Abstract:
Disclosed is a BSLM method of generating U independent alternative symbol sequences by changing a phase of symbol sequences by phase sequences and transmitting the alternative symbol sequences with a PAPR to lower the PARR of a transmission signal of a transmitter in an OFDM system. The transmitter uses m-sequences as the phase sequence and a receiver uses an ML decoder to distinguish the phase sequence. The BSLM method includes rotating phases in blocks according to the m-sequences used in the respective symbol sequences, selecting a signal sequence with a minimum PAPR among U alternative signal sequences generated by the m-sequences in the transmitter, and partitioning U alternative symbol sequences into U/2 blocks in the transmitter. Block partitioning and predetermined phase rotation are used to embed side information in an alternative signal sequence, and the BSLM scheme may be extended to a BPTS. Since low decoding complexity is given while detecting side information, the same BER performance is obtained as that of the conventional BSLM scheme in the AWGN channel.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to blind selective mapping (BSLM) and blind partial transmit sequence (BPTS) schemes with low decoding complexity for peak to average power ratio (PAPR) reduction of an orthogonal frequency division multiplexing (OFDM) signal. More particularly, the present invention relates to BSLM and BPTS schemes with low decoding complexity for PAPR reduction of an OFDM signal, in which m-sequences are used as phase sequences and a maximum likelihood (ML) decoder with low decoding complexity is used, so that a receiver can decide the phase sequence used in the transmitter, thereby decreasing decoding complexity while minimally degrading bit error rate (BER) performance. 
     2. Description of the Related Art 
     Orthogonal frequency division multiplexing (OFDM) is an efficient method for high speed data transmission in multipath fading. In addition, it is possible to significantly enhance the throughput by adapting a data rate per subcarrier from decomposition of a wideband channel. However, a major disadvantage of OFDM is the high peak to average power ratio (PAPR) of the transmitter&#39;s output signal, where the range of PAPR is proportional to the number of subcarriers. Due to the high PAPR feature, an OFDM signal can suffer significant inter-modulation and undesired out-of-band radiation when it passes through nonlinear devices, e.g., a high power amplifier (HPA). 
     Several PAPR reduction schemes have been proposed such as clipping and filtering, coding, selected mapping (SLM), partial transmit sequence (PTS), and tone reservation (TR). SLM and PTS, symbol scrambling techniques, can reduce the peak power of OFDM signals without signal distortion. The key idea of SLM is that the OFDM signal with the smallest PAPR is selected for transmission from several alternative OFDM signals which are obtained by applying inverse fast Fourier transform (IFFT) to each alternative symbol sequence that is an input symbol sequence multiplied by one of the phase sequences. In the PTS scheme, the input symbol sequence is partitioned into a number of disjointed subsequences. IFFT is applied to respective subsequences and the signals of subsequences are summed after they are multiplied by a set of rotating factors. 
     In SLM and PTS, the side information must be transmitted to enable the receiver to search the original OFDM symbol sequence. Such side information causes slight degradation in bandwidth efficiency. Moreover, incorrect detection of side information at to the receiver results in critical degradation of bit error rate (BER). For this reason, the side information must be highly protected so as not to affect the error performance of OFDM systems. Several blind SLM (BSLM) and blind PTS (BPTS) schemes to eliminate the need for side information have been studied. Maximum Likelihood (ML) decoders, which exhibit good BER performance, are derived for the BSLM and BPTS schemes. However, the conventional BSLM and BPTS schemes result in large decoding complexity at the receiver. 
     In the OFDM systems, an input symbol sequence X=[X 0 , X 1 , . . . , X N-1 ] is given as a vector of complex-valued symbols with the time duration Ts. After splitting the serial data into parallel data streams, all substreams are summed by applying IFFT. 
     The discrete time OFDM signal after IFFT is given as Expression 1: 
     
       
         
           
             
               
                 x 
                 k 
               
               = 
               
                 
                   1 
                   
                     N 
                   
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       0 
                     
                     
                       N 
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       X 
                       n 
                     
                     ⁢ 
                     
                       ⅇ 
                       
                         j2π 
                         ⁢ 
                         
                           n 
                           N 
                         
                         ⁢ 
                         k 
                       
                     
                   
                 
               
             
             , 
             
                 
             
             ⁢ 
             
               0 
               ≤ 
               k 
               ≤ 
               
                 N 
                 - 
                 1 
               
             
           
         
       
     
     where X n  is the input data symbol loaded on the n-th subcarrier and N is the number of the subcarriers. X is a vector of N constellation symbols from a given constellation Q. The size of Q is q. 
     Let x=[x 0 , x 1 , . . . , x N-1 ] be an OFDM signal sequence. The PAPR of x is defined by Expression 2: 
     
       
         
           
             
               PAPR 
               ⁡ 
               
                 ( 
                 x 
                 ) 
               
             
             ⁢ 
             
               = 
               Δ 
             
             ⁢ 
             
               
                 
                   max 
                   
                     0 
                     ≤ 
                     k 
                     ≤ 
                     
                       N 
                       - 
                       1 
                     
                   
                 
                 ⁢ 
                 
                   
                      
                     
                       x 
                       k 
                     
                      
                   
                   2 
                 
               
               
                 E 
                 ⁢ 
                 
                   { 
                   
                     
                        
                       
                         x 
                         k 
                       
                        
                     
                     2 
                   
                   } 
                 
               
             
           
         
       
     
     where E{·} denotes the expectation. When all of 1 sequences are an input to N-point IFFT, the output signal is an impulse sequence with amplitude             at k=0. Thus, the theoretical maximum of the PAPR for N number of subcarriers is 10 log(N) dB.
     Below, the conventional BSLM will be described. 
     For an input symbol sequence X, SLM generates U independent alternative symbol sequences and transmits the OFDM signal sequence with the minimum PAPR. U alternative symbol sequences are generated by multiplying X by U phase sequences. Suppose that U phase sequences are given as Expression 3:
 
 P   u   =[e   jφ     0       u     , e   jφ     1       u     , . . . , e   jφ     N-1       u   ]
 
where φ n   u ε[0, 2π) and uε{1, 2, . . . U}. Let a{circle around (×)}b represent the componentwise multiplication of vectors a and b. For an input symbol sequence X, x ũ =IFFT(X{circle around (×)}P ũ ) with the minimum PAPR among U alternative signal sequences x u =IFFT(X{circle around (×)}P u ), 1≦u≦U is selected for transmission. The index ũ of the selected phase sequence should be transmitted to the receiver in the SLM scheme.
 
     In order to eliminate transmission of the side information, a BSLM scheme wherein the indices are embedded into alternative signal sequences without loss of data rate was proposed in the related art. BSLM enables the receiver to distinguish the selected phase sequence from others without transmitting any side information. 
     It is assumed that the receiver receives r=x ũ +n, where r=┌r 1 , r 2 , . . . , r N-1 ┐=FFT(R) and R is a received symbol sequence. To derive the input symbol sequence without the side information ũ from r at the receiver, U phase sequences in the BSLM scheme should have the following properties: 
     1. The set of P u s is fixed and known a priori; and 
     2. X{circle around (×)}P u  and X{circle around (×)}P u  are sufficiently different for u≠v. 
     To decode the input symbol sequence at the receiver, each phase φ n   u  for all n and u must satisfy the condition of X n e jφ     n       u   ∉Q. The set of P u  should be duly chosen to ensure this condition. If u≠ũ and channel noise is not considered in decoding OFDM symbols, each element of R{circle around (×)}P u * will not be a symbol in the constellation Q where P u * is the conjugate of P u . Therefore, by using this result, a simplified ML decoder for the BSLM scheme can be derived to recover the OFDM symbol without the side information. Specially, when the Euclidian distance between any P u  and P v  is very large, BER performance of the ML decoder is expected to be very good. 
     The received symbol R n  after the FFT demodulation at the receiver may be written as Expression 4:
 
 R   n   =G   n   X   n e jφ     n       ũ     +{circumflex over (N)}   n  
 
     where G n  is the frequency response of the fading channel at the n-th subcarrier and {circumflex over (N)} n  is an additive white complex Gaussian noise (AWGN) sample at the nth subcarrier. 
     Without the side information of ũ, the optimal ML decoder computes the decision metric for decoding the received symbol sequence R. The optimal metric of the ML decoder is given as Expression 5: 
     
       
         
           
             
               D 
               opt 
             
             = 
             
               
                 min 
                 
                   
                     
                       [ 
                       
                         
                           
                             X 
                             ^ 
                           
                           0 
                         
                         , 
                         
                           
                             X 
                             ^ 
                           
                           1 
                         
                         , 
                         … 
                         ⁢ 
                         
                             
                         
                         , 
                         
                           
                             X 
                             ^ 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                       
                       ] 
                     
                     ⊆ 
                     
                       Q 
                       N 
                     
                   
                   
                     
                       P 
                       u 
                     
                     , 
                     
                       u 
                       ∈ 
                       
                         { 
                         
                           1 
                           , 
                           2 
                           , 
                           … 
                           ⁢ 
                           
                               
                           
                           , 
                           U 
                         
                         } 
                       
                     
                   
                 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     n 
                     = 
                     0 
                   
                   
                     N 
                     - 
                     1 
                   
                 
                 ⁢ 
                 
                   
                      
                     
                       
                         
                           R 
                           n 
                         
                         ⁢ 
                         
                           ⅇ 
                           
                             - 
                             
                               jϕ 
                               n 
                               u 
                             
                           
                         
                       
                       - 
                       
                         
                           G 
                           n 
                         
                         ⁢ 
                         
                           
                             X 
                             ^ 
                           
                           n 
                         
                       
                     
                      
                   
                   2 
                 
               
             
           
         
       
     
     where |·| denote the absolute value of a complex number. This algorithm has to search all q N  symbols for q-ary constellations that are repeated for each of P 1 , P 2 , . . . , P U . 
     Consequently, the overall decoding complexity to compute Expression 5 is Uq N |·| 2  operations. 
     Since decoding complexity from real additions is negligible as compared to that of |·| 2  operations, the additions will be ignored. Since the decoder exhaustively searches all input symbol sequences, it can be performed only for small N. Thus, a suboptimal decoding method with reduced complexity should be derived. 
     Assume that              n  is detected into the nearest constellation point {circumflex over (X)} n . That is, a soft decision value is made for each subcarrier unlike the optimal ML decoder and the nearest constellation point from            n  is saved for each subcarrier. This process is repeated for 1≦u≦U. Therefore, the suboptimal decision metric of the ML decoder may be written as Expression 6:
     
       
         
           
             
               D 
               so 
             
             = 
             
               
                 min 
                 
                   
                     P 
                     u 
                   
                   , 
                   
                     u 
                     ∈ 
                     
                       { 
                       
                         1 
                         , 
                         2 
                         , 
                         … 
                         ⁢ 
                         
                             
                         
                         , 
                         U 
                       
                       } 
                     
                   
                 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     n 
                     = 
                     0 
                   
                   
                     N 
                     - 
                     1 
                   
                 
                 ⁢ 
                 
                   
                     min 
                     
                       
                         
                           X 
                           ^ 
                         
                         n 
                       
                       ∈ 
                       Q 
                     
                   
                   ⁢ 
                   
                     
                        
                       
                         
                           
                             R 
                             n 
                           
                           ⁢ 
                           
                             ⅇ 
                             
                               - 
                               
                                 jϕ 
                                 n 
                                 u 
                               
                             
                           
                         
                         - 
                         
                           
                             G 
                             n 
                           
                           ⁢ 
                           
                             
                               X 
                               ^ 
                             
                             n 
                           
                         
                       
                        
                     
                     2 
                   
                 
               
             
           
         
       
