Abstract:
A 64-ary QAM (Quadrature Amplitude Modulation) demodulation apparatus and method for receiving an input signal R k (X k ,Y k ) comprised of a k th  quadrature-phase signal Y k  and a k th  in-phase signal X k , and generating soft decision values Λ(s k,5 ), Λ(s k,4 ), Λ(s k,3 ), Λ(s k,2 ), Λ(s k,1 ) and Λ(s k,0 ) for the input signal R k (X k , Y k ) are disclosed. A first soft decision value generator receives the quadrature-phase signal Y k  of the received signal R k  and a distance value 2a between six demodulated symbols on the same axis, and generates soft decision values Λ(s k,5 ), Λ(s k,4 ) and Λ(s k,3 ) for sixth, fifth, and fourth demodulated symbols. A second soft decision value generator receives the in-phase signal X k  of the received signal R k  and the distance value 2a between the six demodulated symbols on the same axis, and generates soft decision values Λ(s k,2 ), Λ(s k,1 ) and Λ(s k,0 ) for third, second and first demodulated symbol.

Description:
PRIORITY  
         [0001]    This application claims priority to an application entitled “Apparatus and Method for Calculating Soft Decision Value Input to Channel Decoder in a Data Communication System” filed in the Korean Industrial Property Office on Sep. 18, 2001 and assigned Serial No. 2001-57622, the entire contents of which are incorporated herein by reference.  
         BACKGROUND OF THE INVENTION  
         [0002]    1. Field of the Invention  
           [0003]    The present invention relates generally to a demodulation apparatus and method for a data communication system employing multi-level modulation, and in particular, to an apparatus and method for calculating an input value to a channel decoder in a demodulator for a data communication system employing 64-ary QAM (Quadrature Amplitude Modulation).  
           [0004]    2. Description of the Related Art  
           [0005]    In general, a data communication system employs multi-level modulation in order to increase spectral efficiency. The multi-level modulation includes various modulation techniques. Herein, reference will be made to 64-ary QAM, one of the multi-level modulation techniques. As known by those skilled in the art, a 64-ary QAM channel encoder modulates a signal coded by binary encoding and transmits the coded signal to a receiver. The receiver then receives the transmitted modulated signal and decodes the modulated signal through soft decision values decoding in a channel decoder. To perform the decoding, a demodulator of the receiver includes a mapping algorithm for generating soft decision values (or soft values), because the received modulated signal is comprised of an in-phase signal component and a quadrature-phase signal component. Therefore, the demodulator of the receiver includes a mapping algorithm for generating soft decision values each corresponding to output bits of the channel encoder from a 2-dimensional received signal.  
           [0006]    The mapping algorithm is classified into a simple metric procedure proposed by Nokia, and a dual minimum metric procedure proposed by Motorola. Both algorithms calculate LLR (Log Likelihood Ratio) values for the output bits and use the calculated LLR values as input soft decision values to the channel decoder. The simple metric procedure, which employs a mapping algorithm given by modifying a complex LLR calculation formula into a simple approximate formula, has a simple LLR calculation formula. However, LLR distortion caused by the use of the approximate formula leads to performance degradation. The dual minimum metric procedure, which employs a mapping algorithm of calculating LLR with a more accurate approximate formula and uses the calculated LLR as an input soft decision value of the channel decoder, can make up for performance degradation of the simple metric procedure to some extent. However, compared with the simple metric procedure, this procedure needs increased calculations, thus causing a considerable increase in hardware complexity.  
         SUMMARY OF THE INVENTION  
         [0007]    It is, therefore, an object of the present invention to provide an apparatus and method for obtaining a soft decision value without performing complex calculations in a demodulator for a data communication system employing 64-ary QAM.  
           [0008]    It is another object of the present invention to provide an apparatus and method for designing demodulator with a simple circuit to obtain a soft decision value for a data communication system employing 64-ary QAM.  
           [0009]    It is yet another object of the present invention to provide an apparatus and method for obtaining a correct soft decision value with a simple circuit in a demodulator for a data communication system employing 64-ary QAM.  
           [0010]    To achieve the above and other objects, an embodiment of the present invention provides a 64-ary QAM (Quadrature Amplitude Modulation) demodulation apparatus for receiving an input signal R k (X k ,Y k ) comprised of a k th  quadrature-phase signal Y k  and a k th  in-phase signal X k , and for generating soft decision values Λ(s k,5 ), Λ(s k,4 ), Λ(s k,3 ), Λ(s k,2 ), Λ(s k,1 ) and Λ(s k,0 ) for the input signal R k (X k , Y k ) by a soft decision techniques. The apparatus comprises a first soft decision value generator that receives the quadrature-phase signal Y k  of the received signal R k  and a distance value 2a between six demodulated symbols on the same axis, and generates soft decision values Λ(s k,5 ), Λ(s k,4 ) and Λ(s k,3 ) for sixth, fifth and fourth demodulated symbols using the following equations.  
             Z   1k   =|Y   k |−4 a    
             Z   2k   =|Z   1k |−2 a              Λ        (     s     k   ,   5       )       =       Y   k     +     c        (       α   ·     Z     1      k         +     β   ·     Z     2      k           )           ,   where           α   =     {             3           if                   MSB        (     Z     1      k       )         =   0             0           if                   MSB        (     Z     1      k       )         =   1                
        β     =     {             0           if                   MSB        (     Z     2      k       )         =   0               -   1             if                   MSB        (     Z     2      k       )         =   1              and        
        c     =     {               1           if                   MSB        (     Y   k     )         =   0               -   1             if                   MSB        (     Y   k     )         =   1                
          Λ        (     s     k   ,   4       )         =       Z     1      k       +     γ   ·     Z     2      k             ,       where        
        γ     =     {         0           if                   MSB        (     Z     2      k       )         =   1             1           if                   MSB        (     Z     2      k       )         =       0                 and                   MSB        (     Z     1      k       )         =   0                 -   1             if                   MSB        (     Z     2      k       )         =       0                 and                   MSB        (     Z     1      k       )         =   1                                                   
 Λ( S   k,3 )= Z   2k    
           [0011]    where Λ(s k,5 ) indicates the soft decision value for the sixth modulated symbol, Λ(s k,4 ) indicates the soft decision value for the fifth modulated symbol, and Λ(s k,3 ) indicates the soft decision value for the fourth modulated symbol. A second soft decision value generator receives the in-phase signal X k  of the received signal R k  and the distance value 2a between the six demodulated symbols on the same axis, and generates soft decision values Λ(s k,2 ), Λ(s k,1 ) and Λ(s k,0 ) for third, second and first demodulated symbols using the following equations.  
