# Patent

## Title
Wireless communication apparatus

## Abstract

In a wireless receiving apparatus, comprising a plurality of antennas and a plurality of receivers each of which is provided for each antenna, each receiver uses a received signal that was received by the receiving antenna that is connected to it, and received signals that were received by the other receiving antennas, to calculate probability of a signal that was transmitted from a specified transmitter, and judges the signal that was transmitted from the transmitter based on that probability. The receivers calculate the probability using an equation that calculates the probability based on the posteriori probability that the signal transmitted from the transmitter is +1, and the posteriori probability that the signal transmitted from the transmitter is −1.

## Background
BACKGROUND OF THE INVENTION
This invention relates to a wireless communication apparatus, and more particularly to a MIMO (Multiple-Input Multiple-Output)-capable wireless communication apparatus in a transmission system comprising a MIMO configuration.
Among current wireless communication systems, there is increasing interest in space division multiplexing transmission technologies (MIMO transmission systems) in which, by transmitting different data streams in parallel from a plurality of transmission antennas, the transmission capacity can be increased in proportion to the number of transmission antennas. The main feature of a MIMO system is its capability to utilize a multipass so that it becomes advantageous to the user, and effectively takes advantage of the strong points of random fading and multipass delay dispersion (when it can be used).
In a MIMO transmission system, normally signal processing to subtract replicas that occur is employed to remove the interference by other data streams from a data stream of interest (V-BLAST, MSSE-VBLAST, Zero-Forced V-BLAST). These methods can be executed with suitable complexity, however, the BER result is very inferior compared with that from MLD (Maximum Likelihood Decoding).
An iterative MAP receiver (turbo receiver) performs iterative maximum likelihood estimation. This technology provides better BER results than V-BLAST, MSSE-VBLAST or Zero-Forced V-BLAST, and that are very close to BER results by MLD. However, a turbo receiver has its weak points. The main weak point is that it is necessary perform turbo iterative operation a plurality of times before performing final determination for the received signal. The nature of the iterative operations performed by the turbo receiver makes computation complicated, and increases the decoding time.
The receiver of this invention (Single Shot SS receiver) is based on a parallel maximum a posteriori probabilities estimation algorithm, and does not require iterative computation. In the SS receiver of this invention, in order to increase the judgment speed for a plurality of soft-judgment results derived from one data stream, all of the receivers execute computation in parallel.
FIG. 1 shows the configuration of a MIMO system, where TRX is a transmission station and REC is a receiver station.
Data streams D0 to DM−1 of the same number as the number of transmission antennas M pass through data modulation, D/A conversion, quadrature modulation, frequency up-conversion, and other processing by respective transmission apparatus TRX0 to TRXM−1, and are transmitted from the respective transmission antennas ATT0 to ATTM−1. Signals transmitted from the antennas ATT0 to ATTM−1 pass through independent fading channels hnm (m=0 to M−1, n=0 to N−1), and after space division multiplexing, are received by N receiving antennas ATR0 to ATRN−1. Signals received by the receiving antennas pass through frequency down-conversion, quadrature detection, A/D conversion and other processing by receiving apparatus REC0 to RECN−1, and received data streams y0 to yN−1 are generated. Each of these received data streams is in the form of M multiplexed transmitted data streams, so that by performing signal processing of all received data streams, the transmitted data streams are separated and reproduced.
Algorithms for signal processing to separate transmitted data streams D0 to DM−1 from received signals include such linear algorithms as ZF (Zero-Forcing) and MMSE which use the inverse matrix of a channel correlation matrix (see non-patent document 1), and non-linear algorithms of which BLAST (Bell Laboratories Layered Space-Time) is representative. In addition, there are also methods, such as MLD (Maximum Likelihood Decoding), which do not use operations on the inverse matrix of a correlation matrix (see non-patent document 2).
ZF (Zero-Forcing) Algorithm
If a transmitted data stream is represented by an M-dimensional complex matrix, and a received data stream is represented by an N-dimensional complex matrix, then the relation of the following equation occurs.




  
    
      
        
          
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The ZF (Zero-Forcing) algorithm uses the following equation to estimate the transmitted data stream.

{circumflex over (D)}=(H*·H)−1H*D=H+·Y  (2)

Here, H*H is called the channel correlation matrix. H+ denotes the pseudo-inverse matrix; and in order for this pseudo-inverse matrix to exist, it is necessary that N≧M.
MMSE Algorithm
The MMSE (Minimum Mean Square Error) method is another linear estimation approach in which the transmitted data stream (received data vector) D is estimated based on the received data stream Y. In this MMSE algorithm, the following equation is used to determine the matrix G:

ε2=E[(D−{circumflex over (D)})*(D−{circumflex over (D)})]=[(D−G·Y)*(D−G·Y)]  (3)

Then, the following equation is used to estimate the data stream D.

{circumflex over (D)}=(α·I+H*·H)−1H*·Y  (4)

Here, α is the noise dispersion, and I is a matrix which depends on G. The ZF algorithm corresponds to the MMSE algorithm when α=0.
Decision Feedback Decoding
It is expected that if the most reliable element of the transmitted vector data D is decoded and used to improve decoding of the other elements, performance will be improved. This method is called symbol cancellation, and when combined with the ZF method and MMSE method, is known as ZF-VBL and MSSE-VBL.
MLD Algorithm
The MLD algorithm is a method that does not use operations in the inverse matrix of a correlation matrix, but uses the following equation to estimate the transmitted data stream (transmission vector) D.

{circumflex over (D)}=arg min∥Y−H·D∥2  (5)

Here, if the number of signal constellations of modulated data input to M antennas is Q, then there exists QM combinations of transmission vectors D. In QPSK, Q=4, in 16 QAM, Q=16, and in 64 QAM, Q=64. In the MLD algorithm, candidates (replicas) of the QM transmission vectors are generated, and the operation of equation (5) is performed, and the replica for which the result is the smallest is estimated to be the input data.
Above, is the normal case in which the number of transmission antennas is M, and the number of receiver antennas is N, so when M=2 and N=2, Equation (1) becomes as shown below.




  
    
      
        
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The inventors have proposed a turbo receiver for a MIMO communication system that comprises: a transmission station that sends transmission using separate transmission antennas for two different data streams, and a receiving station that divides and outputs the data streams that have been multiplexed in space by signals that are received from the two antennas (see patent document 3).
FIG. 2 is a block diagram showing a MIMO communication system that comprises the turbo receiver proposed above, and as shown in (A) of FIG. 2, the transmission station TRX comprises two transmitters TRX0, TRX1 and two transmission antennas AT0, AT1, and the receiving station REC comprises two receiving antennas AR0, AR1 and two receivers (turbo receivers) REC0, REC1. As shown in (B) of FIG. 2, each of the receivers REC0, REC1 comprises a receiving unit RV and signal-processing unit PR, and the receiving unit RV performs processing such as data modulation, D/A conversion, quadrature conversion frequency up-conversion, and it inputs signals y0, y1 to the signal-processing unit PR, and the signal-processing unit PR determines the bit of the data stream.
Each transmitter TRX0, TRX1 performs QPSK or BPSK modulation and transmission of the information symbol Di (i=0, 1). The first transmitter TRX0 transmits a first data stream, and the second transmitter TRX1 transmits a second data stream.
(1) The first receiver REC0 calculates an estimated bit value for the first data stream D0 according to a signal that is received by the antenna AR0, and the second receiver REC1 calculates an estimated bit value for the first data stream D0 according to a signal that is received by the antenna AR1, then the first receiver REC0 combines these as the estimated bit value ΔlnP0 of the first data stream. Similarly, the second receiver REC1 obtains the estimated bit value ΔlnP1 of the second data stream D1.
(2) The first receiver REC0 refines the estimated bit value ΔlnP0 of the first data stream that was obtained by the aforementioned computation with the estimated bit value ΔlnP1 of the second data stream, and similarly the second receiver REC1 refines the estimated bit value ΔlnP1 of the second data stream with the estimated bit value ΔlnP0 of the first data stream.
(3) The first receiver REC0 and second receiver REC1 repeat the process of steps (1) to (2) a specified number of times.
(4) After a specified number of repetitions, the first receiver REC0 determines the bit D0 of the first data stream based on the aforementioned estimated bit value ΔlnP0 of the first data stream, and similarly, the second receiver REC1 determines the bit D1 of the second data stream based on the estimated bit value ΔlnP1 of the second data stream.
FIG. 3 and FIG. 4 show the simulation results for each of the aforementioned receiving methods (ZF-VBL method, MSSE-VBL method, QR-MLD method, MLD method and turbo method), and show the BER (Bit Error Rate) for 2Eb/No. In the simulation, there were two transmission antennas and two receiving antennas, and transmission was performed using QPSK modulation or 16 QAM modulation without encoding. FIG. 3 shows the case in which QPSK modulation was performed, and FIG. 4 shows the case in which 16 QAM modulation was performed, and both show the case in which there was no interference between channels (ICI-Free). The ICI-Free points correspond with AWGN obtained using simulation software.
The simulation results show that the MLD method gave the best BER characteristic. However, there is a very large amount of computations performed in the MLD method, and there is a problem in that the amount of computations due to the increased number of antennas increases exponentially. The proposed turbo receiver shows a BER characteristic that is very close to that of a receiver using the MDL method. However, in 16 QAM modulation, in order to obtain the same BER characteristic as in the MLD method, it is necessary to increase the number of iterations from two times to four times more than in the case of QPSK modulation; therefore computation becomes more complex and increases, and the time necessary for decoding increases. The complexity of the turbo receiver is less than that of a receiver using the MLD method, however, there is a problem in that the complexity increases proportionately with the number of iterations.
[Non-patent Document 1]
A. van Zelst, “Space Division Multiplexing Algorithms”, 10th Mediterranean Electrotechnical Conference 2000, MELECON 2000, Cyprus, May 2000, Vol. 3, pp. 1218-1221.
[Non-patent Document 2]
Richard van Nee et al. “Maximum Likelihood Decoding in a Space Division Multiplexing System”; Proc. of IEEE Vehicular Technology Conference (VTC) 2000, Tokyo 2000.
[Patent Document 3]
Japanese patent application no. 2004-378888 (Dec. 28, 2004)
SUMMARY OF THE INVENTION
The object of the present invention is to provide a wireless receiving apparatus (wireless receiver) for a MIMO system that is capable of low bit error rate, and that is capable of decreasing complexity and the amount of computation.
Another object of the present invention is to provide a wireless receiver (single shot (SS) receiver) that employs a parallel maximum a posteriori probabilities estimation algorithm and is capable of reducing the bit error rate without repetitive processing.
The present invention accomplishes the aforementioned object by a wireless receiving apparatus that comprises a plurality of receiving antennas that receive signals that are transmitted from a plurality of transmission antennas and input via a plurality of propagation paths having different propagation characteristics. This wireless receiving apparatus comprises receivers that correspond to each of the plurality of antennas, where each receiver comprises: a computation unit that uses a received signal that was received by the receiving antenna that is connected to it, and received signals that were received by the other receiving antennas, to calculate probability of a signal that was transmitted from a specified transmitter, and a judgment unit that judges the signal that was transmitted from the transmitter based on that probability.
The computation unit calculates the probability using an equation that calculates the probability based on the posteriori probability that the signal transmitted from the transmitter is +1, and the posteriori probability that the signal transmitted from the transmitter is −1.
The computation unit comprises: a plurality of non-linear-processing circuits having a transfer function as an amplitude limiter; and a combining circuit that combines the output from the plurality of non-linear-processing circuits and outputs the probability of the transmitted signal. Also, in addition to the non-linear-processing circuits and combining circuit, the computation circuit comprises a signal-generation unit that generates an input signal input to each of the non-linear-processing circuits based on the product of a specified antenna received signal and specified propagation characteristic.
When a transmission station transmits data using QPSK modulation, each of the receivers comprises: two computation units that execute the aforementioned processing for the real-number portion and imaginary-number portion of the received signal; and two judgment units that execute the aforementioned processing for the real-number portion and imaginary-number portion of the received signal.
With this invention, the receivers that correspond to each of the plurality of receiving antennas use the signal that was received by the receiving antenna that is connected to it, and the signals that were received by the other receiving antennas, to calculate the probability of the signal that was transmitted from a specified transmitter, and judges the signal that was transmitted from that transmitter based on that probability, so it is possible to improve the BER characteristic and reduce the number of computations without performing iterative computation.
Also, the probability-computation unit comprises a plurality of non-linear-processing circuits having a transfer function as an amplitude limiter, and a combining circuit that combines the outputs from the plurality of non-linear-processing circuits so the construction of the receivers is simplified.
Moreover, when a transmission station transmits data using QPSK modulation, each of the receivers comprises a respective computation unit for the real-number portion and imaginary-number portion of the received signal, and a judgment unit that judges the bit based on the probability calculated by the computation units, so even in the case of QPSK modulation, it is possible to improve the BER characteristic and output the determined bit value with a small amount of computations.
Other features and advantages of the present invention will be apparent from the following description taken in conjunction with the accompanying drawings.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 shows the configuration of a MIMO system.
FIG. 2 is a block diagram of a MIMO communication system comprising a proposed turbo receiver.
FIG. 3 shows the simulation results for each receiving method (ZF-VBL method, MSSE-VBL method, QR-MLD method, MLD method, turbo method), and is for the case in which QPSK modulation is performed.
FIG. 4 shows the simulation results for each receiving method (ZF-VBL method, MSSE-VBL method, QR-MLD method, MLD method, turbo method), and is for the case in which 16 QAM modulation is performed.
FIG. 5 shows the configuration of the MIMO transmission system of this invention for the case in which both the transmission station and receiving station have two antennas.
FIG. 6 shows the configuration of the wireless receiver of this invention.
FIG. 7 shows the transmission characteristics of a limiter, which is a non-linear element.
FIG. 8 shows the configuration of the generation unit for input signals S0 to S4 that are input to the non-linear element.
FIG. 9 shows the configuration of the signal-processing unit that determines whether or not the real number portion of the first symbol D0 is +1 or −1.
FIG. 10 shows the configuration of the generation unit for input signals S0 to S4 that are input to the non-linear element.
FIG. 11 shows the configuration of the signal-processing unit that determines whether or not the imaginary number portion of the first symbol D0 is +1 or −1.
FIG. 12 shows the BER characteristic for the SS receiver of this invention for the case in which QPSK modulation is performed.
FIG. 13 shows the BER characteristic for the SS receiver of this invention for the case in which 16 QAM modulation is performed.



DESCRIPTION OF THE PREFERRED EMBODIMENTS
(A) Outline of the Invention
FIG. 5 is a block diagram of a MIMO communication system comprising the wireless receiver (wireless receiving apparatus) of this invention, where the transmission station TRX comprises two transmitters TRX0, TRX1, and two transmission antennas AT0, AT1, and the receiving station (wireless receiver) REC comprises two receiving antennas AR0, AR1 and two receivers REC0, REC1. Each of the transmitters TRX0, TRX1 performs QPSK, M-QAM or BPSK modulation and transmits the information symbol Di (i=0, 1). The first transmitter TRX0 transmits a first data stream D0, and the second transmitter TRX1 transmits a second data stream D1. D0 and D1 are both 1 bit when modulated using BPSK modulation, are both 2 bits when modulated using QPSK modulation, and are both 4 bits when modulated using 16-QAM modulation.
The first data stream D0 that is transmitted from the first transmitter TRX0 passes through transmission paths having fading characteristics (channel characteristics) h00, h10, and arrives at the first and second receiving antennas AR0, AR1. Also, the second data stream D1 that is transmitted from the second transmitter TRX1 passes through transmission paths having channel characteristics h01, h11, and arrives at the first and second receiving antennas AR0, AR1.
The first receiver REC0 uses the received signal y0 that is received by the antenna AR0 and the received signal y1 that is received by the other receiving antenna AR1, and based on the difference between the posteriori probability that the signal transmitted from the first transmitter TRX0 is +1 and the posteriori probability that the transmitted signal is −1, calculates the probability of the bit signal that is transmitted from the first transmitter TRX0, and then based on that calculated probability, judges the bit signal that is transmitted from the transmitter TRX0. Similarly, the second receiver REC1 uses the received signal y1 that is received by the antenna AR1 and the received signal y0 that is received by the other receiving antenna AR0, and based on the difference between the posteriori probability that the signal transmitted from the second transmitter TRX1 is +1 and the posteriori probability that the transmitted signal is −1, calculates the probability of the bit signal that is transmitted from the second transmitter TRX1, and then based on that calculated probability, determines the bit signal that is transmitted from the transmitter TRX1.
(B) Embodiments
The MIMO transmission system comprising the wireless receiving apparatus of this invention has the block configuration shown in FIG. 5. As shown in FIG. 6, the wireless receiver REC of this MIMO communication system comprises two receiving antennas AR0, AR1 and two receivers REC0, REC1.
The first receiver REC0 comprises a receiving unit RV0 and signal-processing unit PR0, where the receiving unit RV0 performs processing such as data modulation, D/A conversion, quadrature modulation, frequency up-conversion or the like on the signal received by the first receiving antenna AR0, and inputs the obtained signal y0 to the signal-processing unit PR0 and to the signal-processing unit PR1 of the second receiver REC1. The signal-processing unit PR0 uses the received signal y0 that was received by the antenna AR0, and the received signal y1 that was received by the other receiving antenna AR1, and based on the difference between the posteriori probability that the signal transmitted from the first transmitter TRX0 will be +1, and the posteriori probability that the signal will be −1, calculates the probability of the transmitted signal, and based on that probability, judges the signal transmitted by the transmitter TRX0.
When the transmitter transmits using QPSK modulation, the signal-processing unit PR0 calculates the probability for each real-number portion and imaginary-number portion of a plurality of signals transmitted by the first transmitter TRX0, and based on that probability, judges the real-number portion and imaginary-number portion of a plurality of signals transmitted by the transmitter TRX0, and outputs the judgment results Re(D0), Im(D0).
The second receiver REC1 comprises a receiving unit RV1 and signal-processing unit PR1, where the receiving unit RV1 performs processing such as data modulation, D/A conversion, quadrature modulation, frequency up-conversion or the like on the signal received by the second receiving antenna AR1, and inputs the obtained signal y1 to the signal-processing unit PR1 and to the signal-processing unit PR0 of the first receiver REC0. The signal-processing unit PR1 uses the received signal y1 that was received by the antenna AR1, and the received signal y0 that was received by the other receiving antenna AR0, and based on the difference between the posteriori probability that the signal transmitted from the second transmitter TRX1 will be +1, and the posteriori probability that the signal will be −1, calculates the probability of the transmitted signal, and based on that probability, judges the signal transmitted by the transmitter TRX1.
When the transmitter transmits using QPSK modulation, the signal-processing unit PR1 calculates the probability for each real-number portion and imaginary-number portion of a plurality of signals transmitted by the second transmitter TRX1, and based on that probability, judges the real-number portion and imaginary-number portion of a plurality of signals transmitted by the transmitter TRX1, and outputs the judgment results Re(D1), Im(D1).
Each of the transmitters TRX0, TRX1 performs QPSK modulation and transmits the information symbols Di (i=0, 1). The first transmitter TRX0 transmits a first data stream, and the second transmitter TRX1 transmits a second data stream.
The information symbols Di (i=0, 1) of the first and second data streams have a real-number portion Re(Di) and an imaginary-number portion Im(Di). Therefore, Di=(dRe, dIm), where dRe=Re(Di), and dIm=Im(Di). In the case of QPSK modulation, the information symbol Di is one of the four possible signals Sj*(t) (j=0, 1, 2, 3). The number j of the signal Sj*(t) depends on the pair of data bits dRe=Re(Di), dIm=Im(Di) as shown in Table 1.