     
     The input symbol sequence is recovered from the decoded symbol sequence having D so . Though the complexity of suboptimal decoder is reduced to UqN|·| operations, it is still not practical. 
     SUMMARY OF THE INVENTION 
     The present invention is directed to solving the problems of the conventional blind selective mapping (BSLM) as described above, and an aspect of the invention is to provide a BSLM scheme and a blind partial transmit sequence (BPTS) scheme having fewer operations for lowering a peak to average power ratio (PAPR) of an orthogonal frequency division multiplexing (OFDM) signal. Specifically, the invention is directed to providing a BSLM scheme and a BPTS scheme extended from a similar idea of the BSLM scheme, with low decoding complexity for PAPR reduction of an OFDM signal, and in which m-sequences are used as phase sequences, a maximum likelihood (ML) decoder with low decoding complexity is used so that a receiver can decide the phase sequence used in the transmitter, and block partition and phase rotation are used so that the transmitter can transmit the side information embedded in the OFDM symbol sequence to the receiver, thereby decreasing decoding complexity while minimally degrading bit error rate (BER) performance. 
     According to an aspect, the present invention provides a blind selective mapping (BSLM) method of generating U independent alternative symbol sequences by changing a phase of symbol sequences by phase sequences and transmitting the alternative symbol sequences with a minimum peak to average power ratio (PAPR) in an orthogonal frequency division multiplexing (OFDM) system, wherein the transmitter uses m-sequences as the phase sequence and a receiver uses a maximum likelihood (ML) decoder to distinguish the phase sequence used by the transmitter. The BSLM method includes: partitioning U alternative symbol sequences into U/2 blocks in the transmitter; rotating phases in each block for each alternative symbol sequence that is an input symbol sequence multiplied by the m-sequence; and selecting a signal sequence with a minimum PAPR among U alternative signal sequences generated by the m-sequences in the transmitter. 
     The m-sequences may be expressed by P u =[P 0   u , P 1   u , . . . , P N-1   u ], where P i   u ε{+1, −1}, 0≦i≦N−1, uε{1, 2, . . . , U}, and N is the number of subcarriers. 
     The partitioning may include dividing a u-th alternative symbol sequence X{circle around (×)}P u  into L sub-blocks as [X 1   u , X 2   u , . . . , X L   u ] if X=[X 0 , X 1 , . . . , X N-1 ], where 
               X   v   u     =     [         X       N   ⁡     (     v   -   1     )       L       ⁢     P       N   ⁡     (     v   -   1     )       L         ,       X         N   ⁡     (     v   -   1     )       L     +   1       ⁢     P         N   ⁡     (     v   -   1     )       L     +   1         ,   …   ⁢           ,       X       Nv   L     -   1       ⁢     P       Nv   L     -   1           ]           
is the v-th sub-block with size N/L.
 
     If U L-tuple phase rotation vectors are defined as W u =[w 1   u , w 2   u , . . . , w L   u ], where w v   u ε{0, 1}, 1≦u≦U and 1≦v≦L, each element in the v-th sub-block of the u-th alternative symbol sequence may be multiplied by e θw     v       u   , where 0&lt;θ≦π/2, 1≦u≦L, and 1≦v≦L, such that each element in the sub-blocks with w v   u =0 is modulated by using a signal constellation S 0  and each element in the sub-blocks with w v   u =1 is modulated by using a signal constellation S 1 . 
     The phase rotation vectors may include U bi-orthogonal vectors of length L=U/2. 
     After partitioning, the ML decoder in each sub-block may calculate a value by two different metrics, the two different metrics being given as 
     
       
         
           
             
               D 
               
                 v 
                 , 
                 
                   w 
                   v 
                   u 
                 
               
             
             = 
             
               
                 ∑ 
                 
                   n 
                   = 
                   
                     
                       2 
                       ⁢ 
                       
                         N 
                         ⁡ 
                         
                           ( 
                           
                             v 
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                     U 
                   
                 
                 
                   
                     
                       2 
                       ⁢ 
                       Nv 
                     
                     U 
                   
                   - 
                   1 
                 
               
               ⁢ 
               
                 
                   min 
                   
                     
                       
                         X 
                         ^ 
                       
                       n 
                     
                     ∈ 
                     
                       S 
                       0 
                     
                   
                 
                 ⁢ 
                 
                   
                      
                     
                       
                         
                           R 
                           n 
                         
                         ⁢ 
                         
                           ⅇ 
                           
                             
                               - 
                               jθ 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               w 
                               v 
                               u 
                             
                           
                         
                       
                       - 
                       
                         
                           G 
                           n 
                         
                         ⁢ 
                         
                           
                             X 
                             ^ 
                           
                           n 
                         
                       
                     
                      
                   
                   2 
                 
               
             
           
         
       
     
     where w v   u ε{0, 1}, vε{1, 2, . . . , U/2},              n  is a received symbol after fast Fourier transform (FFT) demodulation at the receiver, G n  is a frequency response of the fading channel at an n-th subcarrier, and {circumflex over (X)} n  is a constellation point of S 0 .
     After the calculation, the ML decoder may calculate and derotate the alternative symbol sequence by a decision metric and finally decides an input symbol sequence, the decision metric being given as 
     
       
         
           
             
               D 
               
                 u 
                 _ 
               
             
             = 
             
               
                 min 
                 
                   W 
                   u 
                 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     v 
                     = 
                     1 
                   
                   
                     U 
                     2 
                   
                 
                 ⁢ 
                 
                   
                     D 
                     
                       v 
                       , 
                       
                         w 
                         v 
                         u 
                       
                     
                   
                   . 
                 
               
             
           
         
       
     
     With this configuration, the invention provides a novel BSLM with low decoding complexity. As compared with a conventional blind method, detection failure of side information can be decreased. One input symbol sequence is multiplied by phase sequences so as to generate alternative symbol sequences, and side information of each alternative symbol sequence is then embedded in the alternative symbol sequence through the block partition and the predetermined phase rotation. 
     Further, the BSLM scheme according to this invention may be extended to a blind partial transmit sequence (PTS). This will be described in more detail hereinafter. 
     According to another aspect, the invention provides a blind partial transmit sequence (PTS) method of partitioning an input symbol sequence X into V disjointed symbol subsequences X v =[X v,0 , X v,1 , . . . , X v,N-1 ], and multiplying each subsequence by a rotating factor b v  with |b v |=1 for PAPR reduction of an orthogonal frequency division multiplexing (OFDM) signal. The blind PTS method includes: choosing the rotating factor b v  from a given set            ={+1, e j3π/4 }.
     If b v  is chosen from a given set            ={+1, e j3π/4 }, a decision metric of a maximum likelihood (ML) decoder in a BPTS scheme may be written as
     
       
         
           
             
               D 
               pts 
             
             = 
             
               
                 min 
                 
                   [ 
                   
                     
                       b 
                       1 
                     
                     , 
                     … 
                     ⁢ 
                     
                         
                     
                     , 
                     
                       b 
                       V 
                     
                   
                   ] 
                 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     v 
                     = 
                     1 
                   
                   V 
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       
                         
                           N 
                           ⁡ 
                           
                             ( 
                             
                               v 
                               - 
                               1 
                             
                             ) 
                           
                         
                         V 
                       
                     
                     
                       
                         Nv 
                         V 
                       
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       min 
                       
                         
                           
                             X 
                             ^ 
                           
                           n 
                         
                         ∈ 
                         
                           S 
                           0 
                         
                       
                     
                     ⁢ 
                     
                       
                          
                         
                           
                             
                               R 
                               n 
                             
                             ⁢ 
                             
                               b 
                               v 
                               
                                 - 
                                 1 
                               
                             
                           
                           - 
                           
                             
                               G 
                               n 
                             
                             ⁢ 
                             
                               
                                 X 
                                 ^ 
                               
                               n 
                             
                           
                         
                          
                       
                       2 
                     
                   
                 
               
             
           
         
       
     
     where S 0  and S 1  are constellation points for b v =+1 and e j3π/4 , respectively,              n  is a received symbol after fast Fourier transform (FFT) demodulation at the receiver, G n  is a frequency response of the fading channel at an n-th subcarrier, and {circumflex over (X)} n  is a constellation point of S 0 .
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The above and other aspects, features and advantages of the invention will become apparent from the following detailed description in conjunction with the accompanying drawings, in which: 
         FIG. 1  shows examples of block partitioning and predetermined phase rotation to embed side information for U=4 and 8; 
         FIG. 2  is a block diagram of an ML receiver for BSLM according to one embodiment of this invention; 
         FIG. 3  shows comparison of theoretical and simulated DFPs in the BSLM according to the embodiment of the invention with QPSK and U=4; 
         FIGS. 4 to 7  show DFP and BER of the conventional and the BSLM scheme according to the embodiment of the inventions with N=64, 256 modulated by QPSK and 16 QAM; 
         FIG. 8  compares BER performance of the decoder proposed in Expression 18 with the conventional PTS that has perfect side information in the AWGN channel; and 
         FIG. 9  shows that, for N=64, the BSLM scheme with predetermined phase rotation and block partitioning according to the embodiment of the invention has the same PAPR reduction performance as that of the conventional SLM scheme. 
     
    
    
     DETAILED DESCRIPTION OF THE EMBODIMENTS 
     Embodiments of the invention will hereinafter be described in more detail with reference to the accompanying drawings. Herein, reference numerals given to elements are consistent throughout the drawings, and thus elements having like reference numerals throughout the drawings must be interpreted as having like configurations and functions. 
     Embodiment 1 
     A novel BSLM scheme using bi-orthogonal vectors according to an embodiment of the invention will be described. 
     Instead of randomly selecting each element φ n   u  of a conventional phase sequence, the BSLM scheme according to this embodiment uses an m-sequence and its cyclic shifts with zero padding as phase sequences, P u =[P 0   u , P 1   u , . . . , P N-1   u ], where P i   u ε{+1, −1}, 0≦i≦N−1, and uε{1, 2, . . . , U}. In fact, P u  corresponds to rows of a cyclic Hadamard matrix constructed from m-sequences. It is known that m-sequences satisfy the optimal condition for the phase sequence in the SLM schemes. 
     In order to embed side information of the phase sequences into alternative symbol sequences without loss of data rate, block partitioning and predetermined phase rotation are used in the BSLM scheme according to this embodiment as follows. Assume that U phase sequences are used. The u-th alternative symbol sequence X{circle around (×)}P u  is divided into L sub-blocks as [X 1   u , X 2   u , . . . , X L   u ], 
     where 
               X   v   u     =     [         X       N   ⁡     (     v   -   1     )       L       ⁢     P       N   ⁡     (     v   -   1     )       L         ,       X         N   ⁡     (     v   -   1     )       L     +   1       ⁢     P         N   ⁡     (     v   -   1     )       L     +   1         ,   …   ⁢           ,       X       Nv   L     -   1       ⁢     P       Nv   L     -   1           ]           
is the v-th sub-block with size N/L. U L-tuple phase rotation vectors are defined as Expression 7:
 
 W   u   =[w   1   u   , w   2   u   , . . . , w   L   u ]
 
where w v   u ε{0, 1}, 1≦u≦U and 1≦v≦L. Each element in the v-th sub-block of the u-th alternative symbol sequence is multiplied by e θw     v       u   , where 0&lt;θ≦π/2 and 1≦v≦L. In other words, the phase of each element in the v-th sub-block of the u-th alternative symbol sequence is rotated by θw v   u . It is equivalent to use two signal constellations such that each element in the sub-blocks with w v   u =0 is modulated by using the signal constellation S 0  and each element in the sub-blocks with w v   u =1 is modulated by using the signal constellation S 1 , which is obtained by rotating the signals in S 0  by θ. Here, the u-th alternative symbol sequence may be written as Expression 8:
 
 X   u   =[X   1   u   e   jθw     1       u     , X   2   u   e   jθw     2       u     , . . . , X   L   u   e   jθw     L       u   ]
 
     Now, the optimal θ is derived to construct U L-tuple phase rotation vectors W u , which maximize the detection probability of the index of the selected phase sequence at the receiver. It is not difficult to derive the following design criteria for the phase rotation vectors: 
     1. The sub-block size should be as large as possible; 
     2. The Euclidian distance between alternative symbol sequences obtained by applying phase rotation vectors should be as large as possible. 
     It is not difficult to check that the U bi-orthogonal vectors of length L=U/2 satisfy the above criteria. It should be noted that U bi-orthogonal vectors of length U/2 are U/2 orthogonal vectors of length U/2 and their bit-complement vectors. Actually, L=log 2 U gives the largest size of sub-block which satisfies the first criterion. However, any U rotation vectors of length L=log 2 U do not satisfy the larger Euclidian distance between alternative symbol sequences obtained by applying rotation vectors compared to U bi-orthogonal vectors of length L=U/2. Since the Euclidian distance between alternative symbol sequences is more important than the sub-block size, U bi-orthogonal vectors of length U/2 may be used as rotation vectors in this embodiment. On the other hand, θ must be chosen so as to maximize the minimum Euclidean distance between S 0  and S 1 . For Quadrature Phase Shift Keying (QPSK) and 16 Quadrature Amplitude Modulation (QAM), θ=π/2 is chosen. 
       FIG. 1  shows examples of block partitioning and predetermined phase rotation to embed side information for U=4 and 8. In  FIG. 1 , the bi-orthogonal vectors for U=4 and 8 are given as Expression 9: 
     
       
         
           
             
                 
             
             ⁢ 
             
               
                 
                   For 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   U 
                 
                 = 
                 4 
               
               , 
               
                 
                   
                     
                       { 
                       
                         
                           ( 
                           00 
                           ) 
                         
                         , 
                         
                           ( 
                           01 
                           ) 
                         
                         , 
                         
                           ( 
                           10 
                           ) 
                         
                         , 
                         
                           ( 
                           11 
                           ) 
                         
                       
                       } 
                     
                     . 
                     
