Abstract:
When a narrow-band digital filter is used for demodulating a signal which has been modulated by using a plenty of sub-carriers, the number of stages is increased, analysis requires a long time, the transmission speed is not increased, and the circuit size is increased. The modulation circuit operation is considered as multiplication.

Description:
DETAILED DESCRIPTION OF INVENTION  
         [0001]    [Engineering Field of this Invention] 
           [0002]    This invention use the carrier of which reflection is not so strong, and is applied to the modulation and demodulation apparatus using quadrature magnitude modulation of many number of sub-carrier and transmit the digital data from each other.  
           [0003]    [Conventional Engineering] 
           [0004]    Such method as transmitting the data using many number of sub-carriers which are modulated by quadrature magnitude modulation is applied to QAM of digital cable TV or to DSL of metal twist-pair and so on. These method concentrate on the frequency of each carriers and demodulate the signal by applying digital filter or FFT using impulse response result as coefficient of filter. For this reason it takes comparatively long time to detect the amplitude of a carrier because of looking until it seems as same continuous wave form.  
           [0005]    [Target to Solve by this Invention] 
           [0006]    To detect accurate result from modulated signal, demodulation circuit such as digital filter or FFT concentrate on frequency of each sub-carrier independently and increase the number of wave form to make difficulty of transmission speed. To make transmission speed high, for example, is to increase the number of carriers and reach so high frequency as to decrease transmission distance greatly.  
           [0007]    [Method to Solve these Difficulty by this Invention] 
           [0008]    By this invention, demodulation circuit do not concentrate on frequency of each sub-carrier independently but pay attention the modulated data consisted by the amount of quadrature amplitude modulation of each sub-carriers, and this construction is seemed to be simultaneous linear equation defining amplitude of each sub-carriers as unknown, so can get the result of amplitude of each sub-carrier by solving this simultaneous linear equation. As simultaneous linear equation can be solved in case of the number of unknown equal to the number of each equation, ideally so as to be able to exchange the number of data by equal number of modulated data. This invention is aimed to build the circuit method by taking the simultaneous linear equation and solving in modulation and demodulation circuit. To take amount of quadrature modulation of many sub-carriers is described using matrix mathematics as following.  
           [0009]    This matrix is square and is constructed equal number of line and column which is two times number of sub-carrier frequency and mean sine wave and cosine wave using one frequency. The elements of this matrix called as modulation matrix are value of trigonometric function, and line means number of sampling of which interval is equal to DA converter frequency, and column means sub-carrier which is sine or cosine of a carrier frequency. The product of this modulation matrix and modulation data matrix of one column responding each sub-carrier is named as the modulated data matrix and is converted by DA converter to analog output respectively. As the lines of modulation matrix are arranged according to DA converting number and is the sine or cosine value of carrier frequency, the product of this modulation matrix and one column modulation data matrix means sum of product of the sine or cosine value at every line and modulation data specified to the sub-carrier, and become modulated data for DA converter input of its every interval. These show that quadrature modulation is described as the simultaneous linear equation.  
           [0010]    From the view of demodulation side about this equation, it seams for modulated data to be the received data detected by AD converter, and modulation data as unknown. As there are modulation data as unknown of two times number kinds of carrier frequency, so there are equations of two times number of carrier frequency. Therefore this simultaneous linear equation can be solved. At demodulation side, the sampling frequency of AD converter is adjusted to the sampling frequency of DA converter of modulation side, and similar data as modulated data is got from AD converter, and demodulated data is got as the product of received data matrix from AD converter and inverse matrix of modulation matrix called demodulation matrix. And more, over-sampling is implemented to this concept, the numbers of over-sampling modulation matrix are created, and the numbers of demodulation matrix, which are the inverse matrix of modulation matrix, become as the numbers of over-sampling. And the over-sampling modulation matrix is composed by inserting each line in over-sampling order position, and the over-sampling demodulation matrix is composed by similar method of insertion. When modulation uses this over-sampling modulation matrix and demodulation uses this over-sampling demodulation matrix, then over-sampling number times demodulated data is got. When modulation uses this over-sampling modulation matrix and demodulation uses individual inverse matrices, then the same numbers of over-sampling demodulated data are got.  
           [0011]    In following, these theory is described by using mathematical equation.  
           [0012]    About the modulation matrix which is the base of this theory,  
           [0013]    Line number: i i=1˜2αn  
           [0014]    Column number: j j=1˜2n  
           [0015]    About above definition  
           [0016]    Number of carrier frequency: n  
           [0017]    Number of over-sampling: α 
           [0018]    According to these definition  
           [0019]    Carrier frequency number: p p=0˜(n−1)  
           [0020]    Original sampling order (without over-sampling): q q=0˜(2n−1)  
           [0021]    Order of over-sampling: r r=1˜α 
           [0022]    Kind of wave: s s=1 indicate cosine wave s=2 indicate sine wave  
           [0023]    About relation of these parameter to line number i and column number j  
             i=αq+r q= 0˜(2 n− 1)  r= 1˜α therefore  i= 1˜2 αn    
             j= 2 p+s p= 0( n− 1)  s= 1 or 2 therefore  j= 1˜2 n    
           [0024]    the element of line number i and column number j is Fj(i) and is defined as  
             F   j ( i )= F   2p+s (α q+r )  
           [0025]    Frequency of frequency number p: fp  
           [0026]    Angle velocity of frequency number p: ωp  
           [0027]    Number of original sampling in one complete wave form: ρ 
           [0028]    interval time of over-sampling: T s  
               ω   p     =     2      π                   f   p                     T   s     =     1     ρ   ×   α   ×     f   0                                     
 
           [0029]    and, the angle of sine and cosine in the element of line number i  
           ω   p     ×     T   s     ×   i     =         2      π                   f   p         ρ   ×   α   ×     f   0         ×     (       α                 q     +   r     )                             
 
           [0030]    Therefore, the element of modulation matrix F j (i) is described following.  
                 F   j          (   i   )       =         F       2      p     +   s            (       α                 q     +   r     )       =       cos        {         2      π                 fp       ρ   ×   α   ×     f   0         ×     (       α                 q     +   r     )       }                   In                 case                 of                 s     =   1                                =       sin        {         2      π                 fp       ρ   ×   α   ×     f   0         ×     (       α                 q     +   r     )       }                   In                 case                 of                 s     =   2                                   
 
           [0031]    The size of modulation matrix is 2αn lines and 2n columns. The size of modulation data matrix is 2n lines and one column, and of which element is described as x j  because the line number of modulation data matrix is related to the column number of modulation matrix for the reason of relating each modulation data to the sub-carrier of sine and cosine individually. Equation of quadrature modulation is described as the product of the modulation matrix and modulation data matrix. The size of modulated data matrix, which is the product of the modulation matrix and modulation data matrix, is 2αn lines and one column, and the element of modulated data matrix is described as d i  according to line number of modulation matrix. As the element of modulated data matrix d i  is the amount of quadrature modulation of each sub-carrier at every sampling time, the equation of modulation is described by matrix as following  
           ( F   j ( i ))×( x   j )=( d   i )  
           [0032]    and this equation is described by elements as following  
           (               F   1          (   1   )       ,       F   2          (   1   )       ,       F   3          (   1   )       ,     ⋯                     F       2      n     -   1            (   1   )         ,       F     2      n            (   1   )       ,                   F   1          (   2   )       ,       F   2          (   2   )       ,       F   3          (   2   )       ,     ⋯                     F       2      n     -   1            (   2   )         ,       F     2      n            (   2   )       ,                   F   1          (   3   )       ,       F   2          (   3   )       ,       F   3          (   3   )       ,     ⋯                     F       2      n     -   1            (   3   )         ,       F     2      n            (   3   )       ,                          ⋮                   F   1          (     2      α                 n     )       ,       F   2          (     2      α                 n     )       ,       F   3          (     2      α                 n     )       ,     ⋯                     F       2      n     -   1            (     2      α                 n     )         ,       F     2      n            (     2      α                 n     )       ,           )     ×     (           x   1               x   2             ⋮             x   j             ⋮             x     2      n             )       =     (           d   1               d   2             ⋮             d   i             ⋮             d     2      α                 n             )                           
 
           [0033]    And this equation is described by simultaneous linear equation is as following  
             F   1 (1) x   1   +F   2 (1) x   2   +F   3 (1) x   3   + . . . +F   2n−1 (1) x   2n−1   +F   2n (1) x   2n   =d   1    
             F   1 (2) x   1   +F   2 (2) x   2   +F   3 (2) x   3   + . . . +F   2n−1 (2) x   2n−1   +F   2n (2) x   2n   =d   2    
             F   1 (3) x   1   +F   2 (3) x   2   +F   3 (3) x   3   + . . . +F   2n−1 (3) x   2n−1   +F   2n (3) x   2n   =d   2    
             F   1 (2 αn ) x   1   +F   2 (2 αn ) x   2   +F   3 (2 αn ) x   3   + . . . +F   2n−1 (2 αn ) x   2n−1   +F   2n (2 αn ) x   2n   =d   2αn    
           [0034]    Depend on this simultaneous linear equation, the product of initially determined F j (i) and modulation data x j  related by j, summed together through all j resulting to d j  as the input data of DA converter at sampling number i, and is converted to analog output. d j  can be got by multiplying and accumulating in every sampling interval. Following is described about the treatment in receiving side. The matrix of which elements F r0,j (g) is picked up from modulation matrix by implementing the over-sampling order number r=r 0 , then is represented as following.  
                       F     r0   ,       2      p     +   s         (              q   )     =       cos        {         2      π                   f   p         ρ   ×   α   ×     f   0         ×     (       α                 q     +     r   0       )       }                   in                 case                 of                 s     =   1                                sin        {         2      π                   f   p         ρ   ×   α   ×     f   0         ×     (       α                 q     +     r   0       )       }                   in                 case                 of                 s     =   2                       q   =     0   ∼     (       2      n     -   1     )                                   
 
