aided wireless networks to maximize the signal-to noise ratio (SNR) for a power path-loss model in outdoor communications and an exponential path-loss model in indoor communications. The random locations of all available RISs are modeled as a Poisson point process (PPP). To quantify the network performance, the outage probabilities and average rates attained by the proposed RIS selection policies are evaluated by deriving the distance distribution of the chosen RIS node as per the selection policies for both power and exponential path-loss models. Feedback could incur heavy signaling overhead. To reduce the overhead, we also propose limited-feedback RIS selection policies by limiting the average number of RISs that feed back their location information to the source. The outage probabilities and average rates obtained by the limited-feedback RIS selection policies are derived for both path-loss models. The numerical results show notable performance gains obtained by the proposed RIS selection policies and demonstrate that the conventional relay selection policies are not suitable for RIS-aided wireless networks.
In this paper, we investigate the performance of an RIS-aided wireless communication system subject to outdated channel state information that may operate in both the near- and far-field regions. In particular, we take two RIS deployment strategies into consideration: (i) the centralized deployment, where all the reflecting elements are installed on a single RIS and (ii) the distributed deployment, where the same number of reflecting elements are placed on multiple RISs. For both deployment strategies, we derive accurate closed-form approximations for the ergodic capacity, and we introduce tight upper and lower bounds for the ergodic capacity to obtain useful design insights. From this analysis, we unveil that an increase of the transmit power, the Rician-K factor, the accuracy of the channel state information and the number of reflecting elements help improve the system performance. Moreover, we prove that the centralized RIS-aided deployment may achieve a higher ergodic capacity as compared with the distributed RIS-aided deployment when the RIS is located near the base station or near the user. In different setups, on the other hand, we prove that the distributed deployment outperforms the centralized deployment. Finally, the analytical results are verified by using Monte Carlo simulations.
In this paper, we develop a unified framework for IRS-aided transceiver designs under general power constraints in multiple-input multiple-output (MIMO) systems which implement interference (pre-)subtraction via Tomlinson-Harashima precoding (THP) or Decision Feedback Equalization (DFE) technologies. Armed with majorization theory, two fundamental classes of performance criteria, namely
-increasing Schur-concave and Schur-convex functions of the logarithm of Mean Square Error (MSE) of the data stream, are investigated in depth. Firstly, we propose a simplified counterpart of the optimal transceiver design under general power constraints, with equivalence guaranteed by Pareto optimization theory and Lagrange duality. Moreover, the optimal semi-closed form solution to this simplified transceiver design can be attained using the modified subgradient method. Next, we prove that for any Schur-concave objective, the optimal nonlinear THP (DFE) design is in essence the linear precoding (equalization). For any Schur-convex objective, the optimal transceiver design results in individual data streams with equal MSEs, and thereby reduces to the Gaussian mutual information maximization based design. Based on the above conclusions, we further propose an efficient alternating optimization algorithm to decouple the optimization of the transmit precoder and the IRS reflection coefficients, where the classical successive convex approximation (SCA) technique is applied to fight against non-convex subproblems. From the low computational complexity perspective, a two-stage scheme is also developed inspired by the capability of the IRS in constructing favorable wireless links. Finally, numerical results show the global optimality of the modified subgradient method and the excellent performance of the proposed alternating optimization algorithm and two-stage scheme.
This letter investigates the exploitation of an intelligent reflecting surface (IRS) to communicate securely in a two-way network consisting of an untrusted user. In particular, the transmit powers and the phase shift at each element of the IRS are optimized to maximize the sum-secrecy rate, such that the IRS-reflected and non-IRS-reflected signals are added destructively at the untrusted user. The proposed iterative algorithm converges rapidly to a feasible solution of high accuracy with a few iterations. Numerical results demonstrate sum-secrecy rate gains up to 120% compared to naive or partially optimized schemes.