                        
                     
                     ⁢ 
                     For 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   U 
                 
                 = 
                 8 
               
               , 
               
                 
                   { 
                   
                     
                       ( 
                       0000 
                       ) 
                     
                     , 
                     
                       ( 
                       0011 
                       ) 
                     
                     , 
                     
                       ( 
                       0110 
                       ) 
                     
                     , 
                     
                       ( 
                       0101 
                       ) 
                     
                     , 
                     
                       ( 
                       1100 
                       ) 
                     
                     , 
                     
                       ( 
                       1010 
                       ) 
                     
                     , 
                     
                       ( 
                       1001 
                       ) 
                     
                     , 
                     
                       ( 
                       1111 
                       ) 
                     
                   
                   } 
                 
                 . 
               
             
           
         
       
     
       FIG. 1  shows the examples of the block partitioning by U bi-orthogonal vectors of length L=U/2 with U=4 and 8, where I and II correspond to w v   u =0 and w v   u =1, respectively. 
     In this embodiment, an ML decoder with low complexity is derived instead of the conventional BSLM decoder. Specifically, two metrics for each sub-block corresponding to S 0  and S 1  may be derived. R n  is allowed to be detected at the nearest constellation point on S 0 , similar to the suboptimal ML decoder of the conventional BSLM scheme. In other words, a soft decision value for each subcarrier is calculated, the nearest constellation point from R n  is saved for each subcarrier, and all of 2N/U metrics in the sub-block are added. This process is repeated for 1≦v≦U/2. On the other hand, the points on the signal constellation S 1  must be considered by the ML decoder. Using R n  derotated by θ, this process is also repeated for 1≦v≦U/2 for decoding the symbols on S 1 . 
     Table 1 shows comparison of decoding complexity of the conventional BSLM and the BSLM according to this embodiment for U=4, 8, and 16. 
     
       
         
               
               
               
               
             
               
               
               
               
               
             
           
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 U = 4 
                 U = 8 
                 U = 16 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Conventional Blind SLM 
                 4qN 
                 8qN 
                 16qN 
               
               
                   
                 New Blind SLM 
                 2qN 
                 2qN 
                  2qN 
               
               
                   
                 DCRR 
                 50% 
                 75% 
                 87.5% 
               
               
                   
                   
               
             
          
         
       
     
     Therefore, the metric of the ML decoder in each sub-block may be written as Expression 10: 
     
       
         
           
             
               D 
               
                 v 
                 , 
                 
                   w 
                   v 
                   u 
                 
               
             
             = 
             
               
                 ∑ 
                 
                   n 
                   = 
                   
                     
                       2 
                       ⁢ 
                       
                         N 
                         ⁡ 
                         
                           ( 
                           
                             v 
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                     U 
                   
                 
                 
                   
                     
                       2 
                       ⁢ 
                       Nv 
                     
                     U 
                   
                   - 
                   1 
                 
               
               ⁢ 
               
                 
                   min 
                   
                     
                       
                         X 
                         ^ 
                       
                       n 
                     
                     ∈ 
                     
                       S 
                       0 
                     
                   
                 
                 ⁢ 
                 
                   
                      
                     
                       
                         
                           R 
                           n 
                         
                         ⁢ 
                         
                           ⅇ 
                           
                             
                               - 
                               jθ 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               w 
                               v 
                               u 
                             
                           
                         
                       
                       - 
                       
                         
                           G 
                           n 
                         
                         ⁢ 
                         
                           
                             X 
                             ^ 
                           
                           n 
                         
                       
                     
                      
                   
                   2 
                 
               
             
           
         
       
     
     where w v   u ε{0, 1} and 
             v   ∈       {     1   ,   2   ,   …   ⁢           ,     U   2       }     .           
Two different metrics for each sub-block in the BSLM scheme according to this embodiment are calculated according to w v   u , that is, D v,0  and D v,1 .
 
     The ML decoder chooses the minimum total metric summed by U/2 metrics of all sub-blocks from U bi-orthogonal vectors. The minimum Euclidian distance solution yields the transmitted alternative symbol sequence. Therefore, the decision metric of the ML decoder is given as Expression 11: 
     
       
         
           
             
               D 
               
                 u 
                 _ 
               
             
             = 
             
               
                 min 
                 
                   W 
                   u 
                 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     v 
                     = 
                     1 
                   
                   
                     U 
                     2 
                   
                 
                 ⁢ 
                 
                   D 
                   
                     v 
                     , 
                     
                       w 
                       v 
                       u 
                     
                   
                 
               
             
           
         
       
     
     After finding the index ū and yielding the alternative symbol sequence, {circumflex over (X)}=[{circumflex over (X)} 0 , {circumflex over (X)} 1 , . . . , {circumflex over (X)} N-1 ], the input symbol sequence is determined by {circumflex over (X)}{circle around (×)}P ū . 
       FIG. 2  shows a block diagram of an ML decoder for the BSLM according to this embodiment of the invention. 
     Regardless of U, total complexity of the BSLM according to this embodiment is only 2qN|·| 2  operations in the OFDM system of q-ary modulation. A decoding complexity reduction ratio (DCRR) of the BSLM according to this embodiment over the conventional BSLM is defined as Expression 12: 
     
       
         
           
             
               
                 
                   DCRR 
                   = 
                   
                     
                       ( 
                       
                         1 
                         - 
                         
                           A 
                           B 
                         
                       
                       ) 
                     
                     × 
                     100 
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       ( 
                       
                         
                           UqN 
                           - 
                           
                             2 
                             ⁢ 
                             qN 
                           
                         
                         UqN 
                       
                       ) 
                     
                     × 
                     100 
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       ( 
                       
                         
                           U 
                           - 
                           2 
                         
                         U 
                       
                       ) 
                     
                     × 
                     100 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       % 
                       ) 
                     
                   
                 
               
             
           
         
       
     
     where A is the decoding complexity of the conventional BSLM and B is the decoding complexity of the BSLM according to this embodiment. Table 1 compares the decoding complexity of the conventional BSLM and the BSLM according to this embodiment for U=4, 8, and 16. 
     Embodiment 2 
     The novel BSLM scheme according to this embodiment may be extended to a partial transmit sequence (PTS). This will be described below in more detail. 
     An input symbol sequence X in the PTS scheme is partitioned into V disjointed symbol subsequences, X v =[X v,0 , X v,1 , . . . , X v,N-1 ], v=1, 2, . . . , V satisfying Expression 13: 
     
       
         
           
             X 
             = 
             
               
                 ∑ 
                 
                   v 
                   = 
                   1 
                 
                 V 
               
               ⁢ 
               
                 X 
                 v 
               
             
           
         
       
     
     Each symbol in the input symbol sequence X belongs to only one symbol subsequence X v  called a sub-block. Due to disjoint X v′s , for each given k, 0≦k≦N−1, X v,k =0 except for subcarriers carrying the data. Then, each subsequence is multiplied by a rotating factor b v  with |b v |=1 as 
     
       
         
           
             
               
                 
                   
                     X 
                     ′ 
                   
                   = 
                   
                     
                       ∑ 
                       
                         v 
                         = 
                         1 
                       
                       V 
                     
                     ⁢ 
                     
                       
                         b 
                         v 
                       
                       ⁢ 
                       
                         X 
                         v 
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     14 
                   
                   ] 
                 
               
             
           
         
       
     
     The rotating factor b v  is chosen from a given set             which is usually          ={+1, −1}. Using the linearity of the IFFT, the OFDM signal sequence in the time domain is given as Expression 15:
     
       
         
           
             
               
                 
                   
                     x 
                     ′ 
                   
                   = 
                   
                     IFFT 
                     ⁢ 
                     
                       { 
                       
                         
                           ∑ 
                           
                             v 
                             = 
                             1 
                           
                           V 
                         
                         ⁢ 
                         
                           
                             b 
                             v 
                           
                           ⁢ 
                           
                             X 
                             v 
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       ∑ 
                       
                         v 
                         = 
                         1 
                       
                       V 
                     
                     ⁢ 
                     
                       
                         b 
                         v 
                       
                       ⁢ 
                       IFFT 
                       ⁢ 
                       
                         { 
                         
                           X 
                           v 
                         
                         } 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       ∑ 
                       
                         v 
                         = 
                         1 
                       
                       V 
                     
                     ⁢ 
                     
                       
                         b 
                         v 
                       
                       ⁢ 
                       
                         
                           X 
                           v 
                         
                         . 
                       
                     
                   
                 
               
             
           
         
       
     
     An optimized vector of rotating factors {tilde over (b)}=[{tilde over (b)} 1 , . . . , {tilde over (b)} v ] for the minimum PAPR should be chosen. This is expressed as Expression 16: 
     
       
         
           
             
               b 
               ~ 
             
             = 
             
               
                 [ 
                 
                   
                     
                       b 
                       ~ 
                     
                     1 
                   
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   
                     
                       b 
                       ~ 
                     
                     V 
                   
                 
                 ] 
               
               = 
               
                 
                   argmin 
                   
                     [ 
                     
                       
                         b 
                         1 
                       
                       , 
                       … 
                       ⁢ 
                       
                           
                       
                       , 
                       
                         b 
                         V 
                       
                     
                     ] 
                   
                 
                 ( 
                 
                   
                     max 
                     
                       0 
                       ≤ 
                       k 
                       ≤ 
                       
                         N 
                         - 
                         1 
                       
                     
                   
                   ⁢ 
                   
                      
                     
                       
                         ∑ 
                         
                           v 
                           = 
                           1 
                         
                         V 
                       
                       ⁢ 
                       
                         
                           b 
                           v 
                         
                         ⁢ 
                         
                           x 
                           
                             v 
                             , 
                             k 
                           
                         
                       
                     
                      
                   
                 
                 ) 
               
             
           
         
       
     
     Thus, the optimum transmit sequence is given as Expression 17: 
     
       
         
           
             
               x 
               ′ 
             
             = 
             
               
                 ∑ 
                 
                   v 
                   = 
                   1 
                 
                 V 
               
               ⁢ 
               
                 
                   
                     b 
                     ~ 
                   
                   v 
                 
                 ⁢ 
                 
                   x 
                   v 
                 
               
             
           
         
       
     
     Expression 17 requires              V-1  iterations to obtain a vector {tilde over (b)}. The receiver of the PTS scheme needs side information of the vector {tilde over (b)} for decoding the OFDM symbol sequence.
     Embedding the side information of the BSLM according to this invention through the use of two signal constellations may be applied to the BPTS. Similar to the BSLM scheme according to this invention,            ={+1, e j3π/4 } is used rather than          ={+1, −1} for the BPTS scheme. If          ={+1, e j3π/4 } is used as in the BSLM according to this invention, PAPR performance of the BPTS is degraded compared with that of the conventional PTS as the distance between alternative OFDM signals is reduced. Let S 0  and S 1  be constellation points for b v =+1 and e j3π/4 , respectively. Due to          ={+1, e j3π/4 }, the side information of the vector {tilde over (b)} is not transmitted to the receiver in the BPTS scheme.
     An ML decoder for the BPTS scheme can also be easily derived. Similar to the ML decoder of the BSLM scheme according to this invention, it is necessary that the received R n  find the nearest constellation point on S 0  and S 1 . This process is repeated for 1≦v≦V. Therefore, the decision metric of the ML decoder in the BPTS may be written as Expression 18: 
     