             Z′   1k   =|X   k |−4 a    
             Z′   2k   =|Z′   1k |−2 a              Λ        (     s     k   ,   2       )       =       X   k     +       c   ′          (         α   ′     ·     Z     1      k     ′       +       β   ′     ·     Z     2      k     ′         )           ,   where             α   ′     =     {             3           if                   MSB        (     Z     1      k     ′     )         =   0             0           if                   MSB        (     Z     1      k     ′     )         =   1                
          β   ′       =     {             0           if                   MSB        (     Z     2      k     ′     )         =   0               -   1             if                   MSB        (     Z     2      k     ′     )         =   1              and        
          c   ′       =     {               1           if                   MSB        (     X   k     )         =   0               -   1             if                   MSB        (     X   k     )         =   1                
          Λ        (     s     k   ,   1       )         =       Z     1      k     ′     +       γ   ′     ·     Z     2      k     ′           ,       where        
          γ   ′       =     {         0           if                   MSB        (     Z     2      k     ′     )         =   1             1           if                   MSB        (     Z     2      k     ′     )         =       0                 and                   MSB        (     Z     1      k     ′     )         =   0                 -   1             if                   MSB        (     Z     2      k     ′     )         =       0                 and                   MSB        (     Z     1      k     ′     )         =   1                                                   
 Λ( s   k,0 )= Z═   2k    
           [0012]    where Λ(s k,2 ) indicates the soft decision value for the third modulated symbol, Λ(s k,1 ) indicates the soft decision value for the second modulated symbol, and Λ(s k,0 ) indicates the soft decision value for the first modulated symbol and the “MSB” means the most significant bit and the “a” means a distance value on the same axis.  
           [0013]    The first soft decision value generator comprises a first operator for calculating Z 1k =|Y k |−4a by receiving the quadrature-phase signal Y k  and the distance value between the demodulated symbols on the same axis, and a second operator for calculating Z 2k =|Z 1k |−2a by receiving the output value Z 1k  of the first operator, and providing the calculated value Z 2k  as the soft decision value Λ(s k,3 ) for the fourth demodulated symbol. The first soft decision value generator further comprises a first MSB (Most Significant Bit) calculator for calculating MSB of the quadrature-phase signal Y k , a second MSB calculator for calculating MSB of the output value Z 1k  of the first operator, and a third MSB calculator for calculating MSB of the output value Z 2k  of the second operator. The first soft decision value generator also comprises a first selector for selecting the output value Z 1k  of the first operator or a value “0” according to an output value of the second MSB calculator, a second selector for selecting an inversed value −Z 2k  of the output value Z 2k  of the second operator or a value “0” according to an output value of the third MSB calculator, a first adder for adding an output value of the second selector to a value determined by multiplying the output value of the first selector by 3, a third selector for selecting an output value of the first adder or an inversed value of the output value of the first adder according to an output value of the first MSB calculator. In addition, the first soft decision value generator comprises a second adder for adding an output value of the third selector to the quadrature-phase signal Y k  and generating the added signal as the soft decision value Λ(S k,5 ) for the sixth demodulated symbol, a fourth selector for selecting the output value Z 2k  of the second operator or an inversed value −Z 2k  of the output value Z 2k  according to the output value of the second MSB calculator, a fifth selector for selecting an output value of the fourth selector or a value “0” according to the output value of the third MSB calculator, and a third adder for adding an output value of the fifth selector to the output value Z 1k  of the first operator and generating the added value as the soft decision value Λ(s k,4 ) for the fifth demodulated symbol.  
           [0014]    The second soft decision value generator comprises a third operator for calculating Z′ 1k =|X k |−4a by receiving the in-phase signal X k  and the distance value between the demodulated symbols on the same axis, and a fourth operator for calculating Z′ 2k =|Z′ 1k |−2a by receiving the output value Z′ 1k  of the third operator, and providing the calculated value Z′ 2k  as the soft decision value Λ(s k,0 ) for the first demodulated symbol. The second soft decision value generator also comprises a fourth MSB calculator for calculating MSB of the in-phase signal X k , a fifth MSB calculator for calculating MSB of the output value Z′ 1k  of the third operator, and a sixth MSB calculator for calculating MSB of the output value Z′ 2k  of the fourth operator. The second soft decision value generator further comprises a sixth selector for selecting the output value Z′ 1k  of the third operator or a value “0” according to an output value of the fifth MSB calculator, a seventh selector for selecting an inversed value −Z′ 2k  of the output value Z 2k  of the fourth operator or a value “0” according to an output value of the sixth MSB calculator, a fourth adder for adding an output value of the seventh selector to a value determined by multiplying the output value of the sixth selector by 3, and an eighth selector for selecting an output value of the fourth adder or an inversed value of the output value of the fourth adder according to an output value of the fourth MSB calculator. In addition, the second soft decision value generator comprises a fifth adder for adding an output value of the eighth selector to the in-phase signal X k  and generating the added signal as the soft decision value Λ(s k,2 ) for the third demodulated symbol, a ninth selector for selecting the output value Z′ 2k  of the fourth operator or an inversed value −Z′ 2k  of the output value Z′ 2k  according to the output value of the fifth MSB calculator, a tenth selector for selecting an output value of the ninth selector or a value “0” according to the output value of the sixth MSB calculator, and a sixth adder for adding an output value of the tenth selector to the output value Z′ 1k  of the third operator and generating the added value as the soft decision value Λ(s k,1 ) for the second demodulated symbol. 
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0015]    The above and other objects, features and advantages of the present invention will become more apparent from the following detailed description when taken in conjunction with the accompanying drawings in which:  
         [0016]    [0016]FIG. 1 illustrates an example of a signal constellation for 64-ary QAM (Quadrature Amplitude Modulation);  
         [0017]    [0017]FIGS. 2 and 3 illustrate an example of processes performed for calculating soft decision values according to an embodiment of the present invention;  
         [0018]    [0018]FIG. 4 illustrates a block diagram of an embodiment of the present invention for calculating soft decision values using a quadrature-phase signal component Y k , an in-phase signal component X k , and a distance value “a”; and  
         [0019]    [0019]FIGS. 5 and 6 illustrate an embodiment of the present invention of calculators for calculating the soft decision values for use in a demodulator in a data communication system employing 64-ary QAM. 
     
