Abstract:
A method and device for mixing and separating a plurality signals. In the mixing method, each of the m signal S i (t) to be mixexd within a time perid [T 0 , T 1 ] is sampled for n samples S i (t j ), j=1,2 . . . n, wherein tε[T 0 , T 1 ], T 0 , T 1  εR, t is time variable. Each sample is multiplied by a coefficient function  i a j (t) which is a linear independent set (i=1,2 . . . m, j=1,2 . . . n), thus obtaining m transformed signals for S i (t):            S   i   0          (   t   )       =       ∑     j   =   1     n                         [   i              a   j          (   t   )            S   i          (     t   j     )       ]     .                               
     By summing above m transformed signals obtains the mixed transformed signals are obtained:          SM        (   t   )       =       ∑     i   =   1     m                       [       S   i   0          (   t   )       ]     .                               
     As to the separating method, the coefficient function  i a j (t) is a linear independent set (i=1,2 . . . m, j=1,2 . . . n), therefore S i (t j ) are the unknown of m×n linear equation set and can be solved by known linear algebraic method.

Description:
FIELD OF THE INVENTION 
     This invention relates to a mixing and separating method for a plurality signals, more particularly, to achieve control circuit for signal processing. 
     BACKGROUND OF THE INVENTION 
     The electronic communication has grown more and more prosperous and the problem of available of communication channel becomes even serious. The bandwidth of a specific communication medium is limited, therefore how to most exploit the avail bandwidth is essential. It is the object of the invention to provide a method for transmitting a plurality of signals in a pair of transmission line or a single channel. 
     SUMMARY OF THE INVENTION 
     Principles of the Invention I 
     The present invention is based on the principle of unique solution condition for a set of N linearly equations, i.e., linearly independence. Based on this principle, each of the m signal S i (t) within period [T 0 , T 1 ] is sampled for n samples S i (t j ), j=1,2 . . . n, wherein tε[T 0 , T 1 ], T 0 , T 1  εR, t is time variable. Each sample is multiplied by a coefficient function  i a j (t) which is a linear independent set (i=1,2 . . . m, j=1,2 . . . n), thus obtaining m transformed signals for S i (t):            S   i   0          (   t   )       =       ∑     j   =   1     n                     [         a   j           i              (   t   )              S   i          (     t   j     )         ]                              
     summing above m transformed signals obtains the mixed transformed signals:          SM        (   t   )       =       ∑     i   =   1     m                     [       S   i   0          (   t   )       ]                              
     The mixed transformed signals have m×n variables S i (t j ) with coefficient  i a j (t). If party A transmits SM(t) during time [T 0 , T 1 ] to party B, party B will obtain message of m×n S i (t j ), (i=1,2 . . . m, j=1,2 . . . n), wherein the bandwidth depends on the max bandwidth of  i a j (t). More particularly, party A can m messages S i (t) (i=1,2 . . . m) to party B during time [T 0 , T 1 ], if the samples (unknowns) S i (t 1 ), S i (t 2 ), S i (t 3 ) . . . S i (t n ), ((i=1,2 . . . m) are sufficient to represent S i (t) (i=1,2 . . . m) during time [T 0 , T 1 ]. 
     The party B can resolve S i (t i ) upon receiving SM(t) if m×n−1 differential means are provided to obtain differential signals SM′(t), SM″(t) . . . SM m×n−1 (t):                    ∑     i   =   1     m                       ∑     j   =   1     n                     [         a   j           i              (   t   )              S   i          (     t   j     )         ]         =     SM        (   t   )              
              ∑     i   =   1     m                       ∑     j   =   1     n                     [         a   j   ′       i              (   t   )              S   i          (     t   j     )         ]         =       SM   ′          (   t   )              
        ⋮        
              ∑     i   =   1     m                       ∑     j   =   1     n                     [         a   j     (       m   ×   n     -   1     )         i              (   t   )              S   i          (     t   j     )         ]         =       SM       m   ×   n     -   1            (   t   )                 (   1   )                                
     Eq( 1) is an m×n equation set, wherein functions  i a j (t) (i=1,2 . . . m, j=1,2 . . . n) are linear independent. Therefore, S i (t i ) in Eq(1) has unique solution because the Wronskin (determinant)of functions  i a j (t) is not equal to zero. 
     Therefore, the S i (t i ) can be calculated by choosing a specific time t 0  within [T 0 , T 1 ] and obtain  i a j   (u) (t 0 ) and SM (u) (t 0 ). Moreover, each S i (t) can be calculated (i=1,2 . . . m,j=1,2 . . . n, u=0, 1,2 . . . m×n−1). 