TABLE 1





j
Signal
D
dRe = Re(Di)
dIm = Im(Di)







0
S0*(t)
(+1 +1)
(+1)
(+1)


1
S1*(t)
(+1 −1)
(+1)
(−1)


2
S2*(t)
(−1 +1)
(−1)
(+1)


3
S3*(t)
(−1 −1)
(−1)
(−1)









The thing to notice about signal Sj*(t) is the relationship S0*(t)=−S3*(t), and S1*(t)=−S2*(t). Assuming QPSK transmission, the received signals y0(t), y1(t) that are received by the first and second receivers REC0, REC1 can be expressed in a linear combination with the signals transmitted by the transmitters TRX0, TRX1 as in the following equations.

y0(t)=h00·D0+h01·D1  (7)

y1(t)=h10·D0+h11·D1  (7)′

A new notation Sij*(t) is adopted for the transmitted signal. The index ‘i’ in Sij*(t) indicates the number (0 or 1) of the transmitter, and the index ‘j’ indicates the information symbol Sj*(t) (j=0, 1, 2, 3) in Table 1 that is transmitted by the ith transmitter.
By combining the information symbols Di(i=0, 1) transmitted by the first and second transmitters TRX1, TRX2, there are 16 possible signals for the input of each of the receivers REC0, REC1. Table 2 shows all of the possible signals for the input of the first receiver REC0, and Table 3 shows all of the possible signals for the input of the second receiver REC1.







TABLE 2







REC0











Sj(t)
D0
D1







 S0 = h00 · S00* + h01 · S10*
(+1 +1)
(+1 +1)



 S1 = h00 · S00* + h01 · S11*
(+1 +1)
(+1 −1)



 S2 = h00 · S00* + h01 · S12*
(+1 +1)
(−1 +1)



 S3 = h00 · S00* + h01 · S13*
(+1 +1)
(−1 −1)



 S4 = h00 · S01* + h01 · S10*
(+1 −1)
(+1 +1)



 S5 = h00 · S01* + h01 · S11*
(+1 −1)
(+1 −1)



 S6 = h00 · S01* + h01 · S12*
(+1 −1)
(−1 +1)



 S7 = h00 · S01* + h01 · S13*
(+1 −1)
(−1 −1)



 S8 = h00 · S02* + h01 · S10*
(−1 +1)
(+1 +1)



 S9 = h00 · S02* + h01 · S11*
(−1 +1)
(+1 −1)



S10 = h00 · S02* + h01 · S12*
(−1 +1)
(−1 +1)



S11 = h00 · S02* + h01 · S13*
(−1 +1)
(−1 −1)



S12 = h00 · S03* + h01 · S10*
(−1 −1)
(+1 +1)



S13 = h00 · S03* + h01 · S11*
(−1 −1)
(+1 −1)



S14 = h00 · S03* + h01 · S12*
(−1 −1)
(−1 +1)



S15 = h00 · S03* + h01 · S13*
(−1 −1)
(−1 −1)

















TABLE 3







REC1











Sj(t)
D0
D1







S16 = h10 · S00* + h11 · S10*
(+1 +1)
(+1 +1)



S17 = h10 · S00* + h11 · S11*
(+1 +1)
(+1 −1)



S18 = h10 · S00* + h11 · S12*
(+1 +1)
(−1 +1)



S19 = h10 · S00* + h11 · S13*
(+1 +1)
(−1 −1)



S20 = h10 · S01* + h11 · S10*
(+1 −1)
(+1 +1)



S21 = h10 · S01* + h11 · S11*
(+1 −1)
(+1 −1)



S22 = h10 · S01* + h11 · S12*
(+1 −1)
(−1 +1)



S23 = h10 · S01* + h11 · S13*
(+1 −1)
(−1 −1)



S24 = h10 · S02* + h11 · S10*
(−1 +1)
(+1 +1)



S25 = h10 · S02* + h11 · S11*
(−1 +1)
(+1 −1)



S26 = h10 · S02* + h11 · S12*
(−1 +1)
(−1 +1)



S27 = h10 · S02* + h11 · S13*
(−1 +1)
(−1 −1)



S28 = h10 · S03* + h11 · S10*
(−1 −1)
(+1 +1)



S29 = h10 · S03* + h11 · S11*
(−1 −1)
(+1 −1)



S30 = h10 · S03* + h11 · S12*
(−1 −1)
(−1 +1)



S31 = h10 · S03* + h11 · S11*
(−1 −1)
(−1 −1)










(a) Determining the Real-Number Portion dRe=Re(D0) of the Symbol D0 Received by the First Receiver REC0

From Table 2, the posteriori probability that the first receiver REC0 will receive Sj (j=0, 1, . . . , 15) is given by the following equation according to the Bayes' mixed rule.




  
    
      
        
          
            
              
                
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                        y
                        0
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      P
                      a
                    
                    ⁡
                    
                      (
                      
                        S
                        j
                      
                      )
                    
                  
                  ·
                  
                    P
                    ⁡
                    
                      (
                      
                        
                          
                            y
                            0
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        /
                        
                          S
                          j
                        
                      
                      )
                    
                  
                
                
                  P
                  ⁡
                  
                    (
                    
                      
                        y
                        0
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    )
                  
                
              
            
          
        
        
          
            
              ≡
              
              ⁢
              
                
                  
                    k
                    0
                  
                  ·
                  
                    
                      P
                      a
                    
                    ⁡
                    
                      (
                      
                        S
                        j
                      
                      )
                    
                  
                  ·
                  exp
                
                ⁢
                
                  {
                  
                    
                      -
                      
                        1
                        
                          N
                          0
                        
                      
                    
                    ⁢
                    
                      
                        ∫
                        0
                        T
                      
                      ⁢
                      
                        
                          
                            [
                            
                              
                                
                                  y
                                  0
                                
                                ⁡
                                
                                  (
                                  t
                                  )
                                
                              
                              -
                              
                                S
                                j
                              
                            
                            ]
                          
                          2
                        
                        ⁢
                        
                          d
                          t
                        
                      
                    
                  
                  }
                
              
            
          
        
      
    
    
      
        (
        8
        )
      
    
  




Where,

    
    
        k0 is the normalization factor;
        j=0, 1, . . . , 15;
        y0(t) is a combined signal obtained by combining a signal Sj and a white Gauss noise n(t) that contains the spectral power strength N0(y0(t)=Sj+n(t));
        Pi(Sj/y0(t)) is the posteriori probability that the signal Sj(t) will be received by the ith receiver (probability that the received signal y0(t) is Sj(t));
        Pi(dRe/y0(t)) is the posteriori probability that dRe will be received by the ith receiver (probability that the received signal y0(t) is dRe);
        Pa(Sj) is the priori probability that the transmission signal is Sj(t);
        P(y0(t)/Sj) is a conditional probability, and is the probability that a transmitted code word was Sj when the received word was y0(t);
        P(y0(t)) is the probability that y0(t) will be received.
    
    


Similarly, from Table 3 the posteriori probability that the second receiver REC1 will receive Sj (j=16, 17, . . . , 31) is given by the following equation.




  
    
      
        
          
            
              
                
                  P
                  2
                
                ⁡
                
                  [
                  
                    
                      d
                      Re
                    
                    /
                    
                      
                        y
                        1
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                  ]
                
              
              =
              
              ⁢
              
                
                  P
                  2
                
                ⁡
                
                  [
                  
                    
                      S
                      j
                    
                    /
                    
                      
                        y
                        1
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      P
                      a
                    
                    ⁡
                    
                      (
                      
                        S
                        j
                      
                      )
                    
                  
                  ·
                  
                    P
                    ⁡
                    
                      (
                      
                        
                          
                            y
                            1
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        /
                        
                          S
                          j
                        
                      
                      )
                    
                  
                
                
                  P
                  ⁡
                  
                    (
                    
                      
                        y
                        1
                      
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    )
                  
                
              
            
          
        
        
          
            
              ≡
              
              ⁢
              
                
                  k
                  0
                
                ⁢
                
                  
                    
                      P
                      a
                    
                    ⁡
                    
                      (
                      
                        S
                        j
                      
                      )
                    
                  
                  ·
                  exp
                
                ⁢
                
                  {
                  
                    
                      -
                      
                        1
                        
                          N
                          0
                        
                      
                    
                    ⁢
                    
                      
                        ∫
                        0
                        T
                      
                      ⁢
                      
                        
                          
                            [
                            
                              
                                
                                  y
                                  1
                                
                                ⁡
                                
                                  (
                                  t
                                  )
                                
                              
                              -
                              
                                S
                                j
                              
                            
                            ]
                          
                          2
                        
                        ⁢
                        
                          d
                          t
                        
                      
                    
                  
                  }
                
              
            
          
        
      
    
    
      
        (
        9
        )
      
    
  




The probability that the received bit dRe=+1 can be expressed as the logical AND operation (AND) of the posteriori probabilities P(Sj) that the signal Sj will be received by the first and second receiver REC0, REC1, and can be written as below.




  
    
      
        
          P
          ⁡
          
            (
            
              
                d
                Re
              
              =
              
                +
                1
              
            
            )
          
        
        =
        
          
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  0
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  0
                
                )
              
            
            ⁢
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  16
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  16
                
                )
              
            
          
          +
          
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  1
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  1
                
                )
              
            
            ⁢
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  17
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  17
                
                )
              
            
          
          +
          
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  2
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  2
                
                )
              
            
            ⁢
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  18
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  18
                
                )
              
            
          
          +
          
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  3
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  3
                
                )
              
            
            ⁢
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  19
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  19
                
                )
              
            
          
          +
          
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  4
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  4
                
                )
              
            
            ⁢
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  20
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  20
                
                )
              
            
          
          +
          
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  5
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  5
                
                )
              
            
            ⁢
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  21
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  21
                
                )
              
            
          
          +
          
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  6
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  6
                
                )
              
            
            ⁢
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  22
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  22
                
                )
              
            
          
          +
          
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  7
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  7
                
                )
              
            
            ⁢
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  23
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  23
                
                )
              
            
          
        
      
    
    
      
        (
        10
        )
      
    
  




Similarly, the probability that the received bit dRe=−1 can be written as below.




  
    
      
        
          P
          ⁡
          
            (
            
              
                d
                Re
              
              =
              
                -
                1
              
            
            )
          
        
        =
        
          
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  8
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  8
                
                )
              
            
            ⁢
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  24
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  24
                
                )
              
            
          
          +
          
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  9
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  9
                
                )
              
            
            ⁢
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  25
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  25
                
                )
              
            
          
          +
          
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  10
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  10
                
                )
              
            
            ⁢
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  26
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  26
                
                )
              
            
          
          +
          
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  11
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  11
                
                )
              
            
            ⁢
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  27
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  27
                
                )
              
            
          
          +
          
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  12
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  12
                
                )
              
            
            ⁢
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  28
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  28
                
                )
              
            
          
          +
          
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  13
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  13
                
                )
              
            
            ⁢
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  29
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  29
                
                )
              
            
          
          +
          
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  14
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  14
                
                )
              
            
            ⁢
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  30
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  30
                
                )
              
            
          
          +
          
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  15
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  15
                
                )
              
            
            ⁢
            
              
                P
                a
              
              ⁡
              
                (
                
                  S
                  31
                
                )
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  S
                  31
                
                )
              
            
          
        
      
    
    
      
        (
        11
        )
      
    
  




In equations (10) and (11), Pa(Sj) is the priori probability that the transmitted/received signal is Sj(t), and P(Sj) is the probability of receiving the signal Sj(t) during mixed input of y0(t) and y1(t). Here, supposing that the bit in the stream is statically independent, and the probabilities are equal (P(+1)=P(−1)=1⁄2), then the priori probability Pa(Sj) that the transmitted/received signal is Sj(t) is 0.25 (=0.5×0.5).
According to equations (10) and (11), judging a group of information symbols D0, D1 for both receivers REC0, REC1 is performed in parallel at the same time. Based on Table 2 and Table 3 the signals are joined by the logical AND operation of equations (10) and (11). For example, supposing that a group of information symbols D0, D1 equivalent to (+1+1) are transferred, the signal corresponding to the receiver input becomes the AND of S0 of the first receiver REC0 and S16 of the second receiver REC1. In Equation (10) and (11), this logical AND operation is expressed as the product of probabilities, P(S0) AND P(S16), or P(S0)·P(S16). From Table 2 and Table 3, similar results can be obtained for all possible bit D0, D1 combinations and signals.
Supposing that the priori probability Pa(Sj) that the transmitted/received signal is Sj(t) is equal for all signals, then equations (10) and (11) can be rewritten as below.




  
    
      
        
          P
          ⁡
          
            (
            
              
                d
                Re
              
              =
              
                +
                1
              
            
            )
          
        
        =
        
          
            {
            
              
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      0
                    
                    )
                  
                
                ⁢
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      16
                    
                    )
                  
                
              
              +
              
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      1
                    
                    )
                  
                
                ⁢
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      17
                    
                    )
                  
                
              
              +
              
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      2
                    
                    )
                  
                
                ⁢
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      18
                    
                    )
                  
                
              
              +
              
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      3
                    
                    )
                  
                
                ⁢
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      19
                    
                    )
                  
                
              
            
            }
          
          +
          
            {
            
              
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      4
                    
                    )
                  
                
                ⁢
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      20
                    
                    )
                  
                
              
              +
              
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      5
                    
                    )
                  
                
                ⁢
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      21
                    
                    )
                  
                
              
              +
              
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      6
                    
                    )
                  
                
                ⁢
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      22
                    
                    )
                  
                
              
              +
              
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      7
                    
                    )
                  
                
                ⁢
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      23
                    
                    )
                  
                
              
            
            }
          
        
      
    
    
      
        (
        12
        )
      
    
  
  
    
      
        
          P
          ⁡
          
            (
            
              
                d
                Re
              
              =
              
                -
                1
              
            
            )
          
        
        =
        
          
            {
            
              
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      8
                    
                    )
                  
                
                ⁢
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      24
                    
                    )
                  
                
              
              +
              
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      9
                    
                    )
                  
                
                ⁢
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      25
                    
                    )
                  
                
              
              +
              
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      10
                    
                    )
                  
                
                ⁢
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      26
                    
                    )
                  
                
              
              +
              
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      11
                    
                    )
                  
                
                ⁢
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      27
                    
                    )
                  
                
              
            
            }
          
          +
          
            {
            
              
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      12
                    
                    )
                  
                
                ⁢
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      28
                    
                    )
                  
                
              
              +
              
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      13
                    
                    )
                  
                
                ⁢
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      29
                    
                    )
                  
                
              
              +
              
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      14
                    
                    )
                  
                
                ⁢
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      30
                    
                    )
                  
                
              
              +
              
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      15
                    
                    )
                  
                
                ⁢
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      31
                    
                    )
                  
                
              
            
            }
          
        
      
    
    
      
        (
        13
        )
      
    
  




The code of the real-number portion dRe of the received information symbol received by the maximum-likelihood receiver can be determined based on the comparison result of the posteriori probabilities or the logarithms of the posteriori probabilities. In other words, it is possible to determine the code (0 or 1) by comparing which is larger, the probability that the real-number portion dRe of the received information symbol is 1 or the probability that it is 0, or by comparing the difference of the logarithms of these with the threshold. Moreover, the code of the real portion dRe of the received information symbol is found from the following equation.




  
    
      
        
          
            
              P
              ⁡
              
                (
                
                  
                    d
                    Re
                  
                  =
                  
                    
                      +
                      1
                    
                    /
                    
                      (
                      
                        
                          
                            y
                            0
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        ,
                        
                          
                            y
                            1
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                      
                      )
                    
                  
                
                )
              
            
            
              P
              ⁡
              
                (
                
                  
                    d
                    Re
                  
                  =
                  
                    
                      -
                      1
                    
                    /
                    
                      (
                      
                        
                          
                            y
                            0
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        ,
                        
                          
                            y
                            1
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                      
                      )
                    
                  
                
                )
              
            
          
          ⋛
          1
        
        ⁢
        
          

        
        ⁢
        or
      
    
    
      
        (
        14
        )
      
    
  
  
    
      
        
          
            ln
            ⁢
            
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  
                    d
                    Re
                  
                  =
                  
                    
                      +
                      1
                    
                    /
                    
                      (
                      
                        
                          
                            y
                            0
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        ,
                        
                          
                            y
                            1
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                      
                      )
                    
                  
                
                )
              
            
          
          -
          
            ln
            ⁢
            
              
            
            ⁢
            
              P
              ⁡
              
                (
                
                  
                    d
                    Re
                  
                  =
                  
                    
                      -
                      1
                    
                    /
                    
                      (
                      
                        
                          
                            y
                            0
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                        ,
                        
                          
                            y
                            1
                          
                          ⁡
                          
                            (
                            t
                            )
                          
                        
                      
                      )
                    
                  
                
                )
              
            
          
        
        ⋛
        0
      
    
    
      
        (
        15
        )
      
    
  




Here, ln indicates the logarithm operation. Also, y0(t) and y1(t) are combined signals obtained by combining a signal line Sj and the white Gauss noises n0(t), n1(t) having spectral power strength N0 (y0(t)=Sj+n0(t), y1(t)=Sj+n1(t)). Equation (15) is the decision rule for determine the code for the real-number portion dRe of the received information symbol.

Here, by considering the algebraic identity of the following equation
ln




  
    
      
        
          (
          
            
              e
              X
            
            +
            
              e
              Y
            
          
          )
        
        =
        
          
            
              X
              +
              Y
            
            2
          
          +
          
            ln
            ⁢
            
              
            
            ⁢
            2
          
          +
          
            ln
            ⁢
            
              
            
            ⁢
            
              cosh
              ⁡
              
                (
                
                  
                    X
                    -
                    Y
                  
                  2
                
                )
              
            
          
        
      
    
    
      
        (
        16
        )
      
    
  




the ln function of equations (12) and (13) are given by the following equations.





