# Patent

## Title
Power allocation optimization under constraints of throughput requirements and interference limit for cognitive radio networks

## Abstract

The present invention provides a method of power control for a plurality of secondary users (SUs), or unlicensed users, in a cognitive radio network having a plurality of primary users (PUs), or licensed users. The method comprises determining an optimal power level for each of the SUs such that a total throughput of all the SUs is maximized while satisfying an individual throughput requirement of each SU and an interference limit constraint of the PUs. In particular, a case having two SUs is considered.

## Background
CROSS-REFERENCE TO RELATED APPLICATIONS
This application claims the benefit of U.S. Provisional Patent Application No. 62/197,016, filed on Jul. 26, 2015, which is incorporated by reference herein in its entirety.


BACKGROUND
Field of the Invention
The present invention relates to power control for a cognitive radio communication system.
List of References
There follows a list of references that are occasionally cited in the specification. Each of the disclosures of these references is incorporated by reference herein in its entirety.

    [1] Y. C. Liang, K. C. Chen, G. Y. Li, and P. Mahonen: Cognitive Radio Networking and Communications: An Overview. IEEE Transactions on Vehicular Technology, vol. 60, no. 7, pp. 3386-3407, (2011)
    [2] Y. Liu, L. X. Cai, H. Luo, and X. Shen: Deploying Cognitive Cellular Networks under Dynamic Resource Management. IEEE Wireless Communications, Vol. 20, No. 2, pp. 82-88, (2013)
    [3] A. Gjendemsio, D. Gesbert, G. E. Oien, and S. G. Kiani: Optimal Power Allocation and Scheduling for Two-Cell Capacity Maximization. Proceedings of International Symposium on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks (2006)
    [4] C. S. Chen, G. E. Oien: Optimal Power Allocation for Two-Cell Sum Rate Maximization under Minimum Rate Constraints. Proceedings of IEEE International Symposium on Wireless Communication Systems (2008)
    [5] V. Chandrasekhar, Z. K. Shen: Optimal Uplink Power Control in Two-Cell Systems with Rise-Over-Thermal Constraints. IEEE Communications Letters, vol. 12, no. 3, pp. 173-175, (2008)
    [6] C. H. Yu, K. Doppler, C. B. Ribeiro, and O. Tirkkonen: Resource Sharing Optimization for Device-to-Device Communication Underlaying Cellular Networks. IEEE Transactions on Wireless Communications, vol. 10, no. 8, pp. 2752-2763, (2011)
    [7] Y. Wu, T. Y. Zhang, Danny H. K. Tsang: Joint Pricing and Power Allocation for Dynamic Spectrum Access Networks with Stackelberg Game Model. IEEE Transactions on Wireless Communications, vol. 10, no. 1, pp. 12-19, (2011)
    [8] Y. Wu, Q. H. Zhu, J. W Huang, Danny H. K. Tsang: Revenue Sharing based Resource Allocation for Dynamic Spectrum Access Networks. IEEE Journal on Selected Areas in Communications, vol. 32, no. 11, pp. 2280-2297, (2014)
    [9] J. W. Huang, R. Berry, and M. L. Honig: Auction-Based Spectrum Sharing. ACM MONET, vol. 11, no. 3, pp. 405-418, (2006)


Description of Related Art
With a rapid growth of wireless services in the past decades, the issue of spectrum congestion has attracted a lot of research interest. Spectrum sharing via cognitive radio (CR), in which unlicensed users are eligible to share the spectrum with the licensed users, has been considered as a promising approach to improve the spectrum utilization and to relieve the spectrum congestion problem [1], [2]. In the art, a licensed user and an unlicensed user are also called as a primary user (PU) and a secondary user (SU), respectively. The premise of the spectrum sharing via CR lies in that the transmissions of SUs cannot cause a harmful influence to the PU, which consequently necessitates a careful design of power control such that (i) each SU can obtain a satisfactory throughput, and (ii) the aggregate interference caused by the SUs' transmissions does not exceed a tolerable threshold set by the PU.
To the best of the inventors' knowledge, it is an open topic in the art to analytically derive the SUs' optimal power allocations when the PU's interference limit, a critical property for spectrum sharing cognitive radio, is taken into account. Although there exist previous works investigating the power allocations for multiple SUs subject to the PUs' interference limit constraints by using algorithmic approaches [7]-[9], these works seldom can derive the optimal power allocations analytically.
There is a need in the art for analytically deriving optimal power allocations for SUs under PUs' interference limit constraints so as to apply the derived power allocations for cognitive radio networks.
SUMMARY OF THE INVENTION
An aspect of the present invention is to provide a method of power control for a plurality of SUs in a cognitive radio network having a plurality of PUs. The method comprises determining an optimal power level for each of the SUs such that a total throughput of all the SUs is maximized while satisfying an individual throughput requirement of each SU and an interference limit constraint of the PUs.
Other aspects of the present invention are disclosed as illustrated by the embodiments hereinafter.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 depicts different cases of the feasible region of Problem (P1) according to the different locations of Q relative to l1 and l2.
FIG. 2 depicts different subcases in Case I according to the different locations of line l0 relative to points X1 and Y2.
FIG. 3 depicts Subcase II. 2 and Subcase II.3 in Case II according to the different locations of line l0 relative to points Y1 and Y2, whereas Subcase II.1 is given in FIG. 1(b).
FIG. 4 depicts sum throughput F(p1,p2) with different interference limits (i.e., Γ) of the PU for the case of A≠0, where the channel power gains are g11=0.2, g22=0.4, g12=g21=0.01, g1B=0.15 and g2B=0.1, and the SUs' minimum SINR requirements are γ1=γ2=2.
FIG. 5 depicts sum throughput F(p1,p2) with different levels of cross interference (i.e., g12 and g21) for the case of A≠0, where the rest of the channel gains are g11=0.2, g22=0.4, g1B=0.15 and g2B=0.1, and the SUs' minimum SINR requirements are γ1=γ2=2. The PU's interference limit is Γ=0.1.
FIG. 6 depicts sum throughput F(p1,p2) with different interference limits (i.e. Γ) of the PU for the case of A=0, where the channel power gains are g11=g22=0.2, g12=g21=0.02 and g1B=g2B=0.1. the SUs' minimum SINR requirements are γ1=γ2=2.



DETAILED DESCRIPTION
The present work aims at analytically deriving optimal power allocations for maximizing the SUs' total throughput while taking into account each SU's individual throughput requirement and the PU's interference limit constraint.
To analytically derive the optimal power allocations, we focus on the case of two SUs sharing the spectrum of a PU. In particular, the problem is similar to the two-users transmit power control problem in wireless interference channel, which has been studied in [3]-[6] but without taking into account the PU's interference limit. Specifically, [3] investigated the power allocations for two mobile users in two different cells but only considered each user's peak transmit power constraint. As an extension of [3], [4] further considered each user's minimum throughput requirement. Both [3] and [4] showed that the optimal power allocations exhibited a so-called binary property. Reference [5] formulated a rise-over-thermal constraint (which can be treated as the interference limit at each mobile user) and revealed that the optimal power allocations only resided on one of the vertexes of the feasible region. Reference [6] investigated the power allocations for two interfering device-to-device users with both having the maximum and minimum rate constraints. Again, the optimal power allocations were found only at the vertexes of the feasible region.
Different from the previous works [7]-[9], we find that besides the vertexes of the feasible region, the SUs' optimal power allocations might also reside on the boundary of the feasible region that corresponds to the PU's interference limit (which thus are more general than the previous works). We validate this result via extensive numerical simulations and also show how the PU's interference limit influences the derived optimal power allocations.
A. System Model, Problem Formulation, and Solvability of the Problem
A.1 System Model and Problem Formulation
We consider a scenario of two SUs sharing the licensed spectrum of one PU. Each SU comprises a transmitter and a receiver, and the PU performs the uplink transmission to the base station (BS). Let p1 and p2 denote transmit powers of SU1 and SU2, respectively. We consider the two-user throughput maximization problem as follows:




  
    
      
        
          
            
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Specifically, gii denotes the channel power gain from the transmitter of SU i to its receiver, gij denotes the channel power gain from the transmitter of SU i to the receiver of SU j, and giB denotes the channel power gain from the transmitter of SU i to the BS. Besides, ni denotes the power of the background noise experienced by the receiver of SU i (Notice that ni can also incorporate the interference from the PU to SU i), and W denotes the bandwidth of the PU's channel. Without loss of generality, W is assumed to be unity (i.e. W=1) hereinafter for easy presentation. Constraint (1) ensures that the total interference from the two SUs cannot exceed the PU's interference limit, which is denoted by Γ. Constraints (2) and (3) ensure that each SU achieves its required signal to interference plus noise ratio denoted by γi (or equivalently, each SU i achieves its required throughput log2(1+γi) measured by the channel capacity formula). Finally, constraint (4) ensures that each SU i's transmit power cannot exceed the upper bound pimax. Hereinafter, we use (p*1,p*2) to denote the optimal power allocations of the two SUs of Problem (P1).

A.2 Feasible Region and Feasibility of Problem (P1)

It is apparent that (p*1,p*2) strongly depends on the feasible region comprised of (1)-(4). To geographically characterize the feasible region of (P1), we first define the following five lines:




  
    
      
        
          
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where lines l0, l1, and l2 are obtained by making constraints (1), (2), and (3) strictly binding, and lines lm1 and lm2 are obtained by fixing p1=p1max and p2=p2max, respectively.

The above five lines together constrain the feasible region of Problem (P1) as shown in FIG. 1. Using the five lines, we further provide a set of critical points to be used in the rest of this work in Table 1. Specifically, Table 1 shows the coordinates of each critical point (in the first column) and the two lines whose intersection point corresponds to this critical point (in the second column). For instance, point Z0 in Table 1 represents the pair of minimum power allocations ({circumflex over (p)}1,{circumflex over (p)}2)=((Z0)h,(Z0)v) such that the SUs meet constraints (2) and (3) together exactly, where we use (Z0)h (or (Z0)v) to denote the horizontal-coordinate (or the vertical-coordinate) of point Z0, with the subscript h representing “horizontal” and subscript v representing “vertical”. The same notations will be used in the rest of this work.







TABLE 1







List of Critical Points










Critical Points and their Coordinates
Lines







Q = (p1max, p2max)
lm1, lm2



 







  
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              max
            
          
        
        
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            1
            ⁢
            B
          
        
      
      ,
      
        p
        2
        max
      
    
    )
  




l0, lm2



 







  
    X
    1
  
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              1
            
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              n
              1
            
          
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            ⁢
            
              γ
              1
            
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              2
              max
            
          
        
        
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          11
        
      
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        p
        2
        max
      
    
    )
  




l1, lm2



 







  
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    2
  
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          -
          
            
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            γ
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        2
        max
      
    
    )
  




l2, lm2



 







  
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        1
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l0, lm1



 







  
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    1
  
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        p
        1
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              11
            
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              1
            
          
        
        
          
            γ
            1
          
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            g
            21
          
        
      
    
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l1, lm1



 







  
    Y
    2
  
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    (
    
      
        p
        1
        max
      
      ,
      
        
          
            
              n
              2
            
            ⁢
            
              γ
              2
            
          
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              12
            
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              γ
              2
            
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              p
              1
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    )
  




l2, lm1



 







  
    Z
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l1, l2



 







  
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          +
          
            
              g
              
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              g
              21
            
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              γ
              1
            
          
        
      
    
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l0, l1



 







  
    Z
    2
  
  =
  
    (
    
      
        
          
            
              g
              22
            
            ⁢
            Γ
          
          -
          
            
              g
              
                2
                ⁢
                B
              
            
            ⁢
            
              n
              2
            
            ⁢
            
              γ
              2
            
          
        
        
          
            
              g
              22
            
            ⁢
            
              g
              
                1
                ⁢
                B
              
            
          
          +
          
            
              g
              
                2
                ⁢
                B
              
            
            ⁢
            
              g
              12
            
            ⁢
            
              γ
              2
            
          
        
      
      ,
      
        
          
            
              g
              12
            
            ⁢
            
              γ
              2
            
            ⁢
            Γ
          
          +
          
            
              g
              
                1
                ⁢
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            ⁢
            
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              2
            
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              γ
              2
            
          
        
        
          
            
              g
              22
            
            ⁢
            
              g
              
                1
                ⁢
                B
              
            
          
          +
          
            
              g
              
                2
                ⁢
                B
              
            
            ⁢
            
              g
              12
            
            ⁢
            
              γ
              2
            
          
        
      
    
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l0, l2










As used herein, “Z0≦Q” is used to represent element-wise no greater than, i.e. (Z0)h≦(Q)h and (Z0)v≦(Q)v.
Using point Z0, we can characterize the feasibility of Problem (P1) as follows.
Proposition 1. Problem (P1) is feasible if both conditions (C1) and (C2) below are met:

0≦Z0≦Q,  (C1): and

(Z0)v≦ƒl0((Z0)h).  (C2):

Proof:
Condition (C1) means that Z0 is feasible to the transmit power capacities of the SUs in constraint (4), and condition (C2) means that Z0 is feasible to the PU's interference limit in constraint (1). Recall that function ƒl0(•) has been defined in (5).
Remark 1.
Condition (C1) in Proposition 1 is equivalent to

g11g22−γ1γ2g12g21>0.  (9)

That is, the interference channel power gains should be weaker enough to accommodate the throughput requirements of both SUs. In particular, constraint (9) means that the slope of line l1 in (6) should be greater than that of l2 in (7) such that their intersection (i.e. point Z0) exists.

B. Optimal Power Allocations of Problem (P1)

In this section, we focus on solving Problem (P1) analytically, under the condition that (P1) is feasible (that is, conditions (C1) and (C2) are met). In particular, to account for the most general case of the feasible region, we assume that ƒl0((Q)h)≦(Q)v holds. Otherwise, the PU's interference limit will not influence the feasible region of Problem (P1) at all. In this case, the optimal power allocations have been given in [4]. Graphically, it means that point Q is above line l0 (that is, points X0 and Y0 always exist as shown in FIGS. 1 and 2), and the interference limit Γ in constraint (1) indeed influences the feasible region.
A deep look at Problem (P1) shows that its feasible region can take the following three different cases (depending on the different positions of point Q relative to line l1 and line l2).
Case I: ƒl1((Q)h)≧(Q)v and ƒl2((Q)h)≦(Q)v hold, meaning that point Q is between lines l1 and l2, and points X1 and Y2 appear, as shown in FIG. 1(a). Note that functions ƒl1(•) and ƒl2(•) have been defined in (6) and (7), respectively.
Case II: ƒl1((Q)h)≦(Q)v and ƒl2((Q)h)≦(Q)v hold, meaning that point Q is above lines l1 and l2, and points Y1 and Y2 appear, as shown in FIG. 1(b).
Case III: ƒl1((Q)h)≧(Q)v and ƒl2((Q)h)≧(Q)v hold, meaning that point Q is below lines l1 and l2, and points X1 and X2 appear, as shown in FIG. 1(c).
We next provide the optimal power allocations of Problem (P1) for each of the above three cases. However, even we suppose that Case I happens, it is still difficult to derive the optimal power allocations analytically, since it strongly depends on the location of l0 representing the PU's interference limit. To categorize the impact of l0, we further consider the following four different subcases of Case I, depending on how line l0 locates with respect to points X1 and Y2.
Subcase I.1: ƒl0((X1)h)≧(X1)v and ƒl0((Y2)h)≧(Y2)v, meaning that both points X1 and Y2 are below line l0, and points X0 and Y0 appear as shown in FIG. 2(a).
Subcase I.2: ƒl0((X1)h)≦(X1)v and ƒl0((Y2)h)≦(Y2)v, meaning that both points X1 and Y2 are above line l0, and points Z1 and Z2 appear as shown in FIG. 2(b).
Subcase I.3: ƒl0((X1)h)≧(X1)v and ƒl0((Y2)h)≦(Y2)v, meaning that point X1 is below line l0, and point Y2 is above line l0, and points X0 and Z2 appear as shown in FIG. 2(c).
Subcase I.4: ƒl0((X1)h)≦(X1)v and ƒl0((Y2)h)≧(Y2)v, meaning that point X1 is above line l0, and point Y2 is below line l0, and points Z1 and Y0 appear as shown in FIG. 2(d).
B.1 Optimal Power Allocations for Subcase I.1
In this subsection, we focus on analytically deriving the optimal power allocations for Subcase I.1. We first present the following result.
Lemma 1. The optimal power allocations (p*1,p*2) of Problem (P1) for Subcase I.1 (shown in FIG. 2(a)) reside within the vertexes X0, X1, Y0, and Y2, or on the line segment of l0 between points X0 and Y0.
Proof.
Please refer to Appendix I.
To further characterize (p*1,p*2) that might appear on the line segment of l0 between points X0 and Y0, we use (1) to substitute p2 by p1 and equivalently transform the objective function F(p1,p2) as:




  
    
      
        
          
            F
            ^
          
          ⁡
          
            (
            
              p
              1
            
            )
          
        
        =
        
          
            
              log
              2
            
            ⁡
            
              (
              
                
                  
                    
                      g
                      21
                    
                    ⁢
                    Γ
                  
                  +
                  
                    
                      g
                      
                        2
                        ⁢
                        B
                      
                    
                    ⁢
                    
                      n
                      1
                    
                  
                  -
                  
                    
                      g
                      21
                    
                    ⁢
                    
                      g
                      
                        1
                        ⁢
                        B
                      
                    
                    ⁢
                    
                      p
                      1
                    
                  
                  +
                  
                    
                      g
                      11
                    
                    ⁢
                    
                      g
                      
                        2
                        ⁢
                        B
                      
                    
                    ⁢
                    
                      p
                      1
                    
                  
                
                
                  
                    
                      g
                      21
                    
                    ⁢
                    Γ
                  
                  +
                  
                    
                      g
                      
                        2
                        ⁢
                        B
                      
                    
                    ⁢
                    
                      n
                      1
                    
                  
                  -
                  
                    
                      g
                      21
                    
                    ⁢
                    
                      g
                      
                        1
                        ⁢
                        B
                      
                    
                    ⁢
                    
                      p
                      1
                    
                  
                
              
              )
            
          
          +
          
            
              
                log
                2
              
              ⁡
              
                (
                
                  
                    
                      
                        g
                        
                          2
                          ⁢
                          B
                        
                      
                      ⁢
                      
                        n
                        2
                      
                    
                    +
                    
                      
                        g
                        12
                      
                      ⁢
                      
                        g
                        
                          2
                          ⁢
                          B
                        
                      
                      ⁢
                      
                        p
                        1
                      
                    
                    +
                    
                      
                        g
                        22
                      
                      ⁢
                      Γ
                    
                    -
                    
                      
                        g
                        22
                      
                      ⁢
                      
                        g
                        
                          1
                          ⁢
                          B
                        
                      
                      ⁢
                      
                        p
                        1
                      
                    
                  
                  
                    
                      
                        g
                        
                          2
                          ⁢
                          B
                        
                      
                      ⁢
                      
                        n
                        2
                      
                    
                    +
                    
                      
                        g
                        12
                      
                      ⁢
                      
                        g
                        
                          2
                          ⁢
                          B
                        
                      
                      ⁢
                      
                        p
                        1
                      
                    
                  
                
                )
              
            
            .
          
        
      
    
    
      
        (
        10
        )
      
    
  




Moreover, the following result holds for the above {circumflex over (F)}(p1).
Lemma 2. Function {circumflex over (F)}(p1) has three different extremal points, which can be expressed as follows:



  
    
      
        p
        1
        
          o
          1
        
      
      =
      
        
          
            
              
                -
                B
              
              +
              
                
                  
                    B
                    2
                  
                  -
                  
                    4
                    ⁢
                    A
                    ⁢
                    
                      
                    
                    ⁢
                    C
                  
                
              
            
            
              2
              ⁢
              A
            
          
          ⁢
          
            
          
          ⁢
          if
          ⁢
          
            
          
          ⁢
          A
        
        ≠
        0
      
    
    ,
    
      

    
    ⁢
    
      
        p
        1
        
          o
          2
        
      
      =
      
        
          
            
              
                -
                B
              
              -
              
                
                  
                    B
                    2
                  
                  -
                  
                    4
                    ⁢
                    A
                    ⁢
                    
                      
                    
                    ⁢
                    C
                  
                
              
            
            
              2
              ⁢
              A
            
          
          ⁢
          
            
          
          ⁢
          if
          ⁢
          
            
          
          ⁢
          A
        
        ≠
        
          0
          ⁢
          
            
          
          ⁢
          and
        
      
    
  




  
    
      
        p
        1
        
          o
          3
        
      
      =
      
        
          
            
              -
              C
            
            B
          
          ⁢
          
            
          
          ⁢
          if
          ⁢
          
            
          
          ⁢
          A
        
        =
        0
      
    
    ,
  



where parameters A, B, and C are given by (11), (12) and (13), respectively, as shown below, where the superscripts o1, o2 and o3 represent the three different cases of the extremal points:





  
    
      
        A
        =
        
          
            
              g
              
                2
                ⁢
                B
              
            
            
              ln
              ⁢
              
                
              
              ⁢
              2
            
          
          ⁡
          
            [
            
              
                
                  g
                  11
                
                ⁢
                
                  g
                  12
                
                ⁢
                
                  
                    g
                    
                      2
                      ⁢
                      B
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          n
                          1
                        
                        ⁢
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                      
                      +
                      
                        
                          g
                          21
                        
                        ⁢
                        Γ
                      
                    
                    )
                  
                
                ⁢
                
                  (
                  
                    
                      
                        g
                        12
                      
                      ⁢
                      
                        g
                        
                          2
                          ⁢
                          B
                        
                      
                    
                    -
                    
                      
                        g
                        22
                      
                      ⁢
                      
                        g
                        
                          1
                          ⁢
                          B
                        
                      
                    
                  
                  )
                
              
              -
              
                
                  g
                  22
                
                ⁢
                
                  g
                  21
                
                ⁢
                
                  
                    g
                    
                      1
                      ⁢
                      B
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          n
                          2
                        
                      
                      +
                      
                        
                          g
                          12
                        
                        ⁢
                        Γ
                      
                    
                    )
                  
                
                ⁢
                
                  (
                  
                    
                      
                        g
                        21
                      
                      ⁢
                      
                        g
                        
                          1
                          ⁢
                          B
                        
                      
                    
                    -
                    
                      
                        g
                        11
                      
                      ⁢
                      
                        g
                        
                          2
                          ⁢
                          B
                        
                      
                    
                  
                  )
                
              
            
            ]
          
        
      
    
    
      
        (
        11
        )
      
    
  
  
    
      
        B
        =
        
          
            
              
                g
                
                  2
                  ⁢
                  B
                
              
              ⁡
              
                (
                
                  
                    
                      n
                      1
                    
                    ⁢
                    
                      g
                      
                        2
                        ⁢
                        B
                      
                    
                  
                  +
                  
                    
                      g
                      21
                    
                    ⁢
                    Γ
                  
                
                )
              
            
            
              ln
              ⁢
              
                
              
              ⁢
              2
            
          
          ×
          
            [
            
              
                
                  g
                  11
                
                ⁢
                
                  
                    g
                    
                      2
                      ⁢
                      B
                    
                  
                  ⁡
                  
                    (
                    
                      
                        2
                        ⁢
                        
                          g
                          12
                        
                        ⁢
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          n
                          2
                        
                      
                      +
                      
                        
                          g
                          12
                        
                        ⁢
                        
                          g
                          22
                        
                        ⁢
                        Γ
                      
                      -
                      
                        
                          g
                          22
                        
                        ⁢
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          n
                          2
                        
                      
                    
                    )
                  
                
              
              -
              
                
                  
                    g
                    22
                  
                  ⁡
                  
                    (
                    
                      
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          n
                          2
                        
                      
                      +
                      
                        
                          g
                          12
                        
                        ⁢
                        Γ
                      
                    
                    )
                  
                
                ⁢
                
                  (
                  
                    
                      
                        g
                        11
                      
                      ⁢
                      
                        g
                        
                          2
                          ⁢
                          B
                        
                      
                    
                    -
                    
                      2
                      ⁢
                      
                        g
                        21
                      
                      ⁢
                      
                        g
                        
                          1
                          ⁢
                          B
                        
                      
                    
                  
                  )
                
              
            
            ]
          
        
      
    
    
      
        (
        12
        )
      
    
  
  
    
      
        C
        =
        
          
            
              
                g
                
                  2
                  ⁢
                  B
                
              
              ⁡
              
                (
                
                  
                    
                      n
                      1
                    
                    ⁢
                    
                      g
                      
                        2
                        ⁢
                        B
                      
                    
                  
                  +
                  
                    
                      g
                      21
                    
                    ⁢
                    Γ
                  
                
                )
              
            
            
              ln
              ⁢
              
                
              
              ⁢
              2
            
          
          ×
          
             
            
              [
              
                
                  
                    g
                    11
                  
                  ⁢
                  
                    g
                    
                      2
                      ⁢
                      B
                    
                  
                  ⁢
                  
                    
                      n
                      2
                    
                    ⁡
                    
                      (
                      
                        
                          
                            n
                            2
                          
                          ⁢
                          
                            g
                            
                              2
                              ⁢
                              B
                            
                          
                        
                        +
                        
                          
                            g
                            22
                          
                          ⁢
                          Γ
                        
                      
                      )
                    
                  
                
                -
                
                  
                    
                      g
                      22
                    
                    ⁡
                    
                      (
                      
                        
                          
                            g
                            
                              1
                              ⁢
                              B
                            
                          
                          ⁢
                          
                            n
                            2
                          
                        
                        +
                        
                          
                            g
                            12
                          
                          ⁢
                          Γ
                        
                      
                      )
                    
                  
                  ⁢
                  
                    (
                    
                      
                        
                          n
                          1
                        
                        ⁢
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                      
                      +
                      
                        
                          g
                          21
                        
                        ⁢
                        Γ
                      
                    
                    )
                  
                
              
              ]
            
          
        
      
    
    
      
        (
        13
        )
      
    
  




Proof:
Please refer to Appendix II.
Lemma 2 and the equivalence between {circumflex over (F)}(p1) and F(p1,p2) imply that the derived extremal points p1o1, p1o2, and p1o3 could also possibly maximize F(p1,p2), depending on whether they fall within the line segment of l0 between points X0 and Y0 (i.e.




