Page 89 - ITU Journal Future and evolving technologies Volume 2 (2021), Issue 7 – Terahertz communications
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ITU Journal on Future and Evolving Technologies, Volume 2 (2021), Issue 7




                                                                                  
                                               ∗
                                                                                                      
          The gradient of Lagrangian with respect to      ,    is given by  Next,    is set to 2 . By inserting    = 2 and Φ   ,    =
                                                                                                 
                                                                        
                                                                  (2  −1)2   −    into the above equation, the result in (17) can
                      2           2                                     
                  ̂     ∫ (‖Fa(   ,   )‖ )                     be obtained.
                                  ∞
                                
                     0
                                                ,  
               ∗
                             ∗
                   ,    =         ,    −      
                     2  +1           −1    (  −1)   2
                               | ∑          |                  APPENDIX C
                                      ,  
                 = ∫            =0  ∗             −      
                                                        ,  
                    2  −1               ,                      Proof of Theorem 3
                         
                     2  +1          −1
                               −  2                            The objective function in (22) can be reformulated as
                 = ∫      ∑                  (  −  )      −      
                                                     ,  
                    2  −1       =0
                                                                             0 +  −1
                               −1     (  −  )(2  +1)  (  −  )(2  −1)                     ,    w ‖ 2
                    −  2        −  2    (  −  )            −            ∑ ∑ max  w    ∈W ‖H              2
                 =           ∑                                        =1    =   0
                            =0               (   −   )
                                                                             0 +  −1
                   −                                              = ∑ ∑ max          tr((H   ,    ) H   ,    w w )
                                                                                                 
                                                                                                             
                           ,  
                               −1                                     =1         w    ∈W                            
                    −  2        2 sin((   −   )  )                        =   0
                 =           ∑            −                                  0 +  −1
                                     −             ,  
                            =0                                    = ∑ ∑ max      w    ∈W
                    ̂
                 =        −       ,                                   =1    =   0
                                  ,  
                      ,     ,  
                                                      (45)            1     ̂     ̂   ̂      ̂    
                                                                                               
                                                                                     
          where    ̂    ,    = ∑       −1 2 sin((  −  )  )  , which is not dependent  − (tr((H   ,    − W )(H   ,    − W ) ))
                                                                      2
                                 −  
                          =0
                                                                      1
                                                                                    1
          on   . Therefore, there exists a Lagrange multiplier    that  + tr(H ̂  H ̂     ) + tr(W W ),
                                                                                         ̂
                                                                                           ̂   
          satis ies the KKT conditions. Hence, F is a local optimum   2      ,      ,    2        
          for the optimization problem (14) and (42).                                                       (49)
                                                                              ̂
                                                                                                      
                                                               where H ̂   ,    and W are de ined as (H   ,    ) H   ,    and
                                                                                 
                                                                                                       
                                                                                                               
                                                                                                       ̂
                                                                                                          ̂   
                                                                     
          APPENDIX B                                           w w , respectively. Since tr(H ̂   ,   H ̂     ) and tr(W W ) are
                                                                                                          
                                                                                               ,  
                                                                   
                                                                                                            
                                                                     
                                                               constant, the optimization problem in (22) is equivalent
          Proof of Theorem 2                                   to
                                                                                     0 +  −1
                                                                                                     ̂
                                                                    min W   ∑ ∑ min      w    ∈W   (H ̂   ,   , W )  (50)
                                                                                                       
          Proof. According  to  [16],  the  receive  power                    =1    =   0
             
                             2
          |a (   , Φ)a (   , Φ )| is given by
             
                            ,  
                      
                 
                         
                                                               APPENDIX D
                                      2
                                   sin (   (Φ  − Φ))
                
                                2
            |a (   , Φ)a (   , Φ )| =          ,      (46)     Proof of Theorem 4
                            
                               ,  
                        
                
                   
                                        2
                                     sin (Φ   ,    − Φ)
                                                               Since         (W) is  ixed to 1, W can always be decomposed
          Inserting Φ   ,    =    (2  −1)2   −    to the above equation, we ob‑  as
                                                                                             
          tain:                                                                   W = w w .                 (51)
                                                                                             
                                                                                          
                                 2
              2
            sin (   (Φ   ,    − Φ))  =  sin (   (   (2  −1)2   −    − Φ))  Hence, the optimization problem in (26) is equivalent to
                                      
                   
                                              
               2
                                                                                              
                                      2
             sin ((Φ   ,    − Φ))  sin (Φ   ,    − Φ)  (47)                   max w     ∑ w Hw    
                                                                                              
                                   2
                                sin (   Φ)                                            H∈ℋ                   (52)
                            =              .                                    s.t.  w w = 1.
                                                                                         
                                 2
                               sin (Φ   ,    − Φ)                                            
                                                               The optimal solution for w is given by w  , where
                                                                                                            
          Hence,     maximizing    the    receive   power      w     is the eigenvector of ∑     H corresponding to its
             
          |a (   , Φ)a (   , Φ )| 2  over the codebook in the                           H∈ℋ   
                 
             
                      
                            ,  
                         
               ℎ  layer is equivalent to  inding the minimum of  largest eigenvalue.
            2
          sin (Φ   ,    − Φ), which corresponds to minimizing
          |Φ   ,    − Φ| with 1 ≤    ≤ 2   −1 . In other words, the beam  REFERENCES
          coverage of w(  ,   ) is a set of the steering vector with LOS  [1] Ian F Akyildiz, Josep Miquel Jornet, and Chong
          spatial angle close to Φ . Since the distance between      Han. “Terahertz band: Next frontier for wireless
                                ,  
          Φ   ,    and Φ   ,  −1  is  4      , the beam coverage of w(  ,   ) is  communications”. In: Physical Communication 12
                          2
          obtained as                                                (2014), pp. 16–32.
                                     2        2  
                      (w(  ,   )) = [Φ   ,    −  2     , Φ   ,    +  2     ] .  (48)
                                             © International Telecommunication Union, 2021                    77
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