CN115379462A - Three-dimensional deployment method for 6G intelligent reflector auxiliary network - Google Patents

Three-dimensional deployment method for 6G intelligent reflector auxiliary network Download PDF

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CN115379462A
CN115379462A CN202210766373.4A CN202210766373A CN115379462A CN 115379462 A CN115379462 A CN 115379462A CN 202210766373 A CN202210766373 A CN 202210766373A CN 115379462 A CN115379462 A CN 115379462A
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irs
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张鸿涛
刘江徽
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Beijing University of Posts and Telecommunications
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    • HELECTRICITY
    • H04ELECTRIC COMMUNICATION TECHNIQUE
    • H04WWIRELESS COMMUNICATION NETWORKS
    • H04W16/00Network planning, e.g. coverage or traffic planning tools; Network deployment, e.g. resource partitioning or cells structures
    • H04W16/18Network planning tools
    • HELECTRICITY
    • H04ELECTRIC COMMUNICATION TECHNIQUE
    • H04BTRANSMISSION
    • H04B7/00Radio transmission systems, i.e. using radiation field
    • H04B7/14Relay systems
    • H04B7/145Passive relay systems
    • HELECTRICITY
    • H04ELECTRIC COMMUNICATION TECHNIQUE
    • H04WWIRELESS COMMUNICATION NETWORKS
    • H04W16/00Network planning, e.g. coverage or traffic planning tools; Network deployment, e.g. resource partitioning or cells structures
    • H04W16/24Cell structures
    • H04W16/26Cell enhancers or enhancement, e.g. for tunnels, building shadow
    • HELECTRICITY
    • H04ELECTRIC COMMUNICATION TECHNIQUE
    • H04WWIRELESS COMMUNICATION NETWORKS
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Abstract

Aiming at the problems of inaccurate position optimization and non-uniform parameter models in the existing intelligent reflecting surface deployment research work, the invention establishes a universal three-dimensional deployment model of intelligent reflecting surface auxiliary base station coverage extension, unifies various existing two-dimensional/three-dimensional intelligent reflecting surface models, defines intelligent reflecting surface deployment related parameters such as horizontal turning angle, vertical turning angle, distance from a base station, vertical height and the like, and lays a foundation for network performance accurate analysis; deducing a closed expression of the coverage area of the base station under an intelligent reflector three-dimensional deployment model, and considering a Rice channel gain, a line-of-sight and a non-line-of-sight statistical channel model; an intelligent reflecting surface three-dimensional deployment algorithm for maximizing the coverage area of the base station is designed; forming a convex optimization problem by the coverage maximization problem, and solving the problem by introducing auxiliary variables and utilizing a Lagrange multiplier method; and finally, under the condition of designing the optimal intelligent reflecting surface phase, simulating and analyzing the influence of various parameters on the coverage area of the base station, and giving a deployment guidance suggestion of different intelligent reflecting surface parameters.

Description

Three-dimensional deployment method for 6G intelligent reflector auxiliary network
Technical Field
The invention relates to the technical field of wireless communication, in particular to a method for deploying an Intelligent Reflection Surface (IRS) auxiliary network in the sixth generation mobile communication (6 generation, 6G).
Background
The sixth generation wireless communication systems target large-scale, instantaneous connectivity, data-driven intelligent networks, enabling ubiquitous wireless connectivity. Highly complex networks, high cost hardware and increasing energy consumption are key issues facing future wireless communications. Therefore, new transmission technologies are needed to support new applications and services. Among many alternatives, IRS has the ability to actively change the wireless communication environment, which has been the focus of wireless communication research to alleviate various challenges encountered in 6G wireless networks.
IRS passes through reconfigurable space electromagnetic wave modulator, regulates and control intelligently to the wireless propagation environment between the transmitter, compares in large-scale antenna transceiver and relay node, and the main advantage of intelligent plane of reflection has: the shape is plastic, the weight is light, and the device is easy to be arranged on the surfaces of various scatterers in a wireless propagation environment; reconstructing a Line Of Sight (LoS) link through passive reflection to serve remote users; the passive control of the electromagnetic wave is realized by utilizing the regulation and control of the physical properties of the electromagnetic material, and a high-power-consumption device of a radio frequency link is not needed.
