Abstract:
A noise elimination method is characterized in that when transmitting signals in the same direction on at least two distributed constant lines, a resistance of a terminating resistor at a far-end is set so that voltages propagated to the far-end become equal between two kinds of propagation modes on coupled distributed constant lines. The two kinds of propagation modes are a common mode which propagates with respect to a ground plane and a differential mode which propagates between the coupled lines.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention generally relates to noise elimination methods and transmission circuits for eliminating a crosstalk noise which is generated among a plurality of signal lines, and more particularly to a noise elimination method and a transmission circuit which eliminate a far-end crosstalk noise of a bus transmission by inserting a terminating resistor which has a specific value at a far-end. 
     2. Description of the Related Art 
     In electronic equipments such as personal computers, signal transmissions among LSI circuits in most cases are made in units of 32 bits or 64 bits. In such signals, a plurality of bits make transitions at the same timing, thereby causing signal interference among the bits and in many cases generating the crosstalk noise. The value of this crosstalk noise becomes larger as the number of signals which make the transitions simultaneously becomes larger. In addition, the crosstalk noise becomes a large value even in the case of a short line as the signal rise/fall time becomes shorter. 
     FIGS. 1A and 1B are diagrams for explaining a backward near-end crosstalk for explaining a background of the present invention. FIG. 1A shows a driving line  80 , a driver  81 , a receiver  82 , a passive line  90 , a driver  91 , and a  35  receiver  92 . FIG. 1B additionally shows an internal resistance  83  of the driver  81 , a terminating resistor  84 , an internal resistance  93  of the driver  91 , and a terminating resistor  94  for the case shown in FIG.  1 A. 
     In a case where two lines on which the signal transmitting directions are opposite to each other as shown in FIG. 1A, the backward near-end crosstalk refers to the noise which is introduced on the passive line  90  near the driver  81  due to the signal on the driving line  80 . 
     FIG. 2 is a diagram showing the magnitude of the backward near-end crosstalk which is introduced in the transmission circuit shown in FIGS. 1A and 1B. In FIG. 2, it is assumed that an internal resistance  83  of the driver  81  has a value r=10Ω, and a resistance R of the terminating resistor  84  is infinitely large. In FIG. 2, the ordinate indicates the magnitude of the voltage, and the abscissa indicates the time. In FIG. 2. a thin solid line v 1 (near) indicates a voltage change on the driving line  80  on the side of the driver  81  (near-end), a thin dotted line v 1 (far) indicates a voltage change on the driving line  80  on the side of the receiver  82  (far-end), a bold solid line v 2 (near) indicates a voltage change on the passive line  90  on the side of the driver  91  (near-end), and a bold dotted line v 2 (far) indicates a voltage change on the passive line  90  on the side of the receiver  92  (far-end). 
     The backward near-end crosstalk becomes a considerably large value when the value r of the internal resistance  83  of the driver  81  is sufficiently small compared to the characteristic impedance of the passive line  90 . For this reason, the value r of the internal resistance  83  is conventionally set large so as to eliminate the backward near-end crosstalk noise. 
     The terminating resistor  94  is provided to make a waveform matching with respect to the output signal, and the resistance of this terminating resistor  94  is set to a value approximately equal to the characteristic impedance of the passive line  90 . 
     In other words, in a case where the characteristic impedance of the line is 50Ω, the terminating resistor  94  is set to approximately 50Ω. 
     Conventionally, when signals are transmitted on a plurality of lines in the same direction, no measures were taken with respect to the noise generated at the far-end on the opposite end from the driving side (hereinafter referred to as a forward far-end crosstalk noise) because the amplitude (voltage) of the forward far-end crosstalk noise is small compared to the backward near-end crosstalk noise and the effects of the forward far-end crosstalk noise with respect to the transmission line are small. 
     Although no measures are conventionally take with respect to the forward far-end crosstalk noise, there is a tendency for the physical distance among the signals to become smaller, due to the increased operation speed of the circuits and the reduced size and weight of the equipments. As a result, there is a tendency for the crosstalk noise to be generated more easily. More particularly, when making a parallel signal transmission of multiple bits such as 32 bits or 64 bits, there exists a case where the logic amplitude changes from a “0” state to a “1” state in all of the bits with the exception of one bit, and in such a case, the effects of the lines on which the logic amplitude of the bits which changed to the “1” state appear at the far-end of the signal line on which the logic amplitude of the bit remained at the “0” state. In some cases, such effects appearing at the far-end become large and no longer negligible. In order to simultaneously achieve the increased operation speed and reduced size and weight of the equipment, it is an object to overcome this crosstalk noise from the point of view of electronic packaging. 
     But conventionally, in order to reduce the crosstalk noise described above, it was either necessary to increase the physical distance among the signals or to reduce the number of signals which make the transition simultaneously. For this reason, it was either necessary to sacrifice the packaging or mounting density or to sacrifice the performance by relaxing the signal timings. 
     SUMMARY OF THE INVENTION 
     Accordingly, it is a general object of the present invention to provide a novel and useful noise elimination method and transmission circuit, in which the problems described above are eliminated. 
     Another and more specific object of the present invention is to provide a noise elimination method and a transmission circuit which can eliminate a far-end crosstalk of a bus transmission when transmitting signals in the same direction, by a simple means. 
     Still another object of the present invention is to provide a noise elimination method characterized in that when transmitting signals in the same direction on at least two distributed constant lines, a resistance of a terminating resistor at a far-end is set so that voltages propagated to the far-end become equal between two kinds of propagation modes on coupled distributed constant lines, where the two kinds of propagation modes are a common mode which propagates with respect to a ground plane and a differential mode which propagates between the coupled lines. According to the present invention, it is possible to effectively eliminate the forward far-end crosstalk noise by use of a simple construction. 
     A further object of the present invention is to provide a noise elimination method characterized in that when first and second driving sources are coupled to respective ends of at least two distributed constant lines on which signals can be transmitted two ways, and a signal is to be transmitted from the first driving source to the other end or from the second driving source to the other end, a resistance of a terminating resistor is set so that an approximately reciprocal relationship exists between an internal resistance of the first or second driving source normalized by a characteristic impedance of the line, and a terminating resistance at a far-end with respect to the first or second driving source normalized by the characteristic impedance of the line. According to the present invention, it is possible to effectively eliminate the forward far-end crosstalk noise by use of a simple construction. 
     Another object of the present invention is to provide a transmission circuit having at least two distributed constant lines for transmitting signals in the same direction, characterized in that a terminating resistor is coupled at a far-end of the distributed constant lines, and the terminating resistor has a terminating resistance which is set so that an approximately reciprocal relationship exists between the terminating resistance which is normalized by a characteristic impedance of the line and an internal resistance of a driving source which is normalized by the characteristic impedance of the line. According to the present invention, it is possible to effectively eliminate the forward far-end crosstalk noise by use of a simple construction. 
     Still another object of the present invention is to provide a transmission circuit having at least two distributed constant lines for transmitting signals two ways, and driving sources of the signals on both end of the lines, characterized in that a terminating resistor is coupled to a far-end of the distributed constant lines with respect to each driving source, and the terminating resistor has a terminating resistance which is set so that an approximately reciprocal relationship exists between the terminating resistance which is normalized by a characteristic impedance of the line and an internal resistance of the driving source which is normalized by the characteristic impedance of the line. According to the present invention, it is possible to effectively eliminate the forward far-end crosstalk noise by use of a simple construction. 
     A further object of the present invention is to provide a transmission circuit coupled to at least two distributed constant lines for transmitting signals in the same direction, characterized by a terminating resistor coupled to a far-end of the distributed constant lines to reduce a far-end crosstalk noise. According to the present invention, it is possible to effectively eliminate the forward far-end crosstalk noise by use of a simple construction. 
     Another object of the present invention is to provide a transmission circuit coupled to at least two distributed constant lines for transmitting signals in the same direction, characterized by a terminating resistor having a resistance which makes voltages propagated on the distributed constant lines equal between a common mode and a differential mode. According to the present invention, it is possible to effectively eliminate the forward far-end crosstalk noise by use of a simple construction. 
     Other objects and further features of the present invention will be apparent from the following detailed description when read in conjunction with the accompanying drawings. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIGS. 1A and 1B respectively are diagrams for explaining a backward near-end crosstalk for explaining a background of the present invention; 
     FIG. 2 is a diagram showing the magnitude of a backward near-end crosstalk generated in the transmission circuit shown in FIGS.  1 A and  1 B: 
     FIG. 3 is a diagram showing a coupling distributed constant line for explaining the operating principle of the present invention; 
     FIG. 4 is a diagram showing signal changes caused by a terminating resistor at a far-end in a common mode and a differential mode; 
     FIGS. 5A and 5B respectively are diagrams for explaining elimination of a forward far-end crosstalk noise in a first embodiment of the present invention; 
     FIG. 6 is a diagram for explaining the elimination of the forward far-end crosstalk noise in a second embodiment of the present invention; 
     FIGS. 7A through 7E respectively are diagrams showing embodiments of the terminating resistor; 
     FIGS. 8A and 8B respectively are diagrams showing embodiments of the terminating resistor; 
     FIG. 9 is a diagram showing an analyzed result of the forward far-end crosstalk; 
     FIG. 10 is a diagram showing an analyzed result of the forward far-end crosstalk; 
     FIG. 11 is a diagram showing an analyzed result of the forward far-end crosstalk; 
     FIG. 12 is a diagram showing an analyzed result of the forward far-end crosstalk; 
     FIG. 13 is a diagram showing an analyzed result of the forward far-end crosstalk; 
     FIG. 14 is a diagram showing timings of the forward far-end crosstalk; 
     FIG. 15 is a diagram a change in the absolute value of the forward far-end crosstalk with respect to a drivability of a driver; 
     FIG. 16 is a diagram a change in the absolute value of the forward far-end crosstalk with respect to a drivability of a driver; 
     FIG. 17 is a diagram a change in the absolute value of the forward far-end crosstalk with respect to a drivability of a driver; 
     FIGS. 18A and 18B respectively are diagrams for explaining a simulation of the forward far-end crosstalk which is generated; 
     FIGS. 19A and 19B respectively are diagrams for explaining a simulation of the forward far-end crosstalk which is generated; 
     FIG. 20 is a diagram showing the relationship of the magnitude of a resistance R when a terminating resistor is changed with respect to an optimum value and a forward far-end crosstalk reduction; 
     FIG. 21 is a diagram showing the relationship of the magnitude of the resistance R when the terminating resistor is changed with respect to the optimum value and the forward far-end crosstalk reduction; 
     FIG. 22 is a diagram showing the relationship of the magnitude of the resistance R when the terminating resistor is changed with respect to the optimum value and the forward far-end crosstalk reduction; and 
     FIG. 23 is a perspective view showing a transmission circuit provided on an IC chip. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     First, a description will be given of the operating principle of the present invention. 
     It is both extremely difficult and complicated to solve the crosstalk noise by mathematical formulas. However, the present inventor positively used mathematical formulas and found that the crosstalk noise generated at a far-end can be made zero theoretically, by selecting a terminating resistance at the far-end to an optimum value. 
     Two coupling distributed constant lines shown in FIG. 3 are considered, and signal propagations on lines  1  and  2  are solved by Laplace transform based on basic formulas of transmission. Inductances and capacitances of the lines  1  and  2  are taken into consideration as parameters between the lines  1  and  2 . The inductances and the capacitances of the lines  1  and  2  themselves will be referred to as self inductances and self capacitances, and are respectively denoted by L 11  and C 11  with respect to the line  1  and by L 22  and C 22  with respect to the line  2 . In addition, the inductance and the capacitance between the lines  1  and  2  will be referred to as a mutual inductance and a mutual capacitance, and are respectively denoted by L 12  and C 12 . 
     In this case, the basic formulas of the transmission can be described by the following formulas (1) by taking L and C as matrixes, where v 1  denotes a voltage propagating on the line  1 , and v 2  denotes a voltage propagating on the line  2 .                  -     ∂     ∂   x                         (           v   1               v   2           )       =           ∂     ∂   t                       (           L   11           L   12               L   21           L   22           )                     (           i   1               i   2           )       -       ∂     ∂   x                       (           i   1               i   2           )         =       ∂     ∂   t                       (           C   11           C   12               C   21           C   22           )                     (           v   1               v   2           )                 (   1   )                                
     The following formula (2) is obtained by subjecting the above formulas (1) to a Laplace transform so as to describe the formula (2) solely in terms of V.                         2            x   2              (           V   1               V   2           )       -         s   2          (           L   11           L   12               L   21           L   22           )            (           C   11           C   12               C   21           C   22           )                     (           V   1               V   2           )         =   0           (   2   )                                
     In order to simplify matters, it will be assumed for the sake of convenience that the lines  1  and  2  shown in FIG. 3 have the same characteristic, and that the following relationships stand. 
     