       
         
           
             
               D 
               pts 
             
             = 
             
               
                 min 
                 
                   [ 
                   
                     
                       b 
                       1 
                     
                     , 
                     … 
                     ⁢ 
                     
                         
                     
                     , 
                     
                       b 
                       V 
                     
                   
                   ] 
                 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     v 
                     = 
                     1 
                   
                   V 
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       
                         
                           N 
                           ⁡ 
                           
                             ( 
                             
                               v 
                               - 
                               1 
                             
                             ) 
                           
                         
                         V 
                       
                     
                     
                       
                         Nv 
                         V 
                       
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       min 
                       
                         
                           
                             X 
                             ^ 
                           
                           n 
                         
                         ∈ 
                         
                           S 
                           0 
                         
                       
                     
                     ⁢ 
                     
                       
                          
                         
                           
                             
                               R 
                               n 
                             
                             ⁢ 
                             
                               b 
                               v 
                               
                                 - 
                                 1 
                               
                             
                           
                           - 
                           
                             
                               G 
                               n 
                             
                             ⁢ 
                             
                               
                                 X 
                                 ^ 
                               
                               n 
                             
                           
                         
                          
                       
                       2 
                     
                   
                 
               
             
           
         
       
     
     where b v ε{1, e j3π/4 }. The vector of rotating factors can be determined using Expression 18, and an input symbol sequence is then recovered by {circumflex over (X)}=[{circumflex over (X)} 0 , {circumflex over (X)} 1 , . . . , {circumflex over (X)} N-1 ] obtained from results of the decision metric. Similar to the BSLM according to this invention, total complexity of the BPTS according to this embodiment is also only 2qN|·| 2  operations in the OFDM system of q-ary modulation. 
     Analysis of detection failure probability (DFP) in this embodiment will hereinafter be described. 
     Detection failure of side information can seriously degrade BER of the OFDM system. Therefore, analysis of the detection failure probability (DFP) in the BSLM according to this invention is very important. It can be seen that that, for U=4 and 8, the decision metric for the decoder in Expression 11 is well operated over AWGN without detection failure of the side information in the high SNR region. It can also be seen from the simulation results in the next section that the BSLM scheme according to this invention has good performance of the side information detection over the AWGN channels. 
     The total BER P e  at the receiver is given as Expression 19:
 
 P   e   =P   b (1 −P   df )+ P   b|df   P   df  
 
     where P b  is the BER when correct detection of side information is performed, P b|df  is the BER when detection of side information fails, and P df  is the DFP. When detection of the side information fails in Quadrature Phase Shift Keying (QPSK), P b|df =0.5. Assume that a noise sample {circumflex over (N)} n  has a mean of 0 and variance η 2 . Since real value and imaginary value of {circumflex over (N)} n  on a subcarrier are statistically independent, {circumflex over (N)} n  can be expressed as Expression 20:
 
 N   re,n   Re{{circumflex over (N)}   n   }=Re{R   n   e   −jθw     v       ũ     }−Re{G   n   X   n },
 
 N   im,n   =Im{{circumflex over (N)}   n   }=Im{R   n   e   −jθw     v       ũ     }−Im{G   n   X   n }
 
     where G n =1 from the AWGN channel. 
     Assume that all elements of X have (1, 0) for QPSK and the alternative signal sequence which has no rotation with the minimum PAPR is selected for transmission. 
     That is, all subcarriers of the OFDM signal sequence use the S 0  constellation. Then R n  can be given as
 
 R   n =1 +N   re,n   +jN   im,n   [Expression 21]
 
     All D u  values calculated from Expressions 10 and 11 are given as Expression 22:
 
 D   1   =D   1,0   +D   2,0  
 
 D   2   =D   1,0   +D   2,1  
 
 D   3   =D   1,1   +D   2,0  
 
 D   4   =D   1,1   +D   2,1  
 
     Expression 23:
 
for  U= 4 and for  U= 8
 
 D   1   =D   1,0   +D   2,0   +D   3,0   +D   4,0  
 
 D   2   =D   1,0   +D   2,0   +D   3,1   +D   4,1  
 
 D   3   =D   1,0   +D   2,1   +D   3,1   +D   4,0  
 
 D   4   =D   1,0   +D   2,1   +D   3,0   +D   4,1  
 
 D   5   =D   1,1   +D   2,1   +D   3,0   +D   4,0  
 
 D   6   =D   1,1   +D   2,0   +D   3,1   +D   4,0  
 
 D   7   =D   1,1   +D   2,0   +D   3,0   +D   4,1  
 
 D   8   =D   1,1   +D   2,1   +D   3,1   +D   4,1 .
 
     The detection success probability can be expressed as 
     
       
         
           
             
               
                 
                   
                     Pr 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             D 
                             1 
                           
                           &lt; 
                           
                             D 
                             2 
                           
                         
                         , 
                         
                           
                             D 
                             1 
                           
                           &lt; 
                           
                             D 
                             3 
                           
                         
                         , 
                         
                           
                             D 
                             1 
                           
                           &lt; 
                           
                             D 
                             4 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         Pr 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 
                                   D 
                                   1 
                                 
                                 - 
                                 
                                   D 
                                   2 
                                 
                               
                               &lt; 
                               0 
                             
                             , 
                             
                               
                                 
                                   D 
                                   1 
                                 
                                 - 
                                 
                                   D 
                                   3 
                                 
                               
                               &lt; 
                               0 
                             
                             , 
                             
                               
                                 
                                   D 
                                   1 
                                 
                                 - 
                                 
                                   D 
                                   4 
                                 
                               
                               &lt; 
                               0 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       for 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       U 
                     
                     = 
                     4 
                   
                 
               
               
                 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   24 
                 
               
             
             
               
                 
                   
                     Pr 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             D 
                             1 
                           
                           &lt; 
                           
                             D 
                             2 
                           
                         
                         , 
                         
                           
                             D 
                             1 
                           
                           &lt; 
                           
                             D 
                             3 
                           
                         
                         , 
                         
                           
                             D 
                             1 
                           
                           &lt; 
                           
                             D 
                             4 
                           
                         
                         , 
                         
                           
                             D 
                             1 
                           
                           &lt; 
                           
                             D 
                             5 
                           
                         
                         , 
                         
                           
                             D 
                             1 
                           
                           &lt; 
                           
                             D 
                             6 
                           
                         
                         , 
                         
                           
                             D 
                             1 
                           
                           &lt; 
                           
                             D 
                             7 
                           
                         
                         , 
                         
                           
                             D 
                             1 
                           
                           &lt; 
                           
                             D 
                             8 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         Pr 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 
                                   D 
                                   1 
                                 
                                 - 
                                 
                                   D 
                                   2 
                                 
                               
                               &lt; 
                               0 
                             
                             , 
                             
                               
                                 
                                   D 
                                   1 
                                 
                                 - 
                                 
                                   D 
                                   3 
                                 
                               
                               &lt; 
                               0 
                             
                             , 
                             
                               
                                 
                                   D 
                                   1 
                                 
                                 - 
                                 
                                   D 
                                   4 
                                 
                               
                               &lt; 
                               0 
                             
                             , 
                             
                               
                                 
                                   D 
                                   1 
                                 
                                 - 
                                 
                                   D 
                                   5 
                                 
                               
                               &lt; 
                               0 
                             
                             , 
                             
                               
                                 
                                   D 
                                   1 
                                 
                                 - 
                                 
                                   D 
                                   6 
                                 
                               
                               &lt; 
                               0 
                             
                             , 
                             
                               
                                 
                                   D 
                                   1 
                                 
                                 - 
                                 
                                   D 
                                   7 
                                 
                               
                               &lt; 
                               0 
                             
                             , 
                             
                               
                                 
                                   D 
                                   1 
                                 
                                 - 
                                 
                                   D 
                                   8 
                                 
                               
                               &lt; 
                               0 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       for 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       U 
                     
                     = 
                     8. 
                   
                 
               
               
                 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   25 
                 
               
             
           
         
       
     
     Using Expressions 22 and 23, 24 and 25 can be rewritten as 
     
       
         
           
             
               
                 
                   
                     Pr 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             D 
                             1 
                           
                           &lt; 
                           
                             D 
                             2 
                           
                         
                         , 
                         
                           
                             D 
                             1 
                           
                           &lt; 
                           
                             D 
                             3 
                           
                         
                         , 
                         
                           
                             D 
                             1 
                           
                           &lt; 
                           
                             D 
                             4 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       Pr 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               
                                 D 
                                 
                                   1 
                                   , 
                                   0 
                                 
                               
                               + 
                               
                                 D 
                                 
                                   2 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   1 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   2 
                                   , 
                                   1 
                                 
                               
                             
                             &lt; 
                             0 
                           
                           , 
                           
                             
                               
                                 D 
                                 
                                   1 
                                   , 
                                   0 
                                 
                               
                               + 
                               
                                 D 
                                 
                                   2 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   1 
                                   , 
                                   1 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   2 
                                   , 
                                   0 
                                 
                               
                             
                             &lt; 
                             0 
                           
                           , 
                           
                             
                               
                                 D 
                                 
                                   1 
                                   , 
                                   0 
                                 
                               
                               + 
                               
                                 D 
                                 
                                   2 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   1 
                                   , 
                                   1 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   2 
                                   , 
                                   1 
                                 
                               
                             
                             &lt; 
                             0 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         Pr 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 
                                   D 
                                   
                                     1 
                                     , 
                                     0 
                                   
                                 
                                 - 
                                 
                                   D 
                                   
                                     1 
                                     , 
                                     1 
                                   
                                 
                               
                               &lt; 
                               0 
                             
                             , 
                             
                               
                                 
                                   D 
                                   
                                     2 
                                     , 
                                     0 
                                   
                                 
                                 - 
                                 
                                   D 
                                   
                                     2 
                                     , 
                                     1 
                                   
                                 
                               
                               &lt; 
                               0 
                             
                             , 
                             
                               
                                 
                                   D 
                                   
                                     1 
                                     , 
                                     0 
                                   
                                 
                                 + 
                                 
                                   D 
                                   
                                     2 
                                     , 
                                     0 
                                   
                                 
                                 - 
                                 
                                   D 
                                   
                                     1 
                                     , 
                                     1 
                                   
                                 
                                 - 
                                 
                                   D 
                                   
                                     2 
                                     , 
                                     1 
                                   
                                 
                               
                               &lt; 
                               0 
                             
                           
                           ) 
                         
                       
                       = 
                       
                         Pr 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 
                                   D 
                                   
                                     1 
                                     , 
                                     0 
                                   
                                 
                                 - 
                                 
                                   D 
                                   
                                     1 
                                     , 
                                     1 
                                   
                                 
                               
                               &lt; 
                               0 
                             
                             , 
                             
                               
                                 
                                   D 
                                   
                                     2 
                                     , 
                                     0 
                                   
                                 
                                 - 
                                 
                                   D 
                                   
                                     2 
                                     , 
                                     1 
                                   
                                 
                               
                               &lt; 
                               0 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   26 
                 
               
             
             
               
                 
                   
                     Pr 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             D 
                             1 
                           
                           &lt; 
                           
                             D 
                             2 
                           
                         
                         , 
                         
                           
                             D 
                             1 
                           
                           &lt; 
                           
                             D 
                             3 
                           
                         
                         , 
                         
                           
                             D 
                             1 
                           
                           &lt; 
                           
                             D 
                             4 
                           
                         
                         , 
                         
                           
                             D 
                             1 
                           
                           &lt; 
                           
                             D 
                             5 
                           
                         
                         , 
                         
                           
                             D 
                             1 
                           
                           &lt; 
                           
                             D 
                             6 
                           
                         
                         , 
                         
                           
                             D 
                             1 
                           
                           &lt; 
                           
                             D 
                             7 
                           
                         
                         , 
                         
                           
                             D 
                             1 
                           
                           &lt; 
                           
                             D 
                             8 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       Pr 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               
                                 D 
                                 
                                   1 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   1 
                                   , 
                                   1 
                                 
                               
                               + 
                               
                                 D 
                                 
                                   2 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   2 
                                   , 
                                   1 
                                 
                               
                             
                             &lt; 
                             0 
                           
                           , 
                           
                             
                               
