CN105071840B - A kind of joint implementation method of the control of AF bidirectional relay systems node transmitting power and intermediate position deployment - Google Patents

A kind of joint implementation method of the control of AF bidirectional relay systems node transmitting power and intermediate position deployment Download PDF

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CN105071840B
CN105071840B CN201510425375.7A CN201510425375A CN105071840B CN 105071840 B CN105071840 B CN 105071840B CN 201510425375 A CN201510425375 A CN 201510425375A CN 105071840 B CN105071840 B CN 105071840B
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CN105071840A (en
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吉晓东
包志华
章国安
曹张华
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Nantong University
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    • HELECTRICITY
    • H04ELECTRIC COMMUNICATION TECHNIQUE
    • H04BTRANSMISSION
    • H04B7/00Radio transmission systems, i.e. using radiation field
    • H04B7/02Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
    • H04B7/022Site diversity; Macro-diversity
    • H04B7/026Co-operative diversity, e.g. using fixed or mobile stations as relays
    • HELECTRICITY
    • H04ELECTRIC COMMUNICATION TECHNIQUE
    • H04WWIRELESS COMMUNICATION NETWORKS
    • H04W52/00Power management, e.g. TPC [Transmission Power Control], power saving or power classes
    • H04W52/04TPC
    • H04W52/18TPC being performed according to specific parameters
    • H04W52/22TPC being performed according to specific parameters taking into account previous information or commands
    • H04W52/225Calculation of statistics, e.g. average, variance
    • HELECTRICITY
    • H04ELECTRIC COMMUNICATION TECHNIQUE
    • H04WWIRELESS COMMUNICATION NETWORKS
    • H04W52/00Power management, e.g. TPC [Transmission Power Control], power saving or power classes
    • H04W52/04TPC
    • H04W52/18TPC being performed according to specific parameters
    • H04W52/24TPC being performed according to specific parameters using SIR [Signal to Interference Ratio] or other wireless path parameters
    • H04W52/241TPC being performed according to specific parameters using SIR [Signal to Interference Ratio] or other wireless path parameters taking into account channel quality metrics, e.g. SIR, SNR, CIR, Eb/lo

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Abstract

The invention discloses the joint implementation method that a kind of AF two-way relay communication systems node transmitting power and relay node position are disposed, to minimize system emission power as target, using system QoS requirement as confined condition, according to statistical channel status information and aims of systems speed, the position of transmission power and relay node to network node carries out Joint regulation, under conditions of system QoS requirement is met, the minimum of system total transmission power is realized.

Description

Joint implementation method for node transmission power control and relay position deployment of AF (automatic frequency hopping) bidirectional relay system
Technical Field
The invention relates to a joint implementation method for node transmission power control and relay position deployment of an AF (amplitude-and-Forward) bidirectional relay system, belonging to the technical field of wireless communication.
Background
Researchers in the early century propose a cooperative diversity (relay) technology aiming at the problem that a small mobile terminal cannot be configured with multiple antennas. Unlike traditional point-to-point communications, cooperative diversity techniques allow different user nodes in a wireless network to share each other's antennas and other network resources, hopefully greatly improving wireless network capacity and multiplexing gain. Meanwhile, the method has great development potential in the aspects of resisting channel fading, covering shadow areas, expanding the effective coverage radius of a wireless cellular system, enhancing the data rate of a specific area and the like, and has become the key direction of 4G system evolution. However, due to the half-duplex limitation of practical relay communication systems, the conventional one-way cooperative relay technique also brings loss of spectrum efficiency while improving the wireless communication performance. For this reason, researchers have proposed a cooperative relay mechanism called bidirectional relay based on the Amplify-and-Forward (AF) and Decode-and-Forward (DF) protocols for the classical three-node network. As a special cooperative transmission form, the bidirectional relay can significantly improve the network throughput and improve the spectrum utilization rate, and provides an effective technical means for efficient data communication in a wireless communication network (such as a cellular mobile communication network and a wireless sensor network), and has been paid high attention by the academic and industrial circles.
The power control is an important link self-adaptive technology, and the overall transmission performance of the system can be effectively improved, the energy utilization rate is improved, and the aims of environmental protection, energy conservation and high efficiency are fulfilled by effectively controlling the transmitting power of a user. In general, power control corresponds to two types of optimization problems: 1) Targeting system QoS (Quality of Service) and system power as constraint condition; 2) The system power is used as a target, and the system QoS is used as a constraint condition. Extensive and intensive research work has been carried out by the academia on the first class of power control optimization problems. With the proposal of the concept of 'green radio', how to save energy and reduce emission, reduce the energy consumption of a wireless communication system, improve the battery service cycle of a mobile terminal and attract more and more attention of scientific and technical personnel.