       
         
           L 
           11 
           =L 
           22 
           =L 
         
       
     
     
       
         
           L 
           12 
           =L 
           21 
           =L 
           m 
         
       
     
     
       
         
           C 
           11 
           =C 
           22 
           =C 
         
       
     
     
       
         
           C 
           12 
           =C 
           21 
           =C 
           m 
         
       
     
     In this case, a coefficient matrix of the second term in the formula (2) can be described by the following formula (3).                  (           L   11           L   12               L   21           L   22           )                     (           C   11           C   12               C   21           C   22           )       =         (         L         L   m               L   m         L         )                     (         C         C   m               C   m         C         )       =       (           LC   +       L   m          C   m                 LC   m     +       L   m        C                   LC   m     +       L   m        C             LC   +       L   m          C   m               )     =       1     u   2            (         1       ξ           ξ       1         )                   (   3   )                                
     Here, the following formula (4) stands.                         ξ   =       (         L   m     L     +       C   m     C       )     /     (     1   +         L   m          C   m       LC       )                     u   2     =     1     LC   +       L   m          C   m                         (   4   )                                  
     The following formula (5) is obtained by eliminating V 2 .                         V   1               4              x   4         -     2                     (     s   u     )     2                            V   1               2              x   2           +         (     s   u     )     4          (     1   -     ξ   2       )                     V   1         =   0           (   5   )                                
     The following formula (6) can be obtained by describing the coefficient as a function of D.                φ                   (   D   )       =       D   4     -     2                     (     s   u     )     2          D   2       +         (     s   u     )     4          (     1   -     ξ   2       )                 (   6   )                                
     The root of φ=(D) can be described by the following formula (7).              D   =       ±     s   u              1   ±   ξ                 (composite  arbitrary)(7)                                
     When the following formula (8) is substituted into the formula (7), the following formula ( 9 ) can be obtained, where the suffixes “C” and “D” respectively indicate a common (or also called even) mode and a differential (or also called odd) mode, u C  and u D  denote propagation velocities in the respective modes.                  u   C     =     1   /         (     L   +     L   m       )          (     C   +     C   m       )             ,       u   D     =     1   /         (     L   -     L   m       )          (     C   -     C   m       )                     (   8   )                 D   =     ±     s     u   C           ,     ±     s     u   D                 (   9   )                                
     Impedances Z C  and Z D  also exist in the common mode and the differential mode, respectively, with respect to the specific impedance as indicated by the following formulas (10).                  Z   C     =         L   +     L   m         C   +     C   m             ,       Z   D     =         L   -     L   m         C   -     C   m                     (   10   )                                
     When the currents and voltages are obtained, the following formulas (11) are obtained.                  V   1          (   s   )       =           A   1          (   s   )                   -     x     u   C            s         +         A   2          (   s   )                   x     u   C          s         +         A   3          (   s   )                   -     x     u   D            s         +         A   4          (   s   )                   x     u   D          s                   (   11   )                   V   2          (   s   )       =           A   1          (   s   )                   -     x     u   C            s         +         A   2          (   s   )                   x     u   C          s         -         A   3          (   s   )                   -     x     u   D            s         -         A   4          (   s   )                   x     u   D          s                                       I   1          (   s   )       =             A   1          (   s   )         Z   C                   -     x     u   C            s         -           A   2          (   s   )         Z   C                   x     u   C          s         +           A   3          (   s   )         Z   D                   -     x     u   D            s         -           A   4          (   s   )         Z   D                   x     u   D          s                                       I   2          (   s   )       =             A   1          (   s   )         Z   C                   -     x     u   C            s         -           A   2          (   s   )         Z   C                   x     u   C          s         -           A   3          (   s   )         Z   D                   -     x     u   D            s         +           A   4          (   s   )         Z   D                   x     u   D          s                                                    
     Under a boundary condition x=0, the following relationships exist. 
     
       
         
           V 
           1 
           =V 
           0 
           −R 
           1 
           I 
           1 
           , V 
           2 
           =−R 
           N 
           I 
           2 
         
       
     
