PATENT ABSTRACT
An improved design method of a two-stage FRM filter includes the following steps: constructing an improved two-stage FRM filter; calculating passband and stopband edge parameters of a prototype filter, passband and stopband edge parameters of a second-stage masking filter and passband and stopband edge parameters of a first-stage masking filter in Case A and Case B, respectively; calculating the complexity of the FRM filter according to the obtained parameters, and finding out one or more sets [M, P, Q] having the lowest complexity within a search range; and optimizing the improved FRM filter. The improved design method of a two-stage FRM filter has the following beneficial effect: as compared to a conventional design method of a two-stage FRM filter, the complexity of a narrow-band FIR (Finite Impulse Response) filter can be reduced through design using the improved method, and power consumption is thus reduced in hardware implementation.

PATENT DESCRIPTION
FIELD OF THE INVENTION 
       [0001]    The present invention relates to an improved design method of a two-stage FRM filter. 
       BACKGROUND OF THE INVENTION 
       [0002]    Frequency response masking (FRM) is an efficient method for designing an FIR (Finite Impulse Response) filter having a narrow transition band characteristic. When a filter has a narrow transition band, multi-stage FRM may be utilized to further reduce its complexity. 
         [0003]    A filter is composed of a prototype filter H a (z) and two masking filters H ma (z), H mc (z). The transition band of the filter is provided by an interpolation filter H a (z M ) or its complement 
         [0000]    
       
         
           
             ( 
             
               
                 z 
                 
                   - 
                   
                     
                       M 
                        
                       
                         ( 
                         
                           
                             N 
                             a 
                           
                           - 
                           1 
                         
                         ) 
                       
                     
                     2 
                   
                 
               
               - 
               
                 
                   H 
                   a 
                 
                  
                 
                   ( 
                   
                     z 
                     M 
                   
                   ) 
                 
               
             
             ) 
           
         
       
     
         [0000]    where M is an interpolation factor for H a (z). The purpose of using two masking filters H ma (z) and H mc (z) is to remove unnecessary periodic sub-bands. 
         [0004]    Two-stage FRM filters are widely used in practice. The structure of a two-stage FRM filter is as shown in  FIG. 2 . A transition band shaping filter is represented by G(z). A constraint condition for interpolation factors M, P, Q is as follows: 
         [0000]        M=kP=kQ   (3).
 
         [0005]    At present, various improvements made to the two-stage FRM filters are based on the assumption of the equation (3). The satisfaction of the equation (3) leads to a direct problem that the second-stage output must be a periodic amplitude response. However, the second-stage output does not have to be periodic as long as it provides a desired transition band for a target filter. Therefore, the complexity of an FRM filter can be further reduced if the values of the three factors can be found in a wider range. 
       SUMMARY OF THE INVENTION 
       [0006]    The present invention aims at solving the above problem and provides an improved design method of a two-stage FRM filter. 
         [0007]    In order to achieve the above objective, the present invention employs the following technical scheme. 
         [0008]    An improved design method of a two-stage FRM filter comprises the following steps: 
         [0009]    (1) constructing an improved two-stage FRM filter having a transfer function H(z) as follows: 
         [0000]        H ( z )= G ( z ) H   ma   (1) ( z )+(1− G ( z )) H   mc   (1) ( z ),
 
         [0000]    wherein G(z)=H a   (2) (z M )H ma   (2) (z P )+(1−H a   (2) (z M )H mc   (2) (z Q ), without any constraint among interpolation factors M, P, Q;
 
H a   (2) (z M ) represents a prototype filter, while H ma   (1) (z) and H mc   (2) (z) represent first-stage masking filters, respectively, and H ma   (2) (z P ) and H mc   (2) (z Q ) represent second-stage masking filters, respectively;
 
         [0010]    (2) searching within a search range for [M, P, Q], and for a certain set [M, P, Q], calculating passband and stopband edge parameters of the prototype filter H a   (2) (z), passband and stopband edge parameters of the second-stage masking filters and passband and stopband edge parameters of the first-stage masking filters in Case A and Case B, respectively, on the basis that a transition band of the whole filter is provided by H a   (2) (z M ) or a complement of H a   (2) (z M ); and 
         [0000]    calculating the complexity of the FRM filter according to the obtained parameters, and finding out one or more sets [M, P, Q] having the lowest complexity within the search range; 
         [0011]    (3) optimizing the improved FRM filter according to the calculated filter parameters. A way of calculating the passband edge θ a  and the stopband edge φ a  of the prototype filter H a   (2) (z M ) in the step (2) is as follows: 
         [0000]    when the transition band of the whole filter is H a   (2) (z M ), i.e., in the Case A: 
         [0000]        m=└ω   p   M/ 2π┘,
 
         [0000]      θ a =ω p   M− 2 mπ,  
 
         [0000]      φ a =ω s   M− 2 mπ;  
 
         [0000]    when the transition band of the whole filter is the complement of H a   (2) (z M ), i.e., in the Case B: 
         [0000]        m=┌ω   s   M/ 2π┐,
 
         [0000]      θ a =2 mπ−ω   s   M,  
 
         [0000]      φ a =2 mπ−ω   p   M;  
 
         [0000]    wherein └x┘ represents a largest integer not more than x; ┌x┐ represents a smallest integer not less than x; whether the result satisfies 0&lt;θ a &lt;φ a &lt;π is determined in the two cases respectively, and if not, the results are abandoned. 
         [0012]    Passbands of H a   (2) (Mω) from 0 to π in the step (2) are orderly represented by 0, 2, . . . , 
         [0000]    
       
         
           
             
               2 
               × 
               
                 ⌊ 
                 
                   M 
                   2 
                 
                 ⌋ 
               
             
             , 
           
         
       
     
         [0000]    and passbands of 1−H a   (2) (Mω) from 0 to π are orderly represented by 1, 3, . . . , 
         [0000]    
       
         
           
             
               2 
               × 
               
                 ⌊ 
                 
                   
                     M 
                     - 
                     1 
                   
                   2 
                 
                 ⌋ 
               
             
             ; 
           
         
       
     
         [0000]    assuming that the passband labeled as 2m of H a   (2) (Mω) provides the transition band, the passband of the masking filter from which the transition band is extracted is defined as an effective passband; and in order to reduce the complexity of the two first-stage masking filters, the following restrictive conditions are established:
 
the passband 2m should be at least completely extracted; the passband 2(m+1) should be completely fall outside an effective passband range.
 
         [0013]    When the transition band of the whole filter is provided by H a   (2) (z M ), i.e., in the Case A, a way of calculating the passband edge ω pma   (2)  and the stopband edge ω sma   (2)  of the second-stage masking filter H ma   (2) (z) is as follows: 
         [0014]    (1) when masking is provided by H ma   (2) (z P ), the case is denoted as Case p =A: 
         [0000]    
       
         
           
             { 
             
               
                 
                   
                     
                       
                         ω 
                         pma 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       = 
                       
                         max 
                          
                         
                           ( 
                           
                             
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 p 
                               
                               - 
                               
                                 
                                   ω 
                                   1 
                                 
                                  
                                 P 
                               
                             
                             , 
                             
                               
                                 
                                   ω 
                                   2 
                                 
                                  
                                 P 
                               
                               - 
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 p 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         ω 
                         sma 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       = 
                       
                         
                           
                             ω 
                             3 
                           
                            
                           P 
                         
                         - 
                         
                           2 
                            
                           π 
                            
                           
                               
                           
                            
                           p 
                         
                       
                     
                   
                 
               
               ; 
             
           
         
       
     
         [0015]    (2) when masking is provided by the complement of H ma   (2) (z P ), the case is denoted as Case p =B: 
         [0000]    
       
         
           
             { 
             
               
                 
                   
                     
                       
                         ω 
                         pma 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       = 
                       
                         
                           2 
                            
                           π 
                            
                           
                               
                           
                            
                           p 
                         
                         - 
                         
                           
                             ω 
                             2 
                           
                            
                           P 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         ω 
                         sma 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       = 
                       
                         min 
                          
                         
                           ( 
                           
                             
                               
                                 
                                   ω 
                                   1 
                                 
                                  
                                 P 
                               
                               - 
                               
                                 2 
                                  
                                 
                                   π 
                                    
                                   
                                     ( 
                                     
                                       p 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                             , 
                             
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 p 
                               
                               - 
                               
                                 
                                   ω 
                                   2 
                                 
                                  
                                 P 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               ; 
             
           
         
       
     
         [0000]    wherein ω 1  is a left edge of the passband 2m of the prototype filter, while ω 2  is a right edge of the passband 2m of the prototype filter, and ω 3  is a left edge of the passband 2(m+1) of the prototype filter; P is the interpolation factor, which is a given value;
 
p is an integer, and should satisfy the following condition:
 
         [0000]    
       
         
           
             
               0 
               ≤ 
               p 
               ≤ 
               
                 ⌊ 
                 
                   P 
                   2 
                 
                 ⌋ 
               
             
             ; 
           
         
       
     
         [0000]    if no p satisfying the condition exists, the set of parameters is abandoned. 
         [0016]    When the transition band of the whole filter is provided by H a   (2) (z M ), i.e., in the Case A, a way of calculating the passband edge ω pmc   (2)  and the stopband edge ω smc   (2)  of the second-stage masking filter H mc   (2) (z) is as follows: 
         [0017]    (1) when masking is provided by H mc   (2) (z Q ), the case is denoted as Case q =A: 
         [0000]    
       
         
           
             { 
             
               
                 
                   
                     
                       
                         ω 
                         pma 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       = 
                       
                         max 
                          
                         
                           ( 
                           
                             
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 q 
                               
                               - 
                               
                                 
                                   ω 
                                   4 
                                 
                                  
                                 P 
                               
                             
                             , 
                             
                               
                                 
                                   ω 
                                   5 
                                 
                                  
                                 P 
                               
                               - 
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 q 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         ω 
                         sma 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       = 
                       
                         min 
                          
                         
                           ( 
                           
                             
                               
                                 
                                   ω 
                                   6 
                                 
                                  
                                 Q 
                               
                               - 
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 q 
                               
                             
                             , 
                             
                               
                                 2 
                                  
                                 
                                   π 
                                    
                                   
                                     ( 
                                     
                                       q 
                                       + 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                               - 
                               
                                 
                                   ω 
                                   7 
                                 
                                  
                                 Q 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               ; 
             
           
         
       
     
         [0018]    (2) when masking is provided by the complement of H mc   (2) (z Q ), the case is denoted as Case q =B: 
         [0000]    
       
         
           
             { 
             
               
                 
                   
                     
                       
                         ω 
                         pmc 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       = 
                       
                         max 
                          
                         
                           ( 
                           
                             
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 q 
                               
                               - 
                               
                                 
                                   ω 
                                   6 
                                 
                                  
                                 Q 
                               
                             
                             , 
                             
                               
                                 
                                   ω 
                                   7 
                                 
                                  
                                 Q 
                               
                               - 
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 q 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         ω 
                         smc 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       = 
                       
                         min 
                          
                         
                           ( 
                           
                             
                               
                                 
                                   ω 
                                   4 
                                 
                                  
                                 Q 
                               
                               - 
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 
                                   ( 
                                   