  
    
      
        
          ln
          ⁢
          
            {
            
              P
              ⁡
              
                (
                
                  
                    d
                    Re
                  
                  =
                  
                    +
                    1
                  
                
                )
              
            
            }
          
        
        =
        
          
            0.5
            ⁢
            
              (
              
                ln
                ⁡
                
                  (
                  
                    
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            0
                          
                          )
                        
                      
                      ⁢
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            16
                          
                          )
                        
                      
                    
                    +
                    
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            1
                          
                          )
                        
                      
                      ⁢
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            17
                          
                          )
                        
                      
                    
                    +
                    
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            2
                          
                          )
                        
                      
                      ⁢
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            18
                          
                          )
                        
                      
                    
                    +
                    
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            3
                          
                          )
                        
                      
                      ⁢
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            19
                          
                          )
                        
                      
                    
                  
                  )
                
              
              )
            
          
          +
          
            0.5
            ⁢
            
              (
              
                ln
                ⁡
                
                  (
                  
                    
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            4
                          
                          )
                        
                      
                      ⁢
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            20
                          
                          )
                        
                      
                    
                    +
                    
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            5
                          
                          )
                        
                      
                      ⁢
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            21
                          
                          )
                        
                      
                    
                    +
                    
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            6
                          
                          )
                        
                      
                      ⁢
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            22
                          
                          )
                        
                      
                    
                    +
                    
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            7
                          
                          )
                        
                      
                      ⁢
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            23
                          
                          )
                        
                      
                    
                  
                  )
                
              
              )
            
          
          +
          
            ln
            ⁢
            
              {
              
                cosh
                ⁡
                
                  [
                  
                    
                      
                        
                          
                            0.5
                            ⁢
                            
                              (
                              
                                ln
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        P
                                        ⁡
                                        
                                          (
                                          
                                            S
                                            0
                                          
                                          )
                                        
                                      
                                      ⁢
                                      
                                        P
                                        ⁡
                                        
                                          (
                                          
                                            S
                                            16
                                          
                                          )
                                        
                                      
                                    
                                    +
                                    
                                      
                                        P
                                        ⁡
                                        
                                          (
                                          
                                            S
                                            1
                                          
                                          )
                                        
                                      
                                      ⁢
                                      
                                        P
                                        ⁡
                                        
                                          (
                                          
                                            S
                                            17
                                          
                                          )
                                        
                                      
                                    
                                    +
                                    
                                      
                                        P
                                        ⁡
                                        
                                          (
                                          
                                            S
                                            2
                                          
                                          )
                                        
                                      
                                      ⁢
                                      
                                        P
                                        ⁡
                                        
                                          (
                                          
                                            S
                                            18
                                          
                                          )
                                        
                                      
                                    
                                    +
                                    
                                      
                                        P
                                        ⁡
                                        
                                          (
                                          
                                            S
                                            3
                                          
                                          )
                                        
                                      
                                      ⁢
                                      
                                        P
                                        ⁡
                                        
                                          (
                                          
                                            S
                                            19
                                          
                                          )
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              )
                            
                          
                          -
                        
                      
                    
                    
                      
                        
                          0.5
                          ⁢
                          
                            (
                            
                              ln
                              ⁡
                              
                                (
                                
                                  
                                    
                                      P
                                      ⁡
                                      
                                        (
                                        
                                          S
                                          4
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      P
                                      ⁡
                                      
                                        (
                                        
                                          S
                                          20
                                        
                                        )
                                      
                                    
                                  
                                  +
                                  
                                    
                                      P
                                      ⁡
                                      
                                        (
                                        
                                          S
                                          5
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      P
                                      ⁡
                                      
                                        (
                                        
                                          S
                                          21
                                        
                                        )
                                      
                                    
                                  
                                  +
                                  
                                    
                                      P
                                      ⁡
                                      
                                        (
                                        
                                          S
                                          6
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      P
                                      ⁡
                                      
                                        (
                                        
                                          S
                                          22
                                        
                                        )
                                      
                                    
                                  
                                  +
                                  
                                    
                                      P
                                      ⁡
                                      
                                        (
                                        
                                          S
                                          7
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      P
                                      ⁡
                                      
                                        (
                                        
                                          S
                                          23
                                        
                                        )
                                      
                                    
                                  
                                
                                )
                              
                            
                            )
                          
                        
                      
                    
                  
                  ]
                
              
              }
            
          
        
      
    
    
      
        (
        17
        )
      
    
  
  
    
      
        
          ln
          ⁢
          
            {
            
              P
              ⁡
              
                (
                
                  
                    d
                    Re
                  
                  =
                  
                    -
                    1
                  
                
                )
              
            
            }
          
        
        =
        
          
            0.5
            ⁢
            
              (
              
                ln
                ⁡
                
                  (
                  
                    
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            8
                          
                          )
                        
                      
                      ⁢
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            24
                          
                          )
                        
                      
                    
                    +
                    
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            9
                          
                          )
                        
                      
                      ⁢
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            25
                          
                          )
                        
                      
                    
                    +
                    
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            10
                          
                          )
                        
                      
                      ⁢
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            26
                          
                          )
                        
                      
                    
                    +
                    
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            11
                          
                          )
                        
                      
                      ⁢
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            27
                          
                          )
                        
                      
                    
                  
                  )
                
              
              )
            
          
          +
          
            0.5
            ⁢
            
              (
              
                ln
                ⁡
                
                  (
                  
                    
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            12
                          
                          )
                        
                      
                      ⁢
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            28
                          
                          )
                        
                      
                    
                    +
                    
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            13
                          
                          )
                        
                      
                      ⁢
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            29
                          
                          )
                        
                      
                    
                    +
                    
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            14
                          
                          )
                        
                      
                      ⁢
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            30
                          
                          )
                        
                      
                    
                    +
                    
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            31
                          
                          )
                        
                      
                      ⁢
                      
                        P
                        ⁡
                        
                          (
                          
                            S
                            15
                          
                          )
                        
                      
                    
                  
                  )
                
              
              )
            
          
          +
          
            ln
            ⁢
            
              {
              
                cosh
                ⁡
                
                  [
                  
                    
                      
                        
                          
                            0.5
                            ⁢
                            
                              (
                              
                                ln
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        P
                                        ⁡
                                        
                                          (
                                          
                                            S
                                            8
                                          
                                          )
                                        
                                      
                                      ⁢
                                      
                                        P
                                        ⁡
                                        
                                          (
                                          
                                            S
                                            24
                                          
                                          )
                                        
                                      
                                    
                                    +
                                    
                                      
                                        P
                                        ⁡
                                        
                                          (
                                          
                                            S
                                            9
                                          
                                          )
                                        
                                      
                                      ⁢
                                      
                                        P
                                        ⁡
                                        
                                          (
                                          
                                            S
                                            25
                                          
                                          )
                                        
                                      
                                    
                                    +
                                    
                                      
                                        P
                                        ⁡
                                        
                                          (
                                          
                                            S
                                            10
                                          
                                          )
                                        
                                      
                                      ⁢
                                      
                                        P
                                        ⁡
                                        
                                          (
                                          
                                            S
                                            26
                                          
                                          )
                                        
                                      
                                    
                                    +
                                    
                                      
                                        P
                                        ⁡
                                        
                                          (
                                          
                                            S
                                            11
                                          
                                          )
                                        
                                      
                                      ⁢
                                      
                                        P
                                        ⁡
                                        
                                          (
                                          
                                            S
                                            27
                                          
                                          )
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              )
                            
                          
                          -
                        
                      
                    
                    
                      
                        
                          0.5
                          ⁢
                          
                            (
                            
                              ln
                              ⁡
                              
                                (
                                
                                  
                                    
                                      P
                                      ⁡
                                      
                                        (
                                        
                                          S
                                          12
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      P
                                      ⁡
                                      
                                        (
                                        
                                          S
                                          28
                                        
                                        )
                                      
                                    
                                  
                                  +
                                  
                                    
                                      P
                                      ⁡
                                      
                                        (
                                        
                                          S
                                          13
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      P
                                      ⁡
                                      
                                        (
                                        
                                          S
                                          29
                                        
                                        )
                                      
                                    
                                  
                                  +
                                  
                                    
                                      P
                                      ⁡
                                      
                                        (
                                        
                                          S
                                          14
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      P
                                      ⁡
                                      
                                        (
                                        
                                          S
                                          30
                                        
                                        )
                                      
                                    
                                  
                                  +
                                  
                                    
                                      P
                                      ⁡
                                      
                                        (
                                        
                                          S
                                          15
                                        
                                        )
                                      
                                    
                                    ⁢
                                    
                                      P
                                      ⁡
                                      
                                        (
                                        
                                          S
                                          31
                                        
                                        )
                                      
                                    
                                  
                                
                                )
                              
                            
                            )
                          
                        
                      
                    
                  
                  ]
                
              
              }
            
          
        
      
    
    
      
        (
        18
        )
      
    
  




However, the following equation is established.

ln{exp(A)+exp(B)+exp(C)+exp(D)}=Lin+NLin  (19)

Where, in the equation above

Lin=0.25[(A+B)+(C+D)]+ln(2)+0.5[ln{cos h[0.5(A−B)]}+ln{cos h[0.5(C−D)]}]+ln(2)  (20a)

Arg=0.25[(A+B)−(C+D)]+0.5[ln{cos h[0.5(A−B)]}−ln{cos h[0.5(C−D)]}]  (20b)

NLin=ln{cos h[0.5 Arg]}  (20c)

Also, assuming that

P(S0)P(S16)=A, P(S1)P(S17)=B, 

P(S2)P(S18)=C, P(S3)P(S19)=D  (21)

then the first term on the right side of equation (17) is given by the equation below.





  
    
      
        
          (
          
            ln
            ⁡
            
              (
              
                
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        0
                      
                      )
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        16
                      
                      )
                    
                  
                
                +
                
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        1
                      
                      )
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        17
                      
                      )
                    
                  
                
                +
                
                  P
                  ⁢
                  
                    (
                    
                      S
                      2
                    
                    )
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        18
                      
                      )
                    
                  
                
                +
                
                  P
                  ⁢
                  
                    (
                    
                      S
                      3
                    
                    )
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        19
                      
                      )
                    
                  
                
              
              )
            
          
          )
        
        =
        
          
            0.25
            ⁢
            
              (
              
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        0
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        16
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        1
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        17
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        2
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        18
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        3
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        19
                      
                      )
                    
                  
                
              
              )
            
          
          +
          
            ln
            ⁢
            
              
            
            ⁢
            2
          
          +
          
            0.5
            ⁢
            
              (
              
                ln
                ⁢
                
                  {
                  
                    cosh
                    ⁡
                    
                      [
                      
                        0.5
                        ⁢
                        
                          
                        
                        ⁢
                        
                          (
                          
                            
                          
                          ⁢
                          
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    0
                                  
                                  )
                                
                              
                            
                            +
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    16
                                  
                                  )
                                
                              
                            
                            -
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    1
                                  
                                  )
                                
                              
                            
                            -
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    17
                                  
                                  )
                                
                              
                            
                          
                          )
                        
                      
                      ]
                    
                  
                  }
                
              
              )
            
          
          +
          
            0.5
            ⁢
            
              (
              
                ln
                ⁢
                
                  {
                  
                    cosh
                    ⁡
                    
                      [
                      
                        0.5
                        ⁢
                        
                          (
                          
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    2
                                  
                                  )
                                
                              
                            
                            +
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    18
                                  
                                  )
                                
                              
                            
                            -
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    3
                                  
                                  )
                                
                              
                            
                            -
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    19
                                  
                                  )
                                
                              
                            
                          
                          )
                        
                      
                      ]
                    
                  
                  }
                
              
              )
            
          
          +
          
            ln
            ⁢
            
              
            
            ⁢
            2
          
          +
          
            ln
            ⁢
            
              {
              
                cosh
                ⁡
                
                  [
                  
                    0.5
                    ⁢
                    
                      (
                      
                        
                          
                            
                              
                                0.25
                                ⁢
                                
                                  (
                                  
                                    
                                      
                                        
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  0
                                                
                                                )
                                              
                                            
                                          
                                          +
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  16
                                                
                                                )
                                              
                                            
                                          
                                          +
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  1
                                                
                                                )
                                              
                                            
                                          
                                          +
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  17
                                                
                                                )
                                              
                                            
                                          
                                          -
                                        
                                      
                                    
                                    
                                      
                                        
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  2
                                                
                                                )
                                              
                                            
                                          
                                          -
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  18
                                                
                                                )
                                              
                                            
                                          
                                          -
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  3
                                                
                                                )
                                              
                                            
                                          
                                          -
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  19
                                                
                                                )
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              
                                0.5
                                ⁢
                                
                                  (
                                  
                                    ln
                                    ⁢
                                    
                                      {
                                      
                                        cosh
                                        ⁡
                                        
                                          [
                                          
                                            0.5
                                            ⁢
                                            
                                              (
                                              
                                                
                                                  ln
                                                  ⁢
                                                  
                                                    
                                                  
                                                  ⁢
                                                  
                                                    P
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        S
                                                        0
                                                      
                                                      )
                                                    
                                                  
                                                
                                                +
                                                
                                                  ln
                                                  ⁢
                                                  
                                                    
                                                  
                                                  ⁢
                                                  
                                                    P
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        S
                                                        16
                                                      
                                                      )
                                                    
                                                  
                                                
                                                -
                                                
                                                  ln
                                                  ⁢
                                                  
                                                    
                                                  
                                                  ⁢
                                                  
                                                    P
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        S
                                                        1
                                                      
                                                      )
                                                    
                                                  
                                                
                                                -
                                                
                                                  ln
                                                  ⁢
                                                  
                                                    
                                                  
                                                  ⁢
                                                  
                                                    P
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        S
                                                        17
                                                      
                                                      )
                                                    
                                                  
                                                
                                              
                                              )
                                            
                                          
                                          ]
                                        
                                      
                                      }
                                    
                                  
                                  )
                                
                              
                              -
                            
                          
                        
                        
                          
                            
                              0.5
                              ⁢
                              
                                (
                                
                                  ln
                                  ⁢
                                  
                                    {
                                    
                                      cosh
                                      ⁡
                                      
                                        [
                                        
                                          0.5
                                          ⁢
                                          
                                            (
                                            
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      2
                                                    
                                                    )
                                                  
                                                
                                              
                                              +
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      18
                                                    
                                                    )
                                                  
                                                
                                              
                                              -
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      3
                                                    
                                                    )
                                                  
                                                
                                              
                                              -
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      19
                                                    
                                                    )
                                                  
                                                
                                              
                                            
                                            )
                                          
                                        
                                        ]
                                      
                                    
                                    }
                                  
                                
                                )
                              
                            
                          
                        
                      
                      )
                    
                  
                  ]
                
              
              }
            
          
        
      
    
    
      
        (
        22
        )
      
    
  




Also assuming that

P(S4)P(S20)=A, P(S5)P(S21)=B, 

P(S6)P(S22)=C, P(S7)P(S23)=D  (23)

then the second term on the right side of equation (17) is given by the equation below.





  
    
      
        
          (
          
            ln
            ⁡
            
              (
              
                
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        4
                      
                      )
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        20
                      
                      )
                    
                  
                
                +
                
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        5
                      
                      )
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        21
                      
                      )
                    
                  
                
                +
                
                  P
                  ⁢
                  
                    (
                    
                      S
                      6
                    
                    )
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        22
                      
                      )
                    
                  
                
                +
                
                  P
                  ⁢
                  
                    (
                    
                      S
                      7
                    
                    )
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        23
                      
                      )
                    
                  
                
              
              )
            
          
          )
        
        =
        
          
            0.25
            ⁢
            
              (
              
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        4
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        20
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        5
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        21
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        6
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        22
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        7
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        23
                      
                      )
                    
                  
                
              
              )
            
          
          +
          
            ln
            ⁢
            
              
            
            ⁢
            2
          
          +
          
            0.5
            ⁢
            
              (
              
                ln
                ⁢
                
                  {
                  
                    cosh
                    ⁡
                    
                      [
                      
                        0.5
                        ⁢
                        
                          
                        
                        ⁢
                        
                          (
                          
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    4
                                  
                                  )
                                
                              
                            
                            +
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    20
                                  
                                  )
                                
                              
                            
                            -
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    5
                                  
                                  )
                                
                              
                            
                            -
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    21
                                  
                                  )
                                
                              
                            
                          
                          )
                        
                      
                      ]
                    
                  
                  }
                
              
              )
            
          
          +
          
            0.5
            ⁢
            
              (
              
                ln
                ⁢
                
                  {
                  
                    cosh
                    ⁡
                    
                      [
                      
                        0.5
                        ⁢
                        
                          (
                          
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    6
                                  
                                  )
                                
                              
                            
                            +
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    22
                                  
                                  )
                                
                              
                            
                            -
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    7
                                  
                                  )
                                
                              
                            
                            -
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    23
                                  
                                  )
                                
                              
                            
                          
                          )
                        
                      
                      ]
                    
                  
                  }
                
              
              )
            
          
          +
          
            ln
            ⁢
            
              
            
            ⁢
            2
          
          +
          
            ln
            ⁢
            
              {
              
                cosh
                ⁡
                
                  [
                  
                    0.5
                    ⁢
                    
                      (
                      
                        
                          
                            
                              
                                0.25
                                ⁢
                                
                                  (
                                  
                                    
                                      
                                        
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  4
                                                
                                                )
                                              
                                            
                                          
                                          +
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  20
                                                
                                                )
                                              
                                            
                                          
                                          +
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  5
                                                
                                                )
                                              
                                            
                                          
                                          +
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  21
                                                
                                                )
                                              
                                            
                                          
                                          -
                                        
                                      
                                    
                                    
                                      
                                        
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  6
                                                
                                                )
                                              
                                            
                                          
                                          -
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  22
                                                
                                                )
                                              
                                            
                                          
                                          -
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  7
                                                
                                                )
                                              
                                            
                                          
                                          -
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  23
                                                
                                                )
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              
                                0.5
                                ⁢
                                
                                  (
                                  
                                    ln
                                    ⁢
                                    
                                      {
                                      
                                        cosh
                                        ⁡
                                        
                                          [
                                          
                                            0.5
                                            ⁢
                                            
                                              (
                                              
                                                
                                                  ln
                                                  ⁢
                                                  
                                                    
                                                  
                                                  ⁢
                                                  
                                                    P
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        S
                                                        4
                                                      
                                                      )
                                                    
                                                  
                                                
                                                +
                                                
                                                  ln
                                                  ⁢
                                                  
                                                    
                                                  
                                                  ⁢
                                                  
                                                    P
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        S
                                                        20
                                                      
                                                      )
                                                    
                                                  
                                                
                                                -
                                                
                                                  ln
                                                  ⁢
                                                  
                                                    
                                                  
                                                  ⁢
                                                  
                                                    P
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        S
                                                        5
                                                      
                                                      )
                                                    
                                                  
                                                
                                                -
                                                
                                                  ln
                                                  ⁢
                                                  
                                                    
                                                  
                                                  ⁢
                                                  
                                                    P
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        S
                                                        21
                                                      
                                                      )
                                                    
                                                  
                                                
                                              
                                              )
                                            
                                          
                                          ]
                                        
                                      
                                      }
                                    
                                  
                                  )
                                
                              
                              -
                            
                          
                        
                        
                          
                            
                              0.5
                              ⁢
                              
                                (
                                
                                  ln
                                  ⁢
                                  
                                    {
                                    
                                      cosh
                                      ⁡
                                      
                                        [
                                        
                                          0.5
                                          ⁢
                                          
                                            (
                                            
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      6
                                                    
                                                    )
                                                  
                                                
                                              
                                              +
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      22
                                                    
                                                    )
                                                  
                                                
                                              
                                              -
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      7
                                                    
                                                    )
                                                  
                                                
                                              
                                              -
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      23
                                                    
                                                    )
                                                  
                                                
                                              
                                            
                                            )
                                          
                                        
                                        ]
                                      
                                    
                                    }
                                  
                                
                                )
                              
                            
                          
                        
                      
                      )
                    
                  
                  ]
                
              
              }
            
          
        
      
    
    
      
        (
        24
        )
      
    
  




Also, assuming that

P(S8)P(S24)=A, P(S9)P(S25)=B, 

P(S10)P(S26)=C, P(S11)P(S27)=D  (25)

then the first term on the right side of equation (18) is given by the equation below.