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT  
       [0020]    An embodiment of the present invention will be described below with reference to the accompanying drawings. In the following description, well-known functions or constructions are not described in detail.  
         [0021]    An embodiment of the present invention provides an apparatus and method for obtaining a soft decision value input to a channel decoder, calculated by the dual minimum metric procedure, without a mapping table or complex calculations in a demodulator for a data communication system employing 64-ary QAM.  
         [0022]    An algorithm for generating multi-dimensional soft decision values from a 2-dimentional received signal will be described below. An output sequence of a binary channel encoder is divided into m bits, and mapped to corresponding signal points among M (=2 m ) signal points according to a Gray coding rule. This can be represented by  
         [0023]    Equation (1)  
                   s     k   ,     m   -   1              s     k   ,     m   -   2            ⋯                   s     k   ,   0              -&gt;   f          I   k       ,     Q   k             Equation                   (   1   )                                 
 
         [0024]    In Equation (1), s k,i  (i=0,1, . . . ,m−1) indicates an i th  bit in the output sequence of the binary channel encoder, mapped to a k th  symbol, and I k  and Q k  indicate an in-phase signal component and a quadrature-phase signal component of the k th  symbol, respectively. For 64-ary QAM, m=6 and a corresponding signal constellation is illustrated in FIG. 1.  
         [0025]    A complex output of a symbol demodulator in the receiver, comprised of I k  and Q k , is defined as  
         [0026]    Equation (2)  
           R   k   ≡X   k   +jY   k   =g   k ( I   k   +jQ   k )+(η k   I   +jη   k   Q )  
         [0027]    In Equation (2), X k  and Y k  indicate an in-phase signal component and a quadrature-phase signal component of the output of the symbol demodulator, respectively. Further, g k  is a complex coefficient indicating gains of the transmitter, the transmission media and the receiver. In addition, η k   I  and η k   Q  are Gaussian noises with an average 0 and a divergence σ n   2 , and they are statistically independent of each other.  
         [0028]    LLR related to the sequence s k,i  (i=0,1, . . . ,m−1) can be calculated by Equation (3), and the calculated LLR can be used as a soft decision value input to the channel decoder.  
         [0029]    Equation (3)  
                     Λ        (     s     k   ,   i       )       =            K                 log          Pr        {         s     k   ,   i       =     0        X   k         ,     Y   k       }         Pr        {         s     k   ,   i       =     1        X   k         ,     Y   k       }                         i   =          0     ,   1   ,   …              ,     m   -   1                   Equation                   (   3   )                                 
 
         [0030]    In Equation (3), k is a constant, and Pr{A|B} indicates a conditional probability defined as a probability that an event A will occur when an event B occurs. However, since Equation (3) is non-linear and accompanies relatively many calculations, an algorithm capable of approximating Equation (3) is required for actual realization. In the case of a Gaussian noise channel with g k =1 in Equation (2), Equation (3) can be approximated by the dual minimum metric procedure as follows.  
         [0031]    Equation (4)  
                     Λ        (     s     k   ,   i       )       =            K                 log            ∑     z   k            exp        {         -   1     /     σ   η   2                     R   k     -     
                       z   k          (       s     k   ,   i       =   0     )              2       }             ∑     z   k            exp        {         -   1     /     σ   η   2                     R   k     -     
                       z   k          (       s     k   ,   i       =   1     )              2       }                         ≈            K                 log          exp        {         -   1     /     σ   η   2          min                 R   k     -     
                       z   k          (       s     k   ,   i       =   0     )              2       }         exp        {         -   1     /     σ   η   2          min                 R   k     -     
                       z   k          (       s     k   ,   i       =   1     )              2       }                       =              K   ′          ⌊       min                 R   k     -       z   k          (       s     k   ,   i       =   1     )              2       -                                min                 R   k     -       z   k          (       s     k   ,   i       =   0     )              2       ⌋                   Equation                   (   4   )                                 
 
         [0032]    In Equation (4), K′=(1/σ n   2 )K, and z k (s k,i =0) and z k (s k,i =1) indicate actual values of I k +jQ k  for s k,i =0 and s k,i =1, respectively. In order to calculate Equation (4), it is necessary to determine z k (s k,i =0) and z k (s k,i =1) minimizing |R k −z k (s k,i =0)| 2  and |R k −z k (s k,i =1)| 2 , for a 2-dimensional received signal R k . Equation (4) approximated by the dual minimum metric procedure can be rewritten as  
         [0033]    Equation (5)  
                     Λ        (     s     k   ,   i       )       =              K   ′          ⌊       min                 R     k                  -       z   k          (       s     k   ,   i       =   1     )              2       -                                min                 R   k     -       z   k          (       s     k   ,   i       =   0     )              2       ⌋                 =                K   ′          (       2        n     k   ,   i         -   1     )       [                R   k     -       z   k          (       s     k   ,   i       =     n     k   ,   i         )              2     -                              min                 R   k     -       z   k          (       s     k   ,   i       =       n   _       k   ,   i         )              2       ]                   Equation                   (   5   )                                 
 
         [0034]    In Equation (5), n k,i  indicates an i th  bit value of a reverse mapping sequence for a signal point nearest to R k , and {overscore (n)} k,i  indicates a negation for n k,i . The nearest signal point is determined by ranges of an in-phase signal component and a quadrature-phase signal component of R k . A first term in the brackets of Equation (5) can be rewritten as  
         [0035]    Equation (6)  
         | R   k   −z   k ( s   k,i   =n   k,i )| 2 =( X   k   −U   k ) 2 +( Y   k   −V   k ) 2    
         [0036]    In Equation (6), U k  and V k  denote an in-phase signal component and a quadrature-phase signal component of a nearest signal point mapped by {n k,m-1 , . . . , n k,i , . . . , n k,1 , n k,0 }, respectively. Further, a second term in the brackets of Equation (5) can be written as  
         [0037]    Equation (7)  
         min| R   k   −z   k (s k,i   ={overscore (n)}   k,i )| 2 =( X   k   −U   k,i ) 2 +( Y   k   −V   k,i ) 2    
         [0038]    In Equation (7), U k,i  and V k,i  denote an in-phase signal component and a quadrature-phase signal component of a signal point mapped by a reverse mapping sequence {m k,m-1 , . . . , m k,i (={overscore (n)} k,i ), . . . , m k,1 , m k,0 } of z k  minimizing |R k −z k (s k,i ={overscore (n)} k,i )| 2 , respectively. Equation (5) is rewritten as Equation (8) by Equation (6) and Equation (7).  
         [0039]    Equation (8)  
                     Λ        (     s     k   ,   i       )       =                    (           K   ′          (       2        n     k   ,   i         -   1     )       [     {       X   k     -     U   k           )     )     2     +       (       Y   k     -     V   k       )     2       }     -                            {         (       X   k     -     U     k   ,   i         )     2     +       (       Y   k     -     V     k   ,   i         )     2       }     ]                 =                K   ′          (       2        n     k   ,   i         -   1     )       [         (       U   k     +     U     k   ,   i       -     2        X   k         )          (       U   k     -     U     k   ,   i         )       +                              (       V   k     +     V     k   ,   i       -     2        Y   k         )          (       V   k     -     V     k   ,   i         )       ]                 Equation                   (   8   )                                 
 