                                 D 
                                 
                                   1 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   1 
                                   , 
                                   1 
                                 
                               
                               + 
                               
                                 D 
                                 
                                   3 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   3 
                                   , 
                                   1 
                                 
                               
                             
                             &lt; 
                             0 
                           
                           , 
                           
                             
                               
                                 D 
                                 
                                   1 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   1 
                                   , 
                                   1 
                                 
                               
                               + 
                               
                                 D 
                                 
                                   4 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   4 
                                   , 
                                   1 
                                 
                               
                             
                             &lt; 
                             0 
                           
                           , 
                           
                             
                               
                                 D 
                                 
                                   2 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   2 
                                   , 
                                   1 
                                 
                               
                               + 
                               
                                 D 
                                 
                                   3 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   3 
                                   , 
                                   1 
                                 
                               
                             
                             &lt; 
                             0 
                           
                           , 
                           
                             
                               
                                 D 
                                 
                                   2 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   2 
                                   , 
                                   1 
                                 
                               
                               + 
                               
                                 D 
                                 
                                   4 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   4 
                                   , 
                                   1 
                                 
                               
                             
                             &lt; 
                             0 
                           
                           , 
                           
                             
                               
                                 D 
                                 
                                   3 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   3 
                                   , 
                                   1 
                                 
                               
                               + 
                               
                                 D 
                                 
                                   4 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   4 
                                   , 
                                   1 
                                 
                               
                             
                             &lt; 
                             0 
                           
                           , 
                           
                             
                               
                                 D 
                                 
                                   1 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   1 
                                   , 
                                   1 
                                 
                               
                               + 
                               
                                 D 
                                 
                                   2 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   2 
                                   , 
                                   1 
                                 
                               
                               + 
                               
                                 D 
                                 
                                   3 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   3 
                                   , 
                                   1 
                                 
                               
                               + 
                               
                                 D 
                                 
                                   4 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   4 
                                   , 
                                   1 
                                 
                               
                             
                             &lt; 
                             0 
                           
                         
                         ) 
                       
                     
                     . 
                   
                 
               
               
                 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   27 
                 
               
             
           
         
       
     
     Since each sub-block is statistically independent, Expressions 26 and 27 are modified as 
     
       
         
           
             
               
                 
                   
                     Pr 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             
                               D 
                               
                                 1 
                                 , 
                                 0 
                               
                             
                             - 
                             
                               D 
                               
                                 1 
                                 , 
                                 1 
                               
                             
                           
                           &lt; 
                           0 
                         
                         , 
                         
                           
                             
                               D 
                               
                                 2 
                                 , 
                                 0 
                               
                             
                             - 
                             
                               D 
                               
                                 2 
                                 , 
                                 1 
                               
                             
                           
                           &lt; 
                           0 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         Pr 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 D 
                                 
                                   1 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   1 
                                   , 
                                   1 
                                 
                               
                             
                             &lt; 
                             0 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         Pr 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 D 
                                 
                                   2 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   2 
                                   , 
                                   1 
                                 
                               
                             
                             &lt; 
                             0 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         Pr 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 D 
                                 
                                   1 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   1 
                                   , 
                                   1 
                                 
                               
                             
                             &lt; 
                             0 
                           
                           ) 
                         
                       
                       2 
                     
                   
                 
               
               
                 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   28 
                 
               
             
             
               
                 
                   Pr 
                   ( 
                   
                     
                       
                         
                           D 
                           
                             1 
                             , 
                             0 
                           
                         
                         - 
                         
                           D 
                           
                             1 
                             , 
                             1 
                           
                         
                         + 
                         
                           D 
                           
                             2 
                             , 
                             0 
                           
                         
                         - 
                         
                           D 
                           
                             2 
                             , 
                             1 
                           
                         
                       
                       &lt; 
                       0 
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           D 
                           
                             1 
                             , 
                             0 
                           
                         
                         - 
                         
                           D 
                           
                             1 
                             , 
                             1 
                           
                         
                         + 
                         
                           D 
                           
                             3 
                             , 
                             0 
                           
                         
                         - 
                         
                           D 
                           
                             3 
                             , 
                             1 
                           
                         
                       
                       &lt; 
                       0 
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           D 
                           
                             1 
                             , 
                             0 
                           
                         
                         - 
                         
                           D 
                           
                             1 
                             , 
                             1 
                           
                         
                         + 
                         
                           D 
                           
                             4 
                             , 
                             0 
                           
                         
                         - 
                         
                           D 
                           
                             4 
                             , 
                             1 
                           
                         
                       
                       &lt; 
                       0 
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           D 
                           
                             2 
                             , 
                             0 
                           
                         
                         - 
                         
                           D 
                           
                             2 
                             , 
                             1 
                           
                         
                         + 
                         
                           D 
                           
                             3 
                             , 
                             0 
                           
                         
                         - 
                         
                           D 
                           
                             3 
                             , 
                             1 
                           
                         
                       
                       &lt; 
                       0 
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           D 
                           
                             2 
                             , 
                             0 
                           
                         
                         - 
                         
                           D 
                           
                             2 
                             , 
                             1 
                           
                         
                         + 
                         
                           D 
                           
                             4 
                             , 
                             0 
                           
                         
                         - 
                         
                           D 
                           
                             4 
                             , 
                             1 
                           
                         
                       
                       &lt; 
                       0 
                     
                     , 
                     
                       
                         D 
                         
                           3 
                           , 
                           0 
                         
                       
                       - 
                     
                   
                 
               
               
                 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   29 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         
                           D 
                           
                             3 
                             , 
                             1 
                           
                         
                         + 
                         
                           D 
                           
                             4 
                             , 
                             0 
                           
                         
                         - 
                         
                           D 
                           
                             4 
                             , 
                             1 
                           
                         
                       
                       &lt; 
                       0 
                     
                     , 
                     
                       
                         D 
                         
                           1 
                           , 
                           0 
                         
                       
                       - 
                       
                         D 
                         
                           1 
                           , 
                           1 
                         
                       
                       + 
                       
                         D 
                         
                           2 
                           , 
                           0 
                         
                       
                       - 
                       
                         D 
                         
                           2 
                           , 
                           1 
                         
                       
                       + 
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           D 
                           
                             3 
                             , 
                             0 
                           
                         
                         - 
                         
                           D 
                           
                             3 
                             , 
                             1 
                           
                         
                         + 
                         
                           D 
                           
                             4 
                             , 
                             0 
                           
                         
                         - 
                         
                           D 
                           
                             4 
                             , 
                             1 
                           
                         
                       
                       &lt; 
                       0 
                     
                     ) 
                   
                   = 
                   
                     Pr 
                     ( 
                     
                       
                         
                           D 
                           
                             1 
                             , 
                             0 
                           
                         
                         - 
                         
                           D 
                           
                             1 
                             , 
                             1 
                           
                         
                       
                       &lt; 
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       0 
                       , 
                       
                         
                           
                             D 
                             
                               2 
                               , 
                               0 
                             
                           
                           - 
                           
                             D 
                             
                               2 
                               , 
                               1 
                             
                           
                         
                         &lt; 
                         0 
                       
                       , 
                       
                         
                           
                             D 
                             
                               3 
                               , 
                               0 
                             
                           
                           - 
                           
                             D 
                             
                               3 
                               , 
                               1 
                             
                           
                         
                         &lt; 
                         0 
                       
                       , 
                       
                         
                           
                             D 
                             
                               4 
                               , 
                               0 
                             
                           
                           - 
                           
                             D 
                             
                               4 
                               , 
                               1 
                             
                           
                         
                         &lt; 
                         0 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                       
                   
                   + 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   Pr 
                   ( 
                   
                     
                       
                         
                           D 
                           
                             1 
                             , 
                             0 
                           
                         
                         - 
                         
                           D 
                           
                             1 
                             , 
                             1 
                           
                         
                       
                       &gt; 
                       0 
                     
                     , 
                     
                       
                         
                           D 
                           
                             2 
                             , 
                             0 
                           
                         
                         - 
                         
                           D 
                           
                             2 
                             , 
                             1 
                           
                         
                       
                       &lt; 
                       
                         - 
                         
                           ( 
                           
                             
                               D 
                               
                                 1 
                                 , 
                                 0 
                               
                             
                             - 
                             
                               D 
                               
                                 1 
                                 , 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                     , 
                     
                       
                         D 
                         
                           3 
                           , 
                           0 
                         
                       
                       - 
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           D 
                           
                             3 
                             , 
                             1 
                           
                         
                         &lt; 
                         
                           - 
                           
                             ( 
                             
                               
                                 D 
                                 
                                   1 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   1 
                                   , 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                       
                       , 
                       
                         
                           
                             D 
                             
                               4 
                               , 
                               0 
                             
                           
                           - 
                           
                             D 
                             
                               4 
                               , 
                               1 
                             
                           
                         
                         &lt; 
                         
                           - 
                           
                             ( 
                             
                               
                                 D 
                                 
                                   1 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   1 
                                   , 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     ) 
                   
                   + 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   Pr 
                   ( 
                   
                     
                       
                         
                           D 
                           
                             2 
                             , 
                             0 
                           
                         
                         - 
                         
                           D 
                           
                             2 
                             , 
                             1 
                           
                         
                       
                       &gt; 
                       0 
                     
                     , 
                     
                       
                         
                           D 
                           
                             1 
                             , 
                             0 
                           
                         
                         - 
                         
                           D 
                           
                             1 
                             , 
                             1 
                           
                         
                       
                       &lt; 
                       
                         - 
                         
                           ( 
                           
                             
                               D 
                               
                                 2 
                                 , 
                                 0 
                               
                             
                             - 
                             
                               D 
                               
                                 2 
                                 , 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                     , 
                     
                       
                         D 
                         
                           3 
                           , 
                           0 
                         
                       
                       - 
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                         
                     
                     ⁢ 
                     
                       
                         D 
                         
                           3 
                           , 
                           1 
                         
                       
                       &lt; 
                       
                           
                       
                       ⁢ 
                       
                         
                           
                             - 
                             
                               ( 
                               
                                 
                                   D 
                                   
                                     2 
                                     , 
                                     0 
                                   
                                 
                                 - 
                                 
                                   D 
                                   
                                     2 
                                     , 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             D 
                             
                               4 
                               , 
                               0 
                             
                           
                         
                         - 
                         
                           D 
                           
                             4 
                             , 
                             1 
                           
                         
                       
                       &lt; 
                       
                           
                       
                       ⁢ 
                       
                         - 
                         
                           ( 
                           
                             
                               D 
                               
                                 2 
                                 , 
                                 0 
                               
                             
                             - 
                             
                               D 
                               
                                 2 
                                 , 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                     ) 
                   
                   + 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     Pr 
                     ( 
                     
                       
                         
                           
                             D 
                             
                               3 
                               , 
                               0 
                             
                           
                           - 
                           
                             D 
                             
                               3 
                               , 
                               1 
                             
                           
                         
                         &gt; 
                         0 
                       
                       , 
                       
                         
                           
                             D 
                             
                               1 
                               , 
                               0 
                             
                           
                           - 
                           
                             D 
                             
                               1 
                               , 
                               1 
                             
                           
                         
                         &lt; 
                         
                           - 
                           
                             ( 
                             
                               
                                 D 
                                 
                                   3 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   3 
                                   , 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                       
                       , 
                       
                         
                           D 
                           
                             2 
                             , 
                             0 
                           
                         
                         - 
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           D 
                           
                             2 
                             , 
                             1 
                           
                         
                         &lt; 
                         
                           - 
                           
                             ( 
                             
                               
                                 D 
                                 
                                   3 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   3 
                                   , 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                       
                       , 
                       
                         
                           
                             D 
                             
                               4 
                               , 
                               0 
                             
                           
                           - 
                           
                             D 
                             
                               4 
                               , 
                               1 
                             
                           
                         
                         &lt; 
                         
                           - 
                           
                             ( 
                             
                               
                                 D 
                                 
                                   4 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   4 
                                   , 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     ) 
                   
                   + 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   Pr 
                   ( 
                   
                     
                       
                         
                           D 
                           
                             4 
                             , 
                             0 
                           
                         
                         - 
                         
                           D 
                           
                             4 
                             , 
                             1 
                           
                         
                       
                       &gt; 
                       0 
                     
                     , 
                     
                       
                         