  [
  
    
      
        Γ
        -
        
          
            g
            
              2
              ⁢
              B
            
          
          ⁢
          
            p
            2
            max
          
        
      
      
        g
        
          1
          ⁢
          B
        
      
    
    ,
    
      p
      1
      max
    
  
  ]




according to Table 1). Based on this deduction, we define the three new points, bounded by points X0 and Y0, as follows:





  
    O
    1
    
      I
      ⁢
      .1
    
  
  =
  
    {
    
      
        
          
            
              (
              
                
                  p
                  1
                  
                    o
                    1
                  
                
                ,
                
                  
                    f
                    
                      l
                      0
                    
                  
                  ⁡
                  
                    (
                    
                      p
                      1
                      
                        o
                        1
                      
                    
                    )
                  
                
              
              )
            
          
          
            
              
                if
                ⁢
                
                  
                
                ⁢
                
                  
                    (
                    
                      X
                      0
                    
                    )
                  
                  h
                
              
              <
              
                p
                1
                
                  o
                  1
                
              
              <
              
                
                  (
                  
                    Y
                    0
                  
                  )
                
                h
              
            
          
        
        
          
            φ
          
          
            otherwise
          
        
      
      ;
      
        

      
      ⁢
      
        
          O
          2
          
            I
            ⁢
            .1
          
        
        =
        
          {
          
            
              
                
                  
                    (
                    
                      
                        p
                        1
                        
                          o
                          2
                        
                      
                      ,
                      
                        
                          f
                          
                            l
                            0
                          
                        
                        ⁡
                        
                          (
                          
                            p
                            1
                            
                              o
                              2
                            
                          
                          )
                        
                      
                    
                    )
                  
                
                
                  
                    
                      if
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          (
                          
                            X
                            0
                          
                          )
                        
                        h
                      
                    
                    <
                    
                      p
                      1
                      
                        o
                        2
                      
                    
                    <
                    
                      
                        (
                        
                          Y
                          0
                        
                        )
                      
                      h
                    
                  
                
              
              
                
                  φ
                
                
                  otherwise
                
              
            
            ;
            
              

            
            ⁢
            
              
                O
                3
                
                  I
                  ⁢
                  .1
                
              
              =
              
                {
                
                  
                    
                      
                        
                          (
                          
                            
                              p
                              1
                              
                                o
                                3
                              
                            
                            ,
                            
                              
                                f
                                
                                  l
                                  0
                                
                              
                              ⁡
                              
                                (
                                
                                  p
                                  1
                                  
                                    o
                                    3
                                  
                                
                                )
                              
                            
                          
                          )
                        
                      
                      
                        
                          
                            if
                            ⁢
                            
                              
                            
                            ⁢
                            
                              
                                (
                                
                                  X
                                  0
                                
                                )
                              
                              h
                            
                          
                          <
                          
                            p
                            1
                            
                              o
                              3
                            
                          
                          <
                          
                            
                              (
                              
                                Y
                                0
                              
                              )
                            
                            h
                          
                        
                      
                    
                    
                      
                        φ
                      
                      
                        otherwise
                      
                    
                  
                  .
                
              
            
          
        
      
    
  




Note that points X0 and Y0 are provided in Table 1, and p1o1, p1o2, and p1o3, are given in Lemma 2.

Using Lemma 1 and Lemma 2, we can derive (p*1,p*2) of Problem (P1) under Subcase I.1 as follows.
Theorem 1. (Optimal Power Allocation for Subcase I.1): The optimal power allocations (p*1,p*2) of Problem (P1) under Subcase I.1 are:




  
    (
    
      
        p
        1
        *
      
      ,
      
        p
        2
        *
      
    
    )
  
  =
  
    {
    
      
        
          
            arg
            ⁢
            
              
            
            ⁢
            
              
                max
                
                  
                    (
                    
                      
                        p
                        1
                      
                      ,
                      
                        p
                        2
                      
                    
                    )
                  
                  ∈
                  
                    {
                    
                      
                        X
                        0
                      
                      ,
                      
                        X
                        1
                      
                      ,
                      
                        Y
                        0
                      
                      ,
                      
                        Y
                        2
                      
                      ,
                      
                        O
                        1
                        
                          I
                          ⁢
                          .1
                        
                      
                      ,
                      
                        O
                        2
                        
                          I
                          ⁢
                          .1
                        
                      
                    
                    }
                  
                
              
              ⁢
              
                
              
              ⁢
              
                F
                ⁡
                
                  (
                  
                    
                      p
                      1
                    
                    ,
                    
                      p
                      2
                    
                  
                  )
                
              
            
          
        
        
          
            
              
                if
                ⁢
                
                  
                
                ⁢
                A
              
              ≠
              0
            
            ,
          
        
      
      
        
          
            arg
            ⁢
            
              
            
            ⁢
            
              
                max
                
                  
                    (
                    
                      
                        p
                        1
                      
                      ,
                      
                        p
                        2
                      
                    
                    )
                  
                  ∈
                  
                    {
                    
                      
                        X
                        0
                      
                      ,
                      
                        X
                        1
                      
                      ,
                      
                        Y
                        0
                      
                      ,
                      
                        Y
                        2
                      
                      ,
                      
                        O
                        3
                        
                          I
                          ⁢
                          .1
                        
                      
                    
                    }
                  
                
              
              ⁢
              
                
              
              ⁢
              
                F
                ⁡
                
                  (
                  
                    
                      p
                      1
                    
                    ,
                    
                      p
                      2
                    
                  
                  )
                
              
            
          
        
        
          
            
              if
              ⁢
              
                
              
              ⁢
              A
            
            =
            0.
          
        
      
    
  




Proof:
Please refer to Appendix III.
B.2 Optimal Power Allocations for Subcase I.2
Similar to Subcase I.1, we can also characterize the optimal power allocations of Problem (P1) under Subcase I.2. Specifically, we also define the following three points, which are bounded by points Z1 and Z2:




  
    O
    1
    
      I
      ⁢
      .2
    
  
  =
  
    {
    
      
        
          
            
              (
              
                
                  p
                  1
                  
                    o
                    1
                  
                
                ,
                
                  
                    f
                    
                      l
                      0
                    
                  
                  ⁡
                  
                    (
                    
                      p
                      1
                      
                        o
                        1
                      
                    
                    )
                  
                
              
              )
            
          
          
            
              
                if
                ⁢
                
                  
                
                ⁢
                
                  
                    (
                    
                      Z
                      1
                    
                    )
                  
                  h
                
              
              <
              
                p
                1
                
                  o
                  1
                
              
              <
              
                
                  (
                  
                    Z
                    2
                  
                  )
                
                h
              
            
          
        
        
          
            φ
          
          
            otherwise
          
        
      
      ;
      
        

      
      ⁢
      
        
          O
          2
          
            I
            ⁢
            .2
          
        
        =
        
          {
          
            
              
                
                  
                    (
                    
                      
                        p
                        1
                        
                          o
                          2
                        
                      
                      ,
                      
                        
                          f
                          
                            l
                            0
                          
                        
                        ⁡
                        
                          (
                          
                            p
                            1
                            
                              o
                              2
                            
                          
                          )
                        
                      
                    
                    )
                  
                
                
                  
                    
                      if
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          (
                          
                            Z
                            1
                          
                          )
                        
                        h
                      
                    
                    <
                    
                      p
                      1
                      
                        o
                        2
                      
                    
                    <
                    
                      
                        (
                        
                          Z
                          2
                        
                        )
                      
                      h
                    
                  
                
              
              
                
                  φ
                
                
                  otherwise
                
              
            
            ;
            
              

            
            ⁢
            
              
                O
                3
                
                  I
                  ⁢
                  .2
                
              
              =
              
                {
                
                  
                    
                      
                        
                          (
                          
                            
                              p
                              1
                              
                                o
                                3
                              
                            
                            ,
                            
                              
                                f
                                
                                  l
                                  0
                                
                              
                              ⁡
                              
                                (
                                
                                  p
                                  1
                                  
                                    o
                                    3
                                  
                                
                                )
                              
                            
                          
                          )
                        
                      
                      
                        
                          
                            if
                            ⁢
                            
                              
                            
                            ⁢
                            
                              
                                (
                                
                                  Z
                                  1
                                
                                )
                              
                              h
                            
                          
                          <
                          
                            p
                            1
                            
                              o
                              3
                            
                          
                          <
                          
                            
                              (
                              
                                Z
                                2
                              
                              )
                            
                            h
                          
                        
                      
                    
                    
                      
                        φ
                      
                      
                        otherwise
                      
                    
                  
                  .
                
              
            
          
        
      
    
  




Note that p1o1, p1o2, and p1o3 have been given in Lemma 2. However, different from Subcase I.1, Subcase I.2 (as shown in FIG. 2(b)), the line segment l0 is constrained by points Z1 and Z2 (given in Table 1). Hence, the optimal power allocations of Problem (P1) under Subcase I.2 are as follows.

Theorem 2. (Optimal Power Allocation for Subcase I.2): The optimal power allocations (p*1,p*2) of Problem (P1) under Subcase I.2 are:




  
    (
    
      
        p
        1
        *
      
      ,
      
        p
        2
        *
      
    
    )
  
  =
  
    {
    
      
        
          
            arg
            ⁢
            
              
            
            ⁢
            
              
                max
                
                  
                    (
                    
                      
                        p
                        1
                      
                      ,
                      
                        p
                        2
                      
                    
                    )
                  
                  ∈
                  
                    {
                    
                      
                        Z
                        1
                      
                      ,
                      
                        Z
                        2
                      
                      ,
                      
                        O
                        1
                        
                          I
                          ⁢
                          .2
                        
                      
                      ,
                      
                        O
                        2
                        
                          I
                          ⁢
                          .2
                        
                      
                    
                    }
                  
                
              
              ⁢
              
                
              
              ⁢
              
                F
                ⁡
                
                  (
                  
                    
                      p
                      1
                    
                    ,
                    
                      p
                      2
                    
                  
                  )
                
              
            
          
        
        
          
            
              
                if
                ⁢
                
                  
                
                ⁢
                A
              
              ≠
              0
            
            ,
          
        
      
      
        
          
            arg
            ⁢
            
              
            
            ⁢
            
              
                max
                
                  
                    (
                    
                      
                        p
                        1
                      
                      ,
                      
                        p
                        2
                      
                    
                    )
                  
                  ∈
                  
                    {
                    
                      
                        Z
                        1
                      
                      ,
                      
                        Z
                        2
                      
                      ,
                      
                        O
                        3
                        
                          I
                          ⁢
                          .2
                        
                      
                    
                    }
                  
                
              
              ⁢
              
                
              
              ⁢
              
                F
                ⁡
                
                  (
                  
                    
                      p
                      1
                    
                    ,
                    
                      p
                      2
                    
                  
                  )
                
              
            
          
        
        
          
            
              if
              ⁢
              
                
              
              ⁢
              A
            
            =
            0.
          
        
      
    
  




Proof:
The proof is similar to that for Theorem 1 in Appendix III. It is easy for an ordinary skilled person to develop the proof by reference to Appendix III.
B.3 Optimal Power Allocations for Subcase I.3
Similar to Subcase I.1 and Subcase I.2, we define the following three points, which are bounded by points X0 and Z2 in Subcase I.3:




  
    O
    1
    
      I
      ⁢
      .3
    
  
  =
  
    {
    
      
        
          
            
              (
              
                
                  p
                  1
                  
                    o
                    1
                  
                
                ,
                
                  
                    f
                    
                      l
                      0
                    
                  
                  ⁡
                  
                    (
                    
                      p
                      1
                      
                        o
                        1
                      
                    
                    )
                  
                
              
              )
            
          
          
            
              
                if
                ⁢
                
                  
                
                ⁢
                
                  
                    (
                    
                      X
                      0
                    
                    )
                  
                  h
                
              
              <
              
                p
                1
                
                  o
                  1
                
              
              <
              
                
                  (
                  
                    Z
                    2
                  
                  )
                
                h
              
            
          
        
        
          
            φ
          
          
            otherwise
          
        
      
      ;
      
        

      
      ⁢
      
        
          O
          2
          
            I
            ⁢
            .3
          
        
        =
        
          {
          
            
              
                
                  
                    (
                    
                      
                        p
                        1
                        
                          o
                          2
                        
                      
                      ,
                      
                        
                          f
                          
                            l
                            0
                          
                        
                        ⁡
                        
                          (
                          
                            p
                            1
                            
                              o
                              2
                            
                          
                          )
                        
                      
                    
                    )
                  
                
                
                  
                    
                      if
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          (
                          
                            X
                            0
                          
                          )
                        
                        h
                      
                    
                    <
                    
                      p
                      1
                      
                        o
                        2
                      
                    
                    <
                    
                      
                        (
                        
                          Z
                          2
                        
                        )
                      
                      h
                    
                  
                
              
              
                
                  φ
                
                
                  otherwise
                
              
            
            ;
            
              

            
            ⁢
            
              
                O
                3
                
                  I
                  ⁢
                  .3
                
              
              =
              
                {
                
                  
                    
                      
                        
                          (
                          
                            
                              p
                              1
                              
                                o
                                3
                              
                            
                            ,
                            
                              
                                f
                                
                                  l
                                  0
                                
                              
                              ⁡
                              
                                (
                                
                                  p
                                  1
                                  
                                    o
                                    3
                                  
                                
                                )
                              
                            
                          
                          )
                        
                      
                      
                        
                          
                            if
                            ⁢
                            
                              
                            
                            ⁢
                            
                              
                                (
                                
                                  X
                                  0
                                
                                )
                              
                              h
                            
                          
                          <
                          
                            p
                            1
                            
                              o
                              3
                            
                          
                          <
                          
                            
                              (
                              
                                Z
                                2
                              
                              )
                            
                            h
                          
                        
                      
                    
                    
                      
                        φ
                      
                      
                        otherwise
                      
                    
                  
                  .
                
              
            
          
        
      
    
  




Using the above-defined O1I.3, O2I.3, and O3I.3, one can derive the optimal power allocations of Problem (P1) under Subcase I.3 as follows.

Theorem 3. (Optimal Power Allocation for Subcase I.3): The optimal power allocations (p*1,p*2) of Problem (P1) under Subcase I.3 are:




  
    (
    
      
        p
        1
        *
      
      ,
      
        p
        2
        *
      
    
    )
  
  =
  
    {
    
      
        
          
            arg
            ⁢
            
              
            
            ⁢
            
              
                max
                
                  
                    (
                    
                      
                        p
                        1
                      
                      ,
                      
                        p
                        2
                      
                    
                    )
                  
                  ∈
                  
                    {
                    
                      
                        X
                        0
                      
                      ,
                      
                        X
                        1
                      
                      ,
                      
                        Z
                        2
                      
                      ,
                      
                        O
                        1
                        
                          I
                          ⁢
                          .3
                        
                      
                      ,
                      
                        O
                        2
                        
                          I
                          ⁢
                          .3
                        
                      
                    
                    }
                  
                
              
              ⁢
              
                
              
              ⁢
              
                F
                ⁡
                
                  (
                  
                    
                      p
                      1
                    
                    ,
                    
                      p
                      2
                    
                  
                  )
                
              
            
          
        
        
          
            
              
                if
                ⁢
                
                  
                
                ⁢
                A
              
              ≠
              0
            
            ,
          
        
      
      
        
          
            arg
            ⁢
            
              
            
            ⁢
            
              
                max
                
                  
                    (
                    
                      
                        p
                        1
                      
                      ,
                      
                        p
                        2
                      
                    
                    )
                  
                  ∈
                  
                    {
                    
                      
                        X
                        0
                      
                      ,
                      
                        X
                        1
                      
                      ,
                      
                        Z
                        2
                      
                      ,
                      
                        O
                        3
                        
                          I
                          ⁢
                          .3
                        
                      
                    
                    }
                  
                
              
              ⁢
              
                
              
              ⁢
              
                F
                ⁡
                
                  (
                  
                    
                      p
                      1
                    
                    ,
                    
                      p
                      2
                    
                  
                  )
                
              
            
          
        
        
          
            
              if
              ⁢
              
                
              
              ⁢
              A
            
            =
            0.
          
        
      
    
  




Proof:
The proof is similar to that for Theorem 1 given in Appendix III.
B.4 Optimal Power Allocations for Subcase I.4
Similar to Subcases I.1, I.2, and I.3, we define the following three points, which are now bounded by points Z1 and Y0 in Subcase I.4:




  
    O
    1
    
      I
      ⁢
      .4
    
  
  =
  
    {
    
      
        
          
            
              (
              
                
                  p
                  1
                  
                    o
                    1
                  
                
                ,
                
                  
                    f
                    
                      l
                      0
                    
                  
                  ⁡
                  
                    (
                    
                      p
                      1
                      
                        o
                        1
                      
                    
                    )
                  
                
              
              )
            
          
          
            
              
                if
                ⁢
                
                  
                
                ⁢
                
                  
                    (
                    
                      Z
                      1
                    
                    )
                  
                  h
                
              
              <
              
                p
                1
                
                  o
                  1
                
              
              <
              
                
                  (
                  
                    Y
                    0
                  
                  )
                
                h
              
            
          
        
        
          
            φ
          
          
            otherwise
          
        
      
      ;
      
        

      
      ⁢
      
        
          O
          2
          
            I
            ⁢
            .4
          
        
        =
        
          {
          
            
              
                
                  
                    (
                    
                      
                        p
                        1
                        
                          o
                          2
                        
                      
                      ,
                      
                        
                          f
                          
                            l
                            0
                          
                        
                        ⁡
                        
                          (
                          
                            p
                            1
                            
                              o
                              2
                            
                          
                          )
                        
                      
                    
                    )
                  
                
                
                  
                    
                      if
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          (
                          
                            Z
                            1
                          
                          )
                        
                        h
                      
                    
                    <
                    
                      p
                      1
                      
                        o
                        2
                      
                    
                    <
                    
                      
                        (
                        
                          Y
                          0
                        
                        )
                      
                      h
                    
                  
                
              
              
                
                  φ
                
                
                  otherwise
                
              
            
            ;
            
              

            
            ⁢
            
              
                O
                3
                
                  I
                  ⁢
                  .4
                
              
              =
              
                {
                
                  
                    
                      
                        
                          (
                          
                            
                              p
                              1
                              
                                o
                                3
                              
                            
                            ,
                            
                              
                                f
                                
                                  l
                                  0
                                
                              
                              ⁡
                              
                                (
                                
                                  p
                                  1
                                  
                                    o
                                    3
                                  
                                
                                )
                              
                            
                          
                          )
                        
                      
                      
                        
                          
                            if
                            ⁢
                            
                              
                            
                            ⁢
                            
                              
                                (
                                
                                  Z
                                  1
                                
                                )
                              
                              h
                            
                          
                          <
                          
                            p
                            1
                            
                              o
                              3
                            
                          
                          <
                          
                            
                              (
                              
                                Y
                                0
                              
                              )
                            
                            h
                          
                        
                      
                    
                    
                      
                        φ
                      
                      
                        otherwise
                      
                    
                  
                  .
                
              
            
          
        
      
    
  




Using the points O1I.4, O2I.4, and O3I.4 defined above, we provide the following Theorem 4.

Theorem 4. (Optimal Power Allocation for Subcase I.4): The optimal power allocations (p*1,p*2) of Problem (P1) under Subcase I.4 are:




  
    (
    
      
        p
        1
        *
      
      ,
      
        p
        2
        *
      
    
    )
  
  =
  
    {
    
      
        
          
            arg
            ⁢
            
              
            
            ⁢
            
              
                max
                
                  
                    (
                    
                      
                        p
                        1
                      
                      ,
                      
                        p
                        2
                      
                    
                    )
                  
                  ∈
                  
                    {
                    
                      
                        Y
                        0
                      
                      ,
                      
                        Y
                        2
                      
                      ,
                      
                        Z
                        1
                      
                      ,
                      
                        O
                        1
                        
                          I
                          ⁢
                          .4
                        
                      
                      ,
                      
                        O
                        2
                        
                          I
                          ⁢
                          .4
                        
                      
                    
                    }
                  
                
              
              ⁢
              
                
              
              ⁢
              
                F
                ⁡
                
                  (
                  
                    
                      p
                      1
                    
                    ,
                    
                      p
                      2
                    
                  
                  )
                
              
            
          
        
        
          
            
              
                if
                ⁢
                
                  
                
                ⁢
                A
              
              ≠
              0
            
            ,
          
        
      
      
        
          
            arg
            ⁢
            
              
            
            ⁢
            
              
                max
                
                  
                    (
                    
                      
                        p
                        1
                      
                      ,
                      
                        p
                        2
                      
                    
                    )
                  
                  ∈
                  
                    {
                    
                      
                        Y
                        0
                      
                      ,
                      
                        Y
                        2
                      
                      ,
                      
                        Z
                        1
                      
                      ,
                      
                        O
                        3
                        
                          I
                          ⁢
                          .4
                        
                      
                    
                    }
                  
                
              
              ⁢
              
                
              
              ⁢
              
                F
                ⁡
                
                  (
                  
                    
                      p
                      1
                    
                    ,
                    
                      p
                      2
                    
                  
                  )
                
              
            
          
        
        
          
            
              if
              ⁢
              
                
              
              ⁢
              A
            
            =
            0.
          