  
    
      
        
          (
          
            ln
            ⁡
            
              (
              
                
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        8
                      
                      )
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        24
                      
                      )
                    
                  
                
                +
                
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        9
                      
                      )
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        25
                      
                      )
                    
                  
                
                +
                
                  P
                  ⁢
                  
                    (
                    
                      S
                      10
                    
                    )
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        26
                      
                      )
                    
                  
                
                +
                
                  P
                  ⁢
                  
                    (
                    
                      S
                      11
                    
                    )
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        27
                      
                      )
                    
                  
                
              
              )
            
          
          )
        
        =
        
          
            0.25
            ⁢
            
              (
              
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        8
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        24
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        9
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        25
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        10
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        26
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        11
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        27
                      
                      )
                    
                  
                
              
              )
            
          
          +
          
            ln
            ⁢
            
              
            
            ⁢
            2
          
          +
          
            0.5
            ⁢
            
              (
              
                ln
                ⁢
                
                  {
                  
                    cosh
                    ⁡
                    
                      [
                      
                        0.5
                        ⁢
                        
                          
                        
                        ⁢
                        
                          (
                          
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    8
                                  
                                  )
                                
                              
                            
                            +
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    24
                                  
                                  )
                                
                              
                            
                            -
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    9
                                  
                                  )
                                
                              
                            
                            -
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    25
                                  
                                  )
                                
                              
                            
                          
                          )
                        
                      
                      ]
                    
                  
                  }
                
              
              )
            
          
          +
          
            0.5
            ⁢
            
              (
              
                ln
                ⁢
                
                  {
                  
                    cosh
                    ⁡
                    
                      [
                      
                        0.5
                        ⁢
                        
                          (
                          
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    10
                                  
                                  )
                                
                              
                            
                            +
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    26
                                  
                                  )
                                
                              
                            
                            -
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    11
                                  
                                  )
                                
                              
                            
                            -
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    27
                                  
                                  )
                                
                              
                            
                          
                          )
                        
                      
                      ]
                    
                  
                  }
                
              
              )
            
          
          +
          
            ln
            ⁢
            
              
            
            ⁢
            2
          
          +
          
            ln
            ⁢
            
              {
              
                cosh
                ⁡
                
                  [
                  
                    0.5
                    ⁢
                    
                      (
                      
                        
                          
                            
                              
                                0.25
                                ⁢
                                
                                  (
                                  
                                    
                                      
                                        
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  8
                                                
                                                )
                                              
                                            
                                          
                                          +
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  24
                                                
                                                )
                                              
                                            
                                          
                                          +
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  9
                                                
                                                )
                                              
                                            
                                          
                                          +
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  25
                                                
                                                )
                                              
                                            
                                          
                                          -
                                        
                                      
                                    
                                    
                                      
                                        
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  10
                                                
                                                )
                                              
                                            
                                          
                                          -
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  26
                                                
                                                )
                                              
                                            
                                          
                                          -
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  11
                                                
                                                )
                                              
                                            
                                          
                                          -
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  26
                                                
                                                )
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              
                                0.5
                                ⁢
                                
                                  (
                                  
                                    ln
                                    ⁢
                                    
                                      {
                                      
                                        cosh
                                        ⁡
                                        
                                          [
                                          
                                            0.5
                                            ⁢
                                            
                                              (
                                              
                                                
                                                  ln
                                                  ⁢
                                                  
                                                    
                                                  
                                                  ⁢
                                                  
                                                    P
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        S
                                                        8
                                                      
                                                      )
                                                    
                                                  
                                                
                                                +
                                                
                                                  ln
                                                  ⁢
                                                  
                                                    
                                                  
                                                  ⁢
                                                  
                                                    P
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        S
                                                        24
                                                      
                                                      )
                                                    
                                                  
                                                
                                                -
                                                
                                                  ln
                                                  ⁢
                                                  
                                                    
                                                  
                                                  ⁢
                                                  
                                                    P
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        S
                                                        9
                                                      
                                                      )
                                                    
                                                  
                                                
                                                -
                                                
                                                  ln
                                                  ⁢
                                                  
                                                    
                                                  
                                                  ⁢
                                                  
                                                    P
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        S
                                                        25
                                                      
                                                      )
                                                    
                                                  
                                                
                                              
                                              )
                                            
                                          
                                          ]
                                        
                                      
                                      }
                                    
                                  
                                  )
                                
                              
                              -
                            
                          
                        
                        
                          
                            
                              0.5
                              ⁢
                              
                                (
                                
                                  ln
                                  ⁢
                                  
                                    {
                                    
                                      cosh
                                      ⁡
                                      
                                        [
                                        
                                          0.5
                                          ⁢
                                          
                                            (
                                            
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      10
                                                    
                                                    )
                                                  
                                                
                                              
                                              +
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      26
                                                    
                                                    )
                                                  
                                                
                                              
                                              -
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      11
                                                    
                                                    )
                                                  
                                                
                                              
                                              -
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      27
                                                    
                                                    )
                                                  
                                                
                                              
                                            
                                            )
                                          
                                        
                                        ]
                                      
                                    
                                    }
                                  
                                
                                )
                              
                            
                          
                        
                      
                      )
                    
                  
                  ]
                
              
              }
            
          
        
      
    
    
      
        (
        26
        )
      
    
  




Moreover, assuming that

P(S12)P(S28)=A, P(S13)P(S29)=B, 

P(S14)P(S30)=C, P(S15)P(S31)=D  (27)

then the second term on the right side of equation (18) is given by the equation below.





  
    
      
        
          (
          
            ln
            ⁡
            
              (
              
                
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        12
                      
                      )
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        28
                      
                      )
                    
                  
                
                +
                
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        13
                      
                      )
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        29
                      
                      )
                    
                  
                
                +
                
                  P
                  ⁢
                  
                    (
                    
                      S
                      14
                    
                    )
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        30
                      
                      )
                    
                  
                
                +
                
                  P
                  ⁢
                  
                    (
                    
                      S
                      15
                    
                    )
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        31
                      
                      )
                    
                  
                
              
              )
            
          
          )
        
        =
        
          
            0.25
            ⁢
            
              (
              
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        12
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        28
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        13
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        29
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        14
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        30
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        15
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        31
                      
                      )
                    
                  
                
              
              )
            
          
          +
          
            ln
            ⁢
            
              
            
            ⁢
            2
          
          +
          
            0.5
            ⁢
            
              (
              
                ln
                ⁢
                
                  {
                  
                    cosh
                    ⁡
                    
                      [
                      
                        0.5
                        ⁢
                        
                          (
                          
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    12
                                  
                                  )
                                
                              
                            
                            +
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    28
                                  
                                  )
                                
                              
                            
                            -
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    13
                                  
                                  )
                                
                              
                            
                            -
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    29
                                  
                                  )
                                
                              
                            
                          
                          )
                        
                      
                      ]
                    
                  
                  }
                
              
              )
            
          
          +
          
            0.5
            ⁢
            
              (
              
                ln
                ⁢
                
                  {
                  
                    cosh
                    ⁡
                    
                      [
                      
                        0.5
                        ⁢
                        
                          (
                          
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    14
                                  
                                  )
                                
                              
                            
                            +
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    30
                                  
                                  )
                                
                              
                            
                            -
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    15
                                  
                                  )
                                
                              
                            
                            -
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    31
                                  
                                  )
                                
                              
                            
                          
                          )
                        
                      
                      ]
                    
                  
                  }
                
              
              )
            
          
          +
          
            ln
            ⁢
            
              
            
            ⁢
            2
          
          +
          
            

          
          ⁢
          
            ln
            ⁢
            
              {
              
                cosh
                ⁡
                
                  [
                  
                    0.5
                    ⁢
                    
                      (
                      
                        
                          
                            
                              
                                0.25
                                ⁢
                                
                                  (
                                  
                                    
                                      
                                        
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  12
                                                
                                                )
                                              
                                            
                                          
                                          +
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  28
                                                
                                                )
                                              
                                            
                                          
                                          +
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  13
                                                
                                                )
                                              
                                            
                                          
                                          +
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  29
                                                
                                                )
                                              
                                            
                                          
                                          -
                                        
                                      
                                    
                                    
                                      
                                        
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  14
                                                
                                                )
                                              
                                            
                                          
                                          -
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  30
                                                
                                                )
                                              
                                            
                                          
                                          -
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  15
                                                
                                                )
                                              
                                            
                                          
                                          -
                                          
                                            ln
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              P
                                              ⁡
                                              
                                                (
                                                
                                                  S
                                                  31
                                                
                                                )
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              
                                0.5
                                ⁢
                                
                                  (
                                  
                                    ln
                                    ⁢
                                    
                                      {
                                      
                                        cosh
                                        ⁡
                                        
                                          [
                                          
                                            0.5
                                            ⁢
                                            
                                              (
                                              
                                                
                                                  ln
                                                  ⁢
                                                  
                                                    
                                                  
                                                  ⁢
                                                  
                                                    P
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        S
                                                        12
                                                      
                                                      )
                                                    
                                                  
                                                
                                                +
                                                
                                                  ln
                                                  ⁢
                                                  
                                                    
                                                  
                                                  ⁢
                                                  
                                                    P
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        S
                                                        28
                                                      
                                                      )
                                                    
                                                  
                                                
                                                -
                                                
                                                  ln
                                                  ⁢
                                                  
                                                    
                                                  
                                                  ⁢
                                                  
                                                    P
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        S
                                                        13
                                                      
                                                      )
                                                    
                                                  
                                                
                                                -
                                                
                                                  ln
                                                  ⁢
                                                  
                                                    
                                                  
                                                  ⁢
                                                  
                                                    P
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        S
                                                        29
                                                      
                                                      )
                                                    
                                                  
                                                
                                              
                                              )
                                            
                                          
                                          ]
                                        
                                      
                                      }
                                    
                                  
                                  )
                                
                              
                              -
                            
                          
                        
                        
                          
                            
                              0.5
                              ⁢
                              
                                (
                                
                                  ln
                                  ⁢
                                  
                                    {
                                    
                                      cosh
                                      ⁡
                                      
                                        [
                                        
                                          0.5
                                          ⁢
                                          
                                            (
                                            
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      14
                                                    
                                                    )
                                                  
                                                
                                              
                                              +
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      30
                                                    
                                                    )
                                                  
                                                
                                              
                                              -
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      15
                                                    
                                                    )
                                                  
                                                
                                              
                                              -
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      31
                                                    
                                                    )
                                                  
                                                
                                              
                                            
                                            )
                                          
                                        
                                        ]
                                      
                                    
                                    }
                                  
                                
                                )
                              
                            
                          
                        
                      
                      )
                    
                  
                  ]
                
              
              }
            
          
        
      
    
    
      
        (
        28
        )
      
    
  




Considering that the following equations

ln P(S0)−ln P(S1)=4Im{y0(t)h*01}−4 Im{h00S*00h*01}  (29a)

ln P(S16)−ln P(S17)=4Im{y1(t)h*11}−4 Im{h10S*00h*11}  (29b)

ln P(S2)−ln P(S3)=4Im{y0(t)h*01}−4 Im{h00S*00h*01}  (29c)

ln P(S18)−ln P(S19)=4Im{y1(t)h*11}−4 Im{h10S*00h*11}  (29d)

ln P(S0)+ln P(S16)+ln P(S1)+ln P(S17)−ln P(S2)−ln P(S18)−ln P(S3)−ln P(S19)=2(4Re{y0(t)h*01+y1(t)h*11}−4Re{h00h*01+h10h*11})  (29e)

are established in equation (22), and that the same relationships are established in equations (24), (26) and (28), then finally the decision rule of equation (15) is given by the following equation. Or in other words the decision rule becomes as below.

ln P(dRe=+1/(y0(t), y1(t)))−ln P(dRe=−1/(y0(t),y1(t)))=ln P0(I)=4Re(y0(t)h*00+y1(t)h*10)+0.5(L1−L2+L3+L4)+lnch{0.5(f1+f2)}−ln ch{0.5(f3+f4)}  (30)

The method for finding equation (30) will be explained in appendix.





  
    
      
        In
        ⁢
        
          
        
        ⁢
        equation
        ⁢
        
          
        
        ⁢
        
          (
          30
          )
        
      
    
    
      
        
      
    
  
  
    
      
        
          L
          1
        
        =
        
          (
          
            
              
                
                  
                    ln
                    ⁢
                    
                      
                    
                    ⁢
                    ch
                    ⁢
                    
                      {
                      
                        0.5
                        ⁢
                        
                          (
                          
                            
                              4
                              ⁢
                              
                                Im
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        
                                          y
                                          0
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        01
                                        *
                                      
                                    
                                    +
                                    
                                      
                                        
                                          y
                                          1
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        11
                                        *
                                      
                                    
                                  
                                  )
                                
                              
                            
                            -
                            
                              (
                              
                                
                                  4
                                  ⁢
                                  
                                    Re
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                +
                                
                                  4
                                  ⁢
                                  
                                    Im
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                          )
                        
                      
                      }
                    
                  
                  -
                
              
            
            
              
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  ch
                  ⁢
                  
                    {
                    
                      0.5
                      ⁢
                      
                        (
                        
                          
                            4
                            ⁢
                            
                              Im
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                        y
                                        0
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      01
                                      *
                                    
                                  
                                  +
                                  
                                    
                                      
                                        y
                                        1
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      11
                                      *
                                    
                                  
                                
                                )
                              
                            
                          
                          +
                          
                            (
                            
                              
                                4
                                ⁢
                                
                                  Re
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          11
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              +
                              
                                4
                                ⁢
                                
                                  Im
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                        )
                      
                    
                    }
                  
                
              
            
          
          )
        
      
    
    
      
        (
        
          31
          ⁢
          a
        
        )
      
    
  
  
    
      
        
          L
          2
        
        =
        
          (
          
            
              
                
                  
                    ln
                    ⁢
                    
                      
                    
                    ⁢
                    ch
                    ⁢
                    
                      {
                      
                        0.5
                        ⁢
                        
                          (
                          
                            
                              4
                              ⁢
                              
                                Im
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        
                                          y
                                          0
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        01
                                        *
                                      
                                    
                                    +
                                    
                                      
                                        
                                          y
                                          1
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        11
                                        *
                                      
                                    
                                  
                                  )
                                
                              
                            
                            -
                            
                              (
                              
                                
                                  4
                                  ⁢
                                  
                                    Re
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                -
                                
                                  4
                                  ⁢
                                  
                                    Im
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                          )
                        
                      
                      }
                    
                  
                  -
                
              
            
            
              
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  ch
                  ⁢
                  
                    {
                    
                      0.5
                      ⁢
                      
                        (
                        
                          
                            4
                            ⁢
                            
                              Im
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                        y
                                        0
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      01
                                      *
                                    
                                  
                                  +
                                  
                                    
                                      
                                        y
                                        1
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      11
                                      *
                                    
                                  
                                
                                )
                              
                            
                          
                          +
                          
                            (
                            
                              
                                4
                                ⁢
                                
                                  Re
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              -
                              
                                4
                                ⁢
                                
                                  Im
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                        )
                      
                    
                    }
                  
                
              
            
          
          )
        
      
    
    
      
        (
        
          31
          ⁢
          b
        
        )
      
    
  
  
    
      
        
          L
          3
        
        =
        
          (
          
            
              
                
                  
                    ln
                    ⁢
                    
                      
                    
                    ⁢
                    ch
                    ⁢
                    
                      {
                      
                        0.5
                        ⁢
                        
                          (
                          
                            
                              4
                              ⁢
                              
                                Re
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        
                                          y
                                          0
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        01
                                        *
                                      
                                    
                                    +
                                    
                                      
                                        
                                          y
                                          1
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        11
                                        *
                                      
                                    
                                  
                                  )
                                
                              
                            
                            -
                            
                              (
                              
                                
                                  4
                                  ⁢
                                  
                                    Re
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                -
                                
                                  4
                                  ⁢
                                  
                                    Im
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                          )
                        
                      
                      }
                    
                  
                  -
                
              
            
            
              
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  ch
                  ⁢
                  
                    {
                    
                      0.5
                      ⁢
                      
                        (
                        
                          
                            4
                            ⁢
                            
                              Re
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                        y
                                        0
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      01
                                      *
                                    
                                  
                                  +
                                  
                                    
                                      
                                        y
                                        1
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      11
                                      *
                                    
                                  
                                
                                )
                              
                            
                          
                          +
                          
                            (
                            
                              
                                4
                                ⁢
                                
                                  Re
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              -
                              
                                4
                                ⁢
                                
                                  Im
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                        )
                      
                    
                    }
                  
                
              
            
          
          )
        
      
    
    
      
        (
        
          31
          ⁢
          c
        
        )
      
    
  
  
    
      
        
          L
          4
        
        =
        
          (
          
            
              
                
                  
                    ln
                    ⁢
                    
                      
                    
                    ⁢
                    ch
                    ⁢
                    
                      {
                      
                        0.5
                        ⁢
                        
                          (
                          
                            
                              4
                              ⁢
                              
                                Re
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        
                                          y
                                          0
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        01
                                        *
                                      
                                    
                                    +
                                    
                                      
                                        
                                          y
                                          1
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        11
                                        *
                                      
                                    
                                  
                                  )
                                
                              
                            
                            -
                            
                              (
                              
                                
                                  4
                                  ⁢
                                  
                                    Re
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                +
                                
                                  4
                                  ⁢
                                  
                                    Im
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                          )
                        
                      
                      }
                    
                  
                  -
                
              
            
            
              
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  ch
                  ⁢
                  
                    {
                    
                      0.5
                      ⁢
                      
                        (
                        
                          
                            4
                            ⁢
                            
                              Re
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                        y
                                        0
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      01
                                      *
                                    
                                  
                                  +
                                  
                                    
                                      
                                        y
                                        1
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      11
                                      *
                                    
                                  
                                
                                )
                              
                            
                          
                          +
                          
                            (
                            
                              
                                4
                                ⁢
                                
                                  Re
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              +
                              
                                4
                                ⁢
                                
                                  Im
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                        )
                      
                    
                    }
                  
                
              
            
          
          )
        
      
    
    
      
        (
        
          31
          ⁢
          d
        
        )
      
    
  
  
    
      
        
          f
          1
        
        =
        
          (
          
            
              
                
                  
                    ln
                    ⁢
                    
                      
                    
                    ⁢
                    ch
                    ⁢
                    
                      {
                      
                        0.5
                        ⁢
                        
                          (
                          
                            
                              4
                              ⁢
                              
                                Im
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        
                                          y
                                          0
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        01
                                        *
                                      