In a traditional wireless network, an electromagnetic environment is not controlled by the network, the network performance is limited by the environment, and the intelligent super surface can convert the environment into an intelligent reconfigurable electromagnetic space by benefiting from the advantages, so that paradigm change is brought to information transmission processing. For example, in a conventional urban cellular network, due to the obstruction of wireless signals by large buildings and other obstacles, communication links are blocked, base station signals are not easy to reach, and users cannot obtain good service. The intelligent reflecting surface can be deployed between the base station and the coverage blind area, and the transmission signal reaches the user in the coverage hole through effective reflection/projection, so that effective connection is established between the base station and the user, and the coverage of the user in the hole area is ensured. In addition, the intelligent reflecting surface can also be applied to the fields of cell edge interference suppression, line-of-sight multi-stream transmission, large-scale antenna transceivers, user scenes and the like, and can also be combined with the large-scale antenna transceivers to further enhance the network performance. The wide application scenario makes intelligent super-surface a promising research point for future wireless systems.
The intelligent super-surface deployment problem has the characteristic of diversity, and the research targets for different problems are not very same. In existing intelligent super-surface work, it is typically deployed on the side of the network near the base station or user to help improve the performance of communications with the serving base station. The user-side intelligent super-surface deployment strategy is generally used for hot spots, cell edges and mobile user scenes so as to improve local coverage.
However, existing intelligent super-surface works, mostly assuming that the intelligent super-surface is deployed in a fixed location, without taking advantage of its deployment flexibility. It is well known that different intelligent super-surface deployment parameters, such as intelligent super-surface orientation, distance from a base station, intelligent super-surface height, etc., can result in different intelligent super-surface channels, thereby significantly affecting system capacity and spectral efficiency. In addition, the deployment of the intelligent super surface also considers the information density of space users, namely the intelligent super surface is preferentially deployed in a hot spot area with a large number of users or on the boundary of two cells to eliminate the same-frequency interference of channels between the two cells and simultaneously expand the cell coverage. From this perspective, intelligent super-surface deployment is inefficient in most of the existing work. Therefore, how to achieve optimal deployment of intelligent hypersurfaces in wireless networks remains an important open question.
Disclosure of Invention
Aiming at the problems of inaccurate position optimization and non-uniform parameter models in the existing intelligent reflecting surface deployment research work, the general intelligent reflecting surface three-dimensional deployment method for maximizing the base station coverage area is provided, the Rice channel gain and a LoS/None-LoS (NLoS) statistical channel model are considered, a closed expression of the coverage area on the intelligent reflecting surface parameters is deduced, and the optimal deployment algorithm of the intelligent reflecting surface is designed.
The scheme for intelligent reflecting surface deployment and three-dimensional parameter setting in the dense urban area comprises the following steps:
and 200, acquiring related parameters, and establishing a three-dimensional deployment model covered by the intelligent reflecting surface according to the distance between the base station and the user.
The center position of the intelligent super surface is determined according to the horizontal distance and the height between the super surface and the base station, and the orientation of the intelligent super surface is determined according to the horizontal and vertical rotation angles, so that the structure shown in the attached figure 2 is obtained.
Wherein the user and the base station are on an xOy plane, and the base station and the center of the intelligent reflecting surface are on an xOz plane;
Figure BDA0003722317650000031
represents the horizontal distance of the base station from the user;
Figure BDA0003722317650000032
representing the horizontal distance between the base station and the intelligent reflecting surface;
Figure BDA0003722317650000033
representing the horizontal distance between the intelligent reflecting surface and the user;
h BS indicating the altitude of the base station;
h IRS indicating the altitude of the base station;
φ 1 representing the horizontal rotation angle of the intelligent reflecting surface;
φ 2 representing a vertical corner of the intelligent reflecting surface;
beta represents the horizontal azimuth angle of the user compared to the base station
The position of the intelligent reflecting surface is phi 12 ,
Figure BDA0003722317650000034
h IRS And determining and optimizing the position of the intelligent reflecting surface, namely optimizing the four parameters. After the model is established, the next step is carried out.
Step 210, setting the phase of the reflection factor of the electromagnetic unit of the intelligent super-surface to maximize the signal-to-noise ratio of the signal received by the user.