         [0040]    A process of calculating input soft decision values to the channel decoder by a demodulator in accordance with Equation (8) in a data communication system employing 64-ary QAM will be described below. First, Table 1 and Table 2 are used to calculate {n k,5 , n k,4 , n k,3 , n k,2 , n k,1 , n k,0 }, U k  and V k  from two signal components X k  and Y k  of a 64-ary QAM-modulated received signal R k .  
                       TABLE 1                       Condition of Y k     (n k,5 , n k,4 , n k,3 )   V k                     Y k  &gt; 6a   (0, 0, 0)     7a       4a &lt; Y k  &lt; 6a   (0, 0, 1)     5a       2a &lt; Y k  &lt; 4a   (0, 1, 1)     3a       0 &lt; Y k  &lt; 2a   (0, 1, 0)      a       −2a &lt; Y k  &lt; 0   (1, 1, 0)    −a       −4a &lt; Y k  &lt; −2a   (1, 1, 1)   −3a       −6a &lt; Y k  &lt; −4a   (1, 0, 1)   −5a       Y k  &lt; −6a   (1, 0, 0)   −7a                  
 
         [0041]    [0041]                       TABLE 2                       Condition of Y k     (n k,2 , n k,1 , n k,0 )   U k                     X k  &gt; 6a   (0, 0, 0)     7a       4a &lt; X k  &lt; 6a   (0, 0, 1)     5a       2a &lt; X k  &lt; 4a   (0, 1, 1)     3a       0 &lt; X k  &lt; 2a   (0, 1, 0)      a       −2a &lt; X k  &lt; 0   (1, 1, 0)    −a       −4a &lt; X k  &lt; −2a   (1, 1, 1)   −3a       −6a &lt; X k  &lt; −4a   (1, 0, 1)   −5a       X k  &lt; −6a   (1, 0, 0)   −7a                    
         [0042]    Table 1 illustrates (n k,5 , n k,4 , n k,3 ) and V k  for the case where a quadrature-phase signal component Y k  of the received signal R k  appears in each of 8 regions parallel to a horizontal axis in FIG. 1. For the sake of convenience, 7 boundary values, that is, result values at Y k =−6a, Y k =−4a, Y k =−2a, Y k =0, Y k =2a, Y k =4a and Y k =6a, are omitted from Table 1. Where “a” means a distance value on the same axis and the “a” indicating a distance value, can have a different value according to a modulating/demodulating method. Table 2 illustrates (n k,2 , n k,1 , n k,0 ) and U k  for the case where an in-phase signal component X k  of the received signal R k  appears in each of 8 regions parallel to a vertical axis in FIG. 1. For the sake of convenience, 7 boundary values, that is, result values at X k =−6a, X k =−4a, X k =−2a, X k =0, X k =2a, X k =4a and X k =6a, are omitted from Table 2.  
         [0043]    Table 3 illustrates a sequence {m k,5 , m k,4 , m k,3 , m k,2 , m k,1, m   k,0 } minimizing |R k −z k (s k,i ={overscore (n)}k,i)| 2 , calculated for i (where i∈{0, 1, 2, 3, 4, 5}), in terms of a function {n k,5 , n k,4 , n k,3 , n k,2 , n k,1 , n k,0 }, and also illustrates in-phase and quadrature-phase signal components U k,i  and V k,i  of the corresponding z k .  
                                   TABLE 3                                       i   {m k,5 , m k,4 , m k,3 , m k,2 , m k,1 , m k,0 }   V k,1     U k,1             5   {{overscore (n)} k,5 , 1, 0, n k,2 , n k,1 , n k,0 }   V k,5     U k             4   {n k,5 , {overscore (n)} k,4 , 1, n k,2 , n k,1 , n k,0 }   V k,4     U k             3   {n k,5 , n k,4 , {overscore (n)} k,3 , n k,2 , n k,1 , n k,0 }   V k,3     U k             2   {n k,5 , n k,4 , n k,3 , {overscore (n)} k,2 , 1, 0}   V k      U k,2             1   { n k,5 , n k,4 , n k,3 , n k,2 , {overscore (n)} k,1 , 1}   V k     U k,1             0   { n k,5 , n k,4 , n k,3 , n k,2 , n k,1 , {overscore (n)} k,0 }   V k      U k,0                        
 
         [0044]    Table 4 and Table 5 illustrate V k,i  and U k,i  corresponding to (m k,5 , m k,4, m   k,3 ) and (m k,2 , m k,1 , m k,0 ) calculated in Table 3, for all combinations of (n k,5 , n k,4 , n k,3 ) and (n k,2 , n k,1 , n k,0 ), respectively.  
                                   TABLE 4                                   (n k,5 , n k,4 , n k,3 )   V k,5     V k,4     V k,3                             (0, 0, 0)   −a     3a     5a           (0, 0, 1)   −a     3a     7a           (0, 1, 1)   −a     5a      a           (0, 1, 0)   −a     5a     3a           (1, 1, 0)     a   −5a   −3a           (1, 1, 1)     a   −5a    −a           (1, 0, 1)     a   −3a   −7a           (1, 0, 0)     a   −3a   −5a                      
 
         [0045]    [0045]                                   TABLE 5                                   (n k,2 , n k,1 , n k,0 )   U k,2     U k,1     U k,0                             (0, 0, 0)   −a     3a     5a           (0, 0, 1)   −a     3a     7a           (0, 1, 1)   −a     5a      a           (0, 1, 0)   −a     5a     3a           (1, 1, 0)     a   −5a   −3a           (1, 1, 1)     a   −5a    −a           (1, 0, 1)     a   −3a   −7a           (1, 0, 0)     a   −3a   −5a                        
         [0046]    Table 6 and Table 7 illustrate results given by down-scaling, in a ratio of K′×4a, input soft decision values of the channel decoder obtained by substituting V k,1  and U k,i  of Table 4 and Table 5 into Equation (8).  
                                   TABLE 6                                   Condition of Y k     Λ(s k,5 )   Λ(s k,4 )   Λ(s k,3 )                           Y k  &gt; 6a    4Y k  − 12a    2Y k  − 10a   Y k  − 6a           4a &lt; Y k  &lt; 6a   3Y k  − 6a    Y k  − 4a   Y k  − 6a           2a &lt; Y k  &lt; 4a   2Y k  − 2a    Y k  − 4a   −Y k  + 2a             0 &lt; Y k  &lt; 2a   Y k     2Y k  − 6a   −Y k  + 2a             −2a &lt; Y k  &lt; 0   Y k     −2Y k  − 6a     Y k  + 2a           −4a &lt; Y k  &lt; −2a   2Y k  + 2a    −Y k  − 4a     Y k  + 2a           −6a &lt; Y k  &lt; −4a   3Y k  + 6a    −Y k  − 4a     −Y k  − 6a             Y k  &lt; −6a    4Y k  + 12a   −2Y k  − 10a   −Y k  − 6a                        
 
