PATENT ABSTRACT
A near ideal single frequency and band pass filter comprises an input unit, a filter combination, and an output unit. The input unit accepts an input signal which can be expressed with a mathematical formula f(t). The filter combination comprises a plurality of filters, wherein each filter can generate a control signal and performs the signal filtering process based on the control signal. The control signal can be expressed with a mathematical formula hde(θ). The filter combination processes the input signal based on a set of a plurality of control signals by means of a formula.

PATENT DESCRIPTION
BACKGROUND OF INVENTION 
     1. Field of the Invention 
     The present invention refers to a filter and a method for filtering communication signals, and more particularly to an ideal-like band-pass filter and single-frequency filter which is characterized in Frequency Response Formula and Phase-Shift Response Formula. 
     2. Description of the Prior Art 
     For those conventional filters, no matter analog (not periodical) or digital (periodical), their frequency response formulae and phase-shift response formulae always generate different amplitudes of vibrations and different phase-shifts according to different frequencies of sine signals. 
     Conventional methods for designing filters mostly are related to the application of differential or difference equations. Firstly, a linear system representing the relationship between input and output signals is established, and then the output signal of this system can be calculated. This system can be established by electric circuitries, but also can be simulated by mathematics. Because the operations of circuitries (or operations of mathematical simulations) in the system always involve in actions of delayed signals (or advanced signals), which are actually resulted from the operations of capacitors (or inductors). That&#39;s why conventional filters always generate different amplitudes of vibrations and different phase-shifts according to different frequencies of sine signals. 
     Until now, the ways to avoid the phenomenon of aforementioned variations of amplitudes and phase-shifts are still focal points in designing the filters. 
     In addition, for conventional band-pass filters, the leading and rear edges of each curve of responsive frequency formulae are still not steep enough, which result in unclear borderlines between the responsive frequencies and the unresponsive frequencies, and thus require further improvements. 
     SUMMARY OF INVENTION 
     One objective of the present invention is to provide an ideal-like filter and a method for filtering communication signals, which is characterized in Frequency Response Formula and Phase-Shift Response Formula. For band-pass filters, the height of each top point in a range of responsive frequency is almost equal to 1 (where the error can be controlled to be less than ±10 −4 ). And, the width between the responsive frequency and the unresponsive frequency is also very narrow at the leading and rear edges of each curve of responsive frequency. For single-frequency filters, the range of responsive frequency is very narrow that it just likes a pulse wave having a height equal to 1. For both band-pass and single-frequency filters, their response formulae are equal to zero (absolute zero), such that no phase-shift will occur no matter what their frequencies are. Therefore, the filter in accordance with the present invention can be considered “ideal-like”. 
     In order to achieve the aforementioned objective, the present invention discloses a filter for filtering communication signals, which comprises: 
     an input unit for accepting an input signal which can be expressed with a mathematical formula f(t); 
     a filter combination which further comprising a plurality of filters, each filter being able to generate a control signal and perform a signal filtering process based on the control signal, the control signal being expressed with a mathematical formula hde(θ); wherein, the filter combination processes the input signal based on a set of a plurality of control signals by means of the following formula: 
     
       
         
           
             
               GL 
               ⁡ 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       
                         v 
                         1 
                       
                       ∼ 
                       
                         v 
                         m 
                       
                     
                     ; 
                     
                       
                         u 
                         1 
                       
                       ∼ 
                       
                         u 
                         n 
                       
                     
                   
                   , 
                   S 
                 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           
                             
                               GS 
                               ⁡ 
                               
                                 ( 
                                 
                                   t 
                                   , 
                                   
                                     
                                       v 
                                       1 
                                     
                                     ∼ 
                                     
                                       v 
                                       m 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ; 
                           
                         
                         
                           
                             S 
                             = 
                             1 
                           
                         
                       
                       
                         
                           
                             
                               
                                 GS 
                                 _ 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   t 
                                   , 
                                   
                                     
                                       u 
                                       1 
                                     
                                     ∼ 
                                     
                                       u 
                                       n 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ; 
                           
                         
                         
                           
                             S 
                             = 
                             2 
                           
                         
                       
                       
                         
                           
                             
                               
                                 GS 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     , 
                                     
                                       
                                         v 
                                         1 
                                       
                                       ∼ 
                                       
                                         v 
                                         m 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 
                                   GS 
                                   _ 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     , 
                                     
                                       
                                         u 
                                         1 
                                       
                                       ∼ 
                                       
                                         u 
                                         n 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ; 
                           
                         
                         
                           
                             
                               S 
                               ≠ 
                               1 
                             
                             , 
                             
                               2 
                               ; 
                             
                           
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       i 
                       ) 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     when 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     S 
                   
                   = 
                   1 
                 
                 , 
                 
                   
                     calculate 
                     ⁢ 
                     
                       
                         
                           : 
                         
                         ⁢ 
                         
                             
                         
                         [ 
                         
                           GL 
                           ⁡ 
                           
                             ( 
                             
                               t 
                               ; 
                               
                                 
                                   v 
                                   1 
                                 
                                 ∼ 
                                 
                                   v 
                                   m 
                                 
                               
                               ; 
                               
                                 
                                   u 
                                   1 
                                 
                                 ∼ 
                                 
                                   u 
                                   n 
                                 
                               
                               ; 
                               S 
                             
                             ) 
                           
                         
                         ] 
                       
                       
                         1 
                         m 
                       
                     
                   
                   ; 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         ( 
                         ii 
                         ) 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       when 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       S 
                     
                     = 
                     2 
                   
                 
                 , 
                 
                   
                     calculate 
                     ⁢ 
                     
                       
                         
                           : 
                         
                         ⁢ 
                         
                             
                         
                         [ 
                         
                           GL 
                           ⁡ 
                           
                             ( 
                             
                               t 
                               ; 
                               
                                 
                                   v 
                                   1 
                                 
                                 ∼ 
                                 
                                   v 
                                   m 
                                 
                               
                               ; 
                               
                                 
                                   u 
                                   1 
                                 
                                 ∼ 
                                 
                                   u 
                                   n 
                                 
                               
                               ; 
                               S 
                             
                             ) 
                           
                         
                         ] 
                       
                       
                         1 
                         n 
                       
                     
                   
                   ; 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         ( 
                         iii 
                         ) 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       when 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       S 
                     
                     ≠ 
                     1 
                   
                 
                 , 
                 2 
                 , 
                 
                   
                     calculate 
                     ⁢ 
                     
                       
                         
                           : 
                         
                         ⁢ 
                         
                             
                         
                         [ 
                         
                           GL 
                           ⁡ 
                           
                             ( 
                             
                               t 
                               ; 
                               
                                 
                                   v 
                                   1 
                                 
                                 ∼ 
                                 
                                   v 
                                   m 
                                 
                               
                               ; 
                               
                                 
                                   u 
                                   1 
                                 
                                 ∼ 
                                 
                                   u 
                                   n 
                                 
                               
                               ; 
                               S 
                             
                             ) 
                           
                         
                         ] 
                       
                       
                         1 
                         
                           m 
                           + 
                           n 
                         
                       
                     
                   
                   ; 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   wherein 
                 
                 , 
                 
                   
 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     
                       
                         G 
                         r 
                       
                       ⁡ 
                       
                         ( 
                         
                           t 
                           , 
                           v 
                         
                         ) 
                       
                     
                     = 
                     
                       Fl 
                       ⁢ 
                       
                         { 
                         
                           
                             f 
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                           , 
                           
                             
                               hde 
                               ⁡ 
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                             ; 
                             v 
                           
                         
                         } 
                       
                     
                   
                   ; 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         
                           G 
                           r 
                         
                         _ 
                       
                       ⁡ 
                       
                         ( 
                         
                           t 
                           , 
                           v 
                         
                         ) 
                       
                     
                     = 
                     
                       Fl 
                       ⁢ 
                       
                         { 
                         
                           
                             f 
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                           , 
                           
                             
                               
                                 hde 
                                 _ 
                               
                               ⁡ 
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                             ; 
                             v 
                           
                         
                         } 
                       
                     
                   
                   ; 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         G 
                         s 
                       
                       ⁡ 
                       
                         ( 
                         
                           t 
                           , 
                           
                             
                               v 
                               1 
                             
                             ∼ 
                             
                               v 
                               m 
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         ∏ 
                         
                           i 
                           = 
                           1 
                         
                         m 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           G 
                           r 
                         
                         ⁡ 
                         
                           ( 
                           
                             t 
                             , 
                             
                               v 
                               i 
                             
                           
                           ) 
                         
                       
                     
                   
                   ; 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         
                           G 
                           s 
                         
                         _ 
                       
                       ⁡ 
                       
                         ( 
                         
                           t 
                           , 
                           
                             
                               u 
                               1 
                             
                             ∼ 
                             
                               u 
                               n 
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         ∏ 
                         
                           i 
                           = 
                           1 
                         
                         n 
                       
                       ⁢ 
                       
                         
                           
                             
                                 
                             
                             ⁢ 
                             
                               G 
                               r 
                             
                           
                           _ 
                         
                         ⁢ 
                         
                           ( 
                           
                             t 
                             , 
                             
                               u 
                               i 
                             
                           
                           ) 
                         
                       
                     
                   
                   ; 
                 
               
             
           
         
       
     
     wherein, v 1 ˜v m  and u 1 ˜u n  are practice values for v of each filter of the filter combination,  hde (θ)=1−hde(θ); and 
     an output unit for outputting an output signal which can be expressed with the mathematical formula GL(t,v 1 ˜v m ;u 1 ˜u n ,S). 
     In a preferred embodiment, wherein: 
     
       
         
           
             
               
                 hde 
                 ⁡ 
                 
                   ( 
                   θ 
                   ) 
                 
               
               = 
               
                 
                   A 
                   0 
                 
                 + 
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       1 
                     
                     k 
                   
                   ⁢ 
                   
                     [ 
                     
                       
                         A 
                         n 
                       
                       ⁢ 
                       
                         Cos 
                         ⁡ 
                         
                           ( 
                           
                             n 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                           ) 
                         
                       
                     
                     ] 
                   
                 
               
             
             ; 
           
         
       
     
     A n  (n=0, 1, . . . , k; where k is a positive integral) is a constant; 
     the filter can perform signal processes to a sine signal S(ω,t) by means of the following formula: 
     
       
         
           
             
               
                 
                   
                     Sde 
                     ⁡ 
                     
                       ( 
                       
                         ω 
                         , 
                         
                           t 
                           0 
                         
                       
                       ) 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       S 
                       ⁡ 
                       
                         ( 
                         
                           ω 
                           , 
                           
                             t 
                             0 
                           
                         
                         ) 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           A 
                           0 
                         
                         + 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               1 
                             
                             k 
                           
                           ⁢ 
                           
                             
                               A 
                               n 
                             
                             ⁢ 
                             
                               Cos 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     n 
                                     v 
                                   
                                   ⁢ 
                                   ω 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       S 
                       ⁡ 
                       
                         ( 
                         
                           ω 
                           , 
                           
                             t 
                             0 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       hde 
                       ⁡ 
                       
                         ( 
                         
                           ω 
                           v 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                     ⁢ 
                     
                       Fl 
                       ⁢ 
                       
                         { 
                         
                           
                             S 
                             ⁡ 
                             
                               ( 
                               
                                 ω 
                                 , 
                                 
                                   t 
                                   0 
                                 
                               
                               ) 
                             
                           
                           , 
                           
                             
                               hde 
                               ⁡ 
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                             ; 
                             v 
                           
                         
                         } 
                       
                     
                   
                   ; 
                 
               
             
           
         
       
     
     wherein, the value of A n  of the above equation is equal to the value of A n  of hde(θ), in addition, 
               v   =     ω   θ       ,         
and θ is a constant.
 