                                    
                                    +
                                    
                                      
                                        
                                          y
                                          1
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        11
                                        *
                                      
                                    
                                  
                                  )
                                
                              
                            
                            -
                            
                              (
                              
                                
                                  4
                                  ⁢
                                  
                                    Im
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                +
                                
                                  4
                                  ⁢
                                  
                                    Re
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                          )
                        
                      
                      }
                    
                  
                  -
                
              
            
            
              
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  ch
                  ⁢
                  
                    {
                    
                      0.5
                      ⁢
                      
                        (
                        
                          
                            4
                            ⁢
                            
                              Im
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                        y
                                        0
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      01
                                      *
                                    
                                  
                                  +
                                  
                                    
                                      
                                        y
                                        1
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      11
                                      *
                                    
                                  
                                
                                )
                              
                            
                          
                          +
                          
                            (
                            
                              
                                4
                                ⁢
                                
                                  Re
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              -
                              
                                4
                                ⁢
                                
                                  Im
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                        )
                      
                    
                    }
                  
                
              
            
          
          )
        
      
    
    
      
        (
        
          30
          ⁢
          e
        
        )
      
    
  
  
    
      
        
          f
          2
        
        =
        
          (
          
            
              
                
                  
                    ln
                    ⁢
                    
                      
                    
                    ⁢
                    ch
                    ⁢
                    
                      {
                      
                        0.5
                        ⁢
                        
                          (
                          
                            
                              4
                              ⁢
                              
                                Re
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        
                                          y
                                          0
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        01
                                        *
                                      
                                    
                                    +
                                    
                                      
                                        
                                          y
                                          1
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        11
                                        *
                                      
                                    
                                  
                                  )
                                
                              
                            
                            +
                            
                              (
                              
                                
                                  4
                                  ⁢
                                  
                                    Im
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                -
                                
                                  4
                                  ⁢
                                  
                                    Re
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                          )
                        
                      
                      }
                    
                  
                  -
                
              
            
            
              
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  ch
                  ⁢
                  
                    {
                    
                      0.5
                      ⁢
                      
                        (
                        
                          
                            4
                            ⁢
                            
                              Re
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                        y
                                        0
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      01
                                      *
                                    
                                  
                                  +
                                  
                                    
                                      
                                        y
                                        1
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      11
                                      *
                                    
                                  
                                
                                )
                              
                            
                          
                          -
                          
                            (
                            
                              
                                4
                                ⁢
                                
                                  Im
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              +
                              
                                4
                                ⁢
                                
                                  Re
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                        )
                      
                    
                    }
                  
                
              
            
          
          )
        
      
    
    
      
        (
        
          31
          ⁢
          f
        
        )
      
    
  
  
    
      
        
          f
          3
        
        =
        
          (
          
            
              
                
                  
                    ln
                    ⁢
                    
                      
                    
                    ⁢
                    ch
                    ⁢
                    
                      {
                      
                        0.5
                        ⁢
                        
                          (
                          
                            
                              4
                              ⁢
                              
                                Im
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        
                                          y
                                          0
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        01
                                        *
                                      
                                    
                                    +
                                    
                                      
                                        
                                          y
                                          1
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        11
                                        *
                                      
                                    
                                  
                                  )
                                
                              
                            
                            -
                            
                              (
                              
                                
                                  4
                                  ⁢
                                  
                                    Re
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                -
                                
                                  4
                                  ⁢
                                  
                                    Im
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                          )
                        
                      
                      }
                    
                  
                  -
                
              
            
            
              
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  ch
                  ⁢
                  
                    {
                    
                      0.5
                      ⁢
                      
                        (
                        
                          
                            4
                            ⁢
                            
                              Im
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                        y
                                        0
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      01
                                      *
                                    
                                  
                                  +
                                  
                                    
                                      
                                        y
                                        1
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      11
                                      *
                                    
                                  
                                
                                )
                              
                            
                          
                          +
                          
                            (
                            
                              
                                4
                                ⁢
                                
                                  Re
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              +
                              
                                4
                                ⁢
                                
                                  Im
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                        )
                      
                    
                    }
                  
                
              
            
          
          )
        
      
    
    
      
        (
        
          31
          ⁢
          g
        
        )
      
    
  
  
    
      
        
          f
          4
        
        =
        
          (
          
            
              
                
                  
                    ln
                    ⁢
                    
                      
                    
                    ⁢
                    ch
                    ⁢
                    
                      {
                      
                        0.5
                        ⁢
                        
                          (
                          
                            
                              4
                              ⁢
                              
                                Re
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        
                                          y
                                          0
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        01
                                        *
                                      
                                    
                                    +
                                    
                                      
                                        
                                          y
                                          1
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        11
                                        *
                                      
                                    
                                  
                                  )
                                
                              
                            
                            +
                            
                              (
                              
                                
                                  4
                                  ⁢
                                  
                                    Re
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                +
                                
                                  4
                                  ⁢
                                  
                                    Im
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                          )
                        
                      
                      }
                    
                  
                  -
                
              
            
            
              
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  ch
                  ⁢
                  
                    {
                    
                      0.5
                      ⁢
                      
                        (
                        
                          
                            4
                            ⁢
                            
                              Re
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                        y
                                        0
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      01
                                      *
                                    
                                  
                                  +
                                  
                                    
                                      
                                        y
                                        1
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      11
                                      *
                                    
                                  
                                
                                )
                              
                            
                          
                          -
                          
                            (
                            
                              
                                4
                                ⁢
                                
                                  Im
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              -
                              
                                4
                                ⁢
                                
                                  Re
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                        )
                      
                    
                    }
                  
                
              
            
          
          )
        
      
    
    
      
        (
        
          31
          ⁢
          h
        
        )
      
    
  




In the above equations, ln in the term ln ch is the logarithm, and ch is cosh (hyperbolic cosine).

The non-linear function having the pattern of the following equation




  
    
      
        
          F
          ⁡
          
            (
            
              x
              ,
              
                Δ
                ⁢
                
                  
                
                ⁢
                E
              
            
            )
          
        
        =
        
          
            ln
            ⁢
            
              
            
            ⁢
            cosh
            ⁢
            
              {
              
                
                  x
                  -
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    E
                  
                
                
                  2
                  ⁢
                  
                    
                  
                
              
              }
            
          
          -
          
            ln
            ⁢
            
              
            
            ⁢
            cosh
            ⁢
            
              {
              
                
                  x
                  +
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    E
                  
                
                2
              
              }
            
          
        
      
    
    
      
        (
        32
        )
      
    
  




that exists in equations (31a) to (31h) can be described as a limiter having a linear domain. The limiter level of this non-linear function depends on the S/N ratio (signal to noise spectral power strength ratio) and energy difference ΔE. FIG. 7 shows the transmission characteristics of the non-linear function F(x, ΔE) when the energy difference ΔE is used as a parameter.

Here, new notation will be employed. In other words, the first notation is as below.

S0=Re{k(y0·h*00+y1h*10)}, S1=Im{k(y0·h*01+y1h*11)}

S2=Re{k(y0·h*01+y1h*11)}, S3=Im{k(y0·h*00+y1h*10)}  (33)

As shown in (A) of FIG. 8, the signals S0, S3 in this notation can be calculated using a multiplier 1a that multiplies the received signal y0 by k·h00, a multiplier 1b that multiplies the received signal y1 by k·h10*, an adder 2 that adds the products from each of the multipliers, a real-number-extraction unit 3a that extracts the real-number portion from the added result, and an imaginary-number-extraction unit 3b that extracts the imaginary portion. Also, as shown in (B) of FIG. 8, the signals S1, S2 can be calculated using a multiplier 4a that multiplies the received signal y0 by k·h01*, a multiplier 4b that multiplies the received signal y1 by k·h11*, an adder 5 that adds the products from each of the multipliers, a real-number-extraction unit 6a that extracts the real-number portion from the added result, and an imaginary-number-extraction unit 6b that extracts the imaginary portion.

The second notation is as below.

Im ΔE=Im(h00·h*01+h10h*11), Re ΔE=Re(h00·h*01+h10h*11),

ΔE0=Re ΔE+Im ΔE, ΔE1=Re ΔE−Im ΔE  (34)

FIG. 9 shows the configuration of the signal-processing unit PR0 that calculates lnP0(l) based on equation (30), then uses this lnP0(l) to determine whether or not the real-number portion of the first symbol D0 is +1 or −1, and outputs the judgment result dRe=Re(D0). However, the normalization factor k is taken to be 4/N0 (N0 is the spectrum strength of the Gauss noise).
The signal-processing unit PR0 shown in FIG. 9 is created taking into consideration the notation of (33) and (34) above, and the non-linear function of equation (32), where 7a to 7d, and 8a to 8e in blocks 7 and 8 are limiters having the specified energy difference shown in FIG. 7.
Block 7 calculates the second term on the right of equation (30) (L1−L2+L3+L4), and block 8 calculates the third and fourth terms on the right of equation (30), and the adder 9 calculates and outputs the following.

ln P0(I)=ln P(dRe=+1/(y0(t),y1(t)))−ln P(dRe=−1/(y0(t),y1(t)))  (35)

In block 7, 7e is a subtractor, 7f to 7g are adders, and 7h is a multiplier, in block 8, 8f to 8k are adders/subtractors. The judgment unit 10 determines whether or not the real-number portion of the first symbol D0 is +1 or −1 based on the probability lnP0(l), and outputs the judgment result dRe=Re(D0).

(b) Determining the Real-Number Portion dRe=Re(D1) of the Second Symbol D1 

Determining the real-number portion dRe=Re(D1) of the second symbol D1 can be performed in the same way as determining the real-number portion of the first symbol D0. The following equations can be used as S0 to S3 in FIG. 9.

S0=Re{k(y0·h*01+y1h*11)}, S1=Im{K(y0·h*00+y1h*10)}

S2=Re{k(y0·h*00+y1h*10)}, S3=Im{k(y0·h*01+y1h*11)}  (36)

Also, the following equations can be used as ImΔE, ReΔE, ΔE0 and ΔE1.

Im ΔE=Im(h01·h*00+h11h*10), Re ΔE=Re(h01·h*00+h11h*10),

ΔE0=Re ΔE+Im ΔE, ΔE1=Re ΔE−Im ΔE  (37)

(c) Determining the Imaginary Portion dIm=Im(D0) of the Symbol D0 Received by the First Receiver REC0

The probability P(dIm=+1) that the received bit dIm=+1, and the probability P(dIm=−1) that the received bit dIm=−1 can be expressed by the following equations from Table 2 and Table 3.

P(dIm=+1)=P(S0)P(S16)+P(S1)P(S17)+P(S2)P(S18)+P(S3)P(S19)+

P(S8)P(S24)+P(S9)P(S25)+P(S10)P(S26)+P(S11)P(S27)  (38)

P(dIm=−1)=P(S4)P(S20)+P(S5)P(S21)+P(S6)P(S22)+P(S7)P(S23)+

P(S12)P(S28)+P(S13)P(S29)+P(S14)P(S30)+P(S15)P(S13)  (39)

By finding the ln function of equations (38) and (39), the decision rule of equation (15) can be given by the following equation. In other words, the decision rule is as below.

ln P(dIm=+1/(y0(t),y1(t)))−ln P(dIm=−1/(y0(t),y1(t)))=ln P0(Q)=

4Im(y0(t)h*00+y1(t)h*10)+0.5(L1−L2+L3−L4)+lnch{0.5(f1+f2)}−lnch{0.5(f3+f4)}  (40)





  
    
      
        In
        ⁢
        
          
        
        ⁢
        equation
        ⁢
        
          
        
        ⁢
        
          (
          40
          )
        
      
    
    
      
        
      
    
  
  
    
      
        
          L
          1
        
        =
        
          (
          
            
              
                
                  
                    ln
                    ⁢
                    
                      
                    
                    ⁢
                    ch
                    ⁢
                    
                      {
                      
                        0.5
                        ⁢
                        
                          (
                          
                            
                              4
                              ⁢
                              
                                Im
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        
                                          y
                                          0
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        01
                                        *
                                      
                                    
                                    +
                                    
                                      
                                        
                                          y
                                          1
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        11
                                        *
                                      
                                    
                                  
                                  )
                                
                              
                            
                            -
                            
                              (
                              
                                
                                  4
                                  ⁢
                                  
                                    Im
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                +
                                
                                  4
                                  ⁢
                                  
                                    Re
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                          )
                        
                      
                      }
                    
                  
                  -
                
              
            
            
              
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  ch
                  ⁢
                  
                    {
                    
                      0.5
                      ⁢
                      
                        (
                        
                          
                            4
                            ⁢
                            
                              Im
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                        y
                                        0
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      01
                                      *
                                    
                                  
                                  +
                                  
                                    
                                      
                                        y
                                        1
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      11
                                      *
                                    
                                  
                                
                                )
                              
                            
                          
                          +
                          
                            (
                            
                              
                                4
                                ⁢
                                
                                  Im
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              -
                              
                                4
                                ⁢
                                
                                  Re
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                        )
                      
                    
                    }
                  
                
              
            
          
          )
        
      
    
    
      
        (
        
          41
          ⁢
          a
        
        )
      
    
  
  
    
      
        
          L
          2
        
        =
        
          (
          
            
              
                
                  
                    ln
                    ⁢
                    
                      
                    
                    ⁢
                    ch
                    ⁢
                    
                      {
                      
                        0.5
                        ⁢
                        
                          (
                          
                            
                              4
                              ⁢
                              
                                Im
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        
                                          y
                                          0
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        01
                                        *
                                      
                                    
                                    +
                                    
                                      
                                        
                                          y
                                          1
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        11
                                        *
                                      
                                    
                                  
                                  )
                                
                              
                            
                            -
                            
                              (
                              
                                
                                  4
                                  ⁢
                                  
                                    Re
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                -
                                
                                  4
                                  ⁢
                                  
                                    Im
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                          )
                        
                      
                      }
                    
                  
                  -
                
              
            
            
              
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  ch
                  ⁢
                  
                    {
                    
                      0.5
                      ⁢
                      
                        (
                        
                          
                            4
                            ⁢
                            
                              Im
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                        y
                                        0
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      01
                                      *
                                    
                                  
                                  +
                                  
                                    
                                      
                                        y
                                        1
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      11
                                      *
                                    
                                  
                                
                                )
                              
                            
                          
                          -
                          
                            (
                            
                              
                                4
                                ⁢
                                
                                  Re
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              +
                              
                                4
                                ⁢
                                
                                  Im
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                        )
                      
                    
                    }
                  
                
              
            
          
          )
        
      
    
    
      
        (
        
          41
          ⁢
          b
        
        )
      
    
  
  
    
      
        
          L
          3
        
        =
        
          (
          
            
              
                
                  
                    ln
                    ⁢
                    
                      
                    
                    ⁢
                    ch
                    ⁢
                    
                      {
                      
                        0.5
                        ⁢
                        
                          (
                          
                            
                              4
                              ⁢
                              
                                Re
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        
                                          y
                                          0
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        01
                                        *
                                      
                                    
                                    +
                                    
                                      
                                        
                                          y
                                          1
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        11
                                        *
                                      
                                    
                                  
                                  )
                                
                              
                            
                            -
                            
                              (
                              
                                
                                  4
                                  ⁢
                                  
                                    Re
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                -
                                
                                  4
                                  ⁢
                                  
                                    Im
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                          )
                        
                      
                      }
                    
                  
                  -
                
              
            
            
              
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  ch
                  ⁢
                  
                    {
                    
                      0.5
                      ⁢
                      
                        (
                        
                          
                            4
                            ⁢
                            
                              Re
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                        y
                                        0
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      01
                                      *
                                    
                                  
                                  +
                                  
                                    
                                      
                                        y
                                        1
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      11
                                      *
                                    
                                  
                                
                                )
                              
                            
                          
                          +
                          
                            (
                            
                              
                                4
                                ⁢
                                
                                  Re
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              -
                              
                                4
                                ⁢
                                
                                  Im
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                        )
                      
                    
                    }
                  
                
              
            
          
          )
        
      
    
    
      
        (
        
          41
          ⁢
          c
        
        )
      
    
  
  
    
      
        
          L
          4
        
        =
        
          (
          
            
              
                
                  
                    ln
                    ⁢
                    
                      
                    
                    ⁢
                    ch
                    ⁢
                    
                      {
                      
                        0.5
                        ⁢
                        
                          (
                          
                            
                              4
                              ⁢
                              
                                Re
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        
                                          y
                                          0
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        01
                                        *
                                      
                                    
                                    +
                                    
                                      
                                        
                                          y
                                          1
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        11
                                        *
                                      
                                    
                                  
                                  )
                                
                              
                            
                            +
                            
                              (
                              
                                
                                  4
                                  ⁢
                                  
                                    Re
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                +
                                
                                  4
                                  ⁢
                                  
                                    Im
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                          )
                        
                      
                      }
                    
                  
                  -
                
              
            
            
              
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  ch
                  ⁢
                  
                    {
                    
                      0.5
                      ⁢
                      
                        (
                        
                          
                            4
                            ⁢
                            
                              Re
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                        y
                                        0
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      01
                                      *
                                    
                                  
                                  +
                                  
                                    
                                      
                                        y
                                        1
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      11
                                      *
                                    
                                  
                                
                                )
                              
                            
                          
                          +
                          
                            (
                            
                              
                                4
                                ⁢
                                
                                  Re
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              -
                              
                                4
                                ⁢
                                
                                  Im
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                        )
                      
                    
                    }
                  
                
              
            
          
          )
        
      
    
    
      
        (
        
          41
          ⁢
          d
        
        )
      
    
  
  
    
      
        
          f
          1
        
        =
        
          (
          
            
              
                
                  
                    ln
                    ⁢
                    
                      
                    
                    ⁢
                    ch
                    ⁢
                    
                      {
                      
                        0.5
                        ⁢
                        
                          (
                          
                            
                              4
                              ⁢
                              
                                Im
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        
                                          y
                                          0
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        01
                                        *
                                      
                                    
                                    +
                                    
                                      
                                        
                                          y
                                          1
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        11
                                        *
                                      
                                    
                                  
                                  )
                                
                              
                            
                            -
                            
                              (
                              
                                
                                  4
                                  ⁢
                                  
                                    Re
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                +
                                
                                  4
                                  ⁢
                                  
                                    Im
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
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In the above equations, ln of ln ch is the logarithm, and ch is cos h (hyperbolic cosine).

Here, new notation is employed. In other words, the first notation is as below.

S0=Re{k(y0·h*01+ylh*11)}, S1=Im{k(y0·h*00+ylh*10)}

S2=Re{k(y0·h*00+ylh*10)}, S3=Im{k(y0·h*01+ylh*11)}  (42)

As shown in (A) of FIG. 10, the signals S0, S3 in this notation can be calculated using a multiplier 11a that multiplies the received signal y0 by k·h01*, a multiplier 11b that multiplies the received signal y1 by k·h11*, an adder 12 that adds the products from each of the multipliers, a real-number-extraction unit 13a that extracts the real-number portion from the added result, and an imaginary-number-extract ion unit 13b that extracts the imaginary portion. Also, as shown in (B) of FIG. 10, the signals S1, S2 can be calculated using a multiplier 14a that multiplies the received signal y0 by k·h00*, a multiplier 14b that multiplies the received signal y1 by k·h10*, an adder 15 that adds the products from each of the multipliers, a real-number-extraction unit 16a that extracts the real-number portion from the added result, and an imaginary-number-extraction unit 16b that extracts the imaginary portion.