           [0035]    The first line of this matrix is first r 0  line of modulation matrix, and the other line is picked up from modulation matrix every α line from r 0  to constructing 2n lines. The first line of related modulated data matrix is first r 0  line of modulated data matrix and the other line is picked up from modulated data matrix every α line from r 0  to constructing 2n lines, and of which element is described as d r0,q  as following.  
           
         d 
         r0,q 
         =d 
         (αq+r 
         
           0 
         
         )  
       
           [0036]    The modulation equation of above matrix picking up by sampling order r 0  is described by matrix as following  
           ( F   r0,2p+s ( q ))×( x   2p+s )=( d   r0,     q   )  
           [0037]    and this equation is described by elements as following  
           (               F       r                 0     ,   1            (   0   )       ,       F       r                 0     ,   2            (   0   )       ,       F       r                 0     ,   3            (   0   )       ,     ⋯                     F       r                 0     ,       2      n     -   1              (   0   )         ,       F       r                 0     ,     2      n              (   0   )                       F       r                 0     ,   1            (   1   )       ,       F       r                 0     ,   2            (   1   )       ,       F       r                 0     ,   3            (   1   )       ,     ⋯                     F       r                 0     ,       2      n     -   1              (   1   )         ,       F       r                 0     ,     2      n              (   1   )                              ⋮                           F       r                 0     ,   1            (       2                 n     -   1     )       ,       F       r                 0     ,   2            (       2                 n     -   1     )       ,       F       r                 0     ,   3            (       2                 n     -   1     )       ,   ⋯                                             F       r                 0     ,       2      n     -   1              (       2                 n     -   1     )       ,       F       r                 0     ,     2      n              (       2                 n     -   1     )       ,                   )     ×     (           x   1               x   2             ⋮             x   j             ⋮             x     2      n             )       =     (           d       r                 0     ,   0                 d       r                 0     ,   1               ⋮           ⋮           ⋮             d       r                 0     ,     2      α                 n               )                           
 
           [0038]    And this equation is described by simultaneous linear equation as following  
             F   r0,1 (0) x   1   +F   r0,2 (0) x   2   +F   r0,3 (0) x   3   + . . . F   r0,2n−1 (0) x   2n−1   +F   r0,   2n (0) x   2n   =d   r0,0    
             F   r0,1 (1) x   1   +F   r0,2 (1) x   2   +F   r0,3 (1) x   3   + . . . F   r0,2n−1 (1) x   2n−1   +F   r0,   2n (1) x   2n   =d   r0,1    
             F   r0,1 (2 n− 1) x   1   +F   r0,2 (2 n− 1)x 2   +F   r0,3 (2 n− 1) 3   + . . F   r0,2n−1 (2 n− 1   ) x   2n−1   +F   r0,2n (2 n− 1) x   2n   =d   r0,2n−1    
           [0039]    In this simultaneous linear equation, d r0,0 ˜d r0,2n−1  are similarly got as receiving data by AD converter. For the demodulation side to detect the modulation data x 1 ˜x 2n , the inverse matrix of the modulation matrix of which element is (F r0,2p+s ) is applied to solving this simultaneous linear equation. The element of inverse matrix of modulation matrix is described as G r0,j (q)  
           (               G       r                 0     ,   1            (   0   )       ,       G       r                 0     ,   2            (   0   )       ,       G       r                 0     ,   3            (   0   )       ,     ⋯                     G       r                 0     ,       2      n     -   1              (   0   )         ,       G       r                 0     ,     2      n              (   0   )                       G       r                 0     ,   1            (   1   )       ,       G       r                 0     ,   2            (   1   )       ,       G       r                 0     ,   3            (   1   )       ,     ⋯                     G       r                 0     ,       2      n     -   1              (   1   )         ,       G       r                 0     ,     2      n              (   1   )                              ⋮                     G       r                 0     ,   1            (       2      n     -   1     )       ,       G       r                 0     ,   2            (       2      n     -   1     )       ,       G       r                 0     ,   3            (       2      n     -   1     )       ,   ⋯                                             G       r                 0     ,       2      n     -   1              (       2      n     -   1     )       ,       G       r                 0     ,     2      n              (       2      n     -   1     )       ,             )     ×     (           d       r                 0     ,   0                 d       r                 0     ,   1               ⋮             d       r                 0     ,       2                 n     -   1               )       =     (           x   1               x   2             ⋮             x     2      n             )                           
 
           [0040]    And this equation is described by simultaneous linear equation as following  
             G   r0,1 (0) d   r0,0   +G   r0,2 (0) d   r0,1   +G   r0,3 (0) d   r0,2   + . . . G   r0,2n−1 (0) d   r0,2n−2   +G   r0,2n (0) d   r0,2n−1   =x   1    
             G   r0,1 (1) d   r0,0   +G   r0,2 (1) d   r0,1   +G   r0,3 (1) d   r0,2   + . . . G   r0,2n−1 (1) d   r0,2n−2   +G   r0,2n (0) d   r0,2n−1   =x   2    
             G   r0,1 (2 n−   1)   d   r0,0   +G   r0,2 (2 n− 1) d   r0,1   +G   r0,3 (2 n− 1) d   r0,2   + . . G   r0,2n−1 (2 n− 1   ) d   r0,2n−2   +G   r0,2n (2 n− 1) d   r0,2n−1   =x   2n    
           [0041]    According to this simultaneous linear equation, the inverse matrix of the matrix which is picked up the same over-sampling number line from the over-sampling modulation matrix, and the receiving data coming from AD converter at every r 0  sampling interval, are multiplied and summed together by two times number of carrier frequency accumulators, continuously until the end of one frame of modulation, and then the demodulated data of all sub-carriers are got.  
           [0042]    Mathematically, the size of receiving data matrix is one column matrix, and about the construction of inverse matrix, the line number is relating to column number of modulation matrix, and the column number is relating to line number of modulation matrix and it look like exchange the line and column.  
           [0043]    As the number of over-sampling is α, the number of α inverse matrix are created from modulation matrix and are got the number of α kind of demodulated data by this operation. When demodulation starts from the first line of demodulation matrix synchronizing to the receiving data of the first line of modulation matrix, α kind of demodulated data are equal one after another because of only one kind of modulation data. When demodulation starts from several sampling later than the first line of demodulation matrix but not over a sampling, the number of same demodulation data decrease according to the number of several sampling delay. When demodulation starts from over a sampling after the first line of demodulation matrix, no same demodulation data is got because the one frame time belonging to the one operation of demodulation matrix spread two frame time belonging to first modulated data matrix and of next modulated data matrix and the receiving data are constructed by first modulation data and next modulation data. This property is applied to synchronization of modulation and demodulation. The matrix, of which column is picked up from one column of demodulation matrix and is constructed other column by shifting one over-sampling interval from each other to the end of the line number of demodulation matrix, is created, and demodulation operation is applied to any receiving data from AD converter by this shifted matrix, and synchronize point is found by the columns number of same demodulated data.  
           [0044]    The meaning of demodulation, which use the matrix composed the lines from over-sampling number of inverse matrix placed at over-sampling proper timing under the condition of synchronizing with modulation, is that there are over-sampling number of simultaneous linear equation, and over-sampling number of same modulated data are solved. The products of over-sampling demodulation matrix and receiving data matrix from AD converter are summed together, and over-sampling number times of similar modulated data are got, and this contribute the reduction of electrical circuits of multiplier and accumulator.  
           [0045]    Next explaining the adjusting method in following  
           [0046]    The distortion, which is created by the parameter of line such as twist-pair between terminals, which is created by sampling timing difference between DA converter and AD converter, should be adjusted to get the correct demodulated data. Before the practical communication under the circumstance of decided parameter of line or DA or AD converter, test communication is done to get the parameter of adjustment.  
           [0047]    Modulation data of sub-carrier frequency number p are x 2p+1  for cosine and x 2p+2  for sine and the phase of these wave is shifted θ p  in the receiving data by the parameter of communication line or sampling timing difference of DA and AD converter.  
           [0048]    The phase shifted form of the wave described as following.  
             x   2p+1  cos(ω p   t+θ   p )= x   2p+1  cos θ p  cos ω p   t−x   2p+1  sin θ p  sin ω p   t    
             x   2p+2  sin(ω p   t+θ   p )= x   2p+2  cos θ p  sin ω p   t−x   2p+2  sin θ p  cos ω p   t    
           [0049]    In demodulation side, the amount of these wave is got as the receiving data, and practical demodulated data β 2p+1 , for cosine and β 2p+2  for sine, which is demodulated by the operation of receiving data and demodulation matrix about cosine and sine independently, are got as coefficient of cos ω p t and sin ω p t. And practical demodulated data is described mathematically as following.  
           β 2p+1   =x   2p+1  cos θ p   +x   2p+2  sin θ p    
           β 2p+2   =−x   2p+1  sin θ p   +x   2p+2  cos θ p    
           [0050]    Practical demodulated data of each sampling index r are described as β r,2p+1  and β r,2p+2 , which is detected by the operation of partial demodulation matrix and partial receiving data matrix of each sampling index. This demodulated data are described by use of raff equal symbol because of being distorted by noise and phase shift. And equation is described as following.  
             β   r,2p+1   ≈x   2p+1  cos θ p   +x   2p+2  sin θ p    
           βr, 2 p+ 2   ≅−x   2p+1  sin θ p   +x   2p+2  cos θ p    
           [0051]    Difference is took in spite of raff equal symbol, and amount of square of these difference is described as δ p   2  and is differentiated by θ p  to apply minimum square method.  
                   ∂               ∂     θ   p              δ   p   2              =       2        (         x       2      p     +   1          sin                   θ   p       -       x       2      p     +   2          cos                   θ   p         )            ∑     r   =   1     α                     β     r   ,       2      p     +   1             +                          2        (         x       2      p     +   1          cos                   θ   p       +       x       2      p     +   2          sin                   θ   p         )            ∑     r   =   1     α                     β     r   ,       2      p     +   2                                         
 