In practical wireless relay communication systems, such as wireless sensor networks and wireless local area networks, relay nodes are used to extend coverage areas or enhance transmission quality. Studies have shown that in wireless relay communication systems, optimal system performance can be obtained by deploying relay locations. In fact, the optimal relay location deployment problem in cooperative relay systems is a typical system performance optimization problem. Therefore, there is a question of: where a relay node in a two-way relay communication system should be deployed to obtain the best system performance? Currently, the problems of relay location deployment and node power control in a two-way relay communication system have attracted much attention of science and technology personnel. However, the joint problem of the two is rarely considered from the viewpoint of green energy saving. Therefore, it is necessary to research the joint implementation technology of energy-efficient node power control and relay location deployment meeting the system QoS requirement from the perspective of "green radio" for specific applications.
Disclosure of Invention
The invention provides a joint realization method for node transmitting power control and relay position deployment of an AF bidirectional relay communication system, which utilizes statistical channel state information to jointly adjust the node transmitting power and the relay position deployment according to the target rate of a source node, and realizes the minimization of the total transmitting power of the system under the condition of meeting the QoS requirement of the system. The method is suitable for the two-way relay communication system adopting the amplify-and-forward protocol.
The technical scheme of the invention is as follows:
a joint realization method for AF bidirectional relay system node power control and relay position deployment is characterized in that: the statistical value of the channel state information is utilized, the transmitting power of the system node and the position of the relay node are jointly adjusted according to the target rate of the source node, and the minimization of the total transmitting power of the system is realized under the condition of meeting the QoS requirement of the system;
for a bidirectional relay system adopting an AF protocol, two source nodes A and B in a network exchange information through a relay node R positioned between the two source nodes A and B, wherein one-time information exchange between the source nodes A and B is divided into a multiple access stage and a broadcast stage, and in a time division duplex mode, one-time information exchange between the source nodes A and B occupies two continuous time slots with equal length, wherein the first time slot corresponds to the multiple access stage, the second time slot corresponds to the broadcast stage, and the first time slot is started, the source nodes A and B and the relay node R firstly calculate a parameter eta = z B /z A A value of (2), hereWherein r is A And r B Target rates of A and B, respectively, and then d is calculated according to equations (1) and (2) AR The lower and upper bounds of (a) and (b),
here, d AB Is the distance between the source nodes A and B, d AR Is the distance between the source node a and the relay node R;
then checking the values of eta and alpha, wherein alpha is a path loss factor of the channel, and selecting one of the following 3 conditions according to the values of eta and alpha:
case1, when α =2 and η ≦ 1, d AR =0;
Case2, if α =2 and η&1, d AR =∞;
Case3, otherwise, that is, when α ≠ 2, the distance between the source node a and the relay node R is:
it should be noted that: in a typical wireless propagation environment, the path loss factor α of a channel is generally greater than or equal to 2 and less than or equal to 4, where α =2 represents a small loss wireless propagation environment and α =4 represents a large loss wireless propagation environment;
then, according to d obtained above AR To select one of the following 3 cases:
case1, ifWhen flag =1, the distance between the source node a and the relay node R is:
case2, ifWhen flag =3, the distance between the source node a and the relay node R is:
case3, in other cases, namely,when, flag =2, the distance between the source node a and the relay node R is given by equation (3);
from the distance between the source node A and the relay node R, the distance between the source node B and the relay node R, i.e. d BR =d AB -d AR So that the relay node can depend on its distance from the source nodes a and B, i.e. d AR And d BR To determine its position;
in the following, the nodes a, B and R will determine their respective transmit powers according to the value of flag, which can be divided into the following 3 cases,
case1, if flag =1, the transmission power of nodes a, B and R is:
case2, if flag =2, the transmission power of nodes a, B and R is:
case3, if flag =3, the transmission power of nodes a, B and R is:
here, S Q Which is an acceptable quality of service for the system, is a system parameter,andis an exponential random variable | h AR | 2 And | h BR | 2 Can be obtained by long-term observation at nodes A, B and R, where h is AR And h BR Rayleigh channel gains from a source node a and a relay node R to a relay node B, respectively;