     Principles of the Invention II 
     The above solving procedure requires m×n−1 differential means to solve S i (t), the hardware structure is bulky. However, party B can also take m×n samples after receiving SM(t):                    ∑     i   =   1     m                       ∑     j   =   1     n                     [         a   j           i              (     t   1     )              S   i          (     t   j     )         ]         =     SM        (     t   1     )              
              ∑     i   =   1     m                       ∑     j   =   1     n                     [         a   j           i              (     t   2     )              S   i          (     t   j     )         ]         =     SM        (     t   2     )              
        ⋮        
              ∑     i   =   1     m                       ∑     j   =   1     n                     [         a   j           i              (     t     m   ×   n       )              S   i          (     t   j     )         ]         =     SM        (     t     m   ×   n       )                 (   2   )                                
     wherein t 1 , t 2  . . . t m×n  are all within [T 0 , T 1 ] and t u ≠t v  if u≠v, (u, v=0,1,2 . . . m×n). S i (t) has unique solution because  i a j (t) (i=1,2 . . . m, j=1,2 . . . n) are linear independent in [T 0 , T 1 ]. 
     Principles of the Invention III 
     In above scheme, party requires to take m×n samples within [T 0 , T 1 ] even thought the differential means can be saved. The sample frequency will increase when the number of signal (m) increases. Therefore, the sampling rate of the A/D should be considered to determine the number of signal m. 
     To increase m and keep hardware compact, a compromise is to use m−1 differential means to get m differential signals (including original SM(t)), and to take n samples for each signal within [T 0 , T 1 ] thus obtaining following equation set:                    ∑     i   =   1     m                       ∑     j   =   1     n                     [         a   j           i              (     t   1     )              S   i          (     t   j     )         ]         =     SM        (     t   1     )              
              ∑     i   =   1     m                       ∑     j   =   1     n                     [         a   j           i              (     t   2     )              S   i          (     t   j     )         ]         =     SM        (     t   2     )              
        ⋮        
              ∑     i   =   1     m                       ∑     j   =   1     n                     [         a   j           i              (     t   n     )              S   i          (     t   j     )         ]         =     SM        (     t   n     )              
              ∑     i   =   1     m                       ∑     j   =   1     n                     [         a   j   ′       i              (     t   1     )              S   i          (     t   j     )         ]         =       SM   ′          (     t   1     )              
              ∑     i   =   1     m                       ∑     j   =   1     n                     [         a   j   ′       i              (     t   2     )              S   i          (     t   j     )         ]         =       SM   ′          (     t   2     )              
        ⋮        
              ∑     i   =   1     m                       ∑     j   =   1     n                     [         a   j   ′       i              (     t   n     )              S   i          (     t   j     )         ]         =       SM   ′          (     t   n     )              
        ⋮        
              ∑     i   =   1     m                       ∑     j   =   1     n                     [         a   j     (     m   -   1     )         i              (     t   n     )              S   i          (     t   j     )         ]         =       SM     (     m   -   1     )            (     t   n     )                 (   3   )                                
       i a j (t) (i=1,2 . . . m, j=1,2 . . . n) are linear independent. Therefore, S i (t i ) in Eq(3) has unique solution because the Wronskin (determinant) of functions  i a j (t) is not equal to zero. 
     Party B has a plurality of ways to create m×n linear independent equation set form SM(t) as will be described below.                    ∑     i   =   1     m                       ∑     j   =   1     n                     [         a   j           i              (     t   1     )              S   i          (     t   j     )         ]         =     SM        (     t   1     )              
              ∑     i   =   1     m                       ∑     j   =   1     n                     [         a   j           i              (     t   2     )              S   i          (     t   j     )         ]         =     SM        (     t   2     )              
        ⋮        
              ∑     i   =   1     m                       ∑     j   =   1     n                     [         a   j           i              (     t   n     )              S   i          (     t   j     )         ]         =     SM        (     t   n     )              
              ∑     i   =   1     m                       ∑     j   =   1     n                     [       1   D            a   j           i              (     t   1     )              S   i          (     t   j     )         ]         =       1   D          SM        (     t   1     )                
              ∑     i   =   1     m                       ∑     j   =   1     n                     [       1   D            a   j           i              (     t   2     )              S   i          (     t   j     )         ]         =       1   D          SM        (     t   2     )                
              ∑     i   =   1     m                       ∑     j   =   1     n                     [       1   D            a   j   ′       i              (     t   n     )              S   i          (     t   j     )         ]         =       1   D          SM        (     t   n     )                
        ⋮        
              ∑     i   =   1     m                       ∑     j   =   1     n                     [       1     D     m   -   1                a   j           i              (     t   n     )              S   i          (     t   j     )         ]         =       1     D     m   -   1              SM        (     t   n     )                   (   4   )                                
     wherein          1     D   u              a   j           i              (     t   v     )                              
      and          1     D   u            SM        (     t   v     )                              
      is uth integration of  i a j (t) and SM(t) from 0 to t v ., u=1,2 . . . m−1, v=1,2 . . . n. 