From the model determined in step 200, the base station to user channel gain can be expressed as:
Figure BDA0003722317650000035
wherein gamma is m,n The reflection factor h of the m row and n column reflection unit of the intelligent super surface is represented m,n Representing the channel gain, h, of the signal reflected by the m-th row and n-th column of the reflecting unit of the intelligent super surface D Channel gain representation of the signal-to-user direct link. Further, the two links respectively have LoS and NLoS paths, and are modeled through a Rice model:
Figure BDA0003722317650000041
thus, the signal-to-noise ratio of the user's received signal can be expressed as:
Figure BDA0003722317650000042
and substituting a specific channel gain model, and performing a series of transformations when the phase of the reflection unit satisfies:
Figure BDA0003722317650000043
Figure BDA0003722317650000044
obtaining a maximum value:
Figure BDA0003722317650000045
wherein, P is the transmitting power of the base station;
k 1 a rice channel parameter for a direct link from a base station to a user;
k 2 the parameters of the Leise channel from the base station to the user are reflected by the intelligent reflecting surface;
Figure BDA0003722317650000046
channel gain for LoS link expressed as direct incidence;
Figure BDA0003722317650000047
Channel gain expressed as a direct NLoS link;
Figure BDA0003722317650000048
channel gain expressed as LoS link reflected by the intelligent reflecting surface;
Figure BDA0003722317650000049
expressed as the channel gain of the NLoS link reflected by the intelligent reflecting surface;
cos theta represents the incident angle of the base station to the intelligent reflecting surface;
m and N represent the size of the intelligent reflecting surface, namely M rows of reflecting units of M are provided;
σ 2 is the noise power.
And after the maximum value of the receiving signal-to-noise ratio of the user is obtained, the next step is carried out.
Step 220, calculating the maximum distance of each beta azimuth user of the base station according to the channel state.
Using the model established in step 200, a typical user is used as a reference, the angle of the user relative to the base station is β, and then the maximum linear distance that the base station can cover along the β angle is calculated.
Firstly, four kinds of channel gains are calculated according to the existing conditions
Figure BDA0003722317650000051
Secondly, calculating a cosine value cos theta of an incident angle from the base station to the intelligent reflecting surface; finally, according to the set signal-to-noise ratio threshold value gamma received by the user th And calculating the farthest distance of the typical user from the base station, wherein the distance is the straight-line distance covered by the base station, and then entering the next step.
Step 230, calculating an approximate solution of the coverage range by a discretized summation method, and then calculating an optimal solution of the deployment parameters by using a Lagrange multiplier method.
The coverage area of a base station can be calculated by integrating the angle β, which is expressed as:
Figure BDA0003722317650000052
wherein
Figure BDA0003722317650000053
The deployment parameter phi of the intelligent reflector obtained in step 220 12 ,
Figure BDA0003722317650000054
h IRS And (5) determining. It follows that the coverage area S is a function of phi 12 ,
Figure BDA0003722317650000055
h IRS Is a non-linear function of (a).
Furthermore, according to the model, when
Figure BDA0003722317650000056
h IRS Other variables are affected when changed, e.g. by changing phi 12 And further affects the overall coverage area. However, it is difficult to describe the effect of a variable on a closed-form solution, considering that the coverage area S is a function of φ 12 ,
Figure BDA0003722317650000057
h IRS The design simplifies the area S and decomposes the area S into a plurality of fan-shaped area summation forms with equal angles, wherein the area summation forms are as follows:
Figure BDA0003722317650000061
where K is a parameter of the approximate solution for the estimated area and the opening angle per sector is
Figure BDA0003722317650000062
After the simplification, the problem of solving the optimal deployment parameter becomes an optimization problem constrained by inequality, and the optimal deployment parameter can be solved by using a Lagrange multiplier method and a logarithm barrier function
Figure BDA0003722317650000063
Advantageous effects
The three-dimensional deployment method of the intelligent reflector auxiliary network establishes a universal three-dimensional deployment model of the coverage extension of the intelligent reflector auxiliary base station, unifies various existing two-dimensional/three-dimensional intelligent reflector models, and defines phi 12 ,
Figure BDA0003722317650000064
h IRS And relevant parameters are deployed on the intelligent reflecting surface, so that the spatial position and orientation of the intelligent reflecting surface are accurately described, and a foundation is laid for accurate analysis of network performance. On the basis, a closed expression of the coverage range of the base station under the intelligent reflector three-dimensional deployment model is further deduced, and a Rice channel gain and a LoS/NLoS statistical channel model are considered. Under the condition of designing the optimal intelligent reflector phase, the influence of various parameters on the coverage area of the base station is simulated and analyzed, and the deployment guidance suggestions of different intelligent reflector parameters are given.