           [0052]    The modulation data of the test communication before practical communication is described following  
             x   2p+1   =x   2p+2   =x   test ≢0  
           [0053]    To get θ p  by applying minimum square method  
             ∂               ∂     θ   p              δ   p   2       =   0                         
 
           [0054]    and get the result for θ p  in following  
         tan                   θ   p       =           ∑     r   =   1     α                     β     r   ,       2                 p     +   1           -       ∑     r   =   1     α                     β     r   ,       2                 p     +   2                   ∑     r   =   1     α                     β     r   ,       2                 p     +   1           +       ∑     r   =   1     α                     β     r   ,       2                 p     +   2                                     
 
           [0055]    And cos θ p  or sinθ p  is calculated by tanθ p .  
           [0056]    As modulation data is determined as the mean value of over-sampling number of practical demodulation data, so is described following  
               x       2      p     +   1       ≅              cos                   θ   p     ×     {       1   α            ∑     r   =   1     α                     β     r   ,       2      p     +   1             }       -     sin                   θ   p          {       1   α            ∑     r   =   1     α                     β     r   ,       2      p     +   2             }                       x       2      p     +   2       ≅              sin                   θ   p     ×     {       1   α            ∑     r   =   1     α                     β     r   ,       2      p     +   1             }       +     cos                   θ   p          {       1   α            ∑     r   =   1     α                     β     r   ,       2      p     +   2             }                                     
 
           [0057]    The modulation data x test  before the practical communication is already known at receiving side and described as following.  
           [0058]    In the column number 2p+1  
             x   test ≅cos θ p   ×{overscore (D)}   2p+1 (test)−sin θ p   ×{overscore (D)}   2p+2 (test)  
           [0059]    In the column number 2p+2  
             x   test ≅sin θ p   ×{overscore (D)}   2p+1 (test)+cos θ p   ×{overscore (D)}   2p+2 (test)  
           [0060]    In these equation  
                 D   _         2      p     +   1       =              1   α            ∑     r   =   1     α                     β     r   ,       2      p     +   1                             D   _         2      p     +   2       =              1   α            ∑     r   =   1     α                     β     r   ,       2      p     +   2                                         
 
           [0061]    {overscore (D)} followed by (test) mean the practical demodulated data in test communication and mean value of a demodulated data.  
           [0062]    The demodulated data differ from the modulation data in demodulation side, and is adjusted by the way that ratio of amplitude of test communication and practical communication is equal, and adjustment equation is as following.  
               x       2      p     +   1       -&gt;         x   test     ×     (       cos                   θ   p     ×       D   _         2      p     +   1         -     sin                   θ   p            D   _         2      p     +   2           )           cos                   θ   p     ×         D   _         2      p     +   1            (   test   )         -     sin                   θ   p     ×         D   _         2      p     +   2            (   test   )                           x       2      p     +   2       -&gt;         x   test     ×     (       sin                   θ   p     ×       D   _         2      p     +   1         +     cos                   θ   p            D   _         2      p     +   2           )           sin                   θ   p     ×         D   _         2      p     +   1            (   test   )         +     cos                   θ   p     ×         D   _         2      p     +   2            (   test   )                                         
 
           [0063]    Therefore demodulation data is adjusted from such influence of communication line. 
       
    
    
       [0064]    [0064]FIG. 1 is block diagram of total system of modulation and demodulation. In modulation side, the data of modulation matrix ROM, of which address is specified by the address counter for modulation, and the modulation data are multiplied responding to cosine and sine of all sub-carrier frequency individually and are added all product to the data of DA converter.  
         [0065]    In demodulation side, analog signal is converted by the AD converter to digital signal and is multiplied with the data of demodulation matrix ROM 1 , of which address is specified by the address counter for demodulation, about cosine and sine of all sub-carrier frequency individually in every sampling interval and are accumulated until the end of one modulation block, and adjusted every end of block about phase shift to the adjusted demodulated data.  
         [0066]    About the synchronization between the address counter for modulation and address counter for demodulation, ROM named as ROM 2  is picked up the memory data belonging to one carrier frequency from the demodulation ROM 1 , and the data of the other block of memory are moved some address each other to the end of memory address. The receiving data from AD converter is multiplied and accumulated with the memory data of cosine and sine individually in numbers of shift, and at the end of one modulation block, is adjusted about the phase shift and sent to synchronization circuit. In the synchronization circuit, the demodulated data is arranged according to the order of address shift, and the first data of same data series nearly as a is found out, and reset the address counter for demodulation adjusting the delay between the address counter and the number of shift.  
         [0067]    [0067]FIG. 2 is block diagram of modulation. The maximum number of address counter for modulation is 2αn. Address of modulation ROM is 2αn wide and number of data buss is 2nW(word) wide. 1W of modulation ROM store Fj(i). This ROM send the data of 2nW wide responding i, which is specified as the index of sampling, to the modulation data of 2n. In every clock, the product of each modulation data and specified Fj(i) of 1W wide are summed together for all number of modulation data and is converted by DA converter to analog to the communication line. Before the practical communication, the test communication is made by the modulation data x test    
         [0068]    [0068]FIG. 3 is the block diagram of demodulation to get the demodulated data before adjustment of phase. Maximum address of address counter for demodulation is 2αn. Address of demodulation ROM 1  is 2αn wide and data of it is 2nW wide (number of 2n of 1W wide ROM). The data of 1W of demodulation ROM 1  is specified Gj(i). The analog signal from communication line is converted to digital by AD converter at same sampling interval as the clock of DA converter in modulation side.  
         [0069]    2nW wide data is read out from demodulation ROM 1  at every clock, and every 1W related to cosine or sine of sub-carrier frequency individually is multiplied with the receiving data at this moment, and is accumulated individually until the number of 2αn, and is divided by α as the demodulated data {overscore (D)} 2p+1 , {overscore (D)} 2p+2  before adjustment of phase.  
         [0070]    Using the mean value of demodulated data of cosine and sine of same carrier of number p as {overscore (D)} 2p+1 , {overscore (D)} 2p+2  drawn in FIG. 3 block diagram, FIG. 4 shows the adjustment circuit diagram of phase and magnitude.  
         [0071]    About the basic circuit operation in FIG. 4, when the system is reset or is happen to change the parameter of adjustment according the condition of communication line, by the modulation data x test  which is determined to exchange at initial test communication by both modulation side and demodulation side, the parameter of adjustment is set. And after this operation, practical communication start and use this parameter for adjustment calculation.  
         [0072]    The mean value of demodulation data {overscore (D)} 2p+1  and {overscore (D)} 2p+2  are squared by block of multiplyer1 and multiplyer2 and are added each other by adder2 and are stored by DFF 1  at the end of initial test communication after system reset. And this data is sent to DFF 1 , DFF 2  and DFF 3  at only one time after initial test communication by one clock in the transmission unit frame time as 2n α as numbers conversions of DA and AD, and stored until next reset after first one.  
         [0073]    And by the similar operation, the added value of {overscore (D)} 2p+1  and {overscore (D)} 2p+2  is stored in DFF 2  and difference value of {overscore (D)} 2p+1  and {overscore (D)} 2p+2  is stored in DFF 3 . These three stored data and x test  of initial test data of communication are stored as the parameter of adjustment until next system reset.  
         [0074]    As the parameter of adjustment for multiplier.7 and multiplier.8 are as following.  
         x test    
         [0075]    data stored in DFF 1   {overscore (D)}   2p+1   2 (test)+ {overscore (D)}   2p+2   2 (test)  
         data stored in DFF 2   {overscore (D)}   2p+1 (test)+{overscore (D)} 2p+2 (test)  
         data stored in DFF 3   {overscore (D)}   2p+1  (test)− {overscore (D)}   2p+2 (test)  
         [0076]    In practical communication after the initial test communication, the mean value of demodulated data in every one frame {overscore (D)} 2p+1  is stored in DFF 4  and {overscore (D)} 2p+2  is stored in DFF 5  and is renewed at the interval of 2n α number of clock.  
         [0077]    Stored data in some operation blocks is as following.  
         multiplier.3 ( {overscore (D)}   2p+1 (test)+ {overscore (D)}   2p+2 (test))×{overscore (D)} 2p+1    
         multiplier.4 ( {overscore (D)}   2p+1 (test)+{overscore (D)} 2p+2 (test))× {overscore (D)}   2p+2    
         multiplier.5 ( {overscore (D)}   2p+1 (test)− {overscore (D)}   2p+2 (test))× {overscore (D)}   2p+1    
         multiplier.6 ( {overscore (D)}   2p+1 (test)− {overscore (D)}   2p+2  (test))× {overscore (D)}   2p+2    
         [0078]    and  
         difference.2 ( {overscore (D)}   2p+1 (test)+ {overscore (D)}   2p+2  (test))× {overscore (D)}   2p+1 −( {overscore (D)}   2p+1 (test)− {overscore (D)}   2p+2  (test))× {overscore (D)}   2p+2    
         adder.3 ( {overscore (D)}   2p+1 (test)+ {overscore (D)}   2p+2 (test))× {overscore (D)}   2p+2 +( {overscore (D)}   2p+1 (test)− {overscore (D)}   2p+2 (test))× {overscore (D)}   2p+1    
         [0079]    and modulation data in test communication x test  is took in place  
               x   test     ×     {               (           D   _         2      p     +   1            (   test   )       +         D   _         2      p     +   2            (   test   )         )     ×       D   _         2      p     +   1         -                 (           D   _         2      p     +   1            (   test   )       -         D   _         2      p     +   2            (   test   )         )     ×       D   _         2      p     +   2               }             multiplier   .              7                 x   test     ×     {               (           D   _         2      p     +   1            (   test   )       +         D   _         2      p     +   2            (   test   )         )     ×       D   _         2      p     +   2         +                 (           D   _         2      p     +   1            (   test   )       -         D   _         2      p     +   2            (   test   )         )     ×       D   _         2      p     +   1               }             multiplier   .              8                               
 