     Another alternative is:                    ∑     i   =   1     m                       ∑     j   =   1     n                     [         a   j           i              (     t   1     )              S   i          (     t   j     )         ]         =     SM        (     t   1     )              
              ∑     i   =   1     m                       ∑     j   =   1     n                     [         a   j           i              (     t   2     )              S   i          (     t   j     )         ]         =     SM        (     t   2     )              
        ⋮        
              ∑     i   =   1     m                       ∑     j   =   1     n                     [         a   j           i              (     t   n     )              S   i          (     t   j     )         ]         =     SM        (     t   n     )              
              ∑     i   =   1     m                       ∑     j   =   1     n                     [     Δ          a   j           i              (     t   1     )              S   i          (     t   j     )         ]         =     Δ                   SM        (     t   1     )                
              ∑     i   =   1     m                       ∑     j   =   1     n                     [     Δ          a   j           i              (     t   2     )              S   i          (     t   j     )         ]         =     Δ                   SM        (     t   2     )                
              ∑     i   =   1     m                       ∑     j   =   1     n                     [     Δ          a   j           i              (     t   n     )              S   i          (     t   j     )         ]         =     Δ                   SM        (     t   n     )                
        ⋮        
              ∑     i   =   1     m                       ∑     j   =   1     n                     [       Δ     m   -   1              a   j           i              (     t   n     )              S   i          (     t   j     )         ]         =       Δ     m   -   1            SM        (     t   n     )                   (   5   )                                
     wherein Δ u   i a j (t v ) and Δ u SM(t v ) is uth differential of  i a j (t) and SM(t) at t v , u=1,2 . . . m−1, v=1,2 . . . n. 
     Still another alternative is:                    ∑     i   =   1     m                       ∑     j   =   1     n                     [         a   j           i              (     t   1     )              S   i          (     t   j     )         ]         =     SM        (     t   1     )              
              ∑     i   =   1     m                       ∑     j   =   1     n                     [         a   j           i              (     t   2     )              S   i          (     t   j     )         ]         =     SM        (     t   2     )              
        ⋮        
              ∑     i   =   1     m                       ∑     j   =   1     n                     [         a   j           i              (     t   n     )              S   i          (     t   j     )         ]         =     SM        (     t   n     )              
              ∑     i   =   1     m                       ∑     j   =   1     n                     [       ∇       a   j           i              (     t   1     )                S   i          (     t   j     )         ]         =     ∇     SM        (     t   1     )                
              ∑     i   =   1     m                       ∑     j   =   1     n                     [       ∇       a   j           i              (     t   2     )                S   i          (     t   j     )         ]         =     ∇     SM        (     t   2     )                
              ∑     i   =   1     m                       ∑     j   =   1     n                     [       ∇       a   j           i              (     t   n     )                S   i          (     t   j     )         ]         =     ∇     SM        (     t   n     )                
        ⋮        
              ∑     i   =   1     m                       ∑     j   =   1     n                     [         ∇     m   -   1              a   j           i              (     t   n     )                S   i          (     t   j     )         ]         =       ∇     m   -   1            SM        (     t   n     )                   (   6   )                                
     wherein ∇ u   i a j (t v ) and ∇ u SM(t v ) is uth summation of  i a j (t) and SM(t) from 0 to t v . u=1,2 . . . m−1, v=1,2 . . . n. 
     The determinant in each matrix of Eqs(4)-(6) is not zero because  i a j (t) (i=1,2 . . . m, j=1,2 . . . n) are linear independent in [T 0 , T 1 ]. Therefore, S 1 (t) has unique solution for Eqs (4)-(6). 
     Moreover, party B can mix the operations of differential, integration, difference, summation and sampling to create m×n linear independent equation set. For example, taking differential, integration, difference, summation for number of r1, r2,r3 and r4, and taking sample number of h, such that (r1+r2+r3+r4)h=m×n, party B can create m×n linear independent equation set for solving S i (t j ). However, the other methods are not described here for clarity. 
     Principles of the Invention IV 
     A particular choice of  i a j (t) is described below, wherein  i a j (t) thus selected are orthonormal for t within period [T 0 , T 1 ]            ∫     T   0       T   1                a   l           k              (   x   )              a   j           i              (   x   )                          x         =     {           1   ;                  when                 k     =       i                 and                 l     =   j                   0   ;                  when                 k     ≠     i                 or                 l     ≠   j                                      
     wherein i,k=1,2 . . . m, 1,j=1,2 . . . n. 
     in this situation            ∫     T   0       T   1                   SM        (   x   )                   a   l           k              (   x   )                          x         =         ∫     T   0       T   1              [       ∑     i   =   1     m                       ∑     J   =   1     n                         a   j           i              (   x   )              s   i          (     t   j     )             ]            a   l           k              (   x   )                          x         =       S   k          (     t   l     )                                
     Therefore, when the transformed and mixed signal SM(t) is sent to the receiver during time period [T 0 , T 1 ] the receiver party multiplies the received signal with  k a l (t) (k=1,2 . . . m, 1=1,2 . . . n) and integrates the result between time period [T 0 , T 1 ] to obtain S k (t j ) (the Ith sample for the kth signal). This indicates that each sample value for each signal S i (t j ) can be calculated without the step of solving the linear algebraic equation set. 
     Below describes the way to orthonormalize the function group { i a j (t) (i=1,2 . . . m, j=1,2 . . . n)} within time period [T 0 , T 1 ]. 