        
      
    
  




Proof:
The proof is also similar to that for Theorem 1.
Summarizing the above four subcases, we finish deriving the optimal power allocations for Problem (P1) under Case I.
B.5 Optimal Power Allocations for Case II and Case III
We next derive the optimal power allocations under Case II and Case III by using the similar approach as that for Case I, i.e. categorizing different subcases that differentiate the location of l0. In particular, we will show that each subcase of Case II (or Case III) is equivalent to one subcase of Case I, and thus the analytical optimal power allocations provided before in the previous Subsections A, B, C, and D are also applicable to the subcases of Case II and Case III.
Specifically, we first consider Case II, which includes three subcases, i.e. Subcase II.1 as shown in FIG. 1(b), Subcase II.2 in FIG. 3(a), and Subcase II.3 in FIG. 3(b), depending on how line l0 locates with respect to points Y1 and Y2. (Different from that for Case I, the two points Y1 and Y2 in Case II have the same horizontal-coordinate, and thus there only exist three instead of four subcases in Case II.) In particular, a deep look at Subcase II.1 shows that it is in fact similar to Subcase I.4 (as shown in FIG. 2(d)), and the optimal power allocations in Theorem 4 is applicable. Subcase II.2 is similar to Subcase I.2 (in FIG. 2(b)), and thus the optimal power allocations in Theorem 2 is applicable. Finally, the optimal power allocations under Subcase II.3 are given by (p*1,p*2)=argmax(p1,p2)ε{Y1,Y2}F(p1,p2) according to Lemma 1. We thus finish deriving the optimal power allocations for Case II.
The results for Case II are as follows.
Subcase II.1: ƒlo((Y1)h)≧(Y1)v and ƒlo((Y2)h)≦(Y2)v, meaning that point Y1 is above line l0 and point Y2Y2 is below l0. The optimal power allocation for this subcase is completely similar to Theorem 4. (Notice that all of the points used therein are provided in Table I.)
Subcase II.2: ƒlo((Y1)h)≧(Y1)v and ƒlo((Y2)h)≧(Y2)v, meaning that points Y1 and Y2 are both above line l0. The optimal power allocation for this subcase is completely similar to Theorem 2.
Subcase II.3: ƒlo((Y1)h)≧(Y1)v and ƒlo((Y2)h)≧(Y2)v, meaning that points Y1 and Y2 are both below line l0. The optimal power allocation for this subcase is given by Theorem 5 below.
Theorem 5. (Optimal Power Allocation for Subcase II. 3): The optimal power allocations (p*1,p*2) of Problem (P1) under Subcase II.3 are:




  
    (
    
      
        p
        1
        *
      
      ,
      
        p
        2
        *
      
    
    )
  
  =
  
    arg
    ⁢
    
      
    
    ⁢
    
      
        max
        
          
            (
            
              
                p
                1
              
              ,
              
                p
                2
              
            
            )
          
          ∈
          
            [
            
              
                Y
                1
              
              ,
              
                Y
                2
              
            
            ]
          
        
      
      ⁢
      
        
      
      ⁢
      
        
          F
          ⁡
          
            (
            
              
                p
                1
              
              ,
              
                p
                2
              
            
            )
          
        
        .
      
    
  




Using the same approach as for Case II, an ordinary person skilled in the art can also derive the optimal power allocations for Case III (which in fact also has three different subcases). The results for Case III are as follows.
Subcase III.1: ƒlo((X1)h)≧(X1)v and ƒlo((X2)h)≦(X2)v, meaning that point X1 is below line l0 and point X2 is above l0. The optimal power allocation for this subcase is completely similar to Theorem 3.
Subcase III.2: ƒlo((X1)h)≦(X1)v and ƒlo((X2)h)≦(X2)v, meaning that points X1 and X2 are both above line l0. The optimal power allocation for this subcase is completely similar to Theorem 2.
Subcase III.3: ƒlo((X1)h)≧(X1)v and ƒlo((X2)h)≧(X2)v, meaning that points X1 and X2 are both below line l0. The optimal power allocation for this subcase is given by Theorem 6 below.
Theorem 6. (Optimal Power Allocation for Subcase III.3): The optimal power allocations (p*1,p*2) of Problem (P1) under Subcase III.3 are:




  
    (
    
      
        p
        1
        *
      
      ,
      
        p
        2
        *
      
    
    )
  
  =
  
    arg
    ⁢
    
      
    
    ⁢
    
      
        max
        
          
            (
            
              
                p
                1
              
              ,
              
                p
                2
              
            
            )
          
          ∈
          
            [
            
              
                X
                1
              
              ,
              
                X
                2
              
            
            ]
          
        
      
      ⁢
      
        
      
      ⁢
      
        
          F
          ⁡
          
            (
            
              
                p
                1
              
              ,
              
                p
                2
              
            
            )
          
        
        .
      
    
  




B.6 Appendix I: Proof of Lemma 1

The proof is similar to that in [4], which considered the feasible region comprising each user's peak transmit power constraint, but without considering the PU's interference limit constraint (represented by line segment l0 between points X0 and Y0 as shown in FIG. 2(a)). Thus, using the similar technique as in [4], we can prove that X0, X1, Y0, and Y2 are the candidates for the optimal power allocations of Problem (P1). Moreover, since (i) for some λ>1, there exists




  
    
      
        
          F
          ⁡
          
            (
            
              
                λ
                ⁢
                
                  
                
                ⁢
                
                  p
                  1
                
              
              ,
              
                λ
                ⁢
                
                  
                
                ⁢
                
                  p
                  2
                
              
            
            )
          
        
        =
        
        ⁢
        
          
            
              log
              2
            
            ⁡
            
              (
              
                1
                +
                
                  
                    λ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      g
                      11
                    
                    ⁢
                    
                      p
                      1
                    
                  
                  
                    
                      n
                      1
                    
                    +
                    
                      λ
                      ⁢
                      
                        
                      
                      ⁢
                      
                        g
                        21
                      
                      ⁢
                      
                        p
                        2
                      
                    
                  
                
              
              )
            
          
          +
          
            
              log
              2
            
            ⁡
            
              (
              
                1
                +
                
                  
                    λ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      g
                      22
                    
                    ⁢
                    
                      p
                      2
                    
                  
                  
                    
                      n
                      2
                    
                    +
                    
                      λ
                      ⁢
                      
                        
                      
                      ⁢
                      
                        g
                        12
                      
                      ⁢
                      
                        p
                        1
                      
                    
                  
                
              
              )
            
          
        
      
    
  
  
    
      
        =
        
        ⁢
        
          
            
              log
              2
            
            ⁡
            
              (
              
                1
                +
                
                  
                    
                      
                    
                    ⁢
                    
                      
                        g
                        11
                      
                      ⁢
                      
                        p
                        1
                      
                    
                  
                  
                    
                      
                        n
                        1
                      
                      λ
                    
                    +
                    
                      
                        g
                        21
                      
                      ⁢
                      
                        p
                        2
                      
                    
                  
                
              
              )
            
          
          +
        
      
    
  
  
    
      
        
        ⁢
        
          
            
              
                log
                2
              
              ⁡
              
                (
                
                  1
                  +
                  
                    
                      
                        
                      
                      ⁢
                      
                        
                          g
                          22
                        
                        ⁢
                        
                          p
                          2
                        
                      
                    
                    
                      
                        
                          n
                          2
                        
                        λ
                      
                      +
                      
                        
                      
                      ⁢
                      
                        
                          g
                          12
                        
                        ⁢
                        
                          p
                          1
                        
                      
                    
                  
                
                )
              
            
            >
            
              F
              ⁡
              
                (
                
                  
                    p
                    1
                  
                  ,
                  
                    p
                    2
                  
                
                )
              
            
          
          ,
        
      
    
  




i.e. the total throughput of the two SUs increases when the power allocation pair (p1,p2) increases, and (ii) p1 and p2 are coupled by function g1Bp1+g2Bp2=Γ on line l0. Therefore, the points on the line segment of l0 between X0 and Y0 are also the possible candidates for the optimal power allocations.

Thus, the optimal power allocations of Problem (P1) under Subcase I.1 must reside within the vertices X0, X1, Y0, and Y2, or on the line segment of l0 between points X0 and Y0.
B.7 Appendix II: Proof of Lemma 2
To derive the possible extremal points on line segment l0 between points X0 and Y0, we calculate the first order derivative of {circumflex over (F)}(p1) as follows:




  
    
      
        d
        
          
            F
            ^
          
          ⁡
          
            (
            
              p
              1
            
            )
          
        
      
      
        d
        
          p
          1
        
      
    
    =
    
      
        
          Ap
          1
          2
        
        +
        
          Bp
          1
        
        +
        C
      
      D
    
  
  ,




where parameters A, B, and C have given in (11), (12) and (13), respectively. Meanwhile, D is given by:

D=(n1g2B+g21Γ+g11g2Bp1−g21g1Bp1)×(n1g2B+g21Γ−g21g1Bp1)×(n2g2B+g22Γ+g12g2Bp1−g22g1Bp1)×(n2g2B+g12g2Bp1).

In particular, D>0 always holds, since the four items (in the product form) in D correspond to the numerators and denominators in the log(•) expression in (10). By imposing





  
    
      
        d
        
          
            F
            ^
          
          ⁡
          
            (
            
              p
              1
            
            )
          
        
      
      
        d
        
          p
          1
        
      
    
    =
    0
  
  ,




we obtain a quadratic equation Ap12+Bp1+C=0, whose roots can be analytically given by




  
    
      
        p
        1
        
          o
          1
        
      
      =
      
        
          –B
          +
          
            
              
                B
                2
              
              -
              
                4
                ⁢
                AC
              
            
          
        
        
          2
          ⁢
          A
        
      
    
    ,
    
      
        and
        ⁢
        
          
        
        ⁢
        
          p
          1
          
            o
            2
          
        
      
      =
      
        
          –B
          -
          
            
              
                B
                2
              
              -
              
                4
                ⁢
                AC
              
            
          
        
        
          2
          ⁢
          A
        
      
    
    ,
    
      
    
    ⁢
    
      
        
          if
          ⁢
          
            
          
          ⁢
          A
        
        ≠
        0
      
      ;
    
  




  
    
      
    
    ⁢
    
      
        
          p
          1
          
            o
            3
          
        
        =
        
          
        
        ⁢
        
          -
          
            C
            B
          
        
      
      ,
      
        
          if
          ⁢
          
            
          
          ⁢
          A
        
        =
        0
      
      ,
    
  



which correspond to the extremal points on line segment l0 that might maximize {circumflex over (F)}(p1) (or equivalently, F(p1,p2)).

B.8 Appendix III: Proof of Theorem 3

For the case of A≠0, there exist two extremal points O1I.1=(p1o1,ƒl0(p1o1)), O2I.1=(p1o2,ƒl0(p1o2)) on the line segment of l0 according to Lemma 2. In companion with the previous four points X0,X1,Y0, and Y2 (which have been proved in Lemma 1), the optimal power allocations (p*1,p*2) of Problem (P1) under Subcase I.1 can thus be given by




  
    (
    
      
        p
        1
        *
      
      ,
      
        p
        2
        *
      
    
    )
  
  =
  
    arg
    ⁢
    
      
    
    ⁢
    
      
        max
        
          
            (
            
              
                p
                1
              
              ,
              
                p
                2
              
            
            )
          
          ∈
          
            {
            
              
                X
                0
              
              ,
              
                X
                1
              
              ,
              
                
                  Y
                  0
                
                ⁢
                
                  Y
                  2
                
              
              ,
              
                O
                1
                
                  I
                  ,
                  1
                
              
              ,
              
                O
                2
                
                  I
                  ,
                  1
                
              
            
            }
          
        
      
      ⁢
      
        
          F
          ⁡
          
            (
            
              
                p
                1
              
              ,
              
                p
                2
              
            
            )
          
        
        .
      
    
  




Similarly, for the case of A=0, there exists one extremal point O3I.1=(p1o3, ƒl0(p1o3)) on the line segment of l0. Again, in companion with the previous four points X0,X1,Y0, and Y2, the optimal power allocations (p*1,p*2) under Subcase I.1 can thus be given by




  
    (
    
      
        p
        1
        *
      
      ,
      
        p
        2
        *
      
    
    )
  
  =
  
    arg
    ⁢
    
      
    
    ⁢
    
      
        max
        
          
            (
            
              
                p
                1
              
              ,
              
                p
                2
              
            
            )
          
          ∈
          
            {
            
              
                X
                0
              
              ,
              
                X
                1
              
              ,
              
                
                  Y
                  0
                
                ⁢
                
                  Y
                  2
                
              
              ,
              
                O
                3
                
                  I
                  ,
                  1
                
              
            
            }
          
        
      
      ⁢
      
        
          F
          ⁡
          
            (
            
              
                p
                1
              
              ,
              
                p
                2
              
            
            )
          
        
        .
      
    
  




C. Numerical Results

In this section, we perform numerical simulations to validate our analytical results in Theorems 1, 2, 3, and 4. In particular, we focus on presenting the results that the optimal power allocations (p*1,p*2) occur on line l0 when the PU's interference limit is binding. That is, one of the extremal points given in Lemma 2 turns out to be the optimal power allocations, which is a key advance we have made in this work in comparison with the previous ones. We set the background noise ni=0.01 W and set the SUs' maximum transmit powers p1max=1.0 W and p2max=1.5 W.
FIG. 4 shows the values of F(p1,p2) by enumerating different (p1,p2) for three cases of the PU's interference limit Γ (i.e., Γ=0.1, 0.15, and 0.2). In this figure, we set g11=0.2, g22=0.4, g12=g21=0.01, g1B=0.15, g2B=0.1, and γ1=γ2=2, which yield A≠0 according to (11). In each subfigure thereof, we mark out the optimal power allocations (p*1,p*2) (denoted by “Ana:”), which are analytically derived according to one of Theorems 1, 2, 3, and 4. In particular, only one theorem is applicable to derive (p*1,p*2), depending on which of the specific cases (i.e. Subcases I.1, I.2, I.3, and I.4) occurs. For instance, Subcase I.2 occurs in FIG. 3(a), and we thus use Theorem 2 to derive (p*1,p*2) accordingly. To validate this result, we also use exhaustive search to find the optimal power allocations (in a brute-force manner) and mark out the corresponding result which is denoted by “Ex:”. The comparisons between the two sets of the results show that the analytically derived (p*1,p*2) match those obtained by exhaustive search very well, thus validating Theorems 1, 2, 3, and 4. Besides, in each subfigure, we also mark out the SUs' total interference, i.e. l=p*1g1B+p*2g2B, which matches with Γ well, meaning that the optimal power allocations (p*1,p*2) indeed occur on line l0 (i.e., the PU's interference limit constraint is strictly binding). Moreover, the comparisons among the three subfigures show that when the PU's interference limit increases, the optimal power allocations (p*1,p*2) increase accordingly. That is, the SUs are able to transmit more aggressively and achieve a higher throughput.
FIG. 5 shows F(p1,p2) with different levels of the cross interference (i.e. g12=g21=0.01, 0.02 and 0.03 in the three subfigures, respectively). Here, we set g11=0.2, g22=0.4, g1B=0.15, g2B=0.1, mma1=γ2=2, and Γ=0.1, which again yield A≠0 according to (11). Similar to FIG. 1, the analytically derived optimal power allocations (p*1,p*2) based on Theorems 1, 2, 3, and 4 match those obtained by exhaustive search very well. Besides, the optimal power allocations (p*1,p*2) indeed occur when the PU's interference limit constraint is binding. In particular, the comparison among the three subfigures shows that when the cross interference is weak, both SUs can transmit simultaneously (i.e. sharing the PU's spectrum) for maximizing their total throughput. In comparison, when the cross interference becomes greater, only one favorable SU will be selected to transmit (to avoid excessive interference between the SUs) for maximizing their total throughput. This in fact turns back to the results in the previous work. That is, the optimal power allocations occur on one of the vertices of the feasible region.
Finally, FIG. 1 also shows F(p1,p2) for the case of A=0 (in particular, we set a symmetric case that g11=g22=0.2, g12=g21=0.02, g1B=g2B=0.1, and γ1=γ2=2, which yield A=0 according to (11)). We also vary Γ=0.1, 0.15, and 0.2 in the three subfigures. Again, in all the tested cases, the analytically-derived optimal power allocations match those obtained by exhaustive search, thus validating Theorems 1, 2, 3, and 4 for the case of A=0.
D. The Present Invention
The present invention is developed based on the optimal power allocation scheme as disclosed above.
An aspect of the present invention is to provide a method of power control for a plurality of SUs in a cognitive radio network having a plurality of PUs. In the method, an optimal power level for each of the SUs is determined such that a total throughput of all the SUs is maximized while satisfying an individual throughput requirement of each SU and an interference limit constraint of the PUs.
In one embodiment, the number of SUs is two, and the optimal power level for each of the SUs is determined by maximizing a two-user throughput as defined in P1 subject to constraints given by (1)-(4).
Preferably, the maximizing of the two-user throughput comprises determining feasibility of maximizing the throughput under the constraints by checking if (9) is met.
It is also preferable that the maximizing of the two-user throughput further comprises the following steps. First, lines according to (5)-(8) are constructed. If the feasibility of maximizing the throughput is confirmed, a feasible region being enclosed by the constructed lines is identified. Then coordinates of critical points, as mentioned above and shown in Table I, are determined such that the optimal power level for each of the SUs is determined according to the coordinates of the critical points.
Advantageously, the two-user throughput is maximized according to Theorems 1-6 for Subcases I.1-4, Subcases II.1-3 and Subcases III.1-3.
The embodiments disclosed herein may be implemented using general purpose or specialized computing devices, computer processors, or electronic circuitries including but not limited to digital signal processors, application specific integrated circuits, field programmable gate arrays, and other programmable logic devices configured or programmed according to the teachings of the present disclosure. Computer instructions or software codes running in the general purpose or specialized computing devices, computer processors, or programmable logic devices can readily be prepared by practitioners skilled in the software or electronic art based on the teachings of the present disclosure.
The present invention may be embodied in other specific forms without departing from the spirit or essential characteristics thereof. The present embodiment is therefore to be considered in all respects as illustrative and not restrictive. The scope of the invention is indicated by the appended claims rather than by the foregoing description, and all changes that come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein.

## Claims

1. A method of power control for two secondary users (SUs) in a cognitive radio network having a plurality of primary users (PUs), the method comprising:
determining an optimal power level for each of the two SUs such that a total throughput of the two SUs is maximized while satisfying an individual throughput requirement of each SU and an interference limit constraint of the PUs;
wherein the optimal power level for each of the two SUs is determined by maximizing a two-user throughput of





  
    max
    ⁢
    
      
    
    ⁢
    
      F
      ⁡
      
        (
        
          
            p
            1
          
          ,
          
            p
            2
          
        
        )
      
    
  
  =
  
    
      W
      ⁢
      
        
      
      ⁢
      
        
          log
          2
        
        ⁡
        
          (
          
            1
            +
            
              
                
                  p
                  1
                
                ⁢
                
                  g
                  11
                
              
              
                
                  n
                  1
                
                +
                
                  
                    p
                    2
                  
                  ⁢
                  
                    g
                    21
                  
                
              
            
          
          )
        
      
    
    +
    
      W
      ⁢
      
        
      
      ⁢
      
        
          log
          2
        
        ⁡
        
          (
          
            1
            +
            
              
                
                  p
                  2
                
                ⁢
                
                  g
                  22
                
              
              
                
                  n
                  2
                
                +
                
                  
                    p
                    1
                  
                  ⁢
                  
                    g
                    12
                  
                
              
            
          
          )
        
      
    
  



subject to constraints:




  
    
      
        
          
            p
            1
          
          ⁢
          
            g
            
              1
              ⁢
              B
            
          
        
        +
        
          
            p
            2
          
          ⁢
          
            g
            
              2
              ⁢
              B
            
          
        
      
      ≤
      Γ
    
    ;
  




  
    
      
        
          
            p
            1
          
          ⁢
          
            g
            11
          
        
        
          
            n
            1
          
          +
          
            
              p
              2
            
            ⁢
            
              g
              21
            
          
        
      
      ≥
      
        γ
        1
      
    
    ;
  




  
    
      
        
          
            p
            2
          
          ⁢
          
            g
            22
          
        
        
          
            n
            2
          
          +
          
            
              p
              1
            
            ⁢
            
              g
              12
            
          
        
      
      ≥
      
        γ
        2
      
    
    ;
    and
  




  
    
      0
      ≤
      
        p
        1
      
      ≤
      
        
          p
          1
          
            m
            ⁢
            
              
            
            ⁢
            ax
          
        
        ⁢
        
          
        
        ⁢
        and
        ⁢
        
          
        
        ⁢
        0
      
      ≤
      
        p
        2
      
      ≤
      
        p
        2
        
          ma
          ⁢
          
            
          
          ⁢
          x
        
      
    
    ;
  


where:
pi denotes the transmit power of SU i;
gii denotes a channel power gain from a transmitter of SU i to its receiver;
gij denotes a channel power gain from the transmitter of SU i to a receiver of SU j;
giB denotes a channel power gain from the transmitter of SU i to a base station (BS);
ni denotes the power of background noise experienced by the receiver of SU i;
pimax denotes the upper bound that each SU i's transmit power cannot exceed;
Γ is an interference limit of each of the PUs; and
γi is a signal to interference plus noise ratio required to achieve the individual throughput requirement of SU i.

2. The method of claim 1, wherein the maximizing of the two-user throughput comprises:
determining feasibility of maximizing the throughput under the constraints by checking if a condition g11g22−γ1γ2g12g21>0 is met.

3. The method of claim 2, wherein the maximizing of the two-user throughput further comprises:
constructing lines:




  
    
      
        Line
        ⁢
        
          
        
        ⁢
        
          l
          0
        
        ⁢
        
          :
        
        ⁢
        
          
        
        ⁢
        
          p
          2
        
      
      =
      
        
          
            f
            
              l
              0
            
          
          ⁡
          
            (
            
              p
              1
            
            )
          
        
        =
        
          
            
              -
              
                
                  g
                  
                    1
                    ⁢
                    B
                  
                
                
                  g
                  
                    2
                    ⁢
                    B
                  
                
              
            
            ⁢
            
              p
              1
            
          
          +
          
            Γ
            
              g
              
                2
                ⁢
                B
              
            
          
        
      
    
    ;
  




  
    
      
        Line
        ⁢
        
          
        
        ⁢
        
          l
          1
        
        ⁢
        
          :
        
        ⁢
        
          
        
        ⁢
        
          p
          2
        
      
      =
      
        
          
            f
            
              l
              1
            
          
          ⁡
          
            (
            
              p
              1
            
            )
          
        
        =
        
          
            
              -
              
                
                  g
                  11
                
                
                  
                    γ
                    1
                  
                  ⁢
                  
                    g
                    21
                  
                
              
            
            ⁢
            
              p
              1
            
          
          -
          
            
              n
              1
            
            
              g
              21
            
          
        
      
    
    ;
  




  
    
      
        Line
        ⁢
        
          
        
        ⁢
        
          l
          2
        
        ⁢
        
          :
        
        ⁢
        
          
        
        ⁢
        
          p
          2
        
      
      =
      
        
          
            f
            
              l
              2
            
          
          ⁡
          
            (
            
              p
              1
            
            )
          
        
        =
        
          
            
              
                
                  γ
                  2
                
                ⁢
                
                  g
                  12
                
              
              
                g
                22
              
            
            ⁢
            
              p
              1
            
          
          +
          
            
              
                γ
                2
              
              ⁢
              
                n
                2
              
            
            
              g
              22
            
          
        
      
    
    ;
  




  
    
      
        Line
        ⁢
        
          
        
        ⁢
        
          l
          
            m
            ⁢
            
              
            
            ⁢
            1
          
        
        ⁢
        
          :
        
        ⁢
        
          
        
        ⁢
        
          p
          1
        
      
      =
      
        p
        1
        
          ma
          ⁢
          
            
          
          ⁢
          x
        
      
    
    ;
    and
  




  
    
      
        Line
        ⁢
        
          
        
        ⁢
        
          l
          
            m
            ⁢
            
              
            
            ⁢
            2
          
        
        ⁢
        
          :
        
        ⁢
        
          
        
        ⁢
        
          p
          2
        
      
      =
      
        p
        2
        
          ma
          ⁢
          
            
          
          ⁢
          x
        
      
    
    ;
  


when the feasibility of maximizing the throughput is confirmed, identifying a feasible region being enclosed by the constructed lines; and
determining coordinates of critical points such that the optimal power level for each of the SUs is determined according to the coordinates of the critical points.