The second notation is as below.

Im ΔE=Im(h01·h*00+h11h*10), Re ΔE=Re(h01·h*00+h11h*10),

ΔE0=Re ΔE+Im ΔE, ΔE1=Re ΔE−Im ΔE  (43)

FIG. 11 shows the configuration of the signal-processing unit PR0 that calculates lnP0(Q) based on equation (40), then uses this lnP0(Q) to determine whether or not the imaginary-number portion of the first symbol D0 is +1 or −1, and outputs the judgment result dIm=Im(D0). However, the normalization factor k is taken to be 4/N0 (N0 is the spectrum strength of the Gauss noise).
The signal-processing unit PR0 shown in FIG. 11 is created taking into consideration the notation of (42) and (43) above, and the non-linear function of equation (32), where 17a to 17d, and 18a to 18e in blocks 17 and 18 are limiters having the specified energy difference shown in FIG. 7.
Block 17 calculates the second term on the right of equation (40) (L1−L2+L3+L4), and block 18 calculates the third and fourth terms on the right of equation (40), and the adder 19 calculates and outputs the following.

ln P0(Q)=ln P(dIm=+1/(y0(t),y1(t)))−ln P(dIm=−1/(y0(t),y1(t)))  (44)

In block 17, 17e is a subtractor, 17f to 17g are addesr, and 17h is a multiplier, in block 18, 18f to 18k are adders/subtractors. The judgment unit 20 determines whether or not the imaginary-number portion of the first symbol D0 is +1 or −1 based on the probability lnP0(Q), and outputs the judgment result dIm=Im(D0).

(d) Determining the Imaginary-Number Portion dIm=Im(D1) of the Second Symbol D1 

Determining the imaginary-number portion dIm=Im(D1) of the second symbol D1 can be performed in the same way as determining the imaginary-number portion of the first symbol D0. The following equations can be used as S0 to S3 in FIG. 11.

S0=Im{k·(y0·h*01+y1·h*11)}, S1=Im{k·(y0·h*00+y1·h*10)}

S2=Re{k·(y0·h*00+y1·h*10)}, S3=Re{k·(y0·h*01+y1·h*11)}  (45)

Also, the following equations can be used as ImΔE, ReΔE, ΔE0 and ΔE1.

Im ΔE=Im(k·(h01·h*00+h11·h*10)), Re ΔE=Re(k·(h01·h*00+h11·h*10)), ΔE0=Re ΔE+Im ΔE, ΔE1=Re ΔE−Im ΔE  (46)

(e) BPSK Transmission

The case of QPSK transmission was explained above, however in the case of BPSK transmission, the symbols are only the real-number portion, and a circuit for determining the imaginary-number portion is not necessary.
In other words, Di=(dRe, dIm) becomes just the real-number portion, so dRe=Re(Di), and dIm=0. Also, in the equations above, the imaginary terms Im(•) all become 0.
(f) Simulation
FIG. 12 and FIG. 13 show the BER (Bit Error Rate) for 2Eb/No, which is the BER characteristic of the SS receiver of this invention. In the simulation, two transmission antennas and two receiving antennas are used, and transmission is performed using QPSK modulation or 16QAM modulation without encoding. FIG. 12 shows the case of using QPSK modulation, and FIG. 13 shows the case of using 16QAM modulation, and it was found that a characteristic similar to the BER characteristic of the MLD method is obtained.
From the simulation results shown in FIG. 12, the BER characteristic of the SS receiver of this invention has no SNR loss when compared with the MLD characteristic. Also, as can be seen from the results shown in FIG. 12 and FIG. 13, the SS algorithm of this invention has the same BER characteristic as MLD, and is +6 dB better than the characteristic obtained using the QR-MLD method. With the SS receiver, a characteristic that is +6 dB better than that obtained using QR-MLD method is obtained, and as can be clearly seen from the following analysis, complexity equivalent to that of the QR-MLD method can be expected.
The SS receiver of this invention has better performance than a turbo receiver, and complexity and processing time are improved. Particularly, the advantage of the SS receiver when compared with the turbo receiver is that iterative estimation is not necessary. In order for a turbo receiver to obtain the best performance, 2 to 4 iterations are necessary.
(g) Circuit Complexity
Table 4 to Table 6 show the number of complex computations and the number of real-number computations when the transmitter uses QPSK modulation, and Table 3 shows the case of a MLD receiver, Table 4 shows the case of a QR-MLD receiver, and Table 5 shows the case of the SS receiver of this invention. In the tables, the top row shows the number of complex computations and real-number computations, and the bottom row is the total number of real-number computations when the number of complex computations is replaced with the number of real-number computations. One complex multiplication is according to the following equation

(a+i·b)·(c+i·d)=a·c+i·a·d+i·b·c−b·d  (47)

and corresponds to four real-number multiplications, two real-number additions, and one real-number subtraction. Moreover, complex addition/subtraction corresponds to two real-number additions/subtractions. Furthermore, two real-number multiplications and one addition/subtraction are necessary for a complex norm computation. The complex norm computation when y=Re+jIm is a value calculated using the following equation.

Norm=√{square root over (Re·Re+Im·Im)}








TABLE 4







MLD












Complex
Complex
Complex
Real ADD/



MULT
ADD/Compare
Norm
Compare
















16
16
16
16



Or Real MULT
Or Real ADD



96
112

















TABLE 5







QR-MLD












Complex
Complex
Complex
Real ADD/



MULT
ADD/Compare
Norm
Compare
















 25
16
8
6



Or Real MULT
Or Real ADD



116
121

















TABLE 6







SS











Complex

Real
Real
Real


MULT
Real MULT
ADD/Compare
ADD
ADD














 6
3
16
18 * 2 = 36
14 * 2 =


Or Real MULT

Or Real ADD
16 + 36 + 28
28


27

104









The MLD method is a method that generates candidates (replicas) for QM transmission vectors, and performs the distance computation of equation (5), then estimates that the replica having the minimum distance is the input data, and in this method the number of computations increases exponentially according to the number of antennas M. Here, Q is the modulation method, and in the case of BPSK modulation, Q=2, and in the case of QPSK, Q=4.
Table 9 show the number of complex computations and real-number computations in the case where the transmitter uses 16QAM modulation, where Table 7 is for the case of a MLD receiver, Table 8 is for the case of a QR-MLD receiver, and Table 9 is for the case of the SS receiver of this invention.







TABLE 7







MLD










Complex
Complex
Complex
Real


MULT
ADD/Compare
Norm
ADD/Compare













 256
256
256
256


Or Real MULT
Or Real ADD


1536
1792
















TABLE 8







QR-MLD










Complex
Complex
Complex
Real


MULT
ADD/Compare
Norm
ADD/Compare













 49
40
32
30


Or Real MULT
Or Real ADD


260
289
















TABLE 9







SS










Complex
Real
Real
Real ADD


MULT
MULT
ADD/Compare
(* Non-optimized)













 8
8
21
2 * 1700 = 3400


Or Real MULT

Or Real ADD


40

3500









As can be seen from the above, with the SS receiver of this invention, only a few multiplications are necessary when compared with the MLD method or QR-MLD method. In ASIC or FPGA implementation, an 8-bit real-number multiplier is 10× or more complex than real-number addition, and for M real-number multiplications, the complexity is equivalent to 10×M real-number additions. By using this comparison rule, the complexity of the SS receiver of this invention using QPSK modulation is much less than in the QR-MLD method. However, in the case of 16QAM modulation, the SS receiver of this invention requires more operations than in the QR-MLD method. In other words, when the SS receiver of this invention requires approximately 3,900 additions, then 2,900 additions are required in the QR-MLD method.
Here a non-optimized SS receiver is assumed, so in the future there is a possibility of reducing the complexity of a few SS receivers, and there is a possibility of reducing the number of real-number additions to 400 to 600.
With the invention described above, the respective receiver for each of a plurality of receiving antennas uses a received signal that was received by the receiving antenna that is connected to it, as well as the received signals that were received by other receiving antennas, to calculate the probability of the signal that was transmitted from a specified transmitter, and then by judging the signal that was transmitted by that transmitter based on the calculated probability, is capable of improving the BER characteristic without having to perform iterative computations as in the case of a turbo receiver, so it is possible to improve the amount of computations, computation time and complexity.
Also, since the probability-computation unit comprises a plurality of non-linear processing circuits that have a transfer function as an amplitude limiter, and a combining circuit that combines the output from a plurality of non-linear processing circuits, the SS receiver can be constructed easily.
(c) Appendix
From equations (22), (24), (26) and (28), the linear portion of the decision rule equation of equation (15) can be expressed by equation (48) below.

0.25(ln P(S0)+ln P(S16)+ln P(S1)+ln P(S17)+ln P(S2)+ln P(S18)+ln P(S3)+ln P(S19))+

0.25(ln P(S4)+ln P(S20)+ln P(S5)+ln P(S21)+ln P(S6)+ln P(S22)+ln P(S7)+ln P(S23))−

0.25(ln P(S8)+ln P(S24)+ln P(S9)+ln P(S25)+ln P(S10)+ln P(S26)+ln P(S11)+ln P(S27))−

0.25(ln P(S12)+ln P(S28)+ln P(S13)+ln P(S29)+ln P(S14)+ln P(S30)+ln P(S15)+ln P(S31))=

4 Re(y0(t)h*00+y1(t)h*10)  (48)

The following equations are established for the non-linear higher-order elements in the receiver REC0.
1) First-Order Difference

ln P(S0)−ln P(S1)=4Im{y0(t)h*01}−4Im{S*00h00h*01}

ln P(S2)−ln P(S3)=4Im{y0(t)h*01}−4Im{S*00h00h*01}

ln P(S4)−ln P(S5)=4Im{y0(t)h*01}−4Im{S*01h00h*01}

ln P(S6)−ln P(S7)=4Im{y0(t)h*01}−4Im{S*01h00h*01}

ln P(S8)−ln P(S9)=4Im{y0(t)h*01}−4Im{S*01h00h*01}

ln P(S10)−ln P(S11)=4Im{y0(t)h*01}−4Im{S*01h00h*01}

ln P(S12)−ln P(S13)=4Im{y0(t)h*01}−4Im{S*00h00h*01}

ln P(S14)−ln P(S15)=4Im{y0(t)h*01}−4Im{S*00h00h*01}  (49)

2) Second-Order Difference

0.5((ln P(S0)+ln P(S1))−(ln P(S2)+ln P(S3)))=4 Re{y0(t)h*01}−4 Re{h00S*00h*01}

0.5((ln P(S4)+ln P(S5))−(ln P(S6)+ln P(S7)))=4 Re{y0(t)h*01}−4 Re{h00S*00h*01}

0.5((ln P(S8)+ln P(S9))−(ln P(S10)+ln P(S11)))=4 Re{y0(t)h*01}−4 Re{h00S*01h*01}

0.5((ln P(S12)+ln P(S13))−(ln P(S14)+ln P(S15)))=4 Re{y0(t)h*01}−4 Re{h00S*01h*01}  (50)

Similarly, the following equations are established for the non-linear higher-order elements in the receiver REC1.
1) First-Order Difference

ln P(S16)−ln P(S17)=4 Im{y1(t)h*11}−4 Im{S*00h10h*11}

ln P(S18)−ln P(S19)=4 Im{y1(t)h*11}−4 Im{S*00h10h*11}

ln P(S20)−ln P(S21)=4 Im{y1(t)h*11}−4 Im{S*01h10h*11}

ln P(S22)−ln P(S23)=4 Im{y1(t)h*11}−4 Im{S*01h10h*11}

ln P(S24)−ln P(S25)=4 Im{y1(t)h*11}−4 Im{S*00h10h*11}

ln P(S26)−ln P(S27)=4 Im{y1(t)h*11}−4 Im{S*00h10h*11}

ln P(S28)−ln P(S28)=4 Im{y1(t)h*11}−4 Im{S*01h10h*11}

ln P(S30)−ln P(S31)=4 Im{y1(t)h*11}−4 Im{S*01h10h*11}  (51)

2) Second-Order Difference

0.5((ln P(S16)+ln P(S17))−(ln P(S18)+ln P(S19)))=4 Re{y1(t)h*11}−4 Re{h10S*00h*11}

0.5((ln P(S20)+ln P(S21))−(ln P(S22)+ln P(S23)))=4 Re{y1(t)h*11}−4 Re{h10S*00h*11}

0.5((ln P(S24)+ln P(S25))−(ln P(S26)+ln P(S27)))=4 Re{y1(t)h*11}−4 Re{h10S*01h*11}

0.5((ln P(S28)+ln P(S29))−(ln P(S30)+ln P(S31)))=4 Re{y1(t)h*11}−4 Re{h10S*01h*11}  (52)

Also, equations (17) and (18) are simplified using the following equation.




  
    
      
        
          ln
          ⁢
          
            {
            
              cosh
              ⁡
              
                [
                
                  0.5
                  ⁢
                  
                    (
                    
                      
                        
                          
                            
                              0.25
                              ⁢
                              
                                (
                                
                                  
                                    
                                      
                                        
                                          ln
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          
                                            P
                                            ⁡
                                            
                                              (
                                              
                                                S
                                                0
                                              
                                              )
                                            
                                          
                                        
                                        +
                                        
                                          ln
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          
                                            P
                                            ⁡
                                            
                                              (
                                              
                                                S
                                                16
                                              
                                              )
                                            
                                          
                                        
                                        +
                                        
                                          ln
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          
                                            P
                                            ⁡
                                            
                                              (
                                              
                                                S
                                                1
                                              
                                              )
                                            
                                          
                                        
                                        +
                                        
                                          ln
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          
                                            P
                                            ⁡
                                            
                                              (
                                              
                                                S
                                                17
                                              
                                              )
                                            
                                          
                                        
                                        -
                                      
                                    
                                  
                                  
                                    
                                      
                                        
                                          ln
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          
                                            P
                                            ⁡
                                            
                                              (
                                              
                                                S
                                                2
                                              
                                              )
                                            
                                          
                                        
                                        -
                                        
                                          ln
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          
                                            P
                                            ⁡
                                            
                                              (
                                              
                                                S
                                                18
                                              
                                              )
                                            
                                          
                                        
                                        -
                                        
                                          ln
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          
                                            P
                                            ⁡
                                            
                                              (
                                              
                                                S
                                                3
                                              
                                              )
                                            
                                          
                                        
                                        -
                                        
                                          ln
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          
                                            P
                                            ⁡
                                            
                                              (
                                              
                                                S
                                                19
                                              
                                              )
                                            
                                          
                                        
                                      
                                    
                                  
                                
                                )
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              0.5
                              ⁢
                              
                                (
                                
                                  ln
                                  ⁢
                                  
                                    {
                                    
                                      cosh
                                      ⁡
                                      
                                        [
                                        
                                          0.5
                                          ⁢
                                          
                                            (
                                            
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      0
                                                    
                                                    )
                                                  
                                                
                                              
                                              +
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      16
                                                    
                                                    )
                                                  
                                                
                                              
                                              -
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      1
                                                    
                                                    )
                                                  
                                                
                                              
                                              -
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      17
                                                    
                                                    )
                                                  
                                                
                                              
                                            
                                            )
                                          
                                        
                                        ]
                                      
                                    
                                    }
                                  
                                
                                )
                              
                            
                            -
                          
                        
                      
                      
                        
                          
                            0.5
                            ⁢
                            
                              (
                              
                                ln
                                ⁢
                                
                                  {
                                  
                                    cosh
                                    ⁡
                                    
                                      [
                                      
                                        0.5
                                        ⁢
                                        
                                          (
                                          
                                            
                                              ln
                                              ⁢
                                              
                                                
                                              
                                              ⁢
                                              
                                                P
                                                ⁡
                                                
                                                  (
                                                  
                                                    S
                                                    2
                                                  
                                                  )
                                                
                                              
                                            
                                            +
                                            
                                              ln
                                              ⁢
                                              
                                                
                                              
                                              ⁢
                                              
                                                P
                                                ⁡
                                                
                                                  (
                                                  
                                                    S
                                                    18
                                                  
                                                  )
                                                
                                              
                                            
                                            -
                                            
                                              ln
                                              ⁢
                                              
                                                
                                              
                                              ⁢
                                              
                                                P
                                                ⁡
                                                
                                                  (
                                                  
                                                    S
                                                    3
                                                  
                                                  )
                                                
                                              
                                            
                                            -
                                            
                                              ln
                                              ⁢
                                              
                                                
                                              
                                              ⁢
                                              
                                                P
                                                ⁡
                                                
                                                  (
                                                  
                                                    S
                                                    19
                                                  
                                                  )
                                                
                                              
                                            
                                          
                                          )
                                        
                                      
                                      ]
                                    
                                  
                                  }
                                
                              
                              )
                            
                          
                        
                      
                    
                    )
                  
                
                ]
              
            
            }
          
        
        =
        
          ln
          ⁢
          
            {
            
              cosh
              ⁡
              
                [
                
                  0.5
                  ⁢
                  
                    (
                    
                      0.5
                      ⁢
                      
                        (
                        
                          
                            ln
                            ⁢
                            
                              
                            
                            ⁢
                            
                              P
                              ⁡
                              
                                (
                                
                                  S
                                  0
                                
                                )
                              
                            
                          
                          +
                          
                            ln
                            ⁢
                            
                              
                            
                            ⁢
                            
                              P
                              ⁡
                              
                                (
                                
                                  S
                                  16
                                
                                )
                              
                            
                          
                          -
                          
                            ln
                            ⁢
                            
                              
                            
                            ⁢
                            
                              P
                              ⁡
                              
                                (
                                
                                  S
                                  2
                                
                                )
                              
                            
                          
                          -
                          
                            ln
                            ⁢
                            
                              
                            
                            ⁢
                            
                              P
                              ⁡
                              
                                (
                                
                                  S
                                  18
                                
                                )
                              
                            
                          
                        
                        )
                      
                    
                    )
                  
                
                ]
              
            
            }
          
        
      
    
    
      
        (
        53
        )
      
    
  




Similar simplification is also possible for other elements.

From the above, an equation that subtracts the non-linear element of the third term on the right of equation (28) from the non-linear element of the third term on the right of equation (22) are given by the following equation.