                           D 
                           
                             1 
                             , 
                             0 
                           
                         
                         - 
                         
                           D 
                           
                             1 
                             , 
                             1 
                           
                         
                       
                       &lt; 
                       
                         - 
                         
                           ( 
                           
                             
                               D 
                               
                                 4 
                                 , 
                                 0 
                               
                             
                             - 
                             
                               D 
                               
                                 4 
                                 , 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                     , 
                     
                       
                         D 
                         
                           2 
                           , 
                           0 
                         
                       
                       - 
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           D 
                           
                             2 
                             , 
                             1 
                           
                         
                         &lt; 
                         
                           - 
                           
                             ( 
                             
                               
                                 D 
                                 
                                   4 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   4 
                                   , 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                       
                       , 
                       
                         
                           
                             D 
                             
                               3 
                               , 
                               0 
                             
                           
                           - 
                           
                             D 
                             
                               3 
                               , 
                               1 
                             
                           
                         
                         &lt; 
                         
                           - 
                           
                             ( 
                             
                               
                                 D 
                                 
                                   4 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   4 
                                   , 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     ) 
                   
                   = 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         Pr 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 D 
                                 
                                   1 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   1 
                                   , 
                                   1 
                                 
                               
                             
                             &lt; 
                             0 
                           
                           ) 
                         
                       
                       4 
                     
                     + 
                     
                       4 
                       ⁢ 
                       
                         Pr 
                         ( 
                         
                           
                             
                               
                                 D 
                                 
                                   1 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   1 
                                   , 
                                   1 
                                 
                               
                             
                             &gt; 
                             0 
                           
                           , 
                           
                             
                               D 
                               
                                 2 
                                 , 
                                 0 
                               
                             
                             - 
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         D 
                         
                           2 
                           , 
                           1 
                         
                       
                       &lt; 
                       
                         - 
                         
                           ( 
                           
                             
                               D 
                               
                                 1 
                                 , 
                                 0 
                               
                             
                             - 
                             
                               D 
                               
                                 1 
                                 , 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                     , 
                     
                       
                         
                           D 
                           
                             3 
                             , 
                             0 
                           
                         
                         - 
                         
                           D 
                           
                             3 
                             , 
                             1 
                           
                         
                       
                       &lt; 
                       
                         - 
                         
                           ( 
                           
                             
                               D 
                               
                                 1 
                                 , 
                                 0 
                               
                             
                             - 
                             
                               D 
                               
                                 1 
                                 , 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                     , 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                           D 
                           
                             4 
                             , 
                             0 
                           
                         
                         - 
                         
                           D 
                           
                             4 
                             , 
                             1 
                           
                         
                       
                       &lt; 
                       
                         - 
                         
                           ( 
                           
                             
                               D 
                               
                                 1 
                                 , 
                                 0 
                               
                             
                             - 
                             
                               D 
                               
                                 1 
                                 , 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       
                         
                           Pr 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   D 
                                   
                                     1 
                                     , 
                                     0 
                                   
                                 
                                 - 
                                 
                                   D 
                                   
                                     1 
                                     , 
                                     1 
                                   
                                 
                               
                               &lt; 
                               0 
                             
                             ) 
                           
                         
                         4 
                       
                       + 
                       
                         
 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         4 
                         ⁢ 
                         
                           
                             ∫ 
                             0 
                             ∞ 
                           
                           ⁢ 
                           
                             
                               Pr 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       D 
                                       
                                         1 
                                         , 
                                         0 
                                       
                                     
                                     - 
                                     
                                       D 
                                       
                                         1 
                                         , 
                                         1 
                                       
                                     
                                   
                                   = 
                                   α 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               Pr 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         D 
                                         
                                           2 
                                           , 
                                           0 
                                         
                                       
                                       - 
                                       
                                         D 
                                         
                                           2 
                                           , 
                                           1 
                                         
                                       
                                     
                                     &lt; 
                                     
                                       - 
                                       α 
                                     
                                   
                                   , 
                                   
                                     
 
                                   
                                   ⁢ 
                                   
                                     
                                       
                                         D 
                                         
                                           3 
                                           , 
                                           0 
                                         
                                       
                                       - 
                                       
                                         D 
                                         
                                           3 
                                           , 
                                           1 
                                         
                                       
                                     
                                     &lt; 
                                     
                                       - 
                                       α 
                                     
                                   
                                   , 
                                   
                                     
                                       
                                         
                                           
                                             D 
                                             
                                               4 
                                               , 
                                               0 
                                             
                                           
                                           - 
                                           
                                             D 
                                             
                                               4 
                                               , 
                                               1 
                                             
                                           
                                         
                                         &lt; 
                                         
                                           - 
                                           α 
                                         
                                       
                                       | 
                                       
                                         
                                           D 
                                           
                                             1 
                                             , 
                                             0 
                                           
                                         
                                         - 
                                         
                                           D 
                                           
                                             1 
                                             , 
                                             1 
                                           
                                         
                                       
                                     
                                     = 
                                     α 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               ⅆ 
                               α 
                             
                           
                         
                       
                     
                     = 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           D 
                           
                             4 
                             , 
                             0 
                           
                         
                         - 
                         
                           D 
                           
                             4 
                             , 
                             1 
                           
                         
                       
                       &lt; 
                       
                         - 
                         
                           ( 
                           
                             
                               D 
                               
                                 1 
                                 , 
                                 0 
                               
                             
                             - 
                             
                               D 
                               
                                 1 
                                 , 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       
                         
                           Pr 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   D 
                                   
                                     1 
                                     , 
                                     0 
                                   
                                 
                                 - 
                                 
                                   D 
                                   
                                     1 
                                     , 
                                     1 
                                   
                                 
                               
                               &lt; 
                               0 
                             
                             ) 
                           
                         
                         4 
                       
                       + 
                       
                         
 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         4 
                         ⁢ 
                         
                           
                             ∫ 
                             0 
                             ∞ 
                           
                           ⁢ 
                           
                             
                               Pr 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       D 
                                       
                                         1 
                                         , 
                                         0 
                                       
                                     
                                     - 
                                     
                                       D 
                                       
                                         1 
                                         , 
                                         1 
                                       
                                     
                                   
                                   = 
                                   α 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               Pr 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         D 
                                         
                                           2 
                                           , 
                                           0 
                                         
                                       
                                       - 
                                       
                                         D 
                                         
                                           2 
                                           , 
                                           1 
                                         
                                       
                                     
                                     &lt; 
                                     
                                       - 
                                       α 
                                     
                                   
                                   , 
                                   
                                     
 
                                   
                                   ⁢ 
                                   
                                     
                                       
                                         D 
                                         
                                           3 
                                           , 
                                           0 
                                         
                                       
                                       - 
                                       
                                         D 
                                         
                                           3 
                                           , 
                                           1 
                                         
                                       
                                     
                                     &lt; 
                                     
                                       - 
                                       α 
                                     
                                   
                                   , 
                                   
                                     
                                       
                                         
                                           
                                             D 
                                             
                                               4 
                                               , 
                                               0 
                                             
                                           
                                           - 
                                           
                                             D 
                                             
                                               4 
                                               , 
                                               1 
                                             
                                           
                                         
                                         &lt; 
                                         
                                           - 
                                           α 
                                         
                                       
                                       | 
                                       
                                         
                                           D 
                                           
                                             1 
                                             , 
                                             0 
                                           
                                         
                                         - 
                                         
                                           D 
                                           
                                             1 
                                             , 
                                             1 
                                           
                                         
                                       
                                     
                                     = 
                                     α 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               ⅆ 
                               α 
                             
                           
                         
                       
                     
                     = 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         Pr 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 D 
                                 
                                   1 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   1 
                                   , 
                                   1 
                                 
                               
                             
                             &lt; 
                             0 
                           
                           ) 
                         
                       
                       4 
                     
                     + 
                     
                       4 
                       ⁢ 
                       
                         
                           ∫ 
                           0 
                           ∞ 
                         
                         ⁢ 
                         
                           
                             Pr 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     D 
                                     
                                       1 
                                       , 
                                       0 
                                     
                                   
                                   - 
                                   
                                     D 
                                     
                                       1 
                                       , 
                                       1 
                                     
                                   
                                 
                                 = 
                                 α 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             Pr 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     
                                       D 
                                       
                                         2 
                                         , 
                                         0 
                                       
                                     
                                     - 
                                     
                                       
 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       D 
                                       
                                         2 
                                         , 
                                         1 
                                       
                                     
                                   
                                   &lt; 
                                   
                                     - 
                                     α 
                                   
                                 
                                 , 
                                 
                                   
                                     
                                       D 
                                       
                                         3 
                                         , 
                                         0 
                                       
                                     
                                     - 
                                     
                                       D 
                                       
                                         3 
                                         , 
                                         1 
                                       
                                     
                                   
                                   &lt; 
                                   
                                     - 
                                     α 
                                   
                                 
                                 , 
                                 
                                   
                                     
                                       D 
                                       
                                         4 
                                         , 
                                         0 
                                       
                                     
                                     - 
                                     
                                       D 
                                       
                                         4 
                                         , 
                                         1 
                                       
                                     
                                   
                                   &lt; 
                                   
                                     - 
                                     α 
                                   
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             ⅆ 
                             α 
                           
                         
                       
                     
                   
                   = 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         Pr 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 D 
                                 
                                   1 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   1 
                                   , 
                                   1 
                                 
                               
                             
                             &lt; 
                             0 
                           
                           ) 
                         
                       
                       4 
                     
                     + 
                     
                       4 
                       ⁢ 
                       
                         
                           ∫ 
                           0 
                           ∞ 
                         
                         ⁢ 
                         
                           Pr 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   D 
                                   
                                     1 
                                     , 
                                     0 
                                   
                                 
                                 - 
                                 
                                   D 
                                   
                                     1 
                                     , 
                                     1 
                                   
                                 
                               
                               = 
                               α 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         Pr 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 D 
                                 
                                   2 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   2 
                                   , 
                                   1 
                                 
                               
                             
                             &lt; 
                             
                               - 
                               α 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         Pr 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 D 
                                 
                                   3 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   3 
                                   , 
                                   1 
                                 
                               
                             
                             &lt; 
                             
                                 
                             
                             ⁢ 
                             
                               - 
                               α 
                             
                           
                           ) 
                         
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           Pr 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   D 
                                   
                                     4 
                                     , 
                                     0 
                                   
                                 
                                 - 
                                 
                                   D 
                                   
                                     4 
                                     , 
                                     1 
                                   
                                 
                               
                               &lt; 
                               
                                 - 
                                 α 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           ⅆ 
                           α 
                         
                       
                       = 
                       
                         
                           
                             Pr 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     D 
                                     
                                       1 
                                       , 
                                       0 
                                     
                                   
                                   - 
                                   
                                     D 
                                     
                                       1 
                                       , 
                                       1 
                                     
                                   
                                 
                                 &lt; 
                                 0 
                               
                               ) 
                             
                           
                           4 
                         
                         + 
                         
                           
 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           4 
                           ⁢ 
                           
                             
                               ∫ 
                               0 
                               ∞ 
                             
                             ⁢ 
                             
                               
                                 Pr 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       
                                         D 
                                         
                                           1 
                                           , 
                                           0 
                                         
                                       
                                       - 
                                       
                                         D 
                                         
                                           1 
                                           , 
                                           1 
                                         
                                       
                                     
                                     = 
                                     α 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 
                                   Pr 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         
                                           D 
                                           
                                             
                                               2 
                                               , 
                                               0 
                                             
                                             ⁢ 
                                             
                                                 
                                             
                                           
                                         
                                         - 
                                         
                                           D 
                                           
                                             2 
                                             , 
                                             1 
                                           
                                         
                                       
                                       &lt; 
                                       
                                         - 
                                         α 
                                       
                                     
                                     ) 
                                   
                                 
                                 3 
                               
                               ⁢ 
                               
                                 
                                   ⅆ 
                                   α 
                                 
                                 . 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     Then, the DFP can be expressed as 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       P 
                       
                         df 
                         , 
                         
                           U 
                           = 
                           4 
                         
                       
                     
                     = 
                     
                       1 
                       - 
                       
                         
                           Pr 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   D 
                                   
                                     1 
                                     , 
                                     0 
                                   