                                     q 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                             , 
                             
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 q 
                               
                               - 
                               
                                 
                                   ω 
                                   5 
                                 
                                  
                                 Q 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               ; 
             
           
         
       
     
         [0000]    wherein ω 4  is a left passband edge of the complementary filter passband 2m−1 of the prototype filter; ω 5  is a right stopband edge of the complementary filter passband 2m−1 of the prototype filter; ω 6  is a left passband edge of the complementary filter passband 2m+1 of the prototype filter; ω 7  is a right stopband edge of the complementary filter passband 2m+1 of the prototype filter;
 
q is an integer, and should satisfy the following condition:
 
         [0000]    
       
         
           
             
               0 
               ≤ 
               q 
               ≤ 
               
                 ⌊ 
                 
                   q 
                   2 
                 
                 ⌋ 
               
             
             ; 
           
         
       
     
         [0000]    if no q satisfying the condition exists, the set of parameters is abandoned. 
         [0019]    When the transition band of the whole filter is provided by the complement of H a   (2) (z M ), i.e., in the Case B, a way of calculating the passband edge ω pma   (2)  and the stopband edge ω sma   (2)  of the second-stage masking filter H ma   (2) (z) is as follows: 
         [0020]    (1) when masking is provided by H ma   (2) (z P ), the case is denoted as Case p =A: 
         [0000]    
       
         
           
             { 
             
               
                 
                   
                     
                       
                         ω 
                         pma 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       = 
                       
                         max 
                          
                         
                           ( 
                           
                             
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 p 
                               
                               - 
                               
                                 P 
                                  
                                 
                                     
                                 
                                  
                                 
                                   ω 
                                   4 
                                 
                               
                             
                             , 
                             
                               
                                 
                                   ω 
                                   5 
                                 
                                  
                                 P 
                               
                               - 
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 p 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         ω 
                         sma 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       = 
                       
                         min 
                          
                         
                           ( 
                           
                             
                               
                                 
                                   ω 
                                   6 
                                 
                                  
                                 P 
                               
                               - 
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 p 
                               
                             
                             , 
                             
                               
                                 2 
                                  
                                 
                                   π 
                                    
                                   
                                     ( 
                                     
                                       p 
                                       + 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                               - 
                               
                                 
                                   ω 
                                   7 
                                 
                                  
                                 P 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               ; 
             
           
         
       
     
         [0021]    (2) when masking is provided by the complement of H ma   (2) (z P ), the case is denoted as Case p =B: 
         [0000]    
       
         
           
             { 
             
               
                 
                   
                     
                       
                         ω 
                         pma 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       = 
                       
                         max 
                          
                         
                           ( 
                           
                             
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 p 
                               
                               - 
                               
                                 
                                   ω 
                                   6 
                                 
                                  
                                 P 
                               
                             
                             , 
                             
                               
                                 
                                   ω 
                                   7 
                                 
                                  
                                 P 
                               
                               - 
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 p 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         ω 
                         sma 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       = 
                       
                         min 
                          
                         
                           ( 
                           
                             
                               
                                 
                                   ω 
                                   4 
                                 
                                  
                                 P 
                               
                               - 
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 
                                   ( 
                                   
                                     p 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                             , 
                             
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 p 
                               
                               - 
                               
                                 
                                   ω 
                                   5 
                                 
                                  
                                 P 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               ; 
             
           
         
       
     
         [0000]    wherein ω 4  is a left passband edge of the passband 2(m−1) of the prototype filter; ω 5  is a right stopband edge of the passband 2(m−1) of the prototype filter; ω 6  is a left passband edge of the passband 2m of the prototype filter; ω 7  is a right stopband edge of the passband 2m of the prototype filter;
 
p is an integer, and should satisfy the following condition:
 
         [0000]    
       
         
           
             
               0 
               ≤ 
               p 
               ≤ 
               
                 ⌊ 
                 
                   P 
                   2 
                 
                 ⌋ 
               
             
             ; 
           
         
       
     
         [0000]    if no p satisfying the condition exists, the set of parameters is abandoned. 
         [0022]    When the transition band of the whole filter is provided by the complement of H a   (2) (z M ), i.e., in the Case B, a way of calculating the passband edge ω pmc   (2)  and the stopband edge ω smc   (2)  of the second-stage masking filter H ma   (2) (z) is as follows: 
         [0023]    (1) when masking is provided by H mc   (2) (z Q ), the case is denoted as Case q =A: 
         [0000]    
       
         
           
             { 
             
               
                 
                   
                     
                       
                         ω 
                         pma 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       = 
                       
                         max 
                          
                         
                           ( 
                           
                             
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 q 
                               
                               - 
                               
                                 
                                   ω 
                                   1 
                                 
                                  
                                 Q 
                               
                             
                             , 
                             
                               
                                 
                                   ω 
                                   2 
                                 
                                  
                                 Q 
                               
                               - 
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 q 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         ω 
                         sma 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       = 
                       
                         
                           
                             ω 
                             2 
                           
                            
                           Q 
                         
                         - 
                         
                           2 
                            
                           π 
                            
                           
                               
                           
                            
                           q 
                         
                       
                     
                   
                 
               
               ; 
             
           
         
       
     
         [0024]    (2) when masking is provided by the complement of H mc   (2) (z Q ), the case is denoted as Case q =B: 
         [0000]    
       
         
           
             { 
             
               
                 
                   
                     
                       
                         ω 
                         pma 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       = 
                       
                         
                           2 
                            
                           π 
                            
                           
                               
                           
                            
                           q 
                         
                         - 
                         
                           
                             ω 
                             2 
                           
                            
                           Q 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         ω 
                         sma 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       = 
                       
                         min 
                          
                         
                           ( 
                           
                             
                               
                                 
                                   ω 
                                   1 
                                 
                                  
                                 Q 
                               
                               - 
                               
                                 2 
                                  
                                 
                                   π 
                                    
                                   
                                     ( 
                                     
                                       q 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                             , 
                             
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 q 
                               
                               - 
                               
                                 
                                   ω 
                                   2 
                                 
                                  
                                 Q 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               ; 
             
           
         
       
     
         [0000]    wherein ω 1  is a left stopband edge of the complementary filter passband 2m−1 of the prototype filter; ω 2  is a right passband edge of the complementary filter passband 2m−1 of the prototype filter; ω 3  is a left stopband edge of the complementary filter passband 2m+1 of the prototype filter;
 
q is an integer, and should satisfy the following condition:
 
         [0000]    
       
         
           
             
               0 
               ≤ 
               q 
               ≤ 
               
                 ⌊ 
                 
                   q 
                   2 
                 
                 ⌋ 
               
             
             ; 
           
         
       
     
         [0000]    if no q satisfying the condition exists, the set of parameters is abandoned. 
         [0025]    When the transition band of the whole filter is provided by H a   (2) (z M ), i.e., in the Case A: 
         [0000]    the passband edge ω pma   (1)  of the first-stage masking filter H ma   (1) (z) is equal to ω p ;
 
a way of calculating the stopband edge ω sma   (1)  of the first-stage masking filter H ma   (1) (z) is as follows:
 
         [0000]    
       
         
           
             
               
                 ω 
                 sma 
                 
                   ( 
                   1 
                   ) 
                 
               
               = 
               
                 min 
                  
                 
                   ( 
                   
                     
                       ω 
                       
                         sma 
                          
                         
                             
                         
                          
                         _ 
                          
                         
                             
                         
                          
                         temp 
                          
                         
                             
                         
                          
                         1 
                       
                       
                         ( 
                         1 
                         ) 
                       
                     
                     , 
                     
                       ω 
                       
                         sma 
                          
                         
                             
                         
                          
                         _ 
                          
                         
                             
                         
                          
                         temp 
                          
                         
                             
                         
                          
                         2 
                       
                       
                         ( 
                         1 
                         ) 
                       
                     
                   
                   ) 
                 
               
             
             , 
             
               
 
             
              
             wherein 
           
         
       
       
         
           
             
               ω 
               
                 sma 
                  
                 
                     
                 
                  
                 _ 
                  
                 
                     
                 
                  
                 temp 
                  
                 
                     
                 
                  
                 1 
               
               
                 ( 
                 1 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         t 
                         1 
                       
                     
                     
                       
                         
                           t 
                           1 
                         
                         ∉ 
                         
                           
                             R 
                             stop 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         ω 
                         8 
                       
                     
                     
                       
                         
                           t 
                           1 
                         
                         ∈ 
                         
                           
                             R 
                             stop 
                           
                            
                           
                             ( 
                             
                               k 
                               1 
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 , 
                 
                   
 
                 
                  
                 
                   
                     ω 
                     
                       sma 
                        
                       
                           
                       
                        
                       _ 
                        
                       
                           
                       
                        
                       temp 
                        
                       
                           
                       
                        
                       2 
                     
                     
                       ( 
                       1 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               t 
                               2 
                             
                           
                           
                             
                               
                                 t 
                                 2 
                               
                               ∉ 
                               
                                 
                                   R 
                                   pass 
                                 
                                  
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               ω 
                               9 
                             
                           
                           
                             
                               
                                 t 
                                 2 
                               
                               ∈ 
                               
                                 
                                   R 
                                   pass 
                                 
                                  
                                 
                                   ( 
                                   
                                     k 
                                     2 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       , 
                       
                         
 
                       
                        
                       
                         
                           
                             ω 
                             θ 
                           
                           = 
                           
                             
                               ( 
                               
                                 
                                   2 
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   
                                     k 
                                     1 
                                   
                                 
                                 = 
                                 
                                   ϕ 
                                   a 
                                 
                               
                               ) 
                             
                             / 
                             M 
                           
                         
                         ; 
                         
                           
                             ω 
                             9 
                           
                           = 
                           
                             
                               ( 
                               
                                 
                                   2 
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   
                                     k 
                                     2 
                                   
                                 
                                 - 
                                 
                                   θ 
                                   a 
                                 
                               
                               ) 
                             
                             / 
                             M 
                           
                         
                         ; 
                         
                           
 
                         
                          
                         
                           
                             
                               R 
                               pass 
                             
                              
                             
                               ( 
                               k 
                               ) 
                             
                           
                           = 
                           
                             [ 
                             
                               
                                 
                                   
                                     2 
                                      
                                     π 
                                      
                                     
                                         
                                     
                                      
                                     k 
                                   
                                   - 
                                   
                                     θ 
                                     a 
                                   
                                 
                                 M 
                               
                               , 
                               
                                 
                                   
                                     2 
                                      
                                     π 
                                      
                                     
                                         
                                     
                                      
                                     k 
                                   
                                   + 
                                   
                                     θ 
                                     a 
                                   
                                 
                                 M 
                               
                             
                             ] 
                           
                         
                       
                       , 
                       
                         k 
                         = 
                         0 
                       
                       , 
                       … 
                        
                       
                           
                       
                       , 
                       
                         
                           ⌊ 
                           
                             M 
                             / 
                             2 
                           
                           ⌋ 
                         
                         ; 
                         
                           
 
                         
                          
                         
                           
                             
                               R 
                               stop 
                             
                              
                             
                               ( 
                               k 
                               ) 
                             
                           
                           = 
                           
                             [ 
                             
                               
                                 