     First, m×n functions G 1 (t), G 2  (t) . . . G m×n (t) linearly independent within time period [T 0 , T 1 ] are selected and let              h        (     r   ,   s     )       =       ∫     T   0       T   1                G   r          (   x   )              G   s          (   x   )                          x           ;              r     ,     s   =   1     ,     2                 …                 m   ×   n               A   0     =   1             A   v     =                    h        (     1   ,   1     )             h        (     1   ,   2     )             h        (     1   ,   3     )           …         h        (     1   ,   v     )                 h        (     2   ,   1     )             h        (     2   ,   2     )             h        (     2   ,   3     )           …         h        (     2   ,   v     )                 h        (     3   ,   1     )             h        (     3   ,   2     )             h        (     3   ,   3     )           …         h        (     3   ,   v     )               …       …       …       …       …             h        (     v   ,   1     )             h        (     v   ,   2     )             h        (     v   ,   3     )           …         h        (     v   ,   v     )                                v     =     1.2                 …                 m   ×   n                              
     then establishing the function          A   v     =                h        (     1   ,   1     )             h        (     1   ,   2     )             h        (     1   ,   3     )           …         h        (     1   ,   u     )                 h        (     2   ,   1     )             h        (     2   ,   2     )             h        (     2   ,   3     )           …         h        (     2   ,   u     )               …       …       …       …       …             h        (       u   -   1     ,   1     )             h        (       u   -   1     ,   2     )             h        (     u   -   1.3     )           …         h        (       u   -   1     ,   u     )                   G   1          (   t   )               G   2          (   t   )               G   3          (   t   )           …             G   u          (   t   )       )                        u   =     1.2                 …                 m   ×   n                            
     then the function                l   u          (   t   )       =       1         A     u   -   1            A   u                  P   u          (   t   )           ;                u   =   1       ,     2                 …                 m   ×   n                            
     is orthonormal within time period [T 0 , T 1 ]. 
     Moreover, assume  i a j (t)=Q(I−1)n+j(t) (i=1,2 . . . m, j=1,2 . . . n), that is 
       1 a 2 (t)=Q 1 (t) 
       1 a 2 (t)=Q 2 (t) 
     • 
     • 
     • 
       1 a n (t)=Q n (t) 
       2 a 1 (t)=Q 1+n (t) 
       2 a 2 (t)=Q 2+n (t) 
     • 
     • 
     • 
       2 a n (t)=Q 2n (t) 
       3 a 1 (t)=Q 1+2n (t) 
       3 a 2 (t)=Q 2+2n (t) 
     • 
     • 
     • 
       3 a 2 (t)=Q 3n (t) 
     • 
     • 
     • 
       m a n (t)=Q m×n (t) 
     Apparently, function group { i a j (t) (i=1,2 . . . m, j=1,2 . . . n)} are orthonormal within time period [T 0 , T 1 ]. In above procedure for receiver party to restore S k (t 1 ), the step of solving linear algebraic equation is eliminated. However, m×n integrals are required, this will make the hardware complicated. The present invention provide following approach. 
     By choosing suitable function group {G u (t)|u=1,2 . . . m×n},  k a 1 (t) can be expressed into Power series as following:                  a   l           k              (   t   )       =                    a   l           k              (   b   )       +                    a   l   ′       k              (   b   )                       t        (     t   -   b     )         +                  1     2   !                         a   l   ″       k              (   b   )                         (     t   -   b     )     2       +                                  1     3   !              a   l   ′′′       k              (   b   )              (     t   -   b     )     3                   …                ;                b   ∈              R                                    
     Because the high frequency components of  k a l (t) are limited, the first several terms are sufficiently to represent  k a l (t). Assuming that first M terms are considered here,  k a l (t) can be expressed as:              a   l           k              (   t   )       ≅       ∑     q   =   0       M   -   1                         [       1     q   !              a   l     (   q   )         k              (   b   )              (     t   -   b     )     q       ]                   b         ∈   R                          
     then                        S   k          (     t   l     )       =                  ∫     T   0       T   1                SM        (   x   )       k            a   l          (   x   )                          x                     =                  ∑     q   =   0       M   -   1                       [       1     q   !              a   l     (   q   )         k              (   b   )              ∫     T   0       T   1              SM        (   x   )              (     x   -   b     )     q                        x           ]                     (   7   )                                
     The receiver party only need to store m×n×M data            1     q   !              a   l     (   q   )         k              (   b   )                       (       k   =   1     ,     2                 …                 m     ,     l   =   1     ,     2                 …                 n     ,                q   =   0     ,       1                 …                 M     ;     b              ∈   R         )       ,                          
     then calculate M integrals          ∫     T   0       T   1              SM        (   x   )              (     x   -   b     )     q                        x                              
     upon receiving the SM(t). The m×n samples S k (t 1 ) are restored. In other word, receiver party only requires to prepare M integrators other than m×n. 