     In a preferred embodiment, wherein the constant value of A n  can be obtained by means of calculation using the following steps: 
     firstly, generating a mated square-wave signal having a period of 2π and height of 1, the mated square-wave signal being mathematically represented by the following equation: 
     
       
         
           
             
               
                 
                   
                     he 
                     ⁡ 
                     
                       ( 
                       
                         r 
                         , 
                         θ 
                       
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               1 
                             
                             
                               ; 
                             
                             
                               
                                 
                                   
                                     2 
                                     ⁢ 
                                     p 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     π 
                                   
                                   - 
                                   
                                     r 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     π 
                                   
                                 
                                 ≤ 
                                 θ 
                                 ≤ 
                                 
                                   
                                     2 
                                     ⁢ 
                                     p 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     π 
                                   
                                   + 
                                   
                                     r 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     π 
                                   
                                 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               ; 
                             
                             
                               
                                 θ 
                                 = 
                                 others 
                               
                             
                           
                         
                         ⁢ 
                         
                           
 
                         
                         ⁢ 
                         wherein 
                       
                       , 
                       
                         
 
                       
                       ⁢ 
                       
                         
                           0 
                           &lt; 
                           r 
                           ≤ 
                           
                             1 
                             2 
                           
                         
                         ; 
                         
                           ρ 
                           = 
                           0 
                         
                       
                       , 
                       
                         ± 
                         1 
                       
                       , 
                       
                         ± 
                         2 
                       
                       , 
                       
                         … 
                         ⁢ 
                         
                             
                         
                         ; 
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     1 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   1 
                 
               
             
           
         
       
     
     secondly, using Fourier transform to extend he(r,θ) to item k+1 (i.e., the item having an average value r), and letting it to be hse(k,r,θ), and then obtaining: 
     
       
         
           
             
               
                 
                   
                     
                       hse 
                       ⁡ 
                       
                         ( 
                         
                           k 
                           , 
                           r 
                           , 
                           θ 
                         
                         ) 
                       
                     
                     = 
                     
                       r 
                       + 
                       
                         
                           ∑ 
                           
                             u 
                             = 
                             1 
                           
                           k 
                         
                         ⁢ 
                         
                           
                             a 
                             u 
                           
                           ⁢ 
                           
                             Cos 
                             ⁡ 
                             
                               ( 
                               
                                 k 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   
                     ( 
                     1 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   2 
                 
               
             
           
         
       
     
     wherein, r is average value, and 
     
       
         
           
             
               
                 a 
                 u 
               
               = 
               
                 
                   2 
                   
                     k 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     π 
                   
                 
                 ⁢ 
                 
                   Sin 
                   ⁡ 
                   
                     ( 
                     
                       2 
                       ⁢ 
                       r 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       u 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       π 
                     
                     ) 
                   
                 
               
             
             ; 
           
         
       
     
     thirdly, letting 
               k   =   5     ,     r   =     1   2       ,         
and inputting them into equation (1)-2, and then letting:
 
     
       
         
           
             
               
                 hde 
                 ⁡ 
                 
                   ( 
                   θ 
                   ) 
                 
               
               = 
               
                 
                   
                     { 
                     
                       
                         
                           [ 
                           
                             hse 
                             ( 
                             
                               5 
                               , 
                               
                                 1 
                                 2 
                               
                               , 
                               θ 
                             
                             ) 
                           
                           ] 
                         
                         3 
                       
                       - 
                       1 
                     
                     } 
                   
                   7 
                 
                 + 
                 1 
               
             
             ; 
           
         
       
     
     therefore, the constant value of A n  of hde(θ) being obtained. 
     In a preferred embodiment, wherein a calculation of GL(t;7v;v,7v−2εv;3) can be achieved by performing filtering calculations of  G r   (t,v),  G r   (t,7v−2εv) and G r (t,7v); when εv=1, the calculation of GL(t;7v;v,7v−2εv;3) is a single-frequency filter. 
     In a preferred embodiment, wherein the filter can perform signal processes to the input signal f(t) by using the following mathematic function: 
     
       
         
           
             
               
                 Fl 
                 ⁢ 
                 
                   { 
                   
                     
                       f 
                       ⁡ 
                       
                         ( 
                         
                           t 
                           0 
                         
                         ) 
                       
                     
                     , 
                     
                       
                         hde 
                         ⁡ 
                         
                           ( 
                           θ 
                           ) 
                         
                       
                       ; 
                       v 
                     
                   
                   } 
                 
               
               = 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   m 
                 
                 ⁢ 
                 
                   [ 
                   
                     
                       S 
                       ⁡ 
                       
                         ( 
                         
                           
                             ω 
                             i 
                           
                           , 
                           
                             t 
                             o 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       hde 
                       ⁡ 
                       
                         ( 
                         
                           
                             ω 
                             i 
                           
                           v 
                         
                         ) 
                       
                     
                   
                   ] 
                 
               
             
             ; 
           
         
       
       
         
           
             wherein 
             , 
             
                
             
             ⁢ 
             
               
                 f 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   m 
                 
                 ⁢ 
                 
                   
                     S 
                     ⁡ 
                     
                       ( 
                       
                         
                           ω 
                           i 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                   . 
                 
               
             
           
         
       
     
     In order to achieve the aforementioned objective, the present invention discloses a method for filtering communication signals, which comprises: 
     using an input unit to accept an input signal, the input signal can be expressed by a mathematical formula f(t); 
     using a filter combination for generating a set of a plurality of control signals, and performing a signal filtering process based on the control signals; the control signals being expressed with a mathematical formula hde(θ); wherein, the filter combination processes the input signal based on the control signals by means of the following formula: 
     
       
         
           
             
               GL 
               ⁡ 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       
                         v 
                         1 
                       
                       ∼ 
                       
                         v 
                         m 
                       
                     
                     ; 
                     
                       
                         u 
                         1 
                       
                       ∼ 
                       
                         u 
                         n 
                       
                     
                   
                   , 
                   S 
                 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           
                             
                               GS 
                               ⁡ 
                               
                                 ( 
                                 
                                   t 
                                   , 
                                   
                                     
                                       v 
                                       1 
                                     
                                     ∼ 
                                     
                                       v 
                                       m 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ; 
                           
                         
                         
                           
                             S 
                             = 
                             1 
                           
                         
                       
                       
                         
                           
                             
                               GS 
                               ⁡ 
                               
                                 ( 
                                 
                                   t 
                                   , 
                                   
                                     
                                       u 
                                       1 
                                     
                                     ∼ 
                                     
                                       u 
                                       n 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ; 
                           
                         
                         
                           
                             S 
                             = 
                             2 
                           
                         
                       
                       
                         
                           
                             
                               
                                 GS 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     , 
                                     
                                       
                                         v 
                                         1 
                                       
                                       ∼ 
                                       
                                         v 
                                         m 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 
                                   GS 
                                   _ 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     , 
                                     
                                       
                                         u 
                                         1 
                                       
                                       ∼ 
                                       
                                         u 
                                         n 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ; 
                           
                         
                         
                           
                             
                               S 
                               ≠ 
                               1 
                             
                             , 
                             
                               2 
                               ; 
                             
                           
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       ( 
                       i 
                       ) 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     when 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     S 
                   
                   = 
                   1 
                 
                 , 
                 
                   
                     calculate 
                     ⁢ 
                     
                       
                         
                           : 
                         
                         ⁢ 
                         
                             
                         
                         [ 
                         
                           GL 
                           ⁡ 
                           
                             ( 
                             
                               t 
                               ; 
                               
                                 
                                   v 
                                   1 
                                 
                                 ∼ 
                                 
                                   v 
                                   m 
                                 
                               
                               ; 
                               
                                 
                                   u 
                                   1 
                                 
                                 ∼ 
                                 
                                   u 
                                   n 
                                 
                               
                               ; 
                               S 
                             
                             ) 
                           
                         
                         ] 
                       
                       
                         1 
                         m 
                       
                     
                   
                   ; 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         ( 
                         ii 
                         ) 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       when 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       S 
                     
                     = 
                     2 
                   
                 
                 , 
                 
                   
                     calculate 
                     ⁢ 
                     
                       
                         
                           : 
                         
                         ⁢ 
                         
                             
                         
                         [ 
                         
                           GL 
                           ⁡ 
                           
                             ( 
                             
                               t 
                               ; 
                               
                                 
                                   v 
                                   1 
                                 
                                 ∼ 
                                 
                                   v 
                                   m 
                                 
                               
                               ; 
                               
                                 
                                   u 
                                   1 
                                 
                                 ∼ 
                                 
                                   u 
                                   n 
                                 
                               
                               ; 
                               S 
                             
                             ) 
                           
                         
                         ] 
                       
                       
                         1 
                         n 
                       
                     
                   
                   ; 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         ( 
                         iii 
                         ) 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       when 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       S 
                     
                     ≠ 
                     1 
                   
                 
                 , 
                 2 
                 , 
                 
                   
                     calculate 
                     ⁢ 
                     
                       
                         
                           : 
                         
                         ⁡ 
                         
                           [ 
                           
                             GL 
                             ⁡ 
                             
                               ( 
                               
                                 t 
                                 ; 
                                 
                                   
                                     v 
                                     1 
                                   
                                   ∼ 
                                   
                                     v 
                                     m 
                                   
                                 
                                 ; 
                                 
                                   
                                     u 
                                     1 
                                   
                                   ∼ 
                                   
                                     u 
                                     n 
                                   
                                 
                                 ; 
                                 S 
                               
                               ) 
                             
                           
                           ] 
                         
                       
                       
                         1 
                         
                           m 
                           + 
                           n 
                         
                       
                     
                   
                   ; 
                   
                     
 
                   
                   ⁢ 
                   wherein 
                 
                 , 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     
                       
                         G 
                         r 
                       
                       ⁡ 
                       
                         ( 
                         
                           t 
                           , 
                           v 
                         
                         ) 
                       
                     
                     = 
                     
                       Fl 
                       ⁢ 
                       
                         { 
                         
                           
                             f 
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                           , 
                           
                             
                               hde 
                               ⁡ 
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                             ; 
                             v 
                           
                         
                         } 
                       
                     
                   
                   ; 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         
                           G 
                           r 
                         
                         _ 
                       
                       ⁡ 
                       
                         ( 
                         
                           t 
                           , 
                           v 
                         
                         ) 
                       
                     
                     = 
                     
                       Fl 
                       ⁢ 
                       
                         { 
                         
                           
                             f 
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                           , 
                           
                             
                               
                                 hde 
                                 _ 
                               
                               ⁡ 
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                             ; 
                             v 
                           
                         
                         } 
                       
                     
                   
                   ; 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         G 
                         s 
                       
                       ⁡ 
                       
                         ( 
                         
                           t 
                           , 
                           
                             
                               v 
                               1 
                             
                             ∼ 
                             
                               v 
                               m 
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         ∏ 
                         
                           i 
                           = 
                           1 
                         
                         m 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           G 
                           
                             r 
                             ⁢ 
                             
                                 
                             
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             t 
                             , 
                             
                               v 
                               i 
                             
                           
                           ) 
                         
                       
                     
                   
                   ; 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         
                           G 
                           s 
                         
                         _ 
                       
                       ⁡ 
                       
                         ( 
                         
                           t 
                           , 
                           
                             
                               u 
                               1 
                             
                             ∼ 
                             
                               u 
                               n 
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         ∏ 
                         
                           i 
                           = 
                           1 
                         
                         n 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           
                             G 
                             
                               r 
                               ⁢ 
                               
                                   
                               
                             
                           
                           _ 
                         
                         ⁡ 
                         
                           ( 
                           
                             t 
                             , 
                             
                               u 
                               i 
                             
                           
                           ) 
                         
                       
                     
                   
                   ; 
                 
               
             
           
         
       
     
     wherein, v 1 ˜v m  and u 1 ˜u n  are practice values for v of each filter of the filter combination,  hde (θ)=1−hde(θ); and 
     using an output unit for outputting an output signal which can be expressed with the mathematical formula GL(t,v 1 ˜v m ;u 1 ˜u n ,S). 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The details of the present invention will be more readily understood from a detailed description of the preferred embodiments taken in conjunction with the following figures. 
         FIG. 1  shows a diagram of wave form obtained based on equation (1)-1 according to the present invention. 
         FIG. 2  shows a diagram of wave form obtained based on the function 
               hse   ⁡     (     5   ,     1   2     ,   θ     )           
according to the present invention.
 
         FIG. 3  shows a diagram of wave form obtained based on the function 
                 [     hse   (     5   ,     1   2     ,   θ     )     ]     3         
according to the present invention.
 
         FIG. 4  shows a diagram of wave form obtained based on the function 
                   [     hse   ⁡     (     5   ,     1   2     ,   θ     )       ]     3     -   1         
according to the present invention.
 
         FIG. 5  shows a diagram of wave form obtained based on the function 
                 {         [     hse   (     5   ,     1   2     ,   θ     )     ]     3     -   1     }     7         
according to the present invention.
 
         FIG. 6  shows a diagram of wave form obtained based on the function 
                   {         [     hse   (     5   ,     1   2     ,   θ     )     ]     3     -   1     }     7     +   1         
according to the present invention.
 
         FIG. 7  is an enlarged view of  FIG. 6  showing the wave form in the range that when θ is within the range of 
       
         
           
             
               
                 [ 
                 
                   
                     
                       2 
                       ⁢ 
                       ρπ 
                     
                     - 
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       π 
                     
                   
                   , 
                   
                     
                       2 
                       ⁢ 
                       ρπ 
                     
                     + 
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       π 
                     
                   
                 
                 ] 
               
               . 
             
           
         
       
         FIG. 8  is an enlarged view of  FIG. 6  showing the wave form in the range that when θ is outside the range of 
       
         
           
             
               
                 [ 
                 
                   
                     
                       2 
                       ⁢ 
                       ρπ 
                     
                     - 
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       π 
                     
                   
                   , 
                   
                     
                       2 
                       ⁢ 
                       ρπ 
                     
                     + 
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       π 
                     
                   
                 
                 ] 
               
               . 
             