                                   
                                     2 
                                      
                                     π 
                                      
                                     
                                         
                                     
                                      
                                     
                                       ( 
                                       
                                         k 
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                   
                                   + 
                                   
                                     ϕ 
                                     a 
                                   
                                 
                                 M 
                               
                               , 
                               
                                 
                                   
                                     2 
                                      
                                     π 
                                      
                                     
                                         
                                     
                                      
                                     k 
                                   
                                   + 
                                   
                                     ϕ 
                                     a 
                                   
                                 
                                 M 
                               
                             
                             ] 
                           
                         
                       
                       , 
                       
                         k 
                         = 
                         0 
                       
                       , 
                       … 
                        
                       
                           
                       
                       , 
                       
                         
                           ⌊ 
                           
                             M 
                             / 
                             2 
                           
                           ⌋ 
                         
                         ; 
                       
                     
                   
                 
               
             
           
         
       
     
         [0026]    (1) when H ma   (2) (z P ) is used for masking unnecessary band of an up-branch, 
         [0000]        t   1 =(2π( p+ 1)−ω sma   (2) )/ P;  
 
         [0027]    (2) when the complement of H ma   (2) (z P ) is used for masking the unnecessary band of the up-branch, 
         [0000]        t   1 =(2π p+ω   pma   (2) )/ P;  
 
         [0028]    (3) when H mc   (2) (Z Q ) is used for masking unnecessary band of a down-branch, 
         [0000]        t   2 =(2π( q+ 1)−ω smc   (2) )/ Q;  
 
         [0029]    (4) when a down-branch complement of H mc   (2) (Z Q ) is used for masking unnecessary band, 
         [0000]        t   2 =(2π q+ω   pma   (2) )/ Q;  
 
         [0000]    wherein P, Q, M are interpolation factors; ω 8  is a right endpoint of R stop (k 1 ); k 1  is an integer satisfying t 1 εR stop (k 1 ); ω 9  is a right endpoint of R pass (k 2 ); k 2  is an integer satisfying t 2 εR stop (k 2 ); θ a  is a passband edge of H a   (2) (z); H ma   (1) (z)φ a  is a stopband edge of H a   (2) (z). 
         [0030]    When the transition band of the whole filter is provided by the complement of H a   (2) (z M ), i.e., in the Case B: 
         [0000]    the passband edge ω pma   (1)  of the first-stage masking filter H ma   (1) (z) is equal to ω p ;
 
a way of calculating the stopband edge ω sma   (1)  of the first-stage masking filter H ma   (1) (z) is as follows:
 
         [0000]    
       
         
           
             
               
                 ω 
                 sma 
                 
                   ( 
                   1 
                   ) 
                 
               
               = 
               
                 min 
                  
                 
                   ( 
                   
                     
                       ω 
                       
                         sma 
                          
                         
                             
                         
                          
                         _ 
                          
                         
                             
                         
                          
                         temp 
                          
                         
                             
                         
                          
                         1 
                       
                       
                         ( 
                         1 
                         ) 
                       
                     
                     , 
                     
                       ω 
                       
                         sma 
                          
                         
                             
                         
                          
                         _ 
                          
                         
                             
                         
                          
                         temp 
                          
                         
                             
                         
                          
                         2 
                       
                       
                         ( 
                         1 
                         ) 
                       
                     
                   
                   ) 
                 
               
             
             , 
             
               
 
             
              
             wherein 
           
         
       
       
         
           
             
               ω 
               
                 sma 
                  
                 
                     
                 
                  
                 _ 
                  
                 
                     
                 
                  
                 temp 
                  
                 
                     
                 
                  
                 1 
               
               
                 ( 
                 1 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         t 
                         1 
                       
                     
                     
                       
                         
                           t 
                           1 
                         
                         ∉ 
                         
                           
                             R 
                             stop 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         ω 
                         8 
                       
                     
                     
                       
                         
                           t 
                           1 
                         
                         ∈ 
                         
                           
                             R 
                             stop 
                           
                            
                           
                             ( 
                             
                               k 
                               1 
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 , 
                 
                   
 
                 
                  
                 
                   
                     ω 
                     
                       sma 
                        
                       
                           
                       
                        
                       _ 
                        
                       
                           
                       
                        
                       temp 
                        
                       
                           
                       
                        
                       2 
                     
                     
                       ( 
                       1 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               t 
                               2 
                             
                           
                           
                             
                               
                                 t 
                                 2 
                               
                               ∉ 
                               
                                 
                                   R 
                                   pass 
                                 
                                  
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               ω 
                               9 
                             
                           
                           
                             
                               
                                 t 
                                 2 
                               
                               ∈ 
                               
                                 
                                   R 
                                   pass 
                                 
                                  
                                 
                                   ( 
                                   
                                     k 
                                     2 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       , 
                       
                         
 
                       
                        
                       
                         
                           
                             ω 
                             θ 
                           
                           = 
                           
                             
                               ( 
                               
                                 
                                   2 
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   
                                     k 
                                     1 
                                   
                                 
                                 = 
                                 
                                   θ 
                                   a 
                                 
                               
                               ) 
                             
                             / 
                             M 
                           
                         
                         ; 
                         
                           
                             ω 
                             9 
                           
                           = 
                           
                             
                               ( 
                               
                                 
                                   2 
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   
                                     k 
                                     2 
                                   
                                 
                                 - 
                                 
                                   ϕ 
                                   a 
                                 
                               
                               ) 
                             
                             / 
                             M 
                           
                         
                         ; 
                         
                           
 
                         
                          
                         
                           
                             
                               R 
                               pass 
                             
                              
                             
                               ( 
                               k 
                               ) 
                             
                           
                           = 
                           
                             [ 
                             
                               
                                 
                                   
                                     2 
                                      
                                     π 
                                      
                                     
                                         
                                     
                                      
                                     k 
                                   
                                   - 
                                   
                                     θ 
                                     a 
                                   
                                 
                                 M 
                               
                               , 
                               
                                 
                                   
                                     2 
                                      
                                     π 
                                      
                                     
                                         
                                     
                                      
                                     k 
                                   
                                   + 
                                   
                                     θ 
                                     a 
                                   
                                 
                                 M 
                               
                             
                             ] 
                           
                         
                       
                       , 
                       
                         k 
                         = 
                         0 
                       
                       , 
                       … 
                        
                       
                           
                       
                       , 
                       
                         
                           ⌊ 
                           
                             M 
                             / 
                             2 
                           
                           ⌋ 
                         
                         ; 
                         
                           
 
                         
                          
                         
                           
                             
                               R 
                               stop 
                             
                              
                             
                               ( 
                               k 
                               ) 
                             
                           
                           = 
                           
                             [ 
                             
                               
                                 
                                   
                                     2 
                                      
                                     π 
                                      
                                     
                                         
                                     
                                      
                                     
                                       ( 
                                       
                                         k 
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                   
                                   + 
                                   
                                     ϕ 
                                     a 
                                   
                                 
                                 M 
                               
                               , 
                               
                                 
                                   
                                     2 
                                      
                                     π 
                                      
                                     
                                         
                                     
                                      
                                     k 
                                   
                                   + 
                                   
                                     ϕ 
                                     a 
                                   
                                 
                                 M 
                               
                             
                             ] 
                           
                         
                       
                       , 
                       
                         k 
                         = 
                         0 
                       
                       , 
                       … 
                        
                       
                           
                       
                       , 
                       
                         
                           ⌊ 
                           
                             M 
                             / 
                             2 
                           
                           ⌋ 
                         
                         ; 
                       
                     
                   
                 
               
             
           
         
       
     
         [0031]    (1) when H mc   (2) (z Q ) is used for masking unnecessary band, 
         [0000]        t   1 =(2π( q+ 1)−ω smc   (2) )/ Q;  
 
         [0032]    (2) when the complement of H mc   (2) (z Q ) is used for masking unnecessary band, 
         [0000]        t   1 =(2π q+ω   pma   (2) )/ Q;  
 
         [0033]    (3) when H ma   (2) (z P ) is used for masking unnecessary band, 
         [0000]        t   2 =(2π( p+ 1)−ω sma   (2) )/ P;  
 
         [0034]    (4) when the complement of H ma   (2) (z P ) is used for masking unnecessary band, 
         [0000]        t   2 =(2π p+ω   pma   (2) )/ P;  
 
         [0000]    wherein ω 8  is a right endpoint of R stop (k 1 ); k 1  is an integer satisfying t 1 εR stop (k 1 ); ω 9  is a right endpoint of R pass (k 2 ); k 2  is an integer satisfying t 2 εR stop (k 2 ); P, Q, M are interpolation factors; θ a  is a passband edge of H a   (2) (z); H ma   (1) (z)φ a  is a stopband edge of H a   (2) (z). 
         [0035]    (1) When the transition band of the whole filter is provided by H a   (2) (z M ), i.e., in the Case A: 
         [0000]    the stopband edge ω smc   (1)  of the first-stage masking filter H mc   (1) (z) is equal to ω s ;
 
a way of determining the passband edge ω pmc   (1)  of the first-stage masking filter H mc   (1) (z) is as follows:
 
if t 3 ≧t 4 , then
 
         [0000]    
       
         
           
             
               ω 
               pmc 
               
                 ( 
                 1 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         t 
                         3 
                       
                     
                     
                       
                         
                           t 
                           3 
                         
                         ∉ 
                         
                           
                             R 
                             stop 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         max 
                          
                         
                           ( 
                           
                             
                               
                                 
                                   2 
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         k 
                                         3 
                                       
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   ϕ 
                                   a 
                                 
                               
                               M 
                             
                             , 
                             
                               t 
                               4 
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           t 
                           3 
                         
                         ∈ 
                         
                           
                             R 
                             stop 
                           
                            
                           
                             ( 
                             
                               k 
                               3 
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 ; 
               
             
           
         
       
     
         [0000]    if t 3 &lt;t 4 , then 
         [0000]    
       
         
           
             
               ω 
               pmc 
               
                 ( 
                 1 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         t 
                         4 
                       
                     
                     
                       
                         
                           t 
                           4 
                         
                         ∉ 
                         
                           
                             R 
                             pass 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         max 
                          
                         
                           ( 
                           
                             
                               
                                 
                                   2 
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   
                                     k 
                                     4 
                                   
                                 
                                 - 
                                 
                                   θ 
                                   a 
                                 
                               
                               M 
                             
                             , 
                             
                               t 
                               3 
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           t 
                           4 
                         
                         ∈ 
                         
                           
                             R 
                             pass 
                           
                            
                           
                             ( 
                             
                               k 
                               4 
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 ; 
               
             
           
         
       
     
         [0000]    (a) when H ma   (2) (z P ) is used for masking unnecessary band, 
         [0000]        t   3 =(2π p−ω   pma   (2) )/ P;  
 
         [0000]    (b) when the complement of H ma   (2) (z P ) is used for masking unnecessary band, 
         [0000]        t   3 =(2π( p− 1)+ω sma   (2) )/ P;  
 
         [0000]    (c) when H mc   (2) (z Q ) is used for masking unnecessary band, 
         [0000]        t   4 =(2π q−ω   pmc   (2) )/ Q;  
 