     Moreover, above approach is also applicable to the data compress technology. The m×n samples S k (t 1 ) can be approximately represented by M data:              ∫     T   0       T   1              SM        (   x   )              (     x   -   b     )     q                        x         ;                q   =   0       ,   1   ,         2                 …                 M     -   1     ;                b   ∈   R                              
     In other word, the function group { i a j (t) (i=1,2 . . . m, j=1,2 . . . n)} are orthonormalized within time period [T 0 , T 1 ] by above procedure. The M data;              ∫     T   0       T   1              SM        (   x   )              (     x   -   b     )     q                        x         ;                q   =   0       ,   1   ,         2                 …                 M     -   1     ;                b   ∈   R                              
     are calculated by using known m×n samples S k (t 1 ). Afterward, Eq(7) is employed to restore m×n samples S k (t 1 ), wherein          ∫     T   0       T   1              SM        (   x   )              (     x   -   b     )     q                        x                              
     are the data after compression and          1     q   !              a   l     (   q   )         k              (   b   )                              
     refers to the restoring parameter. It is apparent that the invention is applicable both to transmission of mass data or the compression technology of signal 
     Principles of the Invention V 
     From above description, party A sends message to party B segment by segment with time duration [T 0 , T 1 ]. However, party A need to send a synchronous signal before sending signal containing message. 
     Therefore, the duration [T 0 , T 1 ] is divided into first synchronous period [T 0 , T 1 ′] for sending synchronous signal, and second information period [T 1 ′, T 1 ] for sending information (T 0 &lt;T 1 ′&lt;T 1 ). The decrease of information period due to the incorporation of synchronous period will not influence bandwidth because the maximum bandwidth depends on  i a j (t) 
    
    
     The various objects and advantages of the present invention will be more readily understood from the following detailed description when read in conjunction with the appended drawing, in which: 
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is an analog adding circuit in the invention. 
     FIG. 2 show an inverted amplifier scheme in the invention. 
     FIG. 3 shows the generating circuit for mixed signal SM(t). 
     FIG. 4 is the schematic diagram of the control circuit  12  which outputs the mixing signal and the synchronous signal alternatively. 
     FIG. 5 is the control circuit  17  which make the analog signal be transferred into the digital signal, then be storage into the memory. 
     FIG. 6, is the whole hardware of party A. 
     FIG. 7 is a figure of a differential circuit comprising four differential means. 
     FIG. 8 is the separated circuit  24  which separates the several kind of the original type of the mixing signal. 
     FIG. 9 is the control circuit 22-i which can resolve the parallel equation, and store the resolve result into the memory. 
     FIG.  10 . is the whole hardware of party A. 
     FIG. 11 is the schematic diagram of the all embodiment according by this invention, the number  19  is the party A shown in FIG. 6, the number  26  is the party B shown in FIG.  10 . 
     FIG. 12 is the timing schematic diagram of the R 1 , R 2  and CK which shown in the block  10  of FIG.  4 . 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Embodiment 
     Hereinafter, a preferred embodiment is used to substantially explain the present invention, wherein the communication medium is assumed to be an ideal (distortionless) medium. 
     In this embodiment, the Si(t) in Eq. (3) is solved and below list some important issues. 
     1. The choice of [T 0 , T 1 ′], [T 1 ′, T 1 ], m, n and  i a j (t): 
     Provided that 
     [T 0 , T 1 ′]=[0, ε]; 
     [T 1 ′, T 1 ]=[ε, 5ε],ε={fraction (1/1000)} sec; 
     m=5 
     n=8 
       i a j (t)=G(j+40I−40,t), i=1,2 . . . 5; j=1,2, . . . ,40; 
     wherein G(1,t)=cos [2(400+151)πt]; 1=1,2 . . . 200. 
     and taking 40 samples Si(t1), . . . , Si(t40) for Si(t) (i=1, 2, 3, 4, 5) in [T 0 , T 1 ′]=[0, ε] 
     and further assuming 
     Sa(1)=S 1 (t 1 ) 
     Sa(2)=S 1 (t 2 ) 
     • 
     • 
     Sa(40)=S 1 (t 40 ) 
     Sa(41)=S 2 (t 1 ) 
     Sa(42)=S 2 (t 2 ) 
     • 
     • 
     Sa(80)=S 2 (t 40 ) 
     • 
     • 
     Sa(200)=S 5 (t 40 ). 
     Then the party A (sender) create a mixed transformed signal SM(t)                SM        (   t   )       =       ∑     u   =   1     200                     [       G        (     u   ,   t     )              S   a          (   u   )         ]               (   7   )                                
     the SM(t) in above equation is transformed from party A to party B within [T 1 ′, T 1 ]=[ε, 5ε] and the maximum transmitting frequency is 3.4 kHz which depends on the function group cos [2(400+151)πt], not on the maximum frequency of Si(t). 