Signal detection in colored noise with an unknown covariance matrix has a myriad of applications in diverse scientific/engineering fields. The test statistic is the largest generalized eigenvalue (l.g.e.) of the whitened sample covariance matrix, which is constructed via m-dimensional p signal-plusnoise samples and m-dimensional n noise-only samples. A finite dimensional characterization of this statistic under the alternative hypothesis has hitherto been an open problem. We answer this problem by deriving cumulative distribution function (c.d.f.) of this l.g.e. via the powerful orthogonal polynomial approach, exploiting the deformed Jacobi unitary ensemble (JUE). Two special cases and an asymptotic version of the c.d.f. are also derived. With this new c.d.f., we comprehensively analyze the receiver operating characteristics (ROC) of the detector. Importantly, when the noise-only covariance matrix is nearly rank deficient (i.e., m = n), we show that (a) when m and p increase such that m=p is fixed, at each fixed signal-to-noise ratio (SNR), there exists an optimal ROC profile. We also establish a tight approximation of it; and (b) asymptotically, reliable signal detection is always possible if SNR scales with m.
Employing backscatter communication is a promising solution for Internet of Things (IoT). The novel large intelligent surface (LIS) concept can achieve reliable communication by establishing line-of-sight like channels. This paper thus considers an LIS-aided backscatter system to support high-reliable communications for IoT applications. In this letter, the symbol error probability (SEP) for both intelligent and random phase adjustments at the LIS reflectors is analytically investigated. In particular, we calculate the SEP based on the moment generating function approach and also provide tight SEP upper bounds for either fully correlated or mutually independent channels. Insightful observations of SEP outcomes reveal that having a large number of reflective elements on the LIS has a significantly positive impact on the SEP performance where high reliability can be achieved in moderate signal-to-noise.
The concept of a large intelligent surface (LIS) has recently emerged as a promising wireless communication paradigm that can exploit the entire surface of man-made structures for transmitting and receiving information. An LIS is expected to go beyond massive multiple-input multiple-output (MIMO) system, insofar as the desired channel can be modeled as a perfect line-of-sight. To understand the fundamental performance benefits, it is imperative to analyze its achievable data rate, under practical LIS environments and limitations. In this paper, an asymptotic analysis of the uplink data rate in an LIS-based large antenna-array system is presented. In particular, the asymptotic LIS rate is derived in a practical wireless environment where the estimated channel on LIS is subject to estimation errors, interference channels are spatially correlated Rician fading channels, and the LIS experiences hardware impairments. Moreover, the occurrence of the channel hardening effect is analyzed and the performance bound is asymptotically derived for the considered LIS system. The analytical asymptotic results are then shown to be in close agreement with the exact mutual information as the number of antennas and devices increase without bounds. Moreover, the derived ergodic rates show that hardware impairments, noise, and interference from estimation errors and the non-line-of-sight path become negligible as the number of antennas increases. Simulation results show that an LIS can achieve a performance that is comparable to conventional massive MIMO with improved reliability and a significantly reduced area for antenna deployment.
Large intelligent surface (LIS)-assisted wireless communications have drawn attention worldwide. With the use of low-cost LIS on building walls, signals can be reflected by the LIS and sent out along desired directions by controlling its phases, thereby providing supplementary links for wireless communication systems. In this study, we evaluate the performance of an LIS-assisted large-scale antenna system by formulating a tight approximation of the ergodic capacity and investigate the effect of the phase shifts on the ergodic capacity in different propagation scenarios. In particular, we propose an optimal phase shift design based on the ergodic capacity approximation and statistical channel state information. Furthermore, we derive the requirement on the quantization bits of the LIS to promise an acceptable capacity degradation. Numerical results show that using the proposed phase shift design can achieve the maximum ergodic capacity, and a 2-bit quantizer is sufficient to ensure capacity degradation of no more than 1 bit/s/Hz.