  
    
      
        
          
            0.5
            ⁢
            
              (
              
                ln
                ⁢
                
                  {
                  
                    cosh
                    ⁡
                    
                      [
                      
                        0.5
                        ⁢
                        
                          
                        
                        ⁢
                        
                          (
                          
                            
                          
                          ⁢
                          
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    0
                                  
                                  )
                                
                              
                            
                            +
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    16
                                  
                                  )
                                
                              
                            
                            -
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    1
                                  
                                  )
                                
                              
                            
                            -
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    17
                                  
                                  )
                                
                              
                            
                          
                          )
                        
                      
                      ]
                    
                  
                  }
                
              
              )
            
          
          -
          
            0.5
            ⁢
            
              (
              
                ln
                ⁢
                
                  {
                  
                    cosh
                    ⁡
                    
                      [
                      
                        0.5
                        ⁢
                        
                          (
                          
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    12
                                  
                                  )
                                
                              
                            
                            +
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    28
                                  
                                  )
                                
                              
                            
                            -
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    13
                                  
                                  )
                                
                              
                            
                            -
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    29
                                  
                                  )
                                
                              
                            
                          
                          )
                        
                      
                      ]
                    
                  
                  }
                
              
              )
            
          
        
        =
        
          
            0.5
            ⁢
            
              (
              
                ln
                ⁢
                
                  {
                  
                    cosh
                    ⁡
                    
                      [
                      
                        0.5
                        ⁢
                        
                          (
                          
                            
                              
                                
                                  
                                    4
                                    ⁢
                                    Im
                                    ⁢
                                    
                                      {
                                      
                                        
                                          
                                            y
                                            0
                                          
                                          ⁡
                                          
                                            (
                                            t
                                            )
                                          
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      }
                                    
                                  
                                  -
                                  
                                    4
                                    ⁢
                                    Im
                                    ⁢
                                    
                                      {
                                      
                                        
                                          S
                                          00
                                          *
                                        
                                        ⁢
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      }
                                    
                                  
                                  +
                                
                              
                            
                            
                              
                                
                                  
                                    4
                                    ⁢
                                    Im
                                    ⁢
                                    
                                      {
                                      
                                        
                                          
                                            y
                                            1
                                          
                                          ⁡
                                          
                                            (
                                            t
                                            )
                                          
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                      }
                                    
                                  
                                  -
                                  
                                    4
                                    ⁢
                                    Im
                                    ⁢
                                    
                                      {
                                      
                                        
                                          S
                                          00
                                          *
                                        
                                        ⁢
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                      }
                                    
                                  
                                
                              
                            
                          
                          )
                        
                      
                      ]
                    
                  
                  }
                
              
              )
            
          
          -
          
            0.5
            ⁢
            
              (
              
                ln
                ⁢
                
                  {
                  
                    cosh
                    ⁡
                    
                      [
                      
                        0.5
                        ⁢
                        
                          (
                          
                            
                              
                                
                                  
                                    4
                                    ⁢
                                    Im
                                    ⁢
                                    
                                      {
                                      
                                        
                                          
                                            y
                                            0
                                          
                                          ⁡
                                          
                                            (
                                            t
                                            )
                                          
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      }
                                    
                                  
                                  +
                                  
                                    4
                                    ⁢
                                    Im
                                    ⁢
                                    
                                      {
                                      
                                        
                                          S
                                          00
                                          *
                                        
                                        ⁢
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      }
                                    
                                  
                                  +
                                
                              
                            
                            
                              
                                
                                  
                                    4
                                    ⁢
                                    Im
                                    ⁢
                                    
                                      {
                                      
                                        
                                          
                                            y
                                            1
                                          
                                          ⁡
                                          
                                            (
                                            t
                                            )
                                          
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                      }
                                    
                                  
                                  +
                                  
                                    4
                                    ⁢
                                    Im
                                    ⁢
                                    
                                      {
                                      
                                        
                                          S
                                          00
                                          *
                                        
                                        ⁢
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                      }
                                    
                                  
                                
                              
                            
                          
                          )
                        
                      
                      ]
                    
                  
                  }
                
              
              )
            
          
        
      
    
    
      
        (
        54
        )
      
    
  




Here, by rewriting the equation taking into consideration the following equation,

Im{S*00h00h*01}=Re{h00h01}+Im{h00h01}  (55)

equation (56) is obtained.





  
    
      
        
          L
          1
        
        =
        
          (
          
            
              
                
                  
                    ln
                    ⁢
                    
                      
                    
                    ⁢
                    ch
                    ⁢
                    
                      {
                      
                        0.5
                        ⁢
                        
                          (
                          
                            
                              4
                              ⁢
                              
                                Im
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        
                                          y
                                          0
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        01
                                        *
                                      
                                    
                                    +
                                    
                                      
                                        
                                          y
                                          1
                                        
                                        ⁡
                                        
                                          (
                                          t
                                          )
                                        
                                      
                                      ⁢
                                      
                                        h
                                        11
                                        *
                                      
                                    
                                  
                                  )
                                
                              
                            
                            -
                            
                              (
                              
                                
                                  4
                                  ⁢
                                  
                                    Re
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                +
                                
                                  4
                                  ⁢
                                  
                                    Im
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                          )
                        
                      
                      }
                    
                  
                  -
                
              
            
            
              
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  ch
                  ⁢
                  
                    {
                    
                      0.5
                      ⁢
                      
                        (
                        
                          
                            4
                            ⁢
                            
                              Im
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                        y
                                        0
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      01
                                      *
                                    
                                  
                                  +
                                  
                                    
                                      
                                        y
                                        1
                                      
                                      ⁡
                                      
                                        (
                                        t
                                        )
                                      
                                    
                                    ⁢
                                    
                                      h
                                      11
                                      *
                                    
                                  
                                
                                )
                              
                            
                          
                          +
                          
                            (
                            
                              
                                4
                                ⁢
                                
                                  Re
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          11
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              +
                              
                                4
                                ⁢
                                
                                  Im
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                        )
                      
                    
                    }
                  
                
              
            
          
          )
        
      
    
    
      
        (
        56
        )
      
    
  




Similarly, L={L1, L2, L3, L4} is obtained.
Next, f={f1, f2, f3, f4} will be explained.
When considering the non-linear portion of equation (17)




  
    
      
        ln
        ⁢
        
          {
          
            cosh
            ⁡
            
              [
              
                
                  
                    
                      
                        0.5
                        ⁢
                        
                          (
                          
                            ln
                            ⁡
                            
                              (
                              
                                
                                  
                                    P
                                    ⁡
                                    
                                      (
                                      
                                        S
                                        0
                                      
                                      )
                                    
                                  
                                  ⁢
                                  
                                    P
                                    ⁡
                                    
                                      (
                                      
                                        S
                                        16
                                      
                                      )
                                    
                                  
                                
                                +
                                
                                  
                                    P
                                    ⁡
                                    
                                      (
                                      
                                        S
                                        1
                                      
                                      )
                                    
                                  
                                  ⁢
                                  
                                    P
                                    ⁡
                                    
                                      (
                                      
                                        S
                                        17
                                      
                                      )
                                    
                                  
                                
                                +
                                
                                  
                                    P
                                    ⁡
                                    
                                      (
                                      
                                        S
                                        2
                                      
                                      )
                                    
                                  
                                  ⁢
                                  
                                    P
                                    ⁡
                                    
                                      (
                                      
                                        S
                                        18
                                      
                                      )
                                    
                                  
                                
                                +
                                
                                  
                                    P
                                    ⁡
                                    
                                      (
                                      
                                        S
                                        3
                                      
                                      )
                                    
                                  
                                  ⁢
                                  
                                    P
                                    ⁡
                                    
                                      (
                                      
                                        S
                                        19
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                          )
                        
                      
                      -
                    
                  
                
                
                  
                    
                      0.5
                      ⁢
                      
                        (
                        
                          ln
                          ⁡
                          
                            (
                            
                              
                                
                                  P
                                  ⁡
                                  
                                    (
                                    
                                      S
                                      4
                                    
                                    )
                                  
                                
                                ⁢
                                
                                  P
                                  ⁡
                                  
                                    (
                                    
                                      S
                                      20
                                    
                                    )
                                  
                                
                              
                              +
                              
                                
                                  P
                                  ⁡
                                  
                                    (
                                    
                                      S
                                      5
                                    
                                    )
                                  
                                
                                ⁢
                                
                                  P
                                  ⁡
                                  
                                    (
                                    
                                      S
                                      21
                                    
                                    )
                                  
                                
                              
                              +
                              
                                
                                  P
                                  ⁡
                                  
                                    (
                                    
                                      S
                                      6
                                    
                                    )
                                  
                                
                                ⁢
                                
                                  P
                                  ⁡
                                  
                                    (
                                    
                                      S
                                      22
                                    
                                    )
                                  
                                
                              
                              +
                              
                                
                                  P
                                  ⁡
                                  
                                    (
                                    
                                      S
                                      7
                                    
                                    )
                                  
                                
                                ⁢
                                
                                  P
                                  ⁡
                                  
                                    (
                                    
                                      S
                                      23
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                        )
                      
                    
                  
                
              
              ]
            
          
          }
        
      
    
    
      
        (
        57
        )
      
    
  




the following equation is established.





  
    
      
        
          ln
          ⁡
          
            (
            
              
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      0
                    
                    )
                  
                
                ⁢
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      16
                    
                    )
                  
                
              
              +
              
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      1
                    
                    )
                  
                
                ⁢
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      17
                    
                    )
                  
                
              
              +
              
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      2
                    
                    )
                  
                
                ⁢
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      18
                    
                    )
                  
                
              
              +
              
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      3
                    
                    )
                  
                
                ⁢
                
                  P
                  ⁡
                  
                    (
                    
                      S
                      19
                    
                    )
                  
                
              
            
            )
          
        
        =
        
          
            0.25
            ⁢
            
              (
              
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        0
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        16
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        1
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        17
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        2
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        18
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        3
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        19
                      
                      )
                    
                  
                
              
              )
            
          
          +
          
            ln
            ⁢
            
              
            
            ⁢
            2
          
          +
          
            0.5
            ⁢
            
              (
              
                
                  
                    
                      
                        ln
                        ⁢
                        
                          
                        
                        ⁢
                        cosh
                        ⁢
                        
                          {
                          
                            0.5
                            ⁢
                            
                              (
                              
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    P
                                    ⁡
                                    
                                      (
                                      
                                        S
                                        0
                                      
                                      )
                                    
                                  
                                
                                +
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    P
                                    ⁡
                                    
                                      (
                                      
                                        S
                                        16
                                      
                                      )
                                    
                                  
                                
                                -
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    P
                                    ⁡
                                    
                                      (
                                      
                                        S
                                        1
                                      
                                      )
                                    
                                  
                                
                                -
                                
                                  ln
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    P
                                    ⁡
                                    
                                      (
                                      
                                        S
                                        17
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                          }
                        
                      
                      +
                    
                  
                
                
                  
                    
                      ln
                      ⁢
                      
                        
                      
                      ⁢
                      cosh
                      ⁢
                      
                        {
                        
                          0.5
                          ⁢
                          
                            (
                            
                              
                                ln
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  P
                                  ⁡
                                  
                                    (
                                    
                                      S
                                      2
                                    
                                    )
                                  
                                
                              
                              +
                              
                                ln
                                ⁢
                                
                                  
                                
                                ⁢
                                P
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  (
                                  
                                    S
                                    18
                                  
                                  )
                                
                              
                              -
                              
                                ln
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  P
                                  ⁡
                                  
                                    (
                                    
                                      S
                                      3
                                    
                                    )
                                  
                                
                              
                              -
                              
                                ln
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  P
                                  ⁡
                                  
                                    (
                                    
                                      S
                                      19
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                        }
                      
                    
                  
                
              
              )
            
          
          +
          
            ln
            ⁢
            
              
            
            ⁢
            2
          
          +
          
            ln
            ⁢
            
              
            
            ⁢
            cosh
            ⁢
            
              {
              
                0.5
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      
                        
                          
                            0.25
                            ⁢
                            
                              (
                              
                                
                                  
                                    
                                      
                                        ln
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        
                                          P
                                          ⁡
                                          
                                            (
                                            
                                              S
                                              0
                                            
                                            )
                                          
                                        
                                      
                                      +
                                      
                                        ln
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        
                                          P
                                          ⁡
                                          
                                            (
                                            
                                              S
                                              16
                                            
                                            )
                                          
                                        
                                      
                                      +
                                      
                                        ln
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        
                                          P
                                          ⁡
                                          
                                            (
                                            
                                              S
                                              1
                                            
                                            )
                                          
                                        
                                      
                                      +
                                      
                                        ln
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        
                                          P
                                          ⁡
                                          
                                            (
                                            
                                              S
                                              17
                                            
                                            )
                                          
                                        
                                      
                                      -
                                    
                                  
                                
                                
                                  
                                    
                                      
                                        ln
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        
                                          P
                                          ⁡
                                          
                                            (
                                            
                                              S
                                              2
                                            
                                            )
                                          
                                        
                                      
                                      -
                                      
                                        ln
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        
                                          P
                                          ⁡
                                          
                                            (
                                            
                                              S
                                              18
                                            
                                            )
                                          
                                        
                                      
                                      -
                                      
                                        ln
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        
                                          P
                                          ⁡
                                          
                                            (
                                            
                                              S
                                              3
                                            
                                            )
                                          
                                        
                                      
                                      -
                                      
                                        ln
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        
                                          P
                                          ⁡
                                          
                                            (
                                            
                                              S
                                              19
                                            
                                            )
                                          
                                        
                                      
                                    
                                  
                                
                              
                              )
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          0.50
                          ⁢
                          
                            (
                            
                              
                                
                                  
                                    
                                      ln
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      cosh
                                      ⁢
                                      
                                        {
                                        
                                          0.5
                                          ⁢
                                          
                                            (
                                            
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      0
                                                    
                                                    )
                                                  
                                                
                                              
                                              +
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      16
                                                    
                                                    )
                                                  
                                                
                                              
                                              -
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      1
                                                    
                                                    )
                                                  
                                                
                                              
                                              -
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      17
                                                    
                                                    )
                                                  
                                                
                                              
                                            
                                            )
                                          
                                        
                                        }
                                      
                                    
                                    -
                                  
                                
                              
                              
                                
                                  
                                    
                                      ln
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      cosh
                                      ⁢
                                      
                                        {
                                        
                                          0.5
                                          ⁢
                                          
                                            (
                                            
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      2
                                                    
                                                    )
                                                  
                                                
                                              
                                              +
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      18
                                                    
                                                    )
                                                  
                                                
                                              
                                              -
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      3
                                                    
                                                    )
                                                  
                                                
                                              
                                              -
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      19
                                                    
                                                    )
                                                  
                                                
                                              
                                            
                                            )
                                          
                                        
                                        }
                                      
                                    
                                    ⁢
                                    
                                      
                                    
                                  
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                  ⁢
                  
                    
                  
                  )
                
              
              ⁢
              
                
              
              }
            
          
        
      
    
    
      
        (
        58
        )
      
    
  




Similarly, the following equation is established.




  
    
      
        
          (
          
            ln
            ⁡
            
              (
              
                
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        4
                      
                      )
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        20
                      
                      )
                    
                  
                
                +
                
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        5
                      
                      )
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        21
                      
                      )
                    
                  
                
                +
                
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        6
                      
                      )
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        22
                      
                      )
                    
                  
                
                +
                
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        7
                      
                      )
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        23
                      
                      )
                    
                  
                
              
              )
            
          
          )
        
        =
        
          
            0.25
            ⁢
            
              (
              
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        4
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        20
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        5
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        21
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        6
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        22
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        7
                      
                      )
                    
                  
                
                +
                
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        S
                        23
                      
                      )
                    
                  
                
              
              )
            
          
          +
          
            ln
            ⁢
            
              
            
            ⁢
            2
          
          +
          
            0.5
            ⁢
            
              (
              
                ln
                ⁢
                
                  {
                  
                    cosh
                    ⁡
                    
                      [
                      
                        0.5
                        ⁢
                        
                          (
                          
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    4
                                  
                                  )
                                
                              
                            
                            +
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    20
                                  
                                  )
                                
                              
                            
                            -
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    5
                                  
                                  )
                                
                              
                            
                            -
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    21
                                  
                                  )
                                
                              
                            
                          
                          )
                        
                      
                      ]
                    
                  
                  }
                
              
              )
            
          
          +
          
            0.5
            ⁢
            
              (
              
                ln
                ⁢
                
                  {
                  
                    cosh
                    ⁡
                    
                      [
                      
                        0.5
                        ⁢
                        
                          (
                          
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    6
                                  
                                  )
                                
                              
                            
                            +
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    22
                                  
                                  )
                                
                              
                            
                            -
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    7
                                  
                                  )
                                
                              
                            
                            -
                            
                              ln
                              ⁢
                              
                                
                              
                              ⁢
                              
                                P
                                ⁡
                                
                                  (
                                  
                                    S
                                    23
                                  
                                  )
                                
                              
                            
                          
                          )
                        
                      
                      ]
                    
                  
                  }
                
              
              )
            
          
          +
          
            ln
            ⁢
            
              
            
            ⁢
            2
          
          +
          
            ln
            ⁢
            
              {
              
                cosh
                [
                
                  
                
                ⁢
                
                  0.5
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      
                        
                          
                            
                              
                                
                                  
                                    0.25
                                    ⁢
                                    
                                      (
                                      
                                        
                                          
                                            
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      4
                                                    
                                                    )
                                                  
                                                
                                              
                                              +
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      20
                                                    
                                                    )
                                                  
                                                
                                              
                                              +
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      5
                                                    
                                                    )
                                                  
                                                
                                              
                                              +
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      21
                                                    
                                                    )
                                                  
                                                
                                              
                                              -
                                            
                                          
                                        
                                        
                                          
                                            
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      6
                                                    
                                                    )
                                                  
                                                
                                              
                                              -
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      22
                                                    
                                                    )
                                                  
                                                
                                              
                                              -
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      7
                                                    
                                                    )
                                                  
                                                
                                              
                                              -
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      23
                                                    
                                                    )
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                  +
                                
                              
                            
                            
                              
                                
                                  
                                    0.5
                                    ⁢
                                    
                                      (
                                      
                                        ln
                                        ⁢
                                        
                                          {
                                          
                                            cosh
                                            ⁡
                                            
                                              [
                                              
                                                0.5
                                                ⁢
                                                
                                                  (
                                                  
                                                    
                                                      ln
                                                      ⁢
                                                      
                                                        
                                                      
                                                      ⁢
                                                      
                                                        P
                                                        ⁡
                                                        
                                                          (
                                                          
                                                            S
                                                            4
                                                          
                                                          )
                                                        
                                                      
                                                    
                                                    +
                                                    
                                                      ln
                                                      ⁢
                                                      
                                                        
                                                      
                                                      ⁢
                                                      
                                                        P
                                                        ⁡
                                                        
                                                          (
                                                          
                                                            S
                                                            20
                                                          
                                                          )
                                                        
                                                      
                                                    
                                                    -
                                                    
                                                      ln
                                                      ⁢
                                                      
                                                        
                                                      
                                                      ⁢
                                                      
                                                        P
                                                        ⁡
                                                        
                                                          (
                                                          
                                                            S
                                                            5
                                                          
                                                          )
                                                        
                                                      
                                                    
                                                    -
                                                    
                                                      ln
                                                      ⁢
                                                      
                                                        
                                                      
                                                      ⁢
                                                      
                                                        P
                                                        ⁡
                                                        
                                                          (
                                                          
                                                            S
                                                            21
                                                          
                                                          )
                                                        
                                                      
                                                    
                                                  
                                                  )
                                                
                                              
                                              ]
                                            
                                          
                                          }
                                        
                                      
                                      )
                                    
                                  
                                  -
                                
                              
                            
                          
                        
                      
                      
                        
                          
                            
                              0.5
                              ⁢
                              
                                (
                                
                                  ln
                                  ⁢
                                  
                                    {
                                    
                                      cosh
                                      ⁡
                                      
                                        [
                                        
                                          0.5
                                          ⁢
                                          
                                            (
                                            
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      6
                                                    
                                                    )
                                                  
                                                
                                              
                                              +
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      22
                                                    
                                                    )
                                                  
                                                
                                              
                                              -
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      7
                                                    
                                                    )
                                                  
                                                
                                              
                                              -
                                              
                                                ln
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  P
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      S
                                                      23
                                                    
                                                    )
                                                  
                                                
                                              
                                            
                                            )
                                          
                                        
                                        ]
                                      
                                    
                                    }
                                  
                                
                                )
                              
                            
                            ⁢
                            
                              
                            
                          
                        
                      
                    
                    ⁢
                    
                      
                    
                    )
                  
                
                ⁢
                
                  
                
                ]
              
              ⁢
              
                
              
              }
            
          
        
      
    
    
      
        (
        59
        )
      
    
  




Lin1, Lin2 and Lin in equations (58) and (59) are defined as follows.