     On the other hand, the following relationships exist under a boundary condition x=1. 
       V   1   =R   2   I   1   , V   2   =R   F   I   2   
     The following formulas (12) are simultaneous equations for A 1  through A 4 , and V 1  and V 2  can be obtained by solving the simultaneous equations and substituting the solutions into the original formulas (11). When the obtained V 1  and V 2  are subjected to a Laplace inverse transform, temporal functions v 1 (t) and v 2 (t) are obtained. These results are also a combination of a linear operator and a time lag, and can be obtained by simple calculations similarly as described above.                  A   1     +     A   2     +     A   3     +     A   4       =       V   0     -       R   1          (         A   1       Z   C       -       A   2       Z   C       +       A   3       Z   D       -       A   4       Z   D         )                 (   12   )                   A   1     +     A   2     -     A   3     -     A   4       =     -       R   N          (         A   1       Z   C       -       A   2       Z   C       -       A   3       Z   D       +       A   4       Z   D         )                                       A   1                 -     τ   C          s         +       A   2                 τ   C        s         +       A   3                 -     τ   D          s         +       A   4                 τ   D        s           =       R   2          (           A   1       Z   C                   -     τ   C          s         -         A   2       Z   C                   τ   C        s         +         A   3       Z   D                   -     τ   D          s         -         A   4       Z   D                   τ   D        s           )                                     A   1                 -     τ   C          s         +       A   2                 τ   C        s         -       A   3                 -     τ   D          s         -       A   4                 τ   D        s           =       R   F          (           A   1       Z   C                   -     τ   C          s         -         A   2       Z   C                   τ   C        s         -         A   3       Z   D                   -     τ   D          s         +         A   4       Z   D                   τ   D        s           )                                     (     1   +       R   1       Z   C         )          A   1       +       (     1   -       R   1       Z   C         )          A   2       +       (     1   +       R   1       Z   D         )          A   3       +       (     1   -       R   1       Z   D         )          A   4         =     V   0                                   (     1   +       R   N       Z   C         )          A   1       +       (     1   -       R   N       Z   C         )          A   2       -       (     1   +       R   N       Z   D         )          A   3       -       (     1   -       R   N       Z   D         )          A   4         =   0                                 (     1   -       R   2       Z   C         )                 -     τ   C          s            A   1       +       (     1   +       R   2       Z   C         )                 τ   C        s            A   2       +       (     1   -       R   2       Z   D         )                 -     τ   D          s            A   3       +       (     1   +       R   2       Z   D         )                 τ   D        s            A   4         =   0                                 (     1   -       R   F       Z   C         )                 -     τ   C          s            A   1       +       (     1   +       R   F       Z   C         )                 τ   C        s            A   2       -       (     1   -       R   F       Z   D         )                 -     τ   D          s            A   3       -       (     1   +       R   F       Z   D         )                 τ   D        s            A   4         =   0                                            
     When these results are considered as a function of x, e −(x/u     C     )s , for example, means carrying out an operation f(t−x/u C ) with respect to the temporal function f(t). Since x/u C  describes the time it takes to travel the distance x at the velocity u C , it is a waveform propagating in the x direction. Similarly, it may be seen that a waveform propagating in a direction opposite to the x direction is a composed of signals propagating at the velocities u C  and u D . 
     Hence, the following formula (13) can be obtained, and the following formula (14) can be obtained by denoting the coefficient matrix equation by Δ.                  (           (     1   +       R   1       Z   C         )           (     1   -       R   1       Z   C         )           (     1   +       R   1       Z   D         )           (     1   -       R   1       Z   D         )               (     1   +       R   N       Z   C         )           (     1   -       R   N       Z   C         )           -     (     1   +       R   N       Z   D         )             -     (     1   -       R   N       Z   D         )                   (     1   -       R   2       Z   C         )                 -     τ   C          s                 (     1   +       R   2       Z   C         )                 τ   C        s                 (     1   -       R   2       Z   D         )                 -     τ   D          s                 (     1   +       R   2       Z   D         )                 τ   D        s                     (     1   -       R   F       Z   C         )                 -     τ   C          s                 (     1   +       R   F       Z   C         )                 τ   C        s                 -     (     1   -       R   F       Z   D         )                   -     τ   D          s                 -     (     1   +       R   F       Z   D         )                   τ   D        s               )          (           A   1               A   2               A   3               A   4           )       =     (           V   0             0           0           0         )             (   13   )                     Δ   =                           (     1   +       R   1       Z   C         )           (     1   -       R   1       Z   C         )           (     1   +       R   1       Z   D         )           (     1   -       R   1       Z   D         )               (     1   +       R   N       Z   C         )           (     1   -       R   N       Z   C         )           -     (     1   +       R   N       Z   D         )             -     (     1   -       R   N       Z   D         )                   (     1   -       R   2       Z   C         )                 -     τ   C          s                 (     1   +       R   2       Z   C         )                 τ   C        s                 (     1   -       R   2       Z   D         )                 -     τ   D          s                 (     1   +       R   2       Z   D         )                 τ   D        s                     (     1   -       R   F       Z   C         )                 -     τ   C          s                 (     1   +       R   F       Z   C         )                 τ   C        s                 -     (     1   -       R   F       Z   D         )                   -     τ   D          s                 -     (     1   +       R   F       Z   D         )                   τ   D        s                                =                [         -     {       2        (     1   +         R   1          R   N           Z   C          Z   D           )       +       (       R   1     +     R   N       )          (       1     Z   C       +     1     Z   D         )         }       ×     {       2        (     1   +         R   2          R   F           Z   C          Z   D           )       +       (       R   2     +     R   F       )          (       1     Z   C       +     1     Z   D         )         }       +                                    {       2        (     1   -         R   1          R   N           Z   C          Z   D           )       +       (       R   1     +     R   N       )          (       1     Z   C       -     1     Z   D         )         }     ×     {       2        (     1   -         R   2          R   F           Z   C          Z   D           )       +       (       R   2     +     R   F       )          (       1     Z   C       -     1     Z   D         )         }                 -   2                     τ   D        s         +                                    8        (       R   1     -     R   N       )          (       R   2     -     R   F       )           Z   C          Z   D                     -     (       τ   C     +     τ   D       )          s         +       {       2        (     1   -         R   1          R   N           Z   C          Z   D           )       -       (       R   1     +     R   N       )          (       1     Z   C       -     1     Z   D         )         }     ×                                    {       2        (     1   -         R   2          R   F           Z   C          Z   D           )       -       (       R   2     +     R   F       )          (       1     Z   C       -     1     Z   D         )         }                 -   2                     τ   C        s         -       {       2        (     1   +         R   1          R   N           Z   C          Z   D           )       -       (       R   1     +     R   N       )          (       1     Z   C       +     1     Z   D         )         }     ×                                      {       2        (     1   +         R   2          R   F           Z   C          Z   D           )       -       (       R   2     +     R   F       )          (       1     Z   C       +     1     Z   D         )         }                 -     (       2                   τ   C       +     2                   τ   D         )          s         ]                   (       τ   C     +     τ   D       )        s                     (   14   )                                