     2. The parameter setting in party B 
     The function group cos [2(400+151)πt] is linear independent in tε[ε, 5ε], therefore the below determinant is not zero.        W   =                                G        (     1   ,     X   1       )             G        (     2   ,     X   1       )           ⋯         G        (     200   ,     X   1       )                 G        (     1   ,     X   2       )             G        (     2   ,     X   2       )           ⋯         G        (     200   ,     X   2       )               ⋮       ⋮       ⋯       ⋮             G        (     1   ,     X   40       )             G        (     2   ,     X   40       )           ⋯         G        (     200   ,     X   40       )                   G   ′          (     1   ,     X   1       )               G   ′          (     2   ,     X   1       )           ⋯           G   ′          (     200   ,     X   1       )                   G   ′          (     1   ,     X   2       )               G   ′          (     2   ,     X   2       )           ⋯           G   ′          (     200   ,     X   2       )               ⋮       ⋮       ⋯       ⋮               G   ′          (     1   ,     X   40       )               G   ′          (     2   ,     X   40       )           ⋯           G   ′          (     200   ,     X   40       )                   G   ″          (     1   ,     X   1       )               G   ″          (     2   ,     X   1       )           ⋯           G   ″          (     200   ,     X   1       )               ⋮       ⋮       ⋯       ⋮               G     (   4   )            (     1   ,     X   40       )               G     (   4   )            (     2   ,     X   40       )           ⋯           G     (   4   )            (     200   ,     X   40       )                                                         
     wherein X1,X2 . . . X40 are corresponding to the sampling points within tε[ε, 5ε]. 
     For convenience&#39;s sake, let 
     D(1,j)=G(jX1) 
     D(2,j)=G(j,X2) 
     • 
     • 
     D(40,j)=G(j,X40) 
     D(41,j)=G′(j,X1) 
     D(42,j)=G′(j,X2) 
     • 
     • 
     D(80,j)=G′(j,X40) 
     D(81,j)=G″(j,X1) 
     • 
     • 
     D(200,j )=G (4) (j,X40) 
     W can be written as        W   =                D        (     1   ,   1     )             D        (     1   ,   2     )           …         D        (     1   ,   200     )                 D        (     2   ,   1     )             D        (     2   ,   2     )           …         D        (     2   ,   200     )               ⋮       ⋮       …       ⋮             D        (     200   ,   1     )             D        (     200   ,   2     )           …         D        (     200   ,   200     )                                         
     let          z        (     i   ,   j     )       =                D        (     1   ,   1     )             D        (     1   ,   2     )           …         D        (     1   ,     j   -   1       )             D        (     1   ,     j   +   1       )           …         D        (     1   ,   200     )                 D        (     2   ,   1     )             D        (     2   ,   2     )           …         D        (     2   ,     j   -   1       )             D        (     2   ,     j   +   1       )           …         D        (     2   ,   200     )               ⋮       ⋮       …       ⋮       ⋮       …       ⋮             D        (       i   -   1     ,   1     )             D        (       i   -   1     ,   2     )           …         D        (       i   -   1     ,     j   -   1       )             D        (       i   -   1     ,     j   +   1       )           …         D        (       i   -   1     ,   200     )                 D        (       i   +   1     ,   1     )             D        (       i   +   1     ,   2     )           …         D        (       i   +   1     ,     j   -   1       )             D        (       i   +   1     ,     j   +   1       )           …         D        (       i   +   1     ,   200     )               ⋮       ⋮       …       ⋮       ⋮       …       ⋮             D        (     200   ,   1     )             D        (     200   ,   2     )           …         D        (     200   ,     j   -   1       )             D        (     200   ,     j   +   1       )           …         D        (     200   ,   200     )                                         
     wherein i≧2, 199≧j 
     let          Z        (     1   ,   j     )       =                D        (     2   ,   1     )             D        (     2   ,   2     )           …         D        (     2   ,     j   -   1       )             D        (     2   ,     j   +   1       )           …         D        (     2   ,   200     )                 D        (     3   ,   1     )             D        (     3   ,   2     )           …         D        (     3   ,     j   -   1       )             D        (     3   ,     j   +   1       )           …         D        (     3   ,   200     )               ⋮       ⋮       …       ⋮       ⋮       …       ⋮             D        (     200   ,   1     )             D        (     200   ,   2     )           …         D        (     200   ,     j   -   1       )             D        (     200   ,     j   +   1       )           …         D        (     200   ,   200     )                            Z        (     200   ,   j     )       =                D        (     1   ,   1     )             D        (     1   ,   2     )           …         D        (     1   ,     j   -   1       )             D        (     1   ,     j   +   1       )           …         D        (     1   ,   200     )                 D        (     2   ,   1     )             D        (     2   ,   2     )           …         D        (     2   ,     j   -   1       )             D        (     2   ,     j   +   1       )           …         D        (     2   ,   200     )               ⋮       ⋮       …       ⋮       ⋮       …       ⋮             D        (     199   ,   1     )             D        (     199   ,   2     )           …         D        (     199   ,     j   -   1       )             D        (     199   ,     j   +   1       )           …         D        (     199   ,   200     )                            Z        (     i   ,   1     )       =                D        (     1   ,   2     )             D        (     1   ,   3     )           …         D        (     1   ,   200     )                 D        (     2   ,   2     )             D        (     2   ,   3     )           …         D        (     2   ,   200     )               ⋮       ⋮       …       ⋮             D        (       i   -   1     ,   2     )             D        (       i   -   1     ,   3     )           …         D        (       i   -   1     ,   200     )                 D        (       i   +   1     ,   2     )             D        (       i   +   1     ,   3     )           …         D        (       i   +   1     ,   200     )                 D        (     200   ,   2     )             D        (     200   ,   3     )           …         D        (     200   ,   200     )                            Z        (     i   ,   200     )       =                D        (     1   ,   1     )             D        (     1   ,   2     )           …         D        (     1   ,   199     )                 D        (     2   ,   1     )             D        (     2   ,   2     )           …         D        (     2   ,   199     )               ⋮       ⋮       …       ⋮             D        (       i   -   1     ,   1     )             D        (       i   -   1     ,   2     )           …         D        (       i   -   1     ,   199     )                 D        (       i   +   1     ,   1     )             D        (       i   +   1     ,   2     )           …         D        (       i   +   1     ,   199     )                 D        (     200   ,   1     )             D        (     200   ,   2     )           …         D        (     200   ,   199     )                                         
     moreover let 
     
       
           R ( u,v )=(−1 )u+v   Z ( u,v ) /W   (8) 
       
     
     Wherein u,v=1, 2, . . . ,200 
     The R(u,v) in Eq(8) is the reverse-transform parameter, the value thereof are calculated by computer and then save in memory. 