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In order to clearly and clearly explain the technical steps of the present invention, all the drawings used in the description of the present invention will be briefly described below. It should be noted that the drawings described below are only exemplary words of the present invention, and those skilled in the art can still obtain other drawings in other different scenarios according to the drawings.
FIG. 1 is a flow chart of an embodiment of the present invention;
FIG. 2 is a diagram of a model of an intelligent reflector for assisting a base station to service a user according to the present invention;
FIG. 3 is an explanatory diagram of horizontal and vertical corners of the intelligent reflective surface of the present invention;
FIG. 4 is a geometric plot of angle of incidence versus angle of rotation for the present invention;
FIG. 5 is a plot of coverage area as a function of horizontal rotation angle for the present invention;
FIGS. 6 and 7 are plots of coverage area of the present invention as a function of vertical rotation;
FIG. 8 is a graph of coverage area of the present invention as a function of horizontal distance of the intelligent reflector from the base station;
FIGS. 9 and 10 are graphs of coverage area of the present invention as a function of intelligent reflector height;
Detailed Description
The steps and processes of the present invention are described in detail below with reference to the drawings in the present application, and it is obvious that the example described in the present application is only an example application scenario of the present invention, and other results based on the present disclosure without substantial changes are all within the protection scope of the present invention.
FIG. 2 is an exemplary scenario of the present invention illustrating a step-wise variation of the implementation of the present invention. In the model, the base station is arranged at the center of coordinates, an omnidirectional antenna is adopted, and the intelligent reflecting surface is only required to be arranged on the optimal parameters calculated by the design
Figure BDA0003722317650000071
At the determined position, the maximization of the coverage area enhancement of the base station can be realized.
The invention uses the network scene as a single base station and multi-user scene, does not consider the interference among users, and can provide service for the multi-users by adopting various multi-access modes such as frequency division multiple access, time division multiple access and the like. The mechanism implementation of the present invention is performed by the wireless coverage service provider at the time of deployment of the intelligent reflector.
The specific steps of 3-dimensional deployment of the 6G IRS-assisted network coverage extension are as follows:
and 300, acquiring related parameters, and establishing a three-dimensional deployment model covered by the intelligent reflecting surface according to the distance between the base station and the user.
The center position of the intelligent super surface is determined according to the horizontal distance and the height between the super surface and the base station, and the orientation of the intelligent super surface is determined according to the horizontal and vertical rotation angles, so that the structure shown in the attached figure 2 is obtained.
Wherein the user and the base station are on an xOy plane, and the base station and the center of the intelligent reflecting surface are on an xOz plane;
Figure BDA0003722317650000081
represents the horizontal distance of the base station from the user;
Figure BDA0003722317650000082
representing the horizontal distance between the base station and the intelligent reflecting surface;
Figure BDA0003722317650000083
representing the horizontal distance between the intelligent reflecting surface and the user;
h BS indicating the altitude of the base station;
h IRS indicating the altitude of the base station;
φ 1 representing the horizontal rotation angle of the intelligent reflecting surface;
φ 2 representing a vertical corner of the intelligent reflecting surface;
beta represents the horizontal azimuth of the user compared to the base station.
Step 310, setting the phase of the reflection factor of the electromagnetic unit of the intelligent super-surface to maximize the signal-to-noise ratio of the signal received by the user.
On the basis of formula (3)
Figure BDA0003722317650000084
Equation (3) can be further rewritten as:
Figure BDA0003722317650000085
the mean value of the SNR can then be decomposed into:
Figure BDA0003722317650000086
taking the model of the channel gain into account, further derivation can yield:
Figure BDA0003722317650000087
Figure BDA0003722317650000088
Figure BDA0003722317650000091
Figure BDA0003722317650000092
equation (9) can then be rewritten as:
Figure BDA0003722317650000093
further, it is obtained from the formula (14) in order to
Figure BDA0003722317650000094
To achieve the maximum, it must be satisfied that:
Figure BDA0003722317650000095
step 320, calculating the farthest distance of each beta azimuth user of the base station according to the channel state.
