Abstract:
The bandwidth efficient cooperative two-way amplify-and-forward relaying method allows users in a secondary network to utilize a relay node in the primary users&#39; network while minimizing co-channel interference. In the method, two primary user network sources communicate through a primary user network relay node. A secondary user network source and a secondary user destination agree to act as relays for the primary network sources, all of the above using amplify-and-forward protocol. In return, the primary network relay node allows the secondary user source to communicate through the primary network relay node with the secondary user destination using decode-and-forward protocol. Five symbols, including four primary user symbols and one secondary user symbol, are transmitted in four time slots for a bandwidth efficiency of 1.25. The primary network relay and the secondary users relay transmissions have their power allocated to minimize symbol error rate and maximize sum rate.

Description:
BACKGROUND OF THE INVENTION 
       [0001]    1. Field of the Invention 
         [0002]    The present invention relates to communication methods for networks utilizing relay nodes, and particularly to a bandwidth efficient cooperative two-way amplify-and-forward relaying method that allows users in a secondary network to utilize a relay node in the primary users&#39; network while minimizing co-channel interference. 
         [0003]    2. Description of the Related Art 
         [0004]    Bi-directional relay communications are considered as a promising transmission scheme to increase network throughput and to improve spectral efficiency, especially with half-duplex communication models. The operations in bi-directional relaying communications can be divided into two phases, namely, a transmission phase, in which the two sources transmit their data, and a relaying phase, in which the relay node relays the previously received data. 
         [0005]    The two well-known relaying protocols, namely, amplify-and-forward (AF) and decode-and-forward (DF), are typically employed, resulting in two categories of bi-directional communications known as two-phase and three-phase two-way relaying schemes. In the two phase scheme, the AF relaying protocol is applied, where two symbols are transmitted in two phases (one transmission and one relaying), while the DF relaying protocol is applied in a three-phase scheme, where 2 symbols are transmitted in three phases (two transmission and one relaying). Although the two-phase scheme achieves a relatively better spectral efficiency than the three-phase scheme, the three-phase scheme outperforms the two-phase scheme. 
         [0006]    The performance of the two-way relaying schemes with various transmission protocols and network coding schemes has been investigated. However, none of the proposed amplify-and-forward relaying protocols have proven entirely satisfactory so that it has not been possible to take maximum advantage of the bandwidth efficiency of the amplify-and-forward relay protocol. 
         [0007]    Thus, a bandwidth efficient cooperative two-way amplify-and-forward relaying solving the aforementioned problems is desired. 
       SUMMARY OF THE INVENTION 
       [0008]    The bandwidth efficient cooperative two-way amplify-and-forward relaying method allows users in a secondary network to utilize a relay node in the primary users&#39; network while minimizing co-channel interference. In the method, two primary user network sources communicate through a primary user network relay node. A secondary user network source and a secondary user destination agree to act as relays for the primary network sources, all of the above using amplify-and-forward protocol. In return, the primary network relay node allows the secondary user source to communicate through the primary network relay node with the secondary user destination using decode-and-forward protocol. Five symbols, including four primary user symbols and one secondary user symbol, are transmitted in four time slots for a bandwidth efficiency of 1.25. The primary network relay and the secondary users relay transmissions have their power allocated to minimize symbol error rate and maximize sum rate. 
         [0009]    These and other features of the present invention will become readily apparent upon further review of the following specification and drawings. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0010]      FIG. 1A  is a block diagram showing the entity relations during the transmission phase in a first time slot in a bandwidth efficient cooperative two-way amplify-and-forward relaying method according to the present invention. 
           [0011]      FIG. 1B  is a block diagram showing the entity relations during the transmission phase in a second time slot in a bandwidth efficient cooperative two-way amplify-and-forward relaying method according to the present invention. 
           [0012]      FIG. 2A  is a block diagram showing the entity relations during the relaying phase in a third time slot in a bandwidth efficient cooperative two-way amplify-and-forward relaying method according to the present invention. 
           [0013]      FIG. 2B  is a block diagram showing the entity relations during the relaying phase in a fourth time slot in a bandwidth efficient cooperative two-way amplify-and-forward relaying method according to the present invention. 
           [0014]      FIG. 3  is a plot comparing the bandwidth efficient cooperative two-way amplify-and-forward relaying method according to the present invention and conventional relaying schemes. 
       