4. The method according to claim 3, wherein the maximizing of the two-user throughput further comprises:
when ƒl0((X1)h)≧(X1)v and ƒl0((Y2)h)≧(Y2)v, computing the optimal power levels of SU 1 and SU 2, respectively denoted as p*1 and p*2, for maximizing the two-user throughput by





  
    
  
  ⁢
  
    
      (
      
        
          p
          1
          *
        
        ,
        
          p
          2
          *
        
      
      )
    
    =
    
      {
      
        
          
            
              
                arg
                ⁢
                
                  
                
                ⁢
                
                  
                    max
                    
                      
                        (
                        
                          
                            p
                            1
                          
                          ,
                          
                            p
                            2
                          
                        
                        )
                      
                      ∈
                      
                        {
                        
                          
                            X
                            0
                          
                          ,
                          
                            X
                            1
                          
                          ,
                          
                            Y
                            0
                          
                          ,
                          
                            Y
                            2
                          
                          ,
                          
                            O
                            1
                            
                              I
                              ,
                              1
                            
                          
                          ,
                          
                            O
                            2
                            
                              I
                              ,
                              1
                            
                          
                        
                        }
                      
                    
                  
                  ⁢
                  
                    F
                    ⁡
                    
                      (
                      
                        
                          p
                          1
                        
                        ,
                        
                          p
                          2
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  
                    if
                    ⁢
                    
                      
                    
                    ⁢
                    A
                  
                  ≠
                  0
                
                ,
              
            
          
          
            
              
                arg
                ⁢
                
                  
                
                ⁢
                
                  
                    max
                    
                      
                        (
                        
                          
                            p
                            1
                          
                          ,
                          
                            p
                            2
                          
                        
                        )
                      
                      ∈
                      
                        {
                        
                          
                            X
                            0
                          
                          ,
                          
                            X
                            1
                          
                          ,
                          
                            Y
                            0
                          
                          ,
                          
                            Y
                            2
                          
                          ,
                          
                            O
                            3
                            
                              I
                              ,
                              1
                            
                          
                        
                        }
                      
                    
                  
                  ⁢
                  
                    F
                    ⁡
                    
                      (
                      
                        
                          p
                          1
                        
                        ,
                        
                          p
                          2
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  if
                  ⁢
                  
                    
                  
                  ⁢
                  A
                
                =
                0
              
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            where
            ⁢
            
              :
            
            ⁢
            
              

            
            ⁢
            
              
            
            ⁢
            
              X
              0
            
          
          =
          
            
              
                (
                
                  
                    
                      Γ
                      -
                      
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          p
                          2
                          
                            ma
                            ⁢
                            
                              
                            
                            ⁢
                            x
                          
                        
                      
                    
                    
                      g
                      
                        1
                        ⁢
                        B
                      
                    
                  
                  ,
                  
                    p
                    2
                    
                      ma
                      ⁢
                      
                        
                      
                      ⁢
                      x
                    
                  
                
                )
              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                X
                0
              
            
            =
            
              (
              
                
                  
                    (
                    
                      X
                      0
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      X
                      0
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            X
            1
          
          =
          
            
              
                (
                
                  
                    
                      
                        
                          γ
                          1
                        
                        ⁢
                        
                          n
                          1
                        
                      
                      +
                      
                        
                          g
                          21
                        
                        ⁢
                        
                          γ
                          1
                        
                        ⁢
                        
                          p
                          2
                          
                            ma
                            ⁢
                            
                              
                            
                            ⁢
                            x
                          
                        
                      
                    
                    
                      g
                      11
                    
                  
                  ,
                  
                    p
                    2
                    
                      ma
                      ⁢
                      
                        
                      
                      ⁢
                      x
                    
                  
                
                )
              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                X
                1
              
            
            =
            
              (
              
                
                  
                    (
                    
                      X
                      1
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      X
                      1
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            Y
            0
          
          =
          
            
              
                (
                
                  
                    p
                    1
                    
                      m
                      ⁢
                      
                        
                      
                      ⁢
                      ax
                    
                  
                  ,
                  
                    
                      Γ
                      -
                      
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          p
                          1
                          
                            ma
                            ⁢
                            
                              
                            
                            ⁢
                            x
                          
                        
                      
                    
                    
                      
                        g
                        
                          2
                          ⁢
                          B
                        
                      
                      ⁢
                      
                        
                      
                    
                  
                
                )
              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                Y
                0
              
            
            =
            
              (
              
                
                  
                    (
                    
                      Y
                      0
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      Y
                      0
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            Y
            2
          
          =
          
            
              
                (
                
                  
                    p
                    1
                    
                      ma
                      ⁢
                      
                        
                      
                      ⁢
                      x
                    
                  
                  ,
                  
                    
                      
                        
                          n
                          2
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                      +
                      
                        
                          g
                          12
                        
                        ⁢
                        
                          γ
                          2
                        
                        ⁢
                        
                          p
                          1
                          
                            ma
                            ⁢
                            
                              
                            
                            ⁢
                            x
                          
                        
                      
                    
                    
                      g
                      22
                    
                  
                
                )
              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                Y
                2
              
            
            =
            
              (
              
                
                  
                    (
                    
                      Y
                      2
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      Y
                      2
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            O
            1
            
              
                I
                ,
                1
              
              ⁢
              
                
              
            
          
          =
          
            {
            
              
                
                  
                    
                      (
                      
                        
                          p
                          1
                          
                            o
                            1
                          
                        
                        ,
                        
                          
                            f
                            
                              l
                              0
                            
                          
                          ⁡
                          
                            (
                            
                              p
                              1
                              
                                o
                                1
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                  
                    
                      
                        if
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            (
                            
                              X
                              0
                            
                            )
                          
                          h
                        
                      
                      <
                      
                        p
                        1
                        
                          o
                          1
                        
                      
                      <
                      
                        
                          (
                          
                            Y
                            0
                          
                          )
                        
                        h
                      
                    
                  
                
                
                  
                    φ
                  
                  
                    otherwise
                  
                
              
              ;
              
                

              
              ⁢
              
                
              
              ⁢
              
                
                  O
                  2
                  
                    I
                    ,
                    1
                  
                
                =
                
                  {
                  
                    
                      
                        
                          
                            (
                            
                              
                                p
                                1
                                
                                  o
                                  2
                                
                              
                              ,
                              
                                
                                  f
                                  
                                    l
                                    0
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    p
                                    1
                                    
                                      o
                                      2
                                    
                                  
                                  )
                                
                              
                            
                            )
                          
                        
                        
                          
                            
                              if
                              ⁢
                              
                                
                              
                              ⁢
                              
                                
                                  (
                                  
                                    X
                                    0
                                  
                                  )
                                
                                h
                              
                            
                            <
                            
                              p
                              1
                              
                                o
                                2
                              
                            
                            <
                            
                              
                                (
                                
                                  Y
                                  0
                                
                                )
                              
                              h
                            
                          
                        
                      
                      
                        
                          φ
                        
                        
                          otherwise
                        
                      
                    
                    ;
                    
                      

                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        O
                        3
                        
                          I
                          ,
                          1
                        
                      
                      =
                      
                        {
                        
                          
                            
                              
                                
                                  (
                                  
                                    
                                      p
                                      1
                                      
                                        o
                                        3
                                      
                                    
                                    ,
                                    
                                      
                                        f
                                        
                                          l
                                          0
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          p
                                          1
                                          
                                            o
                                            3
                                          
                                        
                                        )
                                      
                                    
                                  
                                  )
                                
                              
                              
                                
                                  
                                    if
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      
                                        (
                                        
                                          X
                                          0
                                        
                                        )
                                      
                                      h
                                    
                                  
                                  <
                                  
                                    p
                                    1
                                    
                                      o
                                      3
                                    
                                  
                                  <
                                  
                                    
                                      (
                                      
                                        Y
                                        0
                                      
                                      )
                                    
                                    h
                                  
                                
                              
                            
                            
                              
                                φ
                              
                              
                                otherwise
                              
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              p
                              1
                              
                                o
                                1
                              
                            
                            =
                            
                              
                                
                                  
                                    
                                      -
                                      B
                                    
                                    +
                                    
                                      
                                        
                                          B
                                          2
                                        
                                        -
                                        
                                          4
                                          ⁢
                                          A
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          C
                                        
                                      
                                    
                                  
                                  
                                    2
                                    ⁢
                                    A
                                  
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                A
                              
                              ≠
                              0
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              p
                              1
                              
                                o
                                2
                              
                            
                            =
                            
                              
                                
                                  
                                    
                                      -
                                      B
                                    
                                    -
                                    
                                      
                                        
                                          B
                                          2
                                        
                                        -
                                        
                                          4
                                          ⁢
                                          A
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          C
                                        
                                      
                                    
                                  
                                  
                                    2
                                    ⁢
                                    A
                                  
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                A
                              
                              ≠
                              0
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              p
                              1
                              
                                o
                                3
                              
                            
                            =
                            
                              
                                
                                  
                                    -
                                    C
                                  
                                  B
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                A
                              
                              =
                              0
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            A
                            =
                            
                              
                                
                                  g
                                  
                                    2
                                    ⁢
                                    B
                                  
                                
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  2
                                
                              
                              ⁡
                              
                                [
                                
                                  
                                    
                                      g
                                      11
                                    
                                    ⁢
                                    
                                      g
                                      12
                                    
                                    ⁢
                                    
                                      
                                        g
                                        
                                          2
                                          ⁢
                                          B
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              n
                                              1
                                            
                                            ⁢
                                            
                                              g
                                              
                                                2
                                                ⁢
                                                B
                                              
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              21
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            g
                                            12
                                          
                                          ⁢
                                          
                                            g
                                            
                                              2
                                              ⁢
                                              B
                                            
                                          
                                        
                                        -
                                        
                                          
                                            g
                                            22
                                          
                                          ⁢
                                          
                                            g
                                            
                                              1
                                              ⁢
                                              B
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                  -
                                  
                                    
                                      g
                                      22
                                    
                                    ⁢
                                    
                                      g
                                      21
                                    
                                    ⁢
                                    
                                      
                                        g
                                        
                                          1
                                          ⁢
                                          B
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              g
                                              
                                                1
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            g
                                            21
                                          
                                          ⁢
                                          
                                            g
                                            
                                              1
                                              ⁢
                                              B
                                            
                                          
                                        
                                        -
                                        
                                          
                                            g
                                            11
                                          
                                          ⁢
                                          
                                            g
                                            
                                              2
                                              ⁢
                                              B
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                ]
                              
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            B
                            =
                            
                              
                                
                                  
                                    g
                                    
                                      2
                                      ⁢
                                      B
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          n
                                          1
                                        
                                        ⁢
                                        
                                          g
                                          
                                            2
                                            ⁢
                                            B
                                          
                                        
                                      
                                      +
                                      
                                        
                                          g
                                          21
                                        
                                        ⁢
                                        Γ
                                      
                                    
                                    )
                                  
                                
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  2
                                
                              
                              ×
                              
                                [
                                
                                  
                                    
                                      g
                                      11
                                    
                                    ⁢
                                    
                                      
                                        g
                                        
                                          2
                                          ⁢
                                          B
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            2
                                            ⁢
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            
                                              g
                                              
                                                2
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            
                                              g
                                              22
                                            
                                            ⁢
                                            Γ
                                          
                                          -
                                          
                                            
                                              g
                                              22
                                            
                                            ⁢
                                            
                                              g
                                              
                                                1
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                        
                                        )
                                      
                                    
                                  
                                  -
                                  
                                    
                                      
                                        g
                                        22
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              g
                                              
                                                1
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            g
                                            11
                                          
                                          ⁢
                                          
                                            g
                                            
                                              2
                                              ⁢
                                              B
                                            
                                          
                                        
                                        -
                                        
                                          2
                                          ⁢
                                          
                                            g
                                            21
                                          
                                          ⁢
                                          
                                            g
                                            
                                              1
                                              ⁢
                                              B
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                ]
                              
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            C
                            =
                            
                              
                                
                                  
                                    g
                                    
                                      2
                                      ⁢
                                      B
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          n
                                          1
                                        
                                        ⁢
                                        
                                          g
                                          
                                            2
                                            ⁢
                                            B
                                          
                                        
                                      
                                      +
                                      
                                        
                                          g
                                          21
                                        
                                        ⁢
                                        Γ
                                      
                                    
                                    )
                                  
                                
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  2
                                
                              
                              ×
                              
                                
                                  [
                                  
                                    
                                      
                                        g
                                        11
                                      
                                      ⁢
                                      
                                        g
                                        
                                          2
                                          ⁢
                                          B
                                        
                                      
                                      ⁢
                                      
                                        
                                          n
                                          2
                                        
                                        ⁡
                                        
                                          (
                                          
                                            
                                              
                                                n
                                                2
                                              
                                              ⁢
                                              
                                                g
                                                
                                                  2
                                                  ⁢
                                                  B
                                                
                                              
                                            
                                            +
                                            
                                              
                                                g
                                                22
                                              
                                              ⁢
                                              Γ
                                            
                                          
                                          )
                                        
                                      
                                    
                                    -
                                    
                                      
                                        
                                          g
                                          22
                                        
                                        ⁡
                                        
                                          (
                                          
                                            
                                              
                                                g
                                                
                                                  1
                                                  ⁢
                                                  B
                                                
                                              
                                              ⁢
                                              
                                                n
                                                2
                                              
                                            
                                            +
                                            
                                              
                                                g
                                                12
                                              
                                              ⁢
                                              Γ
                                            
                                          
                                          )
                                        
                                      
                                      ⁢
                                      
                                        (
                                        
                                          
                                            
                                              n
                                              1
                                            
                                            ⁢
                                            
                                              g
                                              
                                                2
                                                ⁢
                                                B
                                              
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              21
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                  
                                  ]
                                
                                .

5. The method according to claim 3, wherein the maximizing of the two-user throughput further comprises:
when ƒl0((X1)h)≦(X1)v and ƒl0((Y2)h)≦(Y2)v, computing the optimal power levels of SU 1 and SU 2, respectively denoted as p*1 and p*2, for maximizing the two-user throughput by





  
    
  
  ⁢
  
    
      (
      
        
          p
          1
          *
        
        ,
        
          p
          2
          *
        
      
      )
    
    =
    
      {
      
        
          
            
              
                arg
                ⁢
                
                  
                
                ⁢
                
                  
                    max
                    
                      
                        (
                        
                          
                            p
                            1
                          
                          ,
                          
                            p
                            2
                          
                        
                        )
                      
                      ∈
                      
                        {
                        
                          
                            Z
                            1
                          
                          ,
                          
                            Z
                            2
                          
                          ,
                          
                            O
                            1
                            
                              I
                              ,
                              2
                            
                          
                          ,
                          
                            O
                            2
                            
                              I
                              ,
                              2
                            
                          
                        
                        }
                      
                    
                  
                  ⁢
                  
                    F
                    ⁡
                    
                      (
                      
                        
                          p
                          1
                        
                        ,
                        
                          p
                          2
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  
                    if
                    ⁢
                    
                      
                    
                    ⁢
                    A
                  
                  ≠
                  0
                
                ,
              
            
          
          
            
              
                arg
                ⁢
                
                  
                
                ⁢
                
                  
                    max
                    
                      
                        (
                        
                          
                            p
                            1
                          
                          ,
                          
                            p
                            2
                          
                        
                        )
                      
                      ∈
                      
                        {
                        
                          
                            Z
                            1
                          
                          ,
                          
                            Z
                            2
                          
                          ,
                          
                            O
                            3
                            
                              I
                              ,
                              2
                            
                          
                        
                        }
                      
                    
                  
                  ⁢
                  
                    F
                    ⁡
                    
                      (
                      
                        
                          p
                          1
                        
                        ,
                        
                          p
                          2
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  if
                  ⁢
                  
                    
                  
                  ⁢
                  A
                
                =
                0
              
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            where
            ⁢
            
              :
            
            ⁢
            
              

            
            ⁢
            
              
            
            ⁢
            
              X
              1
            
          
          =
          
            
              
                (
                
                  
                    
                      
                        
                          γ
                          1
                        
                        ⁢
                        
                          n
                          1
                        
                      
                      +
                      
                        
                          g
                          21
                        
                        ⁢
                        
                          γ
                          1
                        
                        ⁢
                        
                          p
                          2
                          
                            ma
                            ⁢
                            
                              
                            
                            ⁢
                            x
                          
                        
                      
                    
                    
                      g
                      11
                    
                  
                  ,
                  
                    p
                    2
                    
                      ma
                      ⁢
                      
                        
                      
                      ⁢
                      x
                    
                  
                
                )
              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                X
                1
              
            
            =
            
              (
              
                
                  
                    (
                    
                      X
                      1
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      X
                      1
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            Y
            2
          
          =
          
            
              
                (
                
                  
                    p
                    1
                    
                      ma
                      ⁢
                      
                        
                      
                      ⁢
                      x
                    
                  
                  ,
                  
                    
                      
                        
                          n
                          2
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                      +
                      
                        
                          g
                          12
                        
                        ⁢
                        
                          γ
                          2
                        
                        ⁢
                        
                          p
                          1
                          
                            ma
                            ⁢
                            
                              
                            
                            ⁢
                            x
                          
                        
                      
                    
                    
                      g
                      22
                    
                  
                
                )
              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                Y
                2
              
            
            =
            
              (
              
                
                  
                    (
                    
                      Y
                      2
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      Y
                      2
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            Z
            1
          
          =
          
            
              
                (
                
                  
                    
                      
                        
                          g
                          21
                        
                        ⁢
                        
                          γ
                          1
                        
                        ⁢
                        Γ
                      
                      +
                      
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          n
                          1
                        
                        ⁢
                        
                          γ
                          1
                        
                      
                    
                    
                      
                        
                          g
                          11
                        
                        ⁢
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                      
                      +
                      
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          g
                          21
                        
                        ⁢
                        
                          γ
                          1
                        
                      
                    
                  
                  ,
                  
                    
                      
                        
                          g
                          11
                        
                        ⁢
                        Γ
                      
                      -
                      
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          n
                          1
                        
                        ⁢
                        
                          γ
                          1
                        
                      
                    
                    
                      
                        
                          g
                          11
                        
                        ⁢
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                      
                      +
                      
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          g
                          21
                        
                        ⁢
                        
                          γ
                          1
                        
                      
                    
                  
                
                )
              
              ⁢
              
                

              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                Z
                1
              
            
            =
            
              (
              
                
                  
                    (
                    
                      Z
                      1
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      Z
                      1
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            Z
            2
          
          =
          
            
              
                (
                
                  
                    
                      
                        
                          g
                          22
                        
                        ⁢
                        Γ
                      
                      -
                      
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          n
                          2
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                    
                    
                      
                        
                          g
                          22
                        
                        ⁢
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                      
                      +
                      
                        
                          g
                          12
                        
                        ⁢
                        
                          g
                          12
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                    
                  
                  ,
                  
                    
                      
                        
                          g
                          12
                        
                        ⁢
                        
                          γ
                          2
                        
                        ⁢
                        Γ
                      
                      +
                      
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          n
                          2
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                    
                    
                      
                        
                          g
                          22
                        
                        ⁢
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                      
                      +
                      
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          g
                          12
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                    
                  
                
                )
              
              ⁢
              
                

              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                Z
                2
              
            
            =
            
              (
              
                
                  
                    (
                    
                      Z
                      2
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      Z
                      2
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            O
            1
            
              
                I
                ,
                2
              
              ⁢
              
                
              
            
          
          =
          
            {
            
              
                
                  
                    
                      (
                      
                        
                          p
                          1
                          
                            o
                            1
                          
                        
                        ,
                        
                          
                            f
                            
                              l
                              0
                            
                          
                          ⁡
                          
                            (
                            
                              p
                              1
                              
                                o
                                1
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                  
                    
                      
                        if
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            (
                            
                              Z
                              1
                            
                            )
                          
                          h
                        
                      
                      <
                      
                        p
                        1
                        
                          o
                          1
                        
                      
                      <
                      
                        
                          (
                          
                            Z
                            2
                          
                          )
                        
                        h
                      
                    
                  
                
                
                  
                    φ
                  
                  
                    otherwise
                  
                
              
              ;
              
                

              
              ⁢
              
                
              
              ⁢
              
                
                  O
                  2
                  
                    I
                    ,
                    2
                  
                
                =
                
                  {
                  
                    
                      
                        
                          
                            (
                            
                              
                                p
                                1
                                
                                  o
                                  2
                                
                              
                              ,
                              
                                
                                  f
                                  
                                    l
                                    0
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    p
                                    1
                                    
                                      o
                                      2
                                    
                                  
                                  )
                                
                              
                            
                            )
                          
                        
                        
                          
                            
                              if
                              ⁢
                              
                                
                              
                              ⁢
                              
                                
                                  (
                                  
                                    Z
                                    1
                                  
                                  )
                                
                                h
                              
                            
                            <
                            
                              p
                              1
                              
                                o
                                2
                              
                            
                            <
                            
                              
                                (
                                
                                  Z
                                  2
                                
                                )
                              
                              h
                            
                          
                        
                      
                      
                        
                          φ
                        
                        
                          otherwise
                        
                      
                    
                    ;
                    
                      

                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        O
                        3
                        
                          I
                          ,
                          2
                        
                      
                      =
                      
                        {
                        
                          
                            
                              
                                
                                  (
                                  
                                    
                                      p
                                      1
                                      
                                        o
                                        3
                                      
                                    
                                    ,
                                    
                                      
                                        f
                                        
                                          l
                                          0
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          p
                                          1
                                          
                                            o
                                            3
                                          
                                        
                                        )
                                      
                                    
                                  
                                  )
                                
                              
                              
                                
                                  
                                    if
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      
                                        (
                                        
                                          Z
                                          1
                                        
                                        )
                                      
                                      h
                                    
                                  
                                  <
                                  
                                    p
                                    1
                                    
                                      o
                                      3
                                    
                                  
                                  <
                                  
                                    
                                      (
                                      
                                        Z
                                        2
                                      
                                      )
                                    
                                    h
                                  
                                
                              
                            
                            
                              
                                φ
                              
                              
                                otherwise
                              
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              p
                              1
                              
                                o
                                1
                              
                            
                            =
                            
                              
                                
                                  
                                    
                                      -
                                      B
                                    
                                    +
                                    
                                      
                                        
                                          B
                                          2
                                        
                                        -
                                        
                                          4
                                          ⁢
                                          A
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          C
                                        
                                      
                                    
                                  
                                  
                                    2
                                    ⁢
                                    A
                                  
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                A
                              
                              ≠
                              0
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              p
                              1
                              
                                o
                                2
                              
                            
                            =
                            
                              
                                
                                  
                                    
                                      -
                                      B
                                    
                                    -
                                    
                                      
                                        
                                          B
                                          2
                                        
                                        -
                                        
                                          4
                                          ⁢
                                          A
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          C
                                        
                                      
                                    
                                  
                                  
                                    2
                                    ⁢
                                    A
                                  
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                A
                              
                              ≠
                              0
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              p
                              1
                              
                                o
                                3
                              
                            
                            =
                            
                              
                                
                                  
                                    -
                                    C
                                  
                                  B
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                A
                              
                              =
                              0
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            A
                            =
                            
                              
                                
                                  g
                                  
                                    2
                                    ⁢
                                    B
                                  
                                
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  2
                                
                              
                              ⁡
                              
                                [
                                
                                  
                                    