Taking a typical user as a reference, the angle of the user compared with the base station is beta, and according to the geometrical relationship among the user, the base station and the intelligent reflecting surface:
Figure BDA0003722317650000096
wherein, according to the cosine theorem:
Figure BDA0003722317650000097
in practice h BS And h UE Are all known and therefore when
Figure BDA0003722317650000098
When determined, d can be calculated by substituting equation (16) BI ,d BU ,d IU Then substituting the model into the channel gain model matched with the environment to calculate
Figure BDA0003722317650000101
In order to further clearly show the relationship between the incident angle θ and other parameters, the present invention designs a more specific angle correlation diagram based on fig. 2, as shown in fig. 3.
The rectangles EBFP, EPLK and PFQL are in the x 'Pz', y 'Pz', x 'Py' plane, respectively. PA represents the normal vector of the intelligent reflecting surface, line PD represents the angle of incidence of the base station signal, point D is on line BF, and line AC is perpendicular to the x 'Py' plane.
Thus, it is possible to obtain:
Figure BDA0003722317650000102
when is phi 12 ,
Figure BDA0003722317650000103
h IRS When all the line segments are determined, the proportional relationship between the line segments is determined, so that cos theta can be calculated.
The receive threshold for a given user is then γ th One user is covered and needs to be satisfied
Figure BDA0003722317650000104
According to this inequality can be obtained
Figure BDA0003722317650000105
Is recorded as
Figure BDA0003722317650000106
Step 330, calculating an approximate solution of the coverage range by a discretized summation method, and then calculating an optimal solution of the deployment parameters by using a Lagrange multiplier method.
First from
Figure BDA0003722317650000107
Randomly selecting initial values meeting the conditions in the value range omega, and obtaining the first derivative and the second derivative of S according to the formula (7)
Figure BDA0003722317650000108
Followed by a recursion according to:
Figure BDA0003722317650000109
iteration
Figure BDA0003722317650000111
Up to
Figure BDA0003722317650000112
Wherein the terminal condition epsilon is given in advance, and the precision of the result can be controlled by controlling the epsilon.
The simulation results are shown in fig. 5-10.
FIG. 5 shows the angle phi of the covered area S with the horizontal 1 The graph shows that the maximum coverage area is always present no matter what value other parameters are taken
Figure BDA0003722317650000113
Is obtained when the compound is used.
FIGS. 6 and 7 show the horizontal rotation phi of the covered area S 2 In most cases, the optimum vertical direction phi 2 =0, this means that the plane of the intelligent reflective surface is perpendicular to the direction of the incident signal. In particular, it is noted in fig. 6 that as the smart reflector is deployed higher, there is a tendency for the coverage area to increase, particularly as the smart reflector element units N are smaller and their optimal vertical-to-angular orientation is achieved
Figure BDA0003722317650000114
When moving. From fig. 7, we can conclude that the coverage area tends to increase as the intelligent reflector is closer to the base station, especially when the unit cell number of the intelligent reflector is smaller, but the trend gradually disappears with the increase of N. Therefore, when the number of the intelligent reflecting surface units is limited or less, the intelligent reflecting surface should be disposed at a position higher than the base station, and the plane thereof should be parallel to the ground as much as possible. Otherwise, its plane should be perpendicular to the direction of the incident signal.
FIG. 8 is a graph showing the variation of the coverage area of the present invention with the horizontal distance from the intelligent reflector to the base station, where phi is shown in each graph to eliminate the effect of the intelligent reflector in the vertical direction 2 Optimum value of (2)
Figure BDA0003722317650000115
It can be found that when the number of the intelligent reflection surface elements N is small, the intelligent reflection surface and the base station
Figure BDA0003722317650000116
When the horizontal distance is close, the coverage area is greatly influenced by the height of the intelligent reflecting surface along with N and
Figure BDA0003722317650000117
increasing, this effect gradually disappears. Furthermore, an increase in N will also spread out away from the optimal deployment location of the base station. Therefore, when the number of smart reflective surface elements is small, the smart reflective surface should be disposed at the edge of the cell, which is highThe altitude is lower than the base station or is arranged at a position close to the base station and higher than the base station. Otherwise, the number of the elements of the intelligent reflecting surface has little influence, and the deployment of the intelligent reflecting surface near the base station is a good strategy.
