Patent ID: 11909566
Assignee: SOOCHOW UNIVERSITY
Field: Digital communication (Electrical engineering)
Classification: CPC H | IPC H

Claim 2:
3. The method for designing a complex-valued channel equalizer according to claim 1, wherein a specific implementation process of “in the adaptive complex-valued L-BFGS algorithm, calculating and comparing the approximation degree between the inverse of quasi-Newton matrix and the inverse of Hessian matrix along a particular vector, and selecting a memory size corresponding to the highest degree of approximation as the optimal memory size at the current iteration” is as follows:
separately calculating, g
     t
    
    c
   
   =
   
    
     [
     
      
       2
       ⁢
       
        
         ∇
         z
        
        *
        
         
          J
          ⁡
          (
          
           z
           ,
           
            z
            *
           
          
          )
         
         T
        
       
      
      ,
       
      
       2
       ⁢
       
        
         ∇
         z
        
        
         
          J
          ⁡
          (
          
           z
           ,
           
            z
            *
           
          
          )
         
         T
        
       
      
     
     ]
    
    T
   
  
  ,
  
   
    
     s
     t
    
    c
   
   =
   
    
     
      z
      t
     
     c
    
    -
    
     z
     
      t
      -
      1
     
    
   
  
  ,
     
  
   
    and
    ⁢
    
      
       
    
    ⁢
    
     
      y
      t
     
     c
    
   
   =
   
    
     
      g
      t
     
     c
    
    -
    
     
      g
      c
     
     
      t
      -
      1, according to the loss function J(z,z*), wherein, z
     c
    
    t
   
   =
   
    
     (
     
      
       z
       t
      
      ,
      
       z
       t
       *
      
     
     )
    
    T
   
  
  ;
 

respectively calculating, γ
    c
   
   t
  
  =
  
   
    
     s
     c
    
    t
   
   -
   
    
     1
     3
    
    ⁢
    
     
      s
      c
     
     
      t
      -
      1, and, ω
    c
   
   t
  
  =
  
   
    
     y
     c
    
    t
   
   -
   
    
     1
     3
    
    ⁢
    
     
      y
      c
     
     
      t
      -
      1, in a multi-step quasi-Newton formula by using, s
     c
    
    y
   
   ⁢
      
   and
   ⁢
      
   
    
     y
     c
    
    t
   
  
  ;
 

according to the formula:, H
    ^
   
   t
   m
  
  =
  
   
    
     (
     
      
       V
       
        t
        -
        2
       
       H
      
      ⁢
         
      …
      ⁢
         
      
       V
       
        t
        -
        m
       
       H
      
     
     )
    
    ⁢
    
     
      
       H
       ^
      
      t
      
       m
       ⁢
       0
      
     
     (
     
      
       V
       
        t
        -
        m
       
      
      ⁢
         
      …
      ⁢
         
      
       V
       
        t
        -
        2
       
      
     
     )
    
   
   +
   
    
     
      
       ρ
       c
      
      
       t
       -
       m
      
     
     (
     
      
       V
       
        t
        -
        2
       
       H
      
      ⁢
         
      …
      ⁢
         
      
       V
       
        t
        -
        m
        +
        1
       
       H
      
     
     )
    
    ⁢
    
     
      γ
      c
     
     
      t
      -
      m
     
    
    ⁢
    
     
      
       γ
       c
      
      
       t
       -
       m
      
      H
     
     (
     
      
       V
       
        t
        -
        m
        +
        1
       
      
      ⁢
         
      …
      ⁢
         
      
       V
       
        t
        -
        2
       
      
     
     )
    
   
   +
   
    
     
      
       ρ
       c
      
      
       t
       -
       m
       +
       1
      
     
     (
     
      
       V
       
        t
        -
        2
       
       H
      
      ⁢
         
      …
      ⁢
         
      
       V
       
        t
        -
        m
        +
        2
       
       H
      
     
     )
    
    ⁢
    
     
      γ
      c
     
     
      t
      -
      m
      +
      1
     
    
    ⁢
    
     
      
       γ
       c
      
      
       
        t
        ⁢
        m
       
       +
       1
      
      H
     
     (
     
      
       V
       
        t
        -
        m
        +
        2
       
      
      ⁢
         
      …
      ⁢
         
      
       V
       
        t
        -
        2
       
      
     
     )
    
   
  
 

calculating a matrix Ĥtm, which is the quasi-Newton matrix and is used to depict the approximation degree between the inverse of the quasi-Newton matrix and the inverse of Hessian matrix; herein, the superscript H denotes conjugate transpose,, ρ
    c
   
   t
  
  =
  
   1
   /
   
    (
    
     
      
       ω
       c
      
      t
      H
     
     ⁢
     
      
       γ
       c
      
      t
     
    
    )
   
  
 

 
  
   
    V
    t
   
   =
   
    I
    -
    
     
      
       ρ
       c
      
      t
     
     ⁢
     
      
       ω
       c
      
      t
     
     ⁢
     
      
       γ
       c
      
      t
     
    
   
  
  ,
 

 
  
   
    
     H
     ^
    
    t
    
     m
     ⁢
     0
    
   
   =
   
    
     
      
       ω
       c
      
      
       t
       -
       1
      
      H
     
     ⁢
     
      
       γ
       c
      
      
       t
       -
       1
      
     
    