                                      g
                                      11
                                    
                                    ⁢
                                    
                                      g
                                      12
                                    
                                    ⁢
                                    
                                      
                                        g
                                        
                                          2
                                          ⁢
                                          B
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              n
                                              1
                                            
                                            ⁢
                                            
                                              g
                                              
                                                2
                                                ⁢
                                                B
                                              
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              21
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            g
                                            12
                                          
                                          ⁢
                                          
                                            g
                                            
                                              2
                                              ⁢
                                              B
                                            
                                          
                                        
                                        -
                                        
                                          
                                            g
                                            22
                                          
                                          ⁢
                                          
                                            g
                                            
                                              1
                                              ⁢
                                              B
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                  -
                                  
                                    
                                      g
                                      22
                                    
                                    ⁢
                                    
                                      g
                                      21
                                    
                                    ⁢
                                    
                                      
                                        g
                                        
                                          1
                                          ⁢
                                          B
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              g
                                              
                                                1
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            g
                                            21
                                          
                                          ⁢
                                          
                                            g
                                            
                                              1
                                              ⁢
                                              B
                                            
                                          
                                        
                                        -
                                        
                                          
                                            g
                                            11
                                          
                                          ⁢
                                          
                                            g
                                            
                                              2
                                              ⁢
                                              B
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                ]
                              
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            B
                            =
                            
                              
                                
                                  
                                    g
                                    
                                      2
                                      ⁢
                                      B
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          n
                                          1
                                        
                                        ⁢
                                        
                                          g
                                          
                                            2
                                            ⁢
                                            B
                                          
                                        
                                      
                                      +
                                      
                                        
                                          g
                                          21
                                        
                                        ⁢
                                        Γ
                                      
                                    
                                    )
                                  
                                
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  2
                                
                              
                              ×
                              
                                [
                                
                                  
                                    
                                      g
                                      11
                                    
                                    ⁢
                                    
                                      
                                        g
                                        
                                          2
                                          ⁢
                                          B
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            2
                                            ⁢
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            
                                              g
                                              
                                                2
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            
                                              g
                                              22
                                            
                                            ⁢
                                            Γ
                                          
                                          -
                                          
                                            
                                              g
                                              22
                                            
                                            ⁢
                                            
                                              g
                                              
                                                1
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                        
                                        )
                                      
                                    
                                  
                                  -
                                  
                                    
                                      
                                        g
                                        22
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              g
                                              
                                                1
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            g
                                            11
                                          
                                          ⁢
                                          
                                            g
                                            
                                              2
                                              ⁢
                                              B
                                            
                                          
                                        
                                        -
                                        
                                          2
                                          ⁢
                                          
                                            g
                                            21
                                          
                                          ⁢
                                          
                                            g
                                            
                                              1
                                              ⁢
                                              B
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                ]
                              
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            C
                            =
                            
                              
                                
                                  
                                    g
                                    
                                      2
                                      ⁢
                                      B
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          n
                                          1
                                        
                                        ⁢
                                        
                                          g
                                          
                                            2
                                            ⁢
                                            B
                                          
                                        
                                      
                                      +
                                      
                                        
                                          g
                                          21
                                        
                                        ⁢
                                        Γ
                                      
                                    
                                    )
                                  
                                
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  2
                                
                              
                              ×
                              
                                
                                  [
                                  
                                    
                                      
                                        g
                                        11
                                      
                                      ⁢
                                      
                                        g
                                        
                                          2
                                          ⁢
                                          B
                                        
                                      
                                      ⁢
                                      
                                        
                                          n
                                          2
                                        
                                        ⁡
                                        
                                          (
                                          
                                            
                                              
                                                n
                                                2
                                              
                                              ⁢
                                              
                                                g
                                                
                                                  2
                                                  ⁢
                                                  B
                                                
                                              
                                            
                                            +
                                            
                                              
                                                g
                                                22
                                              
                                              ⁢
                                              Γ
                                            
                                          
                                          )
                                        
                                      
                                    
                                    -
                                    
                                      
                                        
                                          g
                                          22
                                        
                                        ⁡
                                        
                                          (
                                          
                                            
                                              
                                                g
                                                
                                                  1
                                                  ⁢
                                                  B
                                                
                                              
                                              ⁢
                                              
                                                n
                                                2
                                              
                                            
                                            +
                                            
                                              
                                                g
                                                12
                                              
                                              ⁢
                                              Γ
                                            
                                          
                                          )
                                        
                                      
                                      ⁢
                                      
                                        (
                                        
                                          
                                            
                                              n
                                              1
                                            
                                            ⁢
                                            
                                              g
                                              
                                                2
                                                ⁢
                                                B
                                              
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              21
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                  
                                  ]
                                
                                .

6. The method according to claim 3, wherein the maximizing of the two-user throughput further comprises:
when ƒl0((X1)h)≧(X1)v and ƒl0((Y2)h)≦(Y2)v, computing the optimal power levels of SU 1 and SU 2, respectively denoted as p*1 and p*2, for maximizing the two-user throughput by





  
    
  
  ⁢
  
    
      (
      
        
          p
          1
          *
        
        ,
        
          p
          2
          *
        
      
      )
    
    =
    
      {
      
        
          
            
              
                arg
                ⁢
                
                  
                
                ⁢
                
                  
                    max
                    
                      
                        (
                        
                          
                            p
                            1
                          
                          ,
                          
                            p
                            2
                          
                        
                        )
                      
                      ∈
                      
                        {
                        
                          
                            X
                            0
                          
                          ,
                          
                            X
                            1
                          
                          ,
                          
                            Z
                            2
                          
                          ,
                          
                            O
                            1
                            
                              I
                              ,
                              3
                            
                          
                          ,
                          
                            O
                            2
                            
                              I
                              ,
                              3
                            
                          
                        
                        }
                      
                    
                  
                  ⁢
                  
                    F
                    ⁡
                    
                      (
                      
                        
                          p
                          1
                        
                        ,
                        
                          p
                          2
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  
                    if
                    ⁢
                    
                      
                    
                    ⁢
                    A
                  
                  ≠
                  0
                
                ,
              
            
          
          
            
              
                arg
                ⁢
                
                  
                
                ⁢
                
                  
                    max
                    
                      
                        (
                        
                          
                            p
                            1
                          
                          ,
                          
                            p
                            2
                          
                        
                        )
                      
                      ∈
                      
                        {
                        
                          
                            X
                            0
                          
                          ,
                          
                            X
                            1
                          
                          ,
                          
                            Z
                            2
                          
                          ,
                          
                            O
                            2
                            
                              I
                              ,
                              3
                            
                          
                        
                        }
                      
                    
                  
                  ⁢
                  
                    F
                    ⁡
                    
                      (
                      
                        
                          p
                          1
                        
                        ,
                        
                          p
                          2
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  if
                  ⁢
                  
                    
                  
                  ⁢
                  A
                
                =
                0
              
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            where
            ⁢
            
              :
            
            ⁢
            
              

            
            ⁢
            
              
            
            ⁢
            
              X
              0
            
          
          =
          
            
              
                (
                
                  
                    
                      Γ
                      -
                      
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          p
                          2
                          
                            ma
                            ⁢
                            
                              
                            
                            ⁢
                            x
                          
                        
                      
                    
                    
                      g
                      
                        1
                        ⁢
                        B
                      
                    
                  
                  ,
                  
                    p
                    2
                    
                      ma
                      ⁢
                      
                        
                      
                      ⁢
                      x
                    
                  
                
                )
              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                X
                0
              
            
            =
            
              (
              
                
                  
                    (
                    
                      X
                      0
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      X
                      0
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            X
            1
          
          =
          
            
              
                (
                
                  
                    
                      
                        
                          γ
                          1
                        
                        ⁢
                        
                          n
                          1
                        
                      
                      +
                      
                        
                          g
                          21
                        
                        ⁢
                        
                          γ
                          1
                        
                        ⁢
                        
                          p
                          2
                          
                            ma
                            ⁢
                            
                              
                            
                            ⁢
                            x
                          
                        
                      
                    
                    
                      g
                      11
                    
                  
                  ,
                  
                    p
                    2
                    
                      ma
                      ⁢
                      
                        
                      
                      ⁢
                      x
                    
                  
                
                )
              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                X
                1
              
            
            =
            
              (
              
                
                  
                    (
                    
                      X
                      1
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      X
                      1
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            Y
            2
          
          =
          
            
              
                (
                
                  
                    p
                    1
                    
                      ma
                      ⁢
                      
                        
                      
                      ⁢
                      x
                    
                  
                  ,
                  
                    
                      
                        
                          n
                          2
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                      +
                      
                        
                          g
                          12
                        
                        ⁢
                        
                          γ
                          2
                        
                        ⁢
                        
                          p
                          1
                          
                            ma
                            ⁢
                            
                              
                            
                            ⁢
                            x
                          
                        
                      
                    
                    
                      g
                      22
                    
                  
                
                )
              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                Y
                2
              
            
            =
            
              (
              
                
                  
                    (
                    
                      Y
                      2
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      Y
                      2
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            Z
            2
          
          =
          
            
              
                (
                
                  
                    
                      
                        
                          g
                          22
                        
                        ⁢
                        Γ
                      
                      -
                      
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          n
                          2
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                    
                    
                      
                        
                          g
                          22
                        
                        ⁢
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                      
                      +
                      
                        
                          g
                          12
                        
                        ⁢
                        
                          g
                          12
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                    
                  
                  ,
                  
                    
                      
                        
                          g
                          12
                        
                        ⁢
                        
                          γ
                          2
                        
                        ⁢
                        Γ
                      
                      +
                      
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          n
                          2
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                    
                    
                      
                        
                          g
                          22
                        
                        ⁢
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                      
                      +
                      
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          g
                          12
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                    
                  
                
                )
              
              ⁢
              
                

              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                Z
                2
              
            
            =
            
              (
              
                
                  
                    (
                    
                      Z
                      2
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      Z
                      2
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            O
            1
            
              
                I
                ,
                3
              
              ⁢
              
                
              
            
          
          =
          
            {
            
              
                
                  
                    
                      (
                      
                        
                          p
                          1
                          
                            o
                            1
                          
                        
                        ,
                        
                          
                            f
                            
                              l
                              0
                            
                          
                          ⁡
                          
                            (
                            
                              p
                              1
                              
                                o
                                1
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                  
                    
                      
                        if
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            (
                            
                              X
                              0
                            
                            )
                          
                          h
                        
                      
                      <
                      
                        p
                        1
                        
                          o
                          1
                        
                      
                      <
                      
                        
                          (
                          
                            Z
                            2
                          
                          )
                        
                        h
                      
                    
                  
                
                
                  
                    φ
                  
                  
                    otherwise
                  
                
              
              ;
              
                

              
              ⁢
              
                
              
              ⁢
              
                
                  O
                  2
                  
                    I
                    ,
                    3
                  
                
                =
                
                  {
                  
                    
                      
                        
                          
                            (
                            
                              
                                p
                                1
                                
                                  o
                                  2
                                
                              
                              ,
                              
                                
                                  f
                                  
                                    l
                                    0
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    p
                                    1
                                    
                                      o
                                      2
                                    
                                  
                                  )
                                
                              
                            
                            )
                          
                        
                        
                          
                            
                              if
                              ⁢
                              
                                
                              
                              ⁢
                              
                                
                                  (
                                  
                                    X
                                    0
                                  
                                  )
                                
                                h
                              
                            
                            <
                            
                              p
                              1
                              
                                o
                                2
                              
                            
                            <
                            
                              
                                (
                                
                                  Z
                                  2
                                
                                )
                              
                              h
                            
                          
                        
                      
                      
                        
                          φ
                        
                        
                          otherwise
                        
                      
                    
                    ;
                    
                      

                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        O
                        3
                        
                          I
                          ,
                          3
                        
                      
                      =
                      
                        {
                        
                          
                            
                              
                                
                                  (
                                  
                                    
                                      p
                                      1
                                      
                                        o
                                        3
                                      
                                    
                                    ,
                                    
                                      
                                        f
                                        
                                          l
                                          0
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          p
                                          1
                                          
                                            o
                                            3
                                          
                                        
                                        )
                                      
                                    
                                  
                                  )
                                
                              
                              
                                
                                  
                                    if
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      
                                        (
                                        
                                          X
                                          0
                                        
                                        )
                                      
                                      h
                                    
                                  
                                  <
                                  
                                    p
                                    1
                                    
                                      o
                                      3
                                    
                                  
                                  <
                                  
                                    
                                      (
                                      
                                        Z
                                        2
                                      
                                      )
                                    
                                    h
                                  
                                
                              
                            
                            
                              
                                φ
                              
                              
                                otherwise
                              
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              p
                              1
                              
                                o
                                1
                              
                            
                            =
                            
                              
                                
                                  
                                    
                                      -
                                      B
                                    
                                    +
                                    
                                      
                                        
                                          B
                                          2
                                        
                                        -
                                        
                                          4
                                          ⁢
                                          A
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          C
                                        
                                      
                                    
                                  
                                  
                                    2
                                    ⁢
                                    A
                                  
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                A
                              
                              ≠
                              0
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              p
                              1
                              
                                o
                                2
                              
                            
                            =
                            
                              
                                
                                  
                                    
                                      -
                                      B
                                    
                                    -
                                    
                                      
                                        
                                          B
                                          2
                                        
                                        -
                                        
                                          4
                                          ⁢
                                          A
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          C
                                        
                                      
                                    
                                  
                                  
                                    2
                                    ⁢
                                    A
                                  
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                A
                              
                              ≠
                              0
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              p
                              1
                              
                                o
                                3
                              
                            
                            =
                            
                              
                                
                                  
                                    -
                                    C
                                  
                                  B
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                A
                              
                              =
                              0
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            A
                            =
                            
                              
                                
                                  g
                                  
                                    2
                                    ⁢
                                    B
                                  
                                
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  2
                                
                              
                              ⁡
                              
                                [
                                
                                  
                                    
                                      g
                                      11
                                    
                                    ⁢
                                    
                                      g
                                      12
                                    
                                    ⁢
                                    
                                      
                                        g
                                        
                                          2
                                          ⁢
                                          B
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              n
                                              1
                                            
                                            ⁢
                                            
                                              g
                                              
                                                2
                                                ⁢
                                                B
                                              
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              21
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            g
                                            12
                                          
                                          ⁢
                                          
                                            g
                                            
                                              2
                                              ⁢
                                              B
                                            
                                          
                                        
                                        -
                                        
                                          
                                            g
                                            22
                                          
                                          ⁢
                                          
                                            g
                                            
                                              1
                                              ⁢
                                              B
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                  -
                                  
                                    
                                      g
                                      22
                                    
                                    ⁢
                                    
                                      g
                                      21
                                    
                                    ⁢
                                    
                                      
                                        g
                                        
                                          1
                                          ⁢
                                          B
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              g
                                              
                                                1
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            g
                                            21
                                          
                                          ⁢
                                          
                                            g
                                            
                                              1
                                              ⁢
                                              B
                                            
                                          
                                        
                                        -
                                        
                                          
                                            g
                                            11
                                          
                                          ⁢
                                          
                                            g
                                            
                                              2
                                              ⁢
                                              B
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                ]
                              
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            B
                            =
                            
                              
                                
                                  
                                    g
                                    
                                      2
                                      ⁢
                                      B
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          n
                                          1
                                        
                                        ⁢
                                        
                                          g
                                          
                                            2
                                            ⁢
                                            B
                                          
                                        
                                      
                                      +
                                      
                                        
                                          g
                                          21
                                        
                                        ⁢
                                        Γ
                                      
                                    
                                    )
                                  
                                
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  2
                                
                              
                              ×
                              
                                [
                                
                                  
                                    
                                      g
                                      11
                                    
                                    ⁢
                                    
                                      
                                        g
                                        
                                          2
                                          ⁢
                                          B
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            2
                                            ⁢
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            
                                              g
                                              
                                                2
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            
                                              g
                                              22
                                            
                                            ⁢
                                            Γ
                                          
                                          -
                                          
                                            
                                              g
                                              22
                                            
                                            ⁢
                                            
                                              g
                                              
                                                1
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                        
                                        )
                                      
                                    
                                  
                                  -
                                  
                                    
                                      
                                        g
                                        22
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              g
                                              
                                                1
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            g
                                            11
                                          
                                          ⁢
                                          
                                            g
                                            
                                              2
                                              ⁢
                                              B
                                            
                                          
                                        
                                        -
                                        
                                          2
                                          ⁢
                                          
                                            g
                                            21
                                          
                                          ⁢
                                          
                                            g
                                            
                                              1
                                              ⁢
                                              B
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                ]
                              
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            C
                            =
                            
                              
                                
                                  
                                    g
                                    
                                      2
                                      ⁢
                                      B
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          n
                                          1
                                        
                                        ⁢
                                        
                                          g
                                          
                                            2
                                            ⁢
                                            B
                                          
                                        
                                      
                                      +
                                      
                                        
                                          g
                                          21
                                        
                                        ⁢
                                        Γ
                                      
                                    
                                    )
                                  
                                
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  2
                                
                              
                              ×
                              
                                
                                  [
                                  
                                    
                                      
                                        g
                                        11
                                      
                                      ⁢
                                      
                                        g
                                        
                                          2
                                          ⁢
                                          B
                                        
                                      
                                      ⁢
                                      
                                        
                                          n
                                          2
                                        
                                        ⁡
                                        
                                          (
                                          
                                            
                                              
                                                n
                                                2
                                              
                                              ⁢
                                              
                                                g
                                                
                                                  2
                                                  ⁢
                                                  B
                                                
                                              
                                            
                                            +
                                            
                                              
                                                g
                                                22
                                              
                                              ⁢
                                              Γ
                                            
                                          
                                          )
                                        
                                      
                                    
                                    -
                                    
                                      
                                        
                                          g
                                          22
                                        
                                        ⁡
                                        
                                          (
                                          
                                            
                                              
                                                g
                                                
                                                  1
                                                  ⁢
                                                  B
                                                
                                              
                                              ⁢
                                              
                                                n
                                                2
                                              
                                            
                                            +
                                            
                                              
                                                g
                                                12
                                              
                                              ⁢
                                              Γ
                                            
                                          
                                          )
                                        
                                      
                                      ⁢
                                      
                                        (
                                        
                                          
                                            
                                              n
                                              1
                                            
                                            ⁢
                                            
                                              g
                                              
                                                2
                                                ⁢
                                                B
                                              
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              21
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                  
                                  ]
                                
                                .

7. The method according to claim 3, wherein the maximizing of the two-user throughput further comprises:
when ƒl0((X1)h)≦(X1)v and ƒl0((Y2)h)≧(Y2)v, computing the optimal power levels of SU 1 and SU 2, respectively denoted as p*1 and p*2, for maximizing the two-user throughput by





  
    
  
  ⁢
  
    
      (
      
        
          p
          1
          *
        
        ,
        
          p
          2
          *
        
      
      )
    
    =
    
      {
      
        
          
            
              
                arg
                ⁢
                
                  
                
                ⁢
                
                  
                    max
                    
                      
                        (
                        
                          
                            p
                            1
                          
                          ,
                          
                            p
                            2
                          
                        
                        )
                      
                      ∈
                      
                        {
                        
                          
                            Y
                            0
                          
                          ,
                          
                            Y
                            2
                          
                          ,
                          
                            O
                            1
                            
                              I
                              ,
                              4
                            
                          
                          ,
                          
                            O
                            2
                            
                              I
                              ,
                              4
                            
                          
                        
                        }
                      
                    
                  
                  ⁢
                  
                    F
                    ⁡
                    
                      (
                      
                        
                          p
                          1
                        
                        ,
                        
                          p
                          2
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  
                    if
                    ⁢
                    
                      
                    
                    ⁢
                    A
                  
                  ≠
                  0
                
                ,
              
            
          
          
            
              
                arg
                ⁢
                
                  
                
                ⁢
                
                  
                    max
                    
                      
                        (
                        
                          
                            p
                            1
                          
                          ,
                          
                            p
                            2
                          
                        
                        )
                      
                      ∈
                      
                        {
                        
                          
                            Y
                            0
                          
                          ,
                          
                            Y
                            2
                          
                          ,
                          
                            O
                            3
                            
                              I
                              ,
                              4
                            
                          
                        
                        }
                      
                    
                  
                  ⁢
                  
                    F
                    ⁡
                    
                      (
                      
                        
                          p
                          1
                        
                        ,
                        
                          p
                          2
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  if
                  ⁢
                  
                    
                  
                  ⁢
                  A
                
                =
                0
              
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            where
            ⁢
            
              :
            
            ⁢
            
              

            
            ⁢
            
              
            
            ⁢
            
              X
              1
            
          
          =
          
            
              
                (
                
                  
                    
                      
                        
                          γ
                          1
                        
                        ⁢
                        
                          n
                          1
                        
                      
                      +
                      
                        
                          g
                          21
                        
                        ⁢
                        
                          γ
                          1
                        
                        ⁢
                        
                          p
                          2
                          
                            ma
                            ⁢
                            
                              
                            
                            ⁢
                            x
                          
                        
                      
                    
                    
                      g
                      11
                    
                  
                  ,
                  
                    p
                    2
                    
                      ma
                      ⁢
                      
                        
                      
                      ⁢
                      x
                    
                  
                
                )
              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                X
                1
              
            
            =
            
              (
              
                
                  
                    (
                    
                      X
                      1
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      X
                      1
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            Y
            0
          
          =
          
            
              
                (
                
                  
                    p
                    1
                    
                      m
                      ⁢
                      
                        
                      
                      ⁢
                      ax
                    
                  
                  ,
                  
                    
                      Γ
                      -
                      
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          p
                          1
                          
                            ma
                            ⁢
                            
                              
                            
                            ⁢
                            x
                          
                        
                      
                    
                    
                      
                        g
                        
                          2
                          ⁢
                          B
                        
                      
                      ⁢
                      
                        
                      
                    
                  
                
                )
              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                Y
                0
              
            
            =
            
              (
              
                
                  
                    (
                    
                      Y
                      0
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      Y
                      0
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            Y
            2
          
          =
          
            
              
                (
                
                  
                    p
                    1
                    
                      ma
                      ⁢
                      
                        
                      
                      ⁢
                      x
                    
                  
                  ,
                  
                    
                      
                        
                          n
                          2
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                      +
                      
                        
                          g
                          12
                        
                        ⁢
                        
                          γ
                          2
                        
                        ⁢
                        
                          p
                          1
                          
                            ma
                            ⁢
                            
                              
                            
                            ⁢
                            x
                          
                        
                      
                    
                    
                      g
                      22
                    
                  
                
                )
              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                Y
                2
              
            
            =
            
              (
              
                
                  
                    (
                    
                      Y
                      2
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      Y
                      2
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            Z
            1
          
          =
          
            
              
                (
                
                  
                    
                      
                        
                          g
                          21
                        
                        ⁢
                        
                          γ
                          1
                        
                        ⁢
                        Γ
                      
                      +
                      
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          n
                          1
                        
                        ⁢
                        
                          γ
                          1
                        
                      
                    
                    
                      
                        
                          g
                          11
                        
                        ⁢
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                      
                      +
                      
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          g
                          21
                        
                        ⁢
                        
                          γ
                          1
                        
                      
                    
                  
                  ,
                  
                    
                      
                        
                          g
                          11
                        
                        ⁢
                        Γ
                      
                      -
                      
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          n
                          1
                        
                        ⁢
                        
                          γ
                          1
                        
                      
                    
                    
                      
                        
                          g
                          11
                        
                        ⁢
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                      
                      +
                      
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          g
                          21
                        
                        ⁢
                        
                          γ
                          1
                        
                      
                    
                  
                
                )
              
              ⁢
              
                

              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                Z
                1
              
            
            =
            
              (
              
                
                  
                    (
                    
                      Z
                      1
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      Z
                      1
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            O
            1
            
              
                I
                ,
                4
              
              ⁢
              
                
              
            
          
          =
          
            {
            
              
                
                  
                    