           
         
       
         FIG. 9  shows a frequency response diagram of function hde(θ) when v=8000 Hz according to the present invention. 
         FIG. 10  shows a frequency response diagram of function hde(θ) according to the present invention. 
         FIG. 11  shows a frequency response diagram of function  hde (θ) according to the present invention. 
         FIG. 12  shows a frequency response diagram according to functions Fl{f(t 0 ),hde(θ);8000}, Fl{f(t 0 ),hde(θ),12000}, Fl{f(t 0 ),hde(θ);28000}, and G 1 (t 0 )=Fl{f(t 0 ),hde(θ);8000}Fl{f(t 0 ),hde(θ);12000}Fl{f(t 0 ),hde(θ);28000} of the present invention. 
         FIG. 13  shows a frequency response diagram according to functions Fl{f(t 0 ),  hde (θ);8000}, Fl{f(t 0 ),hde(θ);16000}, Fl{f(t 0 ),hde(θ);44000}, and G 2 (t 0 )=Fl{f(t 0 ),  hde (θ);8000}Fl{f(t 0 ),hde(θ);16000}Fl{f(t 0 ),hde(θ);44000} of the present invention. 
         FIG. 14  shows a frequency response diagram according to functions  G r   (t,v),  G r   (t,7v−2εv), G r (t,7v), and GL(t;7v;v,7v−2εv;3) of the present invention. 
         FIG. 15  shows a block diagram of the ideal-like single-frequency filter according to a preferred embodiment of the present invention. 
         FIG. 16  shows a block diagram of the ideal-like band-pass filter according to a preferred embodiment of the present invention. 
         FIG. 17  shows a block diagram of the combination of ideal-like band-pass filter and single-frequency filter according to a preferred embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION 
     Similar to the conventional filters, the ideal-like band-pass filter and single-frequency filter of the present invention is also characterized in the design of Frequency Response Formula and Phase-Shift Response Formula. 
     (II) Description of Technique 1 
     Mated Periodical Square Waves of Frequency Response Formula 1 
     First, defining a formula for periodical mated square waves as following: 
     
       
         
           
             
               
                 
                   
                     he 
                     ⁡ 
                     
                       ( 
                       
                         r 
                         , 
                         θ 
                       
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               1 
                             
                             
                               ; 
                             
                             
                               
                                 
                                   
                                     2 
                                     ⁢ 
                                     p 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     π 
                                   
                                   - 
                                   
                                     r 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     π 
                                   
                                 
                                 ≤ 
                                 θ 
                                 ≤ 
                                 
                                   
                                     2 
                                     ⁢ 
                                     p 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     π 
                                   
                                   + 
                                   
                                     r 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     π 
                                   
                                 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               ; 
                             
                             
                               
                                 θ 
                                 = 
                                 others 
                               
                             
                           
                         
                         ⁢ 
                         
                           
 
                         
                         ⁢ 
                         wherein 
                       
                       , 
                       
                         
 
                       
                       ⁢ 
                       
                         
                           0 
                           &lt; 
                           r 
                           ≤ 
                           
                             1 
                             2 
                           
                         
                         ; 
                         
                           ρ 
                           = 
                           0 
                         
                       
                       , 
                       
                         ± 
                         1 
                       
                       , 
                       
                         ± 
                         2 
                       
                       , 
                       … 
                     
                   
                 
               
               
                 
                   
                     ( 
                     1 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   1 
                 
               
             
           
         
       
     
     The wave form of equation (1)-1 is shown in  FIG. 1 , which is a formula for periodical mated square waves having a height equal to 1, width equal to 2rπ and period equal to 2π. 
     Using Fourier transform to extend he(r,θ) to item k+1 (the item having an average value r), and letting it to be hse(k,r,θ), and then obtaining: 
     
       
         
           
             
               
                 
                   
                     hse 
                     ⁡ 
                     
                       ( 
                       
                         k 
                         , 
                         r 
                         , 
                         θ 
                       
                       ) 
                     
                   
                   = 
                   
                     r 
                     + 
                     
                       
                         ∑ 
                         
                           u 
                           = 
                           1 
                         
                         k 
                       
                       ⁢ 
                       
                         
                           a 
                           u 
                         
                         ⁢ 
                         
                           Cos 
                           ⁡ 
                           
                             ( 
                             
                               k 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     1 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   2 
                 
               
             
           
         
       
     
     Wherein, 
     r is average value; and 
     
       
         
           
             
               a 
               u 
             
             = 
             
               
                 2 
                 
                   k 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   π 
                 
               
               ⁢ 
               
                 
                   Sin 
                   ⁡ 
                   
                     ( 
                     
                       2 
                       ⁢ 
                       ru 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       π 
                     
                     ) 
                   
                 
                 . 
               
             
           
         
       
     
     Hereunder is an example: 
     In equation (1)-2, if let k=5, 
               r   =     1   2       ,         
then the figure of
 
             hs   (     5   ,     1   2     ,   θ     )         
will become the one shown in  FIG. 2 .
 
     Now, let us calculate the limited series 
             hs   (     5   ,     1   2     ,   θ     )         
by using the following steps:
 
     1. Multiplying 
             hse   (     5   ,     1   2     ,   θ     )         
by itself for 3 times, that is
 
                 [     hse   (     5   ,     1   2     ,   θ     )     ]     3     ,         
and then its wave form will be like the one shown in  FIG. 3 . It can be seen from  FIG. 3  that, when
 
                 2   ⁢   ρπ     -       1   2     ⁢   π       ≤   θ   ≤       2   ⁢   ρπ     +       1   2     ⁢   π             
(wherein, ρ=0, ±1, ±2, . . . ), the value thereof is still various near to value 1 of the height. However, in other ranges, the value will become zero much rapidly than which shown in  FIG. 2 , this is due to the fact that, the absolute value of
 
             he   (     5   ,     1   2     ,   θ     )         
in said “other ranges” is smaller than 1, and therefore this value will approach to zero rapidly after multiplying by itself for 3 times.
 
     2. Subtracting 1 from 
                 [     hse   (     5   ,     1   2     ,   θ     )     ]     3     ,       that   ⁢             ⁢             ⁢       is   ⁢           [     hse   (     5   ,     1   2     ,   θ     )     ]     3       -   1     ,         
and the wave form thereof become the one shown in  FIG. 4 . Under this condition, when θ is belong to the range of
 
               [         2   ⁢   ρπ     -       1   2     ⁢   π       ,       2   ⁢   ρπ     +       1   2     ⁢   π         ]     ,         
its value will be various near to zero (absolute value always smaller than 1).
 
     3. Multiplying 
                 [     hse   (     5   ,     1   2     ,   θ     )     ]     3     -   1         
by itself for 7 times, that is
 
                 {         [     hse   (     5   ,     1   2     ,   θ     )     ]     3     -   1     }     7     ,         
and then its wave form will be like the one shown in  FIG. 5 . That is, when θ is in the range of
 
               [         2   ⁢   ρπ     -       1   2     ⁢   π       ,       2   ⁢   ρπ     +       1   2     ⁢   ρπ         ]     ,         
its value will approach to zero rapidly.
 
     4. Finally, moving 
               {         [     hse   (     5   ,     1   2     ,   θ     )     ]     3     -   1     }     7         
upward for one unit, that is
 
                   {         [     hse   (     5   ,     1   2     ,   θ     )     ]     3     -   1     }     7     +   1     ,         
and the wave form thereof become  FIG. 6 .
 
     Magnifying the wave form of  FIG. 6  near the position where θ is in the range of 
               [         2   ⁢   ρπ     -       1   2     ⁢   π       ,       2   ⁢   ρπ     +       1   2     ⁢   π         ]     ,         
and a magnified wave form like  FIG. 7  will be obtained. In the mean time,  FIG. 8  shows another magnified wave form when θ is outside of the same range.
 
     It can be seen from  FIG. 7  and  FIG. 8  that, limited series 
                 {         [     he   (     5   ,       1   2     ⁢   θ       )     ]     3     -   1     }     7     +   1         
is very near to the value of
 
             he   (       1   2     ,   θ     )         
in the equation (1)-1 when
 
     
       
         
           
             r 
             = 
             
               
                 1 
                 2 
               
               . 
             
           
         
       
     
     The above mentioned 3-times or 7-times self-multiplication is only an example. When the number or self-multiplication increases, the better result will be obtained in approaching 
               he   (       1   2     ,   θ     )     ,         
which should depend on the accuracy required.
 
Generally speaking, we can choose suitable values for variables u, m and n (wherein u, m and n are natural numbers) in the following equation
 
                 {         [     hse   (     u   ,     1   2     ,   θ     )     ]     m     -   1     }     n     *       1   +       (     -   1     )       n   +   1         2           
in order to obtain desired accuracy.
 
     (III) Description of Technique 2 
     Mated Periodical Square Waves of Frequency Response Formula 2 
     Following the above mentioned example, letting: 
     
       
         
           
             
               
                 
                   
                     hde 
                     ⁡ 
                     
                       ( 
                       θ 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         { 
                         
                           
                             
                               [ 
                               
                                 hse 
                                 ( 
                                 
                                   5 
                                   , 
                                   
                                     1 
                                     2 
                                   
                                   , 
                                   θ 
                                 
                                 ) 
                               
                               ] 
                             
                             3 
                           
                           - 
                           1 
                         
                         } 
                       
                       7 
                     
                     + 
                     1 
                   
                 
               
               
                 
                   
                     ( 
                     2 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   1 
                 
               
             
           
         
       
     
     Because hdr(θ) is a mated function, it only contains harmonic cosine signals. Therefore, by extending the above equation according to Cos θ            Cos 2θ          . . . Cos(105θ), the following equation can be obtained:
     
       
         
           
             
               
                 
                   
                     hde 
                     ⁡ 
                     
                       ( 
                       θ 
                       ) 
                     
                   
                   = 
                   
                     
                       A 
                       0 
                     
                     + 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           1 
                         
                         105 
                       
                       ⁢ 
                       
                         [ 
                         
                           
                             A 
                             n 
                           
                           ⁢ 
                           
                             Cos 
                             ⁡ 
                             
                               ( 
                               
                                 n 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     2 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   2 
                 
               
             
           
         
       
     
     Wherein, A n  (n=0, 1, . . . 105) are constants, which are status variables. 
     Equation (2)-2 is a kind of triangular series. It is not a pure Fourier series now, but is another kind of series which can converge much faster. 
     Assuming there exists a sine signal S(ω,t) that:
 
 S (ω, t )= C  Cos(ω t )+ D  Sin(ω, t )
 
     Now, let sampling rate to be v, and then start sampling S(ω,t) to get 105 samples before and after the point t=t 0 , that is, 
               S   ⁡     (     ω   ,       t   0     -     105   v         )       ,     
     ⁢     S   ⁡     (     ω   ,       t   0     -     104   v         )       ,     …   ⁢           ⁢     S   ⁡     (     ω   ,       t   0     -     1   v         )         ,     S   ⁡     (     ω   ,       t   0     +     1   v         )       ,     
     ⁢     S   ⁡     (     ω   ,       t   0     +     2   v         )       ,       …   ⁢           ⁢     S   ⁡     (     ω   ,       t   0     +     105   v         )         ;           
and then, perform the following calculation:
 
     
       
         
           
             
               
                 
                   
                     Sde 
                     ⁡ 
                     
                       ( 
                       
                         ω 
                         , 
                         
                           t 
                           0 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         A 
                         0 
                       
                       ⁢ 
                       
                         S 
                         ⁡ 
                         
                           ( 
                           
                             ω 
                             , 
                             
                               t 
                               0 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             1 
                           
                           105 
                         
                         ⁢ 
                         
                           
                             A 
                             n 
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 S 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     ω 
                                     , 
                                     
                                       
                                         t 
                                         0 
                                       
                                       - 
                                       
                                         n 
                                         v 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 S 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     ω 
                                     , 
                                     
                                       
                                         t 
                                         0 
                                       
                                       + 
                                       
                                         n 
                                         v 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     2 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   3 
                 
               
             
           
         
       
     
     Wherein, A n  is the same as A n  (n=0, 1, 2, . . . , 105) of equation (2)-2, because: 
     
       
         
           
             
               
                 
                   
                     
                       S 
                       ⁡ 
                       
                         ( 
                         
                           ω 
                           , 
                           
                             
                               t 
                               0 
                             
                             - 
                             
                               n 
                               v 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       S 
                       ⁡ 
                       
                         ( 
                         
                           ω 
                           , 
                           
                             
                               t 
                               0 
                             
                             + 
                             
                               n 
                               v 
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                     
                   ⁢ 
                   
                     C 
                     ⁡ 
                     
                       [ 
                       
                         
                           Cos 
                           ⁢ 
                           
                             ( 
                             
                               
                                 ω 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   t 
                                   0 
                                 
                               
                               - 
                               
                                 
                                   n 
                                   v 
                                 
                                 ⁢ 
                                 ω 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           Cos 
                           ⁢ 
                           
                             ( 
                             
                               
                                 ω 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   t 
                                   o 
                                 
                               
                               + 
                               
                                 
                                   n 
                                   v 
                                 
                                 ⁢ 
                                 ω 
                               
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   + 
                     
                   ⁢ 
                   
                     D 
                     ⁡ 
                     
                       [ 
                       
                         
                           Sin 
                           ⁢ 
                           
                             ( 
                             
                               
                                 ω 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   t 
                                   0 
                                 
                               
                               - 
                               
                                 
                                   n 
                                   v 
                                 
                                 ⁢ 
                                 ω 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           Sin 
                           ⁢ 
                           
                             ( 
                             
                               
                                 ω 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   t 
                                   o 
                                 
                               
                               + 
                               
                                 
                                   n 
                                   v 
                                 
                                 ⁢ 
                                 ω 
                               
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       2 
                       ⁢ 
                       C 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         Cos 
                         ⁡ 
                         
                           ( 
                           
                             ω 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               t 
                               0 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         Cos 
                         ⁡ 
                         
                           ( 
                           
                             
                               n 
                               v 
                             
                             ⁢ 
                             ω 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       D 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         Sin 
                         ⁡ 
                         
                           ( 
                           
                             ω 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               t 
                               0 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         Cos 
                         ⁡ 
                         