         [0000]    (d) when the complement of H mc   (2) (z Q ) is used for masking unnecessary band, 
         [0000]        t   4 =(2π( q− 1)+ω sma   (2) )/ Q;  
 
         [0036]    (2) When the transition band of the whole filter is provided by the complement of H a   (2) (z M ), i.e., in the Case B: 
         [0000]    the stopband edge ω smc   (1)  to of the first-stage masking filter H mc   (1) (z) is equal to ω s ;
 
a way of determining the passband edge ω pmc   (1)  of the first-stage masking filter H mc   (1) (z) is as follows:
 
if t 3 ≧t 4 , then
 
         [0000]    
       
         
           
             
               ω 
               pmc 
               
                 ( 
                 1 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         t 
                         3 
                       
                     
                     
                       
                         
                           t 
                           3 
                         
                         ∉ 
                         
                           
                             R 
                             stop 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         max 
                          
                         
                           ( 
                           
                             
                               
                                 
                                   2 
                                    
                                   
                                     π 
                                      
                                     
                                       ( 
                                       
                                         
                                           k 
                                           3 
                                         
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                   
                                 
                                 + 
                                 
                                   ϕ 
                                   a 
                                 
                               
                               M 
                             
                             , 
                             
                               t 
                               4 
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           t 
                           3 
                         
                         ∈ 
                         
                           
                             R 
                             stop 
                           
                            
                           
                             ( 
                             
                               k 
                               3 
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 ; 
               
             
           
         
       
     
         [0000]    if t 3 &lt;4, then 
         [0000]    
       
         
           
             
               ω 
               pmc 
               
                 ( 
                 1 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         t 
                         4 
                       
                     
                     
                       
                         
                           t 
                           4 
                         
                         ∉ 
                         
                           
                             R 
                             pass 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         max 
                          
                         
                           ( 
                           
                             
                               
                                 
                                   2 
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   
                                     k 
                                     4 
                                   
                                 
                                 - 
                                 
                                   θ 
                                   a 
                                 
                               
                               M 
                             
                             , 
                             
                               t 
                               3 
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           t 
                           4 
                         
                         ∈ 
                         
                           
                             R 
                             pass 
                           
                            
                           
                             ( 
                             
                               k 
                               4 
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 ; 
               
             
           
         
       
     
         [0000]    (a) when H mc   (2) (z Q ) is used for masking unnecessary band, 
         [0000]        t   3 =(2π( q− 1)+ω sma   (2) )/ Q;  
 
         [0000]    (b) when the complement of H mc   (2) (z Q ) is used for masking unnecessary band, 
         [0000]        t   3 =(2π( q− 1)+ω sma   (2) )/ Q;  
 
         [0000]    (c) when H ma   (2) (z P ) is used for masking unnecessary band, 
         [0000]        t   4 =(2π( p− 1)+ω sma   (2) )/ P;  
 
         [0000]    (d) when the complement of H ma   (2) (z P ) is used for masking unnecessary band, 
         [0000]        t   4 =(2π( p− 1)+ω sma   (2) )/ P;  
 
         [0000]    wherein t 3  and t 4  are left passband edges of the passband including the transition band and a first passband on the left of the transition band, respectively; k 3  is an integer satisfying t 3 εR stop (k 3 ); k 4  is an integer satisfying t 4 εR pass (k 4 ); P, Q, M are interpolation factors; θ a  is a passband edge of H a   (2) (z); H ma   (1) (z)ω a  is a stopband edge of H a   (2) (z). 
         [0037]    The present invention has the following beneficial effects: 
         [0000]    according to the present invention, the restriction that the interpolation factors must satisfy the constraint condition M=kP=kQ in a conventional structure is broken by constructing an improved two-stage FRM filter structure, and the subfilters are optimized simultaneously by means of a nonlinear joint optimization method. Results indicate that, as compared to a conventional design method of a two-stage FRM filter, the complexity of a narrow-band FIR (Finite Impulse Response) filter can be reduced through design using the improved method, and power consumption is thus reduced in hardware implementation. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0038]      FIG. 1  is a structural schematic diagram of an improved two-stage FRM filter of the present invention. 
           [0039]      FIG. 2  is an amplitude response possibly existing in G(z) of the present invention. 
           [0040]      FIG. 3  is an amplitude response of an up-branch of G(z) of the present invention. 
           [0041]      FIG. 4  is an amplitude response of a down-branch of G(z) of the present invention. 
           [0042]      FIG. 5  is a schematic diagram of calculation of a stopband edge ω sma   (2)  in Case=A according to the present invention. 
           [0043]      FIG. 6  is a schematic diagram of calculation of a passband edge ω pma   (2)  in Case=A according to the present invention. 
       
    
    
     DETAILED DESCRIPTION OF THE EMBODIMENTS 
       [0044]    The present invention will be further illustrated below in combination with the accompanying drawings and embodiment. 
         [0045]    The structure of an improved two-stage FRM filter is as shown in  FIG. 1  in which M, P, Q are interpolation factors. A transfer function H(z) of the filter is expressed as the following formula: 
         [0000]        H ( z )= G ( z ) H   ma   (1) ( z )+(1− G ( z )) H   mc   (1) ( z ),  (4)
 
         [0000]        G ( z )= H   a   (2) ( z   M ) H   ma   (2) ( z   P )+(1− H   a   (2) ( z   M )) H   mc   (2) ( z   Q ).  (5)
 
         [0046]    Z-transformation transfer function of G(z) is expressed as the formula (5), and the possible amplitude response existing therein is as shown in  FIG. 2  in which ω p  and ω s  represent a passband edge and a stopband edge of the designed filter, respectively. It should be noted that the amplitude response of G(z) may be not periodic because of no any constraint among the interpolation factors M, P, Q. The passband and stopband edges ω pma   (1)  and ω sma   (1)  of a filter H ma   (1) (z) are as shown in  FIG. 2 . A distance between ω pma   (1)  and ω sma   (1)  is indicated by d 1 , and a distance between ω pmc   (1)  and ω smc   (1)  is indicated by d 2 . 
         [0047]    It is assumed that the passband ripple and the stopband ripple of the filter are represented by δ p  and δ s , respectively. Since a conventional calculation way cannot be used to determine the passband and stopband edges of each subfilter, a new way is required to obtain them. 
         [0048]    The transition band of the whole filter may be formed by H a   (2) (z M ) or its complement, and therefore, there are two cases: Case A and Case B. In the Case A, the transition band of the filter is provided by H a   (2) (z M ); in the Case B, the transition band of the filter is provided by the complement of H a   (2) (z M ). Besides, masking may be achieved by H ma   (2) (z P ) and H mc   (2) (z Q ) or their complements. The following two parameters are defined to distinguish the above cases: 
         [0000]    when H ma   (2) (z P ) is used, Case p =A, and when H ma   (2) (z P ) is used, Case q =A;
 
when the complement of H ma   (2) (z P ) is used, Case p =B, and when the complement of H ma   (2) (z P ) is used, Case q =B.
 
         [0049]    Depending on the positions of the passband and stopband edges, there exist multiple cases for the improved structure. Now, it is first to focus on the design in Case A since Case A and Case B are similar. A set of interpolation factors [M, P, Q] leading to the lowest complexity of the filter may be found out through global search, and the complexity is decided by the number of multipliers. For the given M, P and Q, illustration will be made on how the parameters of these subfilters are determined below. 
         [0050]    I. Calculation of the Passband and Stopband Edges of a Prototype Filter H a   (2) (z) 
         [0051]    The passband and stopband edges of the prototype filter H a   (2) (z) are determined by a conventional way. The H a   (2) (z) passband edge θ a  and the stopband edge φ a  of H a   (2) (z) are obtained readily: 
       For Case A, 
       [0052]        m=└ω   p   M/ 2π┘,  (6a)
 
         [0000]      θ a =ω p   M− 2 mπ,   (6b)
 
         [0000]      φ a =ω s   M− 2 mπ.   (6c);
 
       For Case B, 
       [0053]        m=┌ω   s   M/ 2π┐,  (7a)
 
         [0000]      θ a =2 mπ−ω   s   M,   (7b)
 
         [0000]      φ a =2 mπ−ω   p   M,   (7c)
 
         [0000]    wherein └x┘ represents a largest integer not more than x. ┌x┐ represents a smallest integer not less than x. whether the results satisfies 0&lt;θ a &lt;φ a &lt;π is determined in the two cases respectively, and if not, the results are abandoned. The condition 0&lt;θ a &lt;φ a &lt;π must be satisfied in both cases, and only one case meets the requirement. 
         [0054]    II. Calculation of the Passband and the Stopband of Second-Stage Masking Filters 
         [0055]    The passband and stopband edges of H ma   (2) (z) and H mc   (2) (z) are represented by ω pma   (2) , ω sma   (2) , ω pmc   (2)  and ω smc   (2) , respectively. In addition, passbands of H a   (2) (Hω) from 0 to π in the step (2) are orderly represented by 0, 2, . . . , 
         [0000]    
       
         
           
             
               2 
               × 
               
                 ⌊ 
                 
                   M 
                   2 
                 
                 ⌋ 
               
             
             , 
           
         
       
     
         [0000]    and passbands of 1−H a   (2) (Mω) from 0 to π are orderly represented by 1, 3, . . . , 
         [0000]    
       
         
           
             2 
             × 
             
               
                 ⌊ 
                 
                   
                     M 
                     - 
                     1 
                   
                   2 
                 
                 ⌋ 
               
               . 
             
           
         
       
     
       1) Case A 
       [0056]    The amplitude response of the up-branch of G(z) is as shown in  FIG. 3 . Assuming that the passband labeled as 2m of H a   (2) (Mω) provides the transition band, the passband of the masking filter from which the transition band is extracted is defined as an “effective passband”. With reference to  FIG. 3 , a restrictive condition is established to prevent d 1  from being too small: the passband 2m should be at least completely extracted. In order to prevent d 2  from being too small: a restrictive condition is also established: the passband 2(m+1) should completely fall outside an effective passband range. The two constraint conditions may also be described as follows: 
         [0000]    1) the left passband edge x 1  of the effective passband should not be greater than the left edge ω 1  of the passband 2m;
 
2) the right passband edge x 2  of the effective passband should not be lower than the right edge ω 2  of the passband 2m; ω 2  
 
3) the right stopband edge x 3  of the effective passband should not be greater than the left edge ω 3  of the passband 2(m+1).
 
         [0057]    Relevant inequations and variable values are as shown in Table I in which 
         [0000]    
       
         
           
             0 
             ≤ 
             p 
             ≤ 
             
               
                 ⌊ 
                 
                   P 
                   2 
                 
                 ⌋ 
               
               . 
             