                      (
                      
                        
                          p
                          1
                          
                            o
                            1
                          
                        
                        ,
                        
                          
                            f
                            
                              l
                              0
                            
                          
                          ⁡
                          
                            (
                            
                              p
                              1
                              
                                o
                                1
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                  
                    
                      
                        if
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            (
                            
                              Z
                              1
                            
                            )
                          
                          h
                        
                      
                      <
                      
                        p
                        1
                        
                          o
                          1
                        
                      
                      <
                      
                        
                          (
                          
                            Y
                            0
                          
                          )
                        
                        h
                      
                    
                  
                
                
                  
                    φ
                  
                  
                    otherwise
                  
                
              
              ;
              
                

              
              ⁢
              
                
              
              ⁢
              
                
                  O
                  2
                  
                    I
                    ,
                    4
                  
                
                =
                
                  {
                  
                    
                      
                        
                          
                            (
                            
                              
                                p
                                1
                                
                                  o
                                  2
                                
                              
                              ,
                              
                                
                                  f
                                  
                                    l
                                    0
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    p
                                    1
                                    
                                      o
                                      2
                                    
                                  
                                  )
                                
                              
                            
                            )
                          
                        
                        
                          
                            
                              if
                              ⁢
                              
                                
                              
                              ⁢
                              
                                
                                  (
                                  
                                    Z
                                    1
                                  
                                  )
                                
                                h
                              
                            
                            <
                            
                              p
                              1
                              
                                o
                                2
                              
                            
                            <
                            
                              
                                (
                                
                                  Y
                                  0
                                
                                )
                              
                              h
                            
                          
                        
                      
                      
                        
                          φ
                        
                        
                          otherwise
                        
                      
                    
                    ;
                    
                      

                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        O
                        3
                        
                          I
                          ,
                          4
                        
                      
                      =
                      
                        {
                        
                          
                            
                              
                                
                                  (
                                  
                                    
                                      p
                                      1
                                      
                                        o
                                        3
                                      
                                    
                                    ,
                                    
                                      
                                        f
                                        
                                          l
                                          0
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          p
                                          1
                                          
                                            o
                                            3
                                          
                                        
                                        )
                                      
                                    
                                  
                                  )
                                
                              
                              
                                
                                  
                                    if
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      
                                        (
                                        
                                          Z
                                          1
                                        
                                        )
                                      
                                      h
                                    
                                  
                                  <
                                  
                                    p
                                    1
                                    
                                      o
                                      3
                                    
                                  
                                  <
                                  
                                    
                                      (
                                      
                                        Y
                                        0
                                      
                                      )
                                    
                                    h
                                  
                                
                              
                            
                            
                              
                                φ
                              
                              
                                otherwise
                              
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              p
                              1
                              
                                o
                                1
                              
                            
                            =
                            
                              
                                
                                  
                                    
                                      -
                                      B
                                    
                                    +
                                    
                                      
                                        
                                          B
                                          2
                                        
                                        -
                                        
                                          4
                                          ⁢
                                          A
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          C
                                        
                                      
                                    
                                  
                                  
                                    2
                                    ⁢
                                    A
                                  
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                A
                              
                              ≠
                              0
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              p
                              1
                              
                                o
                                2
                              
                            
                            =
                            
                              
                                
                                  
                                    
                                      -
                                      B
                                    
                                    -
                                    
                                      
                                        
                                          B
                                          2
                                        
                                        -
                                        
                                          4
                                          ⁢
                                          A
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          C
                                        
                                      
                                    
                                  
                                  
                                    2
                                    ⁢
                                    A
                                  
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                A
                              
                              ≠
                              0
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              p
                              1
                              
                                o
                                3
                              
                            
                            =
                            
                              
                                
                                  
                                    -
                                    C
                                  
                                  B
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                A
                              
                              =
                              0
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            A
                            =
                            
                              
                                
                                  g
                                  
                                    2
                                    ⁢
                                    B
                                  
                                
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  2
                                
                              
                              ⁡
                              
                                [
                                
                                  
                                    
                                      g
                                      11
                                    
                                    ⁢
                                    
                                      g
                                      12
                                    
                                    ⁢
                                    
                                      
                                        g
                                        
                                          2
                                          ⁢
                                          B
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              n
                                              1
                                            
                                            ⁢
                                            
                                              g
                                              
                                                2
                                                ⁢
                                                B
                                              
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              21
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            g
                                            12
                                          
                                          ⁢
                                          
                                            g
                                            
                                              2
                                              ⁢
                                              B
                                            
                                          
                                        
                                        -
                                        
                                          
                                            g
                                            22
                                          
                                          ⁢
                                          
                                            g
                                            
                                              1
                                              ⁢
                                              B
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                  -
                                  
                                    
                                      g
                                      22
                                    
                                    ⁢
                                    
                                      g
                                      21
                                    
                                    ⁢
                                    
                                      
                                        g
                                        
                                          1
                                          ⁢
                                          B
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              g
                                              
                                                1
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            g
                                            21
                                          
                                          ⁢
                                          
                                            g
                                            
                                              1
                                              ⁢
                                              B
                                            
                                          
                                        
                                        -
                                        
                                          
                                            g
                                            11
                                          
                                          ⁢
                                          
                                            g
                                            
                                              2
                                              ⁢
                                              B
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                ]
                              
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            B
                            =
                            
                              
                                
                                  
                                    g
                                    
                                      2
                                      ⁢
                                      B
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          n
                                          1
                                        
                                        ⁢
                                        
                                          g
                                          
                                            2
                                            ⁢
                                            B
                                          
                                        
                                      
                                      +
                                      
                                        
                                          g
                                          21
                                        
                                        ⁢
                                        Γ
                                      
                                    
                                    )
                                  
                                
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  2
                                
                              
                              ×
                              
                                [
                                
                                  
                                    
                                      g
                                      11
                                    
                                    ⁢
                                    
                                      
                                        g
                                        
                                          2
                                          ⁢
                                          B
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            2
                                            ⁢
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            
                                              g
                                              
                                                2
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            
                                              g
                                              22
                                            
                                            ⁢
                                            Γ
                                          
                                          -
                                          
                                            
                                              g
                                              22
                                            
                                            ⁢
                                            
                                              g
                                              
                                                1
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                        
                                        )
                                      
                                    
                                  
                                  -
                                  
                                    
                                      
                                        g
                                        22
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              g
                                              
                                                1
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            g
                                            11
                                          
                                          ⁢
                                          
                                            g
                                            
                                              2
                                              ⁢
                                              B
                                            
                                          
                                        
                                        -
                                        
                                          2
                                          ⁢
                                          
                                            g
                                            21
                                          
                                          ⁢
                                          
                                            g
                                            
                                              1
                                              ⁢
                                              B
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                ]
                              
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            C
                            =
                            
                              
                                
                                  
                                    g
                                    
                                      2
                                      ⁢
                                      B
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          n
                                          1
                                        
                                        ⁢
                                        
                                          g
                                          
                                            2
                                            ⁢
                                            B
                                          
                                        
                                      
                                      +
                                      
                                        
                                          g
                                          21
                                        
                                        ⁢
                                        Γ
                                      
                                    
                                    )
                                  
                                
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  2
                                
                              
                              ×
                              
                                
                                  [
                                  
                                    
                                      
                                        g
                                        11
                                      
                                      ⁢
                                      
                                        g
                                        
                                          2
                                          ⁢
                                          B
                                        
                                      
                                      ⁢
                                      
                                        
                                          n
                                          2
                                        
                                        ⁡
                                        
                                          (
                                          
                                            
                                              
                                                n
                                                2
                                              
                                              ⁢
                                              
                                                g
                                                
                                                  2
                                                  ⁢
                                                  B
                                                
                                              
                                            
                                            +
                                            
                                              
                                                g
                                                22
                                              
                                              ⁢
                                              Γ
                                            
                                          
                                          )
                                        
                                      
                                    
                                    -
                                    
                                      
                                        
                                          g
                                          22
                                        
                                        ⁡
                                        
                                          (
                                          
                                            
                                              
                                                g
                                                
                                                  1
                                                  ⁢
                                                  B
                                                
                                              
                                              ⁢
                                              
                                                n
                                                2
                                              
                                            
                                            +
                                            
                                              
                                                g
                                                12
                                              
                                              ⁢
                                              Γ
                                            
                                          
                                          )
                                        
                                      
                                      ⁢
                                      
                                        (
                                        
                                          
                                            
                                              n
                                              1
                                            
                                            ⁢
                                            
                                              g
                                              
                                                2
                                                ⁢
                                                B
                                              
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              21
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                  
                                  ]
                                
                                .

8. The method according to claim 3, wherein the maximizing of the two-user throughput further comprises:
when ƒl0((Y1)h)≧(Y1)v and ƒl0(Y2)h)≦(Y2)v, computing the optimal power levels of SU 1 and SU 2, respectively denoted as p*1 and p*2, for maximizing the two-user throughput by





  
    
  
  ⁢
  
    
      (
      
        
          p
          1
          *
        
        ,
        
          p
          2
          *
        
      
      )
    
    =
    
      {
      
        
          
            
              
                arg
                ⁢
                
                  
                
                ⁢
                
                  
                    max
                    
                      
                        (
                        
                          
                            p
                            1
                          
                          ,
                          
                            p
                            2
                          
                        
                        )
                      
                      ∈
                      
                        {
                        
                          
                            Y
                            0
                          
                          ,
                          
                            Y
                            2
                          
                          ,
                          
                            Z
                            1
                          
                          ,
                          
                            O
                            1
                            
                              I
                              ,
                              4
                            
                          
                          ,
                          
                            O
                            2
                            
                              I
                              ,
                              4
                            
                          
                        
                        }
                      
                    
                  
                  ⁢
                  
                    F
                    ⁡
                    
                      (
                      
                        
                          p
                          1
                        
                        ,
                        
                          p
                          2
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  
                    if
                    ⁢
                    
                      
                    
                    ⁢
                    A
                  
                  ≠
                  0
                
                ,
              
            
          
          
            
              
                arg
                ⁢
                
                  
                
                ⁢
                
                  
                    max
                    
                      
                        (
                        
                          
                            p
                            1
                          
                          ,
                          
                            p
                            2
                          
                        
                        )
                      
                      ∈
                      
                        {
                        
                          
                            Y
                            0
                          
                          ,
                          
                            Y
                            2
                          
                          ,
                          
                            Z
                            1
                          
                          ,
                          
                            O
                            3
                            
                              I
                              ,
                              4
                            
                          
                        
                        }
                      
                    
                  
                  ⁢
                  
                    F
                    ⁡
                    
                      (
                      
                        
                          p
                          1
                        
                        ,
                        
                          p
                          2
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  if
                  ⁢
                  
                    
                  
                  ⁢
                  A
                
                =
                0
              
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            where
            ⁢
            
              :
            
            ⁢
            
              

            
            ⁢
            
              
            
            ⁢
            
              Y
              0
            
          
          =
          
            
              
                (
                
                  
                    p
                    1
                    
                      m
                      ⁢
                      
                        
                      
                      ⁢
                      ax
                    
                  
                  ,
                  
                    
                      Γ
                      -
                      
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          p
                          1
                          
                            ma
                            ⁢
                            
                              
                            
                            ⁢
                            x
                          
                        
                      
                    
                    
                      
                        g
                        
                          2
                          ⁢
                          B
                        
                      
                      ⁢
                      
                        
                      
                    
                  
                
                )
              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                Y
                0
              
            
            =
            
              (
              
                
                  
                    (
                    
                      Y
                      0
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      Y
                      0
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            Y
            1
          
          =
          
            
              
                (
                
                  
                    p
                    1
                    
                      m
                      ⁢
                      
                        
                      
                      ⁢
                      ax
                    
                  
                  ,
                  
                    
                      
                        
                          g
                          11
                        
                        ⁢
                        
                          p
                          1
                          
                            ma
                            ⁢
                            
                              
                            
                            ⁢
                            x
                          
                        
                      
                      -
                      
                        
                          γ
                          1
                        
                        ⁢
                        
                          n
                          1
                        
                      
                    
                    
                      
                        γ
                        1
                      
                      ⁢
                      
                        g
                        
                          21
                          ⁢
                          
                            
                          
                        
                      
                    
                  
                
                )
              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                Y
                1
              
            
            =
            
              (
              
                
                  
                    (
                    
                      Y
                      1
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      Y
                      1
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            Y
            2
          
          =
          
            
              
                (
                
                  
                    p
                    1
                    
                      ma
                      ⁢
                      
                        
                      
                      ⁢
                      x
                    
                  
                  ,
                  
                    
                      
                        
                          n
                          2
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                      +
                      
                        
                          g
                          12
                        
                        ⁢
                        
                          γ
                          2
                        
                        ⁢
                        
                          p
                          1
                          
                            ma
                            ⁢
                            
                              
                            
                            ⁢
                            x
                          
                        
                      
                    
                    
                      g
                      22
                    
                  
                
                )
              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                Y
                2
              
            
            =
            
              (
              
                
                  
                    (
                    
                      Y
                      2
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      Y
                      2
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            Z
            1
          
          =
          
            
              
                (
                
                  
                    
                      
                        
                          g
                          21
                        
                        ⁢
                        
                          γ
                          1
                        
                        ⁢
                        Γ
                      
                      +
                      
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          n
                          1
                        
                        ⁢
                        
                          γ
                          1
                        
                      
                    
                    
                      
                        
                          g
                          11
                        
                        ⁢
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                      
                      +
                      
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          g
                          21
                        
                        ⁢
                        
                          γ
                          1
                        
                      
                    
                  
                  ,
                  
                    
                      
                        
                          g
                          11
                        
                        ⁢
                        Γ
                      
                      -
                      
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          n
                          1
                        
                        ⁢
                        
                          γ
                          1
                        
                      
                    
                    
                      
                        
                          g
                          11
                        
                        ⁢
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                      
                      +
                      
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          g
                          21
                        
                        ⁢
                        
                          γ
                          1
                        
                      
                    
                  
                
                )
              
              ⁢
              
                

              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                Z
                1
              
            
            =
            
              (
              
                
                  
                    (
                    
                      Z
                      1
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      Z
                      1
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            O
            1
            
              I
              ,
              4
            
          
          =
          
            {
            
              
                
                  
                    
                      (
                      
                        
                          p
                          1
                          
                            o
                            1
                          
                        
                        ,
                        
                          
                            f
                            
                              l
                              0
                            
                          
                          ⁡
                          
                            (
                            
                              p
                              1
                              
                                o
                                1
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                  
                    
                      
                        if
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            (
                            
                              Z
                              1
                            
                            )
                          
                          h
                        
                      
                      <
                      
                        p
                        1
                        
                          o
                          1
                        
                      
                      <
                      
                        
                          (
                          
                            Y
                            0
                          
                          )
                        
                        h
                      
                    
                  
                
                
                  
                    φ
                  
                  
                    otherwise
                  
                
              
              ;
              
                

              
              ⁢
              
                
              
              ⁢
              
                
                  O
                  2
                  
                    I
                    ,
                    4
                  
                
                =
                
                  {
                  
                    
                      
                        
                          
                            (
                            
                              
                                p
                                1
                                
                                  o
                                  2
                                
                              
                              ,
                              
                                
                                  f
                                  
                                    l
                                    0
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    p
                                    1
                                    
                                      o
                                      2
                                    
                                  
                                  )
                                
                              
                            
                            )
                          
                        
                        
                          
                            
                              if
                              ⁢
                              
                                
                              
                              ⁢
                              
                                
                                  (
                                  
                                    Z
                                    1
                                  
                                  )
                                
                                h
                              
                            
                            <
                            
                              p
                              1
                              
                                o
                                2
                              
                            
                            <
                            
                              
                                (
                                
                                  Y
                                  0
                                
                                )
                              
                              h
                            
                          
                        
                      
                      
                        
                          φ
                        
                        
                          otherwise
                        
                      
                    
                    ;
                    
                      

                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        O
                        3
                        
                          I
                          ,
                          4
                        
                      
                      =
                      
                        {
                        
                          
                            
                              
                                
                                  (
                                  
                                    
                                      p
                                      1
                                      
                                        o
                                        3
                                      
                                    
                                    ,
                                    
                                      
                                        f
                                        
                                          l
                                          0
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          p
                                          1
                                          
                                            o
                                            3
                                          
                                        
                                        )
                                      
                                    
                                  
                                  )
                                
                              
                              
                                
                                  
                                    if
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      
                                        (
                                        
                                          Z
                                          1
                                        
                                        )
                                      
                                      h
                                    
                                  
                                  <
                                  
                                    p
                                    1
                                    
                                      o
                                      3
                                    
                                  
                                  <
                                  
                                    
                                      (
                                      
                                        Y
                                        0
                                      
                                      )
                                    
                                    h
                                  
                                
                              
                            
                            
                              
                                φ
                              
                              
                                otherwise
                              
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              p
                              1
                              
                                o
                                1
                              
                            
                            =
                            
                              
                                
                                  
                                    
                                      -
                                      B
                                    
                                    +
                                    
                                      
                                        
                                          B
                                          2
                                        
                                        -
                                        
                                          4
                                          ⁢
                                          A
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          C
                                        
                                      
                                    
                                  
                                  
                                    2
                                    ⁢
                                    A
                                  
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                A
                              
                              ≠
                              0
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              p
                              1
                              
                                o
                                2
                              
                            
                            =
                            
                              
                                
                                  
                                    
                                      -
                                      B
                                    
                                    -
                                    
                                      
                                        
                                          B
                                          2
                                        
                                        -
                                        
                                          4
                                          ⁢
                                          A
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          C
                                        
                                      
                                    
                                  
                                  
                                    2
                                    ⁢
                                    A
                                  
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                A
                              
                              ≠
                              0
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              p
                              1
                              
                                o
                                3
                              
                            
                            =
                            
                              
                                
                                  
                                    -
                                    C
                                  
                                  B
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                A
                              
                              =
                              0
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            A
                            =
                            
                              
                                
                                  g
                                  
                                    2
                                    ⁢
                                    B
                                  
                                
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  2
                                
                              
                              ⁡
                              
                                [
                                
                                  
                                    
                                      g
                                      11
                                    
                                    ⁢
                                    
                                      g
                                      12
                                    
                                    ⁢
                                    
                                      
                                        g
                                        
                                          2
                                          ⁢
                                          B
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              n
                                              1
                                            
                                            ⁢
                                            
                                              g
                                              
                                                2
                                                ⁢
                                                B
                                              
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              21
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            g
                                            12
                                          
                                          ⁢
                                          
                                            g
                                            
                                              2
                                              ⁢
                                              B
                                            
                                          
                                        
                                        -
                                        
                                          
                                            g
                                            22
                                          
                                          ⁢
                                          
                                            g
                                            
                                              1
                                              ⁢
                                              B
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                  -
                                  
                                    
                                      g
                                      22
                                    
                                    ⁢
                                    
                                      g
                                      21
                                    
                                    ⁢
                                    
                                      
                                        g
                                        
                                          1
                                          ⁢
                                          B
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              g
                                              
                                                1
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            g
                                            21
                                          
                                          ⁢
                                          
                                            g
                                            
                                              1
                                              ⁢
                                              B
                                            
                                          
                                        
                                        -
                                        
                                          
                                            g
                                            11
                                          
                                          ⁢
                                          
                                            g
                                            
                                              2
                                              ⁢
                                              B
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                ]
                              
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            B
                            =
                            
                              
                                
                                  
                                    g
                                    
                                      2
                                      ⁢
                                      B
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          n
                                          1
                                        
                                        ⁢
                                        
                                          g
                                          
                                            2
                                            ⁢
                                            B
                                          
                                        
                                      
                                      +
                                      
                                        
                                          g
                                          21
                                        
                                        ⁢
                                        Γ
                                      
                                    
                                    )
                                  
                                
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  2
                                
                              
                              ×
                              
                                [
                                
                                  
                                    
                                      g
                                      11
                                    
                                    ⁢
                                    
                                      
                                        g
                                        
                                          2
                                          ⁢
                                          B
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            2
                                            ⁢
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            
                                              g
                                              
                                                2
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            
                                              g
                                              22
                                            
                                            ⁢
                                            Γ
                                          
                                          -
                                          
                                            
                                              g
                                              22
                                            
                                            ⁢
                                            
                                              g
                                              
                                                1
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                        
                                        )
                                      
                                    
                                  
                                  -
                                  
                                    
                                      
                                        g
                                        22
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              g
                                              
                                                1
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            g
                                            11
                                          
                                          ⁢
                                          
                                            g
                                            
                                              2
                                              ⁢
                                              B
                                            
                                          
                                        
                                        -
                                        
                                          2
                                          ⁢
                                          
                                            g
                                            21
                                          
                                          ⁢
                                          
                                            g
                                            
                                              1
                                              ⁢
                                              B
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                ]
                              
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            C
                            =
                            
                              
                                
                                  
                                    g
                                    
                                      2
                                      ⁢
                                      B
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          n
                                          1
                                        
                                        ⁢
                                        
                                          g
                                          
                                            2
                                            ⁢
                                            B
                                          
                                        
                                      
                                      +
                                      
                                        
                                          g
                                          21
                                        
                                        ⁢
                                        Γ
                                      
                                    
                                    )
                                  
                                
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  2
                                
                              
                              ×
                              
                                
                                  [
                                  
                                    
                                      
                                        g
                                        11
                                      
                                      ⁢
                                      
                                        g
                                        
                                          2
                                          ⁢
                                          B
                                        
                                      
                                      ⁢
                                      
                                        
                                          n
                                          2
                                        
                                        ⁡
                                        
                                          (
                                          
                                            
                                              
                                                n
                                                2
                                              
                                              ⁢
                                              
                                                g
                                                
                                                  2
                                                  ⁢
                                                  B
                                                
                                              
                                            
                                            +
                                            
                                              
                                                g
                                                22
                                              
                                              ⁢
                                              Γ
                                            
                                          
                                          )
                                        
                                      
                                    
                                    -
                                    
                                      
                                        
                                          g
                                          22
                                        
                                        ⁡
                                        
                                          (
                                          
                                            
                                              
                                                g
                                                
                                                  1
                                                  ⁢
                                                  B
                                                
                                              
                                              ⁢
                                              
                                                n
                                                2
                                              
                                            
                                            +
                                            
                                              
                                                g
                                                12
                                              
                                              ⁢
                                              Γ
                                            
                                          
                                          )
                                        
                                      
                                      ⁢
                                      
                                        (
                                        
                                          
                                            
                                              n
                                              1
                                            
                                            ⁢
                                            
                                              g
                                              
                                                2
                                                ⁢
                                                B
                                              
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              21
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                  
                                  ]
                                
                                .