Lin1=0.25(ln P(S0)+ln P(S16)+ln P(S1)+ln P(S17)+ln P(S2)+ln P(S18)+ln P(S3)+ln P(S19))

Lin2=0.25(ln P(S4)+ln P(S20)+ln P(S5)+ln P(S21)+ln P(S6)+ln P(S22)+ln P(S7)+ln P(S23))

Lin=Lin1−Lin2  (60)

Also, the following equations are established for the receivers REC0, REC1.

ln P(S0)−ln P(S4)=4 Im{y0(t)h*00}+4 Im{(S*10)*h00h*01}

ln P(S1)−ln P(S5)=4 Im{y0(t)h*00}+4 Im{(S*11)*h00h*01}

ln P(S2)−ln P(S6)=4 Im{y0(t)h*00}−4 Im{(S*11)*h00h*01}

ln P(S3)−ln P(S7)=4 Im{y0(t)h*00}−4 Im{(S*10)*h00h*01}

ln P(S16)−ln P(S20)=4 Im{y1(t)h*10}+4 Im{(S*10)*h10h*11}

ln P(S17)−ln P(S21)=4 Im{y1(t)h*10}+4 Im{(S*11)*h10h*11}

ln P(S18)−ln P(S22)=4 Im{y1(t)h*10}−4 Im{(S*11)*h10h*11}

ln P(S19)−ln P(S23)=4 Im{y1(t)h*10}−4 Im{(S*10)*h10h*11}  (61)

Equation (62) is found from equation (61).

ln P(S0))+ln P(S1)+ln P(S2)+ln P(S3)−ln P(S4)−ln P(S5)−ln P(S6)−ln P(S7)=4*4 Im{y0(t)h00}

ln P(S16)+ln P(S17)+ln P(S18)+ln P(S19)−ln P(S20)−ln P(S21)−ln P(S22)−ln P(S23)=4*4 Im{y1(t)h10}

Lin=4 Im{y0(t)h00+y1(t)h10}  (62)

Considering the above, after simplifying all of the non-linear elements, the following equations can be obtained.




  
    
      
        
          
            ln
            ⁢
            
              
            
            ⁢
            ch
            ⁢
            
              {
              
                0.5
                ⁢
                
                  (
                  
                    
                      f
                      1
                    
                    +
                    
                      f
                      2
                    
                  
                  )
                
              
              }
            
          
          -
          
            ln
            ⁢
            
              
            
            ⁢
            ch
            ⁢
            
              {
              
                0.5
                ⁢
                
                  (
                  
                    
                      f
                      3
                    
                    +
                    
                      f
                      4
                    
                  
                  )
                
              
              }
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              
                
                  f
                  1
                
                =
                
                ⁢
                
                  (
                  
                    
                      
                        
                          ln
                          ⁢
                          
                            
                          
                          ⁢
                          ch
                          ⁢
                          
                            {
                            
                              0.5
                              ⁢
                              
                                (
                                
                                  
                                    4
                                    ⁢
                                    
                                      Im
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              
                                                y
                                                0
                                              
                                              ⁡
                                              
                                                (
                                                t
                                                )
                                              
                                            
                                            ⁢
                                            
                                              h
                                              01
                                              *
                                            
                                          
                                          +
                                          
                                            
                                              
                                                y
                                                1
                                              
                                              ⁡
                                              
                                                (
                                                t
                                                )
                                              
                                            
                                            ⁢
                                            
                                              h
                                              11
                                              *
                                            
                                          
                                        
                                        )
                                      
                                    
                                  
                                  -
                                  
                                    (
                                    
                                      
                                        4
                                        ⁢
                                        
                                          Im
                                          ⁡
                                          
                                            (
                                            
                                              
                                                
                                                  h
                                                  00
                                                
                                                ⁢
                                                
                                                  h
                                                  01
                                                  *
                                                
                                              
                                              +
                                              
                                                
                                                  h
                                                  10
                                                
                                                ⁢
                                                
                                                  h
                                                  11
                                                  *
                                                
                                              
                                            
                                            )
                                          
                                        
                                      
                                      +
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  4
                                  ⁢
                                  
                                    Re
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                )
                              
                              )
                            
                            }
                          
                          -
                        
                      
                    
                    
                      
                        
                          ln
                          ⁢
                          
                            
                          
                          ⁢
                          ch
                          ⁢
                          
                            {
                            
                              0.5
                              ⁢
                              
                                (
                                
                                  
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                                    ⁢
                                    
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                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              
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                                                0
                                              
                                              ⁡
                                              
                                                (
                                                t
                                                )
                                              
                                            
                                            ⁢
                                            
                                              h
                                              01
                                              *
                                            
                                          
                                          +
                                          
                                            
                                              
                                                y
                                                1
                                              
                                              ⁡
                                              
                                                (
                                                t
                                                )
                                              
                                            
                                            ⁢
                                            
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                                              11
                                              *
                                            
                                          
                                        
                                        )
                                      
                                    
                                  
                                  +
                                  
                                    (
                                    
                                      
                                        4
                                        ⁢
                                        
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                                          ⁡
                                          
                                            (
                                            
                                              
                                                
                                                  h
                                                  00
                                                
                                                ⁢
                                                
                                                  h
                                                  01
                                                  *
                                                
                                              
                                              +
                                              
                                                
                                                  h
                                                  10
                                                
                                                ⁢
                                                
                                                  h
                                                  11
                                                  *
                                                
                                              
                                            
                                            )
                                          
                                        
                                      
                                      -
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                4
                                ⁢
                                
                                  Im
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
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                                          00
                                        
                                        ⁢
                                        
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                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              )
                            
                            )
                          
                          }
                        
                      
                    
                  
                  )
                
              
            
          
          
            
              
                
                  f
                  2
                
                =
                
                ⁢
                
                  (
                  
                    
                      
                        
                          
                            
                              
                                ln
                                ⁢
                                
                                  
                                
                                ⁢
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                                ⁢
                                
                                  {
                                  
                                    0.5
                                    ⁢
                                    
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                                          ⁢
                                          
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                                                    y
                                                    0
                                                  
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                                                    t
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                                                ⁢
                                                
                                                  h
                                                  01
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                                                    1
                                                  
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                                                    )
                                                  
                                                
                                                ⁢
                                                
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                                                  11
                                                  *
                                                
                                              
                                            
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                                              4
                                              ⁢
                                              
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                                                ⁡
                                                
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                                                        10
                                                      
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                                                        *
                                                      
                                                    
                                                  
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                                            -
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          
                            
                              
                                
                                  
                                    
                                      
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                                        ⁢
                                        
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                                          ⁡
                                          
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                                                  *
                                                
                                              
                                            
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                                      )
                                    
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                                -
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
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                                ⁢
                                
                                  
                                
                                ⁢
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                                ⁢
                                
                                  {
                                  
                                    0.5
                                    ⁢
                                    
                                      (
                                      
                                        
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                                          ⁢
                                          
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                                                    y
                                                    0
                                                  
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                                                    t
                                                    )
                                                  
                                                
                                                ⁢
                                                
                                                  h
                                                  01
                                                  *
                                                
                                              
                                              +
                                              
                                                
                                                  
                                                    y
                                                    1
                                                  
                                                  ⁡
                                                  
                                                    (
                                                    t
                                                    )
                                                  
                                                
                                                ⁢
                                                
                                                  h
                                                  11
                                                  *
                                                
                                              
                                            
                                            )
                                          
                                        
                                        -
                                        
                                          (
                                          
                                            
                                              4
                                              ⁢
                                              
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                                                ⁡
                                                
                                                  (
                                                  
                                                    
                                                      
                                                        h
                                                        00
                                                      
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                                                        01
                                                        *
                                                      
                                                    
                                                    +
                                                    
                                                      
                                                        h
                                                        10
                                                      
                                                      ⁢
                                                      
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                                                        11
                                                        *
                                                      
                                                    
                                                  
                                                  )
                                                
                                              
                                            
                                            +
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          
                            
                              
                                
                                  
                                    
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                                      ⁢
                                      
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                                        ⁡
                                        
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                                                11
                                                *
                                              
                                            
                                          
                                          )
                                        
                                      
                                    
                                    )
                                  
                                  )
                                
                                }
                              
                            
                          
                        
                      
                    
                  
                  )
                
              
            
          
          
            
              
                
                  f
                  3
                
                =
                
                ⁢
                
                  (
                  
                    
                      
                        
                          ln
                          ⁢
                          
                            
                          
                          ⁢
                          ch
                          ⁢
                          
                            {
                            
                              0.5
                              ⁢
                              
                                (
                                
                                  
                                    4
                                    ⁢
                                    
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                                      (
                                      
                                        
                                          
                                            
                                              y
                                              0
                                            
                                            ⁡
                                            
                                              (
                                              t
                                              )
                                            
                                          
                                          ⁢
                                          
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                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            
                                              y
                                              1
                                            
                                            ⁡
                                            
                                              (
                                              t
                                              )
                                            
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                  -
                                  
                                    (
                                    
                                      
                                        4
                                        ⁢
                                        
                                          Re
                                          ⁡
                                          
                                            (
                                            
                                              
                                                
                                                  h
                                                  00
                                                
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                                                  h
                                                  01
                                                  *
                                                
                                              
                                              +
                                              
                                                
                                                  h
                                                  10
                                                
                                                ⁢
                                                
                                                  h
                                                  11
                                                  *
                                                
                                              
                                            
                                            )
                                          
                                        
                                      
                                      -
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  4
                                  ⁢
                                  
                                    Im
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
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                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                )
                              
                              )
                            
                            }
                          
                          -
                        
                      
                    
                    
                      
                        
                          ln
                          ⁢
                          
                            
                          
                          ⁢
                          ch
                          ⁢
                          
                            {
                            
                              0.5
                              ⁢
                              
                                (
                                
                                  
                                    4
                                    ⁢
                                    
                                      Im
                                      (
                                      
                                        
                                          
                                            
                                              y
                                              0
                                            
                                            ⁡
                                            
                                              (
                                              t
                                              )
                                            
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            
                                              y
                                              1
                                            
                                            ⁡
                                            
                                              (
                                              t
                                              )
                                            
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                  +
                                  
                                    (
                                    
                                      
                                        4
                                        ⁢
                                        
                                          Re
                                          ⁡
                                          
                                            (
                                            
                                              
                                                
                                                  h
                                                  00
                                                
                                                ⁢
                                                
                                                  h
                                                  01
                                                  *
                                                
                                              
                                              +
                                              
                                                
                                                  h
                                                  10
                                                
                                                ⁢
                                                
                                                  h
                                                  11
                                                  *
                                                
                                              
                                            
                                            )
                                          
                                        
                                      
                                      +
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                4
                                ⁢
                                
                                  Im
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          h
                                          00
                                        
                                        ⁢
                                        
                                          h
                                          01
                                          *
                                        
                                      
                                      +
                                      
                                        
                                          h
                                          10
                                        
                                        ⁢
                                        
                                          h
                                          11
                                          *
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              )
                            
                            )
                          
                          }
                        
                      
                    
                  
                  )
                
              
            
          
          
            
              
                
                  f
                  4
                
                =
                
                ⁢
                
                  (
                  
                    
                      
                        
                          ln
                          ⁢
                          
                            
                          
                          ⁢
                          ch
                          ⁢
                          
                            {
                            
                              0.5
                              ⁢
                              
                                (
                                
                                  
                                    4
                                    ⁢
                                    
                                      Re
                                      (
                                      
                                        
                                          
                                            
                                              y
                                              0
                                            
                                            ⁡
                                            
                                              (
                                              t
                                              )
                                            
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            
                                              y
                                              1
                                            
                                            ⁡
                                            
                                              (
                                              t
                                              )
                                            
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                  +
                                  
                                    (
                                    
                                      
                                        4
                                        ⁢
                                        
                                          Re
                                          ⁡
                                          
                                            (
                                            
                                              
                                                
                                                  h
                                                  00
                                                
                                                ⁢
                                                
                                                  h
                                                  01
                                                  *
                                                
                                              
                                              +
                                              
                                                
                                                  h
                                                  10
                                                
                                                ⁢
                                                
                                                  h
                                                  11
                                                  *
                                                
                                              
                                            
                                            )
                                          
                                        
                                      
                                      +
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  4
                                  ⁢
                                  
                                    Im
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            h
                                            00
                                          
                                          ⁢
                                          
                                            h
                                            01
                                            *
                                          
                                        
                                        +
                                        
                                          
                                            h
                                            10
                                          
                                          ⁢
                                          
                                            h
                                            11
                                            *
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                )
                              
                              )
                            
                            }
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                ln
                                ⁢
                                
                                  
                                
                                ⁢
                                ch
                                ⁢
                                
                                  {
                                  
                                    0.5
                                    ⁢
                                    
                                      (
                                      
                                        
                                          4
                                          ⁢
                                          
                                            Re
                                            (
                                            
                                              
                                                
                                                  
                                                    y
                                                    0
                                                  
                                                  ⁡
                                                  
                                                    (
                                                    t
                                                    )
                                                  
                                                
                                                ⁢
                                                
                                                  h
                                                  01
                                                  *
                                                
                                              
                                              +
                                              
                                                
                                                  
                                                    y
                                                    1
                                                  
                                                  ⁡
                                                  
                                                    (
                                                    t
                                                    )
                                                  
                                                
                                                ⁢
                                                
                                                  h
                                                  11
                                                  *
                                                
                                              
                                            
                                            )
                                          
                                        
                                        -
                                        
                                          (
                                          
                                            
                                              4
                                              ⁢
                                              
                                                Im
                                                ⁡
                                                
                                                  (
                                                  
                                                    
                                                      
                                                        h
                                                        00
                                                      
                                                      ⁢
                                                      
                                                        h
                                                        01
                                                        *
                                                      
                                                    
                                                    +
                                                    
                                                      
                                                        h
                                                        10
                                                      
                                                      ⁢
                                                      
                                                        h
                                                        11
                                                        *
                                                      
                                                    
                                                  
                                                  )
                                                
                                              
                                            
                                            -
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          
                            
                              
                                
                                  
                                    
                                      4
                                      ⁢
                                      
                                        Re
                                        ⁡
                                        
                                          (
                                          
                                            
                                              
                                                h
                                                00
                                              
                                              ⁢
                                              
                                                h
                                                01
                                                *
                                              
                                            
                                            +
                                            
                                              
                                                h
                                                10
                                              
                                              ⁢
                                              
                                                h
                                                11
                                                *
                                              
                                            
                                          
                                          )
                                        
                                      
                                    
                                    )
                                  
                                  )
                                
                                }
                              
                            
                          
                        
                      
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        63
        )
      
    
  




The following equation is obtained from equations (48), (56) and (63).

ln P(dRe=+1/(y0(t),y1(t)))−ln P(dRe=−1/(y0(t),y1(t)))=ln P0(I)=

4 Re(y0(t)h*00+y1(t)h*10)+0.5(L1−L2+L3+L4)+ln ch{0.5(f1+f20)}−lnch{0.5(f3+f4)}  (64)

As many apparently widely different embodiments of the present invention can be made without departing from the spirit and scope thereof, it is to be understood that the invention is not limited to the specific embodiments thereof except as defined in the appended claims.

## Claims

1. A wireless receiving apparatus comprising:
a plurality of receiving antennas that receive signals that are transmitted from a plurality of transmission antennas and input via a plurality of propagation paths having different propagation characteristics, and
receivers that correspond to each of said plurality of antennas; wherein
each receiver comprises:
a computation unit that uses a received signal that was received by the receiving antenna that is connected to it, and received signals that were received by the other receiving antennas, to calculate probability of a signal that was transmitted from a specified transmitter, and
a judgment unit that judges the signal that was transmitted from said transmitter based on said probability wherein
said computation unit calculates said probability using an equation that calculates said probability based on the posteriori probability that the signal transmitted from the transmitter is +1, and the posteriori probability that the signal transmitted from the transmitter is −1.

2. A wireless receiving apparatus comprising:
a plurality of receiving antennas that receive signals that are transmitted from a plurality of transmission antennas and input via a plurality of propagation paths having different propagation characteristics, and
receivers that correspond to each of said plurality of antennas; wherein
each receiver comprises:
a computation unit that uses a received signal that was received by the receiving antenna that is connected to it, and received signals that were received by the other receiving antennas, to calculate probability of a signal that was transmitted from a specified transmitter, and
a judgment unit that judges the signal that was transmitted from said transmitter based on said probability wherein
said computation unit comprises:
a plurality of non-linear-processing circuits having a transfer function as an amplitude limiter; and
a combining circuit that combines the output from the plurality of non-linear-processing circuits and outputs the probability of said transmitted signal.

3. The wireless receiving apparatus of claim 2 wherein
said computation unit further comprises:
a signal-generation unit that generates an input signal input to each of said non-linear-processing circuits based on the product of a specified received signal and specified propagation characteristic.

4. The wireless receiving apparatus of claim 2 wherein the limit level of said non-linear-processing circuits depends on said propagation characteristic.

5. A wireless receiving apparatus comprising:
a plurality of receiving antennas that receive signals that are transmitted from a plurality of transmission antennas and input via a plurality of propagation paths having different propagation characteristics, and
receivers that correspond to each of said plurality of antennas; wherein
each receiver comprises:
a computation unit that uses a received signal that was received by the receiving antenna that is connected to it, and received signals that were received by the other receiving antennas, to calculate probability of a signal that was transmitted from a specified transmitter, and
a judgment unit that judges the signal that was transmitted from said transmitter based on said probability wherein
when a transmission station transmits data using QPSK modulation, said computation units comprises:
first and second computation units that execute the aforementioned processing for the real-number portion and imaginary-number portion of the received signal; and
said judgment unit comprises first and second judgment units that execute the aforementioned processing for the real-number portion and imaginary-number portion of the received signal.