         [0080]    finally, the amount of squared mean value of demodulated data is took in place  
                   x   test     ×     {         (           D   _         2      p     +   1            (   test   )       +         D   _         2      p     +   2            (   test   )         )     ×       D   _         2      p     +   1         -       (           D   _         2      p     +   1            (   test   )       -         D   _         2      p     +   2            (   test   )         )     ×       D   _         2      p     +   2           }               D   _         2      p     +   1     2          (   test   )       +         D   _         2      p     +   2     2          (   test   )           -&gt;     x       2      p     +   1               divider   .              1                     x   test     ×     {         (           D   _         2      p     +   1            (   test   )       +         D   _         2      p     +   2            (   test   )         )     ×       D   _         2      p     +   2         -       (           D   _         2      p     +   1            (   test   )       -         D   _         2      p     +   2            (   test   )         )     ×       D   _         2      p     +   1           }               D   _         2      p     +   1     2          (   test   )       +         D   _         2      p     +   2     2          (   test   )           -&gt;     x       2      p     +   2               divider   .              2                               
 
         [0081]    Therefore, the demodulated data is adjusted.  
         [0082]    Demodulation data is got at every 2αn number of data from AD converter which is the frame of one modulation. And phase adjustment of {overscore (D)} 2p+1 , {overscore (D)} 2p+2  in every carrier may be done at 2αn clock interval. To increase the efficiency of usage of circuit, the amount of circuit decrease by using a time sharing phase adjustment circuit.  
         [0083]    The time sharing phase adjustment circuit is represented in FIG. 5. {overscore (D)} 2p+1  and {overscore (D)} 2p+2  detect and store demodulated data before phase adjustment, and sent one block by selector o to phase adjustment circuit at every one frame. In time sharing phase adjustment circuit, number of n registers in spite of DFF 1 , DFF 2  and DFF 3  which are represented in phase adjustment circuit FIG. 4 for one sub-carrier frequency, are selected one by one by selector  1  synchronized to selector  0 , and are stored as the parameter of every sub-carrier in one round of selector  1  when the parameter is decided in system.  
         [0084]    In practical communication, the parameters responding to the index of sub-carrier are selected by the selector  2  synchronizing to the selector  0  in spite of operation DFF 4  and DFF 5 . The calculation in the circuit is done ideally by the pip-line operation. Therefore phase adjusted demodulation data are sent from the time sharing phase adjustment circuit continuously.  
         [0085]    Demodulation circuit block diagram for synchronization is represented in FIG. 6. Address counter for demodulation is used for this circuit and output 2αn addresses. Demodulation ROM 2  has 2αn addresses and 4nW wide data bus. ROM 2  pick up the memory data G 2p+1 (i) and G 2p+2 (i) belonging to one carrier frequency p from the demodulation ROM 1  and the data of the other block of memory are moved a address each other to the end of memory address. Data of this ROM 2  is 4nW wide which is 2nW numbers of cosine data and 2nW number of sine data. In one clock interval, demodulation ROM 2  output 4nW wide data, with which the data from AD converter is multiplied and accumulated by a number individually and selected as {overscore (D)} 2p+1  and {overscore (D)} 2p+2  to the time sharing phase adjustment circuit. Time sharing phase adjustment circuit is operated not by the half clock but by two times clock of address counter. At least, one out put of the time sharing adjustment circuit is sent to synchronization circuit represented in FIG. 7 to detect frame synchronized signal. Serise of adjusted demodulated data for synchronization are shifted by equal or less than α number of DFF, and each shifted data are compared, and synchronization signal is output in case of all equal data. This synchronization signal is shifted as long as the delay between synchronization circuit and address counter for demodulation, and reset the counter for demodulation to synchronize the counter for demodulation with the counter for modulation of the other side terminal.  
         [0086]    About method of sub-carrier frequency determination described in claim  2 , the cosine and sine wave equation of sub-carrier number p are  
             cos        {         2      π                   f   ρ         ρ   ×   α   ×     f   0              (       α                 q     +   r     )       }                 sin        {         2      π                   f   ρ         ρ   ×   α   ×     f   0              (       α                 q     +   r     )       }                                 
 
         [0087]    and following equation for sub-carrier frequency reference  
           ∑     q   =   0       (       2      n     -   1     )                         cos   2          {         2      π                   f   ρ         ρ   ×   α   ×     f   0              (       α                 q     +   r     )       }         =       ∑     q   =   0       (       2      n     -   1     )                         sin   2          {         2      π                   f   ρ         ρ   ×   α   ×     f   0              (       α                 q     +   r     )       }                               
 
         [0088]    and this equation is solved by many ƒ p  by which is made matrix and inverse matrix. And the difference of range in the inverse matrix elements are not so wide or so near to zero that the proper ƒ p  are selected.  
         [0089]    The transmission speed is as following  
         [0090]    The sampling clock of DA converter is CLK and most high frequency of sub-carrier is ƒ 0  and ρ is sampling numbers in one wave of most high frequency of sub-carrier  
         CLK=ρ×α×ƒ 0 =ραƒ 0    
         [0091]    The number of sampling of one frame in modulation is same as the number of address of modulation ROM, that is 2αn. The bits wide of modulation data are specified as A, then the total bits wide of modulation data are 2nA. Therefore the transmission speed is following.  
           CLK       2      α                 n                    ×   2                 nA     =     CLK   ×     A   α                             
 
         [0092]    Another method of synchronization block diagram is represented in FIG. 8. Although the modulation data are same in first two round address for modulation and same in second two round address of modulation, the modulation data of the sub-carrier which is specified as the use of synchronization should be different in first two round address from in second two round address. The data of demodulation ROM 1  which is used for synchronization is called as synchronization ROM. The address of synchronization ROM is connected to the continuous address counter.  
         [0093]    The product of the data from AD converter and the data of synchronization ROM is accumulated for the one round address of demodulation circuit, and the accumulator, which start accumulation from every address for one round addresses to output result continuously, is installed in synchronization circuit. These value of the accumulator after multiplying are not different from each other for the term of same modulation data, but are different from each other for the term of the different modulation data. In synchronization circuit, this property contribute to make synchronization signal which is output by the comparator indication the equality or the difference between two series adjusted demodulated data. Multiplier and accumulator starting from every address is represented in FIG. 9.  
         [0094]    The data from AD converter and the data of synchronization ROM are multiplied by the circuit of multiplier, and send to the circuit of accumulator. The output of accumulator is sent to DFF 6  by every clock, and is returned to accumulator to be added with next data one after another.  
         [0095]    The carry-out signal, which is output at every one round of address counter, reset DFF 6 , and another DFF 7  store the last accumulated value as the data of accumulation. This accumulating operation is same as in demodulation circuit.  
         [0096]    Until next accumulation from previous this one, the accumulator, which start accumulation from every address for one round addresses, is operated as following. The accumulated data until this time of previous round in DFF 8  is subtracted from the previous accumulated data in DFF 7  and is added by newly accumulated data until this time in DFF 6 , and putout this data at every address.  
         [0097]    To achieve this operation, dual-port RAM, of which highest write address bit is connected through TFF to carry-out of address counter and the lower address is connected to address counter output, and of which read address is similar to write address with inverted highest address, output the previous round data, which are accumulated and stored in DFF 8 , and above subscription is got.  
         [0098]    Watching whether the number of address counter for demodulation is continuous or not at synchronization, the demodulation data is adjusted in case of over-ride by subscripting the product and incase of less number adding the product.  
       EXAMPLE  
       [0099]    Number of carrier frequency n=4  
         [0100]    Number of over-sampling α=4  
         [0101]    Kind of wave s s=1 indicate cosine wave s=2 indicate sine wave  
         [0102]    about modulation matrix  
         [0103]    number of line i i=1˜32  
         [0104]    number of column j j=1˜8  
         [0105]    sub-carrier frequency number p p=0˜3  
         [0106]    Original sampling order q q=0˜7  
         [0107]    Order of over-sampling r r=1˜4  
           i=αq+r= 4 q+r    
           j= 2 p+s    
         [0108]    the elements of matrix  
           F   i ( i )= F   2p+s (4 q+r )  
         [0109]    Number of original sampling in one complete wave form ρ=2.399 
                 F       2      p     +   s            (       4      q     +   r     )       =            cos        {         2      π                   f   p         9.596                   f   0              (       4      q     +   r     )       }                          in                 case                 of                 s     =   1                          sin        {         2      π                   f   p         9.596                   f   0              (       4      q     +   r     )       }                          in                 case                 of                 s     =   2                                 
 