9. The method according to claim 3, wherein the maximizing of the two-user throughput further comprises:
when ƒl0((Y1)h)≦(Y1)v and ƒl0((Y2)h)≦(Y2)v, computing the optimal power levels of SU 1 and SU 2, respectively denoted as p*1 and p*2, for maximizing the two-user throughput by





  
    
  
  ⁢
  
    
      (
      
        
          p
          1
          *
        
        ,
        
          p
          2
          *
        
      
      )
    
    =
    
      {
      
        
          
            
              
                arg
                ⁢
                
                  
                
                ⁢
                
                  
                    max
                    
                      
                        (
                        
                          
                            p
                            1
                          
                          ,
                          
                            p
                            2
                          
                        
                        )
                      
                      ∈
                      
                        {
                        
                          
                            Z
                            1
                          
                          ,
                          
                            Z
                            2
                          
                          ,
                          
                            O
                            1
                            
                              I
                              ,
                              2
                            
                          
                          ,
                          
                            O
                            2
                            
                              I
                              ,
                              2
                            
                          
                        
                        }
                      
                    
                  
                  ⁢
                  
                    F
                    ⁡
                    
                      (
                      
                        
                          p
                          1
                        
                        ,
                        
                          p
                          2
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  
                    if
                    ⁢
                    
                      
                    
                    ⁢
                    A
                  
                  ≠
                  0
                
                ,
              
            
          
          
            
              
                arg
                ⁢
                
                  
                
                ⁢
                
                  
                    max
                    
                      
                        (
                        
                          
                            p
                            1
                          
                          ,
                          
                            p
                            2
                          
                        
                        )
                      
                      ∈
                      
                        {
                        
                          
                            Z
                            1
                          
                          ,
                          
                            Z
                            2
                          
                          ,
                          
                            O
                            3
                            
                              I
                              ,
                              2
                            
                          
                        
                        }
                      
                    
                  
                  ⁢
                  
                    F
                    ⁡
                    
                      (
                      
                        
                          p
                          1
                        
                        ,
                        
                          p
                          2
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  if
                  ⁢
                  
                    
                  
                  ⁢
                  A
                
                =
                0
              
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            where
            ⁢
            
              :
            
            ⁢
            
              

            
            ⁢
            
              
            
            ⁢
            
              Y
              1
            
          
          =
          
            
              
                (
                
                  
                    p
                    1
                    
                      m
                      ⁢
                      
                        
                      
                      ⁢
                      ax
                    
                  
                  ,
                  
                    
                      
                        
                          g
                          11
                        
                        ⁢
                        
                          p
                          1
                          
                            ma
                            ⁢
                            
                              
                            
                            ⁢
                            x
                          
                        
                      
                      -
                      
                        
                          γ
                          1
                        
                        ⁢
                        
                          n
                          1
                        
                      
                    
                    
                      
                        γ
                        1
                      
                      ⁢
                      
                        g
                        21
                      
                    
                  
                
                )
              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                Y
                1
              
            
            =
            
              (
              
                
                  
                    (
                    
                      Y
                      1
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      Y
                      1
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            Y
            2
          
          =
          
            
              
                (
                
                  
                    p
                    1
                    
                      ma
                      ⁢
                      
                        
                      
                      ⁢
                      x
                    
                  
                  ,
                  
                    
                      
                        
                          n
                          2
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                      +
                      
                        
                          g
                          12
                        
                        ⁢
                        
                          γ
                          2
                        
                        ⁢
                        
                          p
                          1
                          
                            ma
                            ⁢
                            
                              
                            
                            ⁢
                            x
                          
                        
                      
                    
                    
                      g
                      22
                    
                  
                
                )
              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                Y
                2
              
            
            =
            
              (
              
                
                  
                    (
                    
                      Y
                      2
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      Y
                      2
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            Z
            1
          
          =
          
            
              
                (
                
                  
                    
                      
                        
                          g
                          21
                        
                        ⁢
                        
                          γ
                          1
                        
                        ⁢
                        Γ
                      
                      +
                      
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          n
                          1
                        
                        ⁢
                        
                          γ
                          1
                        
                      
                    
                    
                      
                        
                          g
                          11
                        
                        ⁢
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                      
                      +
                      
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          g
                          21
                        
                        ⁢
                        
                          γ
                          1
                        
                      
                    
                  
                  ,
                  
                    
                      
                        
                          g
                          11
                        
                        ⁢
                        Γ
                      
                      -
                      
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          n
                          1
                        
                        ⁢
                        
                          γ
                          1
                        
                      
                    
                    
                      
                        
                          g
                          11
                        
                        ⁢
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                      
                      +
                      
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          g
                          21
                        
                        ⁢
                        
                          γ
                          1
                        
                      
                    
                  
                
                )
              
              ⁢
              
                

              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                Z
                1
              
            
            =
            
              (
              
                
                  
                    (
                    
                      Z
                      1
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      Z
                      1
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            Z
            2
          
          =
          
            
              
                (
                
                  
                    
                      
                        
                          g
                          22
                        
                        ⁢
                        Γ
                      
                      -
                      
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          n
                          2
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                    
                    
                      
                        
                          g
                          22
                        
                        ⁢
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                      
                      +
                      
                        
                          g
                          12
                        
                        ⁢
                        
                          g
                          12
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                    
                  
                  ,
                  
                    
                      
                        
                          g
                          12
                        
                        ⁢
                        
                          γ
                          2
                        
                        ⁢
                        Γ
                      
                      +
                      
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          n
                          2
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                    
                    
                      
                        
                          g
                          22
                        
                        ⁢
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                      
                      +
                      
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          g
                          12
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                    
                  
                
                )
              
              ⁢
              
                

              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                Z
                2
              
            
            =
            
              (
              
                
                  
                    (
                    
                      Z
                      2
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      Z
                      2
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            O
            1
            
              
                I
                ,
                2
              
              ⁢
              
                
              
            
          
          =
          
            {
            
              
                
                  
                    
                      (
                      
                        
                          p
                          1
                          
                            o
                            1
                          
                        
                        ,
                        
                          
                            f
                            
                              l
                              0
                            
                          
                          ⁡
                          
                            (
                            
                              p
                              1
                              
                                o
                                1
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                  
                    
                      
                        if
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            (
                            
                              Z
                              1
                            
                            )
                          
                          h
                        
                      
                      <
                      
                        p
                        1
                        
                          o
                          1
                        
                      
                      <
                      
                        
                          (
                          
                            Z
                            2
                          
                          )
                        
                        h
                      
                    
                  
                
                
                  
                    φ
                  
                  
                    otherwise
                  
                
              
              ;
              
                

              
              ⁢
              
                
              
              ⁢
              
                
                  O
                  2
                  
                    I
                    ,
                    2
                  
                
                =
                
                  {
                  
                    
                      
                        
                          
                            (
                            
                              
                                p
                                1
                                
                                  o
                                  2
                                
                              
                              ,
                              
                                
                                  f
                                  
                                    l
                                    0
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    p
                                    1
                                    
                                      o
                                      2
                                    
                                  
                                  )
                                
                              
                            
                            )
                          
                        
                        
                          
                            
                              if
                              ⁢
                              
                                
                              
                              ⁢
                              
                                
                                  (
                                  
                                    Z
                                    1
                                  
                                  )
                                
                                h
                              
                            
                            <
                            
                              p
                              1
                              
                                o
                                2
                              
                            
                            <
                            
                              
                                (
                                
                                  Z
                                  2
                                
                                )
                              
                              h
                            
                          
                        
                      
                      
                        
                          φ
                        
                        
                          otherwise
                        
                      
                    
                    ;
                    
                      

                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        O
                        3
                        
                          I
                          ,
                          2
                        
                      
                      =
                      
                        {
                        
                          
                            
                              
                                
                                  (
                                  
                                    
                                      p
                                      1
                                      
                                        o
                                        3
                                      
                                    
                                    ,
                                    
                                      
                                        f
                                        
                                          l
                                          0
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          p
                                          1
                                          
                                            o
                                            3
                                          
                                        
                                        )
                                      
                                    
                                  
                                  )
                                
                              
                              
                                
                                  
                                    if
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      
                                        (
                                        
                                          Z
                                          1
                                        
                                        )
                                      
                                      h
                                    
                                  
                                  <
                                  
                                    p
                                    1
                                    
                                      o
                                      3
                                    
                                  
                                  <
                                  
                                    
                                      (
                                      
                                        Z
                                        2
                                      
                                      )
                                    
                                    h
                                  
                                
                              
                            
                            
                              
                                φ
                              
                              
                                otherwise
                              
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              p
                              1
                              
                                o
                                1
                              
                            
                            =
                            
                              
                                
                                  
                                    
                                      -
                                      B
                                    
                                    +
                                    
                                      
                                        
                                          B
                                          2
                                        
                                        -
                                        
                                          4
                                          ⁢
                                          A
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          C
                                        
                                      
                                    
                                  
                                  
                                    2
                                    ⁢
                                    A
                                  
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                A
                              
                              ≠
                              0
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              p
                              1
                              
                                o
                                2
                              
                            
                            =
                            
                              
                                
                                  
                                    
                                      -
                                      B
                                    
                                    -
                                    
                                      
                                        
                                          B
                                          2
                                        
                                        -
                                        
                                          4
                                          ⁢
                                          A
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          C
                                        
                                      
                                    
                                  
                                  
                                    2
                                    ⁢
                                    A
                                  
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                A
                              
                              ≠
                              0
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              p
                              1
                              
                                o
                                3
                              
                            
                            =
                            
                              
                                
                                  
                                    -
                                    C
                                  
                                  B
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                A
                              
                              =
                              0
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            A
                            =
                            
                              
                                
                                  g
                                  
                                    2
                                    ⁢
                                    B
                                  
                                
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  2
                                
                              
                              ⁡
                              
                                [
                                
                                  
                                    
                                      g
                                      11
                                    
                                    ⁢
                                    
                                      g
                                      12
                                    
                                    ⁢
                                    
                                      
                                        g
                                        
                                          2
                                          ⁢
                                          B
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              n
                                              1
                                            
                                            ⁢
                                            
                                              g
                                              
                                                2
                                                ⁢
                                                B
                                              
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              21
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            g
                                            12
                                          
                                          ⁢
                                          
                                            g
                                            
                                              2
                                              ⁢
                                              B
                                            
                                          
                                        
                                        -
                                        
                                          
                                            g
                                            22
                                          
                                          ⁢
                                          
                                            g
                                            
                                              1
                                              ⁢
                                              B
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                  -
                                  
                                    
                                      g
                                      22
                                    
                                    ⁢
                                    
                                      g
                                      21
                                    
                                    ⁢
                                    
                                      
                                        g
                                        
                                          1
                                          ⁢
                                          B
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              g
                                              
                                                1
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            g
                                            21
                                          
                                          ⁢
                                          
                                            g
                                            
                                              1
                                              ⁢
                                              B
                                            
                                          
                                        
                                        -
                                        
                                          
                                            g
                                            11
                                          
                                          ⁢
                                          
                                            g
                                            
                                              2
                                              ⁢
                                              B
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                ]
                              
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            B
                            =
                            
                              
                                
                                  
                                    g
                                    
                                      2
                                      ⁢
                                      B
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          n
                                          1
                                        
                                        ⁢
                                        
                                          g
                                          
                                            2
                                            ⁢
                                            B
                                          
                                        
                                      
                                      +
                                      
                                        
                                          g
                                          21
                                        
                                        ⁢
                                        Γ
                                      
                                    
                                    )
                                  
                                
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  2
                                
                              
                              ×
                              
                                [
                                
                                  
                                    
                                      g
                                      11
                                    
                                    ⁢
                                    
                                      
                                        g
                                        
                                          2
                                          ⁢
                                          B
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            2
                                            ⁢
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            
                                              g
                                              
                                                2
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            
                                              g
                                              22
                                            
                                            ⁢
                                            Γ
                                          
                                          -
                                          
                                            
                                              g
                                              22
                                            
                                            ⁢
                                            
                                              g
                                              
                                                1
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                        
                                        )
                                      
                                    
                                  
                                  -
                                  
                                    
                                      
                                        g
                                        22
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              g
                                              
                                                1
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            g
                                            11
                                          
                                          ⁢
                                          
                                            g
                                            
                                              2
                                              ⁢
                                              B
                                            
                                          
                                        
                                        -
                                        
                                          2
                                          ⁢
                                          
                                            g
                                            21
                                          
                                          ⁢
                                          
                                            g
                                            
                                              1
                                              ⁢
                                              B
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                ]
                              
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            C
                            =
                            
                              
                                
                                  
                                    g
                                    
                                      2
                                      ⁢
                                      B
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          n
                                          1
                                        
                                        ⁢
                                        
                                          g
                                          
                                            2
                                            ⁢
                                            B
                                          
                                        
                                      
                                      +
                                      
                                        
                                          g
                                          21
                                        
                                        ⁢
                                        Γ
                                      
                                    
                                    )
                                  
                                
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  2
                                
                              
                              ×
                              
                                
                                  [
                                  
                                    
                                      
                                        g
                                        11
                                      
                                      ⁢
                                      
                                        g
                                        
                                          2
                                          ⁢
                                          B
                                        
                                      
                                      ⁢
                                      
                                        
                                          n
                                          2
                                        
                                        ⁡
                                        
                                          (
                                          
                                            
                                              
                                                n
                                                2
                                              
                                              ⁢
                                              
                                                g
                                                
                                                  2
                                                  ⁢
                                                  B
                                                
                                              
                                            
                                            +
                                            
                                              
                                                g
                                                22
                                              
                                              ⁢
                                              Γ
                                            
                                          
                                          )
                                        
                                      
                                    
                                    -
                                    
                                      
                                        
                                          g
                                          22
                                        
                                        ⁡
                                        
                                          (
                                          
                                            
                                              
                                                g
                                                
                                                  1
                                                  ⁢
                                                  B
                                                
                                              
                                              ⁢
                                              
                                                n
                                                2
                                              
                                            
                                            +
                                            
                                              
                                                g
                                                12
                                              
                                              ⁢
                                              Γ
                                            
                                          
                                          )
                                        
                                      
                                      ⁢
                                      
                                        (
                                        
                                          
                                            
                                              n
                                              1
                                            
                                            ⁢
                                            
                                              g
                                              
                                                2
                                                ⁢
                                                B
                                              
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              21
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                  
                                  ]
                                
                                .

10. The method according to claim 3, wherein the maximizing of the two-user throughput further comprises:
when ƒl0((Y1)h)≧(Y1)v and ƒl0((Y2)h)≧(Y2)v, computing the optimal power levels of SU 1 and SU 2, respectively denoted as p*1 and p*2, for maximizing the two-user throughput by




  
    
      
        (
        
          
            p
            1
            *
          
          ,
          
            p
            2
            *
          
        
        )
      
      =
      
        arg
        ⁢
        
          
        
        ⁢
        
          
            max
            
              
                (
                
                  
                    p
                    1
                  
                  ,
                  
                    p
                    2
                  
                
                )
              
              ∈
              
                [
                
                  
                    Y
                    1
                  
                  ,
                  
                    Y
                    2
                  
                
                ]
              
            
          
          ⁢
          
            F
            ⁡
            
              (
              
                
                  p
                  1
                
                ,
                
                  p
                  2
                
              
              )
            
          
        
      
    
    ;
  




  
    where
    ⁢
    
      :
    
  




  
    
      
        Y
        1
      
      =
      
        
          
            (
            
              
                p
                1
                
                  m
                  ⁢
                  
                    
                  
                  ⁢
                  ax
                
              
              ,
              
                
                  
                    
                      g
                      11
                    
                    ⁢
                    
                      p
                      1
                      
                        ma
                        ⁢
                        
                          
                        
                        ⁢
                        x
                      
                    
                  
                  -
                  
                    
                      γ
                      1
                    
                    ⁢
                    
                      n
                      1
                    
                  
                
                
                  
                    γ
                    1
                  
                  ⁢
                  
                    g
                    21
                  
                
              
            
            )
          
          ⁢
          
            
          
          ⁢
          and
          ⁢
          
            
          
          ⁢
          
            Y
            1
          
        
        =
        
          (
          
            
              
                (
                
                  Y
                  1
                
                )
              
              h
            
            ,
            
              
                (
                
                  Y
                  1
                
                )
              
              v
            
          
          )
        
      
    
    ;
  




  
    
      Y
      2
    
    =
    
      
        
          (
          
            
              p
              1
              
                ma
                ⁢
                
                  
                
                ⁢
                x
              
            
            ,
            
              
                
                  
                    n
                    2
                  
                  ⁢
                  
                    γ
                    2
                  
                
                +
                
                  
                    g
                    12
                  
                  ⁢
                  
                    γ
                    2
                  
                  ⁢
                  
                    p
                    1
                    
                      ma
                      ⁢
                      
                        
                      
                      ⁢
                      x
                    
                  
                
              
              
                g
                22
              
            
          
          )
        
        ⁢
        
          
        
        ⁢
        and
        ⁢
        
          
        
        ⁢
        
          Y
          2
        
      
      =
      
        
          (
          
            
              
                (
                
                  Y
                  2
                
                )
              
              h
            
            ,
            
              
                (
                
                  Y
                  2
                
                )
              
              v
            
          
          )
        
        .

11. The method according to claim 3, wherein the maximizing of the two-user throughput further comprises:
when ƒl0((X1)h)≧(X1)v and ƒl0((X2)h)≦(X2)v, computing the optimal power levels of SU 1 and SU 2, respectively denoted as p*1 and p*2, for maximizing the two-user throughput by





  
    
  
  ⁢
  
    
      (
      
        
          p
          1
          *
        
        ,
        
          p
          2
          *
        
      
      )
    
    =
    
      {
      
        
          
            
              
                arg
                ⁢
                
                  
                
                ⁢
                
                  
                    max
                    
                      
                        (
                        
                          
                            p
                            1
                          
                          ,
                          
                            p
                            2
                          
                        
                        )
                      
                      ∈
                      
                        {
                        
                          
                            X
                            0
                          
                          ,
                          
                            X
                            1
                          
                          ,
                          
                            Z
                            2
                          
                          ,
                          
                            O
                            1
                            
                              I
                              ,
                              3
                            
                          
                          ,
                          
                            O
                            2
                            
                              I
                              ,
                              3
                            
                          
                        
                        }
                      
                    
                  
                  ⁢
                  
                    F
                    ⁡
                    
                      (
                      
                        
                          p
                          1
                        
                        ,
                        
                          p
                          2
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  
                    if
                    ⁢
                    
                      
                    
                    ⁢
                    A
                  
                  ≠
                  0
                
                ,
              
            
          
          
            
              
                arg
                ⁢
                
                  
                
                ⁢
                
                  
                    max
                    
                      
                        (
                        
                          
                            p
                            1
                          
                          ,
                          
                            p
                            2
                          
                        
                        )
                      
                      ∈
                      
                        {
                        
                          
                            X
                            0
                          
                          ,
                          
                            X
                            1
                          
                          ,
                          
                            Z
                            2
                          
                          ,
                          
                            O
                            3
                            
                              I
                              ,
                              3
                            
                          
                        
                        }
                      
                    
                  
                  ⁢
                  
                    F
                    ⁡
                    
                      (
                      
                        
                          p
                          1
                        
                        ,
                        
                          p
                          2
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  if
                  ⁢
                  
                    
                  
                  ⁢
                  A
                
                =
                0
              
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            where
            ⁢
            
              :
            
            ⁢
            
              

            
            ⁢
            
              
            
            ⁢
            
              X
              0
            
          
          =
          
            
              
                (
                
                  
                    
                      Γ
                      -
                      
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          p
                          2
                          
                            ma
                            ⁢
                            
                              
                            
                            ⁢
                            x
                          
                        
                      
                    
                    
                      g
                      
                        1
                        ⁢
                        B
                      
                    
                  
                  ,
                  
                    p
                    2
                    
                      ma
                      ⁢
                      
                        
                      
                      ⁢
                      x
                    
                  
                
                )
              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                X
                0
              
            
            =
            
              (
              
                
                  
                    (
                    
                      X
                      0
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      X
                      0
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            X
            1
          
          =
          
            
              
                (
                
                  
                    
                      
                        
                          γ
                          1
                        
                        ⁢
                        
                          n
                          1
                        
                      
                      +
                      
                        
                          g
                          21
                        
                        ⁢
                        
                          γ
                          1
                        
                        ⁢
                        
                          p
                          2
                          
                            ma
                            ⁢
                            
                              
                            
                            ⁢
                            x
                          
                        
                      
                    
                    
                      g
                      11
                    
                  
                  ,
                  
                    p
                    2
                    
                      ma
                      ⁢
                      
                        
                      
                      ⁢
                      x
                    
                  
                
                )
              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                X
                1
              
            
            =
            
              (
              
                
                  
                    (
                    
                      X
                      1
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      X
                      1
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            X
            2
          
          =
          
            
              
                (
                
                  
                    
                      
                        
                          g
                          22
                        
                        ⁢
                        
                          p
                          2
                          
                            ma
                            ⁢
                            
                              
                            
                            ⁢
                            x
                          
                        
                      
                      -
                      
                        
                          γ
                          2
                        
                        ⁢
                        
                          n
                          2
                        
                      
                    
                    
                      
                        γ
                        2
                      
                      ⁢
                      
                        g
                        12
                      
                    
                  
                  ,
                  
                    p
                    2
                    
                      ma
                      ⁢
                      
                        
                      
                      ⁢
                      x
                    
                  
                
                )
              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                X
                2
              
            
            =
            
              (
              
                
                  
                    (
                    
                      X
                      2
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      X
                      2
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            Z
            2
          
          =
          
            
              
                (
                
                  
                    
                      
                        
                          g
                          22
                        
                        ⁢
                        Γ
                      
                      -
                      
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          n
                          2
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                    
                    
                      
                        
                          g
                          22
                        
                        ⁢
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                      
                      +
                      
                        
                          g
                          12
                        
                        ⁢
                        
                          g
                          12
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                    
                  
                  ,
                  
                    
                      
                        
                          g
                          12
                        
                        ⁢
                        
                          γ
                          2
                        
                        ⁢
                        Γ
                      
                      +
                      
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          n
                          2
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                    
                    
                      
                        
                          g
                          22
                        
                        ⁢
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                      
                      +
                      
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          g
                          12
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                    
                  
                
                )
              
              ⁢
              
                

              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                Z
                2
              
            
            =
            
              (
              
                
                  
                    (
                    
                      Z
                      2
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      Z
                      2
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            O
            1
            
              
                I
                ,
                3
              
              ⁢
              
                
              
            
          
          =
          
            {
            
              
                
                  
                    
                      (
                      
                        
                          p
                          1
                          
                            o
                            1
                          
                        
                        ,
                        
                          
                            f
                            
                              l
                              0
                            
                          
                          ⁡
                          
                            (
                            
                              p
                              1
                              
                                o
                                1
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                  
                    
                      
                        if
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            (
                            
                              X
                              0
                            
                            )
                          
                          h
                        
                      
                      <
                      
                        p
                        1
                        
                          o
                          1
                        
                      
                      <
                      
                        
                          (
                          
                            Z
                            2
                          
                          )
                        
                        h
                      
                    
                  
                
                
                  
                    φ
                  
                  
                    otherwise
                  
                
              
              ;
              
                

              
              ⁢
              
                
              
              ⁢
              
                
                  O
                  2
                  
                    I
                    ,
                    3
                  
                
                =
                
                  {
                  
                    
                      
                        
                          
                            (
                            
                              
                                p
                                1
                                
                                  o
                                  2
                                
                              
                              ,
                              
                                
                                  f
                                  
                                    l
                                    0
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    p
                                    1
                                    
                                      o
                                      2
                                    
                                  
                                  )
                                
                              
                            
                            )
                          
                        
                        
                          
                            
                              if
                              ⁢
                              
                                
                              
                              ⁢
                              
                                
                                  (
                                  
                                    X
                                    0
                                  
                                  )
                                
                                h
                              
                            
                            <
                            
                              p
                              1
                              
                                o
                                2
                              
                            
                            <
                            
                              
                                (
                                
                                  Z
                                  2
                                
                                )
                              
                              h
                            
                          
                        
                      
                      
                        
                          φ
                        
                        
                          otherwise
                        
                      
                    
                    ;
                    
                      

                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        O
                        3
                        
                          I
                          ,
                          3
                        
                      
                      =
                      
                        {
                        
                          
                            
                              
                                
                                  (
                                  
                                    
                                      p
                                      1
                                      
                                        o
                                        3
                                      
                                    
                                    ,
                                    
                                      
                                        f
                                        
                                          l
                                          0
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          p
                                          1
                                          
                                            o
                                            3
                                          
                                        
                                        )
                                      
                                    
                                  
                                  )
                                
                              
                              
                                
                                  
                                    if
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      
                                        (
                                        
                                          X
                                          0
                                        
                                        )
                                      
                                      h
                                    
                                  
                                  <
                                  
                                    p
                                    1
                                    
                                      o
                                      3
                                    
                                  
                                  <
                                  
                                    
                                      (
                                      
                                        Z
                                        2
                                      
                                      )
                                    
                                    h
                                  
                                
                              
                            
                            
                              
                                φ
                              
                              
                                otherwise
                              
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              p
                              1
                              