Future wireless networks are expected to constitute a distributed intelligent wireless communications, sensing, and computing platform, which will have the challenging requirement of interconnecting the physical and digital worlds in a seamless and sustainable manner. Currently, two main factors prevent wireless network operators from building such networks: (1) the lack of control of the wireless environment, whose impact on the radio waves cannot be customized, and (2) the current operation of wireless radios, which consume a lot of power because new signals are generated whenever data has to be transmitted. In this paper, we challenge the usual “more data needs more power and emission of radio waves” status quo, and motivate that future wireless networks necessitate a smart radio environment: a transformative wireless concept, where the environmental objects are coated with artificial thin films of electromagnetic and reconfigurable material (that are referred to as reconfigurable intelligent meta-surfaces), which are capable of sensing the environment and of applying customized transformations to the radio waves. Smart radio environments have the potential to provide future wireless networks with uninterrupted wireless connectivity, and with the capability of transmitting data without generating new signals but recycling existing radio waves. We will discuss, in particular, two major types of reconfigurable intelligent meta-surfaces applied to wireless networks. The first type of meta-surfaces will be embedded into, e.g., walls, and will be directly controlled by the wireless network operators via a software controller in order to shape the radio waves for, e.g., improving the network coverage. The second type of meta-surfaces will be embedded into objects, e.g., smart t-shirts with sensors for health monitoring, and will backscatter the radio waves generated by cellular base stations in order to report their sensed data to mobile phones. These functionalities will enable wireless network operators to offer new services without the emission of additional radio waves, but by recycling those already existing for other purposes. This paper overviews the current research efforts on smart radio environments, the enabling technologies to realize them in practice, the need of new communication-theoretic models for their analysis and design, and the long-term and open research issues to be solved towards their massive deployment. In a nutshell, this paper is focused on discussing how the availability of reconfigurable intelligent meta-surfaces will allow wireless network operators to redesign common and well-known network communication paradigms.
We adopt the concept of the composite parameterization of the unitary group
U(d) to the special unitary group SU(d). Furthermore, we also consider the Haar
measure in terms of the introduced parameters. We show that the well-defined
structure of the parameterization leads to a concise formula for the normalized
Haar measure on U(d) and SU(d). With regard to possible applications of our
results, we consider the computation of high-order integrals over unitary
In this paper, we introduce a physics-consistent analytical characterization of the free-space path-loss of a wireless link in the presence of a reconfigurable intelligent surface. The proposed approach is based on the vector generalization of Green’s theorem. The obtained path-loss model can be applied to two-dimensional homogenized metasurfaces, which are made of sub-wavelength scattering elements and that operate either in reflection or transmission mode. The path-loss is formulated in terms of a computable integral that depends on the transmission distances, the polarization of the radio waves, the size of the surface, and the desired surface transformation. Closed-form expressions are obtained in two asymptotic regimes that are representative of far-field and near-field deployments. Based on the proposed approach, the impact of several design parameters and operating regimes is unveiled.
Intelligent reflecting surfaces (IRSs) have the potential to transform wireless communication channels into smart reconfigurable propagation environments. To realize this new paradigm, the passive IRSs have to be large, especially for communication in far-field scenarios, so that they can compensate for the large end-to-end path-loss, which is caused by the multiplication of the individual path-losses of the transmitter-to-IRS and IRS-to-receiver channels. However, optimizing a large number of sub-wavelength IRS elements imposes a significant challenge for online transmission. To address this issue, in this article, we develop a physics-based model and a scalable optimization framework for large IRSs. The basic idea is to partition the IRS unit cells into several subsets, referred to as tiles, model the impact of each tile on the wireless channel, and then optimize each tile in two stages, namely an offline design stage and an online optimization stage. For physics-based modeling, we borrow concepts from the radar literature, model each tile as an
reflector, and derive its impact on the wireless channel for a given phase shift by solving the corresponding integral equations for the electric and magnetic vector fields. In the offline design stage, the IRS unit cells of each tile are jointly designed for the support of different transmission modes, where each transmission mode effectively corresponds to a given configuration of the phase shifts that the unit cells of the tile apply to an impinging electromagnetic wave. In the online optimization stage, the best transmission mode of each tile is selected such that a desired quality-of-service (QoS) criterion is maximized. We consider an exemplary downlink system and study the minimization of the base station (BS) transmit power subject to QoS constraints for the users. Since the resulting mixed-integer programming problem for joint optimization of the BS beamforming vectors and the tile transmission modes is non-convex, we derive two efficient suboptimal solutions, which are based on alternating optimization and a greedy approach, respectively. We show that the proposed modeling and optimization framework can be used to efficiently optimize large IRSs comprising thousands of unit cells.