    
    
       [0015]    Similar reference characters denote corresponding features consistently throughout the attached drawings. 
       DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
       [0016]    The bandwidth efficient cooperative two-way amplify-and-forward relaying has a primary user (PU) network that includes two PU sources that are communicating with each other via a single relay. On the other end, a secondary user (SU) source transmits its data to a SU destination via the same PU relay node. The PU network considers the SU network pairs (i.e., source and destination) as two additional relay nodes which help the original PU relay node in improving the PU network performance. As a reward for its cooperation, the PU network allows the SU network to communicate simultaneously via the PU relay node by applying decode-and-forward (DF) protocol. The proposed system transmits four PU symbols and one SU symbol in four time slots, which achieves a bandwidth efficiency of 1:25. Two power allocation optimization problems were formulated; one to minimize the average symbol error rate of both primary and secondary systems, while the other problem is to maximize the total achievable sum rate. A Lagrangian multiplier method is used to find the optimal solutions for both problems under the constraint of maximum allowable power budget. 
         [0017]    The proposed relaying scheme considers multiuser joint detection at relay node and is based on a cooperative cognitive system between a PU network and a SU network. Further, there are several assumptions, such as that there is no direct link between sources and destinations, multiuser maximum likelihood detection, and that the SU pairs source and destination via the PU relay node. The SU network serves as relay nodes the PU network to mitigate its interference and improve system performance. A complete cooperative PU network consists of one PU source, one PU relay and one PU destination. The proposed work consider the SU network pairs (i.e., source and destination) as two extra relay nodes to improve PU network performance. Finally, the SU source communicates with its destination via the cooperation of the PU relay node following the well-known DF protocol while the PU network deals with SU transmission as an interference signal. 
         [0018]    The operation of the proposed scheme that enables the transmission of four PU symbols and one SU symbol in four time slots is presented in  FIGS. 1A, 1B, 2A, and 2B  respectively. The channel gain between X and R is denoted by h XR , and the channel gains between X and A and B are denoted by h XA  and h XB , respectively, with an average of v X   2 . Similarly, the channel gains between Y and R, and A and B, are denoted by h YR , h YA  and h YB , respectively with an average of v Y   2 . The channel gain between A and R, are h AR  with average channel gain v A   2 . Finally, the channel gain between R and B is h RB  with an average of v R   2 . 
         [0019]    For notational simplicity, all the channels are assumed to be independent and identically distributed (i.i.d) flat Rayleigh fading channels. For PU transmission, AF protocol is applied by the three relays since it is relatively less complex and relatively more flexible in handling interference than DF protocol. The operation of the proposed scheme can be divided into two phases. Namely, the transmission phase and the relaying phase. 
         [0020]    In the first time slot shown in  FIG. 1A , the PU sources X and Y transmit their modulated symbols denoted by x 1  and y 1  with transmission powers of P X  and P Y , respectively. At the same time, SU source A transmits its data a 1  with power P A  which interferes with PU data at R. Since there is no direct link between the SU network pairs A and B, the SU receiver B receives the PU transmission with no interference. Then, the received signals at R and B during the first time slot are given by: 
         [0000]        z   R   (1) =√{square root over ( P   X )} h   XR   x   1 +√{square root over ( P   Y )} h   YR   y   1 +√{square root over ( P   A )} h   AR   a   1   +w   R   (1)   (1)
 
         [0000]        z   B   (1) =√{square root over ( P   X )} h   XB   x   1 +√{square root over ( P   Y )} h   YB   y   1   +w   B   (1) ,  (2)
 