    
     
      
       ω
       c
      
      
       t
       -
       1
      
     
     ⁢
     
      
       ω
       c
      
      
       t
       -
       1
      
     
    
   
  
  ;
 

given an upper bound M of memory size, calculating the approximation degree between Ĥtm and the inverse of Hessian matrix for m=1, 2, . . . , M according to a formula, e
    m
   
   =
   
    
     
      (
      
       
        
         
          H
          ^
         
         t
         m
        
        ⁢
        
         
          ω
          c
         
         t
        
       
       -
       
        
         γ
         c
        
        t
       
      
      )
     
     H
    
    ⁢
    
     (
     
      
       
        
         H
         ^
        
        t
        m
       
       ⁢
       
        
         ω
         c
        
        t
       
      
      -
      
       
        γ
        c
       
       t
      
     
     )
    
   
  
  ;
 

based on the foregoing calculated em, selecting a corresponding value of m achieving the minimum of em as the optimal memory size m* of the complex-valued L-BFGS algorithm;
when the optimal memory size is determined, calculating a search direction at the current iteration with the help of, H
    ^
   
   t
  
  =
  
   
    
     (
     
      
       V
       
        t
        -
        1
       
       H
      
      ⁢
         
      ⋯
      ⁢
         
      
       V
       
        t
        -
        m
       
       H
      
     
     )
    
    ⁢
    
     
      
       H
       ^
      
      t
      0
     
     (
     
      
       V
       
        t
        -
        m
       
      
      ⁢
         
      ⋯
      ⁢
         
      
       V
       
        t
        -
        1
       
      
     
     )
    
   
   +
   
    
     
      
       ρ
       c
      
      
       t
       -
       m
      
     
     (
     
      
       V
       
        t
        -
        1
       
       H
      
      ⁢
         
      ⋯
      ⁢
         
      
       V
       
        t
        -
        m
        +
        1
       
       H
      
     
     )
    
    ⁢
    
     
      s
      c
     
     
      t
      -
      m
     
    
    ⁢
    
     
      
       s
       c
      
      
       t
       -
       m
      
      H
     
     (
     
      
       V
       
        t
        -
        m
        +
        1
       
      
      ⁢
         
      ⋯
      ⁢
         
      
       V
       
        t
        -
        1
       
      
     
     )
    
   
   +
   
    
     
      
       ρ
       c
      
      
       t
       -
       m
       +
       1
      
     
     (
     
      
       V
       
        t
        -
        1
       
       H
      
      ⁢
         
      ⋯
      ⁢
         
      
       V
       
        t
        -
        m
        +
        2
       
       H
      
     
     )
    
    ⁢
    
     
      s
      c
     
     
      t
      -
      m
      +
      1
     
    
    ⁢
    
     
      
       s
       c
      
      
       t
       -
       m
       +
       1
      
      H
     
     (
     
      
       V
       
        t
        -
        m
        +
        2
           
       
      
      ⁢
      ⋯
      ⁢
         
      
       V
       
        t
        -
        1
       
      
     
     )
    
   
   +
   ⋯
   +
   
    
     
      ρ
      c
     
     
      t
      -
      1
     
    
    ⁢
    
     
      s
      c
     
     
      t
      -
      1
     
    
    ⁢
    
     
      s
      c
     
     
      t
      -
      1
     
     H
    
   
  
 

calculating a search direction by, p
     c
    
    t
    T
   
   =
   
    
     -
     
      
       H
       ^
      
      t
     
    
    ⁢
    
     
      g
      c
     
     t
    
   
  
  ;, and then, using a line search condition:, J
   ⁡
   (
   
    
     
      z
      c
     
     t
    
    +
    
     
      α
      t
     
     ⁢
     
      
       p
       c
      
      t
     
    
   
   )
  
  ≤
  
   
    J
    ⁡
    (
    
     
      z
      c
     
     t
    
    )
   
   +
   
    
     k
     1
    
    ⁢
    
     α
     t
    
    ⁢
    
     
      p
      c
     
     t
    
    ⁢
    
     
      ∂
      
       J
       ⁡
       (
       
        
         z
         c
        
        t
       
       )
      
     
     
      ∂
      
       
        z
        c
       
       t
      
     
    
   
  
 

 
  
   
    
     p
     c
    
    t
    T
   
   ⁢
   
    
     ∂
     
      J
      ⁡
      (
      
       
        
         z
         c
        
        t
       
       +
       
        
         α
         t
        
        ⁢
        
         
          p
          c
         
         t
        
       
      
      )
     
    
    
     ∂
     
      
       z
       c
      
      t
     
    
   
  
  ≥
  
   
    k
    2
   
   ⁢
   
    
     p
     c
    
    t
    T
   
   ⁢
   
    
     ∂
     
      J
      ⁡
      (
      
       
        z
        c
       
       t
      
      )
     
    
    
     ∂
     
      
       z
       c
      
      t
     
    
   
  
 

calculating a step size, wherein k1 takes value from the interval (0,0.5), and k2 takes value from the interval (k1, 1), to obtain a parameter update formula as, z
    c
   
   t
  
  =
  
   
    
     z
     c
    
    
     t
     -
     1
    
   
   +
   
    
     α
     t
    
    ⁢
    
     
      
       p
       c
      
      t
      T
     
     .