Reconfigurable intelligent surfaces have emerged as a promising technology for future wireless networks. Given that a large number of reflecting elements is typically used and that the surface has no signal processing capabilities, a major challenge is to cope with the overhead that is required to estimate the channel state information and to report the optimized phase shifts to the surface. This issue has not been addressed by previous works, which do not explicitly consider the overhead during the resource allocation phase. This work aims at filling this gap, by developing an overhead-aware resource allocation framework for wireless networks where reconfigurable intelligent surfaces are used to improve the communication performance. An overhead model is proposed and incorporated in the expressions of the system rate and energy efficiency, which are then optimized with respect to the phase shifts of the reconfigurable intelligent surface, the transmit and receive filters, the power and bandwidth used for the communication and feedback phases. The bi-objective maximization of the rate and energy efficiency is investigated, too. The proposed framework characterizes the trade-off between optimized radio resource allocation policies and the related overhead in networks with reconfigurable intelligent surfaces.
We consider a fading channel in which a multi-antenna transmitter communicates with a multi-antenna receiver through a reconfigurable intelligent surface (RIS) that is made of N reconfigurable passive scatterers impaired by phase noise. The beamforming vector at the transmitter, the combining vector at the receiver, and the phase shifts of the N scatterers are optimized in order to maximize the signal-to-noise-ratio (SNR) at the receiver. By assuming Rayleigh fading (or line-of-sight propagation) on the transmitter-RIS link and Rayleigh fading on the RIS-receiver link, we prove that the SNR is a random variable that is equivalent in distribution to the product of three (or two) independent random variables whose distributions are approximated by two (or one) gamma random variables and the sum of two scaled non-central chi-square random variables. The proposed analytical framework allows us to quantify the robustness of RIS-aided transmission to fading channels. For example, we prove that the amount of fading experienced on the transmitter-RIS-receiver channel linearly decreases with N1. This proves that RISs of large size can be effectively employed to make fading less severe and wireless channels more reliable.
Reconfigurable intelligent surfaces (RISs) are an emerging transmission technology for application to wireless communications. RISs can be realized in different ways, which include (i) large arrays of inexpensive antennas that are usually spaced half of the wavelength apart; and (ii) metamaterial-based planar or conformal large surfaces whose scattering elements have sizes and inter-distances much smaller than the wavelength. Compared with other transmission technologies, e.g., phased arrays, multi-antenna transmitters, and relays, RISs require the largest number of scattering elements, but each of them needs to be backed by the fewest and least costly components. Also, no power amplifiers are usually needed. For these reasons, RISs constitute a promising software-defined architecture that can be realized at reduced cost, size, weight, and power (C-SWaP design), and are regarded as an enabling technology for realizing the emerging concept of smart radio environments (SREs). In this paper, we (i) introduce the emerging research field of RIS-empowered SREs; (ii) overview the most suitable applications of RISs in wireless networks; (iii) present an electromagnetic-based communication-theoretic framework for analyzing and optimizing metamaterial-based RISs; (iv) provide a comprehensive overview of the current state of research; and (v) discuss the most important research issues to tackle. Owing to the interdisciplinary essence of RIS-empowered SREs, finally, we put forth the need of reconciling and reuniting C. E. Shannon’s mathematical theory of communication with G. Green’s and J. C. Maxwell’s mathematical theories of electromagnetism for appropriately modeling, analyzing, optimizing, and deploying future wireless networks empowered by RISs.