           
         
       
     
         [0000]    For the known set of interpolation factors [M, P, Q], if there exists p allowing ω pma   (2)  and ω sma   (2)  to satisfy the three inequations from (10a) to (10c), other filter parameters will be calculated continuously; or otherwise, this set of interpolation factors is abandoned. After the inequations are solved, upper and lower bounds of ω pma   (2)  and ω sma   (2)  may be obtained. The maximum of ω sma   (2)  and the minimum of ω pma   (2)  are taken, respectively, and then the transition band of the filter H ma   (2) (z) is the widest. The values of ω pma   (2)  and ω sma   (2)  are as follows: 
       For Case p =A, 
       [0058]    
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 ω 
                                 pma 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                               = 
                               
                                 max 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         2 
                                          
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         p 
                                       
                                       - 
                                       
                                         
                                           ω 
                                           1 
                                         
                                          
                                         P 
                                       
                                     
                                     , 
                                     
                                       
                                         
                                           ω 
                                           2 
                                         
                                          
                                         P 
                                       
                                       - 
                                       
                                         2 
                                          
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         p 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             , 
                           
                         
                       
                       
                         
                           
                             
                               ω 
                               sma 
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                             = 
                             
                               
                                 
                                   ω 
                                   3 
                                 
                                  
                                 P 
                               
                               - 
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 p 
                               
                             
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
       For Case p =B, 
       [0059]    
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 ω 
                                 pma 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                               = 
                               
                                 
                                   2 
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   p 
                                 
                                 - 
                                 
                                   
                                     ω 
                                     3 
                                   
                                    
                                   P 
                                 
                               
                             
                             , 
                           
                         
                       
                       
                         
                           
                             
                               ω 
                               sma 
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                             = 
                             
                               min 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         ω 
                                         1 
                                       
                                        
                                       P 
                                     
                                     - 
                                     
                                       2 
                                        
                                       
                                         π 
                                          
                                         
                                           ( 
                                           
                                             p 
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   , 
                                   
                                     
                                       2 
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       p 
                                     
                                     - 
                                     
                                       
                                         ω 
                                         2 
                                       
                                        
                                       P 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0000]    
       
         
               
             
           
               
                 TABLE I 
               
               
                   
               
               
                 Variable Values And Inequations For Calculating The Passband And The 
               
               
                 Stopband Of H ma   (2)  (z) In Case A 
               
               
                   
               
             
             
               
                 
                   
                     
                       
                         
                           
                             Inequations 
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   x 
                                   1 
                                 
                                 ≤ 
                                 
                                   ω 
                                   1 
                                 
                               
                               = 
                               
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     π 
                                      
                                     
                                         
                                     
                                      
                                     m 
                                   
                                   - 
                                   
                                     ϕ 
                                     a 
                                   
                                 
                                 M 
                               
                             
                           
                           
                             
                               ( 
                               
                                 10 
                                  
                                 
                                     
                                 
                                  
                                 a 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   x 
                                   2 
                                 
                                 ≥ 
                                 
                                   ω 
                                   2 
                                 
                               
                               = 
                               
                                 ω 
                                 p 
                               
                             
                           
                           
                             
                               ( 
                               
                                 10 
                                  
                                 
                                     
                                 
                                  
                                 b 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   x 
                                   3 
                                 
                                 ≤ 
                                 
                                   ω 
                                   2 
                                 
                               
                               = 
                               
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     
                                       π 
                                        
                                       
                                         ( 
                                         
                                           m 
                                           + 
                                           1 
                                         
                                         ) 
                                       
                                     
                                   
                                   - 
                                   
                                     ϕ 
                                     a 
                                   
                                 
                                 M 
                               
                             
                           
                           
                             
                               ( 
                               
                                 10 
                                  
                                 
                                     
                                 
                                  
                                 c 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 Case 
                                 p 
                               
                               = 
                               A 
                             
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                               
                                 x 
                                 1 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       p 
                                     
                                     - 
                                     
                                       ω 
                                       pma 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 P 
                               
                             
                           
                           
                             
                               ( 
                               
                                 11 
                                  
                                 
                                     
                                 
                                  
                                 a 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 x 
                                 2 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       p 
                                     
                                     + 
                                     
                                       ω 
                                       pma 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 P 
                               
                             
                           
                           
                             
                               ( 
                               
                                 11 
                                  
                                 
                                     
                                 
                                  
                                 b 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 x 
                                 3 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       p 
                                     
                                     + 
                                     
                                       ω 
                                       sma 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 P 
                               
                             
                           
                           
                             
                               ( 
                               
                                 11 
                                  
                                 
                                     
                                 
                                  
                                 c 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 Case 
                                 p 
                               
                               = 
                               B 
                             
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                               
                                 x 
                                 1 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       
                                         ( 
                                         
                                           p 
                                           - 
                                           1 
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       ω 
                                       sma 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 P 
                               
                             
                           
                           
                             
                               ( 
                               
                                 12 
                                  
                                 
                                     
                                 
                                  
                                 a 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 x 
                                 2 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       p 
                                     
                                     - 
                                     
                                       ω 
                                       sma 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 P 
                               
                             
                           
                           
                             
                               ( 
                               
                                 12 
                                  
                                 
                                     
                                 
                                  
                                 b 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 x 
                                 3 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       p 
                                     
                                     - 
                                     
                                       ω 
                                       pma 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 P 
                               
                             
                           
                           
                             
                               ( 
                               
                                 12 
                                  
                                 
                                     
                                 
                                  
                                 c 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
             
          
         
       
     
         [0060]    The passband and stopband edges of H mc   (2) (z) may be obtained similarly. The amplitude response of the down-branch of G(z) is as shown in  FIG. 4 . To prevent d 1  from being too small, the passband 2(m−1) is fully reserved. To prevent d 2  from being too small, the passband (2m+1) is fully removed. Corresponding inequations and variable values are shown in Table II in which 
         [0000]    
       
         
           
             0 
             ≤ 
             q 
             ≤ 
             
               
                 ⌊ 
                 
                   q 
                   2 
                 
                 ⌋ 
               
               . 
             
           
         
       
     
         [0000]    If there exists the parameter q allowing ω pmc   (2)  and ω smc   (2)  to satisfy the four inequations from (14a) to (14d), other filter parameters will be calculated continuously; or otherwise, this set of interpolation factors is abandoned. To make the transition band of H mc   (2) (z) widest, the maximum of ω smc   (2)  and the minimum of ω pmc   (2)  are taken, respectively. 
       For Case q =A, 
       [0061]    
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 ω 
                                 pmc 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                               = 
                               
                                 max 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         2 
                                          
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         q 
                                       
                                       - 
                                       
                                         
                                           ω 
                                           4 
                                         
                                          
                                         Q 
                                       
                                     
                                     , 
                                     
                                       
                                         
                                           ω 
                                           5 
                                         
                                          
                                         Q 
                                       
                                       - 
                                       
                                         2 
                                          
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         q 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             , 
                           
                         
                       
                       
                         
                           
                             
                               ω 
                               smc 
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                             = 
                             
                               min 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         ω 
                                         6 
                                       
                                        
                                       Q 
                                     
                                     - 
                                     
                                       2 
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       q 
                                     
                                   
                                   , 
                                   
                                     
                                       2 
                                        
                                       
                                         π 
                                          
                                         
                                           ( 
                                           
                                             q 
                                             + 
                                             1 
                                           
                                           ) 
                                         
                                       
                                     
                                     - 
                                     
                                       
                                         ω 
                                         7 
                                       
                                        
                                       Q 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
       For Case q =B, 
       [0062]    
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 ω 
                                 pmc 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                               = 
                               
                                 max 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         2 
                                          
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         q 
                                       
                                       - 
                                       
                                         
                                           ω 
                                           6 
                                         
                                          
                                         Q 
                                       
                                     
                                     , 
                                     
                                       
                                         
                                           ω 
                                           7 
                                         
                                          
                                         Q 
                                       
                                       - 
                                       
                                         2 
                                          
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         q 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             , 
                           
                         
                       
                       
                         
                           
                             
                               ω 
                               smc 
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                             = 
                             
                               min 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       ω 
                                       4 
                                     
                                     - 
                                     
                                       2 
                                        
                                       
                                         π 
                                          
                                         
                                           ( 
                                           
                                             q 
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   , 
                                   
                                     
                                       2 
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       q 
                                     
                                     - 
                                     
                                       
                                         ω 
                                         5 
                                       
                                        
                                       Q 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0000]    
       
         
               
             
           
               
                 TABLE II 
               
               
                   
               
               
                 Variable Values And Inequations For Calculating The Passband And The 
               
               
                 Stopband Of H ma   (2)  (z) In Case A 
               
               
                   
               
             
             
               
                 
                   
                     
                       
                         
                           
                             Inequations 
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   y 
                                   1 
                                 
                                 ≤ 
                                 
                                   ω 
                                   4 
                                 
                               
                               = 
                               
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     
                                       π 
                                        
                                       
                                         ( 
                                         
                                           m 
                                           - 
                                           1 
                                         
                                         ) 
                                       
                                     
                                   
                                   + 
                                   
                                     ϕ 
                                     a 
                                   
                                 
                                 M 
                               
                             
                           
                           
                             
                               ( 
                               
                                 14 
                                  
                                 
                                     
                                 
                                  
                                 a 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   y 
                                   2 
                                 
                                 ≥ 
                                 
                                   ω 
                                   5 
                                 
                               
                               = 
                               
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     π 
                                      
                                     
                                         
                                     
                                      
                                     m 
                                   
                                   - 
                                   
                                     θ 
                                     a 
                                   
                                 
                                 M 
                               
                             
                           
                           
                             
                               ( 
                               
                                 14 
                                  
                                 
                                     
                                 
                                  
                                 b 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   y 
                                   3 
                                 
                                 ≤ 
                                 
                                   ω 
                                   6 
                                 
                               
                               = 
                               
                                 ω 
                                 s 
                               
                             
                           
                           
                             
                               ( 
                               
                                 14 
                                  
                                 
                                     
                                 
                                  
                                 c 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   y 
                                   4 
                                 
                                 ≥ 
                                 
                                   ω 
                                   7 
                                 
                               
                               = 
                               
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     
                                       π 
                                        
                                       
                                         ( 
                                         
                                           m 
                                           + 
                                           1 
                                         
                                         ) 
                                       
                                     
                                   
                                   - 
                                   
                                     θ 
                                     a 
                                   
                                 
                                 M 
                               
                             
                           
                           
                             
                               ( 
                               
                                 14 
                                  
                                 
                                     
                                 
                                  
                                 d 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 Case 
                                 q 
                               
                               = 
                               A 
                             
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                               
                                 y 
                                 1 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       q 
                                     
                                     - 
                                     
                                       ω 
                                       pma 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 Q 
                               
                             
                           
                           
                             
                               ( 
                               
                                 15 
                                  
                                 
                                     
                                 
                                  
                                 a 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 y 
                                 2 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       q 
                                     
                                     + 
                                     
                                       ω 
                                       pma 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 Q 
                               
                             
                           
                           
                             
                               ( 
                               
                                 15 
                                  
                                 
                                     
                                 
                                  
                                 b 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 y 
                                 3 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       q 
                                     
                                     + 
                                     
                                       ω 
                                       ama 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 Q 
                               
                             
                           
                           
                             
                               ( 
                               
                                 15 
                                  
                                 
                                     
                                 
                                  
                                 c 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 y 
                                 4 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         π 
                                          
                                         
                                           ( 
                                           
                                             q 
                                             + 
                                             1 
                                           
                                           ) 
                                         
                                       
                                     
                                     - 
                                     
                                       ω 
                                       ama 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 Q 
                               
                             
                           
                           
                             