         [0110]    f 0 =1.0423 MHz  
         [0111]    f 1 =0.7809 MHz  
         [0112]    f 2 =0.6255 MHz  
         [0113]    f 3 =0.4684 MHz  
         [0114]    basic sampling interval 383.7 nSec  
         [0115]    over-sampling interval 95.9 nSec  
                                                                                                                                                                                                                     demodulation ROM2(conbined ROM for synchronization)                p                0   1   2   3                r                1   2   1   2   1   2   1   2                j            q   r   i   1   2   3   4   5   6   7   8                    0   1   1   9D62   8CD2   8CD2   AC89   AC89   9C5E   9C5E   7AA0           2   2   97C8   97FB   97FB   9FE6   9FE6   AA6B   AA6B   6DA9           3   3   8C8B   9D6B   9D6B   8E22   8E22   B0DB   B0DB   650B           4   4   8051   9AE3   9AE3   7B6D   7B6D   AE2A   AE2A   6216       1   1   5   8CD2   AC89   AC89   9C5E   9C5E   7AA0   7AA0   AF18           2   6   97FB   9FE6   9FE6   AA6B   AA6B   6DA9   6DA9   A655           3   7   9D6B   8E22   8E22   B0DB   B0DB   650B   650B   9945           4   8   9AE3   7B6D   7B6D   AE2A   AE2A   6216   6216   89E8       2   1   9   AC89   9C5E   9C5E   7AA0   7AA0   AF18   AF18   7BD9           2   10   9FE6   AA6B   AA6B   6DA9   6DA9   A655   A655   7DB4           3   11   8E22   B0DB   B0DB   650B   650B   9945   9945   80A2           4   12   7B6D   AE2A   AE2A   6216   6216   89E8   89E8   8461       3   1   13   9C5E   7AA0   7AA0   AF18   AF18   7BD9   7BD9   85E0           2   14   AA6B   6DA9   6DA9   A655   A655   7DB4   7DB4   81E8           3   15   B0DB   650B   650B   9945   9945   80A2   80A2   7EA6           4   16   AE2A   6216   6216   89E8   89E8   8461   8461   7C62       4   1   17   7AA0   AF18   AF18   7BD9   7BD9   85E0   85E0   9D62           2   18   6DA9   A655   A655   7DB4   7DB4   81E8   81E8   97C8           3   19   650B   9945   9945   80A2   80A2   7EA6   7EA6   8C8B           4   20   6216   89E8   89E8   8461   8461   7C62   3E98   EE06       5   1   21   AF18   7BD9   7BD9   85E0   85E0   9D62   9D62   8CD2           2   22   A655   7DB4   7DB4   81E8   81E8   97C8   97C8   97FB           3   23   9945   80A2   80A2   7EA6   7EA6   8C8B   8C8B   9D6B           4   24   89E8   8461   8461   7C62   3E98   EE06   8051   9AE3       6   1   25   7BD9   85E0   85E0   9D62   9D62   8CD2   8CD2   AC89           2   26   7DB4   81E8   81E8   97C8   97C8   97FB   97FB   9FE6           3   27   80A2   7EA6   7EA6   8C8B   8C8B   9D6B   9D6B   8E22           4   28   8461   7C62   3E98   EE06   8051   9AE3   9AE3   7B6D       7   1   29   85E0   9D62   9D62   8CD2   8CD2   AC89   AC89   9C5E           2   30   81E8   97C8   97C8   97FB   97FB   9FE6   9FE6   AA6B           3   31   7EA6   8C8B   8C8B   9D6B   9D6B   8E22   8E22   B0DB           4   32   3E98   EE06   8051   9AE3   9AE3   7B6D   7B6D   AE2A                  
 
         [0116]    [0116]                                                                                                                                                                                                                     demodulation ROM2(conbined ROM for synchronization)                p                0   1   2   3                r                1   2   1   2   1   2   1   2                j            q   r   i   1   2   3   4   5   6   7   8                    0   1   1   9D62   8CD2   8CD2   AC89   AC89   9C5E   9C5E   7AA0           2   2   97C8   97FB   97FB   9FE6   9FE6   AA6B   AA6B   6DA9           3   3   8C8B   9D6B   9D6B   8E22   8E22   B0DB   B0DB   650B           4   4   8051   9AE3   9AE3   7B6D   7B6D   AE2A   AE2A   6216       1   1   5   8CD2   AC89   AC89   9C5E   9C5E   7AA0   7AA0   AF18           2   6   97FB   9FE6   9FE6   AA6B   AA6B   6DA9   6DA9   A655           3   7   9D6B   8E22   8E22   B0DR   B0DB   650B   650B   9945           4   8   9AE3   7B6D   7B6D   AE2A   AE2A   6216   6216   89E8       2   1   9   AC89   9C5E   9C5E   7AA0   7AA0   AF18   AF18   7BD9           2   10   9FE6   AA6B   AA6B   6DA9   6DA9   A655   A655   7DB4           3   11   8E22   B0DB   B0DB   650B   650B   9945   9945   80A2           4   12   7B6D   AE2A   AE2A   6216   6216   89E8   89E8   8461       3   1   13   9C5E   7AA0   7AA0   AF18   AF18   7BD9   7BD9   85E0           2   14   AA6B   6DA9   6DA9   A655   A655   7DB4   7DB4   81E8           3   15   B0DB   650B   650B   9945   9945   80A2   80A2   7EA6           4   16   AE2A   6216   6216   89E8   89E8   8461   8461   7C62       4   1   17   7AA0   AF18   AF18   7BD9   7BD9   85E0   85E0   9D62           2   18   6DA9   A655   A655   7DB4   7DB4   81E8   81E8   97C8           3   19   650B   9945   9945   80A2   80A2   7EA6   7EA6   8C8B           4   20   6216   89E8   89E8   8461   8461   7C62   3E98   EE06       5   1   21   AF18   7BD9   7BD9   85E0   85E0   9D62   9D62   8CD2           2   22   A655   7DB4   7DB4   81E8   81E8   97C8   97C8   97FB           3   23   9945   80A2   80A2   7EA6   7EA6   8C8B   8C8B   9D6B           4   24   89E8   8461   8461   7C62   3E98   EE06   8051   9AE3       6   1   25   7BD9   85E0   85E0   9D62   9D62   8CD2   8CD2   AC89           2   26   7DB4   81E8   81E8   97C8   97C8   97FB   97FB   9FE6           3   27   80A2   7EA6   7EA6   8C8B   8C8B   9D6B   9D6B   8E22           4   28   8461   7C62   3E98   EE06   8051   9AE3   9AE3   7B6D       7   1   29   85E0   9D62   9D62   8CD2   8CD2   AC89   AC89   9C5E           2   30   81E8   97C8   97C8   97FB   97FB   9FE6   9FE6   AA6B           3   31   7EA6   8C8B   8C8B   9D6B   9D6B   8E22   8E22   B0DB           4   32   3E98   EE06   8051   9AE3   9AE3   7B6D   7B6D   AE2A                    
         [0117]    [0117]                                                                                                                                                               demodulation ROM2-4(for synchronization)                p                0   1   2   3                r                1   2   1   2   1   2   1   2                j            q   1   2   3   4   5   6   7   8               0   8051   9AE3   7B6D   AE2A   6216   89E8   8461   7C62       1   9AE3   7B6D   AE2A   6216   89E8   8461   7C62   8051       2   7B6D   AE2A   6216   89E8   8461   7C62   8051   9AE3       3   AE2A   6216   89E8   8461   7C62   8051   9AE3   7B6D       4   6216   89E8   8461   7C62   8051   9AE3   7B6D   AE2A       5   89E8   8461   7C62   8051   9AE3   7B6D   AE2A   6216       6   8461   7C62   8051   9AE3   7B6D   AE2A   6216   89E8       7   7C62   8051   9AE3   7B6D   AE2A   6216   89E8   8461                    
         [0118]    these four demodulation ROM are conbined to one by the method descrived in this example  
                                                                                                                                                               demodulation ROM2-1(for synchronization, p = 0 block of       ROM1 is arranged address incrementally)                p                0   1   2   3                r                1   2   1   2   1   2   1   2                j            q   1   2   3   4   5   6   7   8               0   9D62   8CD2   AC89   9C5E   7AA0   AF18   7BD9   85E0       1   8CD2   AC89   9C5E   7AA0   AF18   7BD9   85E0   9D62       2   AC89   9C5E   7AA0   AF18   7BD9   85E0   9D62   8CD2       3   9C5E   7AA0   AF18   7BD9   85E0   9D62   8CD2   AC89       4   7AA0   AF18   7BD9   85E0   9D62   8CD2   AC89   9C5E       5   AF18   7BD9   85E0   9D62   8CD2   AC89   9C5E   7AA0       6   7BD9   85E0   9D62   8CD2   AC89   9C5E   7AA0   AF18       7   85E0   9D62   8CD2   AC89   9C5E   7AA0   AF18   7BD9                  
 
         [0119]    [0119]                                                                                                                                                               demodulation ROM2-2(for synchronization)                p                0   1   2   3                r                1   2   1   2   1   2   1   2                j            q   1   2   3   4   5   6   7   8               0   97C8   97FB   9FE6   AA6B   6DA9   A655   7DB4   81E8       1   97FB   9FE6   AA6B   6DA9   A655   7DB4   81E8   97C8       2   9FE6   AA6B   6DA9   A655   7DB4   81E8   97C8   97FB       3   AA6B   6DA9   A655   7DB4   81E8   97C8   97FB   9FE6       4   6DA9   A655   7DB4   81E8   97C8   97FB   9FE6   AA6B       5   A655   7DB4   81E8   97C8   97FB   9FE6   AA6B   6DA9       6   7DB4   81E8   97C8   97FB   9FE6   AA6B   6DA9   A655       7   81E8   97C8   97FB   9FE6   AA6B   6DA9   A655   7DB4                    
         [0120]    [0120]                                                                                                                                                               demodulation ROM2-3(for synchronization)                p                0   1   2   3                r                1   2   1   2   1   2   1   2                j            q   1   2   3   4   5   6   7   8               0   8C8B   9D6B   8E22   B0DB   650B   9945   80A2   7EA6       1   9D6B   8E22   B0DB   650B   9945   80A2   7EA6   8C8B       2   8E22   B0DB   650B   9945   80A2   7EA6   8C8B   9D6B       3   B0DB   650B   9945   80A2   7EA6   8C8B   9D6B   8E22       4   650B   9945   80A2   7EA6   8C8B   9D6B   8E22   B0DB       5   9945   80A2   7EA6   8C8B   9D6B   8E22   B0DB   650B       6   80A2   7EA6   8C8B   9D6B   8E22   B0DB   650B   9945       7   7EA6   8C8B   9D6B   8E22   B0DB   650B   9945   80A2                    
         [0121]    to combine these four demodulation ROM of which data placed at timing proper over-sampling poditic  
                                                                                                                                                                                                                     demodulation ROM1(convined data of 4 ROMs)                p                0   1   2   3                r                1   2   1   2   1   2   1   2                j            q   r   i   1   2   3   4   5   6   7   8                    0   1   1   9D62   997B   B890   AF21   B887   AF2C   9D5F   997E           2   2   97C8   8F11   AB0D   95FE   AB0A   9606   97C7   8F13           3   3   8C8B   829F   8FED   77F6   8FF2   77F7   8C8D   829F           4   4   8051   794B   726B   6175   7274   616F   8055   7949       1   1   5   8CD2   960D   9094   A6DE   908D   A6DF   8CD0   960D           2   6   97FB   9CEC   AB83   B772   AB79   B77A   97F7   9CEF           3   7   9D6B   9C08   B8A5   B549   B89B   B554   9D68   9C0B           4   8   9AE3   93BE   B28B   A147   B286   A151   9AE2   93C1       2   1   9   AC89   A7C5   C013   C67C   BC7F   A548   A8F2   8674           2   10   9FE6   89B7   99AC   903D   84E5   7B3E   8B0C   74A5           3   11   8E22   6BC3   6F9D   588F   4E89   54B8   6CF0   67E9           4   12   7B6D   54FB   4BD1   2C95   263E   3ACF   55BA   6341       3   1   13   9C5E   C288   CE26   EE2C   F38D   DC95   C1E6   B0E2           2   14   AA6B   C9D6   DF89   FECC   FED7   DFA4   C9D5   AA92           3   15   B0DB   C21D   DCC6   F3E7   EE98   CE87   C2BD   9C9C           4   16   AE2A   AD31   C685   D00F   C6A7   AD48   AE4C   8A4B       4   1   17   7AA0   4A52   24CB   0B29   0A60   1643   3AAC   6533           2   18   6DA9   3CA4   14B6   0089   008B   14E7   3CD9   6DF2           3   19   650B   3AC7   1689   0AE5   0BB3   256C   4AD0   7AFF           4   20   6216   4505   29FE   28A9   2A23   454E   626F   8A5C       5   1   21   AFI8   BA97   C855   BF2F   BD43   A4C7   944D   7AAC           2   22   A655   9E73   9CC5   8A81   886D   765D   7516   6DB3           3   23   9945   7F37   6E5F   55C5   53DB   4AF7   5916   6510           4   24   89E8   61A6   4436   2907   2791   2935   4491   6216       6   1   25   7BD9   6FD3   63CC   6B74   7B6F   8F46   9374   939A           2   26   7DB4   776C   7737   8187   923E   A0B3   9CB0   968D           3   27   80A2   80A0   849C   9857   A85B   B02F   A455   9870           4   28   8461   8AA7   967B   ADF0   BBE0   BC66   A9B9   9918       7   1   29   85E0   73E0   5AAE   429D   3DB5   4D55   68F1   7E95           2   30   81E8   6D32   5198   3CC8   3CB9   516F   6D11   81D2           3   31   7EA6   6902   4D5F   3D9B   4266   5A69   73B1   85C6           4   32   7C62   67AB   4E60   4503   4E3E   677E   7C40   8A1B                  
 
         [0122]    the circuit block diagram which use demodulation ROM 1  conbined data in number of over-sampling represented in FIG. 5, and output the demodulation data which is accumulated and divided by α.  
                                                                                                                                                               demodulation ROM1-4(data of demodulation matrix4       exchanged to positive Hex. data)                p                0   1   2   3                r                1   2   1   2   1   2   1   2                j            q   1   2   3   4   5   6   7   8               0   8051   794B   726B   6175   7274   616F   8055   7949       1   9AE3   93BE   B28B   A147   B286   A151   9AE2   93C1       2   7B6D   54FB   4BD1   2C95   263E   3ACF   55BA   6341       3   AE2A   AD31   C685   D00F   C6A7   AD48   AE4C   8A4B       4   6216   4505   29FE   28A9   2A23   454E   626F   8A5C       5   89E8   61A6   4436   2907   2791   2935   4491   6216       6   8461   8AA7   967B   ADF0   BBE0   BC66   A9B9   9918       7   7C62   67AB   4E60   4503   4E3E   677E   7C40   8A1B                  
 
         [0123]    before the multiplication, data stored in ROM is exchanged to the number indicating positive or negat  
         [0124]    also the modulation data is exchanged to the number having positive signe.  
         [0125]    where Di is 8 bit modulation data, equation of exchange is  
         2 ×Di −255  
         [0126]    when the result of calculation prosess is output, the data is done by inverse exchange.  
                                                                                                                                                               demodulation ROM1-1(data of demodulation matrix1       exchanged to positive Hex. data)                p                0   1   2   3                r                1   2   1   2   1   2   1   2                j            q   1   2   3   4   5   6   7   8               0   9D62   997B   B890   AF21   B887   AF2C   9D5F   997E       1   8CD2   960D   9094   A6DE   908D   A6DF   8CD0   960D       2   AC89   A7C5   C013   C67C   BC7F   A548   A8F2   8674       3   9C5E   C288   CE26   EE2C   F38D   DC95   C1E6   B0E2       4   7AA0   4A52   24CB   0B29   0A60   1643   3AAC   6533       5   AF18   BA97   C855   BF2F   BD43   A4C7   944D   7AAC       6   7BD9   6FD3   63CC   6B74   7B6F   8F46   9374   939A       7   85E0   73E0   5AAE   429D   3DB5   4D55   68F1   7E95                  
 
         [0127]    [0127]                                                                                                                                                               demodulation ROM1-2(data of demodulation matrix2       exchanged to positive Hex. data)                p                0   1   2   3                r                1   2   1   2   1   2   1   2                j            q   1   2   3   4   5   6   7   8               0   97C8   8F11   AB0D   95FE   AB0A   9606   97C7    8F13       1   97FB   9CEC   AB83   B772   AB79   B77A   97F7   9CEF       2   9FE6   89B7   99AC   903D   84E5   7B3E   8B0C   74A5       3   AA6B   C9D6   DF89   FECC   FED7   DFA4   C9D5   AA92       4   6DA9   3CA4   14B6   0089   008B   14E7   3CD9   6DF2       5   A655   9E73   9CC5   8A81   886D   765D   7516   6DB3       6   7DB4   776C   7737   8187   923E   A0B3   9CB0   968D       7   81E8   6D32   5198   3CC8   3CB9   516F   6D11   81D2                    
         [0128]    [0128]                                                                                                                                                               demodulation ROM1-3(data of demodulation matrix3       exchanged to positive Hex. data)                p                0   1   2   3                r                1   2   1   2   1   2   1   2                j            q   1   2   3   4   5   6   7   8               0   8C8B   829F   8FED   77F6   8FF2   77F7   8C8D   829F       1   9D6B   9C08   B8A5   B549   B89B   B554   9D68   9C0B       2   8E22   6BC3   6F9D   588F   4E89   54B8   6CF0   67E9       3   B0DB   C21D   DCC6   F3E7   EE98   CE87   C2BD   9C9C       4   650B   3AC7   1689   0AE5   0BB3   256C   4AD0   7AFF       5   9945   7F37   6E5F   55C5   53DB   4AF7   5916   6510       6   80A2   80A0   849C   9857   A85B   B02F   A455   9870       7   7EA6   6902   4D5F   3D9B   4266   5A69   73B1   85C6                    
         [0129]    [0129]                                                                                                                                                                                                                     stored data in modulation ROM       (data of modulation matrix exchanged to positive Hex. data)                p                0   1   2   3                r                1   2   1   2   1   2   1   2                j            q   r   i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
         [0130]    and above data are delayed one over-sampling and demodulated using the demodulation matrix 1˜4 the mean value of demodulation data are  
                                                                                 q   r   j   x j                                          0   1   1   3.9847               2   2   7.2244           1   1   3   −2.1027               2   4   −10.1946           2   1   5   1.5695               2   6   11.9217           3   1   7   −1.3175               2   8   −13.1774                      
 
         [0131]    the out put adjusted data by which is mentioned adjustment circuit in this paper using above adjustment parameter  
                                                                                 q   r   j   x j                                          0   1   1   1.0945               2   2   14.9236           1   1   3   −0.679               2   4   −14.8599           2   1   5   1.0464               2   6   15.1295           3   1   7   −1.0984               2   8   −15.1392                      
 
         [0132]    above data is rounded as below  
                                                                                 q   r   j   x j                                          0   1   1   1               2   2   15           1   1   3   −1               2   4   −15           2   1   5   1               2   6   15           3   1   7   −1               2   8   −15                      
 
         [0133]    this demodulation data is same as the modulation data of transmission side.  
         [0134]    above data are changed a little by the noise of line to demodulation circuit  
                                                                 i   d i                                          1   3.4752           2   3.8076           3   0.2641           4   −6.2598           5   −13.2246           6   −18.1845           7   −17.6699           8   −12.0553           9   −3.1223           10   6.2118           11   11.5807           12   11.6888           13   5.5937           14   −2.9461           15   −10.6135           16   −12.4051           17   −6.4104           18   5.6919           19   21.8561           20   34.7286           21   40.1087           22   35.2151           23   19.0815           24   −3.6546           25   −26.993           26   −44.5621           27   −52.648           28   −49.4096           29   −36.3798           30   −18.107           31   0.7936           32   15.1676                      
 
         [0135]    modulation data are as x 1 =x 5 =x 6 =15, x 3 =x 7 =−1, x 4 =x 8 =−15 in practical communication, modulaed data are  
                                                                 i   d i                                          1   3.8853           2   0.4624           3   −6.1088           4   −13.2972           5   −17.9469           6   −17.698           7   −12.1194           8   −3.0384           9   6.1155           10   11.7103           11   11.4849           12   5.6252           13   −3.1445           14   −10.5474           15   −12.4635           16   −6.6923           17   5.9496           18   21.6735           19   34.9428           20   40.4428           21   35.062           22   19.1293           23   −3.5679           24   −26.9999           25   −44.8891           26   −52.7427           27   −49.1863           28   −36.1509           29   −17.9151           30   0.5586           31   15.3163           32   3.3394                      
 
         [0136]    and above data are delayed one over-sampling and demodulated using the demodulation matrix 1˜4 the mean value of demodulation data are  
                                                                                 q   r   j   x j                                          0   1   1   10.9686               2   2   4.0607           1   1   3   12.015               2   4   8.7172           2   1   5   12.6055               2   6   11.1354           3   1   7   13.4383               2   8   12.7259                      
 
         [0137]    the parameters of adjustment data are determined as  
                                                       p   {overscore (D)} 2p+1   2 (test) + {overscore (D)} 2p+2   2 (test)                       0   136.8002           1   220.3448           2   282.8956           3   342.5367                       p   {overscore (D)} 2p+1 (test) + {overscore (D)} 2p+2 (test)                       0   15.0293           1   20.7322           2   23.7409           3   26.1642                       p   {overscore (D)} 2p+1 (test) − {overscore (D)} 2p+2 (test)                       0   6.908           1   3.2977           2   1.4702           3   0.7124                      
 
         [0138]    above data are changed a little by the noise of line to demodulation circuit  
                                                                 i   d i                                          1   79.7745           2   81.1090           3   65.0050           4   37.0460           5   5.6597           6   −21.4443           7   −37.4286           8   −41.5717           9   −36.3611           10   −26.2757           11   −17.1823           12   −11.3325           13   −10.1673           14   −10.4529           15   −9.9276           16   −5.3161           17   3.3021           18   12.8503           19   21.3957           20   23.7096           21   19.4529           22   10.3367           23   −1.1009           24   −9.2484           25   −10.4869           26   −3.8698           27   7.3148           28   18.5145           29   24.5774           30   21.4752           31   9.2918           32   −9.5909                      
 
         [0139]    modulated data for DA converter input  
                                                                 i   d i                                          1   81.1867           2   65.2033           3   37.1970           4   5.5872           5   −21.2066           6   −37.4566           7   −41.6358           8   −36.2772           9   −26.3720           10   −17.0527           11   −11.5364           12   −10.1358           13   −10.6513           14   −9.8615           15   −5.3745           16   3.0202           17   13.1080           18   21.2131           19   23.9238           20   19.7870           21   10.1836           22   −1.0532           23   −9.1617           24   −10.4938           25   −4.1968           26   7.2201           27   18.7377           28   24.8063           29   21.6671           30   9.0569           31   −9.4422           32   79.6386                      
 
         [0140]    the frequency of sub-carriers is determined as below  
         [0141]    r=1 α=4 ρf 0 =2.5×10 and according to below equation  
           ∑     q   =   0       n   -   1                         cos   2            2      π                   f   ρ         ρ   ×   α   ×     f   0             =       ∑     q   =   0       n   -   1                         sin   2            2      π                   f   ρ         ρ   ×   α   ×     f   0                                   
 
         [0142]    ƒ 0 =1.0423 MHz  
         [0143]    ƒ 1 =0.7809 MHz  
         [0144]    ƒ 2 =0.6255 MHz  
         [0145]    ƒ 3 =0.4684 MHz  
         [0146]    therefore ρ=2.399 is got.  
         [0147]    if the bit wide of the modulation data is only one bit, the trancemission speed is  
         2.399   ×   4   ×   1.0423   ×     1   4       =     2.5                 Mbps                           
 
         [0148]    elements data of modulation matris and demodulation matrix 1˜4 should be changed to positive hex number to store in ROM, and the example method about cos θ is descrived as following  
             cos                 θ     +   1     2     ×   65535                             data  of  demodulati  on  matrix     +               (absolute  maximum  negative                     data  of  demodulati  on  matrix)                     (absolute  maximum  negative               data  of  demodulati  on  matrix)           ×   2       ×   65535                         
 
         [0149]    by above equation, cos θ values of modulation matrix and of demodulation matris 1˜4 are changed to positive dezimal number and are changed to hex number and are stored in ROM.  
         [0150]    each address of ROMS are i 1 and q and the number of port is j and 16 bits wide in this example.  
         [0151]    at first in test communication, modulation is done as all modulation data are 15(F)  
         x 1 =x 2 =x 3 = . . . x 8 =15  
         [0152]    [0152]                                                                                                                                                                                                     demodulation matrix 2                p                0   1   2   3                r                1   2   1   2   1   2   1   2                j            q   1   2   3   4   5   6   7   8                    0   0.3351   0.1206   0.8091   0.2909   0.8088   0.2916   0.3351   0.1209       1   0.3403   0.4618   0.8213   1.1147   0.8203   1.1154   0.3400   0.4621       2   0.5350   −0.0107   0.3819   0.1500   −0.1295   −0.3669   0.0219   −0.5293       3   0.7938   1.5670   2.1010   2.8705   2.8715   2.1036   1.5669   0.7976       4   −0.7012   −1.9079   −2.8909   −3.3878   −3.3876   −2.8863   −1.9030   −0.6944       5   0.6936   0.5000   0.4588   0.0094   −0.0421   −0.4867   −0.5184   −0.7003       6   −0.3065   −0.4610   −0.5643   −0.2119   0.1997   0.5554   0.4565   0.3052       7   −0.2032   −0.7132   −1.3929   −1.9051   −1.9064   −1.3964   −0.7161   −0.2052                    
         [0153]    [0153]                                                                                                                                                                                                     demodulation matrix 3                p                0   1   2   3                r                1   2   1   2   1   2   1   2                j            q   1   2   3   4   5   6   7   8                    0   0.0585   −0.1856   0.1414   −0.4483   0.1418   −0.4482   0.0586   −0.1856       1   0.4740   0.4398   1.1442   1.0612   1.1432   1.0622   0.4737   0.4401       2   0.0979   −0.7478   −0.6531   −1.2202   −1.4673   −1.3148   −0.7190   −0.8427       3   0.9522   1.3771   2.0331   2.6026   2.4718   1.6826   1.3924   0.4541       4   −0.9133   −1.9539   −2.8462   −3.1332   −3.1132   −2.4799   −1.5594   −0.3733       5   0.3722   −0.2687   −0.6832   −1.2887   −1.3362   −1.5549   −1.2076   −0.9129       6   −0.2344   −0.2343   −0.1359   0.3499   0.7442   0.9366   0.6446   0.3516       7   −0.2834   −0.8163   −1.4967   −1.8847   −1.7665   −1.1753   −0.5529   −0.1078                    
         [0154]    [0154]                                                                                                                                                                                                     demodulation matrix4                p                0   1   2   3                r                1   2   1   2   1   2   1   2                j            q   1   2   3   4   5   6   7   8                    0   −0.2423   −0.4151   −0.5848   −1.0019   −0.5839   −1.0024   −0.2420   −0.4153       1   0.4116   0.2357   0.9936   0.5686   0.9931   0.5695   0.4114   0.2360       2   −0.3624   −1.3085   −1.5340   −2.3026   −2.4589   −1.9525   −1.2902   −0.9572       3   0.8860   0.8623   1.4855   1.7206   1.4888   0.8646   0.8893   0.0034       4   −0.9862   −1.7021   −2.3676   −2.4008   −2.3642   −4.6954   −0.9780   0.0047       5   −0.0059   −0.9965   −1.7211   −2.3903   −2.4266   −2.3861   −1.7128   −0.9862       6   −0.1421   0.0126   0.3041   0.8816   1.2246   1.2372   0.7773   0.3678       7   −0.3392   −0.8491   −1.4717   −1.7021   −1.4747   −0.8531   −0.3421   −0.0012                    
         [0155]    method of demodulation matrix creation  
         [0156]    for example, matrix of which size is 8 columns is created by picking the line number r=1 
                                                                                                                                                                                                           p                0   1   2   3                r                1   2   1   2   1   2   1   2                j            q   1   2   3   4   5   6   7   8                    0   0.7931   0.6091   0.8820   0.4712   0.9238   0.3830   0.9570   0.2901       1   −0.9912   −0.1326   −0.7723   0.6353   −0.3841   0.9233   0.0990   0.9951       2   0.9253   −0.3791   −0.2922   −0.9564   −0.9228   −0.3852   −0.8811   0.4730       3   −0.6132   0.7900   0.9955   0.0951   0.3863   −0.9224   −0.7747   −0.6324       4   0.1377   −0.9905   −0.4680   0.8837   0.9219   0.3874   0.2870   −0.9579       5   0.3744   0.9273   −0.6380   −0.7700   −0.3885   0.9214   0.9948   −0.1022       6   −0.7868   −0.6172   0.9553   −0.2956   −0.9210   −0.3896   0.4758   0.8796       7   0.9898   0.1428   −0.0916   0.9958   0.3907   −0.9205   −0.6299   0.7767                  
 
         [0157]    the demodulation matrix is inverse matrix of the above matrix  
                                                                                                                                                                                                     demodulation matrix 1                p                0   1   2   3                r                1   2   1   2   1   2   1   2                j            q   1   2   3   4   5   6   7   8                    0   0.4731   0.3770   1.1419   0.9096   1.1411   0.9106   0.4728   0.3773       1   0.0658   0.2928   0.1586   0.7069   0.1580   0.7070   0.0656   0.2929       2   0.8459   0.7289   1.3268   1.4849   1.2389   0.6677   0.7577   −0.0911       3   0.4480   1.3872   1.6732   2.4612   2.5938   2.0283   1.3717   0.9529       4   −0.3821   −1.5709   −2.4948   −3.1259   −3.1454   −2.8527   −1.9564   −0.9097       5   0.9092   1.1926   1.5310   1.3061   1.2585   0.6558   0.2499   −0.3810       6   −0.3523   −0.6481   −0.9441   −0.7554   −0.3619   0.1264   0.2291   0.2326       7   −0.1056   −0.5488   −1.1693   −1.7617   −1.8824   −1.4975   −0.8178   −0.2849                  
 
         [0158]    [0158]                                                                                                                                                                                                                     modulation matrix                p                0   1   2   3                r                1   2   1   2   1   2   1   2                j            q   r   i   1   2   3   4   5   6   7   8                    0   1   1   0.7931   0.6091   0.8820   0.4712   0.9238   0.3830   0.9570   0.2901           2   2   0.2580   0.9661   0.5559   0.8312   0.7067   0.7075   0.8317   0.5552           3   3   −0.3839   0.9234   0.0987   0.9951   0.3818   0.9242   0.6349   0.7726           4   4   −0.8669   0.4985   −0.3819   0.9242   −0.0012   1.0000   0.3834   0.9236       1   1   5   −0.9912   −0.1326   −0.7723   0.6353   −0.3841   0.9233   0.0990   0.9951           2   6   −0.7053   −0.7089   −0.9805   0.1964   −0.7084   0.7058   −0.1939   0.9810           3   7   −0.1276   −0.9918   −0.9574   −0.2888   −0.9247   0.3807   −0.4702   0.8826           4   8   0.5030   −0.8643   −0.7084   −0.7058   −1.0000   −0.0024   −0.7060   0.7082       2   1   9   0.9253   −0.3791   −0.2992   −0.9564   −0.9228   −0.3852   −0.8811   0.4730           2   10   0.9648   0.2629   0.1929   −0.9812   −0.7050   −0.7092   −0.9804   0.1970           3   11   0.6050   0.7962   0.6325   −0.7746   −0.3796   −0.9251   −0.9954   −0.0958           4   12   −0.0051   1.0000   0.9228   −0.3852   0.0036   −1.0000   −0.9248   −0.3805       3   1   13   −0.6132   0.7900   0.9955   0.0951   0.3863   −0.9224   −0.7747   −0.6324           2   14   −0.9675   0.2530   0.8332   0.5530   0.7101   −0.7041   −0.5579   −0.8299           3   15   −0.9214   −0.3886   0.4744   0.8803   0.9256   −0.3785   −0.2931   −0.9561           4   16   −0.4941   −0.8694   0.0036   1.0000   1.0000   0.0048   −0.0032   −1.0000       4   1   17   0.1377   −0.9905   −0.4680   0.8837   0.9219   0.3874   0.2870   −0.9579           2   18   0.7125   −0.7017   −0.8292   0.5589   0.7033   0.7109   0.5526   −0.8335           3   19   0.9925   −0.1225   −0.9948   0.1022   0.3774   0.9261   0.7706   −0.6373           4   20   0.8617   0.5074   −0.9256   −0.3785   −0.0060   1.0000   0.9223   −0.3864       5   1   21   0.3744   0.9273   −0.6380   −0.7700   −0.3885   0.9214   0.9948   −0.1022           2   22   −0.2679   0.9635   −0.1999   −0.9798   −0.7118   0.7024   0.9816   0.1908           3   23   −0.7993   0.6009   0.2854   −0.9584   −0.9265   0.3763   0.8841   0.4673           4   24   −0.9999   −0.0102   0.7033   −0.7109   −1.0000   −0.0072   0.7105   0.7037       6   1   25   −0.7868   −0.6172   0.9553   −0.2956   −0.9210   −0.3896   0.4758   0.8796           2   26   −0.2481   −0.9687   0.9819   0.1894   −0.7016   −0.7126   0.2002   0.9798           3   27   0.3933   −0.9194   0.7768   0.6297   −0.3752   −0.9270   −0.0927   0.9957           4   28   0.8719   −0.4896   0.3885   0.9215   0.0084   −1.0000   −0.3775   0.9260       7   1   29   0.9898   0.1428   −0.0916   0.9958   0.3907   −0.9205   −0.6299   0.7767           2   30   0.6980   0.7161   −0.5500   0.8352   0.7135   −0.7007   −0.8281   0.5605           3   31   0.1174   0.9931   −0.8786   0.4775   0.9274   −0.3740   −0.9551   0.2962           4   32   −0.5118   0.8591   −1.0000   0.0072   1.0000   0.0096   −1.0000   0.0064                    
       EXAMPLE  
       [0159]    [Effect of Invention] 
         [0160]    The effect of this invention applied to DSL of metal twist pair is described below. The parameter of the modulation and demodulation system is differ from example and determined below.  
                                                       Number of carrier frequency   n = 16           Number of over-sampling   α = 8           Bit wide of modulation data   A = 8                                          Numbers of basic sampling in one wave form   ρ           Most high frequency of sub-carrier   ƒ 0                        
 
         [0161]    defined as ρƒ 0 =12.5 MHz. Sampling frequency CLK of DA and AD converter is  
         CLK=ρ×α×ƒ 0 =12.5×8=100 MHz  
         [0162]    Transmission speed is  
         CLK   ×     A   α       =       100   ×     8   8       =     100                 Mbps                             
 
         [0163]    About frequency of sub-carrier concern in the frequency range of 6.0 MHz˜0.09 MHz  
           ∑     q   =   0     31            cos   2            2                 π                   f   p         8   ×   12.5            (       8      q     +   1     )         ≅       ∑     q   =   0     31            sin   2            2                 π                   f   p         8   ×   12.5            (       8      q     +   1     )                               
 
         [0164]    The frequency is determined as  
               f   0     =     0.0950                 MHz               f   1     =     0.473                 MHz               f   2     =     0.852                 MHz               f   3     =     1.231                 MHz                   f   4     =     1.610                 MHz               f   5     =     1.989                 MHz               f   6     =     2.368                 MHz               f   7     =     2.747                 MHz                   f   8     =     3.314                 MHz               f   9     =     3.693                 MHz               f   10     =     4.072                 MHz               f   11     =     4.451                 MHz                   f   12     =     4.830                 MHz               f   13     =     5.208                 MHz               f   14     =     5.588                 MHz               f   15     =     5.966                 MHz                                 
 
       SIMPLE PROVE OF FIG.  
       [0165]    [FIG. 1] Modulation and demodulation total system block diagram.  
         [0166]    [FIG. 2] Block diagram of modulation.  
         [0167]    [FIG. 3] Block diagram of demodulation.  
         [0168]    [FIG. 4] Block diagram of phase and magnitude adjustment.  
         [0169]    [FIG. 5] Block diagram of distributed time phase and magnitude adjustment.  
         [0170]    [FIG. 6] Block diagram of demodulation for synchronization.  
         [0171]    [FIG. 7] Block diagram of adjustment signal.  
         [0172]    [FIG. 8] Block diagram of adjustment signal.  
         [0173]    [FIG. 9] Block diagram of adjustment signal.  
       SIMPLE PROVE OF SYMBOL  
       [0174]    DFF data flip-flop.  
         [0175]    CLK clock.  
         [0176]    ROM read only memory.  
         [0177]    TFF toggle flip-flop.