                                o
                                1
                              
                            
                            =
                            
                              
                                
                                  
                                    
                                      -
                                      B
                                    
                                    +
                                    
                                      
                                        
                                          B
                                          2
                                        
                                        -
                                        
                                          4
                                          ⁢
                                          A
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          C
                                        
                                      
                                    
                                  
                                  
                                    2
                                    ⁢
                                    A
                                  
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                A
                              
                              ≠
                              0
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              p
                              1
                              
                                o
                                2
                              
                            
                            =
                            
                              
                                
                                  
                                    
                                      -
                                      B
                                    
                                    -
                                    
                                      
                                        
                                          B
                                          2
                                        
                                        -
                                        
                                          4
                                          ⁢
                                          A
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          C
                                        
                                      
                                    
                                  
                                  
                                    2
                                    ⁢
                                    A
                                  
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                A
                              
                              ≠
                              0
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              p
                              1
                              
                                o
                                3
                              
                            
                            =
                            
                              
                                
                                  
                                    -
                                    C
                                  
                                  B
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                A
                              
                              =
                              0
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            A
                            =
                            
                              
                                
                                  g
                                  
                                    2
                                    ⁢
                                    B
                                  
                                
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  2
                                
                              
                              ⁡
                              
                                [
                                
                                  
                                    
                                      g
                                      11
                                    
                                    ⁢
                                    
                                      g
                                      12
                                    
                                    ⁢
                                    
                                      
                                        g
                                        
                                          2
                                          ⁢
                                          B
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              n
                                              1
                                            
                                            ⁢
                                            
                                              g
                                              
                                                2
                                                ⁢
                                                B
                                              
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              21
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            g
                                            12
                                          
                                          ⁢
                                          
                                            g
                                            
                                              2
                                              ⁢
                                              B
                                            
                                          
                                        
                                        -
                                        
                                          
                                            g
                                            22
                                          
                                          ⁢
                                          
                                            g
                                            
                                              1
                                              ⁢
                                              B
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                  -
                                  
                                    
                                      g
                                      22
                                    
                                    ⁢
                                    
                                      g
                                      21
                                    
                                    ⁢
                                    
                                      
                                        g
                                        
                                          1
                                          ⁢
                                          B
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              g
                                              
                                                1
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            g
                                            21
                                          
                                          ⁢
                                          
                                            g
                                            
                                              1
                                              ⁢
                                              B
                                            
                                          
                                        
                                        -
                                        
                                          
                                            g
                                            11
                                          
                                          ⁢
                                          
                                            g
                                            
                                              2
                                              ⁢
                                              B
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                ]
                              
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            B
                            =
                            
                              
                                
                                  
                                    g
                                    
                                      2
                                      ⁢
                                      B
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          n
                                          1
                                        
                                        ⁢
                                        
                                          g
                                          
                                            2
                                            ⁢
                                            B
                                          
                                        
                                      
                                      +
                                      
                                        
                                          g
                                          21
                                        
                                        ⁢
                                        Γ
                                      
                                    
                                    )
                                  
                                
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  2
                                
                              
                              ×
                              
                                [
                                
                                  
                                    
                                      g
                                      11
                                    
                                    ⁢
                                    
                                      
                                        g
                                        
                                          2
                                          ⁢
                                          B
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            2
                                            ⁢
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            
                                              g
                                              
                                                2
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            
                                              g
                                              22
                                            
                                            ⁢
                                            Γ
                                          
                                          -
                                          
                                            
                                              g
                                              22
                                            
                                            ⁢
                                            
                                              g
                                              
                                                1
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                        
                                        )
                                      
                                    
                                  
                                  -
                                  
                                    
                                      
                                        g
                                        22
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              g
                                              
                                                1
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            g
                                            11
                                          
                                          ⁢
                                          
                                            g
                                            
                                              2
                                              ⁢
                                              B
                                            
                                          
                                        
                                        -
                                        
                                          2
                                          ⁢
                                          
                                            g
                                            21
                                          
                                          ⁢
                                          
                                            g
                                            
                                              1
                                              ⁢
                                              B
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                ]
                              
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            C
                            =
                            
                              
                                
                                  
                                    g
                                    
                                      2
                                      ⁢
                                      B
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          n
                                          1
                                        
                                        ⁢
                                        
                                          g
                                          
                                            2
                                            ⁢
                                            B
                                          
                                        
                                      
                                      +
                                      
                                        
                                          g
                                          21
                                        
                                        ⁢
                                        Γ
                                      
                                    
                                    )
                                  
                                
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  2
                                
                              
                              ×
                              
                                
                                  [
                                  
                                    
                                      
                                        g
                                        11
                                      
                                      ⁢
                                      
                                        g
                                        
                                          2
                                          ⁢
                                          B
                                        
                                      
                                      ⁢
                                      
                                        
                                          n
                                          2
                                        
                                        ⁡
                                        
                                          (
                                          
                                            
                                              
                                                n
                                                2
                                              
                                              ⁢
                                              
                                                g
                                                
                                                  2
                                                  ⁢
                                                  B
                                                
                                              
                                            
                                            +
                                            
                                              
                                                g
                                                22
                                              
                                              ⁢
                                              Γ
                                            
                                          
                                          )
                                        
                                      
                                    
                                    -
                                    
                                      
                                        
                                          g
                                          22
                                        
                                        ⁡
                                        
                                          (
                                          
                                            
                                              
                                                g
                                                
                                                  1
                                                  ⁢
                                                  B
                                                
                                              
                                              ⁢
                                              
                                                n
                                                2
                                              
                                            
                                            +
                                            
                                              
                                                g
                                                12
                                              
                                              ⁢
                                              Γ
                                            
                                          
                                          )
                                        
                                      
                                      ⁢
                                      
                                        (
                                        
                                          
                                            
                                              n
                                              1
                                            
                                            ⁢
                                            
                                              g
                                              
                                                2
                                                ⁢
                                                B
                                              
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              21
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                  
                                  ]
                                
                                .

12. The method according to claim 3, wherein the maximizing of the two-user throughput further comprises:
when ƒl0((X1)h)≦(X1)v and ƒl0((X2)h)≦(X2)v, computing the optimal power levels of SU 1 and SU 2, respectively denoted as p*1 and p*2, for maximizing the two-user throughput by





  
    
  
  ⁢
  
    
      (
      
        
          p
          1
          *
        
        ,
        
          p
          2
          *
        
      
      )
    
    =
    
      {
      
        
          
            
              
                arg
                ⁢
                
                  
                
                ⁢
                
                  
                    max
                    
                      
                        (
                        
                          
                            p
                            1
                          
                          ,
                          
                            p
                            2
                          
                        
                        )
                      
                      ∈
                      
                        {
                        
                          
                            Z
                            1
                          
                          ,
                          
                            Z
                            2
                          
                          ,
                          
                            O
                            1
                            
                              I
                              ,
                              2
                            
                          
                          ,
                          
                            O
                            2
                            
                              I
                              ,
                              2
                            
                          
                        
                        }
                      
                    
                  
                  ⁢
                  
                    F
                    ⁡
                    
                      (
                      
                        
                          p
                          1
                        
                        ,
                        
                          p
                          2
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  
                    if
                    ⁢
                    
                      
                    
                    ⁢
                    A
                  
                  ≠
                  0
                
                ,
              
            
          
          
            
              
                arg
                ⁢
                
                  
                
                ⁢
                
                  
                    max
                    
                      
                        (
                        
                          
                            p
                            1
                          
                          ,
                          
                            p
                            2
                          
                        
                        )
                      
                      ∈
                      
                        {
                        
                          
                            Z
                            1
                          
                          ,
                          
                            Z
                            2
                          
                          ,
                          
                            O
                            3
                            
                              I
                              ,
                              2
                            
                          
                        
                        }
                      
                    
                  
                  ⁢
                  
                    F
                    ⁡
                    
                      (
                      
                        
                          p
                          1
                        
                        ,
                        
                          p
                          2
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  if
                  ⁢
                  
                    
                  
                  ⁢
                  A
                
                =
                0
              
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            where
            ⁢
            
              :
            
            ⁢
            
              

            
            ⁢
            
              
            
            ⁢
            
              X
              1
            
          
          =
          
            
              
                (
                
                  
                    
                      
                        
                          γ
                          1
                        
                        ⁢
                        
                          n
                          1
                        
                      
                      +
                      
                        
                          g
                          21
                        
                        ⁢
                        
                          γ
                          1
                        
                        ⁢
                        
                          p
                          2
                          
                            ma
                            ⁢
                            
                              
                            
                            ⁢
                            x
                          
                        
                      
                    
                    
                      g
                      11
                    
                  
                  ,
                  
                    p
                    2
                    
                      ma
                      ⁢
                      
                        
                      
                      ⁢
                      x
                    
                  
                
                )
              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                X
                1
              
            
            =
            
              (
              
                
                  
                    (
                    
                      X
                      1
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      X
                      1
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            X
            2
          
          =
          
            
              
                (
                
                  
                    
                      
                        
                          g
                          22
                        
                        ⁢
                        
                          p
                          2
                          
                            ma
                            ⁢
                            
                              
                            
                            ⁢
                            x
                          
                        
                      
                      -
                      
                        
                          γ
                          2
                        
                        ⁢
                        
                          n
                          2
                        
                      
                    
                    
                      
                        γ
                        2
                      
                      ⁢
                      
                        g
                        12
                      
                    
                  
                  ,
                  
                    p
                    2
                    
                      ma
                      ⁢
                      
                        
                      
                      ⁢
                      x
                    
                  
                
                )
              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                X
                2
              
            
            =
            
              (
              
                
                  
                    (
                    
                      X
                      2
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      X
                      2
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            Z
            1
          
          =
          
            
              
                (
                
                  
                    
                      
                        
                          g
                          21
                        
                        ⁢
                        
                          γ
                          1
                        
                        ⁢
                        Γ
                      
                      +
                      
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          n
                          1
                        
                        ⁢
                        
                          γ
                          1
                        
                      
                    
                    
                      
                        
                          g
                          11
                        
                        ⁢
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                      
                      +
                      
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          g
                          21
                        
                        ⁢
                        
                          γ
                          1
                        
                      
                    
                  
                  ,
                  
                    
                      
                        
                          g
                          11
                        
                        ⁢
                        Γ
                      
                      -
                      
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          n
                          1
                        
                        ⁢
                        
                          γ
                          1
                        
                      
                    
                    
                      
                        
                          g
                          11
                        
                        ⁢
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                      
                      +
                      
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          g
                          21
                        
                        ⁢
                        
                          γ
                          1
                        
                      
                    
                  
                
                )
              
              ⁢
              
                

              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                Z
                1
              
            
            =
            
              (
              
                
                  
                    (
                    
                      Z
                      1
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      Z
                      1
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            Z
            2
          
          =
          
            
              
                (
                
                  
                    
                      
                        
                          g
                          22
                        
                        ⁢
                        Γ
                      
                      -
                      
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          n
                          2
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                    
                    
                      
                        
                          g
                          22
                        
                        ⁢
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                      
                      +
                      
                        
                          g
                          12
                        
                        ⁢
                        
                          g
                          12
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                    
                  
                  ,
                  
                    
                      
                        
                          g
                          12
                        
                        ⁢
                        
                          γ
                          2
                        
                        ⁢
                        Γ
                      
                      +
                      
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          n
                          2
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                    
                    
                      
                        
                          g
                          22
                        
                        ⁢
                        
                          g
                          
                            1
                            ⁢
                            B
                          
                        
                      
                      +
                      
                        
                          g
                          
                            2
                            ⁢
                            B
                          
                        
                        ⁢
                        
                          g
                          12
                        
                        ⁢
                        
                          γ
                          2
                        
                      
                    
                  
                
                )
              
              ⁢
              
                

              
              ⁢
              
                
              
              ⁢
              and
              ⁢
              
                
              
              ⁢
              
                Z
                2
              
            
            =
            
              (
              
                
                  
                    (
                    
                      Z
                      2
                    
                    )
                  
                  h
                
                ,
                
                  
                    (
                    
                      Z
                      2
                    
                    )
                  
                  v
                
              
              )
            
          
        
        ;
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            O
            1
            
              
                I
                ,
                1
              
              ⁢
              
                
              
            
          
          =
          
            {
            
              
                
                  
                    
                      (
                      
                        
                          p
                          1
                          
                            o
                            1
                          
                        
                        ,
                        
                          
                            f
                            
                              l
                              0
                            
                          
                          ⁡
                          
                            (
                            
                              p
                              1
                              
                                o
                                1
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                  
                    
                      
                        if
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            (
                            
                              Z
                              1
                            
                            )
                          
                          h
                        
                      
                      <
                      
                        p
                        1
                        
                          o
                          1
                        
                      
                      <
                      
                        
                          (
                          
                            Z
                            2
                          
                          )
                        
                        h
                      
                    
                  
                
                
                  
                    φ
                  
                  
                    otherwise
                  
                
              
              ;
              
                

              
              ⁢
              
                
              
              ⁢
              
                
                  O
                  2
                  
                    I
                    ,
                    1
                  
                
                =
                
                  {
                  
                    
                      
                        
                          
                            (
                            
                              
                                p
                                1
                                
                                  o
                                  2
                                
                              
                              ,
                              
                                
                                  f
                                  
                                    l
                                    0
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    p
                                    1
                                    
                                      o
                                      2
                                    
                                  
                                  )
                                
                              
                            
                            )
                          
                        
                        
                          
                            
                              if
                              ⁢
                              
                                
                              
                              ⁢
                              
                                
                                  (
                                  
                                    Z
                                    1
                                  
                                  )
                                
                                h
                              
                            
                            <
                            
                              p
                              1
                              
                                o
                                2
                              
                            
                            <
                            
                              
                                (
                                
                                  Z
                                  2
                                
                                )
                              
                              h
                            
                          
                        
                      
                      
                        
                          φ
                        
                        
                          otherwise
                        
                      
                    
                    ;
                    
                      

                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        O
                        3
                        
                          I
                          ,
                          1
                        
                      
                      =
                      
                        {
                        
                          
                            
                              
                                
                                  (
                                  
                                    
                                      p
                                      1
                                      
                                        o
                                        3
                                      
                                    
                                    ,
                                    
                                      
                                        f
                                        
                                          l
                                          0
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          p
                                          1
                                          
                                            o
                                            3
                                          
                                        
                                        )
                                      
                                    
                                  
                                  )
                                
                              
                              
                                
                                  
                                    if
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      
                                        (
                                        
                                          Z
                                          1
                                        
                                        )
                                      
                                      h
                                    
                                  
                                  <
                                  
                                    p
                                    1
                                    
                                      o
                                      3
                                    
                                  
                                  <
                                  
                                    
                                      (
                                      
                                        Z
                                        2
                                      
                                      )
                                    
                                    h
                                  
                                
                              
                            
                            
                              
                                φ
                              
                              
                                otherwise
                              
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              p
                              1
                              
                                o
                                1
                              
                            
                            =
                            
                              
                                
                                  
                                    
                                      -
                                      B
                                    
                                    +
                                    
                                      
                                        
                                          B
                                          2
                                        
                                        -
                                        
                                          4
                                          ⁢
                                          A
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          C
                                        
                                      
                                    
                                  
                                  
                                    2
                                    ⁢
                                    A
                                  
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                A
                              
                              ≠
                              0
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              p
                              1
                              
                                o
                                2
                              
                            
                            =
                            
                              
                                
                                  
                                    
                                      -
                                      B
                                    
                                    -
                                    
                                      
                                        
                                          B
                                          2
                                        
                                        -
                                        
                                          4
                                          ⁢
                                          A
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          C
                                        
                                      
                                    
                                  
                                  
                                    2
                                    ⁢
                                    A
                                  
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                A
                              
                              ≠
                              0
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              p
                              1
                              
                                o
                                3
                              
                            
                            =
                            
                              
                                
                                  
                                    -
                                    C
                                  
                                  B
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                if
                                ⁢
                                
                                  
                                
                                ⁢
                                A
                              
                              =
                              0
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            A
                            =
                            
                              
                                
                                  g
                                  
                                    2
                                    ⁢
                                    B
                                  
                                
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  2
                                
                              
                              ⁡
                              
                                [
                                
                                  
                                    
                                      g
                                      11
                                    
                                    ⁢
                                    
                                      g
                                      12
                                    
                                    ⁢
                                    
                                      
                                        g
                                        
                                          2
                                          ⁢
                                          B
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              n
                                              1
                                            
                                            ⁢
                                            
                                              g
                                              
                                                2
                                                ⁢
                                                B
                                              
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              21
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            g
                                            12
                                          
                                          ⁢
                                          
                                            g
                                            
                                              2
                                              ⁢
                                              B
                                            
                                          
                                        
                                        -
                                        
                                          
                                            g
                                            22
                                          
                                          ⁢
                                          
                                            g
                                            
                                              1
                                              ⁢
                                              B
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                  -
                                  
                                    
                                      g
                                      22
                                    
                                    ⁢
                                    
                                      g
                                      21
                                    
                                    ⁢
                                    
                                      
                                        g
                                        
                                          1
                                          ⁢
                                          B
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              g
                                              
                                                1
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            g
                                            21
                                          
                                          ⁢
                                          
                                            g
                                            
                                              1
                                              ⁢
                                              B
                                            
                                          
                                        
                                        -
                                        
                                          
                                            g
                                            11
                                          
                                          ⁢
                                          
                                            g
                                            
                                              2
                                              ⁢
                                              B
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                ]
                              
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            B
                            =
                            
                              
                                
                                  
                                    g
                                    
                                      2
                                      ⁢
                                      B
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          n
                                          1
                                        
                                        ⁢
                                        
                                          g
                                          
                                            2
                                            ⁢
                                            B
                                          
                                        
                                      
                                      +
                                      
                                        
                                          g
                                          21
                                        
                                        ⁢
                                        Γ
                                      
                                    
                                    )
                                  
                                
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  2
                                
                              
                              ×
                              
                                [
                                
                                  
                                    
                                      g
                                      11
                                    
                                    ⁢
                                    
                                      
                                        g
                                        
                                          2
                                          ⁢
                                          B
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            2
                                            ⁢
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            
                                              g
                                              
                                                2
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            
                                              g
                                              22
                                            
                                            ⁢
                                            Γ
                                          
                                          -
                                          
                                            
                                              g
                                              22
                                            
                                            ⁢
                                            
                                              g
                                              
                                                1
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                        
                                        )
                                      
                                    
                                  
                                  -
                                  
                                    
                                      
                                        g
                                        22
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              g
                                              
                                                1
                                                ⁢
                                                B
                                              
                                            
                                            ⁢
                                            
                                              n
                                              2
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              12
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            g
                                            11
                                          
                                          ⁢
                                          
                                            g
                                            
                                              2
                                              ⁢
                                              B
                                            
                                          
                                        
                                        -
                                        
                                          2
                                          ⁢
                                          
                                            g
                                            21
                                          
                                          ⁢
                                          
                                            g
                                            
                                              1
                                              ⁢
                                              B
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                ]
                              
                            
                          
                          ;
                          
                            

                          
                          ⁢
                          
                            C
                            =
                            
                              
                                
                                  
                                    g
                                    
                                      2
                                      ⁢
                                      B
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          n
                                          1
                                        
                                        ⁢
                                        
                                          g
                                          
                                            2
                                            ⁢
                                            B
                                          
                                        
                                      
                                      +
                                      
                                        
                                          g
                                          21
                                        
                                        ⁢
                                        Γ
                                      
                                    
                                    )
                                  
                                
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  2
                                
                              
                              ×
                              
                                
                                  [
                                  
                                    
                                      
                                        g
                                        11
                                      
                                      ⁢
                                      
                                        g
                                        
                                          2
                                          ⁢
                                          B
                                        
                                      
                                      ⁢
                                      
                                        
                                          n
                                          2
                                        
                                        ⁡
                                        
                                          (
                                          
                                            
                                              
                                                n
                                                2
                                              
                                              ⁢
                                              
                                                g
                                                
                                                  2
                                                  ⁢
                                                  B
                                                
                                              
                                            
                                            +
                                            
                                              
                                                g
                                                22
                                              
                                              ⁢
                                              Γ
                                            
                                          
                                          )
                                        
                                      
                                    
                                    -
                                    
                                      
                                        
                                          g
                                          22
                                        
                                        ⁡
                                        
                                          (
                                          
                                            
                                              
                                                g
                                                
                                                  1
                                                  ⁢
                                                  B
                                                
                                              
                                              ⁢
                                              
                                                n
                                                2
                                              
                                            
                                            +
                                            
                                              
                                                g
                                                12
                                              
                                              ⁢
                                              Γ
                                            
                                          
                                          )
                                        
                                      
                                      ⁢
                                      
                                        (
                                        
                                          
                                            
                                              n
                                              1
                                            
                                            ⁢
                                            
                                              g
                                              
                                                2
                                                ⁢
                                                B
                                              
                                            
                                          
                                          +
                                          
                                            
                                              g
                                              21
                                            
                                            ⁢
                                            Γ
                                          
                                        
                                        )
                                      
                                    
                                  
                                  ]
                                
                                .

13. The method according to claim 3, wherein the maximizing of the two-user throughput further comprises:
when ƒl0((X1)h)≧(X1)v and ƒl0((X2)h)≧(X2)v, computing the optimal power levels of SU 1 and SU 2, respectively denoted as p*1 and p*2, for maximizing the two-user throughput by




  
    
      
        (
        
          
            p
            1
            *
          
          ,
          
            p
            2
            *
          
        
        )
      
      =
      
        arg
        ⁢
        
          
        
        ⁢
        
          
            max
            
              
                (
                
                  
                    p
                    1
                  
                  ,
                  
                    p
                    2
                  
                
                )
              
              ∈
              
                [
                
                  
                    X
                    1
                  
                  ,
                  
                    X
                    2
                  
                
                ]
              
            
          
          ⁢
          
            F
            ⁡
            
              (
              
                
                  p
                  1
                
                ,
                
                  p
                  2
                
              
              )
            
          
        
      
    
    ;
  




  
    where
    ⁢
    
      :
    
  




  
    
      
        X
        1
      
      =
      
        
          
            (
            
              
                
                  
                    
                      γ
                      1
                    
                    ⁢
                    
                      n
                      1
                    
                  
                  +
                  
                    
                      g
                      21
                    
                    ⁢
                    
                      γ
                      1
                    
                    ⁢
                    
                      p
                      2
                      
                        ma
                        ⁢
                        
                          
                        
                        ⁢
                        x
                      
                    
                  
                
                
                  g
                  11
                
              
              ,
              
                p
                2
                
                  ma
                  ⁢
                  
                    
                  
                  ⁢
                  x
                
              
            
            )
          
          ⁢
          
            
          
          ⁢
          and
          ⁢
          
            
          
          ⁢
          
            X
            1
          
        
        =
        
          (
          
            
              
                (
                
                  X
                  1
                
                )
              
              h
            
            ,
            
              
                (
                
                  X
                  1
                
                )
              
              v
            
          
          )
        
      
    
    ;
  




  
    
      X
      2
    
    =
    
      
        
          (
          
            
              
                
                  
                    g
                    22
                  
                  ⁢
                  
                    p
                    2
                    
                      ma
                      ⁢
                      
                        
                      
                      ⁢
                      x
                    
                  
                
                -
                
                  
                    γ
                    2
                  
                  ⁢
                  
                    n
                    2
                  
                
              
              
                
                  γ
                  2
                
                ⁢
                
                  g
                  12
                
              
            
            ,
            
              p
              2
              
                ma
                ⁢
                
                  
                
                ⁢
                x
              
            
          
          )
        
        ⁢
        
          
        
        ⁢
        and
        ⁢
        
          
        
        ⁢
        
          X
          2
        
      
      =
      
        
          (
          
            
              
                (
                
                  X
                  2
                
                )
              
              h
            
            ,
            
              
                (
                
                  X
                  2
                
                )
              
              v
            
          
          )
        
        .