                               ( 
                               
                                 15 
                                  
                                 
                                     
                                 
                                  
                                 d 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 Case 
                                 q 
                               
                               = 
                               B 
                             
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                               
                                 y 
                                 1 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         π 
                                          
                                         
                                           ( 
                                           
                                             q 
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                     
                                       ω 
                                       sma 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 Q 
                               
                             
                           
                           
                             
                               ( 
                               
                                 16 
                                  
                                 
                                     
                                 
                                  
                                 a 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 y 
                                 2 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       q 
                                     
                                     - 
                                     
                                       ω 
                                       ama 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 Q 
                               
                             
                           
                           
                             
                               ( 
                               
                                 16 
                                  
                                 
                                     
                                 
                                  
                                 b 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 y 
                                 3 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       q 
                                     
                                     - 
                                     
                                       ω 
                                       pma 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 Q 
                               
                             
                           
                           
                             
                               ( 
                               
                                 16 
                                  
                                 
                                     
                                 
                                  
                                 c 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 y 
                                 4 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       q 
                                     
                                     + 
                                     
                                       ω 
                                       pma 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 Q 
                               
                             
                           
                           
                             
                               ( 
                               
                                 16 
                                  
                                 
                                     
                                 
                                  
                                 d 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
             
          
         
       
     
       2) Case B 
       [0063]    For Case B, the transition band of G(z) is formed by the complement of H a   (2) (z M ). Determination of the passband and stopband edges of H ma   (2) (z) is similar to solving of the passband and stopband edges of H mc   (2) (z) in Case A. Corresponding inequations and parameters are as shown in Table III. The passband and stopband edges of H ma   (2) (z) are shown below: 
       For Case p =A, 
       [0064]    
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 ω 
                                 pma 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                               = 
                               
                                 max 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         2 
                                          
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         p 
                                       
                                       - 
                                       
                                         P 
                                          
                                         
                                             
                                         
                                          
                                         
                                           ω 
                                           4 
                                         
                                       
                                     
                                     , 
                                     
                                       
                                         
                                           ω 
                                           5 
                                         
                                          
                                         P 
                                       
                                       - 
                                       
                                         2 
                                          
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         p 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             , 
                           
                         
                       
                       
                         
                           
                             
                               ω 
                               sma 
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                             = 
                             
                               min 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         ω 
                                         6 
                                       
                                        
                                       P 
                                     
                                     - 
                                     
                                       2 
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       p 
                                     
                                   
                                   , 
                                   
                                     
                                       2 
                                        
                                       
                                         π 
                                          
                                         
                                           ( 
                                           
                                             p 
                                             + 
                                             1 
                                           
                                           ) 
                                         
                                       
                                     
                                     - 
                                     
                                       
                                         ω 
                                         7 
                                       
                                        
                                       P 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
       For Case p =B, 
       [0065]    
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 ω 
                                 pma 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                               = 
                               
                                 max 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         2 
                                          
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         p 
                                       
                                       - 
                                       
                                           
                                       
                                        
                                       
                                         
                                           ω 
                                           6 
                                         
                                          
                                         P 
                                       
                                     
                                     , 
                                     
                                       
                                         
                                           ω 
                                           7 
                                         
                                          
                                         P 
                                       
                                       - 
                                       
                                         2 
                                          
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         p 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             , 
                           
                         
                       
                       
                         
                           
                             
                               ω 
                               sma 
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                             = 
                             
                               min 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         ω 
                                         4 
                                       
                                        
                                       P 
                                     
                                     - 
                                     
                                       2 
                                        
                                       
                                         π 
                                          
                                         
                                           ( 
                                           
                                             p 
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   , 
                                   
                                     
                                       2 
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       p 
                                     
                                     - 
                                     
                                       
                                         ω 
                                         5 
                                       
                                        
                                       P 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
         [0000]    
       
         
               
             
           
               
                 TABLE III 
               
               
                   
               
               
                 Variable Values And Inequations For Calculating The Passband And The 
               
               
                 Stopband Of H ma   (2)  (z) in Case B 
               
               
                   
               
             
             
               
                 
                   
                     
                       
                         
                           
                             Inequations 
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   y 
                                   1 
                                 
                                 ≤ 
                                 
                                   ω 
                                   4 
                                 
                               
                               = 
                               
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     
                                       π 
                                        
                                       
                                         ( 
                                         
                                           m 
                                           - 
                                           1 
                                         
                                         ) 
                                       
                                     
                                   
                                   - 
                                   
                                     θ 
                                     a 
                                   
                                 
                                 M 
                               
                             
                           
                           
                             
                               ( 
                               
                                 19 
                                  
                                 
                                     
                                 
                                  
                                 a 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   y 
                                   2 
                                 
                                 ≥ 
                                 
                                   ω 
                                   5 
                                 
                               
                               = 
                               
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     
                                       π 
                                        
                                       
                                         ( 
                                         
                                           m 
                                           - 
                                           1 
                                         
                                         ) 
                                       
                                     
                                   
                                   + 
                                   
                                     ϕ 
                                     a 
                                   
                                 
                                 M 
                               
                             
                           
                           
                             
                               ( 
                               
                                 19 
                                  
                                 
                                     
                                 
                                  
                                 b 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   y 
                                   3 
                                 
                                 ≤ 
                                 
                                   ω 
                                   6 
                                 
                               
                               = 
                               
                                 ω 
                                 s 
                               
                             
                           
                           
                             
                               ( 
                               
                                 19 
                                  
                                 
                                     
                                 
                                  
                                 c 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   y 
                                   4 
                                 
                                 ≥ 
                                 
                                   ω 
                                   7 
                                 
                               
                               = 
                               
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     π 
                                      
                                     
                                         
                                     
                                      
                                     m 
                                   
                                   + 
                                   
                                     ϕ 
                                     a 
                                   
                                 
                                 M 
                               
                             
                           
                           
                             
                               ( 
                               
                                 19 
                                  
                                 
                                     
                                 
                                  
                                 d 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 Case 
                                 p 
                               
                               = 
                               A 
                             
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                               
                                 y 
                                 1 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       p 
                                     
                                     - 
                                     
                                       ω 
                                       pma 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 P 
                               
                             
                           
                           
                             
                               ( 
                               
                                 20 
                                  
                                 
                                     
                                 
                                  
                                 a 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 y 
                                 2 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       p 
                                     
                                     + 
                                     
                                       ω 
                                       pma 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 P 
                               
                             
                           
                           
                             
                               ( 
                               
                                 20 
                                  
                                 
                                     
                                 
                                  
                                 b 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 y 
                                 3 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       p 
                                     
                                     + 
                                     
                                       ω 
                                       sma 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 P 
                               
                             
                           
                           
                             
                               ( 
                               
                                 20 
                                  
                                 
                                     
                                 
                                  
                                 c 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 y 
                                 4 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         π 
                                          
                                         
                                           ( 
                                           
                                             p 
                                             + 
                                             1 
                                           
                                           ) 
                                         
                                       
                                     
                                     - 
                                     
                                       ω 
                                       sma 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 P 
                               
                             
                           
                           
                             
                               ( 
                               
                                 20 
                                  
                                 
                                     
                                 
                                  
                                 d 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 Case 
                                 p 
                               
                               = 
                               B 
                             
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                               
                                 y 
                                 1 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         π 
                                          
                                         
                                           ( 
                                           
                                             p 
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                     
                                       ω 
                                       sma 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 P 
                               
                             
                           
                           
                             
                               ( 
                               
                                 21 
                                  
                                 
                                     
                                 
                                  
                                 a 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 y 
                                 2 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       p 
                                     
                                     - 
                                     
                                       ω 
                                       sma 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 P 
                               
                             
                           
                           
                             
                               ( 
                               
                                 21 
                                  
                                 
                                     
                                 
                                  
                                 b 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 y 
                                 3 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       p 
                                     
                                     - 
                                     
                                       ω 
                                       pma 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 P 
                               
                             
                           
                           
                             
                               ( 
                               
                                 21 
                                  
                                 
                                     
                                 
                                  
                                 c 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 y 
                                 4 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       p 
                                     
                                     + 
                                     
                                       ω 
                                       pma 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 P 
                               
                             
                           
                           
                             
                               ( 
                               
                                 21 
                                  
                                 
                                     
                                 
                                  
                                 d 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
             
          
         
       
     
         [0066]    Determination of the passband and stopband edges of H mc   (2) (z) is similar to the method of calculating the passband and stopband edges of H ma   (2) (z) in Case A. Corresponding inequations and parameters are shown in Table IV. The passband and stopband edges of H mc   (2) (z) are shown below: 
       For Case q =A, 
       [0067]    
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 ω 
                                 pmc 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                               = 
                               
                                 max 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         2 
                                          
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         q 
                                       
                                       - 
                                       
                                         
                                           ω 
                                           1 
                                         
                                          
                                         Q 
                                       
                                     
                                     , 
                                     
                                       
                                         
                                           ω 
                                           2 
                                         
                                          
                                         Q 
                                       
                                       - 
                                       
                                         2 
                                          
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         q 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             , 
                           
                         
                       
                       
                         
                           
                             
                               ω 
                               smc 
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                             = 
                             
                               
                                 
                                   ω 
                                   3 
                                 
                                  
                                 Q 
                               
                               - 
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 q 
                               
                             
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
       For Case q =B, 
       [0068]    
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 ω 
                                 pmc 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                               = 
                               
                                 
                                   2 
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   q 
                                 
                                 - 
                                 
                                   
                                     ω 
                                     3 
                                   
                                    
                                   Q 
                                 
                               
                             
                             , 
                           
                         
                       
                       
                         
                           
                             
                               ω 
                               smc 
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                             = 
                             
                               min 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         ω 
                                         1 
                                       
                                        
                                       Q 
                                     
                                     - 
                                     
                                       2 
                                        
                                       
                                         π 
                                          
                                         
                                           ( 
                                           
                                             q 
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   , 
                                   
                                     
                                       2 
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       q 
                                     
                                     - 
                                     
                                       
                                         ω 
                                         2 
                                       
                                        
                                       Q 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
         [0000]    
       
         
               
             
           
               
                 TABLE IV 
               
               
                   
               
               
                 Variable Values And Inequations For Calculating The Passband And The 
               
               
                 Stopband Of H ma   (2)  (z) In Case B 
               
               
                   
               
             
             
               
                 
                   
                     
                       
                         
                           
                             Inequations 
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   x 
                                   1 
                                 
                                 ≤ 
                                 
                                   ω 
                                   1 
                                 
                               
                               = 
                               
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     
                                       π 
                                        
                                       
                                         ( 
                                         
                                           m 
                                           - 
                                           3 
                                         
                                         ) 
                                       
                                     
                                   
                                   + 
                                   
                                     θ 
                                     a 
                                   
                                 
                                 M 
                               
                             
                           
                           
                             
                               ( 
                               
                                 24 
                                  
                                 
                                     
                                 
                                  
                                 a 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   x 
                                   2 
                                 
                                 ≥ 
                                 
                                   ω 
                                   2 
                                 
                               
                               = 
                               
                                 ω 
                                 s 
                               
                             
                           
                           
                             
                               ( 
                               
                                 24 
                                  
                                 
                                     