     Based on the above, the unknowns A 1 , A 2 , A 3  and A 4  can be obtained by the following formulas (15).                      A   1     =                    V   0     Δ     [         -     (     1   +       R   N       Z   D         )            {       2        (     1   +         R   2          R   F           Z   C          Z   D           )       +       (       R   2     +     R   F       )          (       1     Z   C       +     1     Z   D         )         }       +                                    (     1   -       R   N       Z   D         )          {       2        (     1   -         R   2          R   F           Z   C          Z   D           )       +       (       R   2     +     R   F       )          (       1     Z   C       -     1     Z   D         )         }                 -   2                     τ   p        s         +                                  2        (     1   -       R   N       Z   C         )              R   2     -     R   F         Z   D                   -     (       τ   C     +     τ   D       )          s         ]                   (       τ   C     +     τ   D       )        s                     (   15   )                       A   2     =                    V   0     Δ     [         -   2          (     1   +       R   N       Z   C         )              R   2     -     R   F         Z   D                   -     (       τ   C     +     τ   D       )          s         +                                    (     1   +       R   N       Z   D         )          {       2        (     1   -         R   2          R   F           Z   C          Z   D           )       -       (       R   2     +     R   F       )          (       1     Z   C       -     1     Z   D         )         }                 -   2                     τ   C        s         -                                    (     1   -       R   N       Z   D         )          {       2        (     1   +         R   2          R   F           Z   C          Z   D           )       -       (       R   2     +     R   F       )          (       1     Z   C       +     1     Z   D         )         }                 -     (       2                   τ   C       +     2        τ   D         )          s         ]                   (       τ   C     +     τ   D       )        s                                             A   3     =                    V   0     Δ     [         -     (     1   +       R   N       Z   C         )            {       2        (     1   +         R   2          R   F           Z   C          Z   D           )       +       (       R   2     +     R   F       )          (       1     Z   C       +     1     Z   D         )         }       +                                  2        (     1   -       R   N       Z   D         )              R   2     -     R   F         Z   C                   -     (                  τ   C     +     τ   D       )          s         +                                    (     1   -       R   N       Z   C         )          {       2        (     1   -         R   2          R   F           Z   C          Z   D           )       -       (       R   2     +     R   F       )          (       1     Z   C       -     1     Z   D         )         }                 -   2                     τ   C        s         ]                   (       τ   C     +     τ   D       )        s                                             A   4     =                    V   0     Δ     [         (     1   +       R   N       Z   C         )          {       2        (     1   -         R   2          R   F           Z   C          Z   D           )       +       (       R   2     +     R   F       )          (       1     Z   C       -     1     Z   D         )         }                 -   2                     τ   D        s         -                                  2        (     1   +       R   N       Z   D         )              R   2     -     R   F         Z   C                 -     (                  τ   C     +       τ   D        s       )           -                                    (     1   -       R   N       Z   C         )          {       2        (     1   +         R   2          R   F           Z   C          Z   D           )       -       (       R   2     +     R   F       )          (       1     Z   C       +     1     Z   D         )                 -     (       2                   τ   C       +     2        τ   D         )          s               ]                 (       τ   C     +     τ   D       )        s                                                        
     When obtaining the forward far-end crosstalk, the resistances of the resistors shown in FIG. 3 are set to R 1 =r, R 2 =R, R N =r and R F =R in the following formulas (16) and (17) for the sake of convenience to simplify matters. In addition, a common one-way time τ C  and a differential one-way time τ D  are both denoted by τ, that is, it is assumed that τ C =τ D =τ.                    Δ   =                  -     {       2        (     1   +         R   1          R   N           Z   C          Z   D           )       +       (       R   1     +     R   N       )          (       1     Z   C       +     1     Z   D         )         }       ×                              {       2        (     1   +         R   2          R   F           Z   C          Z   D           )       +       (       R   2     +     R   F       )          (       1     Z   C       +     1     Z   D         )         }                 =                  -   4          {       (     1   +       r   2         Z   C          Z   D           )     +     r        (       1     Z   C       +     1     Z   D         )         }          {       (     1   +       R   2         Z   C          Z   D           )     +     R        (       1     Z   C       +     1     Z   D         )         }                   =                  -   4          (     1   +     r     Z   C         )          (     1   +     r     Z   D         )          (     1   +     R     Z   C         )          (     1   +     R     Z   D         )                     (   16   )                         A   1                 -   τ                   s         =           V   0     Δ          [       -     (     1   +       R   N       Z   D         )            {       2        (     1   +         R   2          R   F           Z   C          Z   D           )       +       (       R   2     +     R   F       )          (       1     Z   C       +     1     Z   D         )         }       ]                   -   τ                   s                     =       -       2                   V   0       Δ            (     1   +     r     Z   D         )          {       (     1   +       R   2         Z   C          Z   D           )     +     R        (       1     Z   C       +     1     Z   D         )         }                 -   τ                   s                                                 A   2               τ                 s         =                    V   0     Δ     [           -   2          (     1   +       R   N       Z   C         )          R   2       -       R   F       Z   D         +                                      (     1   +       R   N       Z   D         )          {       2        (     1   -         R   2          R   F           Z   C          Z   D           )       -       (       R   2     +     R   F       )          (       1     Z   C       -     1     Z   D         )         }       ]            e       -   τ                   s                   =                    2                   V   0       Δ          (     1   +     r     Z   D         )          {       (     1   -       R   2         Z   C          Z   D           )     -     R        (       1     Z   C       -     1     Z   D         )         }                 -   τ                   s                                               (     V     2      C              )         x   =   1     ,     t   =   τ         =           A   1                 -   τ                   s         +       A   2               τ                 s           =       -       4                   RV   0       Δ            (     1   +     r     Z   D         )          (       R       Z   C          Z   D         +     1     Z   C         )                 -   τ                   s                   (   17   )                         A   3                 -   τ                   s         =           V   0     Δ          [       -     (     1   +       R   N       Z   C         )            {       2        (     1   +         R   2          R   F           Z   C          Z   D           )       +       (       R   2     +     R   F       )          (       1     Z   C       +     1     Z   D         )         }       ]                   -   τ                   s                     =                  -       2                   V   0       Δ            (     1   +     r     Z   C         )          {       (     1   +       R   2         Z   C          Z   D           )     +     R        (       1     Z   C       +     1     Z   D         )         }                 -   τ                   s                                                 A   4               τ                 s         =                      V   0       Δ          (     1   +       R   N       Z   C         )          {       2        (     1   -         R   2          R   F           Z   C          Z   D           )       +       (       R   2     +     R   F       )          (       1     Z   C       -     1     Z   D         )         }                 -   τ                   s                     =                    2                   V   0       Δ          (     1   +     r     Z   C         )          {       (     1   -       R   2         Z   C          Z   D           )     +     R        (       1     Z   C       -     1     Z   D         )         }                 -   τ                   s                                               (     V     2      D              )         x   =   1     ,     t   =   τ         =           -     A   3                   -   τ                   s         -       A   4               τ                 s           =         4                   RV   0       Δ          (     1   +     r     Z   C         )          (       R       Z   C          Z   D         +     1     Z   D         )                 -   τ                   s                                                 (             (             (     V   2            )         x   =   1     ,     t   =   τ         =     V     2      C                )         x   =   1     ,     t   =   τ         +     V     2      D                )         x   =   1     ,     t   =   τ         =         4                   RV   0       Δ          {       (     1   -       R                 r         Z   C          Z   D           )          (       1     Z   D       -     1     Z   C         )                 -   τ                   s                         =           R        (       1     Z   D       -     1     Z   C         )            (       Rr       Z   C          Z   D         -   1     )           (     1   +     r     Z   C         )          (     1   +     r     Z   D         )          (     1   +     R     Z   C         )          (     1   +     R     Z   D         )                     -   τ                   s            V   0                                                      
     Accordingly, it can be seen that the forward far-end crosstalk becomes zero when (Rr)/(Z C Z D )=1. But since each of Z C  and Z D  is equal to a square of the characteristic impedance Z 0  of the other when one of the coupling distributed constant lines  1  and  2  is terminated by a matched termination, the following relationship stands. 
     
       
         
           R/Z 
           0 
           =Z 
           0 
           /r 
         
       
     
     In other words, when normalized by the characteristic impedance of the line, a reciprocal relationship exists between the resistance at the near-end and the terminating resistance at the far-end. 
     FIG. 4 is a diagram showing signal changes caused by the terminating resistor at the far-end in the common mode and the differential mode. In FIG. 4, the ordinate indicates the voltage in arbitrary units, and the abscissa indicates the resistance R in Ω. In addition, V 2C  indicates a voltage propagating on the line  2  in the common mode, V 2D  indicates the voltage propagating on the line  2  in the differential mode, V 2  indicates a forward far-end crosstalk noise, that is, the crosstalk noise generated in the line  2 . As may be seen from FIG. 4, the voltage V 2C  in the common mode and the voltage V 2D  in the differential mode change depending on the resistance R of the terminating resistor at the far-end, and a point exists where V 2C =V 2D . Since r=30Ω, Z C =102Ω, Z D =47Ω in FIG. 4, V 2C =V 2D  at the point where R=160Ω. 
     Therefore, by setting the resistance R of the terminating resistor to R=Z 0   2 /r, the forward far-end crosstalk value theoretically becomes zero. Of course, it is not essential from the practical point of view that the crosstalk value is exactly equal to zero. As will be described later, the inventor has found that sufficient effects are obtainable when the resistance R of the terminating resistor increases or decreases by approximately 50 to 30% with respect to the maximum value of Z 0   2 /r. 
     FIGS. 5A and 5B respectively are diagrams for explaining the elimination of the forward far-end crosstalk noise in a first embodiment of the present invention. FIG. 5A shows a driving line  10 , a driver (driving source)  11 , a receiver  12 , a passive line  20 , a driver (driving source)  21 , and a receiver  22 . FIG. 5B additionally shows an internal resistance  13  of the driver  11 , a terminating resistor  14 , an internal resistance  23  of the driver  21 , and a terminating resistor  24 . 
     When signals are transmitted in the same direction on the two lines  10  and  20  which are close to each other as shown in FIG. 5A, the forward far-end crosstalk refers to the noise which is generated by the signal on the driving line  10  on the side of the receiver  22  on the other passive line  20 . 
     When the value of the internal resistance  13  of the driver  11  which is the driving source is denoted by r, the characteristic impedance of the driving line  10  and the passive line  20  is denoted by Z 0 , and the value of the terminating resistor  24  connected at the far-end of the passive line  20  is denoted by R, the value R is set so as to satisfy R=Z 0   2 /r. In this case, it is possible to make the forward far-end crosstalk value theoretically zero. 
     FIG. 6 is a diagram for explaining the elimination of the forward far-end crosstalk noise in a second embodiment of the present invention. In FIG. 6, signal transmissions on distributed constant lines  30  and  40  can be made from both the left to right and from the right to left. In FIG. 6, the reference numerals  31 ,  34 ,  41  and  44  indicate drivers (driving sources), the reference numerals  33 ,  36 ,  43  and  46  indicate receivers, the reference numerals  32 ,  35 ,  42  and  45  indicate internal resistances of the drivers  31 ,  34 ,  41  and  44 , and the reference numerals  37 ,  38 ,  47  and  48  indicate terminal resistors. 
     In this second embodiment, the first embodiment described above is expanded to the two-way transmission. A case will now be considered where the signal is transmitted from the left to right in FIG.  6 . In this case, the drivers  34  and  44  are set to a high impedance state. When the signal line  30  is regarded as a driving line and the signal line  40  is regarded as a passive line, the circuit construction becomes similar to that shown in FIG.  5 B. Accordingly, when the characteristic impedance of the signal lines  30  and  40  is denoted by Z 0 , the far-end crosstalk noise can be eliminated by setting a value R 1  of the terminal resistor  37  so as to satisfy R 1 =Z 0   2 /r 1 , where r 1  denotes the value of the internal resistance  32  of the driver  31 . 
     On the other hand, when the signal line  40  is regarded as a driving line and the signal line  30  is regarded as a passive line, the far-end crosstalk noise on the signal line  30  due to the signal line  40  can be eliminated by setting a value R 2  of the terminal resistor  47  so as to satisfy R 2 =Z 0   2 /r 2 , where r 2  denotes the value of the internal resistance  42  of the driver  41 . In addition, in order to eliminate the far-end crosstalk noise when making a signal transmission in a reverse direction, from the right to left, values R 3  and R 4  of the terminal resistors  38  and  39  which are connected are set so as to respectively satisfy R 3 =Z 0   2 /r 3  and R 4 =Z 0   2 /r 4 , where r 3  and r 4  respectively denote the values of the internal resistances  35  and  45  of the drivers  34  and  44 . 
     FIGS. 7A through 7E and FIGS. 8A and 8B are diagrams showing embodiments of the terminal resistor. 
     In FIG. 7A, one end of a terminating resistor  50  which is provided to eliminate the forward far-end crosstalk noise described above is grounded, and a terminating voltage is set to a logic amplitude “0”. In this embodiment, only one terminating resistor  50  is required for each line, and the construction is simple. When this terminating resistor  50  is provided, there is an advantage in that no level change occurs on the “0” side of the original signal. 
     In FIG. 7B, one end of a terminating resistor  51  which is provided to eliminate the forward far-end crosstalk noise described above is connected to a power supply voltage Vcc, and a terminating voltage is set to a logic amplitude “1”. In this embodiment, only one terminating resistor  51  is required for each line. When this terminating resistor  51  is provided, there is an advantage in that no level change occurs on the “1” side of the original signal. 
     In FIG. 7C, one end of a terminating resistor  52  which is provided to eliminate the forward far-end crosstalk noise described above is connected to an intermediate voltage V TH  between the logic amplitudes “0” and “1”. This intermediate voltage V TH  satisfies a relationship 0&lt;V TH &lt;Vcc, where Vcc is the power supply voltage. In this embodiment, only one terminating resistor  52  is required for each line. When this terminating resistor  52  is provided, a slight level change occurs on the “0” side and the “1” side of the original signal, but there is an advantage in that the symmetry of the waveform is maintained when the intermediate voltage V TH  is selected exactly to the center between 0 and Vcc. 
     In FIG. 7D, the terminating resistor which is provided to eliminate the forward far-end crosstalk noise described above is formed by two resistors  53  and  54 . One end of the resistor  53  is connected to the power supply voltage Vcc (that is, to the logic amplitude “1”), and one end of the resistor  54  is grounded (that is, connected to he logic amplitude “0”). A node connecting these resistors  53  and  54  is connected to the far-end of the line. When the resistances of the resistors  53  and  54  are respectively denoted by  2 R, this circuit becomes equivalent to a circuit surrounded by a dotted line and shown on the right side in FIG.  7 D. In the circuit surrounded by the dotted line, a resistor  55  having a resistance R is connected between the far-end and a voltage Vcc/2 which is ½ the power supply voltage Vcc. In this case, there is an advantage in that the circuit construction becomes equivalent to terminating to an intermediate voltage, without the need to prepare a terminating voltage. 