after the above operations are completed, the source nodes A and B will use their respective binary information m A And m B By coded modulation into a transmission signal s A And s B And simultaneously sending the signals to the relay node R, the two paths of combined signals received by the relay node R are as follows:
wherein n is R Is gaussian white noise at the relay node R;
at the end of the first time slot, the relay R will be coupled to the received signal s R Performing scaling, i.e. multiplying by a scaling factor
Then, broadcasting to two source nodes in a second time slot, and at the end of the second time slot, the signals received by the source nodes a and B from the relay broadcasting are respectively:
wherein: n is A And n B Denotes white Gaussian noise, h, at source nodes A and B, respectively RA And h RB Channel gains for the relay node R to the source nodes A and B, respectively, assuming that the channels have reciprocity, i.e., h AR =h RA ,h BR =h RB And the receiver can obtain the ideal channel state information, then at the end of the second time slot, the source nodes a and B can use self-interference cancellation technique to transmit their own signal items in the first time slotAndremoving to obtain
Thus, at the end of the second time slot, the mutual information amounts that the source nodes a and B can obtain are: I.C. A A =log 2 (1+γ A ) [ 2 ] and I B =log 2 (1+γ B )/2,γ A And gamma B Received signal-to-noise ratios for source nodes a and B, respectively:
finally, the source nodes A and B will each be coupled to the received signal y A And y B Carrying out self-interference elimination, and then obtaining information sent by the other party through demodulation and decoding to complete information interchange;
assuming that the source nodes A and B and the relay node R can obtain the statistic value of the channel state information through long-term observation, the QoS requirement of the system is S Q Here, the outage probability is used as a system QoS performance index, bi-directionalThe relay system is used as a multi-user system, when any one of the source nodes cannot correctly decode the signal transmitted by the opposite terminal, the system is considered to have an interrupt event, and therefore, the system interrupt probability is as follows:
Q out (E A ,E B ,E R )=Pr(I A <r B ,I B <r B )=Pr(γ A <z BB <z A ) (17)
therefore, for one AF bidirectional relay communication system Q out (E A ,E B ,E R )≤S Q Must be satisfied;
for an AF bidirectional relay system, the joint optimization problem of node power control and relay node position deployment is established by taking the minimum total system transmitting power as a target and taking the system QoS requirement as a condition, and the problem is obtained
Subject to Q out (E A ,E B ,E R )≤S Q (18b)
E A ,E B ,E R >0 (18c)
To understand the problem (18), i.e., the joint optimization problem consisting of formula (18 a), formula (18 b), and formula (18 c), the following two-stage approach is used:
first stage, let d AR And d BR And fixing and researching the minimization problem of the total system transmitting power under the limitation of the system QoS:
subject to Q out (E A ,E B ,E R )≤S Q (19b)
E A ,E B ,E R >0 (19c)
and in the second stage, the sum of the problem solutions in the first stage is taken as an objective function, and the optimal relay position deployment problem is researched:
subject to d AR +d BR ≥d AB (20b)
d AR ,d BR >0 (20c)
in formula (20 a)Is the optimal solution of the problem (19), i.e., the optimal solution of the optimization problem composed of the formula (19 a), the formula (19 b) and the formula (19 c);
in order to solve the problem (19), the probability constraint (19 b) needs to be converted into a deterministic function, and therefore Q needs to be set out (E A ,E B ,E R ) Is taken into (19 b), however, Q out (P A ,P B ,P R ) Is not feasible, where an exact approximate closed-form expression is given
Then, the probability constraint (19 b) is replaced by equation (21), and for the problem (20), that is, the optimization problem composed of equation (20 a), equation (20 b) and equation (20 c), its global optimum point falls on d AR +d BR =d AB In this case, the optimal relay position is located on the connection line between the source nodes a and B, and only the optimal d is obtained AR Value of d AB The best d can be obtained BR
The final solution of the problem (18), i.e. the optimal transmit powers of the nodes A, B and R, and the optimal position of the relay node R, obtained by the above method can be divided into the following 3 cases,
case1, if the inequality (22) is true,can be given by the formula (23),may be given by formula (24);
case2, if the inequality (25) holds,can be given by the formula (26),may be given by formula (27);
case3, otherwise, the inequality (28) holds,can be given by the formula (29),may be given by formula (30);
finally, byCan obtain the product
The invention has the advantages and beneficial effects that:
the invention utilizes the statistic value of the channel state information to jointly adjust the transmitting power of the system node and the position of the relay node according to the target rate of the source node, and realizes the minimization of the total transmitting power of the system under the condition of meeting the QoS requirement of the system. Simulation experiments also show that the combined implementation method has advantages in total transmission power.
Drawings
FIG. 1 is a schematic view of the process of the present invention;
fig. 2 is an optimal relay deployment location;
FIG. 3 is a comparison of total system transmit power;
fig. 4 shows the probability of system outage.
Detailed Description
As shown in fig. 1, two consecutive time slots with equal length are occupied by the source nodes a and B to complete one information exchange. At the beginning of the first time slot, the source nodes A and B and the relay node R need to checkAnd then decides in which interval to calculate the respective transmit power according to equation (6), equation (7) or equation (8). In addition, the relay node R will also be based onDetermines to calculate the distance d between itself and the source node A according to the formula (3), the formula (4) or the formula (5) AR On the basis of this, the distance d between itself and the source node B is calculated BR =d AB -d AR To complete the deployment of relay locations. Then, the source node a and the source node B simultaneously transmit respective information to the relay node R. To accomplish the above, nodes A, B and R need to know d AB 、η、α、λ 1 、λ 2 And S Q The value of (a). In fact, S Q The acceptable interruption probability of the system is a system parameter; lambda [ alpha ] 1 And λ 2 For the statistics of channel state information, under a time division system, the nodes A, B and R can be obtained locally through long-term observation; α is the channel path loss factor and nodes A, B and R can be estimated locally. Thus, nodes A, B and R are able to know the parameters α, λ 1 、λ 2 And S Q The value of (a). d AB Is the distance between source nodes a and B, which can be obtained by estimating the direct link signal; η = z B /z A Wherein z is A And z B Which are exponential functions of the a and B target rates, respectively. Since the target rates of a and B and the distance between them are slowly varying parameters, the nodes, once acquired, will not change for a long period of time. Therefore, to enable the relay R to obtain the parameter d AB And η, the source nodes A and B obtain z respectively B And z A Here, it is proposed that: in the channel estimation phase, the source nodePoints A and B will be z A And z B And d AB Is included in the pilot signal. When the relay obtains the parameter z A 、z B And d AB Then, z is further substituted A And z B To source nodes a and B.
In the second time slot, the relay node R needs to receive the signal s R Scaled (operational flow inherent to the AF protocol) and then broadcast to the source nodes a and B. At the end of the second time slot, the source nodes A and B respectively carry out interference self-elimination on the received signals, and then demodulate and decode the signals to obtain the information sent by the opposite side, thereby finishing the information interchange.
For the joint implementation method provided by the invention, simulation experiments are carried out on the optimal relay deployment position, the total system transmitting power and the system interruption probability, and compared with the traditional node equal power transmitting mode and the non-relay position adjusting mode, the experimental environment is the Matlab environment. Assume the distance d of source nodes A and B AB =1。
Fig. 2 shows the optimal relay positions for different target rates of the source node. The target rate of the source node B is set to be constant, i.e., r B =0.5bit/s.hz, the target rate of the source node a varies from 0 to 1bit/s.hz, i.e. 0<r A Less than or equal to 1bit/s.Hz. Here, four radio propagation environments are considered, i.e., α =2, α =2.5, α =3, and α =4. Because the best relay position will be on the source node a to source node B connection. Only the optimal distance of the source node a to the relay R, i.e.,as shown in fig. 2, in the case of symmetric rates, i.e., r A =r B Regardless of the radio propagation environment, the optimal relay location will be located in the middle of the two source nodes. When η → 0 or η → ∞, i.e., the target rate between source nodes is highly asymmetric, there are casesThis means that the relay should be deployed in the middle of two source nodes. In fact, when r A When =0, i.e.Eta → 0, the two-way relay transmission will degrade to the conventional one-way relay mode. As is well known, in the case of unidirectional relaying, the relaying location should be deployed in the middle between two source nodes. When r is A On → ∞, there is η → 0, in which case the system outage performance will only depend on the large target rate link, i.e. link a to R to B, independent of the other link. At this time, the two-way relay transmission will also be degraded to the one-way relay case. In addition, fig. 2 shows: as α increases, the dynamic range of the optimal relay location also becomes larger, especially when the target rates of the two source nodes are close. This means that the better the radio propagation environment, or the closer the target rates of the two source nodes are, the more important the deployment of the optimal relay position is.
Fig. 3 shows a comparison of the total transmit power for four bi-directional relay transmission modes. Let α =2, target rate r of source node B B =0.5bit/s.hz, target rate 0 for source node a<r A Less than or equal to 1bit/s.Hz. Here, according to r A Dynamic range of (i.e., 0)<r A &lt, r is not less than 0.5bit/s.Hz and not more than 0.5bit/s.Hz A Less than or equal to 1bit/s.Hz, and dividing FIG. 3 into two subgraphs, namely, FIG. 3 (a) and FIG. 3 (b). In FIGS. 3 (a) and 3 (b), the parameter S Q Are respectively set to 10 -3 And 5X 10 -3 . In fig. 3, "equal power transmission" means that the transmission power of all nodes in the system is equal; "no relay location adjustment" means that the relay node is deployed in the middle of two source nodes. As shown in fig. 3, the proposed power control method and the combined power control and relay location deployment implementation method have performance advantages compared to the conventional transmission mode. Compared with the equal power transmission, the method for jointly realizing the power control and the relay position deployment can save more system transmitting power, particularly when the target rate of the source node is not symmetrical. Furthermore, when r is A And on the occasion of → 1bit/s.Hz, the performance difference between the method for realizing the joint implementation of the node transmission power control and the relay position deployment only tends to zero. This is because, when η → 0 or η → ∞,this means that in this case the relay should be deployed closer and closer to the middle of the two source nodes.
Fig. 4 shows the system outage probability for the four bi-directional relay transmission modes of fig. 3. As can be seen from fig. 4, the system outage probability meets the original constraint condition, and the correctness of the proposed power allocation method is verified.

Claims (1)

1. A joint realization method for AF bidirectional relay system node transmitting power control and relay position deployment is characterized in that: the statistical value of the channel state information is utilized, the transmitting power of the system node and the position of the relay node are jointly adjusted according to the target rate of the source node, and the minimization of the total transmitting power of the system is realized under the condition of meeting the QoS requirement of the system;
for a bidirectional relay system adopting an AF protocol, two source nodes A and B in a network exchange information through a relay node R positioned between the two source nodes A and B, wherein one-time information exchange between the source nodes A and B is divided into a multiple access stage and a broadcast stage, and in a time division duplex mode, one-time information exchange between the source nodes A and B occupies two continuous time slots with equal length, wherein the first time slot corresponds to the multiple access stage, the second time slot corresponds to the broadcast stage, and the first time slot is started, the source nodes A and B and the relay node R firstly calculate a parameter eta = z B /z A The value of (a) is,r A and r B Target rates of A and B, respectively, and then d is calculated according to equations (1) and (2) AR The lower and upper bounds of (a) and (b),
d AB is the distance between the source nodes A and B, d AR Is the distance between the source node a and the relay node R;
then checking the values of eta and alpha, wherein alpha is a path loss factor of the channel, and selecting one of the following 3 conditions according to the values of eta and alpha:
case1, when α =2 and η ≦ 1, d AR =0;
Case2, if α =2 and η > 1, d AR =∞;
Case3, otherwise, that is, when α ≠ 2, the distance between the source node a and the relay node R is:
in a typical wireless propagation environment, the path loss factor α of a channel is generally greater than or equal to 2 and less than or equal to 4, where α =2 represents a small loss wireless propagation environment and α =4 represents a large loss wireless propagation environment;
then, according to d obtained above AR To select one of the following 3 cases:
case1, ifWhen flag =1, the distance between the source node a and the relay node R is:
case2, ifWhen flag =3, the distance between the source node a and the relay node R is:
case3, in other cases, namely,when, flag =2, the distance between the source node a and the relay node R is given by equation (3);
the distance from the source node A to the relay node R is obtained as d BR =d AB -d AR So that the relay node can depend on its distance from the source nodes a and B, i.e. d AR And d BR To determine its location;
then, the nodes A, B and R determine their respective transmission powers according to the value of flag, which is divided into the following 3 cases,
case1, if flag =1, the transmission power of nodes a, B and R is:
case2, if flag =2, the transmission power of nodes a, B and R is:
case3, if flag =3, the transmission power of nodes a, B and R is:
here, S Q Which is an acceptable quality of service for the system, is a system parameter,andis an exponential random variable | h AR | 2 And | h BR | 2 Can be obtained by long-term observation at nodes A, B and R, where h is AR And h BR Rayleigh channel gains from a source node a and a relay node R to a relay node B, respectively;
after the above operations are completed, the source nodes A and B will use their respective binary information m A And m B By code modulation into a transmission signal s A And s B And simultaneously sending the signals to the relay node R, the two paths of combined signals received by the relay node R are as follows:
wherein n is R Is gaussian white noise at the relay node R;
at the end of the first time slot, the relay R will be coupled to the received signal s R Performing scaling, i.e. multiplying by a scaling factor
Then, broadcasting to two source nodes in a second time slot, and at the end of the second time slot, the source nodes a and B receive the relay broadcast signals as follows:
wherein: n is A And n B Denotes white Gaussian noise, h, at source nodes A and B, respectively RA And h RB Channel gains for the relay node R to the source nodes A and B, respectively, assuming that the channels have reciprocity, i.e., h AR =h RA ,h BR =h RB And the receiver can obtain the desired channel state information, then, at the end of the second time slot, the sourceNodes A and B can utilize self-interference elimination technology to transmit signal items of the nodes A and B in the first time slotAndremoving to obtain
Thus, at the end of the second time slot, the mutual information amounts that the source nodes a and B can obtain are: I.C. A A =log 2 (1+γ A ) [ 2 ] and I B =log 2 (1+γ B )/2,γ A And gamma B Received signal-to-noise ratio for source nodes a and B, respectively:
finally, the source nodes A and B will each be coupled to the received signal y A And y B Carrying out self-interference elimination, and then obtaining information sent by the other party through demodulation and decoding to complete information interchange;
assuming that the source nodes A and B and the relay node R can obtain the statistic value of the channel state information through long-term observation, the QoS requirement of the system is S Q Here, the interruption probability is used as a system QoS performance index, the bidirectional relay system is used as a multi-user system, and when any one of the source nodes cannot correctly decode a signal transmitted by the opposite end, the system is considered to beWhen an interrupt event occurs, the system interrupt probability is:
Q out (E A ,E B ,E R )=Pr(I A <r B ,I B <r B )=Pr(γ A <z BB <z A ) (17)
therefore, for one AF bidirectional relay communication system Q out (E A ,E B ,E R )≤S Q Must be satisfied;
for an AF bidirectional relay system, the joint optimization problem of node power control and relay node position deployment is established by taking the minimum total system transmitting power as a target and taking the system QoS requirement as a condition, and the method is obtained
Subject to Q out (E A ,E B ,E R )≤S Q (18b)
E A ,E B ,E R >0 (18c)
To understand the problem (18), i.e., the joint optimization problem consisting of equation (18 a), equation (18 b), and equation (18 c), the following two-stage approach is employed:
first stage, let d AR And d BR And fixing and researching the minimization problem of the total system transmitting power under the limitation of the system QoS:
subject to Q out (E A ,E B ,E R )≤S Q (19b)
E A ,E B ,E R >0 (19c)
and in the second stage, the sum of the problem solutions in the first stage is used as an objective function to research the optimal relay position deployment problem:
subject to d AR +d BR ≥d AB (20b)
d AR ,d BR >0 (20c)
in the formula (20 a)Is the optimal solution of the problem (19), i.e. the optimal solution of the optimization problem consisting of the formula (19 a), the formula (19 b) and the formula (19 c);
in order to solve the problem (19), the probability constraint (19 b) needs to be converted into a deterministic function, and therefore Q needs to be changed out (E A ,E B ,E R ) Is taken into (19 b), however, Q out (P A ,P B ,P R ) Is not feasible, where an exact approximate closed-form expression is given
Then, the probability constraint (19 b) is replaced by equation (21), and for the problem (20), that is, the optimization problem composed of equation (20 a), equation (20 b) and equation (20 c), its global optimum point falls on d AR +d BR =d AB That is, the optimal relay position is located on the connection line of the source nodes a and B, only the optimal d needs to be obtained AR Value of d AB The best d can be obtained BR
The final solution of the problem (18), i.e. the optimal transmit powers of the nodes A, B and R, and the optimal position of the relay node R, obtained by the above method can be divided into the following 3 cases,
case1, if inequality (22) holds,can be given by the formula (23),may be given by formula (24);
case2, if the inequality (25) holds,can be given by the formula (26),may be given by formula (27);
case3, otherwise, i.e., inequality (28) holds,can be given by the formula (29),may be given by formula (30);
finally, byCan obtain the product
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