                                 
                                  
                                 b 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   x 
                                   3 
                                 
                                 ≤ 
                                 
                                   ω 
                                   3 
                                 
                               
                               = 
                               
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     π 
                                      
                                     
                                         
                                     
                                      
                                     m 
                                   
                                   + 
                                   
                                     θ 
                                     a 
                                   
                                 
                                 M 
                               
                             
                           
                           
                             
                               ( 
                               
                                 24 
                                  
                                 
                                     
                                 
                                  
                                 c 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 Case 
                                 q 
                               
                               = 
                               A 
                             
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                               
                                 x 
                                 1 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       q 
                                     
                                     - 
                                     
                                       ω 
                                       pma 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 Q 
                               
                             
                           
                           
                             
                               ( 
                               
                                 25 
                                  
                                 
                                     
                                 
                                  
                                 a 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 x 
                                 2 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       q 
                                     
                                     + 
                                     
                                       ω 
                                       pma 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 Q 
                               
                             
                           
                           
                             
                               ( 
                               
                                 25 
                                  
                                 
                                     
                                 
                                  
                                 b 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 x 
                                 3 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       q 
                                     
                                     + 
                                     
                                       ω 
                                       sma 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 Q 
                               
                             
                           
                           
                             
                               ( 
                               
                                 25 
                                  
                                 
                                     
                                 
                                  
                                 c 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 Case 
                                 q 
                               
                               = 
                               B 
                             
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                               
                                 x 
                                 1 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         π 
                                          
                                         
                                           ( 
                                           
                                             q 
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                     
                                       ω 
                                       sma 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 Q 
                               
                             
                           
                           
                             
                               ( 
                               
                                 26 
                                  
                                 
                                     
                                 
                                  
                                 a 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 x 
                                 2 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       q 
                                     
                                     - 
                                     
                                       ω 
                                       sma 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 Q 
                               
                             
                           
                           
                             
                               ( 
                               
                                 26 
                                  
                                 
                                     
                                 
                                  
                                 b 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 x 
                                 3 
                               
                               = 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       q 
                                     
                                     - 
                                     
                                       ω 
                                       pma 
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   / 
                                 
                                  
                                 Q 
                               
                             
                           
                           
                             
                               ( 
                               
                                 26 
                                  
                                 
                                     
                                 
                                  
                                 c 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
             
          
         
       
     
         [0069]    III. Calculation of the Passband and Stopband Edges of First-Stage Masking Filters 
       1) Case A 
       [0070]    For a masking filter H ma   (1) (z), since the transition band of H(z) is provided by H a   (2) (z M ), its passband edge ω pma   (1)  is equal to ω p . Determination of the stopband edges ω sma   (1) , ω sma   (1)  is as shown in  FIG. 5 . 
         [0071]    The stopband edge ω sma   (1)  is the right endpoint of d 2 . Therefore, the first masking filter passband on the right of the transition band is focused. The left stopband cutoff points of the two passbands are denoted as t 1  and t 2 . It is required to find out the position of t 1  in H a   (2) (Mω) and the position of t 2  in 1−H a   (2) (Mω). The passband and stopband regions of H a   (2) (Hω) may be obtained through the following formulas: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         R 
                         pass 
                       
                        
                       
                         ( 
                         k 
                         ) 
                       
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               2 
                                
                               π 
                                
                               
                                   
                               
                                
                               k 
                             
                             - 
                             
                               θ 
                               a 
                             
                           
                           M 
                         
                         , 
                         
                           
                             
                               2 
                                
                               π 
                                
                               
                                   
                               
                                
                               k 
                             
                             + 
                             
                               θ 
                               a 
                             
                           
                           M 
                         
                       
                       ] 
                     
                   
                   , 
                   
                     k 
                     = 
                     0 
                   
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     ⌊ 
                     
                       M 
                       / 
                       2 
                     
                     ⌋ 
                   
                   , 
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         R 
                         stop 
                       
                        
                       
                         ( 
                         k 
                         ) 
                       
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               2 
                                
                               π 
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   k 
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                             + 
                             
                               ϕ 
                               a 
                             
                           
                           M 
                         
                         , 
                         
                           
                             
                               2 
                                
                               π 
                                
                               
                                   
                               
                                
                               k 
                             
                             + 
                             
                               ϕ 
                               a 
                             
                           
                           M 
                         
                       
                       ] 
                     
                   
                   , 
                   
                     k 
                     = 
                     0 
                   
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     ⌊ 
                     
                       M 
                       / 
                       2 
                     
                     ⌋ 
                   
                   , 
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
         [0072]    A temporary value of ω sma   (1)  is obtained according to the position of t 1 , and denoted as ω sma   _   temp1   (1) : 
         [0000]    
       
         
           
             
               
                 
                   
                     ω 
                     
                       sma 
                        
                       
                           
                       
                        
                       _ 
                        
                       
                           
                       
                        
                       temp 
                        
                       
                           
                       
                        
                       1 
                     
                     
                       ( 
                       1 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               t 
                               1 
                             
                           
                           
                             
                               
                                 t 
                                 1 
                               
                               ∉ 
                               
                                 
                                   R 
                                   stop 
                                 
                                  
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               ω 
                               8 
                             
                           
                           
                             
                               
                                 t 
                                 1 
                               
                               ∈ 
                               
                                 
                                   R 
                                   stop 
                                 
                                  
                                 
                                   ( 
                                   
                                     k 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       , 
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
         [0000]    wherein ω 8  is the right endpoint of R stop (k 1 ), and k 1  is an integer satisfying t 1 εR stop (k 1 ). 
         [0073]    Another temporary value of ω sma   (1)  is obtained according to the position of t 2 , and denoted as ω sma   _   temp2   (1) : 
         [0000]    
       
         
           
             
               
                 
                   
                     ω 
                     
                       sma 
                        
                       
                           
                       
                        
                       _ 
                        
                       
                           
                       
                        
                       temp 
                        
                       
                           
                       
                        
                       2 
                     
                     
                       ( 
                       1 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               t 
                               2 
                             
                           
                           
                             
                               
                                 t 
                                 2 
                               
                               ∉ 
                               
                                 
                                   R 
                                   pass 
                                 
                                  
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               ω 
                               9 
                             
                           
                           
                             
                               
                                 t 
                                 2 
                               
                               ∈ 
                               
                                 
                                   R 
                                   pass 
                                 
                                  
                                 
                                   ( 
                                   
                                     k 
                                     2 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       , 
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
         [0000]    wherein ω 9  is the right endpoint of R pass (k 2 ), and k 2  is an integer satisfying t 2 εR stop (k 2 ). 
         [0074]    The value of ω sma   (1)  is obtained through the following formula: 
         [0000]      ω sma   (1) =min(ω sma   _   temp1   (1) ,ω sma   _   temp2   (1) ),  (31)
 
         [0000]    wherein parameters t 1 , t 2 , ω 8  and ω 9  are as shown in Table V. 
         [0000]    
       
         
               
             
               
               
               
               
             
           
               
                 TABLE V 
               
               
                   
               
               
                 Variable Values For Calculating ω sma   (1)  in Case A 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Case p  = A 
                 t 1  = (2π(p + 1) − ω sma   (2) )/P 
                 (32) 
               
               
                   
                 Case p  = B 
                 t 1  = (2πp + ω pma   (2) )/P 
                 (33) 
               
               
                   
                 Case q  = A 
                 t 2  = (2π(q + 1) − ω smc   (2) )/Q 
                 (34) 
               
               
                   
                 Case q  = B 
                 t 2  = (2πq + ω pmc   (2) )/Q 
                 (35) 
               
               
                   
                   
                 ω θ  = (2πk 1  − φ a )/M 
                 (36) 
               
               
                   
                   
                 ω φ  = (2πk 2  + θ a )/M 
                 (37) 
               
               
                   
                   
               
             
          
         
       
     
         [0075]    For a masking filter H mc   (1) (z), its stopband edge ω smc   (1)  is equal to ω s , and only its passband edge ω pmc   (1)  needs to be determined. Calculation of ω pmc   (1)  is as shown in  FIG. 6 . Since ω pmc   (1)  is the left endpoint of d 1 , for H ma   (2) (Pω), only the passband including the transition band is focused, and for H mc   (2) (Qω), only the first passband on the left of the transition band is focused. The left passband edges of the two passbands are defined as t 3  and t 4 . 
         [0000]    If t 3 ≧t 4 , then 
         [0000]    
       
         
           
             
               
                 
                   
                     ω 
                     pmc 
                     
                       ( 
                       1 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               t 
                               3 
                             
                           
                           
                             
                               
                                 t 
                                 3 
                               
                               ∉ 
                               
                                 
                                   R 
                                   stop 
                                 
                                  
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               max 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         2 
                                          
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               k 
                                               3 
                                             
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                       
                                       + 
                                       
                                         ϕ 
                                         a 
                                       
                                     
                                     M 
                                   
                                   , 
                                   
                                     t 
                                     4 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 t 
                                 3 
                               
                               ∈ 
                               
                                 
                                   R 
                                   stop 
                                 
                                  
                                 
                                   ( 
                                   
                                     k 
                                     3 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       ; 
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
         [0000]    if t 3 &lt;t 4 , then 
         [0000]    
       
         
           
             
               
                 
                   
                     ω 
                     pmc 
                     
                       ( 
                       1 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               t 
                               4 
                             
                           
                           
                             
                               
                                 t 
                                 4 
                               
                               ∉ 
                               
                                 
                                   R 
                                   pass 
                                 
                                  
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               max 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         2 
                                          
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         
                                           k 
                                           4 
                                         
                                       
                                       - 
                                       
                                         θ 
                                         a 
                                       
                                     
                                     M 
                                   
                                   , 
                                   
                                     t 
                                     3 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 t 
                                 4 
                               
                               ∈ 
                               
                                 
                                   R 
                                   pass 
                                 
                                  
                                 
                                   ( 
                                   
                                     k 
                                     4 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       , 
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
         [0000]    wherein the values of the parameters t 3  and t 4  are as shown in Table VI. 
         [0000]    
       
         
               
             
               
               
               
               
             
           
               
                 TABLE V 
               
               
                   
               
               
                 Variable Values For Calculating ω pmc   (1)  in Case A 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Case p  = A 
                 t 2  = (2πp − ω pma   (2) )/P 
                 (40) 
               
               
                   
                 Case p  = B 
                 t 2  = (2π(p − 1) + ω sma   (2) )/P 
                 (41) 
               
               
                   
                 Case q  = A 
                 t 4  = (2πq − ω pmc   (2) )/Q 
                 (42) 
               
               
                   
                 Case q  = B 
                 t 4  = (2π(q − 1) + ω smc   (2) )/Q 
                 (43) 
               
               
                   
                   
               
             
          
         
       
     
       2) Case B 
       [0076]    In Case B, the ways of calculating ω sma   (1)  and ω pmc   (1)  are similar to those in Case A with corresponding parameters as shown in Table VII and Table VIII. It should be noted that the passband and stopband edges of all the subfilters must fall into the range from 0 to π; otherwise, the subsequent design of the set of interpolation factors [M, P, Q] is considered to be meaningless. 
         [0000]    
       
         
               
             
               
               
               
               
             
           
               
                 TABLE VII 
               
               
                   
               
               
                 Variable Values For Calculating ω sma   (1)  in Case B 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Case p  = A 
                 t 2  = (2π(p + 1) − ω sma   (2) )/P 
                  44) 
               
               
                   
                 Case p  = B 
                 t 2  = (2πp + ω pma   (2) )/P 
                 (45) 
               
               
                   
                 Case q  = A 
                 t 1  = (2π(q + 1) − ω sma   (2) )/Q 
                 (46) 
               
               
                   
                 Case q  = B 
                 t 1  = (2πq + ω pmc   (2) )/Q 
                 (47) 
               
               
                   
                   
                 ω θ  = (2πk 1  + θ a )/M 
                 (48) 
               
               
                   
                   
                 ω φ  = (2πk 2  − φ a )/M 
                 (49) 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
             
           
               
                 TABLE VIII 
               
               
                   
               
               
                 Variable Values For Calculating ω pmc   (1)  in Case B 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Case p  = A 
                 t 4  = (2πp − ω pma   (2) )/P 
                 (50) 
               
               
                   
                 Case p  = B 
                 t 4  = (2π(p − 1) + ω sma   (2) )/P 
                 (51) 
               
               
                   
                 Case q  = A 
                 t 2  = (2πq − ω pmc   (2) )/Q 
                 (52) 
               
               
                   
                 Case q  = B 
                 t 2  = (2π(q − 1) + ω smc   (2) )/Q 
                 (53) 
               
               
                   
                   
               
             
          
         
       
     
       Optimization Method 
     Step 1 
       [0077]    Some sets having fewer multipliers are selected from all the effective sets of interpolation factors as optimization objects (generally not more than 105% of the lowest number of multipliers. Calculation of the number of multipliers is carried out through the following formula: 
         [0000]        {circumflex over (N)}   mult =( {circumflex over (N)}   a   +{circumflex over (N)}   ma2   +{circumflex over (N)}   mc2 )/2+└( {circumflex over (N)}   ma +2)/2┘+└( {circumflex over (N)}   mc +2)/2┘+3,  (54)
 
         [0000]    wherein {circumflex over (N)} a , {circumflex over (N)} ma2 , {circumflex over (N)} mc2 , {circumflex over (N)} ma  and {circumflex over (N)} mc  are filter orders estimated for H a   (2) (z), H ma   (2) (z), H mc   (2) (z), H ma   (1) (z) and H mc   (1) (z), respectively, and wherein the orders may be obtained) by using the firpmord function in matlab. 
       Step 2 
       [0078]    Each subfilter is designed according to the Parks-McClellan algorithm. As a matter of experience, after optimization, the order N a  of H a   (2) (z) is equal to the estimated order {circumflex over (N)} a  while the orders N ma2 , N mc2 , N ma  and N mc  of the masking filters are 60% of the respective estimated orders. When the order of a subfilter is not an integer, it is rounded. It should be noted that the order of the prototype filter and the orders of the second-stage masking filters must be even numbers, while the orders of the two first-stage masking filters are simultaneously odd numbers or even numbers. 
       Step 3 
       [0079]    The subfilters are simultaneously optimized by using a nonlinear optimization algorithm. A vector  defined as a combination of parameters of the subfilters is as shown below: 
         [0000]    
       
         
           
             
               
                 
                   ∅ 
                   = 
                   
                     
                       [ 
                       
                         
                           
                             h 
                             ma 
                             
                               ( 
                               1 
                               ) 
                             
                           
                            
                           
                             ( 
                             0 
                             ) 
                           
                         
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         
                           
                             h 
                             ma 
                             
                               ( 
                               1 
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               ⌊ 
                               
                                 
                                   N 
                                   ma 
                                 
                                 2 
                               
                               ⌋ 
                             
                             ) 
                           
                         
                         , 
                         
                           
                             h 
                             
                               m 
                                
                               
                                   
                               
                                
                               c 
                             
                             
                               ( 
                               1 
                               ) 
                             
                           
                            
                           
                             ( 
                             0 
                             ) 
                           
                         
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         
                           
                             h 
                             
                               m 
                                
                               
                                   
                               
                                
                               c 
                             
                             
                               ( 
                               1 
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               ⌊ 
                               
                                 
                                   N 
                                   
                                     m 
                                      
                                     
                                         
                                     
                                      
                                     c 
                                   
                                 
                                 2 
                               
                               ⌋ 
                             
                             ) 
                           
                         
                         , 
                         
                           
 
                         
                          
                         
                           
                             h 
                             ma 
                             
                               ( 
                               2 
                               ) 
                             
                           
                            
                           
                             ( 
                             0 
                             ) 
                           
                         
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         
                           
                             h 
                             ma 
                             
                               ( 
                               2 
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 N 
                                 
                                   ma 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                               2 
                             
                             ) 
                           
                         
                         , 
                         
                           
                             h 
                             
                               m 
                                
                               
                                   
                               
                                
                               c 
                             
                             
                               ( 
                               2 
                               ) 
                             
                           
                            
                           
                             ( 
                             0 
                             ) 
                           
                         
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         
                           
                             h 
                             
                               m 
                                
                               
                                   
                               
                                
                               c 
                             
                             
                               ( 
                               2 
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 N 
                                 
                                   m 
                                    
                                   
                                       
                                   
                                    
                                   c 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                               2 
                             
                             ) 
                           
                         
                         , 
                         
                           
 
                         
                          
                         
                           
                             h 
                             a 
                             
                               ( 
                               2 
                               ) 
                             
                           
                            
                           
                             ( 
                             0 
                             ) 
                           
                         
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         
                           
                             h 
                             a 
                             
                               ( 
                               2 
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 N 
                                 a 
                               
                               2 
                             
                             ) 
                           
                         
                       
                       ] 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   55 
                   ) 
                 
               
             
           
         
       
     
         [0080]    To optimize the filter parameters, the following object function should be minimized: 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                     = 
                     
                       
                         max 
                         
                           ω 
                           ∈ 
                           
                             
                               [ 
                               
                                 0 
                                 , 
                                 
                                   ω 
                                   p 
                                 
                               
                               ] 
                             
                             ⋃ 
                             
                               [ 
                               
                                 
                                   ω 
                                   s 
                                 
                                 , 
                                 π 
                               
                               ] 
                             
                           
                         
                       
                        
                       
                           
                       
                        
                       
                          
                         
                           
                             W 
                              
                             
                               ( 
                               ω 
                               ) 
                             
                           
                            
                           
                             [ 
                             
                               
                                 H 
                                  
                                 
                                   ( 
                                   
                                     ∅ 
                                     , 
                                     ω 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 D 
                                  
                                 
                                   ( 
                                   ω 
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                          
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   56 
                   ) 
                 
               
             
           
         
       
     
         [0000]    wherein H(,ω) is the zero phase frequency response of the system function H(z); D(ω) is an ideal zero phase frequency function; W(ω) is a weighted vector. For the passbands ωε[0,ω p ], W(ω) and D(ω) are equal to 1; for the stopbands ωε[ω s ,π], W(ω) is equal to δ p /δ s , and D(ω) is equal to 0. 
         [0081]    This is a problem of minimization and maximization. The fminimax function in the optimization toolbox provided by MathWorks Company can solve this problem. To achieve more efficient optimization, non-uniform frequency points are employed; the closer to the regions of the passband edge and the stopband edge, the denser the frequency points. It is suggested that the frequency points within 10% region closest to the passband edge account for 25% of the total number of points, and it is the same as the stopband edge. During optimization using the fminimax function, the parameter requirements of a filter can be met only when E is not greater than δ p . 
         [0082]    In order to find out the optimal solutions, for each set of interpolation factors, the order of each subfilter needs to be varied near the respective estimated order before optimization. As a matter of experience, the order of H a   (2) (z) is not varied, and variation is only required to be made to the order of each masking filter. For N ma2 , N mc2 , N ma  and N mc , a range of variation is 4; the variation of N ma2  and N mc2  is 2, while the variation of N ma  and N mc  is 1. After optimization, there may exist multiple solutions satisfying filter specifications, and the one having the fewest multipliers is selected as the optimal solution. In case of multiple optimal solutions, the one having the smallest time delay is selected. 
       Example 1 
       [0083]    The specific specifications of a filter having an extremely narrow transition band are as follows: ω p =0.6π, ω s =0.602π, δ p =0.01, δ s =0.01. By using the present design method, the final results are as follows: N a =28, N ma2 =20, N mc2 =16, N ma =17, N mc =29. The three interpolation factors M, P, Q are 69, 9, 9, respectively, which fall into Case B type: Case p =B, and Case q =B. The number of multipliers is 59 with a group time delay being 1070.5. The passband and stopband ripples are 0.01 and 0.00999, respectively. Comparison of results of the method of the present invention with those of other existing methods is as shown in Table IX. 
         [0000]    
       
         
               
             
               
               
               
               
             
               
               
               
               
             
           
               
                 TABLE IX 
               
             
             
               
                   
               
               
                 Results of Designing Filter Using Various Methods in 
               
               
                 Example 1 And Comparison Thereof 
               
             
          
           
               
                   
                   
                 Total 
                 Group 
               
               
                   
                   
                 Number Of 
                 Time 
               
               
                   
                 Method 
                 Multipliers 
                 Delay 
               
               
                   
                   
               
             
          
           
               
                   
                 Conventional 
                 92 
                 1105 
               
               
                   
                 two-stage FRM 
               
               
                   
                 SFFM-FRM case A 
                 86 
                 1638.5 
               
               
                   
                 SFFM-FRM case B 
                 84 
                 2150.5 
               
               
                   
                 Serial-masking FRM 
                 83 
                 1016 
               
               
                   
                 Non-periodical FRM 
                 55 
                 1214 
               
               
                   
                 Improved method 
                 59 
                 1070.5 
               
               
                   
                   
               
             
          
         
       
     
         [0084]    As can be seen from Table IX, when compared with the conventional two-stage FRM, SFFM and serial-masking FRM methods, the method provided by the present invention is lower in complexity. When compared with the conventional two-stage FRM, the complexity is reduced by 35.8%; when compared with the SFFM-FRM, it is reduced by 31.4%; when compared with a serial-masking filter, it is reduced by 28.9%. When compared with the Non-periodical FRM, although the number of multipliers is increased by 7.3%, the group time delay is reduced by 11.8%. The time delay in the result of the present method is the smallest when compared with other methods apart from the Serial-masking FRM. Although the time delay is increased by 5.1% in contrast with the Serial-masking FRM, the number of multipliers is reduced by 28.9%. 
         [0085]    The constraint conditions among the interpolation factors of a conventional FRM filter are broken through a new design method, and the subfilters are optimized simultaneously by means of a nonlinear joint optimization method. Results indicate that, as compared to a conventional design method of a two-stage FRM filter, a filter designed using the improved method is lower in complexity. 
         [0086]    While the specific embodiments of the present invention are described above in conjunction with the accompanying drawings, they are not limitations to the protection scope of the present invention. It should be understood by those skilled in the relevant art that various modifications or variations made by those skilled in the art without creative work on the basis of the technical scheme of the present invention still fall into the protection range of the present invention.