         [0047]    [0047]                                   TABLE 7                                   Condition of X k     Λ(s k,2 )   Λ(s k,1 )   Λ(s k,0 )                           X k  &gt; 6a    4X k  − 12a    2X k  − 10a   X k  − 6a           4a &lt; X k  &lt; 6a   3X k  − 6a    X k  − 4a   X k  − 6a           2a &lt; X k  &lt; 4a   2X k  − 2a    X k  − 4a   −X k  + 2a             0 &lt; X k  &lt; 2a   X k     2X k  − 6a   −X k  + 2a             −2a &lt; X k  &lt; 0   X k     −2X k  − 6a     X k  + 2a           −4a &lt; X k  &lt; −2a   2X k  + 2a    −X k  − 4a     X k  + 2a           −6a &lt; X k  &lt; −4a   3X k  + 6a    −X k  − 4a     −X k  − 6a             X k  &lt; −6a    4X k  + 12a   −2X k  − 10a   −X k  − 6a                          
         [0048]    That is, when a received signal R k  is applied, LLR satisfying a corresponding condition can be output as an input soft decision value by Table 6 and Table 7. If the channel decoder used in the system is not a max-logMAP (logarithmic maximum a posteriori) decoder, a process of up-scaling the LLR of Table 6 and Table 7 in a reverse ratio of the down-scale ratio must be added.  
         [0049]    However, when outputting an input soft decision value of the channel decoder using the mapping table of Table 6 or Table 7, the demodulator should perform an operation of deciding a condition of the received signal and require a memory for storing the output contents according to the corresponding condition. This can be avoided by calculating the input soft decision values to the channel decoder using a formula having a simple condition decision operation instead of the mapping table.  
         [0050]    To this end, the condition decision formulas shown in Table 6 and Table 7 can be expressed as shown in Table 8 and Table 9.  
                                   TABLE 8                       Condition of Y k     Sign of Y k     Sign of Z 1k     Sign of Z 2k     Z 1k     Z 2k                     Y k  &gt; 6a   Y k  ≧ 0   Z 1k  ≧ 0   Z 2k  ≧ 0   Y k  − 4a   Y k  − 6a       4a &lt; Y k  &lt; 6a           Z 2k  &lt; 0   Y k  − 4a   Y k  − 6a       2a &lt; Y k  &lt; 4a       Z 1k  &lt; 0   Z 2k  &lt; 0   Y k  − 4a   −Y k  + 2a         0 &lt; Y k  &lt; 2a           Z 2k  ≧ 0   Y k  − 4a   −Y k  + 2a         −2a &lt; Y k  &lt; 0   Y k  &lt; 0   Z 1k  &lt; 0   Z 2k  ≧ 0   −Y k  − 4a     Y k  + 2a       −4a &lt; Y k  &lt; −2a           Z 2k  &lt; 0   −Y k  − 4a     Y k  + 2a       −6a &lt; Y k  &lt; −4a       Z 1k  ≧ 0   Z 2k  &lt; 0   −Y k  − 4a     −Y k  + 6a         Y k  &lt; −6a           Z 2k  ≧ 0   −Y k  − 4a     −Y k  + 6a                    
 
         [0051]    [0051]                                   TABLE 8                       Condition of X k     Sign of X k     Sign of Z&#39; 1k     Sign of Z&#39; 2k     Z&#39; 1k     Z&#39; 2k                     X k  &gt; 6a   X k  ≧ 0   Z&#39; 1k  ≧ 0   Z&#39; 2k  ≧ 0   X k  − 4a   X k  − 6a       4a &lt; X k  &lt; 6a           Z&#39; 2k  &lt; 0   X k  − 4a   X k  − 6a       2a &lt; X k  &lt; 4a       Z&#39; 1k  &lt; 0   Z&#39; 2k  &lt; 0   X k  − 4a   −X k  + 2a         0 &lt; X k  &lt; 2a           Z&#39; 2k  ≧ 0   X k  − 4a   −X k  + 2a         −2a &lt; X k  &lt; 0   X k  &lt; 0   Z&#39; 1k  &lt; 0   Z&#39; 2k  ≧ 0   −X k  − 4a     X k  + 2a       −4a &lt; X k  &lt; −2a           Z&#39; 2k  &lt; 0   −X k  − 4a     X k  + 2a       −6a &lt; X k  &lt; −4a       Z&#39; 1k  ≧ 0   Z&#39; 2k  &lt; 0   −X k  − 4a     −X k  + 6a         X k  &lt; −6a           Z&#39; 2k  ≧ 0   −X k  − 4a     −X k  + 6a                      
         [0052]    In Table 8, Z 1k =|Y k |−4a and Z 2k =|Z 1k |−2a, and in Table 9, Z′ 1k =|X k |−4a and Z′ 2k =|Z′ 1k |−2a. In Table 8 and Table 9, even the soft decision values at the 7 boundary values, which were omitted from Table 6 and Table 7 for convenience, are taken into consideration.  
         [0053]    In hardware realization, Table 8 and Table 9 can be simplified into Table 10 and Table 11 on condition that a sign of X k , Y k , Z 1k , Z 2k , Z′ 1k  and Z′ 2k  can be expressed by sign bits. Table 10 and Table 11 illustrate LLR values in terms of Y k , Z 1k , Z 2k , and X k , Z′ 1k , Z′ 2k , respectively.  
                                   TABLE 10                       MSB(Y k )   MSB(Z 1k )   MSB(Z 2k )   Λ(s k,5 )   Λ(s k,4 )   Λ(s k,3 )                   0   0   0   Y k  + 3Z 1k     Z 1k  + Z 2k     Z 2k                 1   Y k  + 3Z 1k  −   Z 1k     Z 2k                     Z 2k             1   0   Y k     Z 1k  − Z 2k     Z 2k                 1   Y k  − Z 2k     Z 1k     Z 2k         1   0   0   Y k  + 3Z 1k     Z 1k  + Z 2k     Z 2k                 1   Y k  − 3Z 1k  +   Z 1k     Z 2k                     Z 2k             1   0   Y k     Z 1k  − Z 2k     Z 2k                 1   Y k  + Z 2k     Z 1k     Z 2k                    
 