In this paper, we investigate the two-way communication between two users assisted by a reconfigurable intelligent surface (RIS). The scheme that two users communicate simultaneously over Rayleigh fading channels is considered. The channels between the two users and RIS can either be reciprocal or non-reciprocal. For reciprocal channels, we determine the optimal phases at the RIS to maximize the signal-to-interference-plus-noise ratio (SINR). We then derive exact closed-form expressions for the outage probability and spectral efficiency for single-element RIS. By capitalizing the insights obtained from the single-element analysis, we introduce a gamma approximation to model the product of Rayleigh random variables which is useful for the evaluation of the performance metrics in multiple-element RIS. Asymptotic analysis shows that the outage decreases at (log(ρ)/ρ)
<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</sup>
rate where L is the number of elements, whereas the spectral efficiency increases at log(ρ) rate at large average SINR p. For non-reciprocal channels, the minimum user SINR is targeted to be maximized. For single-element RIS, closed-form solution is derived whereas for multiple-element RIS the problem turns out to be non-convex. The latter one is solved through semidefinite programming relaxation and a proposed greedy-iterative method, which can achieve higher performance and lower computational complexity, respectively.
Reconfigurable intelligent surface (RIS) has drawn considerable attention from the research society recently, which creates favorable propagation conditions by controlling the phase shifts of the reflected waves at the surface, thereby enhancing wireless transmissions. In this paper, we study a downlink multi-user system where the transmission from a multi-antenna base station (BS) to various users is achieved by the RIS reflecting the incident signals of the BS towards the users. Unlike most existing works, we consider the practical case where only a limited number of discrete phase shifts can be realized by the finite-sized RIS. A hybrid beamforming scheme is proposed and the sum-rate maximization problem is formulated. Specifically, the continuous digital beamforming and discrete RIS-based analog beamforming are performed at the BS and the RIS, respectively, and an iterative algorithm is designed to solve this problem. Both theoretical analysis and numerical validations show that the RIS-based system can achieve a good sum-rate performance by setting a reasonable size of RIS and a small number of discrete phase shifts.
This work focuses on the downlink of a single-cell multi-user system in which a base station (BS) equipped with
antennas communicates with
single-antenna users through a reconfigurable intelligent surface (RIS) installed in the line-of-sight (LoS) of the BS. RIS is envisioned to offer unprecedented spectral efficiency gains by utilizing
passive reflecting elements that induce phase shifts on the impinging electromagnetic waves to smartly reconfigure the signal propagation environment. We study the minimum signal-to-interference-plus-noise ratio (SINR) achieved by the optimal linear precoder (OLP), that maximizes the minimum SINR subject to a given power constraint for any given RIS phase matrix, for the cases where the LoS channel matrix between the BS and the RIS is of rank-one and of full-rank. In the former scenario, the minimum SINR achieved by the RIS-assisted link is bounded by a quantity that goes to zero with
. For the high-rank scenario, we develop accurate deterministic approximations for the parameters of the asymptotically OLP, which are then utilized to optimize the RIS phase matrix. Simulation results show that RISs can outperform half-duplex relays with a small number of passive reflecting elements while large RISs are needed to outperform full-duplex relays.
Reconfigurable intelligent surface~(RIS) has drawn a great attention worldwide as it can create favorable propagation conditions by controlling the phase shifts of the reflected signals at the surface to enhance the communication quality. However, the practical RIS only has limited phase shifts, which will lead to the performance degradation. In this letter, we evaluate the performance of an uplink RIS assisted communication system by giving an approximation of the achievable data rate, and investigate the effect of limited phase shifts on the data rate. In particular, we derive the required number of phase shifts under a data rate degradation constraint. Numerical results verify our analysis.
Intelligent reflecting surface (IRS) is a revolutionary and transformative technology for achieving spectrum and energy efficient wireless communication cost-effectively in the future. Specifically, an IRS consists of a large number of low-cost passive elements each being able to reflect the incident signal independently with an adjustable phase shift so as to collaboratively achieve three-dimensional (3D) passive beamforming without the need of any transmit radio-frequency (RF) chains. In this paper, we study an IRS-aided single-cell wireless system where one IRS is deployed to assist in the communications between a multi-antenna access point (AP) and multiple single-antenna users. We formulate and solve new problems to minimize the total transmit power at the AP by jointly optimizing the transmit beamforming by active antenna array at the AP and reflect beamforming by passive phase shifters at the IRS, subject to users’ individual signal-to-interference-plus-noise ratio (SINR) constraints. Moreover, we analyze the asymptotic performance of IRS’s passive beamforming with infinitely large number of reflecting elements and compare it to that of the traditional active beamforming/relaying. Simulation results demonstrate that an IRS-aided MIMO system can achieve the same rate performance as a benchmark massive MIMO system without using IRS, but with significantly reduced active antennas/RF chains. We also draw useful insights into optimally deploying IRS in future wireless systems.
This short note is about the singular value distribution of Gaussian random matrices (i.e. Gaussian Ensemble or GE) of size N. We present a new approach for deriving the p.d.f. of the singular values directly from the singular value decomposition (SVD) form, which also takes advantage of the rotational invariance of GE and the Lie algebra of the orthogonal group. Our method is direct and more general than the conventional approach that relies on the Wishart Ensemble and the combination of QR and Cholesky decomposition. Directly based on this p.d.f., and its interpretation by statistical mechanics, we give the physics proof that in the thermodynamic limit (N→∞), the singular value distribution satisfies the quadrant law, similar to the celebrated semi-circle law established by Wigner more than 40 years ago for the spectral distribution of Gaussian Orthogonal (or Unitary) Ensembles. This quadrant law was also proved earlier and mathematically more rigorously by some authors based on probabilistic estimations and the moment method, but not directly from the p.d.f. formula.
The paper is largely expository, but some new results are included to round out the paper and bring it up to date. The following distributions are quoted in Section 7. 1. Type $_0F_0$, exponential: (i) $\chi^2$, (ii) Wishart, (iii) latent roots of the covariance matrix. 2. Type $_1F_0$, binomial series: (i) variance ratio, $F$, (ii) latent roots with unequal population covariance matrices. 3. Type $_0F_1$, Bessel: (i) noncentral $\chi^2$, (ii) noncentral Wishart, (iii) noncentral means with known covariance. 4. Type $_1F_1$, confluent hypergeometric: (i) noncentral $F$, (ii) noncentral multivariate $F$, (iii) noncentral latent roots. 5. Type $_2F_1$, Gaussian hypergeometric: (i) multiple correlation coefficient, (ii) canonical correlation coefficients. The modifications required for the corresponding distributions derived from the complex normal distribution are outlined in Section 8, and the distributions are listed. The hypergeometric functions $_pF_q$ of matrix argument which occur in the multivariate distributions are defined in Section 4 by their expansions in zonal polynomials as defined in Section 5. Important properties of zonal polynomials and hypergeometric functions are quoted in Section 6. Formulae and methods for the calculation of zonal polynomials are given in Section 9 and the zonal polynomials up to degree 6 are given in the appendix. The distribution of quadratic forms is discussed in Section 10, orthogonal expansions of $_0F_0$ and $_1F_1$ in Laguerre polynomials in Section 11 and the asymptotic expansion of $_0F_0$ in Section 12. Section 13 has some formulae for moments.
Physics-based modeling and scalable optimization of large intelligent reflecting surfaces
V H Poor
On the path-loss of reconfigurable intelligent surfaces: An approach based on Green's theorem applied to vector fields
F H Danufane
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Nonlinear Programming, ser. Athena scientific optimization and computation series
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