         [0000]    where w R  and w B  are AWGN samples with zero-mean and variance σ 2 . 
         [0021]    In the second time slot shown in  FIG. 1B , PU sources X and Y transmit their second PU symbols x 2  and y 2  with transmission powers of PX and PY, respectively to A and B. Simultaneously, the relay R jointly decodes the previous received SU data symbol â 1  and transmits it to the SU receiver B with a transmission power P R . The received signals at A and B during the second time slot are given by: 
         [0000]        z   A   (2) =√{square root over ( P   X )} h   XA   x   2 +√{square root over ( P   Y )} h   YA   y   2 +√{square root over ( P   R )} h   RA   â   1   +w   A   (2)   (3)
 
         [0000]        z   B   (2) =√{square root over ( P   X )} h   XB   x   2 +√{square root over ( P   Y )} h   YB   y   2 +√{square root over ( P   R )} h   RB   â   1   +w   B   (2) ,  (4)
 
         [0000]    where w A  and w B  are AWGN samples with zero-mean and variance σ 2 . By the end of transmission phase, the SU transmission is completed. The SU receiver B decodes the transmitted symbol â 1  from R, which is denoted by            1 . 
         [0022]    During the third time slot shown in  FIG. 2A , PU and SU sources are idle while R transmits the received signal after trying to remove the interfered SU data a 1  by subtracting the decoded SU symbol at R (i.e., â 1 ) from the received signal z R   (1) . On the other hand, the SU receiver B decodes the interfered SU data during the second time slot and applies AF protocol to the remaining signal. Under the assumption of knowing CSI by all relay nodes and destinations, the received signals at X and Y during the third time slot are given by: 
         [0000]        z   X   (3)   =h   RX β R ( z   R   (1) −√{square root over ( P   A )} h   AR   â   1 )+ h   BX β B     2   ( z   B   (2) −√{square root over ( P   R )} h   RB             1 .)+ w   X   (3)   (5)
 
         [0000]        z   Y   (3)   =h   RY β R ( z   R   (1) −√{square root over ( P   A )} h   AR   â   1 )+ h   BY β B     2   ( z   B   (2) −√{square root over ( P   R )} h   RB             1 .)+ w   Y   (3) ,  (6)
 
         [0000]    where w X  and w Y  are AWGN samples with zero-mean and variance σ 2 . The normalized amplification coefficient at R is given by 
         [0000]    
       
         
           
             
               β 
               R 
               2 
             
             = 
             
               
                 
                   λ 
                   R 
                 
                 
                   z 
                   R 
                   
                     ( 
                     1 
                     ) 
                   
                 
               
               . 
             
           
         
       
     
       Similarly, 
       [0023]    
       
         
           
             
               β 
               
                 B 
                 2 
               
               2 
             
             = 
             
               
                 
                   λ 
                   
                     B 
                     2 
                   
                 
                 
                   z 
                   B 
                   
                     ( 
                     2 
                     ) 
                   
                 
               
               . 
             
           
         
       
     
         [0024]    During the fourth time slot shown in  FIG. 2B , PU sources and relay nodes are idle while the SU nodes A and B relay the previously received PU data. The SU source A performs self-interference cancellation for its own data a 1  from its received signal during second time slot, i.e., Z A   (2) , then applies AF protocol to the resultant signal before re-transmitting it to both PU destinations X and Y. On the other hand, the SU destination B applies AF protocol to the previously received signal during first time slot, i.e., Z B   (2) , before re-transmitting to both PU destinations X and Y. The received signals at both PU destinations X and Y during the fourth time slot are given by: 
         [0000]        z   X   (4)   =h   BX β B     1     z   B   (1)   +h   AX β A ( z   A   (2) −√{square root over ( P   R )} h   RA   a   1 )+ w   X   (4)   (7)
 
         [0000]        z   Y   (4)   =h   BY β B     1     z   B   (1)   +h   AY β A ( z   A   (2) −√{square root over ( P   R )} h   RA   a   1   +w   Y   (4) ,  (8)
 
         [0000]    where w X  and w Y  are AWGN samples with zero-mean and variance σ 2 . The normalized amplification coefficient at A is given by 
         [0000]    
       
         
           
             
               β 
               A 
               2 
             
             = 
             
               
                 
                   λ 
                   A 
                 
                 
                   z 
                   A 
                   
                     ( 
                     2 
                     ) 
                   
                 
               
               . 
             