     After party B receiving the signal as (7) from party A, party B uses differential means to obtain SM(t), SM′(t), SM″(t), SM′″(t) and SM (4) (t), and then takes sample to get 200 data including SM(X1), SM(X2), . . . SM(X40), SM′(X1), SM′(X2), . . . SM′(X40), . . . SM (4) (X1), . . . 
     SM (4) (X40) 
     For convenience, let 
     α(1)=SMG(X1) 
     α(2)=SM(X2) 
     • 
     • 
     α(40)=SM(X40) 
     α(41)=SM′(X1) 
     α(42)=SM′(X2) 
     • 
     • 
     α(80)=G′(X40) 
     α(81)=Gα(X1) 
     • 
     α(161)=G (4) (X1) 
     • 
     • 
     α(200)=G (4) (X40) 
     Sa(j) can be calculated by below equation                  S   a          (   j   )       =       ∑     i   =   1     200                     [       α        (   i   )            R        (     i   ,   j     )         ]               (   9   )                                
     wherein i=1,2, . . . 200 
     In (9), Sa(1), Sa(2)..Sa(40) are the samples in S 1 (t) taken by party A, Sa(41), Sa(42) . . . Sa(80) are the samples in S 2 (t) taken by party A, . . . Sa(161), Sa(162) . . . Sa(200) are the samples in S 5 (t) taken by party A. Eq (9) is apparently a reverse-transform formula. 
     3. The hardware of party A (sender) 
     FIG. 1 is an analog adding circuit wherein  010  is a high-gain amplifier,  020  is feedback resistor,  021 - 1 ˜ 021 - 200  are input resistors. If all input resistors have resistance same as that of the feedback resistor, then 
     
       
           e   0 =−( e   1   +e   2   + . . . +e   200 ) 
       
     
     wherein e 1 ,e 2  . . . e 200  are input voltages, and e 0  is output voltage. 
     FIG. 2 show an inverted amplifier scheme wherein output voltage e 0  equal to e in  multiplied by sample data Sa(i), (i=1,2 . . . ,200) and then inverted. 
     Below are features of FIG.  2 . 
     1.  011  is operational amplifier 
     2.  022 - 0 ˜ 022 - 7  are eight serially-connected feedback resistors and the action thereof depend on the on-off state of electronic switch  040 - 0 ˜ 040 - 7 .  022 -j is shorted and has no action when  040 -j is on, and has action when  040 -j is off. The on-off state of  040 -j is controlled by b j .  040 -j is on when b j  is low, (j=0, 1,2 . . . 7). 
     3. Setting Sa(i) (i=1,2 . . . 200) as binary data in byte base, and has value of b 7 X2 7 +b 6 X2 6 + . . . b 1 X 2 +b 0 X2 0 . 
     4. Let resistance of  022 - 0 ˜ 022 - 7  are r, 2r, 2 2 r, . . . 2 7 r, respectively, the resistance of input resistor  023  is r. 
     5. The resistance of feedback resistor will be controlled by Sa(i) (i=1,2 . . . 200) and becomes b 7 X2 7 r+b 6 X2 6 r+ . . . b 1 X 2 r+b 0 Xr=Sa(i)r. 
      Therefore, input/output voltage has below relationship 
     
       
         e 0   =−Sz ( I ) e   in   
       
     
     FIG. 3 shows the generating circuit for mixed signal SM(t) which has following features. 
     1.  06 -l is the generating circuit for functions cos [2(400+15l)πt] (l=1,2, . . . 200, l is time veriable) which generate those functions when signal in RESET off. 
     2.  05 - 1 ˜ 05 - 200  is the circuit shown in FIG.  2 . 
     3.  03  is the circuit shown in FIG.  1 . 
     4.  041 - 1 ,  041 - 2  are electronic switch,  070  is inverter,  041 - 1  OFF and  041 - 2  ON when SW HIGH. 