Fig. 9 and 10 are graphs of the coverage area of the present invention as a function of the height of the intelligent reflective surface. As can be seen from FIG. 9, when the intelligent reflection surface is horizontally distant from the base station
Figure BDA0003722317650000121
When the number N of the elements of the intelligent reflecting surface is large, the optimal height of the intelligent reflecting surface exists, and the optimal solution is equal to the height of the base station. Looking closely at fig. 10, it can be seen that coverage is proportional to the height of the intelligent reflective surface, in the range of 0-90 meters. The closer the intelligent reflector is to the base station, the greater the influence of the height of the intelligent reflector on the coverage rate, and the influence is gradually weakened along with the increase of N. While following with
Figure BDA0003722317650000122
The optimal solution of the intelligent reflecting surface height disappears due to the increase of N and the decrease of N. Therefore, in the case of a large number of intelligent reflective surface elements, it is recommended to dispose the intelligent reflective surface near and at the height of the base station, while in other scenarios, it is recommended to dispose the intelligent reflective surface at the highest allowable position.

Claims (5)

1. A three-dimensional deployment method for a 6G-oriented Intelligent Reflecting Surface (IRS) auxiliary network is characterized by comprising the following steps: firstly, establishing a universal three-dimensional deployment model for coverage extension of an IRS auxiliary base station; then the phase phi of the IRS reflection unit is designed m,n Maximizing the received signal-to-noise ratio of the user; then, calculating the farthest straight line distance between the user and the base station under the condition of a certain signal-to-noise ratio threshold value for the user in each orientation of the base station; finally, a closed expression of the coverage range of the base station under the three-dimensional deployment model is given, an IRS three-dimensional deployment algorithm for maximizing the coverage range of the base station is designed, the coverage range maximization problem is formed into a convex optimization problem, and the convex optimization problem is processed by introducing auxiliary variables and utilizing a Lagrange multiplier methodAnd (5) solving the problem.
2. The three-dimensional deployment model of claim 1, wherein existing two-dimensional and three-dimensional IRS models are unified, and a horizontal rotation angle phi is defined 1 Angle of vertical rotation phi 2 Horizontal distance from base station
Figure FDA0003722317640000011
Vertical height h BS And when the IRS deploys related parameters, the spatial position and orientation of the IRS are accurately described, and a foundation is laid for accurate analysis of network performance.
3. IRS unit phase phi according to claim 1 m,n The design of (2) is characterized in that an IRS phase condition for maximizing the receiving signal-to-noise ratio of a user is given through closed derivation in consideration of a Rice channel gain and line-of-sight, non-line-of-sight statistical channel model
Figure FDA0003722317640000012
Wherein phi is m,n Indicates the phase of the cell at row m and column n on IRS D Is the phase shift value of the direct link,
Figure FDA0003722317640000013
phase shift values for the reflective links through the m row and n column elements on the IRS based on phi m,n Closed-form solution giving the maximum value of the mean value of the signal-to-noise ratio
Figure FDA0003722317640000014
Where P is the base station transmit power, k 1 Rice channel parameter, k, for direct base station-to-user link 2 For the rice channel parameters of the base station to user reflecting the link straight through the IRS,
Figure FDA0003722317640000021
expressed as the channel gain of a direct Line of Sight (LoS) link,
Figure FDA0003722317640000022
expressed as the channel gain of a direct Line of Sight (NLoS) link,
Figure FDA0003722317640000023
expressed as the channel gain of the LoS link reflected by the IRS,
Figure FDA0003722317640000024
expressed as channel gain of NLoS link reflected by IRS, theta represents incident angle of base station to IRS, M, N represents IRS horizontal and vertical unit number, sigma 2 Is the noise power.
4. The method of claim 1, wherein the farthest linear distance from each base station to the user is calculated based on channel conditions
Figure FDA0003722317640000025
And fourthly, calculating the incidence angle of the base station signal, finally carrying in the closed-form solution sum of the maximum value of the average value of the signal to noise ratio, and calculating the maximum distance of the user under the condition that the maximum signal to noise ratio of the user is greater than the threshold value.
5. The IRS three-dimensional deployment algorithm for maximizing the coverage of the base station as claimed in claim 1, wherein the closed solution of the coverage area is converted into an approximate solution capable of convex optimization by a discrete approximation method, and the problem is solved by introducing an auxiliary variable and using a Lagrange multiplier method.
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