                           ( 
                           
                             
                               n 
                               v 
                             
                             ⁢ 
                             ω 
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     2 
                     ⁢ 
                     
                       
                         Cos 
                         ⁡ 
                         
                           ( 
                           
                             
                               n 
                               v 
                             
                             ⁢ 
                             ω 
                           
                           ) 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             C 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Cos 
                             ⁢ 
                             
                               ( 
                               
                                 ω 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   t 
                                   0 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             D 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Sin 
                             ⁢ 
                             
                               ( 
                               
                                 ω 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   t 
                                   0 
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     2 
                     ⁢ 
                     
                       Cos 
                       ⁡ 
                       
                         ( 
                         
                           
                             n 
                             v 
                           
                           ⁢ 
                           ω 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       S 
                       ⁡ 
                       
                         ( 
                         
                           ω 
                           , 
                           
                             t 
                             0 
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
     Input the above result into equation (2)-3 and get: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           Sde 
                           ⁡ 
                           
                             ( 
                             
                               ω 
                               , 
                               
                                 t 
                                 0 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             S 
                             ⁡ 
                             
                               ( 
                               
                                 ω 
                                 , 
                                 
                                   t 
                                   0 
                                 
                               
                               ) 
                             
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 A 
                                 0 
                               
                               + 
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     1 
                                   
                                   105 
                                 
                                 ⁢ 
                                 
                                   
                                     A 
                                     n 
                                   
                                   ⁢ 
                                   
                                     Cos 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           n 
                                           v 
                                         
                                         ⁢ 
                                         ω 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             S 
                             ⁡ 
                             
                               ( 
                               
                                 ω 
                                 , 
                                 
                                   t 
                                   0 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             hde 
                             ⁡ 
                             
                               ( 
                               
                                 ω 
                                 v 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     2 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   4 
                 
               
             
           
         
       
     
     Because ω≧0 in this place, therefore it can be seen from  FIG. 6  that, when 
             0   ≤     ω   v     ≤       π   2     ⁢           ⁢   or   ⁢           ⁢   when                         2   ⁢   i     +   1     2     ≤     ω   v     ≤           2   ⁢   i     +   3     2     ⁢     π   (       i   =   1     ,   2   ,     3   ⁢           ⁢   …       ⁢           )         ,         
that is, when
 
             0   ≤   ω   ≤       π   2     ⁢   v           
or when
 
                       2   ⁢   i     +   1     2     -     π   ⁢           ⁢   v       ≤   ω   ≤           2   ⁢   i     +   3     2     ⁢   π   ⁢           ⁢   v       ,         
the Sde(ω,t 0 )→S(ω,t 0 ); otherwise, when ω is not in the above mentioned ranges, then Sde(ω,t 0 )→0.
 
     Where ω is angle-frequency and can be presented by ω=2πf (where f is frequency). Thereby, it can also be referred as the frequency f of sine signal S(ω,t). When 
               f   ∈       [     0   ,       1   4     ⁢   v       ]     ⁢           ⁢   or   ⁢           ⁢   f     ∈     [             2   ⁢   i     +   1     4     ⁢   v     ,           2   ⁢   i     +   3     4     ⁢   v       ]       ,         
and when t=t 0 , the result of calculation of equation (2)-3 will almost the same as the original S(ω,t 0 ) (same for any t 0 ); however, the result will be near to zero if f is not in the above mentioned range.
 
     From the equation of Sde(ω,t 0 )→S(ω,t 0 ), it can be seen (or be seen from  FIG. 6 ) that, for any frequency of signal in the range of responsive f (that is, 
               f   ∈       [     0   ,       1   4     ⁢   v       ]     ⁢           ⁢   or   ⁢           ⁢   f     ∈     [             2   ⁢   i     +   1     4     ⁢   v     ,           2   ⁢   i     +   3     4     ⁢   v       ]       ,         
where i=1, 2, 3, . . . ), the extracted signal not only has an amplitude which is almost constant, but also has a constant delay of time due to calculation for any responsive frequency. Since there is no problem of amplitude variations in different frequencies, therefore there won&#39;t be any problem of phase-shifts as well.
 
     As a result, the filter in accordance with the present invention can be considered to be an ideal-like filter. 
     However, the responsive frequencies thereof are periodical. Based on the aforementioned descriptions, it is understood that the period is corresponding to the sampling rate vHz, and the total range of the responsive frequencies of each period is 
     
       
         
           
             
               v 
               2 
             
             ⁢ 
             
               Hz 
               . 
             
           
         
       
     
     For example, as shown in  FIG. 9 , let v=8000 Hz, such that the period of responsive frequencies will be 8000 Hz. The ranges of responsive frequencies in the first period are 0 Hz˜2000 Hz and 6000 Hz˜8000 Hz (wherein, the total range of responsive frequencies is 2000 Hz+2000 Hz=4000 Hz, which is equal to a half of 8000 Hz), while the range of unresponsive frequency is 2000 Hz˜6000 Hz (wherein, the total range of unresponsive frequencies is also equal to a half of 8000 Hz, that is 4000 Hz). In the second period, The ranges of responsive frequencies are 8000 Hz˜10000 Hz and 14000 Hz˜16000 Hz (wherein the total range is still 4000 Hz), while the range of unresponsive frequency is 10000 Hz˜14000 Hz (also 4000 Hz). Because the same rules can also be applied to the following periods, thereby descriptions thereof will be omitted. 
     In addition, according to the aforementioned equation (2)-1 and the example of hde(θ), the width between the responsive and unresponsive frequencies (also referred as threshold width hereinafter) is around 10 −4  Tv (where Tv is period and has its unit is Hz) as shown in  FIG. 10 . Therefore, if Tv=8000 Hz, then its threshold width is around 0.8 Hz. Thereby, the borderlines between responsive and unresponsive frequencies are very clear and definite. 
     However, no matter what, the above illustrated method for designing filters is still periodical. And, for the example of v=8000 Hz, the width between responsive and unresponsive frequencies is only 4000 Hz, and thus is not practical enough for commercial applications. Therefore, the following description will illustrate a method to extend the period to Mv (where M is a very large number, and v is the basic sampling rate). 
     (IV) Description of Technique 3 
     Method to Extend Periods 
     The following illustration is divided into 3 sections: 
     Section 1: Compensatory Function 
     Let  hde (θ)=1−hde(θ), and then it can be seen from equation (2)-2 that: 
     
       
         
           
             
               
                 
                   
                     
                       hde 
                       _ 
                     
                     ⁡ 
                     
                       ( 
                       θ 
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         1 
                         - 
                         
                           A 
                           0 
                         
                       
                       ) 
                     
                     - 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           1 
                         
                         105 
                       
                       ⁢ 
                       
                         [ 
                         
                           
                             A 
                             n 
                           
                           ⁢ 
                           
                             Cos 
                             ⁡ 
                             
                               ( 
                               
                                 n 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     3 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   1 
                 
               
             
           
         
       
     
     where  hde (θ) is referred as the compensatory function of hde(θ) hereinafter. 
     Similarly, assume there is a sine signal S(ω,t) that:
 
 S (ω, t )= C  Cos(ω t )+ D  Sin(ω, t )
 
     And then, start sampling S(ω,t) before and after t=t 0  in order to get 105 samples each, and perform the following calculation: 
     
       
         
           
             
               
                 
                   
                     
                       Sde 
                       _ 
                     
                     ⁡ 
                     
                       ( 
                       
                         ω 
                         , 
                         
                           t 
                           0 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           1 
                           - 
                           
                             A 
                             0 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         S 
                         ⁡ 
                         
                           ( 
                           
                             ω 
                             , 
                             
                               t 
                               0 
                             
                           
                           ) 
                         
                       
                     
                     - 
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             1 
                           
                           105 
                         
                         ⁢ 
                         
                           
                             A 
                             n 
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 S 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     ω 
                                     , 
                                     
                                       t 
                                       - 
                                       
                                         n 
                                         v 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 S 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     ω 
                                     , 
                                     
                                       
                                         t 
                                         0 
                                       
                                       + 
                                       
                                         n 
                                         v 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     3 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   2 
                 
               
             
           
         
       
     
     And thus we can obtain: 
     
       
         
           
             
               
                 
                   
                     
                       Sde 
                       _ 
                     
                     ⁡ 
                     
                       ( 
                       
                         ω 
                         , 
                         
                           t 
                           0 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       S 
                       ⁡ 
                       
                         ( 
                         
                           ω 
                           , 
                           
                             t 
                             0 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         hde 
                         _ 
                       
                       ⁡ 
                       
                         ( 
                         
                           ω 
                           v 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     3 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   3 
                 
               
             
           
         
       
     
     From equation (3)-2, it can be seen that the range between responsive and unresponsive frequencies is like the one shown in  FIG. 11 , which is right opposite to the figure shown in  FIG. 9 . 
     Section 2: Filtering Calculation 
     In the calculation of equation (2)-3, we can say that it refers to a filtering calculation for signal S(ω,t) and hde(θ) of equation (2)-3 when t=t 0  and that the period of responsive frequencies is vHz. Let symbol Fl{S(ω,t 0 ),hde(θ);v} to represent this calculation, and it can be obtained from equation (2)-4 that: 
     
       
         
           
             
               
                 
                   
                     Fl 
                     ⁢ 
                     
                       { 
                       
                         
                           S 
                           ⁡ 
                           
                             ( 
                             
                               ω 
                               , 
                               
                                 t 
                                 0 
                               
                             
                             ) 
                           
                         
                         , 
                         
                           
                             hde 
                             ⁡ 
                             
                               ( 
                               θ 
                               ) 
                             
                           
                           ; 
                           v 
                         
                       
                       } 
                     
                   
                   = 
                   
                     
                       S 
                       ⁡ 
                       
                         ( 
                         
                           ω 
                           , 
                           
                             t 
                             0 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       hde 
                       ⁡ 
                       
                         ( 
                         
                           ω 
                           v 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     3 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   5 
                 
               
             
           
         
       
     
     Similarly, in the calculation of equation (3)-2, we can say that it refers to a filtering calculation for signal S(ω,t) and  hde (θ) of equation (3)-2 when t=t 0  and that the period of responsive frequencies is vHz. Let symbol Fl{S(ω,t 0 ),  hde (θ);v} to represent this calculation, and it can be obtained from equation (3)-3 that: 
     
       
         
           
             
               
                 
                   
                     Fl 
                     ⁢ 
                     
                       { 
                       
                         
                           S 
                           ⁡ 
                           
                             ( 
                             
                               ω 
                               , 
                               
                                 t 
                                 0 
                               
                             
                             ) 
                           
                         
                         , 
                         
                           
                             
                               hde 
                               _ 
                             
                             ⁡ 
                             
                               ( 
                               θ 
                               ) 
                             
                           
                           ; 
                           v 
                         
                       
                       } 
                     
                   
                   = 
                   
                     
                       S 
                       ⁡ 
                       
                         ( 
                         
                           ω 
                           , 
                           
                             t 
                             0 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         hde 
                         _ 
                       
                       ⁡ 
                       
                         ( 
                         
                           ω 
                           v 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     3 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   6 
                 
               
             
           
         
       
     
     Assume there is a signal being defined as: 
     
       
         
           
             
               
                 
                   
                     
                       f 
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         m 
                       
                       ⁢ 
                       
                         S 
                         ⁡ 
                         
                           ( 
                           
                             
                               ω 
                               i 
                             
                             , 
                             t 
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     Wherein 
                     , 
                   
                 
               
               
                 
                   
                     ( 
                     3 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   7 
                 
               
             
             
               
                 
                   
                     S 
                     ⁡ 
                     
                       ( 
                       
                         
                           ω 
                           i 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         C 
                         i 
                       
                       ⁢ 
                       
                         Cos 
                         ⁡ 
                         
                           ( 
                           
                             
                               ω 
                               i 
                             
                             ⁢ 
                             t 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         D 
                         i 
                       
                       ⁢ 
                       
                         Sin 
                         ⁡ 
                         
                           ( 
                           
                             
                               ω 
                               i 
                             
                             ⁢ 
                             t 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     3 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   8 
                 
               
             
           
         
       
     
     Then we can also define f(t) to perform filtering calculations with hde(θ) or  hde (θ) when t=t 0  and that the period of responsive frequencies is vHz. And, the symbols Fl{f(t 0 ),hde(θ);v} and Fl{f(t 0 ),  hde (θ);v} are used to represent these two calculations. Base on this definition, the following equations can be obtained easily from equations (3)-5 and (3)-6: 
     
       
         
           
             
               
                 
                   
                     Fl 
                     ⁢ 
                     
                       { 
                       
                         
                           f 
                           ⁡ 
                           
                             ( 
                             
                               t 
                               0 
                             
                             ) 
                           
                         
                         , 
                         
                           
                             hde 
                             ⁡ 
                             
                               ( 
                               θ 
                               ) 
                             
                           
                           ; 
                           v 
                         
                       
                       } 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       m 
                     
                     ⁢ 
                     
                       [ 
                       
                         
                           S 
                           ⁡ 
                           
                             ( 
                             
                               
                                 ω 
                                 i 
                               
                               , 
                               
                                 t 
                                 o 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           hde 
                           ⁡ 
                           
                             ( 
                             
                               
                                 ω 
                                 i 
                               
                               v 
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   
                     ( 
                     3 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   9 
                 