     5. When RS signal disappears and SW HIGH, ST generates signal SM(t) with below form:          SM        (   t   )       =       ∑     i   =   1     200                     [       Sa        (   i   )            cos        [     2        (     400   +   ɛ     )        π                 t     ]         ]                              
     FIG. 4 has following features 
     1. DATA BUS DS 1 ˜DS 5  come from DS 1 ˜DS 5  in FIG. 5 which send Sa(1)˜Sa(200) to data register  11 , wherein DS 1  sends Sa(1)˜Sa(40), DS 2  sends Sa(41)˜Sa(80) . . . DS 5  sends Sa(161)˜Sa(200), RS 1  is the timing control terminal for data. 
     2.  08  is circuit in FIG. 3,  09  is synchronous signal generator and sends synchronous signal when RESET signal is over,  012  is operational amplifier,  024  is feedback resistor of  012 ,  025 - 1  and  025 - 2  are input resistors with same resistance,  042 - 1  and  042 - 2  are electronic switch,  042 - 1  is off and  042 - 2  is on when R 2  is HIGH,  071  is inverter. 
     3.  10  is a timing control circuit which begin to function when RS is excited and generate timing as shown in FIG.  12 . 
     4. Therefore, output S 0  sends synchronous signal within [T 0 , T 1 ′] and mixed transformed signal SM(t) within [T 1 ′, T 1 ] 
     FIG. 5 has following features 
     1. The box  14  enclosed by dashed line has 5 A/D converters  13 - 1 ˜ 13 - 5  to convert the analog signals from D 0 ˜D 4  to digital signals, wherein CL is CLOCK control for sampling. 
       2 . The box  16  enclosed by dashed line has 5 memory means  15 - 1 ˜ 15 - 5  to store the digital data from A/D converters  13 - 1 ˜ 13 - 5 , wherein AD 1 ˜AD 5  are address bus, R and W are read and write control. 
     3. From the control of W end, the digital data from ND converters  13 - 1 ˜ 13 - 5  are stored in 5 memory means  15 - 1 ˜ 15 - 5 , from the control of R end, the stored digital data are read and output from DS 1 ˜DS 5    
     FIG. 6 is the whole hardware of party A, wherein  12  is circuit in FIG. 4,  17  is circuit in FIG.  5 . Moreover,  18  is a well-know timing circuit and the description is omitted. 
     4. The hardware of party B 
     FIGS. 7-10 is block diagram of party B. FIG. 7 is a figure of a differential circuit comprising four differential means  20 - 1 ,  20 - 2 ,  20 - 3  and  20 - 4 , and the description is omitted for they are well known art. 
     FIGS. 8 and 9 have following features 
     1.  22 -i(i=1,2 . . . 5) in FIG. 8 is circuit in FIG. 9 
       2 .  220  in FIG. 9 is a circuit for solving linear equation set and has parametric memory for storing reverse-transform parameter R(u,v) (u,v=1,2 . . . 200), CE is a timing control end. 
     3.  221  in FIG. 9 is memory for storing the result of  220 , wherein Asi(i=1,2 . . . 5) are address bus, DRi(i=1,2 . . . 5) are data bus and R and W are read and write control ends. 
     4. In FIG. 8,  23 - 1 ˜ 23 - 5  are D/A converters, EN is chip enable, R, W, CK are the same as R, W, CK in FIG. 9, data bus DS 1 ˜DS 5  are DS 1 ˜DS 5  in FIG. 9 
     FIG. 10 is the whole hardware of party A and has below features. 
     1.  21  is the differential circuit in FIG. 7,  17  is that shown in FIG. 5,  21  execute first, second, third, fourth order differential operation on SM(t) to get SM′(T), SM″(T), SM′″(T), SM (4)  (T), and send those signal with SM(t) to D 0 ˜D 4  in  17 , those signals are digitized by 5 A/D converters in  17  and stored in memory  15 - 1 ˜ 15 - 5  in  17 . 
     2.  24  is same as that shown in FIG. 8, the terminals around the dashed box are the same as those in FIG.  8 . 
     3.  25  is a timing control circuit, which initial  21  to execute first, second, third, fourth order differential operation on Si and store those signals with Si into  17 , then 5 A/D converters  13 - 1 ˜ 13 - 5  in  17  digitize those signal and store the result in  15 - 1 ˜ 15 - 5 , the data in  15 - 1 ˜ 15 - 5  are fetched to  220  in  22 -i (i=1,2 . . . 5) for solving linear equation set, the solution are stored in  221 , the data in  221  are sent to  23 - 1 ˜ 23 - 5  to convert into analog signal S 1 (5)-S 5 (t), thus reverse transforming the data. 
     4. Circuit  25  is well known and the description thereof is omitted for clarity. 
     Although the present invention has been described with reference to the preferred embodiment thereof, it will be understood that the invention is not limited to the details thereof. Various substitutions and modifications have suggested in the foregoing description, and other will occur to those of ordinary skill in the art. Therefore, all such substitutions and modifications are intended to be embraced within the scope of the invention as defined in the appended claims.