                                 
                                 - 
                                 
                                   D 
                                   
                                     1 
                                     , 
                                     1 
                                   
                                 
                               
                               &lt; 
                               0 
                             
                             ) 
                           
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   30 
                 
               
             
             
               
                 
                   
                     P 
                     
                       df 
                       , 
                       
                         U 
                         = 
                         8 
                       
                     
                   
                   = 
                   
                     1 
                     - 
                     
                       
                         Pr 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 D 
                                 
                                   1 
                                   , 
                                   0 
                                 
                               
                               - 
                               
                                 D 
                                 
                                   1 
                                   , 
                                   1 
                                 
                               
                             
                             &lt; 
                             0 
                           
                           ) 
                         
                       
                       4 
                     
                     - 
                     
                       4 
                       ⁢ 
                       
                         
                           ∫ 
                           0 
                           ∞ 
                         
                         ⁢ 
                         
                           
                             Pr 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     D 
                                     
                                       1 
                                       , 
                                       0 
                                     
                                   
                                   - 
                                   
                                     D 
                                     
                                       1 
                                       , 
                                       1 
                                     
                                   
                                 
                                 = 
                                 α 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               Pr 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       D 
                                       
                                         2 
                                         , 
                                         0 
                                       
                                     
                                     - 
                                     
                                       D 
                                       
                                         2 
                                         , 
                                         1 
                                       
                                     
                                   
                                   &lt; 
                                   
                                     - 
                                     α 
                                   
                                 
                                 ) 
                               
                             
                             3 
                           
                           ⁢ 
                           
                             ⅆ 
                             α 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   31 
                 
               
             
           
         
       
     
     Since DFP for U larger than U=8 can be derived in the same manner as when U=4 and 8, a description of DFP for U larger than 8 will be omitted herein. 
     In order to calculate Pr(D 1,0 −D 1,1 &lt;0) in Expressions 30 and 31, it is necessary know the probability distribution of D v,0 −D v,1  on QPSK. Let m i , 1≦i≦8, be the number of R n  in each sub-block such that 
               arg   ⁢           ⁢     (     R   n     )       ∈       [           (     ⅈ   -   1     )     ⁢   π     4     ,     ⅈπ   4       )     .           
That is, we have
 
     
       
         
           
             
               ∑ 
               
                 i 
                 = 
                 1 
               
               8 
             
             ⁢ 
             
               
                 m 
                 i 
               
               : 
               
                 
                   
                     2 
                     ⁢ 
                     N 
                   
                   U 
                 
                 . 
               
             
           
         
       
     
     Using Expression 21, D v,0  in S 0  and D v,1  in S 1  may be given as 
     
       
         
           
             
               
                 
                   
                     D 
                     
                       v 
                       , 
                       0 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         
                           m 
                           1 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             N 
                             
                               re 
                               , 
                               k 
                             
                             2 
                           
                           + 
                           
                             N 
                             
                               im 
                               , 
                               k 
                             
                             2 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         
                           m 
                           2 
                         
                       
                       ⁢ 
                       
                         { 
                         
                           
                             
                               ( 
                               
                                 
                                   N 
                                   
                                     re 
                                     , 
                                     k 
                                   
                                 
                                 + 
                                 1 
                               
                               ) 
                             
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   N 
                                   
                                     im 
                                     , 
                                     k 
                                   
                                 
                                 - 
                                 1 
                               
                               ) 
                             
                             2 
                           
                         
                         } 
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         
                           m 
                           3 
                         
                       
                       ⁢ 
                       
                         { 
                         
                           
                             
                               ( 
                               
                                 
                                   N 
                                   
                                     re 
                                     , 
                                     k 
                                   
                                 
                                 + 
                                 1 
                               
                               ) 
                             
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   N 
                                   
                                     im 
                                     , 
                                     k 
                                   
                                 
                                 - 
                                 1 
                               
                               ) 
                             
                             2 
                           
                         
                         } 
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         
                           m 
                           4 
                         
                       
                       ⁢ 
                       
                         { 
                         
                           
                             
                               ( 
                               
                                 
                                   N 
                                   
                                     re 
                                     , 
                                     k 
                                   
                                 
                                 + 
                                 2 
                               
                               ) 
                             
                             2 
                           
                           + 
                           
                             N 
                             
                               im 
                               , 
                               k 
                             
                             2 
                           
                         
                         } 
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         
                           m 
                           5 
                         
                       
                       ⁢ 
                       
                         { 
                         
                           
                             
                               ( 
                               
                                 
                                   N 
                                   
                                     re 
                                     , 
                                     k 
                                   
                                 
                                 + 
                                 2 
                               
                               ) 
                             
                             2 
                           
                           + 
                           
                             N 
                             
                               im 
                               , 
                               k 
                             
                             2 
                           
                         
                         } 
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         
                           m 
                           6 
                         
                       
                       ⁢ 
                       
                         { 
                         
                           
                             
                               ( 
                               
                                 
                                   N 
                                   
                                     re 
                                     , 
                                     k 
                                   
                                 
                                 + 
                                 1 
                               
                               ) 
                             
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   N 
                                   
                                     im 
                                     , 
                                     k 
                                   
                                 
                                 + 
                                 1 
                               
                               ) 
                             
                             2 
                           
                         
                         } 
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         
                           m 
                           7 
                         
                       
                       ⁢ 
                       
                         { 
                         
                           
                             
                               ( 
                               
                                 
                                   N 
                                   
                                     re 
                                     , 
                                     k 
                                   
                                 
                                 + 
                                 1 
                               
                               ) 
                             
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   N 
                                   
                                     im 
                                     , 
                                     k 
                                   
                                 
                                 + 
                                 1 
                               
                               ) 
                             
                             2 
                           
                         
                         } 
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         
                           m 
                           8 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             N 
                             
                               re 
                               , 
                               k 
                             
                             2 
                           
                           + 
                           
                             N 
                             
                               im 
                               , 
                               k 
                             
                             2 
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   32 
                 
               
             
             
               
                 
                   
                     D 
                     
                       v 
                       , 
                       1 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         
                           m 
                           1 
                         
                       
                       ⁢ 
                       
                         [ 
                         
                           
                             
                               { 
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       N 
                                       
                                         re 
                                         , 
                                         k 
                                       
                                     
                                   
                                   ) 
                                 
                                 - 
                                 
                                   1 
                                   
                                     2 
                                   
                                 
                               
                               } 
                             
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   N 
                                   
                                     im 
                                     , 
                                     k 
                                   
                                 
                                 - 
                                 
                                   1 
                                   
                                     2 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                         ] 
                       
                     
                     + 
                   
                 
               
               
                 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   33 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         
                           m 
                           2 
                         
                       
                       ⁢ 
                       
                         [ 
                         
                           
                             
                               { 
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       N 
                                       
                                         re 
                                         , 
                                         k 
                                       
                                     
                                   
                                   ) 
                                 
                                 - 
                                 
                                   1 
                                   
                                     2 
                                   
                                 
                               
                               } 
                             
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   N 
                                   
                                     im 
                                     , 
                                     k 
                                   
                                 
                                 - 
                                 
                                   1 
                                   
                                     2 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                         ] 
                       
                     
                     + 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         
                           m 
                           3 
                         
                       
                       ⁢ 
                       
                         [ 
                         
                           
                             
                               { 
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       N 
                                       
                                         re 
                                         , 
                                         k 
                                       
                                     
                                   
                                   ) 
                                 
                                 + 
                                 
                                   1 
                                   
                                     2 
                                   
                                 
                               
                               } 
                             
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   N 
                                   
                                     im 
                                     , 
                                     k 
                                   
                                 
                                 - 
                                 
                                   1 
                                   
                                     2 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                         ] 
                       
                     
                     + 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         
                           m 
                           4 
                         
                       
                       ⁢ 
                       
                         [ 
                         
                           
                             
                               { 
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       N 
                                       
                                         re 
                                         , 
                                         k 
                                       
                                     
                                   
                                   ) 
                                 
                                 + 
                                 
                                   1 
                                   
                                     2 
                                   
                                 
                               
                               } 
                             
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   N 
                                   
                                     im 
                                     , 
                                     k 
                                   
                                 
                                 - 
                                 
                                   1 
                                   
                                     2 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                         ] 
                       
                     
                     + 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         
                           m 
                           5 
                         
                       
                       ⁢ 
                       
                         [ 
                         
                           
                             
                               { 
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       N 
                                       
                                         re 
                                         , 
                                         k 
                                       
                                     
                                   
                                   ) 
                                 
                                 + 
                                 
                                   1 
                                   
                                     2 
                                   
                                 
                               
                               } 
                             
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   N 
                                   
                                     im 
                                     , 
                                     k 
                                   
                                 
                                 + 
                                 
                                   1 
                                   
                                     2 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                         ] 
                       
                     
                     + 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         
                           m 
                           6 
                         
                       
                       ⁢ 
                       
                         [ 
                         
                           
                             
                               { 
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       N 
                                       
                                         re 
                                         , 
                                         k 
                                       
                                     
                                   
                                   ) 
                                 
                                 + 
                                 
                                   1 
                                   
                                     2 
                                   
                                 
                               
                               } 
                             
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   N 
                                   
                                     im 
                                     , 
                                     k 
                                   
                                 
                                 + 
                                 
                                   1 
                                   
                                     2 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                         ] 
                       
                     
                     + 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         
                           m 
                           7 
                         
                       
                       ⁢ 
                       
                         [ 
                         
                           
                             
                               { 
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       N 
                                       
                                         re 
                                         , 
                                         k 
                                       
                                     
                                   
                                   ) 
                                 
                                 - 
                                 
                                   1 
                                   
                                     2 
                                   
                                 
                               
                               } 
                             
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   N 
                                   
                                     im 
                                     , 
                                     k 
                                   
                                 
                                 + 
                                 
                                   1 
                                   
                                     2 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                         ] 
                       
                     
                     + 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         1 
                       
                       
                         m 
                         8 
                       
                     
                     ⁢ 
                     
                       [ 
                       
                         
                           
                             { 
                             
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     N 
                                     
                                       re 
                                       , 
                                       k 
                                     
                                   
                                 
                                 ) 
                               
                               - 
                               
                                 1 
                                 
                                   2 
                                 
                               
                             
                             } 
                           
                           2 
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 N 
                                 
                                   im 
                                   , 
                                   k 
                                 
                               
                               + 
                               
                                 1 
                                 
                                   2 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     Using Expressions 32 and 33, Expression 34 is obtained: 
     
       
         
           
             
               
                 
                   
                     
                       D 
                       
                         v 
                         , 
                         0 
                       
                     
                     - 
                     
                       D 
                       
                         v 
                         , 
                         1 
                       
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         
                           m 
                           1 
                         
                       
                       ⁢ 
                       
                         { 
                         
                           
                             
                               ( 
                               
                                 
                                   2 
                                 
                                 - 
                                 2 
                               
                               ) 
                             
                             ⁢ 
                             
                               N 
                               
                                 re 
                                 , 
                                 k 
                               
                             
                           
                           + 
                           
                             
                               2 
                             
                             ⁢ 
                             
                               N 
                               
                                 im 
                                 , 
                                 k 
                               
                             
                           
                           + 
                           
                             2 
                           
                           - 
                           2 
                         
                         } 
                       
                     
                     + 
                   
                 
               
             
             
               
                 
                     
                   ⁢ 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         
                           m 
                           2 
                         
                       
                       ⁢ 
                       
                         { 
                         
                           
                             
                               2 
                             
                             ⁢ 
                             
                               N 
                               
                                 re 
                                 , 
                                 k 
                               
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   2 
                                 
                                 - 
                                 2 
                               
                               ) 
                             
                             ⁢ 
                             
                               N 
                               
                                 im 
                                 , 
                                 k 
                               
                             
                           
                           + 
                           
                             2 
                           
                         
                         } 
                       
                     
                     + 
                   
                 
               
             
             
               
                 
                     
                   ⁢ 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         
                           m 
                           3 
                         
                       
                       ⁢ 
                       
                         { 
                         
                           
                             
                               - 
                               
                                 2 
                               
                             
                             ⁢ 
                             
                               N 
                               
                                 re 
                                 , 
                                 k 
                               
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   2 
                                 
                                 - 
                                 2 
                               
                               ) 
                             