     In FIG. 7E, the terminating resistor which is provided to eliminate the forward far-end crosstalk noise described above is formed by a non-inverting gate circuit  60 . An input and an output of this non-inverting gate circuit  60  are connected directly or indirectly via a resistor  63  as shown. In addition, the far-end of the line and the input of the non-inverting gate circuit  60  are connected via a resistor  62 . When the resistance of the resistor  62  is denoted by R 11 , the resistance of the resistor  63  is denoted by R 12 , and the output resistance of the non-inverting gate circuit  60  is denoted by r 11 , the resistance R of the terminating resistor as a whole can be described by R=R 11 +r 11 +R 12 . 
     When the line is simply terminated as in the above described embodiments shown in FIGS. 7A through 7D, the power consumption increases. However, by employing the construction of the embodiment shown in FIG. 7E, it is possible to eliminate the power consumption caused by the terminating resistor in the steady state. Furthermore, by selecting the output resistance r 11  of the non-inverting gate circuit  60  equal to the resistance R of the terminating resistor, it is possible to obtain an effect whereby the connections of the resistors  62  and  63  shown in FIG. 7E may be omitted. In addition, by employing the construction in which the input of the non-inverting gate circuit  60  is not directly connected to the line but is connected to the line through the resistor  62 , the construction becomes strong against electrostatic discharge failure, and the waveform will not be distorted by the electrostatic capacitance of the non-inverting gate circuit  60 . 
     In FIG. 8A, the resistance of a terminating resistor  70  which is provided to eliminate the forward far-end crosstalk noise described above is selectable by an external control input  71 . When forming the circuit construction shown in FIG. 7E in the form of an integrated circuit, it is necessary to use different parts such that the resistance of the terminating resistor is different depending on the drivability of the driver. But by providing a plurality of kinds of resistances and making one of the resistances selectable depending on the control input  71 , it becomes possible to use only one kind of part and cope with the different drivability of the driver. In addition, even in a case where a resistor (damping resistor) is inserted in series with espect to the driver after the circuit is constructed and the equivalent internal resistance of the driving source changes, it is unnecessary to change the part, and it becomes possible to realize an optimum noise elimination by simply changing the setting by the control input  71 . 
     FIG. 8B shows an embodiment of the construction for varying the resistance of the terminating resistor  70  shown in FIG. 8A depending on the control input  71 . In FIG. 8B, outputs of tristate gates  72 A through  72 C can be controlled to a high impedance state or an active state, based respectively on control inputs  71 A through  71 C. If it is assumed that drivabilities of 1 mA, 2 mA and 4 mA are respectively obtained when the tristate gates  72 A through  72 C are active, it is possible to obtain resistances depending on the currents of 1 mA to 7 mA, based on a combination of the control inputs  71 A through  71 C. Of course, the circuit construction for making the resistance of the terminating resistor variable is not limited to the circuit construction shown in FIG.  8 B. 
     FIGS. 9 through 13 are diagrams showing analyzed results of the forward far-end crosstalk. 
     FIG. 9 shows signal waveforms appearing at the near-end and the far-end of the driving line  10  and the passive line  20  of the transmission circuit shown in FIG. 5B, in a case where the value r of the internal resistance  13  of the driver  11  is 10Ω and the resistance R of the terminating resistor  24  is infinitely large, that is, when the terminating resistor  24  is not connected. The drivability of the driver  11  is approximately 24 mA, and the characteristic impedance Z 0  of the driving line  10  and the passive line  20  is 69Ω. 
     In FIG. 9, the ordinate indicates the magnitude of the voltage, and the abscissa indicates the time. In FIG. 9, a thin solid line v 1 (near) indicates a voltage change on the driving line  10  on the side of the driver  11  (near-end), a thin dotted line v 1 (far) indicates a voltage change on the driving line  10  on the side of the receiver  12  (far-end), a bold solid line v 2 (near) indicates a voltage change on the passive line  20  on the side of the driver  21  (near-end), and a bold dotted line v 2 (far) indicates a voltage change on the passive line  20  on the side of the receiver  22  (far-end). The same designations are used in FIGS. 10,  11 ,  12  and  13  which will be described hereinafter. 
     As may be seen from the analyzed results shown in FIG. 9, the forward far-end crosstalk does clearly appear when the resistance R of the terminating resistor  24  is infinitely large, although the forward far-end crosstalk is not as large as the backward near-end crosstalk described above in conjunction with FIG.  2 . 
     In a normal transmission circuit, when connecting the terminating resistor, the resistance of the terminating resistor is in general matched to the characteristic impedance Z 0  so as to eliminate the signal reflection. Hence, when the resistance R of the terminating resistor  24  shown in FIG. 5B is set to R=Z 0 =69Ω, and the signal waveforms appearing at the near-end and the far-end of the driving line  10  and the passive line  20  are analyzed, the analyzed results shown in FIG. 10 are obtained. In this case shown in FIG. 10, the forward far-end crosstalk appears at the far-end of the passive line  20 , as indicated by the bold dotted line v 2 (far). 
     In the present invention, in the transmission circuit having the same construction as that described above, the resistance R of the terminating resistor  24  is selected to R=Z 0   2 /r. In other words, the resistance R is set as follows. 
       R=Z   0   2   /r =69 2 /10(Ω)=475(Ω) 
     In this case, the signal waveforms appearing at the near-end and the far-end of the driving line  10  and the passive line  20  become as shown in FIG.  11 . As may be seen from FIG. 11, virtually no forward far-end crosstalk is generated at the far-end of the passive line  20 . A whisker-like noise is generated theoretically (based on calculations) at the far-end of the passive line  20 , but this noise only has a width of approximately 50 ps, and such a noise signal of 100 ps or less can completely be neglected since such a small noise signal will actually disappear due to rounding of the waveform. 
     FIG. 12 shows signal waveforms similar to those shown in FIG. 8, with respect to a case where the value r of the internal resistance  13  of the driver  11  is 20Ω in the transmission circuit shown in FIG.  5 B. The resistance R of the terminating resistor  24  is set as follows. 
     
       
           R=Z   0   2   /r =69 2 /20(Ω)=237(Ω) 
       
     
     In this case, the forward far-end crosstalk also becomes zero. 
     FIG. 13 shows signal waveforms similar to those shown in FIG. 11, with respect to a case where the value r of the internal resistance  13  of the driver  11  is 30Ω in the transmission circuit shown in FIG.  5 B. The resistance R of the terminating resistor  24  is set as follows. 
     
       
           R=Z   0   2   /r =69 2 /30(Ω)=158(Ω) 
       
     
     In this case, the forward far-end crosstalk also becomes zero. 
     Next, a description will be given of how the absolute value of the forward far-end crosstalk changes with respect to the drivability of the driver  11 , by referring to FIGS. 14 through 17. 
     For the sake of convenience, timings of the forward far-end crosstalk are named 1T, 3T and 5T as shown in FIG. 14. 1T indicates a noise value after the time required to travel the line one way, 3T indicates a noise value after the time required to travel the line one way and after the time required to travel the line on both the going and returning ways also elapses, and 5T indicates a noise value after the time required to travel the line on both the going and returning ways elapses after the timing of 3T. 
     In FIGS. 15 through 17, the abscissa indicates the drivability of the driver in mA, and the ordinate indicates the magnitude of the crosstalk when the magnitude is normalized by 1. The drivability of the driver can be described by the following. 
     
       
         Drivability (mA)=400(mV)/(1.5 ×r (Ω)) 
       
     
     FIG. 15 shows a case where the resistance R of the terminating resistor is 475Ω and corresponds to the case shown in FIG.  11 . FIG. 16 shows a case where the resistance R of the terminating resistor is 237Ω and corresponds to the case shown in FIG.  12 . FIG. 17 shows a case where the resistance R of the terminating resistor is 158Ω and corresponds to the case shown in FIG.  13 . The crosstalk value at the timings 1T, 3T and 5T changes depending on the drivability of the driver, as shown in FIGS. 15 through 17. 
     FIGS. 18A and 18B and FIGS. 19A and 19B respectively are diagrams for explaining simulations of the forward far-end crosstalk which is generated, by use of a software circuit simulator. The simulation results shown in FIGS. 18A and 19A respectively correspond to the analyzed result shown in FIG. 12 described above. 
     FIG. 18A shows the simulation result which is obtained with respect to two distributed constant lines formed by the driving line  10  and the passive line  20  shown in FIG. 18B. A pattern length of the line was set to 14 cm. The characteristic impedance Z 0  of the line was set to 73Ω, and the internal resistance r of the driving source was set to 20Ω. 
     In FIG. 18A, v 10  indicates an output signal of the driving source on the driving line  10 , v 11  indicates a signal observed at an observation point P 1  on the driving line  10  when the resistance R of the terminating resistor is set infinitely large, and v 12  indicates a signal change observed at the observation point P 1  on the driving line  10  when the resistance R of the terminating resistor is set to 279Ω a value close to (Z 0   2 /r). 
     In addition, v 21  indicates a signal observed at an observation point P 2  on the passive line  20  when the resistance R of the terminating resistor is set infinitely large, and v 22  indicates a signal change obverted at the observation point P 2  on the passive line  20  when the resistance R of the terminating resistor is set to 279Ω a value close to (Z 0   2 /r). 
     As may be seen from FIG. 18A, virtually no crosstalk noise appears at the far-end of the passive line  20  if the resistance R of the terminating resistor is set to a value close to (Z 0   2 /r). 
     FIG. 19A shows the simulation result which is obtained with respect to the transmission circuit shown in FIG.  19 B. In FIG. 19B,  5  driving lines  10  are arranged on both sides of the passive line  20 , that is, a total of 10 driving lines  10  are provided. Otherwise, the conditions of this simulation are the same as those used in FIGS. 18A and 18B. Of course, a crosstalk value at an observation point P 4  on the passive line  20  shown in FIG. 19B becomes larger than the crosstalk value observed in FIG.  18 B. However, when the signal v 22  which is obtained when the resistance R of the terminating resistor is set to 279Ω is compared with the signal v 21  which is obtained when the resistance R of the terminating resistor is set infinitely large, that is, R=∞, the crosstalk value is negligibly small. In the signal v 22 , a slight fluctuation in the negative direction appears in correspondence with the rise of the signal v 12 , but this slight fluctuation only occurs for an extremely short time, and no problems are introduced thereby from the practical point of view. 
     Therefore, by setting the resistance R of the terminating resistor to R=(Z 0   2 /r), it is possible to make the forward far-end crosstalk noise zero. However, when applying the present invention, it is not essential from the practical point of view that the resistance R is set exactly to the above value. For this reason, a description will now be given of the relationship of the error in the terminating resistor and the change in the crosstalk reducing effect. 
     FIGS. 20 through 22 are diagrams showing the relationship of the resistance R and the forward far-end crosstalk reduction with respect to a case where the terminating resistor is changed from the optimum value R=(Z 0   2 /r). FIG. 20 shows a case where the internal resistance r of the driver is 10Ω, FIG. 21 shows a case where the internal resistance r of the driver is 20Ω, and FIG. 22 shows a case where the internal resistance r of the driver is 30Ω. 
     In FIGS. 20 through 22, the abscissa indicates the magnitude of the resistance R of the terminating resistor normalized by the optimum value (Z 0   2 /r), and the ordinate indicates the crosstalk value which is normalized by the crosstalk value which is obtained when the resistance R is infinitely large, that is, R=∞. For example, 0.2 on the scale of the ordinate indicates that the crosstalk noise value can be reduced by up to 20%, that is, reduced to a maximum of ⅕, as compared to the case where no measures are taken to reduce the crosstalk noise. In FIGS. 20 through 22, 1T, 3T and 5T indicate the noise values at the timings described above in conjunction with FIG.  14 . 
     If the crosstalk value can be reduced by up to 20%, this noise elimination measure is sufficient from the practical point of view. Hence, when this is used as a judging value, tolerable values are in the range of 0.7 times to 1.5 times with respect to the maximum value of R=(Z 0   2 /r). 
     Accordingly, it may be regarded that the arrangement falls within the technical range of the present invention if the resistance R of the terminating resistor connected at the far-end of the passive line falls at least within the following range. The resistance R of the terminating resistor which is matched to the characteristic impedance of the line is considerably smaller than a value within this range. 
     
       
         ( Z   0   2   /r )×0.7 ≦R ≦( Z   0   2   /r )×1.5 
       
     
     Furthermore, if the resistance R of the terminating resistor connected at the far-end of the passive line falls within the following range, the crosstalk value becomes less than or equal to 10% of the crosstalk value which is obtained when no terminating resistor is connected. 
     FIG. 23 is a perspective view showing a transmission circuit provided in an IC chip. In FIG. 23, a transmission circuit  100  according to the present invention is provided within an IC chip  101 . In addition, the IC chip  101  is provided on a board  102 , that is, a circuit board provided within a communication unit or an information processing apparatus such as a personal computer. Of course, IC chips and elements other than the IC chip  101  may also be provided on the board  102 , but such other IC chips and elements are not directly related to the subject matter of the present invention, and an illustration thereof will be omitted. In addition, the board  102  may of course be constructed to be arranged externally to the apparatus. 
     Therefore, according to the present invention, it is possible to effectively eliminate the forward far-end crosstalk noise by use of a simple construction. This effect of eliminating the forward far-end crosstalk noise cannot be achieved by other methods such as increasing the pattern gap or reducing the line impedance. According to such other methods, it may be possible to slightly reduce the crosstalk noise, however, it is not only difficult to reduce the crosstalk to a value close to zero, but from the practical point of view, other problems are newly introduced. 
     Further, the present invention is not limited to these embodiments, but various variations and modifications may be made without departing from the scope of the present invention.