           
         
       
     
       Similarly, 
       [0025]    
       
         
           
             
               β 
               
                 B 
                 1 
               
               2 
             
             = 
             
               
                 
                   λ 
                   
                     B 
                     1 
                   
                 
                 
                   z 
                   B 
                   
                     ( 
                     1 
                     ) 
                   
                 
               
               . 
             
           
         
       
     
         [0026]    After the completion of the proposed system phases, the PU nodes apply self-interference cancellation on their received signals to remove their own data before the decoding process. Then, the received signals at both X and Y during the third time slot after self-interference cancellation are given by: 
         [0000]        {tilde over (z)}   X   (3)   =z   X   (3) −√{square root over ( P   X )} h   XR   x   1 −√{square root over ( P   X )} h   XB   x   2   (9)
 
         [0000]        {tilde over (z)}   Y   (3)   =z   Y   (3) −√{square root over ( P   Y )} h   YR   y   1 −√{square root over ( P   Y )} h   YB   y   2 .  (10)
 
         [0027]    Similarly, the received signals at both X and Y during the fourth time slot after self-interference cancellation are given by: 
         [0000]        {tilde over (z)}   X   (4)   =z   X   (4) −√{square root over ( P   X )} h   XB   x   1 −√{square root over ( P   X )} h   XA   x   2   (11)
 
         [0000]        {tilde over (z)}   Y   (4)   =z   Y   (4) −√{square root over ( P   Y )} h   YB   y   1 −√{square root over ( P   Y )} h   YA   y   2 .  (12)
 
         [0028]    From the previous equations and the presence of two PU destinations in this model, the matrix model for the proposed system at PU node X can be written as: 
         [0000]        {tilde over (z)}   X   =H   X   y+{tilde over (w)}   X ,  (13)
 
         [0000]    where {tilde over (z)} X =[{tilde over (z)} X   (3)  {tilde over (z)} X   (4) ] T , y=[y 1  y 2 ] T , the channel matrix H X  is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       H 
                       X 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               
                                 β 
                                 R 
                               
                                
                               
                                 h 
                                 RX 
                               
                                
                               
                                 h 
                                 RY 
                               
                             
                           
                           
                             
                               
                                 β 
                                 
                                   B 
                                   2 
                                 
                               
                                
                               
                                 h 
                                 BX 
                               
                                
                               
                                 h 
                                 YB 
                               
                             
                           
                         
                         
                           
                             
                               
                                 β 
                                 
                                   B 
                                   1 
                                 
                               
                                
                               
                                 h 
                                 BX 
                               
                                
                               
                                 h 
                                 YB 
                               
                             
                           
                           
                             
                               
                                 β 
                                 A 
                               
                                
                               
                                 h 
                                 AX 
                               
                                
                               
                                 h 
                                 YA 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and the noise vector at X is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       w 
                       ~ 
                     
                     X 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   β 
                                   R 
                                 
                                  
                                 
                                   
                                     h 
                                     RX 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             P 
                                             A 
                                           
                                         
                                          
                                         
                                           
                                             h 
                                             AR 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 a 
                                                 1 
                                               
                                               - 
                                               
                                                 
                                                   a 
                                                   ^ 
                                                 
                                                 1 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         w 
                                         R 
                                         
                                           ( 
                                           1 
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   β 
                                   
                                     B 
                                     2 
                                   
                                 
                                  
                                 
                                   
                                     h 
                                     BX 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             P 
                                             R 
                                           
                                         
                                          
                                         
                                           
                                             h 
                                             RB 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 
                                                   a 
                                                   ^ 
                                                 
                                                 1 
                                               
                                               - 
                                               
                                                 
                                                   