         [0054]    [0054]                                   TABLE 11                       MSB(Y k )   MSB(Z&#39; 1k )   MSB(Z&#39; 2k )   Λ(s k,2 )   Λ(s k,1 )   Λ(s k,0 )                   0   0   0   X k  + 3Z&#39; 1k     Z&#39; 1k  + Z&#39; 2k     Z&#39; 2k                 1   X k  + 3Z&#39; 1k  − Z&#39; 2k     Z&#39; 1k     Z&#39; 2k             1   0   X k     Z&#39; 1k  − Z&#39; 2k     Z&#39; 2k                 1   X k  − Z&#39; 2k     Z&#39; 1k     Z&#39; 2k         1   0   0   X k  + 3Z&#39; 1k     Z&#39; 1k  + Z&#39; 2k     Z&#39; 2k                 1   X k  − 3Z&#39; 1k  + Z&#39; 2k     Z&#39; 1k     Z&#39; 2k             1   0   X k     Z&#39; 1k  − Z&#39; 2k     Z&#39; 2k                 1   X k  − Z&#39; 2k     Z&#39; 1k     Z&#39; 2k                      
         [0055]    In Table 10 and Table 11 MSB(x) denotes a most significant bit (MSB) of a given value x.  
         [0056]    From Table 10, soft decision values Λ(s k,5 ), Λ(s k,4 ) and Λ(s k,3 ) at i=5, 4 and 3 are respectively expressed as  
         [0057]    Equation (9)  
                   Λ        (     s     k   ,   5       )       =       Y   k     +     c        (       α   ·     Z     1      k         +     β   ·     Z     2      k           )           ,   where          
          α   =     {             3           if                   MSB        (     Z     1      k       )         =   0             0           if                   MSB        (     Z     1      k       )         =   1                
        β     =     {             0           if                   MSB        (     Z     2      k       )         =   0               -   1             if                   MSB        (     Z     2      k       )         =   1              and        
        c     =     {         1           if                   MSB        (     Y   k     )         =   0               -   1             if                   MSB        (     Y   k     )         =   1                               Equation                   (   9   )                                 
 
         [0058]    Equation (10)  
                 Λ        (     s     k   ,   4       )       =       Z     1      k       +     γ   ·     Z     2      k             ,       where        
        γ     =     {         0           if                   MSB        (     Z     2      k       )         =   1             1           if                   MSB        (     Z     2      k       )         =       0                 and                   MSB        (     Z     1      k       )         =   0                 -   1             if                   MSB        (     Z     2      k       )         =       0                 and                   MSB        (     Z     1      k       )         =   1                         Equation                   (   10   )                                 
 
         [0059]    Equation (11)  
         Λ( s   k,3 )= Z   2k    
         [0060]    From Table 11, soft decision values Λ(s k,2 ), Λ(s k,1 ) and Λ(s k,0 ) at i=2, 1 and 0 are respectively expressed as  
         [0061]    Equation (12)  
                   Λ        (     s     k   ,   2       )       =       X   k     +       c   ′          (         α   ′     ·     Z     1      k     ′       +       β   ′     ·     Z     2      k     ′         )           ,   where          
            α   ′     =     {             3           if                   MSB        (     Z     1      k     ′     )         =   0             0           if                   MSB        (     Z     1      k     ′     )         =   1                
          β   ′       =     {             0           if                   MSB        (     Z     2      k     ′     )         =   0               -   1             if                   MSB        (     Z     2      k     ′     )         =   1              and        
          c   ′       =     {         1           if                   MSB        (     X   k     )         =   0               -   1             if                   MSB        (     X   k     )         =   1                               Equation                   (   12   )                                 
 
         [0062]    Equation (13)  
                 Λ        (     s     k   ,   1       )       =       Z     1      k     ′     +       γ   ′     ·     Z     2      k     ′           ,       where        
          γ   ′       =     {         0           if                   MSB        (     Z     2      k     ′     )         =   1             1           if                   MSB        (     Z     2      k     ′     )         =       0                 and                   MSB        (     Z     1      k     ′     )         =   0                 -   1             if                   MSB        (     Z     2      k     ′     )         =       0                 and                   MSB        (     Z     1      k     ′     )         =   1                         Equation                   (   13   )                                 
 