                             ⁢ 
                             
                               N 
                               
                                 im 
                                 , 
                                 k 
                               
                             
                           
                           - 
                           
                             2 
                           
                         
                         } 
                       
                     
                     + 
                   
                 
               
             
             
               
                 
                     
                   ⁢ 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         
                           m 
                           4 
                         
                       
                       ⁢ 
                       
                         { 
                         
                           
                             
                               ( 
                               
                                 2 
                                 - 
                                 
                                   2 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               N 
                               
                                 re 
                                 , 
                                 k 
                               
                             
                           
                           + 
                           
                             
                               2 
                             
                             ⁢ 
                             
                               N 
                               
                                 im 
                                 , 
                                 k 
                               
                             
                           
                           + 
                           2 
                           - 
                           
                             2 
                           
                         
                         } 
                       
                     
                     + 
                   
                 
               
             
             
               
                 
                     
                   ⁢ 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         
                           m 
                           5 
                         
                       
                       ⁢ 
                       
                         { 
                         
                           
                             
                               ( 
                               
                                 2 
                                 - 
                                 
                                   2 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               N 
                               
                                 re 
                                 , 
                                 k 
                               
                             
                           
                           - 
                           
                             
                               2 
                             
                             ⁢ 
                             
                               N 
                               
                                 im 
                                 , 
                                 k 
                               
                             
                           
                           + 
                           2 
                           - 
                           
                             2 
                           
                         
                         } 
                       
                     
                     + 
                   
                 
               
             
             
               
                 
                     
                   ⁢ 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         
                           m 
                           6 
                         
                       
                       ⁢ 
                       
                         { 
                         
                           
                             
                               - 
                               
                                 2 
                               
                             
                             ⁢ 
                             
                               N 
                               
                                 re 
                                 , 
                                 k 
                               
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 2 
                                 - 
                                 
                                   2 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               N 
                               
                                 im 
                                 , 
                                 k 
                               
                             
                           
                           - 
                           
                             2 
                           
                         
                         } 
                       
                     
                     + 
                   
                 
               
             
             
               
                 
                     
                   ⁢ 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         
                           m 
                           7 
                         
                       
                       ⁢ 
                       
                         { 
                         
                           
                             
                               2 
                             
                             ⁢ 
                             
                               N 
                               
                                 re 
                                 , 
                                 k 
                               
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 2 
                                 - 
                                 
                                   2 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               N 
                               
                                 im 
                                 , 
                                 k 
                               
                             
                           
                           + 
                           
                             2 
                           
                         
                         } 
                       
                     
                     + 
                   
                 
               
             
             
               
                 
                     
                   ⁢ 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         1 
                       
                       
                         m 
                         8 
                       
                     
                     ⁢ 
                     
                       { 
                       
                         
                           
                             ( 
                             
                               
                                 2 
                               
                               - 
                               2 
                             
                             ) 
                           
                           ⁢ 
                           
                             N 
                             
                               re 
                               , 
                               k 
                             
                           
                         
                         - 
                         
                           
                             2 
                           
                           ⁢ 
                           
                             N 
                             
                               im 
                               , 
                               k 
                             
                           
                         
                         + 
                         
                           2 
                         
                         - 
                         2 
                       
                       } 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         
                           
                             2 
                             ⁢ 
                             
                               N 
                               ⁡ 
                               
                                 ( 
                                 
                                   v 
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                           
                           U 
                         
                       
                       
                         
                           
                             2 
                             ⁢ 
                             Nv 
                           
                           U 
                         
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       { 
                       
                         
                           
                             α 
                             n 
                           
                           ⁢ 
                           
                             N 
                             
                               re 
                               , 
                               n 
                             
                           
                         
                         + 
                         
                           
                             b 
                             n 
                           
                           ⁢ 
                           
                             N 
                             
                               im 
                               , 
                               n 
                             
                           
                         
                         + 
                         
                           c 
                           n 
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         
                           
                             2 
                             ⁢ 
                             
                               N 
                               ⁡ 
                               
                                 ( 
                                 
                                   v 
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                           
                           U 
                         
                       
                       
                         
                           
                             2 
                             ⁢ 
                             Nv 
                           
                           U 
                         
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       z 
                       n 
                     
                   
                 
               
             
           
         
       
     
     where a n , b n , c n ε{+√{square root over (2)}, −√{square root over (2)},2−√{square root over (2)}, √{square root over (2)}−2} and z n =a n N re,n +b n N im,n +c n . 
     Since Z n  values are statistically independent, D v,0 −D v,1  for large N may be assumed to have a Gaussian distribution by central limit theorem. It is not easy to calculate mean and variance of D v,0 −D v,1 . However, the mean and variance can be obtained through numerical analysis. Using the mean and variance of D v,0 −D v,1  by numerical analysis, the DFP can be calculated by Expression 30 for U=4 and Expression 31 for U=8 as 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       P 
                       
                         df 
                         , 
                         
                           U 
                           = 
                           4 
                         
                       
                     
                     = 
                     
                       1 
                       - 
                       
                         
                           Q 
                           ⁡ 
                           
                             ( 
                             
                               
                                 E 
                                 ⁢ 
                                 
                                   { 
                                   
                                     
                                       D 
                                       
                                         1 
                                         , 
                                         0 
                                       
                                     
                                     - 
                                     
                                       D 
                                       
                                         1 
                                         , 
                                         1 
                                       
                                     
                                   
                                   } 
                                 
                               
                               
                                 σ 
                                 
                                   
                                     D 
                                     
                                       1 
                                       , 
                                       0 
                                     
                                   
                                   - 
                                   
                                     D 
                                     
                                       1 
                                       , 
                                       1 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   35 
                 
               
             
             
               
                 
                   
                     P 
                     
                       df 
                       , 
                       
                         U 
                         = 
                         8 
                       
                     
                   
                   = 
                   
                     1 
                     - 
                     
                       
                         Q 
                         ⁡ 
                         
                           ( 
                           
                             
                               E 
                               ⁢ 
                               
                                 { 
                                 
                                   
                                     D 
                                     
                                       1 
                                       , 
                                       0 
                                     
                                   
                                   - 
                                   
                                     D 
                                     
                                       1 
                                       , 
                                       1 
                                     
                                   
                                 
                                 } 
                               
                             
                             
                               σ 
                               
                                 
                                   D 
                                   
                                     1 
                                     , 
                                     0 
                                   
                                 
                                 - 
                                 
                                   D 
                                   
                                     1 
                                     , 
                                     1 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                       4 
                     
                     - 
                     
                       4 
                       ⁢ 
                       
                         
                           ∫ 
                           0 
                           ∞ 
                         
                         ⁢ 
                         
                           
                             1 
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                               
                               ⁢ 
                               
                                 σ 
                                 
                                   
                                     D 
                                     
                                       1 
                                       , 
                                       0 
                                     
                                   
                                   - 
                                   
                                     D 
                                     
                                       1 
                                       , 
                                       1 
                                     
                                   
                                 
                               
                             
                           
                           ⁢ 
                           
                             exp 
                             ⁡ 
                             
                               ( 
                               
                                 - 
                                 
                                   
                                     
                                       ( 
                                       
                                         α 
                                         - 
                                         
                                           E 
                                           ⁢ 
                                           
                                             { 
                                             
                                               
                                                 D 
                                                 
                                                   1 
                                                   , 
                                                   0 
                                                 
                                               
                                               - 
                                               
                                                 D 
                                                 
                                                   1 
                                                   , 
                                                   1 
                                                 
                                               
                                             
                                             } 
                                           
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                   
                                     2 
                                     ⁢ 
                                     
                                       σ 
                                       
                                         
                                           D 
                                           
                                             1 
                                             , 
                                             0 
                                           
                                         
                                         - 
                                         
                                           D 
                                           
                                             1 
                                             , 
                                             1 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               Q 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     α 
                                     + 
                                     
                                       E 
                                       ⁢ 
                                       
                                         { 
                                         
                                           
                                             D 
                                             
                                               1 
                                               , 
                                               0 
                                             
                                           
                                           - 
                                           
                                             D 
                                             
                                               1 
                                               , 
                                               1 
                                             
                                           
                                         
                                         } 
                                       
                                     
                                   
                                   
                                     σ 
                                     
                                       
                                         D 
                                         
                                           1 
                                           , 
                                           0 
                                         
                                       
                                       - 
                                       
                                         D 
                                         
                                           1 
                                           , 
                                           1 
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                             3 
                           
                           ⁢ 
                           
                             ⅆ 
                             α 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   36 
                 
               
             
           
         
       
     
     DFP for M-QAM may be derived similar to that for QPSK.  FIG. 3  compares theoretical results by Expression 35 with simulation results of DFP for U=4. This shows that analysis of the DFP is suitable for the BSLM according to this invention. If N is large, the negligible detection failure probability in a low SNR region can be observed. Therefore, the decoder of the BSLM according to this invention ensures detection of nearly perfect side information over the AWGN channel. 
     Last, the simulation results will be described below in more detail. 
     Simulation is done in the AWGN channel, QPSK and 16-QAM with phase rotation θ=π/4, and N=64 and 256. Further, U=4, 8, and 16 are considered. The decoding algorithm using Expressions 10 and 11 is used for the BSLM scheme according to this invention. Further, the decoder in Expression 18 is considered for the BPTS scheme according to this invention. 
       FIGS. 4 to 7  show DFP of the conventional and the BSLM scheme according to this inventions for N=64, 256 modulated by QPSK and 16 QAM. For small N, such as 64, there exists noticeable DFP degradation for both the conventional BSLM scheme and the BSLM according to this invention at the low SNR region because the number of subcarriers is too small to obtain reliable decision metrics. When N=256, DFP decreases fast and no detection failure of side information for N=1024 is observed in the BSLM according to this invention. 
     The value of U affects DFP of side information. If U increases, DFP of both BSLM schemes is degraded in the low SNR. 
     When the conventional BSLM scheme uses large U, the Euclidian distance between the used phase sequence and the others is reduced and thus the DFP of the phase sequence is worsened. Similarly, the large U degrades DFP of the BSLM scheme according to this invention for the same reason. Due to the effect of block partitioning and predetermined phase rotation, DFP of the BSLM scheme according to this invention is better than that of the conventional BSLM scheme in the low SNR for QPSK. However, for 16 QAM and N=256, DFP of the BSLM scheme according to this invention slowly converges to 0 compared with that of the conventional BSLM scheme, because the Euclidean distance between S 0  and S 1  for 16 QAM is not maximized by θ=π/4. 
       FIGS. 4 to 7  also compare BER performance of the conventional and the BSLM scheme according to this inventions for N=64, 256 modulated by QPSK and 16 QAM. Even though the increased erroneous side information detection degrades BER, BER degradation exists at low SNR for small N and large U. In addition, DFP P df  converges to 0 at the BER of 10 −2  in both BSLM schemes. Since the practical OFDM systems require better BER than 10 −2 , BER degradation by detection failure in a low SNR can be negligible. In addition, BER in both BSLM schemes for N=256 is not degraded from the detection failure of side information, which is almost negligible. 
       FIG. 8  compares BER performance of the decoder proposed in Expression 18 with that of the conventional PTS that has perfect side information in the AWGN channel. Similar to the BSLM according to this invention, the BPTS almost perfectly recovers the received data in the AWGN channel. However, BER degradation also exists in the low SNR for small N and large V similar to the BSLM. 
       FIG. 9  shows that, for N=64, the BSLM scheme with predetermined phase rotation and block partitioning according to this invention has the same PAPR reduction performance as that of the conventional SLM scheme. 
     As apparent from the above description, in the BSLM scheme according to the embodiment of the invention, block partitioning and predetermined phase rotation are used to embed side information in an alternative signal sequence, and the BSLM scheme may be extended to the BPTS. Since low decoding complexity is given while detecting the side information, the same BER performance is obtained as that of the conventional BSLM scheme in the AWGN channel. 
     Further, the performance of PAPR reduction is the same as in the conventional SLM and PTS schemes. 
     Although some embodiments have been provided to illustrate the invention, it will be apparent to those skilled in the art that the embodiments are given by way of illustration, and that various modifications and equivalent embodiments can be made without departing from the spirit and scope of the invention. The scope of the invention should be limited only by the accompanying claims and equivalents thereof.