                                                     a 
                                                     ^ 
                                                   
                                                   ^ 
                                                 
                                                 1 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         w 
                                         B 
                                         
                                           ( 
                                           2 
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               + 
                               
                                 w 
                                 X 
                                 
                                   ( 
                                   3 
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   β 
                                   
                                     B 
                                     1 
                                   
                                 
                                  
                                 
                                   h 
                                   BX 
                                 
                                  
                                 
                                   w 
                                   B 
                                   
                                     ( 
                                     1 
                                     ) 
                                   
                                 
                               
                               + 
                               
                                 
                                   β 
                                   A 
                                 
                                  
                                 
                                   
                                     h 
                                     AX 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             P 
                                             R 
                                           
                                         
                                          
                                         
                                           
                                             h 
                                             RA 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 
                                                   a 
                                                   ^ 
                                                 
                                                 1 
                                               
                                               - 
                                               
                                                 a 
                                                 1 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         w 
                                         A 
                                         
                                           ( 
                                           2 
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               + 
                               
                                 w 
                                 X 
                                 
                                   ( 
                                   4 
                                   ) 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
         [0029]    Similarly, the matrix model for the proposed system at PU node Y can be written as: 
         [0000]        {tilde over (z)}   Y   =H   Y   x+{tilde over (w)}   Y ,  (16)
 
         [0000]    where {tilde over (z)} Y =[{tilde over (z)} Y   (3)  {tilde over (z)} Y   (4) ] T , y=[x 1  x 2 ] T , the channel matrix H Y  is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       H 
                       Y 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               
                                 β 
                                 R 
                               
                                
                               
                                 h 
                                 RY 
                               
                                
                               
                                 h 
                                 XR 
                               
                             
                           
                           
                             
                               
                                 β 
                                 
                                   B 
                                   2 
                                 
                               
                                
                               
                                 h 
                                 BY 
                               
                                
                               
                                 h 
                                 XB 
                               
                             
                           
                         
                         
                           
                             
                               
                                 β 
                                 
                                   B 
                                   1 
                                 
                               
                                
                               
                                 h 
                                 BY 
                               
                                
                               
                                 h 
                                 XB 
                               
                             
                           
                           
                             
                               
                                 β 
                                 A 
                               
                                
                               
                                 h 
                                 AY 
                               
                                
                               
                                 h 
                                 XA 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and the noise vector at Y is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       w 
                       ~ 
                     
                     Y 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   β 
                                   R 
                                 
                                  
                                 
                                   
                                     h 
                                     RY 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             P 
                                             A 
                                           
                                         
                                          
                                         
                                           
                                             h 
                                             AR 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 a 
                                                 1 
                                               
                                               - 
                                               
                                                 
                                                   a 
                                                   ^ 
                                                 
                                                 1 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         w 
                                         R 
                                         
                                           ( 
                                           1 
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   β 
                                   
                                     B 
                                     2 
                                   
                                 
                                  
                                 
                                   
                                     h 
                                     BY 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             P 
                                             R 
                                           
                                         
                                          
                                         
                                           
                                             h 
                                             RB 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 
                                                   a 
                                                   ^ 
                                                 
                                                 1 
                                               
                                               - 
                                               
                                                 
                                                   
                                                     a 
                                                     ^ 
                                                   
                                                   ^ 
                                                 
                                                 1 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         w 
                                         B 
                                         
                                           ( 
                                           2 
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               + 
                               
                                 w 
                                 Y 
                                 
                                   ( 
                                   3 
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   β 
                                   
                                     B 
                                     1 
                                   
                                 
                                  
                                 
                                   h 
                                   BY 
                                 
                                  
                                 
                                   w 
                                   B 
                                   
                                     ( 
                                     1 
                                     ) 
                                   
                                 
                               
                               + 
                               
                                 
                                   β 
                                   A 
                                 
                                  
                                 
                                   
                                     h 
                                     AY 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             P 
                                             R 
                                           
                                         
                                          
                                         
                                           
                                             h 
                                             RA 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 
                                                   a 
                                                   ^ 
                                                 
                                                 1 
                                               
                                               - 
                                               
                                                 a 
                                                 1 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         w 
                                         A 
                                         
                                           ( 
                                           2 
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               + 
                               
                                 w 
                                 Y 
                                 
                                   ( 
                                   4 
                                   ) 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
         [0030]    Note that, for a relay selection scheme, the best relay is selected with maximum channel gains for both PU sources (i.e., X and Y). Then, all the previous equations in relaying phase are valid with setting the unselected relay channel coefficients to zero. 
         [0031]    A power allocation optimization problem was formulated to minimize the sum SER of both PU and SU networks of the proposed system by controlling the SU transmission power (i.e., P A  and P R ) and the three relays amplifying factors (i.e., λ A , λ R , λ B     1   , and λ B     2   ). The goal is to find the values of those parameters that minimize the overall SER. Then, an optimization problem has been formulated in which the target function can be minimizing the total sum SER of the PU and SU networks. Such that: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           minimize 
                            
                           
                               
                           
                            
                           
                             SER 
                             PU 
                           
                         
                         + 
                         
                           SER 
                           SU 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             
                               subject 
                                
                               
                                   
                               
                                
                               to 
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   i 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   P 
                                   i 
                                 
                               
                             
                             + 
                             
                               
                                 ∑ 
                                 j 
                               
                                
                               
                                   
                               
                                
                               
                                 λ 
                                 j 
                               
                             
                           
                           ≤ 
                           
                             
                               P 
                               _ 
                             
                             total 
                           
                         
                         , 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where i=A and R, while j=A, R, B1 and B2. Lagrangian multipliers method with the power constraint in (19) is used. The Lagrangian function ∫(.) can be expressed as: 
         [0000]      ∫( P   i ,λ i )= SER   PU   +SER   SU +Λ 1 (Σ i   P   i +Σ j λ j   − P     total )  (20)
 
         [0000]    where Λ 1  denotes the Lagrangian multipliers. 
         [0032]    A power allocation optimization problem for maximizing the average achievable sum rate of the proposed system was also formulated. The average achievable sum rate is a function of SU transmission power (i.e., P A  and P R ) and the three relays amplifying factors (i.e., λA, λR, λ B     1   , and λ B     2   ). The goal is to find the optimal values which maximize the average achievable sum rate. Then, an optimization problem has been formulated such that: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           maximize 
                            
                           
                               
                           
                            
                           
                             PU 
                           
                         
                         + 
                         
                           SU 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             subject 
                              
                             
                                 
                             
                              
                             to 
                              
                             
                                 
                             
                              
                             
                               
                                 ∑ 
                                 i 
                               
                                
                               
                                   
                               
                                
                               
                                 P 
                                 i 
                               
                             
                           
                           + 
                           
                             
                               ∑ 
                               j 
                             
                              
                             
                                 
                             
                              
                             
                               λ 
                               j 
                             
                           
                         
                         ≤ 
                         
                           
                             
                               P 
                               _ 
                             
                             total 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
         [0033]    Following the same steps in solving equation (19), the optimal solution for rate maximization can be obtained. 
         [0034]    Referring to  FIG. 3 , numerical examples are presented to verify the performance of proposed scheme. Since the proposed scheme transmits 5 data symbols in 4 time slots with bandwidth efficiency equal to 1:25. The proposed system performance was compared with the conventional two-way AF relaying scheme. For a fair comparison, the total power budget is set to be the same. A SER performance comparison between the proposed system and the conventional TWR is presented in  FIG. 3 . Results show that the conventional TWR model achieves better SER performance compared to the proposed work. As the SNR goes higher, the proposed system in both cases of spatial multiplexing and relay selection outperforms the conventional TWR scheme, which encourages the PU system to cooperate with the SU network. 
         [0035]    It is to be understood that the present invention is not limited to the embodiments described above, but encompasses any and all embodiments within the scope of the following claims.