         [0063]    Equation (14)  
         Λ( s   k,0 )= Z′   2k    
         [0064]    That is, in the data communication system employing 64-ary QAM, it is possible to actually calculate 6 soft decision values, which are outputs of the demodulator for one received signal and inputs of the channel decoder, using the dual minimum metric procedure of Equation (4), through the simple conditional formulas of Equation (9) to Equation (14). This process is illustrated in FIGS. 2 and 3. FIGS. 2 and 3 illustrate an example of processes performed for calculating soft decision values according to an embodiment of the present invention.  
         [0065]    First, a process of calculating soft decision values Λ(s k,5 ), Λ(s k,4 ) and Λ(s k,3 ) will be described with reference to FIG. 2. In step  200 , a demodulator determines whether an MSB value of a quadrature-phase signal component Y k  is 0. As a result of the determination, if an MSB value of the quadrature-phase signal component Y k  is 0, the demodulator proceeds to step  204  and sets a value of a parameter c to 1. Otherwise, the demodulator proceeds to step  202  and sets a value of the parameter c to −1. After determining a value of the parameter c, the demodulator sets a value of Z 1k  to |Y k |−4a in step  206 . Thereafter, the demodulator determines in step  208  whether MSB of the Z 1k  determined in step  206  is 0. As a result of the determination, if MSB of the Z 1k  is 0, the demodulator proceeds to step  212  and sets a value of a parameter α to 3. Otherwise, the demodulator proceeds to step  210  and sets a value of the parameter α to 0. After setting a value of the parameter α, the demodulator sets a value of Z 2k  to |Z 1k |−2a in step  214 . Thereafter, the demodulator determines in step  216  whether MSB of the Z 2k  is 0. As a result of the determination, if MSB of the Z 2k  is 0, the demodulator proceeds to step  220  and sets a value of a parameter β to 0. Otherwise, the demodulator proceeds to step  218  and sets a value of the parameter β to −1 and a value of a parameter γ to 0. After step  220 , the demodulator determines in step  222  whether MSB of the Z 1k  is 0. As a result of the determination, if MSB of the Z 1k  is 0, the demodulator proceeds to step  224  and sets a value of the parameter γ to 1. Otherwise, the demodulator proceeds to step  226  and sets a value of the parameter γ to −1. Based on the determined values of the parameters α, β, γ and c, the demodulator calculates the soft decision values Λ(s k,5 ), Λ(s k,4 ) and Λ(s k,3 ) in step  228 .  
         [0066]    Next, a process for calculating soft decision values Λ(s k,2 ), Λ(s k,1 ) and Λ(s k,0 ) will be described with reference to FIG. 3. In step  300 , the demodulator determines whether an MSB value of an in-phase signal component X k  is 0. As a result of the determination, if an MSB value of the in-phase signal component X k  is 0, the demodulator proceeds to step  304  and sets a value of a parameter c′ to 1. Otherwise, the demodulator proceeds to step  302  and sets a value of the parameter c′ to −1. After determining a value of the parameter c′, the demodulator sets a value of Z′ 1k  to |X k |−4a in step  306 . Thereafter, the demodulator determines in step  308  whether MSB of the Z 1k  determined in step  306  is 0. As a result of the determination, if MSB of the Z 1k  is 0, the demodulator proceeds to step  312  and sets a value of a parameter α′ to 3. Otherwise, the demodulator proceeds to step  310  and sets a value of the parameter α′ to 0. After setting a value of the parameter α′, the demodulator sets a value of Z′ 2k  to |Z′ 1k |−2a in step  314 . Thereafter, the demodulator determines in step  316  whether MSB of the Z′ 2k  is 0. As a result of the determination, if MSB of the Z′ 2k  is 0, the demodulator proceeds to step  320  and sets a value of a parameter β′ to 0. Otherwise, the demodulator proceeds to step  318  and sets a value of the parameter β′ to −1 and a value of a parameter γ′ to 0. After step  320 , the demodulator determines in step  322  whether MSB of the Z′ 1k  is 0. As a result of the determination, if MSB of the Z′ 1k  is 0, the demodulator proceeds to step  324  and sets a value of the parameter γ′ to 1. Otherwise, the demodulator proceeds to step  326  and sets a value of the parameter γ′to −1. Based on the determined values of the parameters α′, β′, γ′ and c′, the demodulator calculates the soft decision values Λ(s k,2 ), Λ(s k,1 ) and Λ(s k,0 ) in step  328 .  
         [0067]    The process for calculating the soft decision values by the dual minimum metric procedure as described in conjunction with FIGS. 2 and 3 can be divided into (i) a first step of determining the parameters α, β, γ and c by analyzing the quadrature-phase signal component Y k  and a value “a” and determining the parameters α′, β′, γ′ and c′ by analyzing the in-phase signal component X k  and a value “a”, and (ii) a second step of calculating soft decision values using a received signal and the parameters determined in the first step. This process can be realized by a block diagram illustrated in FIG. 4.  
         [0068]    [0068]FIG. 4 illustrates a block diagram for calculating soft decision values using a quadrature-phase signal component Y k , an in-phase signal component X k , and a value “a”. The processes of FIGS. 2 and 3 will be described in brief with reference to FIG. 3. A quadrature-phase signal analyzer  410  determines parameters α, β, γ and c using the quadrature-phase signal Y k  and the value “a” through the process of FIG. 2. A first soft decision value output unit  420  calculates soft decision values Λ(s k,5 ), Λ(s k,4 ) and Λ(s k,3 ) using the determined parameters α, β, γ and c. Similarly, an in-phase signal analyzer  430  determines parameters α′, β′, γ′ and c′ using the in-phase signal X k  and the value “a” through the process of FIG. 3. A second soft decision value output unit  440  calculates soft decision values Λ(s k,2 ), Λ(s k,1 ) and Λ(s k,0 ) using the determined parameters α′, β′, γ′ and c′.  
         [0069]    [0069]FIGS. 5 and 6 illustrate calculators for calculating soft decision values input to a channel decoder for use in a channel demodulator in a data communication system employing  64 -ary QAM. FIG. 5 illustrates a calculator for calculating soft decision values Λ(s k,5 ), Λ(s k,4 ) and Λ(s k,3 ), and FIG. 6 illustrates a calculator for calculating soft decision values Λ(s k,2 ), Λ(s k,1 ) and Λ(s k,0 ).  
         [0070]    First, an example of a structure and operation of an apparatus for calculating the soft decision values Λ(s k,5 ), Λ(s k,4 ) and Λ(s k,3 ) will be described with reference to FIG. 5. A quadrature-phase signal Y k  and a value “a” are applied to a first operator  501 . Further, the quadrature-phase signal Y k  is applied to a second adder  519  and a first MSB calculator  529 . The first operator  501  calculates Z 1k =|Y k |−4a as described in step  206  of FIG. 2. The first MSB calculator  529  calculates MSB of the received quadrature-phase signal Y k . The output of the first operator  501  is applied to a second operator  503 , an input terminal “0” of a first multiplexer  505 , a second MSB calculator  531 , and a third adder  527 . The second MSB calculator  531  calculates MSB of the Z 1k  and provides its output to a select terminal of the first multiplexer  505  and a select terminal of a fourth multiplexer  523 . A value “0” is always applied to an input terminal “1” of the first multiplexer  505 . The first multiplexer  505  selects the input terminal “0” or the input terminal “1” thereof according to a select signal from the second MSB calculator  531 .  
         [0071]    The second operator  503  calculates Z 2k =|Z 1k |−2a as described in step  214  of FIG. 2, and provides the calculated value Z 2k  to a second multiplier  509 , a third MSB calculator  533 , a fourth multiplier  521 , and an input terminal “0” of the fourth multiplexer  523 . The value Z 2k  becomes a soft decision value Λ(s k,3 ). The second multiplier  509  multiplies the output value of the second operator  503  by a value “−1,” and provides its output to an input terminal “1” of a second multiplexer  511 . An input terminal “0” of the second multiplexer  511  always has a value “0.” 
         [0072]    Meanwhile, the third MSB calculator  533  calculates MSB of the Z 2k , and provides its output to a select terminal of the second multiplexer  511  and a select terminal of a fifth multiplexer  525 . The second multiplexer  511  selects the input terminal “0” or the input terminal “1” thereof according to a select signal from the third MSB calculator  533 . The output of the second multiplexer  511  is applied to a first adder  513 .  