               
             
             
               
                 
                   
                     Fl 
                     ⁢ 
                     
                       { 
                       
                         
                           f 
                           ⁡ 
                           
                             ( 
                             
                               t 
                               0 
                             
                             ) 
                           
                         
                         , 
                         
                           
                             
                               hde 
                               _ 
                             
                             ⁡ 
                             
                               ( 
                               θ 
                               ) 
                             
                           
                           ; 
                           v 
                         
                       
                       } 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       m 
                     
                     ⁢ 
                     
                       [ 
                       
                         
                           S 
                           ⁡ 
                           
                             ( 
                             
                               
                                 ω 
                                 i 
                               
                               , 
                               
                                 t 
                                 o 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           
                             hde 
                             _ 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 ω 
                                 i 
                               
                               v 
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   
                     ( 
                     3 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   10 
                 
               
             
           
         
       
     
     Section 3: Method for Extending Periods of Responsive Frequencies 
     The following two examples are illustrated before introducing the general method. 
     Example 1 
     (1) Firstly, perform filtering calculation of Fl{f(t 0 ),hde(θ);8000}, and the condition of its responsive frequencies will be like the one shown in axis V 1  of  FIG. 12  (that is, being responsive when the height is 1, and unresponsive when the height is 0). 
     (2) Secondly, perform calculation of Fl{f(t 0 ),hde(θ);12000}, and the condition of its responsive frequencies will be like the one shown in axis V 2  of  FIG. 12 . 
     (3) Thirdly, perform calculation of Fl{f(t 0 ),hde(θ);28000}, and the condition of its responsive frequencies will be like the one shown in axis V 3  of  FIG. 12 . 
     (4) Finally, multiply the results of the above three calculations, and let its result to be G 1 (t 0 ), that is:
 
 G   1 ( t   0 )= Fl{f ( t   0 ), hde (θ);8000 }Fl{f ( t   0 ), hde (θ);24000 }Fl{f ( t   0 ), hde (θ);56000}
 
     It is obvious that, the condition of the responsive frequencies of G 1 (t 0 ) is like the one shown in axis V 4  of  FIG. 12 . In can be seen from axis V 4  that, the signal in the range 0˜2000 Hz is responsive at beginning, and then, the next responsive range will directly jump to 46000 Hz˜50000 Hz. 
     (5) The cube root of the calculated G 1 (t 0 ) is 
                 [       G   1     ⁡     (     t   0     )       ]       1   3       .         
Therefore, signals in f(t 0 ) having frequencies among 0˜2000 Hz will be extracted (the frequencies of next extracted signals will directly jump to 46000 Hz˜50000 Hz).
 
     Example 2 
     (1) Perform filtering calculations for Fl{f(t 0 ),  hde (θ);8000}, Fl{f(t 0 ),hde(θ);32000} and Fl{f(t 0 ),hde(θ);88000} separately, and the conditions of their responsive frequencies will be like axes V 1 , V 2  and V 3  of  FIG. 13  respectively. 
     (2) Let the multiplication of these three results of calculations to be G 2 (t 0 ), that is:
 
 G   2 ( t   0 )= Fl{f ( t   0 ),   hde   (θ);8000 }Fl{f ( t   0 ); hde (θ);32000 }Fl{f ( t   0 ); hde (θ);88000}
 
     The condition of the responsive frequencies of G 2 (t 0 ) will be like axis V 4  of  FIG. 13 . In can be seen from this axis V 4  that, the signal in the range 2000 Hz˜6000 Hz is responsive at beginning, and then, the next responsive range will directly jump to 60000 Hz˜70000 Hz. 
     (3) Finally, calculate 
                 [       G   2     ⁡     (     t   0     )       ]       1   3       .         
Therefore, signals in f(t 0 ) having frequencies among 2000 Hz˜6000 Hz will be extracted (the frequencies of next extracted signals will directly jump to 60000 Hz˜70000 Hz).
 
     Example 3 
     General Method 
     (1) Firstly, let&#39;s simplify these mathematic symbols by defining: 
     
       
         
           
             
               
                 
                   
                     
                       G 
                       r 
                     
                     ⁡ 
                     
                       ( 
                       
                         t 
                         , 
                         v 
                       
                       ) 
                     
                   
                   = 
                   
                     Fl 
                     ⁢ 
                     
                       { 
                       
                         
                           f 
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                         , 
                         
                           
                             hde 
                             ⁡ 
                             
                               ( 
                               θ 
                               ) 
                             
                           
                           ; 
                           v 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   
                     ( 
                     3 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   11 
                 
               
             
             
               
                 
                   
                     
                       
                         G 
                         _ 
                       
                       r 
                     
                     ⁡ 
                     
                       ( 
                       
                         t 
                         , 
                         v 
                       
                       ) 
                     
                   
                   = 
                   
                     Fl 
                     ⁢ 
                     
                       { 
                       
                         
                           f 
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                         , 
                         
                           
                             
                               hde 
                               _ 
                             
                             ⁡ 
                             
                               ( 
                               θ 
                               ) 
                             
                           
                           ; 
                           v 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   
                     ( 
                     3 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   12 
                 
               
             
             
               
                 
                   
                     
                       G 
                       s 
                     
                     ⁡ 
                     
                       ( 
                       
                         t 
                         , 
                         
                           
                             v 
                             1 
                           
                           ~ 
                           
                             v 
                             m 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∏ 
                       
                         i 
                         = 
                         1 
                       
                       m 
                     
                     ⁢ 
                     
                       
                         G 
                         r 
                       
                       ⁡ 
                       
                         ( 
                         
                           t 
                           , 
                           
                             v 
                             i 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     3 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   13 
                 
               
             
             
               
                 
                   
                     
                       
                         G 
                         s 
                       
                       _ 
                     
                     ⁡ 
                     
                       ( 
                       
                         t 
                         , 
                         
                           
                             u 
                             1 
                           
                           ~ 
                           
                             u 
                             m 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∏ 
                       
                         i 
                         = 
                         1 
                       
                       n 
                     
                     ⁢ 
                     
                       
                         
                           G 
                           r 
                         
                         _ 
                       
                       ⁡ 
                       
                         ( 
                         
                           t 
                           , 
                           
                             u 
                             i 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     3 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   14 
                 
               
             
             
               
                 
                   
                     G 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       L 
                       ⁡ 
                       
                         ( 
                         
                           t 
                           , 
                           
                             
                               
                                 v 
                                 i 
                               
                               ~ 
                               
                                 v 
                                 m 
                               
                             
                             ; 
                             
                               
                                 u 
                                 1 
                               
                               ~ 
                               
                                 u 
                                 m 
                               
                             
                           
                           , 
                           S 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               G 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 S 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     , 
                                     
                                       
                                         v 
                                         1 
                                       
                                       ~ 
                                       
                                         v 
                                         m 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ; 
                           
                         
                         
                           
                             S 
                             = 
                             1 
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   G 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   S 
                                 
                                 _ 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   t 
                                   , 
                                   
                                     
                                       u 
                                       1 
                                     
                                     ~ 
                                     
                                       u 
                                       n 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ; 
                           
                         
                         
                           
                             S 
                             = 
                             2 
                           
                         
                       
                       
                         
                           
                             
                               G 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 S 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     , 
                                     
                                       
                                         v 
                                         1 
                                       
                                       ~ 
                                       
                                         v 
                                         m 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 
                                   G 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   S 
                                 
                                 _ 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   t 
                                   , 
                                   
                                     
                                       u 
                                       1 
                                     
                                     ~ 
                                     
                                       u 
                                       m 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ; 
                           
                         
                         
                           
                             
                               S 
                               ≠ 
                               1 
                             
                             , 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     3 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   15 
                 
               
             
           
         
       
     
     (2) When v 1 ˜v m , u 1 ˜u m  and S are selected, let&#39;s firstly perform calculation of equation (3)-15 for GL(t;v 1 ˜v m ;u 1 ˜u m ;S), and then perform one of the following calculations: 
     
       
         
           
             
               
                 
                   ( 
                   i 
                   ) 
                 
               
               
                 
                   
                     
                       when 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       S 
                     
                     = 
                     1 
                   
                   , 
                   
                     calculate 
                     ⁢ 
                     
                       
                         
                           : 
                         
                         ⁢ 
                         
                             
                         
                         [ 
                         
                           G 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             L 
                             ⁡ 
                             
                               ( 
                               
                                 t 
                                 ; 
                                 
                                   
                                     v 
                                     1 
                                   
                                   ~ 
                                   
                                     v 
                                     m 
                                   
                                 
                                 ; 
                                 
                                   
                                     u 
                                     1 
                                   
                                   ~ 
                                   
                                     u 
                                     m 
                                   
                                 
                                 ; 
                                 S 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                       
                         1 
                         m 
                       
                     
                   
                 
               
             
             
               
                 
                   ( 
                   ii 
                   ) 
                 
               
               
                 
                   
                     
                       when 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       S 
                     
                     = 
                     2 
                   
                   , 
                   
                     calculate 
                     ⁢ 
                     
                       
                         
                           : 
                         
                         ⁢ 
                         
                             
                         
                         [ 
                         
                           G 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             L 
                             ⁡ 
                             
                               ( 
                               
                                 t 
                                 ; 
                                 
                                   
                                     v 
                                     1 
                                   
                                   ~ 
                                   
                                     v 
                                     m 
                                   
                                 
                                 ; 
                                 
                                   
                                     u 
                                     1 
                                   
                                   ~ 
                                   
                                     u 
                                     m 
                                   
                                 
                                 ; 
                                 S 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                       
                         1 
                         n 
                       
                     
                   
                 
               
             
             
               
                 
                   ( 
                   iii 
                   ) 
                 
               
               
                 
                   
                     
                       when 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       S 
                     
                     ≠ 
                     1 
                   
                   , 
                   2 
                   , 
                   
                     calculate 
                     ⁢ 
                     
                       
                         
                           : 
                         
                         ⁢ 
                         
                             
                         
                         [ 
                         
                           G 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             L 
                             ⁡ 
                             
                               ( 
                               
                                 t 
                                 ; 
                                 
                                   
                                     v 
                                     1 
                                   
                                   ~ 
                                   
                                     v 
                                     m 
                                   
                                 
                                 ; 
                                 
                                   
                                     u 
                                     1 
                                   
                                   ~ 
                                   
                                     u 
                                     m 
                                   
                                 
                                 ; 
                                 S 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                       
                         1 
                         
                           m 
                           + 
                           n 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     (3) It can be seen from the above description that, the beginning frequency of the first range of responsive frequencies and the beginning frequency of the next range of responsive frequencies are both decided by the values of S, v 1 ˜v m  and u 1 ˜u m . And this is the general method to extend the period of responsive frequencies. 
     (V) Description of Technique 4 
     Single-Frequency Filter 
     By using the aforementioned method for extending period of responsive frequencies, the single-frequency filter can also be designed. This method can be described by the following five points. 
     1. Firstly, perform filtering calculation for  G r   (t,v) by equation (3)-12, and the condition of its responsive frequencies will be like the one shown in axis V 1  of  FIG. 14 . 
     2. Select a very small positive value ε, and then perform filtering calculation for  G r   (t,7v−2εv), and the condition of its responsive frequencies will be like the one shown in axis V 2  of  FIG. 14 . 
     3. Then, perform filtering calculation for G r (t,7v) by using equation (3)-11, and the condition of its responsive frequencies will be like the one shown in axis V 3  of  FIG. 14 . 
     4. Select S=3 (now S≠1,2), and then perform calculation for GL(t,7v;v,7v−2εv,3) by equation (3)-15, and then, the beginning frequency of its responsive frequencies will be 
                 (       7   4     -   ɛ     )     ⁢   v   ⁢           ⁢   Hz     ,         
the width will be εHz; and the next/second responsive frequencies will start from
 
               (       35   4     -     3   ⁢   ɛ       )     ⁢   v   ⁢           ⁢   Hz         
with a width of 3 εHz. The condition of its responsive frequencies will be like the one shown in axis V 4  of  FIG. 14 .
 
     5. Calculations: 
     
       
         
           
             
               [ 
               
                 G 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   L 
                   ⁡ 
                   
                     ( 
                     
                       t 
                       , 
                       
                         
                           7 
                           ⁢ 
                           v 
                         
                         ; 
                         v 
                       
                       , 
                       
                         
                           7 
                           ⁢ 
                           v 
                         
                         - 
                         
                           2 
                           ⁢ 
                           ɛ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           v 
                         
                       
                       , 
                       3 
                     
                     ) 
                   
                 
               
               ] 
             
             
               1 
               3 
             
           
         
       
     
     And the result of calculation is that, at beginning, the signals contained in f(t) having a frequency in the range of 
               (       7   4     -   ɛ     )     ⁢     vHz   ~     7   4       ⁢   vHz         
will be extracted, and then, the next range of the extracted signals will directly jump to the frequencies between
 
     
       
         
           
             
               ( 
               
                 
                   35 
                   4 
                 
                 - 
                 
                   3 
                   ⁢ 
                   ɛ 
                 
               
               ) 
             
             ⁢ 
             
               
                 vHz 
                 ~ 
                 35 
               
               4 
             
             ⁢ 
             
               vHz 
               . 
             
           
         
       
     
     No matter what is the value of v, we only need to select ε in order to make εv=1, and then the beginning frequency of the first range of extracted signals will be 1 Hz. And this is so called single-frequency filter. 
     (VI) Description of Technique 5 
     Filter Using Analog Calculations 
     The filters mentioned above, no matter band-pass filters or single frequency filters, all need to sample the signals to be filtered before performing the calculations, and thus the range of responsive frequencies are also directly corresponding to the sampling rate of signals. 
     However, if the frequency of signals to be filtered is extremely high and even higher than the sampling rate of conventional A/D converters, then the calculating methods described above will not be suitable for achieving the objectives of the present invention. For example, the carrier frequencies used in microwave communications are counted by GHz units, however, currently there is no such technology to produce A/D converters with sampling rates in GHz. 
     Here is a method to resolve the problem of extremely high sampling rates of A/D converters. The descriptions of such novel method is divided in three sections as follows: 
     (1) Euler-Maclaurin Formula: 
     Assuming f(t) is continuous in tε[a,b], and all its sub-order values exist, then: 
     
       
         
           
             
               
                 
                   
                     
                       
                         b 
                         - 
                         a 
                       
                       m 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           0 
                         
                         
                           m 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           
                             a 
                             + 
                             
                               i 
                               ⁢ 
                               
                                 
                                   b 
                                   - 
                                   a 
                                 
                                 m 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         ∫ 
                         a 
                         b 
                       
                       ⁢ 
                       
                         
                           f 
                           ⁡ 
                           
                             ( 
                             x 
                             ) 
                           
                         
                         ⁢ 
                         
                           ⅆ 
                           x 
                         
                       
                     
                     - 
                     
                       
                         
                           b 
                           - 
                           a 
                         
                         
                           2 
                           ⁢ 
                           m 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             f 
                             ⁡ 
                             
                               ( 
                               b 
                               ) 
                             
                           
                           - 
                           
                             f 
                             ⁡ 
                             
                               ( 
                               a 
                               ) 
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       R 
                       ⁡ 
                       
                         ( 
                         
                           
                             m 
                             
                               b 
                               - 
                               a 
                             
                           
                           , 
                           L 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     6 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   1 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   1 
                 
               
             
             
               
                 
                   or 
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         
                           b 
                           - 
                           a 
                         
                         m 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             0 
                           
                           
                             m 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           f 
                           ⁡ 
                           
                             ( 
                             
                               a 
                               + 
                               
                                 i 
                                 ⁢ 
                                 
                                   
                                     b 
                                     - 
                                     a 
                                   
                                   m 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           ∫ 
                           a 
                           b 
                         
                         ⁢ 
                         
                           
                             f 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                             ⅆ 
                             x 
                           
                         
                       
                       + 
                       
                         
                           
                             b 
                             - 
                             a 
                           
                           
                             2 
                             ⁢ 
                             m 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               f 
                               ⁡ 
                               
                                 ( 
                                 b 
                                 ) 
                               
                             
                             + 
                             
                               f 
                               ⁡ 
                               
                                 ( 
                                 a 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                       + 
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             
                               m 
                               
                                 b 
                                 - 
                                 a 
                               
                             
                             , 
                             L 
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     wherein 
                     ⁢ 
                     
                       : 
                     
                   
                 
               
               
                 
                   
                     ( 
                     6 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   1 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   2 
                 
               
             
             
               
                 
                   
                     R 
                     ⁡ 
                     
                       ( 
                       
                         
                           m 
                           
                             b 
                             - 
                             a 
                           
                         
                         , 
                         L 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       lim 
                       
                         L 
                         -&gt; 
                         ∞ 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         L 
                       
                       ⁢ 
                       
                         { 
                         
                           
                             
                               ( 
                               
                                 - 
                                 1 
                               
                               ) 
                             
                             
                               i 
                               + 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   b 
                                   - 
                                   a 
                                 
                                 m 
                               
                               ) 
                             
                             
                               2 
                               ⁢ 
                               i 
                             
                           
                           ⁢ 
                           
                             
                               
                                 B 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     2 
                                     ⁢ 
                                     i 
                                   
                                   ) 
                                 
                               
                               
                                 2 
                                 ⁢ 
                                 i 
                               
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     f 
                                     
                                       ( 
                                       
                                         
                                           2 
                                           ⁢ 
                                           i 
                                         
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     b 
                                     ) 
                                   
                                 
                                 - 
                                 
                                   
                                     f 
                                     
                                       ( 
                                       
                                         
                                           2 
                                           ⁢ 
                                           i 
                                         
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     a 
                                     ) 
                                   
                                 
                               
                               ] 
                             
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     6 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   2 
                 
               
             
           
         
       
     
     wherein B(2i) is a Bernoulli value. 
     Either equation (6)-1-1 or equation (6)-1-2 will be Euler-Maclaurin Formula. 
     In the above equations, 
               b   -   a     m         
is the distance between any two samples of signals obtained when f(t) is in tε[a,b], and thus it is a reciprocal of sampling rate v 0 , that is:
 
     
       
         
           
             
               
                 
                   
                     1 
                     
                       v 
                       0 
                     
                   
                   = 
                   
                     
                       b 
                       - 
                       a 
                     
                     m 
                   
                 
               
               
                 
                   
                     ( 
                     6 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   3 
                 
               
             
           
         
       
     
     Therefore, equations (6)-1-1, (6)-1-2 and (6)-2 can be rewritten into: 
     
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         0 
                       
                       
                         m 
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       f 
                       ⁡ 
                       
                         ( 
                         
                           a 
                           + 
                           
                             i 
                             
                               v 
                               0 
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         v 
                         0 
                       
                       ⁢ 
                       
                         
                           ∫ 
                           a 
                           b 
                         
                         ⁢ 
                         
                           
                             f 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                             ⅆ 
                             x 
                           
                         
                       
                     
                     - 
                     
                       
                         1 
                         2 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             f 
                             ⁡ 
                             
                               ( 
                               b 
                               ) 
                             
                           
                           - 
                           
                             f 
                             ⁡ 
                             
                               ( 
                               a 
                               ) 
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         v 
                         0 
                       
                       ⁢ 
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             
                               v 
                               0 
                             
                             , 
                             L 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     6 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   4 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   1 
                 
               
             
             
               
                 
                   or 
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           0 
                         
                         m 
                       
                       ⁢ 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           
                             a 
                             + 
                             
                               i 
                               
                                 v 
                                 0 
                               
                             
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         
                           v 
                           0 
                         
                         ⁢ 
                         
                           
                             ∫ 
                             b 
                             a 
                           
                           ⁢ 
                           
                             
                               f 
                               ⁡ 
                               
                                 ( 
                                 x 
                                 ) 
                               
                             
                             ⁢ 
                             
                               ⅆ 
                               x 
                             
                           
                         
                       
                       + 
                       
                         
                           1 
                           2 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               f 
                               ⁡ 
                               
                                 ( 
                                 b 
                                 ) 
                               
                             
                             + 
                             
                               f 
                               ⁡ 
                               
                                 ( 
                                 a 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                       + 
                       
                         
                           v 
                           0 
                         
                         ⁢ 
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               v 
                               , 
                               L 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     6 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   4 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   2 
                 
               
             
             
               
                 
                   
                     R 
                     ⁡ 
                     
                       ( 
                       
                         
                           v 
                           0 
                         
                         , 
                         k 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       lim 
                       
                         L 
                         -&gt; 
                         ∞ 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         L 
                       
                       ⁢ 
                       
                         { 
                         
                           
                             
                               ( 
                               
                                 - 
                                 1 
                               
                               ) 
                             
                             
                               i 
                               + 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 1 
                                 
                                   v 
                                   0 
                                 
                               
                               ) 
                             
                             
                               2 
                               ⁢ 
                               i 
                             
                           
                           ⁢ 
                           
                             
                               
                                 B 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     2 
                                     ⁢ 
                                     i 
                                   
                                   ) 
                                 
                               
                               
                                 
                                   ( 
                                   
                                     2 
                                     ⁢ 
                                     i 
                                   
                                   ) 
                                 
                                 ! 
                               
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     f 
                                     
                                       ( 
                                       
                                         
                                           2 
                                           ⁢ 
                                           i 
                                         
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     b 
                                     ) 
                                   
                                 
                                 - 
                                 
                                   
                                     f 
                                     
                                       ( 
                                       
                                         
                                           2 
                                           ⁢ 
                                           i 
                                         
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     a 
                                     ) 
                                   
                                 
                               
                               ] 
                             
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     6 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   5 
                 
               
             
           
         
       
     
     (2) Coefficient Functions of Frequency Response Formulas: 
     The functions represented by equations (1)-2 and (2)-2 are frequency response formulas (both mated). Wherein equation (1)-2 is an original function before transformation, and equation (2)-2 is a function transformed by decreasing the number of selected grades and then self-multiplying it for two times (one for 3 times and one for 7 times). The values r, a u (u=1, 2, . . . ) and A n  (n=0, 1, 2 . . . ) of equations (1)-2 and (2)-2 are all coefficients. Because these coefficients are all related to their locations (e.g., u and n), and are all members of the frequency response formulas, and thus are called as coefficient functions of frequency response formulas (also referred as CFFRF hereinafter). 
     CFFRF can be described in mathematic formulas. Firstly, the mathematic formula of CFFRF of equation (1)-2 is described. 
     Let&#39;s change the variable u shown in equation (1)-2 into variable n, and rewrite the equation as below: 
                       h   ⁢           ⁢   s   ⁢           ⁢     e   ⁡     (     k   ,   r   ,   θ     )         =     r   +       ∑     n   =   1     k     ⁢       a   n     ⁢     Cos   ⁡     (     n   ⁢           ⁢   θ     )               ⁢     
     ⁢         Let   ⁢           ⁢     α   ⁡     (   0   )         =   r     ,         α   ⁡     (   n   )       -       1   2     ⁢     a   n     ⁢           ⁢   and   ⁢           ⁢     α   ⁡     (     -   n     )           =       1   2     ⁢     a   n         ,               (   1   )     ⁢     -     ⁢   2               
and then equation (1)-2 can be rewritten into:
 
     
       
         
           
             
               
                 
                   
                     h 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     s 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       e 
                       ⁡ 
                       
                         ( 
                         
                           k 
                           , 
                           r 
                           , 
                           θ 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         
                           - 
                           k 
                         
                       
                       k 
                     
                     ⁢ 
                     
                       
                         a 
                         ⁡ 
                         
                           ( 
                           n 
                           ) 
                         
                       
                       ⁢ 
                       
                         Cos 
                         ⁡ 
                         
                           ( 
                           
                             n 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     6 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   6 
                 
               
             
           
         
       
     
     wherein, ∝(0) is the average value of one period shown in  FIG. 1 , and the value ∝(n) will be: 
     
       
         
           
             
               
                 
                   
                     α 
                     ⁡ 
                     
                       ( 
                       n 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         a 
                         n 
                       
                     
                     = 
                     
                       
                         1 
                         
                           n 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         Sin 
                         ⁡ 
                         
                           ( 
                           
                             rn 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             π 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     6 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   6 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   1 
                 
               
             
           
         
       
     
     In equation (6)-6, when k→∞, the wave form will become the one shown in  FIG. 1 . Wherein, if the value k is big enough, then the wave form will be close to the one shown in  FIG. 1 . 
     The value ∝(0) can be rewritten into: 
     
       
         
           
             
               
                 
                   
                     α 
                     ⁡ 
                     
                       ( 
                       0 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         2 
                         ⁢ 
                         π 
                       
                     
                     ⁢ 
                     
                       
                         ∫ 
                         
                           
                             - 
                             r 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                         
                         
                           r 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         x 
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     6 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   6 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   2 
                 
               
             
           
         
       
     
     wherein, equations (6)-6-1 and (6)-6-2 are the CFFRF of equation (1)-2. 
     Secondly, the mathematic formula of CFFRF of equation (2)-2 is discussed. Let 
                 A   ⁡     (   0   )       =     A   0       ,       A   ⁡     (   n   )       =         1   2     ⁢     A   n     ⁢           ⁢   and   ⁢           ⁢     A   ⁡     (     -   n     )         =       1   2     ⁢     A   n           ,   and         
then equation (2)-2 can be rewritten into:
 
     
       
         
           
             
               
                 
                   
                     h 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     d 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       e 
                       ⁡ 
                       
                         ( 
                         θ 
                         ) 
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         
                           - 
                           105 
                         
                       
                       105 
                     
                     ⁢ 
                     
                       [ 
                       
                         
                           A 
                           ⁡ 
                           
                             ( 
                             n 
                             ) 
                           
                         
                         ⁢ 
                         
                           Cos 
                           ⁡ 
                           
                             ( 
                             
                               n 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   
                     ( 
                     6 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   7 
                 
               
             
             
               
                 
                   because 
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         1 
                         
                           2 
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         
                           ∫ 
                           
                             - 
                             π 
                           
                           π 
                         
                         ⁢ 
                         
                           
                             Cos 
                             ⁡ 
                             
                               ( 
                               nx 
                               ) 
                             
                           
                           ⁢ 
                           
                             Cos 
                             ⁡ 
                             
                               ( 
                               mx 
                               ) 
                             
                           
                           ⁢ 
                           
                             ⅆ 
                             x 
                           
                         
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   1 
                                   ; 
                                 
                               
                               
                                 
                                   n 
                                   = 
                                   m 
                                 
                               
                             
                             
                               
                                 
                                   0 
                                   ; 
                                 
                               
                               
                                 
                                   n 
                                   ≠ 
                                   m 
                                 
                               
                             
                           
                           ⁢ 
                           
                             
 
                           
                           ⁢ 
                           therefore 
                           ⁢ 
                           
                             : 
                           
                           ⁢ 
                           
                             
 
                           
                           ⁢ 
                           
                             A 
                             ⁡ 
                             
                               ( 
                               n 
                               ) 
                             
                           
                         
                         = 
                         
                           
                             1 
                             
                               2 
                               ⁢ 
                               π 
                             
                           
                           ⁢ 
                           
                             
                               ∫ 
                               
                                 - 
                                 π 
                               
                               π 
                             
                             ⁢ 
                             
                               h 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 Cos 
                                 ⁡ 
                                 
                                   ( 
                                   nx 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 x 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     6 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   8 
                 
               
             
           
         
       
     
     wherein, equation (6)-8 is the mathematic formula of CFFRF of equation (6)-7. 
     (3) Analog Calculation Equivalent to Digital Calculation: 
     Assume there is a sine signal S(ω,t)=C Cos(ωt)+D Sin(ω,t) Let&#39;s take t=t i  to be the original point, and then obtain k samples to S(ω,t) before and after the original point by using the sampling rate v, that is, totally 2k+1 samples will be obtained including the sample S(ω,t i ) when t=t i . Then, performing the following calculation: 
     
       
         
           
             
               
                 
                   
                     
                       SD 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       
                         ω 
                         , 
                         
                           t 
                           i 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       r 
                       · 
                       
                         S 
                         ⁡ 
                         
                           ( 
                           
                             ω 
                             , 
                             
                               t 
                               i 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             1 
                           
                           k 
                         
                         ⁢ 
                         
                           
                             a 
                             n 
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 S 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     ω 
                                     , 
                                     
                                       
                                         t 
                                         i 
                                       
                                       - 
                                       
                                         n 
                                         v 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 S 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     ω 
                                     , 
                                     
                                       
                                         t 
                                         i 
                                       
                                       + 
                                       
                                         n 
                                         v 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     6 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   9 
                 
               
             
           
         
       
     
     According to the description illustrated in the section of “(III) Description of Technique 2”, the equation (6)-9 can be transformed into: 
     
       
         
           
             
               
                 
                   
                     
                       SD 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       
                         ω 
                         , 
                         
                           t 
                           i 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       S 
                       ⁡ 
                       
                         ( 
                         
                           ω 
                           , 
                           
                             t 
                             i 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     h 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     s 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       e 
                       ⁡ 
                       
                         ( 
                         
                           k 
                           , 
                           r 
                           , 
                           
                             n 
                             v 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     6 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   10 
                 
               
             
           
         
       
     
     It can be seen from the definition of hse(k,r,θ) according to equation (1)-2 that, if the number of grades taken for k is large enough to let the value of hse(k,r,θ) similar to the value of he(r,θ) (wherein the definition of he(r,θ) is based on equation (1)-1), and then, when 
                   2   ⁢   ρπ     -     r   ⁢           ⁢   π       ≤     ω   v     ≤       2   ⁢   ρπ     +     r   ⁢           ⁢   π   ⁢           ⁢   or   ⁢           ⁢     v   ⁡     (       2   ⁢   ρπ     -     r   ⁢           ⁢   π       )           ≤   ω   ≤     v   ⁡     (       2   ⁢   ρπ     +     r   ⁢           ⁢   π       )         ,         
the value of S(ω,t i ) can be extracted, otherwise it will be cut off.
 
     It also can be seen from aforementioned ∝(0)=r and 
               α   ⁡     (   n   )       =       α   ⁡     (     -   n     )       =       1   2     ⁢     a   n               
(based on equation (6)-6-1) that, equation (6)-9 can also be transformed into:
 
     
       
         
           
             
               
                 SD 
                 1 
               
               ⁡ 
               
                 ( 
                 
                   ω 
                   , 
                   
                     t 
                     i 
                   
                 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   n 
                   = 
                   
                     - 
                     k 
                   
                 
                 k 
               
               ⁢ 
               
                 [ 
                 
                   
                     α 
                     ⁡ 
                     
                       ( 
                       n 
                       ) 
                     
                   
                   ⁢ 
                   
                     S 
                     ⁡ 
                     
                       ( 
                       
                         ω 
                         , 
                         
                           
                             t 
                             i 
                           
                           + 
                           
                             n 
                             v 
                           
                         
                       
                       ) 
                     
                   
                 
                 ] 
               
             
           
         
       
     
     Let&#39;s rewrite the above equation by changing the starting point of symbol “Σ” to “0”: 
     
       
         
           
             
               
                 
                   
                     
                       SD 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       
                         ω 
                         , 
                         
                           t 
                           i 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         0 
                       
                       
                         2 
                         ⁢ 
                         k 
                       
                     
                     ⁢ 
                     
                       [ 
                       
                         
                           α 
                           ⁡ 
                           
                             ( 
                             
                               n 
                               - 
                               k 
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           S 
                           ⁡ 
                           
                             ( 
                             
                               ω 
                               , 
                               
                                 
                                   t 
                                   i 
                                 
                                 + 
                                 
                                   
                                     n 
                                     - 
                                     k 
                                   
                                   v 
                                 
                               
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   
                     ( 
                     6 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   11 
                 
               
             
           
         
       
     
     And let: 
                   Y   1     ⁡     (   n   )       =       α   ⁡     (     n   -   k     )       ⁢     S   ⁡     (     ω   ,       t   i     +       n   -   k     v         )           ,         
therefore, it can be obtained by equation (6)-4-2 that:
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               SD 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 ω 
                                 , 
                                 
                                   t 
                                   i 
                                 
                               
                               ) 
                             
                           
                           = 
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                               
                                 2 
                                 ⁢ 
                                 k 
                               
                             
                             ⁢ 
                             
                               
                                 Y 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               
                                 ∫ 
                                 0 
                                 
                                   2 
                                   ⁢ 
                                   k 
                                 
                               
                               ⁢ 
                               
                                 
                                   
                                     Y 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   ⅆ 
                                   x 
                                 
                               
                             
                             + 
                             
                               
                                 1 
                                 2 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     Y 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         2 
                                         ⁢ 
                                         k 
                                       
                                       ) 
                                     
                                   
                                   + 
                                   
                                     Y 
                                     ⁡ 
                                     
                                       ( 
                                       0 
                                       ) 
                                     
                                   
                                 
                                 ] 
                               
                             
                             + 
                             
                               
                                 R 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   1 
                                   , 
                                   L 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     Because 
                     ⁢ 
                     
                       : 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             
                               ∫ 
                               0 
                               
                                 2 
                                 ⁢ 
                                 k 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   Y 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 x 
                               
                             
                           
                           = 
                           
                             
                               ∫ 
                               0 
                               
                                 2 
                                 ⁢ 
                                 k 
                               
                             
                             ⁢ 
                             
                               
                                 α 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     x 
                                     - 
                                     k 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 S 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     ω 
                                     , 
                                     
                                       
                                         t 
                                         i 
                                       
                                       + 
                                       
                                         
                                           x 
                                           - 
                                           k 
                                         
                                         v 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 x 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               ∫ 
                               
                                 - 
                                 k 
                               
                               k 
                             
                             ⁢ 
                             
                               
                                 α 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 S 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     ω 
                                     , 
                                     
                                       
                                         t 
                                         i 
                                       
                                       + 
                                       
                                         x 
                                         v 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 x 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               ∫ 
                               
                                 - 
                                 k 
                               
                               k 
                             
                             ⁢ 
                             
                               
                                 α 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     - 
                                     x 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 S 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     ω 
                                     , 
                                     
                                       
                                         t 
                                         i 
                                       
                                       - 
                                       
                                         x 
                                         v 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 x 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     6 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   12 
                 
               
             
           
         
       
     
     It is understood by equation (6)-6-1 that ∝(x)=∝(−x), and thereby: 
     
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       0 
                       
                         2 
                         ⁢ 
                         k 
                       
                     
                     ⁢ 
                     
                       
                         
                           Y 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         x 
                       
                     
                   
                   = 
                   
                     
                       ∫ 
                       
                         - 
                         k 
                       
                       k 
                     
                     ⁢ 
                     
                       
                         α 
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                       ⁢ 
                       
                         S 
                         ⁡ 
                         
                           ( 
                           
                             ω 
                             , 
                             
                               
                                 t 
                                 i 
                               
                               - 
                               
                                 x 
                                 v 
                               
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         x 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     v 
                     ⁢ 
                     
                       
                         ∫ 
                         
                           - 
                           
                             k 
                             v 
                           
                         
                         
                           k 
                           v 
                         
                       
                       ⁢ 
                       
                         
                           α 
                           ⁡ 
                           
                             ( 
                             vx 
                             ) 
                           
                         
                         ⁢ 
                         
                           S 
                           ⁡ 
                           
                             ( 
                             
                               ω 
                               , 
                               
                                 
                                   t 
                                   i 
                                 
                                 - 
                                 x 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           ⅆ 
                           x 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     If the value t i  in the above equation is replaced by value t, and put the result back to equation (6)-12, then the following equation is obtained: 
     
       
         
           
             
               
                 
                   
                     
                       
                         SD 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           ω 
                           , 
                           t 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         v 
                         ⁢ 
                         
                           
                             ∫ 
                             
                               - 
                               
                                 k 
                                 v 
                               
                             
                             
                               k 
                               v 
                             
                           
                           ⁢ 
                           
                             
                               α 
                               ⁡ 
                               
                                 ( 
                                 vx 
                                 ) 
                               
                             
                             ⁢ 
                             
                               S 
                               ⁡ 
                               
                                 ( 
                                 
                                   ω 
                                   , 
                                   
                                     t 
                                     - 
                                     x 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               ⅆ 
                               x 
                             
                           
                         
                       
                       + 
                       
                         
                           1 
                           2 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 Y 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   k 
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 Y 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                       + 
                       
                         
                           R 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           
                             1 
                             , 
                             L 
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     wherein 
                     ⁢ 
                     
                       : 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         Y 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           k 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         α 
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                       ⁢ 
                       
                         S 
                         ⁡ 
                         
                           ( 
                           
                             ω 
                             , 
                             
                               t 
                               + 
                               
                                 k 
                                 v 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         Y 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         0 
                         ) 
                       
                     
                     = 
                     
                       
                         α 
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                       ⁢ 
                       
                         S 
                         ⁡ 
                         
                           ( 
                           
                             ω 
                             , 
                             
                               t 
                               - 
                               
                                 k 
                                 v 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         R 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           1 
                           , 
                           L 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         lim 
                         
                           L 
                           -&gt; 
                           ∞ 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           L 
                         
                         ⁢ 
                         
                           { 
                           
                             
                               
                                 ( 
                                 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               
                                 i 
                                 + 
                                 ! 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   B 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       2 
                                       ⁢ 
                                       i 
                                     
                                     ) 
                                   
                                 
                                 
                                   
                                     ( 
                                     
                                       2 
                                       ⁢ 
                                       i 
                                     
                                     ) 
                                   
                                   ! 
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     
                                       Y 
                                       1 
                                       
                                         ( 
                                         
                                           
                                             2 
                                             ⁢ 
                                             i 
                                           
                                           - 
                                           1 
                                         
                                         ) 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         2 
                                         ⁢ 
                                         k 
                                       
                                       ) 
                                     
                                   
                                   - 
                                   
                                     
                                       Y 
                                       1 
                                       
                                         ( 
                                         
                                           
                                             2 
                                             ⁢ 
                                             i 
                                           
                                           - 
                                           1 
                                         
                                         ) 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       0 
                                       ) 
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                           } 
                         
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     6 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   13 
                 
               
             
           
         
       
     
     The integration at right side of equation (6)-13 is the Convolution integration of values ∝(vt) and S(ω,t) within the range of 
     
       
         
           
             
               [ 
               
                 
                   - 
                   
                     k 
                     v 
                   
                 
                 , 
                 
                   k 
                   v 
                 
               
               ] 
             
             . 
           
         
       
     
     The calculation performed for the right side of equation (6)-13 is an analog calculation, which is equivalent to the digital calculation performed for the right side of equation (6)-9. 
     When the angle-frequency of S(ω,t) is extremely high and the sampling rate of A/D converter cannot match such speed, then the analog calculation of the right side of equation (6)-13 can be used for calculation. 
     At beginning, if the following equation is used for calculation instead of equation (6)-9: 
     
       
         
           
             
               
                 
                   
                     
                       SD 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         ω 
                         , 
                         
                           t 
                           i 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         A 
                         0 
                       
                       ⁢ 
                       
                         S 
                         ⁡ 
                         
                           ( 
                           
                             ω 
                             , 
                             
                               t 
                               i 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             1 
                           
                           105 
                         
                         ⁢ 
                         
                           
                             A 
                             n 
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 S 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     ω 
                                     , 
                                     
                                       
                                         t 
                                         i 
                                       
                                       - 
                                       
                                         n 
                                         v 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 S 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     ω 
                                     , 
                                     
                                       
                                         t 
                                         i 
                                       
                                       + 
                                       
                                         n 
                                         v 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     6 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   14 
                 
               
             
           
         
       
     
     Thereby, it can be determined that: 
     
       
         
           
             
               
                 
                   
                     
                       SD 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         ω 
                         , 
                         
                           t 
                           i 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       v 
                       ⁢ 
                       
                         
                           ∫ 
                           
                             - 
                             
                               105 
                               v 
                             
                           
                           
                             105 
                             v 
                           
                         
                         ⁢ 
                         
                           
                             A 
                             ⁡ 
                             
                               ( 
                               vx 
                               ) 
                             
                           
                           ⁢ 
                           
                             S 
                             ⁡ 
                             
                               ( 
                               
                                 ω 
                                 , 
                                 
                                   t 
                                   - 
                                   x 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             ⅆ 
                             x 
                           
                         
                       
                     
                     + 
                     
                       
                         1 
                         2 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               Y 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               210 
                               ) 
                             
                           
                           - 
                           
                             
                               Y 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                           + 
                           
                             
                               R 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 1 
                                 , 
                                 L 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   
                     ( 
                     6 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   15 
                 
               
             
           
         
       
     
     Wherein: 
     A(vx) is the definition defined in equation (6)-8; 
     
       
         
           
             
               
                 Y 
                 2 
               
               ⁡ 
               
                 ( 
                 210 
                 ) 
               
             
             = 
             
               
                 A 
                 ⁡ 
                 
                   ( 
                   105 
                   ) 
                 
               
               ⁢ 
               
                 S 
                 ⁡ 
                 
                   ( 
                   
                     ω 
                     , 
                     
                       t 
                       + 
                       
                         105 
                         v 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 Y 
                 2 
               
               ⁡ 
               
                 ( 
                 0 
                 ) 
               
             
             = 
             
               
                 A 
                 ⁡ 
                 
                   ( 
                   105 
                   ) 
                 
               
               ⁢ 
               
                 S 
                 ⁡ 
                 
                   ( 
                   
                     ω 
                     , 
                     
                       t 
                       - 
                       
                         105 
                         v 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 R 
                 2 
               
               ⁡ 
               
                 ( 
                 
                   1 
                   , 
                   L 
                 
                 ) 
               
             
             = 
             
               
                 lim 
                 
                   j 
                   = 
                   1 
                 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   L 
                 
                 ⁢ 
                 
                   { 
                   
                     
                       
                         ( 
                         
                           - 
                           1 
                         
                         ) 
                       
                       
                         i 
                         + 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         
                           B 
                           ⁡ 
                           
                             ( 
                             
                               2 
                               ⁢ 
                               i 
                             
                             ) 
                           
                         
                         
                           
                             ( 
                             
                               2 
                               ⁢ 
                               i 
                             
                             ) 
                           
                           ! 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               Y 
                               2 
                               
                                 ( 
                                 
                                   
                                     2 
                                     ⁢ 
                                     i 
                                   
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                             ⁡ 
                             
                               ( 
                               210 
                               ) 
                             
                           
                           - 
                           
                             
                               Y 
                               2 
                               
                                 ( 
                                 
                                   
                                     2 
                                     ⁢ 
                                     i 
                                   
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                   } 
                 
               
             
           
         
       
       
         
           
             
               
                 Y 
                 2 
               
               ⁡ 
               
                 ( 
                 x 
                 ) 
               
             
             = 
             
               
                 A 
                 ⁡ 
                 
                   ( 
                   
                     x 
                     - 
                     105 
                   
                   ) 
                 
               
               ⁢ 
               
                 S 
                 ⁡ 
                 
                   ( 
                   
                     ω 
                     , 
                     
                       t 
                       + 
                       
                         
                           x 
                           - 
                           105 
                         
                         v 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             Finally 
             , 
             
               if 
               ⁢ 
               
                   
               
               ⁢ 
               let 
               ⁢ 
               
                 : 
               
             
           
         
       
       
         
           
             
               F 
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   j 
                   = 
                   1 
                 
                 m 
               
               ⁢ 
               
                 S 
                 ⁡ 
                 
                   ( 
                   
                     
                       ω 
                       j 
                     
                     , 
                     t 
                   
                   ) 
                 
               
             
           
         
       
     
     then we can use the value F(t−x) shown above to replace the value S(ω,t−x) shown in either equation (6)-13 or equation (6)-15 so as to achieve the objective of ideal-like analog band-pass filter according to the present invention. 
     The mathematic formula of the digital band-pass filter as described above is: 
     
       
         
           
             
               
                 
                   a 
                   0 
                 
                 ⁢ 
                 
                   F 
                   ⁡ 
                   
                     ( 
                     t 
                     ) 
                   
                 
               
               + 
               
                 
                   1 
                   2 
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       1 
                     
                     k 
                   
                   ⁢ 
                   
                     
                       a 
                       n 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           F 
                           ⁡ 
                           
                             ( 
                             
                               t 
                               - 
                               
                                 n 
                                 v 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           F 
                           ⁡ 
                           
                             ( 
                             
                               t 
                               + 
                               
                                 n 
                                 v 
                               
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             = 
             
               
                 v 
                 ⁢ 
                 
                   
                     ∫ 
                     
                       - 
                       
                         k 
                         v 
                       
                     
                     
                       k 
                       v 
                     
                   
                   ⁢ 
                   
                     
                       α 
                       ⁡ 
                       
                         ( 
                         vx 
                         ) 
                       
                     
                     ⁢ 
                     
                       F 
                       ⁡ 
                       
                         ( 
                         
                           t 
                           - 
                           x 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       x 
                     
                   
                 
               
               + 
               
                 
                   1 
                   2 
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       Y 
                       ⁡ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           k 
                         
                         ) 
                       
                     
                     + 
                     
                       Y 
                       ⁡ 
                       
                         ( 
                         0 
                         ) 
                       
                     
                   
                   ] 
                 
               
               + 
               
                 R 
                 ⁡ 
                 
                   ( 
                   
                     v 
                     , 
                     L 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
                 
             
             ⁢ 
             
               wherein 
               , 
               
                 
                   Y 
                   ⁡ 
                   
                     ( 
                     x 
                     ) 
                   
                 
                 = 
                 
                   
                     α 
                     ⁡ 
                     
                       ( 
                       vx 
                       ) 
                     
                   
                   ⁢ 
                   
                     F 
                     ⁡ 
                     
                       ( 
                       
                         t 
                         - 
                         x 
                       
                       ) 
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
             ⁢ 
             
               
                 R 
                 ⁡ 
                 
                   ( 
                   
                     v 
                     , 
                     L 
                   
                   ) 
                 
               
               = 
               
                 
                   lim 
                   
                     L 
                     -&gt; 
                     ∞ 
                   
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       i 
                       = 
                       1 
                     
                     L 
                   
                   ⁢ 
                   
                     { 
                     
                       
                         
                           ( 
                           
                             - 
                             1 
                           
                           ) 
                         
                         
                           i 
                           + 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             1 
                             v 
                           
                           ) 
                         
                         
                           2 
                           ⁢ 
                           i 
                         
                       
                       ⁢ 
                       
                         
                           
                             B 
                             ⁡ 
                             
                               ( 
                               
                                 2 
                                 ⁢ 
                                 i 
                               
                               ) 
                             
                           
                           
                             2 
                             ⁢ 
                             i 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 Y 
                                 
                                   
                                     2 
                                     ⁢ 
                                     i 
                                   
                                   - 
                                   1 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   k 
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 Y 
                                 
                                   ( 
                                   
                                     
                                       2 
                                       ⁢ 
                                       i 
                                     
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                     
                     } 
                   
                 
               
             
           
         
       
     
     Wherein, ∝(x) is a coefficient function obtained by Fourier transformation of a periodical function of mated square-wave signal, wherein: 
     
       
         
           
             
               α 
               ⁡ 
               
                 ( 
                 0 
                 ) 
               
             
             = 
             
               a 
               0 
             
           
         
       
       
         
           
             
               α 
               ⁡ 
               
                 ( 
                 n 
                 ) 
               
             
             = 
             
               
                 α 
                 ⁡ 
                 
                   ( 
                   
                     - 
                     n 
                   
                   ) 
                 
               
               = 
               
                 
                   1 
                   2 
                 
                 ⁢ 
                 
                   a 
                   n 
                 
               
             
           
         
       
     
     In addition, because F(t) is a sine function, therefore the aforementioned values R 1 (1,L) and R 2 (1,L) must be convergence values, and its accuracy can be determined by the value of L. Generally speaking, when L=10, then the error will be less than ±0.1%. 
     (VII) Description of Preferred Embodiments 
     1. Embodiment of Band-Pass Filter 
     Consider axes V 1 , V 2  and V 3  shown in  FIG. 13  to be the examples of this embodiment, whose periods of responsive frequencies are 8000 Hz, 16000 Hz and 44000 Hz respectively. 
       FIG. 15  and  FIG. 16  are block diagrams of the circuitries of this embodiment, which are illustrated in the following points: 
     (1) In  FIG. 15 , the dotted line represents a filter  01  of the present invention in which the period of responsive frequencies is vHz (where v has three kinds of values and will be illustrated in  FIG. 16 ). Wherein, the calculator  010  is for performing filtering calculation for f(t) and hde(θ) or  hde (θ) when the period of responsive frequencies is vHz; and the memory  011  is for storing parameters A 0 ˜A 105  for calculations of equation (2)-2 or equation (3)-1. 
     (2)  FIG. 16  illustrates the block diagrams of the whole band-pass filter  08  of the present invention, which can be illustrated by the following points: 
     (i) The end “In” is the input of signals f(t) which are going to be filtered; and the end “out” is the output of filtered signals. 
     (ii) Each of the filters  01 - 1 ,  01 - 2  and  01 - 3  is the same as the filter  01  of  FIG. 15 . However, the period of responsive frequencies of filter  01 - 1  is 8000 Hz, the period of filter  01 - 2  is 16000 Hz, and the period of filter  01 - 3  is 44000 Hz. 
     (iii) The blocks  02 - 1 ,  02 - 2  and  02 - 3  are A/D converters, where CK 1 , CK 2  and CK 3  are their inputs of signals for sampling, and their sampling rates are 8000 Hz, 16000 Hz and 44000 Hz respectively. 
     (iv) The blocks  03 - 1 ,  03 - 2  and  03 - 3  are sampling memories for temporarily storing the digital data transformed by A/D converters  02 - 1 ,  02 - 2  and  02 - 3  respectively. 
     (v) The block  04  is a multiplier for multiplying the calculated results of the filters  01 - 1 ,  01 - 2  and  01 - 3 . 
     (vi) The block  05  is a cubic root calculator for calculating the cubic root of the result of the multiplier  04 . 
     (vii) The block  06  is a D/A converter for converting the digital signals of calculated result of the calculator  05  into analog signals for output. 
     (viii) The block  07  is a clock generator for generating clock signals with frequencies of 8000 Hz, 16000 HZ and 44000 Hz for sampling purposes. 
     (ix) The area  08  defined by dotted lines is a primary circuit block in accordance with the present embodiment, that is, a filter combination. 
     The above description is an illustration of an embodiment of the band-pass filter in accordance with the present invention. 
     2. Embodiment of Single-Frequency Filter 
       FIG. 17  is a block diagram of the circuitries of this embodiment, which are illustrated in the following points: 
     (i) Firstly, select 
             ɛ   =       1   8000     ⁢           ⁢   Hz           
and v=8000 Hz for  FIG. 14 , as a result, the periods of responsive frequencies of axes V 1 , V 2  and V 3  shown in  FIG. 14  will be 8000 Hz, 55999 Hz and 56000 Hz respectively.
 
     (ii) The block  07   a  in  FIG. 17  is a clock generator which is similar with the clock generator  07  shown in  FIG. 16 . However, the clock generator  07   a  in this embodiment will generate clock signals having frequencies 8000 Hz (output from CK 1 ), 55999 Hz (output from CK 2 ) and 56000 Hz (output from CK 3 ). 
     (iii) The block  08   a  is a filter similar to the band-pass filter  08  shown in  FIG. 16 . However, the periods of responsive frequencies of the filters in this embodiment are 8000 Hz, 55999 Hz and 56000 Hz, which are different from the ones shown in  FIG. 16 . 
     (iv) It is obvious that, the output signal of the end “out” of the filter  08   a  will have a beginning frequency of 13999 Hz˜14000 Hz, and then the next range of frequency which is passable from the end “out” will jump to 69997 Hz˜70000 Hz directly (it can be referred to  FIG. 14  when v is 8000 and ε is 
     
       
         
           
             
               1 
               8000 
             
             ⁢ 
             
               
                 ) 
               
               . 
             
           
         
       
     
     The above description is an illustration of an embodiment of the single-frequency filter in accordance with the present invention. 
     Because the detailed layout of circuitries of each block of these two embodiments are known by ordinary people skilled in the art, and thus no further descriptions for these blocks are provided herewith. 
     In addition, according to the embodiments of equations (6)-13 and (6)-15, the only difference is to use analog calculations in stead of digital calculations previously described. Because the circuits used for such analog calculations can be realized by ordinary persons skilled in the art according to the embodiments previously illustrated in the present invention, no further descriptions will be provided herein. 
     The preferred embodiments of the invention have been disclosed above, which however should not be construed as a limitation on the actual application of the invention. Hence all modifications and alterations made by those familiar with the skill without departing from the spirits of the invention and appended claims shall remain within the protected scope of the invention.