         [0073]    The output of the first multiplexer  505  is applied to a first multiplier  507 . The first multiplier  507  triples the output value of the first multiplexer  505 , and provides its output to the first adder  513 . The first adder  513  adds the output of the second multiplexer  511  to the output of the first multiplier  507 , and provides its output to a third multiplier  515  and an input terminal “0” of a third multiplexer  517 . The third multiplier  515  multiplies the output of the first adder  513  by a value “−1,” and provides its output to an input terminal “1” of the third multiplexer  517 . The third multiplexer  517  selects the input terminal “0” or the input terminal “1” thereof according to a select signal provided from the first MSB calculator  529 . The output of the third multiplexer  517  is applied to the second adder  519 . The second adder  519  adds the quadrature-phase signal component Y k  to the output of the third multiplexer  517 . The output of the second adder  519  becomes the soft decision value Λ(s k,5 )  
         [0074]    Further, the fourth multiplier  521  multiplies the value Z 2k  by a value “−1” and provides its output to an input terminal “1” of the fourth multiplexer  523 . The fourth multiplexer  523  selects the input terminal “0” or the input terminal “1” thereof according to a select signal provided from the second MSB calculator  531 . The output of the fourth multiplexer  523  is applied to an input terminal “0” of the fifth multiplexer  525 . A value “0” is always applied to an input terminal “1” of the fifth multiplexer  525 . The fifth multiplexer  525  selects the input terminal “0” or the input terminal “1” thereof according to a select signal provided from the third MSB calculator  533 . The output of the fifth multiplexer  525  is applied to the third adder  527 . The third adder  527  adds the output of the fifth multiplexer  525  to the output Z 1k  of the first operator  501 . The output value of the third adder  527  becomes the soft decision value Λ(s k,4 ).  
         [0075]    In this manner, the circuit of FIG. 5 can calculate the soft decision values Λ(s k,5 ), Λ(s k,4 ) and Λ(s k,3 ) from the quadrature-phase signal component Y k  and the value “a”.  
         [0076]    Next, an example of a structure and operation of an apparatus for calculating the soft decision values Λ(s k,2 ), Λ(s k,1 ) and Λ(s k,0 ) will be described with reference to FIG. 6. An in-phase signal X k  and a value “a” are applied to a third operator  601 . Further, the in-phase signal X k  is applied to a fifth adder  619  and a fourth MSB calculator  629 . The third operator  601  calculates Z′ 1k =|X k |−4a as described in step  306  of FIG. 3. The fourth MSB calculator  629  calculates MSB of the received in-phase signal X k . The output of the third operator  601  is applied to a fourth operator  603 , an input terminal “0” of a sixth multiplexer  605 , a fifth MSB calculator  631 , and a sixth adder  627 . The fifth MSB calculator  631  calculates MSB of the Z′ 1k  and provides its output to a select terminal of the sixth multiplexer  605  and a select terminal of a ninth multiplexer  623 . A value “0” is always applied to an input terminal “1” of the sixth multiplexer  605 . The sixth multiplexer  605  selects the input terminal “0” or the input terminal “1” thereof according to a select signal from the fifth MSB calculator  631 .  
         [0077]    The fourth operator  603  calculates Z′ 2k =|Z′ 1k |−2a as described in step  314  of FIG. 3, and provides the calculated value Z′ 2k  to a sixth multiplier  609 , a sixth MSB calculator  633 , an eighth multiplier  621 , and an input terminal “0” of the ninth multiplexer  623 . The value Z′ 2k  becomes a soft decision value Λ(s k,0 ). The sixth multiplier  609  multiplies the output value of the fourth operator  603  by a value “−1,” and provides its output to an input terminal “1” of a seventh multiplexer  611 . An input terminal “0” of the seventh multiplexer  611  always has a value “0.” 
         [0078]    Meanwhile, the sixth MSB calculator  633  calculates MSB of the Z′ 2k , and provides its output to a select terminal of the seventh multiplexer  611  and a select terminal of a tenth multiplexer  625 . The seventh multiplexer  611  selects the input terminal “0” or the input terminal “1” thereof according to a select signal from the sixth MSB calculator  633 . The output of the seventh multiplexer  611  is applied to a fourth adder  613 .  
         [0079]    The output of the sixth multiplexer  605  is applied to a fifth multiplier  607 . The fifth multiplier  607  triples the output value of the sixth multiplexer  605 , and provides its output to the fourth adder  613 . The fourth adder  613  adds the output of the seventh multiplexer  611  to the output of the fifth multiplier  607 , and provides its output to a seventh multiplier  615  and an input terminal “0” of an eighth multiplexer  617 . The seventh multiplier  615  multiplies the output of the fourth adder  613  by a value “−1,” and provides its output to an input terminal “1” of the eighth multiplexer  617 . The eighth multiplexer  617  selects the input terminal “0” or the input terminal “1” thereof according to a select signal provided from the fourth MSB calculator  629 . The output of the eighth multiplexer  617  is applied to the fifth adder  619 . The fifth adder  619  adds the in-phase signal component X k  to the output of the eighth multiplexer  617 . The output of the fifth adder  619  becomes the soft decision value Λ(s k,2 ).  
         [0080]    Further, the eighth multiplier  621  multiplies the value Z′ 2k  by a value “−1” and provides its output to an input terminal “1” of the ninth multiplexer  623 . The ninth multiplexer  623  selects the input terminal “0” or the input terminal “1” thereof according to a select signal provided from the fifth MSB calculator  631 . The output of the ninth multiplexer  623  is applied to an input terminal “0” of the tenth multiplexer  625 . A value “0” is always applied to an input terminal “1” of the tenth multiplexer  625 . The tenth multiplexer  625  selects the input terminal “0” or the input terminal “1” thereof according to a select signal provided from the sixth MSB calculator  633 . The output of the tenth multiplexer  625  is applied to the sixth adder  627 . The sixth adder  627  adds the output of the tenth multiplexer  625  to the output Z′ 1k  of the third operator  601 . The output value of the sixth adder  627  becomes the soft decision value Λ(s k,1 ). In this manner, the circuit of FIG. 6 can calculate the soft decision values Λ(s k,2 ), Λ(s k,1 ) and Λ(s k,0 ) from the in-phase signal component X k  and the value “a”. According to the foregoing description, a conventional soft decision value calculator using the dual minimum metric procedure realized by Equation (4) needs one hundred or more squaring operations and comparison operations. However, the calculators according to an embodiment of the present invention as exemplified in FIGS. 5 and 6 and realized using Equation (9) to Equation (14) are comprised of 10 adders (first to fourth operators are also realized by adders), 8 multipliers and 10 multiplexers, contributing to a remarkable reduction in operation time and complexity of the calculator. Table 12 below illustrates a comparison made between the conventional calculator realized by Equation (4) and the novel calculator realized by Equations (9) to (14) in terms of the type and number of operations, for i∈{0, 1, 2, 3, 4, 5}.  
                                     TABLE 12                           Equation (4)   Equations (9) to (14)            Operation   No of Operations   Operation   No of Operations               Addition   3 × 64 + 6 = 198   Addition   10       Squaring   2 × 64 = 128   Multiplication    8       Comparison   31 × 2 × 6 = 372   Multiplexing   10                  
 
         [0081]    In summary, the embodiment of the present invention described above derives Table 6 to Table 11 from Equation (6) to Equation (8) and the process of Table 1 to Table 5, in order to reduce a time delay and complexity, which may occur when Equation (4), the known dual minimum metric procedure, or Equation (5) obtained by simplifying the dual minimum metric procedure is actually realized using the 64-ary QAM. Further, the embodiment of the present invention provides Equation (9) to Equation (14), new formulas used to realize the dual minimum metric procedure in the 64-ary QAM. In addition, the present invention provides a hardware device realized based on Equation (9) and Equation (14).  
         [0082]    As described above, in deriving a soft decision value needed as an input of a channel decoder using the dual minimum metric procedure, the novel 64-ary QAM demodulator for a data communication system can perform simple and rapid calculations while obtaining the same result as when the exiting formula is used. A soft decision value calculator realized by hardware remarkably reduces an operation time and complexity of the demodulator.  
         [0083]    While the invention has been shown and described with reference to an embodiment thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims.