Abstract:
The present invention relates to a method of detecting a space-time code in a mobile communication system. When detecting a space-time code B for a plurality of transmitting antennas, an ML (maximum likelihood) metric, which is generally calculated by using a specific structure of a matrix B, is defined as a new ML metric by reducing a variable of the ML metric. Two dependent sphere decoding processes are performed and an intersection of the decoding results is calculated, and a signal is detected from the ML metric. Therefore, since detection of the space-time code B for the plurality of transmitting antennas is performed by using the newly defined ML metric, an amount of matrix calculated can be reduced, and a calculation amount can be reduced by performing the two dependent sphere decoding processes. Further, since a detection order of symbols is changed according to a channel situation, it is possible to improve the performance of the system.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application claims priority to and the benefit of Korean Patent Application No. 10-2006-0123915 filed in the Korean Intellectual Property Office on Dec. 7, 2006, the entire contents of which are incorporated herein by reference. 
     BACKGROUND OF THE INVENTION 
     (a) Field of the Invention 
     The present invention relates to a mobile communication system, and more particularly, to a method of detecting a space-time code. 
     (b) Description of the Related Art 
     In general, WiBro, which is a field of a mobile communication system, means a portable Internet system that has two meanings of “wireless” and “wideband”. A Wibro system has been defined as a system that satisfies the standard established by IEEE 802.16 and five performance criteria, by the Ministry of Information and Communication on Jul. 29, 2004. Here, the five performance criteria include that {circumflex over ( 1 )} when a mobile terminal moves at a speed of 60 km/h, a data transmission rate of at least 512 kbps during a downlink and a data transmission rate of at least 128 kbps during an uplink are ensured at cell interfaces, {circumflex over ( 2 )} a channel bandwidth is 9 MHz or more, {circumflex over ( 3 )} a roaming process of communication equipment needs to be performed between service providers, {circumflex over ( 4 )} a frequency reuse coefficient is 1, and {circumflex over ( 5 )} a TDD method is used to minimize interference between frequencies, while transmitting/receiving time slots synchronize with each other. 
     In the IEEE 802.16e, when there are four transmitting antennas in a system, three usable space-time codes are defined. The three space-time codes are represented by three matrices A, B, and C, respectively, and each space-time code has a different symbol transmission rate and diversity gain. Among them, particularly, the space-time code B has a superior trade-off performance for the symbol transmission rate and the diversity gain, as follows. 
     
       
         
           
             B 
             = 
             
               [ 
               
                 
                   
                     
                       
                         s 
                         1 
                       
                       - 
                       
                         
                           s 
                           2 
                           * 
                         
                         ⁢ 
                         
                           s 
                           5 
                         
                       
                       - 
                       
                         s 
                         7 
                         * 
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           s 
                           2 
                         
                         ⁢ 
                         
                           s 
                           1 
                           * 
                         
                         ⁢ 
                         
                           s 
                           6 
                         
                       
                       - 
                       
                         s 
                         8 
                         * 
                       
                     
                   
                 
                 
                   
                     
                       
                         s 
                         3 
                       
                       - 
                       
                         
                           s 
                           4 
                           * 
                         
                         ⁢ 
                         
                           s 
                           7 
                         
                         ⁢ 
                         
                           s 
                           5 
                           * 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         s 
                         4 
                       
                       ⁢ 
                       
                         s 
                         3 
                         * 
                       
                       ⁢ 
                       
                         s 
                         8 
                       
                       ⁢ 
                       
                         s 
                         6 
                         * 
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
     In this case, the vertical axis indicates an antenna. It can be understood that four symbols are simultaneously transmitted from four antennas. The horizontal axis indicates time or a carrier frequency. 
     When symbols are transmitted by using the given space-time code, the symbols are simultaneously received by a receiving terminal. Thus, the entire performance of a system depends on an efficient detection method in the receiving terminal. Among detection methods that have been proposed in the related art, a maximum likelihood (ML) detection method is most superior in terms of performance. 
     However, the ML detection method is very complicated. In particular, in the case of when a magnitude of a constellation increases (e.g., in the case of 16-QAM or 64-QAM), since complexity of the system is increased, it is not possible to perform the ML detection in real time. 
     In order to provide optimal ML performance while the complexity of the system is reduced, a sphere decoding scheme has been proposed. In the sphere decoding scheme, since a test is performed on constellation points that exist near the received signals, complexity is alleviated as compared with the case of the simple ML detection method according to the related art. 
     However, when considering a downlink, the size of a mobile terminal is generally restricted. Therefore, it is difficult for a plurality of receiving antennas to be located at one mobile terminal. Actually, in the WiBro system, even though two or more receiving antennas are defined in one mobile terminal, a plurality of mobile terminals have a number of receiving antennas that is smaller than a number of transmitting terminals. 
     In particular, a scheme that is entitled “A new approach for fast generalized sphere decoding in MIMO systems” that was proposed by Z. Yang, C. Liu and J. He and disclosed in IEEE Signal Processing Letters, has been known as the fastest sphere decoding scheme when the number of receiving antennas is smaller than the number of transmitting antennas. However, as in the WiBro system, in the case where four transmitting antennas exist and the magnitude of the constellation is large, an algorithm is very complex. As a result, it is difficult to use the scheme in the system. 
     Examples of a method of reducing the complexity of a system and providing suboptimal performance include ZF (Zero-Forcing), MMSE (Minimum Mean Squared Error), SIC (ZF with Successive Interference Cancellation), MMSE with SIC, and the like. However, since these methods cannot provide the ML performance, they are inferior to the sphere decoding scheme that can ensure the ML performance. 
     The above information disclosed in this Background section is only for enhancement of understanding of the background of the invention and therefore it may contain information that does not form the prior art that is already known in this country to a person of ordinary skill in the art. 
     SUMMARY OF THE INVENTION 
     The present invention has been made in an effort to provide a method of detecting a space-time code, having advantages of detecting the space-time code in a mobile communication system where a plurality of transmitting antennas are provided. 
     An exemplary embodiment of the present invention provides a method of detecting a space-time code for a plurality of transmitting antennas. The method includes receiving signals transmitted from the plurality of transmitting antennas and determining channel statuses of the received signals for the transmitting antennas to determine a detection order of symbol groups, the received signals being divided into a first symbol group and a second symbol group each of which includes a pair of a first symbol and a second symbol, each symbol including imaginary and real parts each having one or more bits; calculating an ML (maximum likelihood) metric for the first symbol group if a detection on the first symbol group is determined according to the channel statuses; performing sphere decoding on the calculated ML metric and sequentially detecting an imaginary part of one bit and a real part of one bit for the first symbol and the second symbol of the first symbol group; and detecting the second symbol group as a single symbol on the basis of the detected first symbol group. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic diagram illustrating a system according to an exemplary embodiment of the present invention. 
         FIG. 2  is a flowchart illustrating a detection method according to an exemplary embodiment of the present invention. 
         FIG. 3  is an exemplary view illustrating comparison results between running times in a 4-QAM method according to an exemplary embodiment of the present invention and a method according to the related art. 
         FIG. 4  is an exemplary view illustrating comparison results between CPU clocks in a 4-QAM method according to an exemplary embodiment of the present invention and a method according to the related art. 
         FIG. 5  is an exemplary view illustrating comparison results between running times in a 16-QAM method according to an exemplary embodiment of the present invention and a method according to the related art. 
         FIG. 6  is an exemplary view illustrating comparison results between CPU clocks in a 16-QAM method according to an exemplary embodiment of the present invention and a method according to the related art. 
     
    
    
     DETAILED DESCRIPTION OF THE EMBODIMENTS 
     In the following detailed description, only certain exemplary embodiments of the present invention have been shown and described, simply by way of illustration. As those skilled in the art would realize, the described embodiments may be modified in various different ways, all without departing from the spirit or scope of the present invention. Accordingly, the drawings and description are to be regarded as illustrative in nature and not restrictive. Like reference numerals designate like elements throughout the specification. 
     In addition, unless explicitly described to the contrary, the word “comprise” and variations such as “comprises” or “comprising” will be understood to imply the inclusion of stated elements but not the exclusion of any other elements. 
     Before describing an exemplary embodiment of the present invention, an assumption about a space-time code B will be described. 
     A space-time code B is composed of a 4×4 matrix. But symbols that are completely different from symbols that are transmitted in first and second columns are transmitted in third and fourth columns, and a receiving terminal receives a space-time code at a different time slot or frequency. Thus, it is assumed that space-time codes are generated only in consideration of the first and second columns, as represented by Equation 1. 
     
       
         
           
             
               
                 
                   X 
                   = 
                   
                      
                     
                       
                         
                           
                             
                               x 
                               1 
                             
                             - 
                             
                               x 
                               2 
                               * 
                             
                           
                         
                       
                       
                         
                           
                             
                               x 
                               2 
                             
                             ⁢ 
                             
                               x 
                               1 
                               * 
                             
                           
                         
                       
                       
                         
                           
                             
                               x 
                               3 
                             
                             - 
                             
                               x 
                               4 
                               * 
                             
                           
                         
                       
                       
                         
                           
                             
                               x 
                               4 
                             
                             ⁢ 
                             
                               x 
                               3 
                               * 
                             
                           
                         
                       
                     
                      
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                   ) 
                 
               
             
           
         
       
     
     On the assumption, a space-time codes detecting system according to an exemplary embodiment of the present invention will be described with reference to  FIG. 1 . 
       FIG. 1  is a schematic diagram illustrating a wireless communication system according to an exemplary embodiment of the present invention. 
     As shown in  FIG. 1 , the wireless communication system comprises a transmitter with 4 transmitting antenna and a receiver with n R  receiving antenna. The receiver has a detector. 
     In an exemplary embodiment of the present invention, a case that the number of receiving antennas is one will be described, then a case that the number of receiving antennas is n R  will be described. 
     In case that one receiving antenna exists in a receiver, signals that are received in first and second time slots in the receiver are represented by Equation 2.
 
 r 1 =h   1   x   1   +h   2   x   2   +h   3   x   3   +h   4   x   4   +n   1  
 
 r 2 =−h   1   x   2   *+h   2   x   1   *−h   3   x   4   *+h   4   x   3   *+n   2   (Equation 2)
 
     In this case, h j  indicates a channel gain between a j-th transmitting antenna and a receiving antenna. In addition, n i  indicates white noise in an i-th time slot. In this case, ML (maximum likelihood) detection is performed as represented by Equation 3. 
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         x 
                         1 
                         opt 
                       
                       , 
                       
                         x 
                         2 
                         opt 
                       
                       , 
                       
                         x 
                         3 
                         opt 
                       
                       , 
                       
                         x 
                         4 
                         opt 
                       
                     
                     ) 
                   
                   = 
                   
                     arg 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         min 
                         
                           
                             x 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                           , 
                           
                             x 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                           , 
                           
                             x 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             3 
                           
                           , 
                           
                             
                               x 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               4 
                             
                             ∈ 
                             S 
                           
                         
                       
                       ⁢ 
                       
                         M 
                         _ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     3 
                   
                   ) 
                 
               
             
           
         
       
     
     In this case, S indicates a constellation, and an ML metric is given by Equation 4.
 
   M =|r   1   −h   1   x   1   −h   2   x   2   −h   3   x   3   −h   4   x   4 | 2   +|r   2   +h   1   x   2   *−h   2   x   1   *+h   3   x   4   *−h   4   x   3 *| 2   (Equation 4)
 
     From Equation 4, the ML metric can be represented in Equation 5, which is an equation that is newly derived according to the exemplary embodiment of the present invention. In Equation 5, a matrix operation can be briefly performed by reducing the number of variables that are included in respective terms at the right side.
 
 M=∥α   0   −D   A   X   A ∥ 2 +∥β0 −D   B   X   B ∥ 2   (Equation 5)
 
     A variety of vectors and matrices in Equation 5 are defined as follows. 
     
       
         
           
             
               α 
               0 
             
             = 
             
               
                 [ 
                 
                   
                     ℜ 
                     ⁡ 
                     
                       [ 
                       
                         α 
                         0 
                       
                       ] 
                     
                   
                   , 
                   
                     ?? 
                     ⁡ 
                     
                       [ 
                       
                         α 
                         0 
                       
                       ] 
                     
                   
                 
                 ] 
               
               T 
             
           
         
       
       
         
           
             
               β 
               0 
             
             = 
             
               
                 [ 
                 
                   
                     ℜ 
                     ⁡ 
                     
                       [ 
                       
                         β 
                         0 
                       
                       ] 
                     
                   
                   , 
                   
                     ?? 
                     ⁡ 
                     
                       [ 
                       
                         β 
                         0 
                       
                       ] 
                     
                   
                 
                 ] 
               
               T 
             
           
         
       
       
         
           
             
               D 
               A 
             
             = 
             
               [ 
               
                 
                   
                     
                       
                         ℜ 
                         ⁡ 
                         
                           [ 
                           
                             α 
                             1 
                           
                           ] 
                         
                       
                       - 
                       
                         
                           ?? 
                           ⁡ 
                           
                             [ 
                             
                               α 
                               1 
                             
                             ] 
                           
                         
                         ⁢ 
                         
                           ℜ 
                           ⁡ 
                           
                             [ 
                             
                               α 
                               3 
                             
                             ] 
                           
                         
                       
                       - 
                       
                         
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                             [ 
                             
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                               3 
                             
                             ] 
                           
                         
                         ⁢ 
                         
                           ℜ 
                           ⁡ 
                           
                             [ 
                             
                               α 
                               4 
                             
                             ] 
                           
                         
                       
                       - 
                       
                         ?? 
                         ⁡ 
                         
                           [ 
                           
                             α 
                             4 
                           
                           ] 
                         
                       
                     
                   
                 
                 
                   
                     
                       
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                         ⁡ 
                         
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                             1 
                           
                           ] 
                         
                       
                       ⁢ 
                       
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                         ⁡ 
                         
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                           ] 
                         
                       
                       ⁢ 
                       
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                         ⁡ 
                         
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                       ⁢ 
                       
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                         ⁡ 
                         
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                             4 
                           
                           ] 
                         
                       
                     
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               D 
               B 
             
             = 
             
               [ 
               
                 
                   
                     
                       
                         ℜ 
                         ⁡ 
                         
                           [ 
                           
                             β 
                             2 
                           
                           ] 
                         
                       
                       - 
                       
                         
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                             [ 
                             
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                         ⁢ 
                         
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                           ⁡ 
                           
                             [ 
                             
                               β 
                               3 
                             
                             ] 
                           
                         
                       
                       - 
                       
                         
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                           ⁡ 
                           
                             [ 
                             
                               β 
                               3 
                             
                             ] 
                           
                         
                         ⁢ 
                         
                           ℜ 
                           ⁡ 
                           
                             [ 
                             
                               β 
                               4 
                             
                             ] 
                           
                         
                       
                       - 
                       
                         ?? 
                         ⁡ 
                         
                           [ 
                           β4 
                           ] 
                         
                       
                     
                   
                 
                 
                   
                     
                       
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                         ⁡ 
                         
                           [ 
                           
                             β 
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                           ] 
                         
                       
                       ⁢ 
                       
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                       ⁢ 
                       
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                           ] 
                         
                       
                       ⁢ 
                       
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                         ⁡ 
                         
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                             4 
                           
                           ] 
                         
                       
                     
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               
                 
                   
                     X 
                     A 
                   
                   := 
                   
                     
                       [ 
                       
                         
                           
                             
                               x 
                               1 
                               A 
                             
                           
                         
                         
                           
                             
                               x 
                               2 
                               A 
                             
                           
                         
                         
                           
                             
                               x 
                               3 
                               A 
                             
                           
                         
                         
                           
                             
                               x 
                               4 
                               A 
                             
                           
                         
                         
                           
                             
                               x 
                               5 
                               A 
                             
                           
                         
                         
                           
                             
                               x 
                               6 
                               A 
                             
                           
                         
                       
                       ] 
                     
                     := 
                     
                       [ 
                       
                         
                           
                             
                               ℜ 
                               ⁡ 
                               
                                 [ 
                                 
                                   x 
                                   1 
                                 
                                 ] 
                               
                             
                           
                         
                         
                           
                             
                               ?? 
                               ⁡ 
                               
                                 [ 
                                 
                                   x 
                                   1 
                                 
                                 ] 
                               
                             
                           
                         
                         
                           
                             
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                               ⁡ 
                               
                                 [ 
                                 
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                                   3 
                                 
                                 ] 
                               
                             
                           
                         
                         
                           
                             
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                                 [ 
                                 
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                                   3 
                                 
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                               ⁡ 
                               
                                 [ 
                                 
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                               ⁡ 
                               
                                 [ 
                                 
                                   x 
                                   4 
                                 
                                 ] 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             α 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                           = 
                           
                             
                               
                                 
                                    
                                   
                                     h 
                                     1 
                                   
                                    
                                 
                                 2 
                               
                               + 
                               
                                 
                                    
                                   
                                     h 
                                     2 
                                   
                                    
                                 
                                 2 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             α 
                             0 
                           
                           = 
                           
                             
                               
                                 
                                   r 
                                   1 
                                 
                                 ⁢ 
                                 
                                   h 
                                   1 
                                   * 
                                 
                               
                               + 
                               
                                 
                                   r 
                                   2 
                                   * 
                                 
                                 ⁢ 
                                 
                                   h 
                                   2 
                                 
                               
                             
                             
                               α 
                               1 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             α 
                             3 
                           
                           = 
                           
                             
                               
                                 
                                   h 
                                   1 
                                   * 
                                 
                                 ⁢ 
                                 
                                   h 
                                   3 
                                 
                               
                               + 
                               
                                 
                                   h 
                                   2 
                                 
                                 ⁢ 
                                 
                                   h 
                                   4 
                                   * 
                                 
                               
                             
                             
                               α 
                               1 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             α 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             4 
                           
                           = 
                           
                             
                               
                                 
                                   h 
                                   1 
                                   * 
                                 
                                 ⁢ 
                                 
                                   h 
                                   4 
                                 
                               
                               + 
                               
                                 
                                   h 
                                   1 
                                 
                                 ⁢ 
                                 
                                   h 
                                   3 
                                   * 
                                 
                               
                             
                             
                               α 
                               1 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   
                     X 
                     B 
                   
                   := 
                   
                     
                       [ 
                       
                         
                           
                             
                               x 
                               1 
                               B 
                             
                           
                         
                         
                           
                             
                               x 
                               2 
                               B 
                             
                           
                         
                         
                           
                             
                               x 
                               3 
                               B 
                             
                           
                         
                         
                           
                             
                               x 
                               4 
                               B 
                             
                           
                         
                         
                           
                             
                               x 
                               5 
                               B 
                             
                           
                         
                         
                           
                             
                               x 
                               6 
                               B 
                             
                           
                         
                       
                       ] 
                     
                     := 
                     
                       [ 
                       
                         
                           
                             
                               ℜ 
                               ⁡ 
                               
                                 [ 
                                 
                                   x 
                                   2 
                                 
                                 ] 
                               
                             
                           
                         
                         
                           
                             
                               ?? 
                               ⁡ 
                               
                                 [ 
                                 
                                   x 
                                   2 
                                 
                                 ] 
                               
                             
                           
                         
                         
                           
                             
                               ℜ 
                               ⁡ 
                               
                                 [ 
                                 
                                   x 
                                   3 
                                 
                                 ] 
                               
                             
                           
                         
                         
                           
                             
                               ?? 
                               ⁡ 
                               
                                 [ 
                                 
                                   x 
                                   3 
                                 
                                 ] 
                               
                             
                           
                         
                         
                           
                             
                               ℜ 
                               ⁡ 
                               
                                 [ 
                                 
                                   x 
                                   4 
                                 
                                 ] 
                               
                             
                           
                         
                         
                           
                             
                               ?? 
                               ⁡ 
                               
                                 [ 
                                 
                                   x 
                                   4 
                                 
                                 ] 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             β 
                             2 
                           
                           = 
                           
                             α 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           
                             β 
                             0 
                           
                           = 
                           
                             
                               
                                 
                                   r 
                                   1 
                                 
                                 ⁢ 
                                 
                                   h 
                                   2 
                                   * 
                                 
                               
                               - 
                               
                                 
                                   r 
                                   2 
                                   * 
                                 
                                 ⁢ 
                                 
                                   h 
                                   1 
                                 
                               
                             
                             
                               β 
                               2 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             β 
                             3 
                           
                           = 
                           
                             
                               
                                 
                                   h 
                                   2 
                                   * 
                                 
                                 ⁢ 
                                 
                                   h 
                                   3 
                                 
                               
                               - 
                               
                                 
                                   h 
                                   1 
                                 
                                 ⁢ 
                                 
                                   h 
                                   4 
                                   * 
                                 
                               
                             
                             
                               β 
                               2 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             β 
                             4 
                           
                           = 
                           
                             
                               
                                 
                                   h 
                                   2 
                                   * 
                                 
                                 ⁢ 
                                 
                                   h 
                                   4 
                                 
                               
                               + 
                               
                                 
                                   h 
                                   1 
                                 
                                 ⁢ 
                                 
                                   h 
                                   3 
                                   * 
                                 
                               
                             
                             
                               β 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     Further, the ML metric of Equation 5 can be represented by Equation 6.
 
 M=∥F   A (ρ A   −X   A )∥ 2   +∥F   B (ρ B   −X   B )∥ 2   (Equation 6)
 
     In this case, the conditions ρ A =D A   T (D A D A   T ) −1 α 0  and ρ B =D B   T (D B D B   T ) −1 β 0  are satisfied. 
     Further, the two matrices F A  and F B  are obtained by applying the normalized Cholesky decomposition to the two matrices G A =(D A ) T D A  and G B =(D B ) T D B . Further, Equation 6 can be represented as Equation 7. 
     
       
         
           
             
               
                 
                   
                     
                       
                         M 
                         = 
                           
                         ⁢ 
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               2 
                             
                             ⁢ 
                             
                               
                                  
                                 
                                   
                                     ∑ 
                                     
                                       j 
                                       = 
                                       i 
                                     
                                     6 
                                   
                                   ⁢ 
                                   
                                     
                                       F 
                                       ij 
                                       A 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           ρ 
                                           j 
                                           A 
                                         
                                         - 
                                         
                                           x 
                                           j 
                                           A 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                  
                               
                               2 
                             
                           
                           + 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               2 
                             
                             ⁢ 
                             
                               
                                  
                                 
                                   
                                     ∑ 
                                     
                                       j 
                                       = 
                                       i 
                                     
                                     6 
                                   
                                   ⁢ 
                                   
                                     
                                       F 
                                       ij 
                                       B 
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           ρ 
                                           j 
                                           B 
                                         
                                         - 
                                         
                                           x 
                                           j 
                                           B 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                  
                               
                               2 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                                
                               
                                 
                                   ∑ 
                                   
                                     j 
                                     = 
                                     1 
                                   
                                   6 
                                 
                                 ⁢ 
                                 
                                   
                                     F 
                                     
                                       1 
                                       ⁢ 
                                       j 
                                     
                                     A 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         ρ 
                                         j 
                                         A 
                                       
                                       - 
                                       
                                         x 
                                         j 
                                         A 
                                       
                                     
                                     ) 
                                   
                                 
                               
                                
                             
                             2 
                           
                           + 
                           
                             
                                
                               
                                 
                                   ∑ 
                                   
                                     j 
                                     = 
                                     2 
                                   
                                   6 
                                 
                                 ⁢ 
                                 
                                   
                                     F 
                                     
                                       2 
                                       ⁢ 
                                       j 
                                     
                                     A 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         ρ 
                                         j 
                                         A 
                                       
                                       - 
                                       
                                         x 
                                         j 
                                         A 
                                       
                                     
                                     ) 
                                   
                                 
                               
                                
                             
                             2 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             
                                
                               
                                 
                                   ∑ 
                                   
                                     j 
                                     = 
                                     1 
                                   
                                   6 
                                 
                                 ⁢ 
                                 
                                   
                                     F 
                                     
                                       1 
                                       ⁢ 
                                       j 
                                     
                                     B 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         ρ 
                                         j 
                                         B 
                                       
                                       - 
                                       
                                         x 
                                         j 
                                         B 
                                       
                                     
                                     ) 
                                   
                                 
                               
                                
                             
                             2 
                           
                           + 
                           
                             
                                
                               
                                 
                                   ∑ 
                                   
                                     j 
                                     = 
                                     2 
                                   
                                   6 
                                 
                                 ⁢ 
                                 
                                   
                                     F 
                                     
                                       2 
                                       ⁢ 
                                       j 
                                     
                                     B 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         ρ 
                                         j 
                                         B 
                                       
                                       - 
                                       
                                         x 
                                         j 
                                         B 
                                       
                                     
                                     ) 
                                   
                                 
                               
                                
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     7 
                   
                   ) 
                 
               
             
           
         
       
     
     Here, ρ j   A  indicates the j-th element of ρ A  and F ij   A  indicates the (i, j)-th element of F A . In this way, ρ j   B  and F ij   B  are defined. 
     In order to apply sphere decoding, the given ML metric M is restricted to be smaller than a constant C that is defined in advance. In this case, two necessary conditions are represented by Equations 8 and 9, respectively. 
     
       
         
           
             
               
                 
                   
                     
                        
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             2 
                           
                           6 
                         
                         ⁢ 
                         
                           
                             F 
                             
                               2 
                               ⁢ 
                               j 
                             
                             A 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 ρ 
                                 j 
                                 A 
                               
                               - 
                               
                                 x 
                                 j 
                                 A 
                               
                             
                             ) 
                           
                         
                       
                        
                     
                     2 
                   
                   ≤ 
                   C 
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     8 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                        
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             2 
                           
                           6 
                         
                         ⁢ 
                         
                           
                             F 
                             
                               2 
                               ⁢ 
                               j 
                             
                             B 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 ρ 
                                 j 
                                 B 
                               
                               - 
                               
                                 x 
                                 j 
                                 B 
                               
                             
                             ) 
                           
                         
                       
                        
                     
                     2 
                   
                   ≤ 
                   C 
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     9 
                   
                   ) 
                 
               
             
           
         
       
     
     In the exemplary embodiment of the present invention, two dependent sphere decoding numerical formulas exist. In the exemplary embodiment of the present invention, it is assumed that x 3  and x 4  are detected and then x1 and x2 are detected. Further, it is assumed that each symbol has a 4-PSK constellation. That is, each symbol may have any one of four values of {1, −1, j, −j}. 
     Thus, each of x A   j  and x B   j  has a value of either +1 or −1. A sphere decoding scheme that is represented by Equation 8 will now be described. 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           2 
                         
                         6 
                       
                       ⁢ 
                       
                         
                           F 
                           
                             2 
                             ⁢ 
                             j 
                           
                           A 
                         
                         ⁢ 
                         
                           ρ 
                           j 
                           A 
                         
                       
                     
                     - 
                     
                       C 
                     
                   
                   ≤ 
                   
                     
                       ∑ 
                       
                         j 
                         = 
                         2 
                       
                       6 
                     
                     ⁢ 
                     
                       
                         F 
                         
                           2 
                           ⁢ 
                           j 
                         
                         A 
                       
                       ⁢ 
                       
                         x 
                         j 
                         A 
                       
                     
                   
                   ≤ 
                   
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           2 
                         
                         6 
                       
                       ⁢ 
                       
                         
                           F 
                           
                             2 
                             ⁢ 
                             j 
                           
                           A 
                         
                         ⁢ 
                         
                           ρ 
                           j 
                           A 
                         
                       
                     
                     + 
                     
                       C 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     10 
                   
                   ) 
                 
               
             
           
         
       
     
     If solving Equation 1 by applying a method that has been proposed by Z. Yang, x A   6  can be estimated as represented by Equation 11.
 
x 6   A ε{−1,1}∩[LB 6   A ,UB 6   A ]  (Equation 11)
 
     However, when F A   2,6  is a positive number, 
     
       
         
           
             
               L 
               ⁢ 
               
                   
               
               ⁢ 
               
                 B 
                 6 
                 A 
               
             
             = 
             
               
                 
                   
                     Q 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     A 
                   
                   - 
                   
                     C 
                   
                   - 
                   
                     2 
                     ⁢ 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           2 
                         
                         5 
                       
                       ⁢ 
                       
                          
                         
                           F 
                           
                             2 
                             ⁢ 
                             j 
                           
                           A 
                         
                          
                       
                     
                   
                 
                 
                    
                   
                     F 
                     
                       2 
                       , 
                       6 
                     
                     A 
                   
                    
                 
               
               - 
               1 
             
           
         
       
       
         
           
             
               U 
               ⁢ 
               
                   
               
               ⁢ 
               
                 B 
                 6 
                 A 
               
             
             = 
             
               
                 
                   
                     Q 
                     A 
                   
                   + 
                   
                     C 
                   
                 
                 
                    
                   
                     F 
                     
                       2 
                       , 
                       6 
                     
                     A 
                   
                    
                 
               
               - 
               1. 
             
           
         
       
     
     Further, when F A   2,6  is a negative number, 
     
       
         
           
             
               L 
               ⁢ 
               
                   
               
               ⁢ 
               
                 B 
                 6 
                 A 
               
             
             = 
             
               
                 - 
                 
                   
                     
                       Q 
                       A 
                     
                     + 
                     
                       C 
                     
                   
                   
                      
                     
                       F 
                       
                         2 
                         , 
                         6 
                       
                       A 
                     
                      
                   
                 
               
               + 
               1 
             
           
         
       
       
         
           
             
               U 
               ⁢ 
               
                   
               
               ⁢ 
               
                 B 
                 6 
                 A 
               
             
             = 
             
               
                 - 
                 
                   
                     
                       Q 
                       A 
                     
                     - 
                     
                       C 
                     
                     - 
                     
                       2 
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             2 
                           
                           5 
                         
                         ⁢ 
                         
                            
                           
                             F 
                             
                               2 
                               ⁢ 
                               j 
                             
                             A 
                           
                            
                         
                       
                     
                   
                   
                      
                     
                       F 
                       
                         2 
                         , 
                         6 
                       
                       A 
                     
                      
                   
                 
               
               + 
               1. 
             
           
         
       
     
     In the same method, if solving Equation 9, the condition for x B   6  can be obtained as follows.
 
 x   6   A   =x   6   B   ℑ|x   4 |
 
     Thus, the condition for an imaginary part of x 4  is given by Equation 12.
 
ℑ[x 4 ]ε{−1,1}∩[max(LB 6   A ,LB 6   B ),min(UB 6   A ,UB 6   B )]  (Equation 12)
 
     If repeating the above-described method, the condition for x B   6  can be obtained as follows.
 
X 5   A ε{−1,1}∩[LB 5   A ,UB 5   A ]
 
     When F A   2,5  is a positive number, 
     
       
         
           
             
               L 
               ⁢ 
               
                   
               
               ⁢ 
               
                 B 
                 5 
                 A 
               
             
             = 
             
               
                 
                   
                     Q 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     A 
                   
                   - 
                   
                     C 
                   
                   - 
                   
                     2 
                     ⁢ 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           2 
                         
                         4 
                       
                       ⁢ 
                       
                          
                         
                           F 
                           
                             2 
                             ⁢ 
                             j 
                           
                           A 
                         
                          
                       
                     
                   
                   - 
                   
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         u 
                         _ 
                       
                       6 
                       A 
                     
                   
                 
                 
                    
                   
                     F 
                     
                       2 
                       , 
                       5 
                     
                     A 
                   
                    
                 
               
               - 
               1 
             
           
         
       
       
         
           
             
               U 
               ⁢ 
               
                   
               
               ⁢ 
               
                 B 
                 5 
                 A 
               
             
             = 
             
               
                 
                   
                     Q 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     A 
                   
                   + 
                   
                     C 
                   
                   - 
                   
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         u 
                         _ 
                       
                       6 
                       A 
                     
                   
                 
                 
                    
                   
                     F 
                     
                       2 
                       , 
                       5 
                     
                     A 
                   
                    
                 
               
               - 
               1 
             
           
         
       
     
     When F A   2,5  is a negative number, 
     
       
         
           
             
               L 
               ⁢ 
               
                   
               
               ⁢ 
               
                 B 
                 5 
                 A 
               
             
             = 
             
               
                 - 
                 
                   
                     
                       Q 
                       A 
                     
                     + 
                     
                       C 
                     
                     - 
                     
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           u 
                           _ 
                         
                         6 
                         A 
                       
                     
                   
                   
                      
                     
                       F 
                       
                         2 
                         , 
                         5 
                       
                       A 
                     
                      
                   
                 
               
               + 
               1 
             
           
         
       
       
         
           
             
               U 
               ⁢ 
               
                   
               
               ⁢ 
               
                 B 
                 5 
                 A 
               
             
             = 
             
               
                 - 
                 
                   
                     
                       Q 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       A 
                     
                     - 
                     
                       C 
                     
                     - 
                     
                       2 
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             2 
                           
                           4 
                         
                         ⁢ 
                         
                            
                           
                             F 
                             
                               2 
                               ⁢ 
                               j 
                             
                             A 
                           
                            
                         
                       
                     
                     - 
                     
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           u 
                           _ 
                         
                         6 
                         A 
                       
                     
                   
                   
                      
                     
                       F 
                       
                         2 
                         , 
                         5 
                       
                       A 
                     
                      
                   
                 
               
               + 
               1. 
             
           
         
       
     
     In the same method, if calculating Equation 9, the condition for x B   5  can be obtained, and the condition for a real part of x 4  is given by Equation 13, similar to Equation 12.
 
           [x 4 ]ε{−1,1}∩[max(LB 5   A ,LB 5   B ),min(UB 5   A ,UB 5   B )]  (Equation 13)

     In this way, the conditions for imaginary and real parts of x 3  can be obtained as represented by Equations 14 and 15.
 
ℑ[x 3 ]ε{−1,1}∩[max(LB 4   A ,LB 4   B ),min(UB 4   A ,UB 4   B )]  (Equation 14)
 
           [x 3 ]ε{−1,1}∩[max(LB 3   A ,LB 3   B ),min(UB 3   A ,UB 3   B )]  (Equation 15)

     In order to display estimated values for imaginary and real parts of symbols that are calculated by using Equations 12 to 15, a bar - is displayed above each symbol. Finally, the estimated values for x 3  and x 4  are represented by Equation 16.
 
    x   3   =             [ x   3 ] + j ·  ℑ[ x   3 ] 
 
    x   4   =           [ x   4 ] + j ·  ℑ[ x   4 ]   (Equation 16)

     As described above, after detecting x 3  and x 4 , x 1  and x 2  are detected as follows. In the exemplary embodiment of the present invention, different from a general method, an Alamouti decoding scheme instead of the sphere decoding scheme is applied to two remaining symbols. 
     First, if substituting the estimated values for x 3  and x 4  given by Equation 16 for Equation 2, Equation 17 is obtained.
 
 {tilde over (r)}   1   =h   1   x   1   +h   2   x   2   +n   1  
 
 {tilde over (r)} 2 =−h   1   x   2   *+h   2   x   1   *+n   2  
 
 {tilde over (r)}   1   =r   1   −h   3     x     3   −h   4     x     4  
 
 {tilde over (r)}   2   =r   2   +h   3     x     4   *−h   4     x     4 .  (Equation 17)
 
     Equation 17 shows a received signal in the case where an Alamouti decoding scheme is applied once. Thus, extremely simple detection can be performed, and the results that are obtained by detecting x 1  and x 2  are as follows. 
     
       
         
           
             
               
                 
                   
                     
                       
                         x 
                         _ 
                       
                       ⁢ 
                       1 
                     
                     = 
                     
                       sign 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             ^ 
                           
                           1 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         x 
                         _ 
                       
                       2 
                     
                     = 
                     
                       sign 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             ^ 
                           
                           2 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         x 
                         ^ 
                       
                       1 
                     
                     = 
                     
                       
                         
                           
                             
                               r 
                               ~ 
                             
                             1 
                           
                           ⁢ 
                           
                             h 
                             1 
                             * 
                           
                         
                         + 
                         
                           
                             
                               r 
                               ~ 
                             
                             2 
                             * 
                           
                           ⁢ 
                           
                             h 
                             2 
                           
                         
                       
                       
                         
                           
                              
                             
                               h 
                               1 
                             
                              
                           
                           2 
                         
                         + 
                         
                           
                              
                             
                               h 
                               2 
                             
                              
                           
                           2 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         x 
                         ^ 
                       
                       2 
                     
                     = 
                     
                       
                         
                           
                             
                               
                                 r 
                                 ~ 
                               
                               1 
                             
                             ⁢ 
                             
                               h 
                               2 
                               * 
                             
                           
                           + 
                           
                             
                               
                                 r 
                                 ~ 
                               
                               2 
                               * 
                             
                             ⁢ 
                             
                               h 
                               1 
                             
                           
                         
                         
                           
                             
                                
                               
                                 h 
                                 1 
                               
                                
                             
                             2 
                           
                           + 
                           
                             
                                
                               
                                 h 
                                 2 
                               
                                
                             
                             2 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     18 
                   
                   ) 
                 
               
             
           
         
       
     
     In Equation 18, the sign means a signum function. 
     From Equation 18, the estimated values for real and imaginary parts of x 1  and x 2  can be simply calculated as follows.
 
             | x   1 | =         [   x     1 ]
 
  ℑ[x 1 ] =ℑ[  x   1 ]
 
           [x 2 ] =         [  x   2 ]
 
  ℑ[x 1 ] =ℑ[  x   1 ]  (Equation 19)

     Finally, estimated values for real and imaginary parts of X A  and X B  are obtained. In the description below, in order to indicate the estimated values, a bar - is additionally displayed. By using the estimated values that are calculated above, C′ is calculated as represented by Equation 20.
 
 C′=∥α   0   −D   A     X     A ∥ 2 +∥β 0   −D   B     X     B ∥ 2   (Equation 20)
 
     If the condition C′&gt;C is satisfied, it means that there is no constellation point in a given radius C, like the existing sphere coding scheme. Thus, the C value is increased by a predetermined method. However, if the condition C′&lt;C is satisfied, a constellation point exists. 
     Thus, first, Equation 7 is restricted to be smaller than or equal to the constant C. As a result, the two necessary conditions are derived. Then, the constant C is converted into the constant C′. The above-described processes are repeated until the constellation point is not discovered. In this case, the finally discovered constellation point becomes an ML estimated value. 
     The description has been given on the assumption that the number of receiving antennas is one. The description will be made on the assumption that n r  receiving antennas exist. First, the ML metric, which has been represented by Equation 5, is represented by Equation 21. 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         M 
                         = 
                         
                           
                             
                                
                               
                                 
                                   
                                     α 
                                     → 
                                   
                                   0 
                                 
                                 - 
                                 
                                   
                                     
                                       D 
                                       → 
                                     
                                     A 
                                   
                                   ⁢ 
                                   
                                     X 
                                     A 
                                   
                                 
                               
                                
                             
                             2 
                           
                           + 
                           
                             
                                
                               
                                 
                                   
                                     β 
                                     → 
                                   
                                   0 
                                 
                                 - 
                                 
                                   
                                     
                                       D 
                                       → 
                                     
                                     B 
                                   
                                   ⁢ 
                                   
                                     X 
                                     B 
                                   
                                 
                               
                                
                             
                             2 
                           
                         
                       
                       ⁢ 
                       
                         
 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       where 
                       ⁢ 
                       
                         
 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           
                             α 
                             → 
                           
                           0 
                         
                         = 
                         
                           
                             [ 
                             
                               
                                 ℜ 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     α 
                                     0 
                                     
                                       ( 
                                       1 
                                       ) 
                                     
                                   
                                   ] 
                                 
                               
                               , 
                               
                                 ?? 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     α 
                                     0 
                                     
                                       ( 
                                       1 
                                       ) 
                                     
                                   
                                   ] 
                                 
                               
                               , 
                               … 
                               ⁢ 
                               
                                   
                               
                               , 
                               
                                 ℜ 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     α 
                                     0 
                                     
                                       n 
                                       r 
                                     
                                   
                                   ] 
                                 
                               
                               , 
                               
                                 ?? 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     α 
                                     0 
                                     
                                       n 
                                       r 
                                     
                                   
                                   ] 
                                 
                               
                             
                             ] 
                           
                           T 
                         
                       
                       ⁢ 
                       
                         
 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           
                             β 
                             → 
                           
                           0 
                         
                         = 
                         
                           
                             [ 
                             
                               
                                 ℜ 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     β 
                                     0 
                                     
                                       ( 
                                       1 
                                       ) 
                                     
                                   
                                   ] 
                                 
                               
                               , 
                               
                                 ?? 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     β 
                                     0 
                                     
                                       ( 
                                       1 
                                       ) 
                                     
                                   
                                   ] 
                                 
                               
                               , 
                               … 
                               ⁢ 
                               
                                   
                               
                               , 
                               
                                 ℜ 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     β 
                                     0 
                                     
                                       n 
                                       r 
                                     
                                   
                                   ] 
                                 
                               
                               , 
                               
                                 ?? 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     β 
                                     0 
                                     
                                       n 
                                       r 
                                     
                                   
                                   ] 
                                 
                               
                             
                             ] 
                           
                           T 
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         
                           D 
                           → 
                         
                         A 
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   ℜ 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       α 
                                       1 
                                       
                                         ( 
                                         1 
                                         ) 
                                       
                                     
                                     ] 
                                   
                                 
                                 - 
                                 
                                   
                                     ?? 
                                     ⁡ 
                                     
                                       [ 
                                       
                                         α 
                                         1 
                                         
                                           ( 
                                           1 
                                           ) 
                                         
                                       
                                       ] 
                                     
                                   
                                   ⁢ 
                                   
                                     ℜ 
                                     ⁡ 
                                     
                                       [ 
                                       
                                         α 
                                         3 
                                         
                                           ( 
                                           1 
                                           ) 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 - 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     ?? 
                                     ⁡ 
                                     
                                       [ 
                                       
                                         α 
                                         3 
                                         
                                           ( 
                                           1 
                                           ) 
                                         
                                       
                                       ] 
                                     
                                   
                                   ⁢ 
                                   
                                     ℜ 
                                     ⁡ 
                                     
                                       [ 
                                       
                                         α 
                                         4 
                                         
                                           ( 
                                           1 
                                           ) 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 - 
                                 
                                   ?? 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       α 
                                       4 
                                       
                                         ( 
                                         1 
                                         ) 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ?? 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       α 
                                       1 
                                       
                                         ( 
                                         1 
                                         ) 
                                       
                                     
                                     ] 
                                   
                                 
                                 ⁢ 
                                 
                                   ℜ 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       α 
                                       1 
                                       
                                         ( 
                                         1 
                                         ) 
                                       
                                     
                                     ] 
                                   
                                 
                                 ⁢ 
                                 
                                   ?? 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       α 
                                       3 
                                       
                                         ( 
                                         1 
                                         ) 
                                       
                                     
                                     ] 
                                   
                                 
                                 ⁢ 
                                 
                                   ℜ 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       α 
                                       3 
                                       
                                         ( 
                                         1 
                                         ) 
                                       
                                     
                                     ] 
                                   
                                 
                                 ⁢ 
                                 
                                   ?? 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       α 
                                       4 
                                       
                                         ( 
                                         1 
                                         ) 
                                       
                                     
                                     ] 
                                   
                                 
                                 ⁢ 
                                 
                                   ℜ 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       α 
                                       4 
                                       
                                         ( 
                                         1 
                                         ) 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       
                                         ℜ 
                                         ⁡ 
                                         
                                           [ 
                                           
                                             α 
                                             1 
                                             
                                               
                                                 n 
                                                 r 
                                               
                                               ⁢ 
                                               
                                                   
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                       - 
                                       
                                         
                                           ?? 
                                           ⁡ 
                                           
                                             [ 
                                             
                                               α 
                                               1 
                                               
                                                 
                                                   ( 
                                                   
                                                     n 
                                                     r 
                                                   
                                                   ) 
                                                 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                               
                                             
                                             ] 
                                           
                                         
                                         ⁢ 
                                         
                                           ℜ 
                                           ⁡ 
                                           
                                             [ 
                                             
                                               α 
                                               3 
                                               
                                                 
                                                   ( 
                                                   
                                                     n 
                                                     r 
                                                   
                                                   ) 
                                                 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                               
                                             
                                             ] 
                                           
                                         
                                       
                                       - 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         
                                           ?? 
                                           ⁡ 
                                           
                                             [ 
                                             
                                               α 
                                               3 
                                               
                                                 
                                                   ( 
                                                   
                                                     n 
                                                     r 
                                                   
                                                   ) 
                                                 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                               
                                             
                                             ] 
                                           
                                         
                                         ⁢ 
                                         
                                           ℜ 
                                           ⁡ 
                                           
                                             [ 
                                             
                                               α 
                                               4 
                                               
                                                 
                                                   ( 
                                                   
                                                     n 
                                                     r 
                                                   
                                                   ) 
                                                 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                               
                                             
                                             ] 
                                           
                                         
                                       
                                       - 
                                       
                                         ?? 
                                         ⁡ 
                                         
                                           [ 
                                           
                                             α 
                                             4 
                                             
                                               
                                                 ( 
                                                 
                                                   n 
                                                   r 
                                                 
                                                 ) 
                                               
                                               ⁢ 
                                               
                                                   
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         ?? 
                                         ⁡ 
                                         
                                           [ 
                                           
                                             α 
                                             1 
                                             
                                               
                                                 n 
                                                 r 
                                               
                                               ⁢ 
                                               
                                                   
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                       ⁢ 
                                       
                                         ℜ 
                                         ⁡ 
                                         
                                           [ 
                                           
                                             α 
                                             1 
                                             
                                               
                                                 ( 
                                                 
                                                   n 
                                                   r 
                                                 
                                                 ) 
                                               
                                               ⁢ 
                                               
                                                   
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                       ⁢ 
                                       
                                         ?? 
                                         ⁡ 
                                         
                                           [ 
                                           
                                             α 
                                             3 
                                             
                                               
                                                 ( 
                                                 
                                                   n 
                                                   r 
                                                 
                                                 ) 
                                               
                                               ⁢ 
                                               
                                                   
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                       ⁢ 
                                       
                                         ℜ 
                                         ⁡ 
                                         
                                           [ 
                                           
                                             α 
                                             3 
                                             
                                               
                                                 ( 
                                                 
                                                   n 
                                                   r 
                                                 
                                                 ) 
                                               
                                               ⁢ 
                                               
                                                   
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                       ⁢ 
                                       
                                         ?? 
                                         ⁡ 
                                         
                                           [ 
                                           
                                             α 
                                             4 
                                             
                                               
                                                 ( 
                                                 
                                                   n 
                                                   r 
                                                 
                                                 ) 
                                               
                                               ⁢ 
                                               
                                                   
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                       ⁢ 
                                       
                                         ℜ 
                                         ⁡ 
                                         
                                           [ 
                                           
                                             α 
                                             4 
                                             
                                               
                                                 ( 
                                                 
                                                   n 
                                                   r 
                                                 
                                                 ) 
                                               
                                               ⁢ 
                                               
                                                   
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         
                           D 
                           → 
                         
                         B 
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   ℜ 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       β 
                                       1 
                                       
                                         ( 
                                         1 
                                         ) 
                                       
                                     
                                     ] 
                                   
                                 
                                 - 
                                 
                                   
                                     ?? 
                                     ⁡ 
                                     
                                       [ 
                                       
                                         β 
                                         1 
                                         
                                           ( 
                                           1 
                                           ) 
                                         
                                       
                                       ] 
                                     
                                   
                                   ⁢ 
                                   
                                     ℜ 
                                     ⁡ 
                                     
                                       [ 
                                       
                                         β 
                                         3 
                                         
                                           ( 
                                           1 
                                           ) 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 - 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     ?? 
                                     ⁡ 
                                     
                                       [ 
                                       
                                         β 
                                         3 
                                         
                                           ( 
                                           1 
                                           ) 
                                         
                                       
                                       ] 
                                     
                                   
                                   ⁢ 
                                   
                                     ℜ 
                                     ⁡ 
                                     
                                       [ 
                                       
                                         β 
                                         4 
                                         
                                           ( 
                                           1 
                                           ) 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 - 
                                 
                                   ?? 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       β 
                                       4 
                                       
                                         ( 
                                         1 
                                         ) 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ?? 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       β 
                                       1 
                                       
                                         ( 
                                         1 
                                         ) 
                                       
                                     
                                     ] 
                                   
                                 
                                 ⁢ 
                                 
                                   ℜ 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       β 
                                       1 
                                       
                                         ( 
                                         1 
                                         ) 
                                       
                                     
                                     ] 
                                   
                                 
                                 ⁢ 
                                 
                                   ?? 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       β 
                                       3 
                                       
                                         ( 
                                         1 
                                         ) 
                                       
                                     
                                     ] 
                                   
                                 
                                 ⁢ 
                                 
                                   ℜ 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       β 
                                       3 
                                       
                                         ( 
                                         1 
                                         ) 
                                       
                                     
                                     ] 
                                   
                                 
                                 ⁢ 
                                 
                                   ?? 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       β 
                                       4 
                                       
                                         ( 
                                         1 
                                         ) 
                                       
                                     
                                     ] 
                                   
                                 
                                 ⁢ 
                                 
                                   ℜ 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       β 
                                       4 
                                       
                                         ( 
                                         1 
                                         ) 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       
                                         ℜ 
                                         ⁡ 
                                         
                                           [ 
                                           
                                             β 
                                             1 
                                             
                                               
                                                 n 
                                                 r 
                                               
                                               ⁢ 
                                               
                                                   
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                       - 
                                       
                                         
                                           ?? 
                                           ⁡ 
                                           
                                             [ 
                                             
                                               β 
                                               1 
                                               
                                                 
                                                   ( 
                                                   
                                                     n 
                                                     r 
                                                   
                                                   ) 
                                                 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                               
                                             
                                             ] 
                                           
                                         
                                         ⁢ 
                                         
                                           ℜ 
                                           ⁡ 
                                           
                                             [ 
                                             
                                               β 
                                               3 
                                               
                                                 
                                                   ( 
                                                   
                                                     n 
                                                     r 
                                                   
                                                   ) 
                                                 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                               
                                             
                                             ] 
                                           
                                         
                                       
                                       - 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         
                                           ?? 
                                           ⁡ 
                                           
                                             [ 
                                             
                                               β 
                                               3 
                                               
                                                 
                                                   ( 
                                                   
                                                     n 
                                                     r 
                                                   
                                                   ) 
                                                 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                               
                                             
                                             ] 
                                           
                                         
                                         ⁢ 
                                         
                                           ℜ 
                                           ⁡ 
                                           
                                             [ 
                                             
                                               β 
                                               4 
                                               
                                                 
                                                   ( 
                                                   
                                                     n 
                                                     r 
                                                   
                                                   ) 
                                                 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                               
                                             
                                             ] 
                                           
                                         
                                       
                                       - 
                                       
                                         ?? 
                                         ⁡ 
                                         
                                           [ 
                                           
                                             β 
                                             4 
                                             
                                               
                                                 ( 
                                                 
                                                   n 
                                                   r 
                                                 
                                                 ) 
                                               
                                               ⁢ 
                                               
                                                   
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         ?? 
                                         ⁡ 
                                         
                                           [ 
                                           
                                             β 
                                             1 
                                             
                                               
                                                 n 
                                                 r 
                                               
                                               ⁢ 
                                               
                                                   
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                       ⁢ 
                                       
                                         ℜ 
                                         ⁡ 
                                         
                                           [ 
                                           
                                             β 
                                             1 
                                             
                                               
                                                 ( 
                                                 
                                                   n 
                                                   r 
                                                 
                                                 ) 
                                               
                                               ⁢ 
                                               
                                                   
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                       ⁢ 
                                       
                                         ?? 
                                         ⁡ 
                                         
                                           [ 
                                           
                                             β 
                                             3 
                                             
                                               
                                                 ( 
                                                 
                                                   n 
                                                   r 
                                                 
                                                 ) 
                                               
                                               ⁢ 
                                               
                                                   
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                       ⁢ 
                                       
                                         ℜ 
                                         ⁡ 
                                         
                                           [ 
                                           
                                             β 
                                             3 
                                             
                                               
                                                 ( 
                                                 
                                                   n 
                                                   r 
                                                 
                                                 ) 
                                               
                                               ⁢ 
                                               
                                                   
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                       ⁢ 
                                       
                                         ?? 
                                         ⁡ 
                                         
                                           [ 
                                           
                                             β 
                                             4 
                                             
                                               
                                                 ( 
                                                 
                                                   n 
                                                   r 
                                                 
                                                 ) 
                                               
                                               ⁢ 
                                               
                                                   
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                       ⁢ 
                                       
                                         ℜ 
                                         ⁡ 
                                         
                                           [ 
                                           
                                             β 
                                             4 
                                             
                                               
                                                 ( 
                                                 
                                                   n 
                                                   r 
                                                 
                                                 ) 
                                               
                                               ⁢ 
                                               
                                                   
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         α 
                         1 
                         
                           ( 
                           k 
                           ) 
                         
                       
                       = 
                       
                         
                           
                             
                                
                               
                                 h 
                                 
                                   k 
                                   , 
                                   1 
                                 
                               
                                
                             
                             2 
                           
                           + 
                           
                             
                                
                               
                                 h 
                                 
                                   k 
                                   , 
                                   2 
                                 
                               
                                
                             
                             2 
                           
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         α 
                         0 
                         
                           ( 
                           k 
                           ) 
                         
                       
                       = 
                       
                         
                           
                             
                               r 
                               
                                 k 
                                 , 
                                 1 
                               
                             
                             ⁢ 
                             
                               h 
                               
                                 k 
                                 , 
                                 1 
                               
                               * 
                             
                           
                           + 
                           
                             
                               r 
                               
                                 k 
                                 , 
                                 2 
                               
                               * 
                             
                             ⁢ 
                             
                               h 
                               
                                 k 
                                 , 
                                 2 
                               
                             
                           
                         
                         
                           α 
                           1 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         α 
                         3 
                         
                           ( 
                           k 
                           ) 
                         
                       
                       = 
                       
                         
                           
                             
                               h 
                               
                                 k 
                                 , 
                                 1 
                               
                               * 
                             
                             ⁢ 
                             
                               h 
                               
                                 k 
                                 , 
                                 3 
                               
                             
                           
                           + 
                           
                             
                               h 
                               
                                 k 
                                 , 
                                 2 
                               
                             
                             ⁢ 
                             
                               h 
                               
                                 k 
                                 , 
                                 4 
                               
                               * 
                             
                           
                         
                         
                           α 
                           1 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         α 
                         4 
                         
                           ( 
                           k 
                           ) 
                         
                       
                       = 
                       
                         
                           
                             
                               h 
                               
                                 k 
                                 , 
                                 1 
                               
                               * 
                             
                             ⁢ 
                             
                               h 
                               
                                 k 
                                 , 
                                 4 
                               
                             
                           
                           + 
                           
                             
                               h 
                               
                                 k 
                                 , 
                                 1 
                               
                             
                             ⁢ 
                             
                               h 
                               
                                 k 
                                 , 
                                 3 
                               
                               * 
                             
                           
                         
                         
                           α 
                           1 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         β 
                         2 
                         
                           ( 
                           k 
                           ) 
                         
                       
                       = 
                       
                         α 
                         1 
                         
                           ( 
                           k 
                           ) 
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         β 
                         0 
                         k 
                       
                       = 
                       
                         
                           
                             
                               r 
                               
                                 k 
                                 , 
                                 1 
                               
                             
                             ⁢ 
                             
                               h 
                               
                                 k 
                                 , 
                                 2 
                               
                               * 
                             
                           
                           - 
                           
                             
                               r 
                               
                                 k 
                                 , 
                                 2 
                               
                               * 
                             
                             ⁢ 
                             
                               h 
                               
                                 k 
                                 , 
                                 1 
                               
                             
                           
                         
                         
                           β 
                           2 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         β 
                         3 
                         
                           ( 
                           k 
                           ) 
                         
                       
                       = 
                       
                         
                           
                             
                               h 
                               
                                 k 
                                 , 
                                 2 
                               
                               * 
                             
                             ⁢ 
                             
                               h 
                               
                                 k 
                                 , 
                                 3 
                               
                             
                           
                           - 
                           
                             
                               h 
                               
                                 k 
                                 , 
                                 1 
                               
                             
                             ⁢ 
                             
                               h 
                               
                                 k 
                                 , 
                                 4 
                               
                               * 
                             
                           
                         
                         
                           β 
                           2 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         β 
                         4 
                         
                           ( 
                           k 
                           ) 
                         
                       
                       = 
                       
                         
                           
                             
                               h 
                               
                                 k 
                                 , 
                                 2 
                               
                               * 
                             
                             ⁢ 
                             
                               h 
                               
                                 k 
                                 , 
                                 4 
                               
                             
                           
                           + 
                           
                             
                               h 
                               
                                 k 
                                 , 
                                 1 
                               
                             
                             ⁢ 
                             
                               h 
                               
                                 k 
                                 , 
                                 3 
                               
                               * 
                             
                           
                         
                         
                           β 
                           2 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     21 
                   
                   ) 
                 
               
             
           
         
       
     
     and k indicates an index of a receiving antenna. 
     In this case, if a constant Z is defined as Z=min(2n r , 6), the ML metric is given as follows. 
     
       
         
           
             
               
                 
                   M 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         Z 
                       
                       ⁢ 
                       
                         
                            
                           
                             
                               ∑ 
                               
                                 j 
                                 = 
                                 1 
                               
                               6 
                             
                             ⁢ 
                             
                               
                                 F 
                                 ij 
                                 A 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     ρ 
                                     j 
                                     A 
                                   
                                   - 
                                   
                                     x 
                                     j 
                                     A 
                                   
                                 
                                 ) 
                               
                             
                           
                            
                         
                         2 
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         Z 
                       
                       ⁢ 
                       
                          
                         
                           
                             ∑ 
                             
                               j 
                               = 
                               1 
                             
                             6 
                           
                           ⁢ 
                           
                             
                               F 
                               ij 
                               B 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     ρ 
                                     j 
                                     B 
                                   
                                   - 
                                   
                                     x 
                                     j 
                                     B 
                                   
                                 
                                  
                               
                               2 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         
                           Z 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                            
                           
                             
                               ∑ 
                               
                                 j 
                                 = 
                                 1 
                               
                               6 
                             
                             ⁢ 
                             
                               
                                 F 
                                 ij 
                                 A 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     ρ 
                                     j 
                                     A 
                                   
                                   - 
                                   
                                     x 
                                     j 
                                     A 
                                   
                                 
                                 ) 
                               
                             
                           
                            
                         
                         2 
                       
                     
                     + 
                     
                       
                          
                         
                           
                             ∑ 
                             
                               j 
                               = 
                               Z 
                             
                             6 
                           
                           ⁢ 
                           
                             
                               F 
                               
                                 4 
                                 ⁢ 
                                 j 
                               
                               A 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   ρ 
                                   j 
                                   A 
                                 
                                 - 
                                 
                                   x 
                                   j 
                                   A 
                                 
                               
                               ) 
                             
                           
                         
                          
                       
                       2 
                     
                   
                 
               
             
             
               
                 
                     
                   ⁢ 
                   
                     
                       
                         
                            
                           
                             
                               ∑ 
                               
                                 j 
                                 = 
                                 Z 
                               
                               6 
                             
                             ⁢ 
                             
                               
                                 F 
                                 
                                   4 
                                   ⁢ 
                                   j 
                                 
                                 A 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     ρ 
                                     j 
                                     A 
                                   
                                   - 
                                   
                                     x 
                                     j 
                                     A 
                                   
                                 
                                 ) 
                               
                             
                           
                            
                         
                         2 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           
                             Z 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                              
                             
                               
                                 ∑ 
                                 
                                   j 
                                   = 
                                   1 
                                 
                                 6 
                               
                               ⁢ 
                               
                                 
                                   F 
                                   ij 
                                   B 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       ρ 
                                       j 
                                       B 
                                     
                                     - 
                                     
                                       x 
                                       j 
                                       B 
                                     
                                   
                                   ) 
                                 
                               
                             
                              
                           
                           2 
                         
                       
                     
                     + 
                     
                       
                          
                         
                           
                             ∑ 
                             
                               j 
                               = 
                               Z 
                             
                             6 
                           
                           ⁢ 
                           
                             
                               F 
                               
                                 4 
                                 ⁢ 
                                 j 
                               
                               B 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   ρ 
                                   j 
                                   B 
                                 
                                 - 
                                 
                                   x 
                                   j 
                                   B 
                                 
                               
                               ) 
                             
                           
                         
                          
                       
                       2 
                     
                   
                 
               
             
           
         
       
     
     If the given numerical formula is solved by the above-described method, XAj(j=3, 4, 5, and 6) is estimated as represented by Equation 22.
 
x j   A ε{−1,1}∩[LB j   A ,UB j   A ]
 
j=3,4,5,6  (Equation 22)
 
     If F A   Zj  is a positive number, 
     
       
         
           
             
               L 
               ⁢ 
               
                   
               
               ⁢ 
               
                 B 
                 j 
                 A 
               
             
             = 
             
               
                 
                   
                     
                       Q 
                       ⁢ 
                       
                           
                       
                     
                     A 
                   
                   - 
                   
                     C 
                   
                   - 
                   
                     2 
                     ⁢ 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           Z 
                         
                         
                           i 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                          
                         
                           F 
                           
                             Z 
                             , 
                             j 
                           
                           A 
                         
                          
                       
                     
                   
                   - 
                   
                     2 
                     ⁢ 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           
                             j 
                             + 
                             1 
                           
                         
                         6 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           u 
                           _ 
                         
                         k 
                         A 
                       
                     
                   
                 
                 
                   F 
                   
                     2 
                     , 
                     5 
                   
                   A 
                 
               
               - 
               1 
             
           
         
       
       
         
           
             
               U 
               ⁢ 
               
                   
               
               ⁢ 
               
                 B 
                 j 
                 A 
               
             
             = 
             
               
                 
                   
                     Q 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     A 
                   
                   + 
                   
                     C 
                   
                   - 
                   
                     2 
                     ⁢ 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           
                             j 
                             + 
                             1 
                           
                         
                         6 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           u 
                           _ 
                         
                         k 
                         A 
                       
                     
                   
                 
                 
                    
                   
                     F 
                     
                       Z 
                       , 
                       j 
                     
                     A 
                   
                    
                 
               
               - 
               1. 
             
           
         
       
     
     If F A   Zj  is a negative number, 
     
       
         
           
             
               LB 
               j 
               A 
             
             = 
             
               
                 - 
                 
                   
                     
                       Q 
                       A 
                     
                     + 
                     
                       C 
                     
                     - 
                     
                       2 
                       ⁢ 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             
                               j 
                               + 
                               1 
                             
                           
                           6 
                         
                         ⁢ 
                         
                           
                             u 
                             _ 
                           
                           k 
                           A 
                         
                       
                     
                   
                   
                      
                     
                       F 
                       
                         Z 
                         , 
                         j 
                       
                       A 
                     
                      
                   
                 
               
               + 
               1 
             
           
         
       
       
         
           
             
               UB 
               j 
               A 
             
             = 
             
               
                 - 
                 
                   
                     QA 
                     - 
                     
                       C 
                     
                     - 
                     
                       2 
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             Z 
                           
                           
                             i 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                            
                           
                             F 
                             
                               Z 
                               , 
                               j 
                             
                             A 
                           
                            
                         
                       
                     
                     - 
                     
                       2 
                       ⁢ 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             
                               j 
                               + 
                               1 
                             
                           
                           6 
                         
                         ⁢ 
                         
                           
                             u 
                             _ 
                           
                           k 
                           A 
                         
                       
                     
                   
                   
                      
                     
                       F 
                       
                         Z 
                         , 
                         j 
                       
                       A 
                     
                      
                   
                 
               
               + 
               1. 
             
           
         
       
     
     In the same method, X B   j  (j=3, 4, 5, and 6) is estimated, and estimate values for imaginary and real parts of x 3  and x 4  can be calculated by using Equations 12 to 15. The following processes are the same as those in the case in which one receiving antenna exists. 
     The description has been given to the case of 4-QAM. However, the result that is obtained above can be generalized to the case of arbitrary M-QAM. First, for better comprehension and ease of description, a case of 16-QAM is exemplified. In this case, as a value of each of x A   j  and x B   j , any one of {−3, −1, 1, 3} is selected. 
     First, if estimating x A   j , the following conversion is performed as proposed by Z. Yang, C. Liu, and J. He. If a value of F A   2,j  is a positive number, with respect to x A   j  corresponding to j, the following conversion is performed. 
     
       
         
           
             
               b 
               j 
               A 
             
             = 
             
               
                 3 
                 + 
                 
                   x 
                   j 
                   A 
                 
               
               2 
             
           
         
       
     
     If a value of F A   2,j  is a negative number, with respect to x A   j  corresponding to j, the following conversion is performed. 
     
       
         
           
             
               b 
               j 
               A 
             
             = 
             
               
                 3 
                 - 
                 
                   x 
                   j 
                   A 
                 
               
               2 
             
           
         
       
     
     If representing the value of x A   j  in a form of a binary number, it is as follows.
 
 b   j   A =2 b   1j   A   +b   0j   A  
 
b ij   A ε{0,1}
 
     If using the numerical formula, an estimated value for x A   j  can be calculated as follows. 
     
       
         
           
             
               b 
               
                 0 
                 , 
                 6 
               
               A 
             
             ∈ 
             
               
                 { 
                 
                   0 
                   , 
                   1 
                 
                 } 
               
               ⋂ 
               
                 [ 
                 
                   
                     
                       
                         Q 
                         A 
                       
                       - 
                       
                         C 
                       
                       - 
                       
                         6 
                         ⁢ 
                         
                           
                             ∑ 
                             
                               j 
                               = 
                               2 
                             
                             5 
                           
                           ⁢ 
                           
                              
                             
                               F 
                               
                                 2 
                                 ⁢ 
                                 j 
                               
                               A 
                             
                              
                           
                         
                       
                       - 
                       
                         4 
                         ⁢ 
                         
                            
                           
                             F 
                             
                               2 
                               , 
                               6 
                             
                             A 
                           
                            
                         
                       
                     
                     
                       2 
                       ⁢ 
                       
                          
                         
                           F 
                           
                             2 
                             , 
                             6 
                           
                           A 
                         
                          
                       
                     
                   
                   , 
                   
                     
                       
                         Q 
                         A 
                       
                       + 
                       
                         C 
                       
                     
                     
                       2 
                       ⁢ 
                       
                          
                         
                           F 
                           
                             2 
                             , 
                             6 
                           
                           A 
                         
                          
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     In this way, an estimated value for b B   j  can be calculated as follows. 
     
       
         
           
             
               b 
               
                 0 
                 , 
                 6 
               
               B 
             
             ∈ 
             
               
                 { 
                 
                   0 
                   , 
                   1 
                 
                 } 
               
               ⋂ 
               
                 [ 
                 
                   
                     
                       
                         Q 
                         B 
                       
                       - 
                       
                         C 
                       
                       - 
                       
                         6 
                         ⁢ 
                         
                           
                             ∑ 
                             
                               j 
                               = 
                               2 
                             
                             5 
                           
                           ⁢ 
                           
                              
                             
                               F 
                               
                                 2 
                                 ⁢ 
                                 j 
                               
                               B 
                             
                              
                           
                         
                       
                       - 
                       
                         4 
                         ⁢ 
                         
                            
                           
                             F 
                             
                               2 
                               , 
                               6 
                             
                             B 
                           
                            
                         
                       
                     
                     
                       2 
                       ⁢ 
                       
                          
                         
                           F 
                           
                             2 
                             , 
                             6 
                           
                           B 
                         
                          
                       
                     
                   
                   , 
                   
                     
                       
                         Q 
                         B 
                       
                       + 
                       
                         C 
                       
                     
                     
                       2 
                       ⁢ 
                       
                          
                         
                           F 
                           
                             2 
                             , 
                             6 
                           
                           B 
                         
                          
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     If F A   2,6  and F B   2,6  have the same code, X A   6  and X B   6  have the same value. As a result, b A   6  and b B   6  have the same value. Thus, the conditions b A   0,6 =b B   0,6  and b A   1,6 =b B   1,6  are satisfied. 
     However, if F A   2,6  and F B   2,6  have different codes, respectively, the conditions b A   0,6 =b B   0,6  and b A   1,6 =b B   1,6  are not satisfied. For example, when it is assumed that F A   2,6  is a positive number and F B   2,6  is a negative number, the following numerical formula is satisfied. 
     
       
         
           
             
               b 
               6 
               A 
             
             = 
             
               
                 
                   3 
                   + 
                   
                     x 
                     6 
                     A 
                   
                 
                 2 
               
               = 
               
                 
                   
                     3 
                     + 
                     
                       x 
                       6 
                       B 
                     
                   
                   2 
                 
                 = 
                 
                   
                     3 
                     - 
                     
                       
                         3 
                         - 
                         
                           x 
                           6 
                           B 
                         
                       
                       2 
                     
                   
                   = 
                   
                     3 
                     - 
                     
                       b 
                       6 
                       B 
                     
                   
                 
               
             
           
         
       
     
     Here, any one of {0, 1, 2, 3} is selected as a value of each of the constants b A   6  and b B   6 . Accordingly, b B   6  can be considered as a complement of 1 with respect to b A   6 . Thus, using b A   0,6  and b B   0,6 , b 0,6  can be estimated as represented by Equation 23. 
     
       
         
           
             
               
                 
                   
                     
                       b 
                       _ 
                     
                     0.6 
                   
                   := 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 b 
                                 
                                   0 
                                   , 
                                   6 
                                 
                                 A 
                               
                               ⋂ 
                               
                                 b 
                                 
                                   0 
                                   , 
                                   6 
                                 
                                 B 
                               
                             
                             , 
                             
                               
                                 … 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ifsign 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       F 
                                       
                                         2 
                                         , 
                                         6 
                                       
                                       A 
                                     
                                     ) 
                                   
                                 
                               
                               = 
                               
                                 sign 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     F 
                                     
                                       2 
                                       , 
                                       6 
                                     
                                     B 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 b 
                                 
                                   0 
                                   , 
                                   6 
                                 
                                 A 
                               
                               ⋂ 
                               
                                 b 
                                 
                                   0 
                                   , 
                                   6 
                                 
                                 B 
                               
                             
                             , 
                             
                               … 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               otherwise 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     23 
                   
                   ) 
                 
               
             
           
         
       
     
     In this case, a dot • indicates a complement of 1. 
     In the same method, b 0,6  can be estimated as follows. 
     
       
         
           
             
               
                 
                   
                     
                       b 
                       _ 
                     
                     
                       1 
                       , 
                       6 
                     
                   
                   := 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 b 
                                 
                                   1 
                                   , 
                                   6 
                                 
                                 A 
                               
                               ⋂ 
                               
                                 b 
                                 
                                   1 
                                   , 
                                   6 
                                 
                                 B 
                               
                             
                             , 
                             
                               
                                 … 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ifsign 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       F 
                                       
                                         2 
                                         , 
                                         6 
                                       
                                       A 
                                     
                                     ) 
                                   
                                 
                               
                               = 
                               
                                 sign 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     F 
                                     
                                       2 
                                       , 
                                       6 
                                     
                                     B 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 b 
                                 
                                   1 
                                   , 
                                   6 
                                 
                                 A 
                               
                               ⋂ 
                               
                                 b 
                                 
                                   1 
                                   , 
                                   6 
                                 
                                 B 
                               
                             
                             , 
                             
                               … 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               otherwise 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     24 
                   
                   ) 
                 
               
             
           
         
       
     
     In Equation 24, 
     
       
         
           
             
               b 
               
                 1 
                 , 
                 6 
               
               B 
             
             ∈ 
             
               
                 { 
                 
                   0 
                   , 
                   1 
                 
                 } 
               
               ⋂ 
               
                 [ 
                 
                   
                     
                       
                         
                           
                             
                               Q 
                               A 
                             
                             - 
                             
                               C 
                             
                             - 
                             
                               6 
                               ⁢ 
                               
                                 
                                   ∑ 
                                   
                                     j 
                                     = 
                                     2 
                                   
                                   5 
                                 
                                 ⁢ 
                                 
                                    
                                   
                                     F 
                                     
                                       2 
                                       ⁢ 
                                       j 
                                     
                                     A 
                                   
                                    
                                 
                               
                             
                             - 
                             
                               2 
                               ⁢ 
                               
                                 
                                   b 
                                   _ 
                                 
                                 
                                   0 
                                   , 
                                   6 
                                 
                               
                               ⁢ 
                               
                                  
                                 
                                   F 
                                   
                                     2 
                                     , 
                                     6 
                                   
                                   A 
                                 
                                  
                               
                             
                           
                           
                             4 
                             ⁢ 
                             
                                
                               
                                 F 
                                 
                                   2 
                                   , 
                                   6 
                                 
                                 A 
                               
                                
                             
                           
                         
                         , 
                       
                     
                   
                   
                     
                       
                         
                           
                             Q 
                             A 
                           
                           + 
                           
                             C 
                           
                           - 
                           
                             2 
                             ⁢ 
                             
                               
                                 b 
                                 _ 
                               
                               
                                 0 
                                 , 
                                 6 
                               
                             
                             ⁢ 
                             
                                
                               
                                 F 
                                 
                                   2 
                                   , 
                                   6 
                                 
                                 A 
                               
                                
                             
                           
                         
                         
                           4 
                           ⁢ 
                           
                              
                             
                               F 
                               
                                 2 
                                 , 
                                 6 
                               
                               A 
                             
                              
                           
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     If F A   2,6  and F B   2,6  have the same code, 
     
       
         
           
             
               b 
               
                 1 
                 , 
                 6 
               
               B 
             
             ∈ 
             
               
                 { 
                 
                   0 
                   , 
                   1 
                 
                 } 
               
               ⋂ 
               
                 [ 
                 
                   
                     
                       
                         
                           
                             
                               Q 
                               B 
                             
                             - 
                             
                               C 
                             
                             - 
                             
                               6 
                               ⁢ 
                               
                                 
                                   ∑ 
                                   
                                     j 
                                     = 
                                     2 
                                   
                                   5 
                                 
                                 ⁢ 
                                 
                                    
                                   
                                     F 
                                     
                                       2 
                                       ⁢ 
                                       j 
                                     
                                     B 
                                   
                                    
                                 
                               
                             
                             - 
                             
                               2 
                               ⁢ 
                               
                                 
                                   b 
                                   _ 
                                 
                                 
                                   0 
                                   , 
                                   6 
                                 
                               
                               ⁢ 
                               
                                  
                                 
                                   F 
                                   
                                     2 
                                     , 
                                     6 
                                   
                                   B 
                                 
                                  
                               
                             
                           
                           
                             4 
                             ⁢ 
                             
                                
                               
                                 F 
                                 
                                   2 
                                   , 
                                   6 
                                 
                                 B 
                               
                                
                             
                           
                         
                         , 
                       
                     
                   
                   
                     
                       
                         
                           
                             Q 
                             B 
                           
                           + 
                           
                             C 
                           
                           - 
                           
                             2 
                             ⁢ 
                             
                               
                                 b 
                                 _ 
                               
                               
                                 0 
                                 , 
                                 6 
                               
                             
                             ⁢ 
                             
                                
                               
                                 F 
                                 
                                   2 
                                   , 
                                   6 
                                 
                                 B 
                               
                                
                             
                           
                         
                         
                           4 
                           ⁢ 
                           
                              
                             
                               F 
                               
                                 2 
                                 , 
                                 6 
                               
                               B 
                             
                              
                           
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     If F A   2,6  and F B   2,6  have different codes, respectively, in a numerical formula for b B   1,6 , the value of  b 0,6    needs to be changed to a complement of 1. An imaginary part for X 4  is estimated as represented by Equation 25.
 
  ℑ[ x 4] =sign( F   2,6   A )(2(2   b     1,6   +  b     0,6 )−3)  (Equation 25)
 
     In the same method, a real part of X 4  and imaginary and real parts of X 3  can be estimated. That is, imaginary and real parts of X 4  and imaginary and real parts of X 3  are sequentially estimated. At this time, imaginary and real parts, each of which is composed of a plurality of bits, are estimated in an order from the LSB (least significant bit) to the MSB (most significant bit). 
     Then, x^ 1  and x^ 2  are estimated by using Equations 17 and 18. In order to obtain x^ 1  and x^ 2  that are ML estimated values, points that are close to x^ 1  and x^ 2 , are selected from among all points in a constellation. 
     The case of 16-QAM has been described. The case of M-QAM will now be described. 
     In the case of M-QAM, as a value of each of X A   j  and X B   j , any one of {−−√{square root over (M)}+1, −√{square root over (M)}+3, . . . , −1, 1, . . . , √{square root over (M)}−3, √{square root over (M)}−1} is selected. In this case, if a value of F A   2,j  is a positive number, a conversion like Equation 26 is performed on X A   J  that corresponds to j. 
     
       
         
           
             
               
                 
                   
                     b 
                     j 
                     A 
                   
                   = 
                   
                     
                       
                         M 
                       
                       - 
                       1 
                       + 
                       
                         x 
                         j 
                         A 
                       
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     26 
                   
                   ) 
                 
               
             
           
         
       
     
     If a value of F A   2j  is a negative number, a conversion like Equation 27 is performed on X A   j  that corresponds to j. 
     
       
         
           
             
               
                 
                   
                     b 
                     j 
                     A 
                   
                   = 
                   
                     
                       
                         M 
                       
                       - 
                       1 
                       - 
                       
                         x 
                         j 
                         A 
                       
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     27 
                   
                   ) 
                 
               
             
           
         
       
     
     A detection method, which is executed after the conversion is performed, is the same as the method that is described in the 16-QAM. 
     Finally, in order to improve detection performance, detection ordering is performed. In the above-described method, x 3  and x 4  are estimated and then x 1  and x 2  are estimated. According to this method, when a channel through which x 3  and x 4  pass is superior to a channel through which x 1  and x 2  pass, superior performance is obtained. However, when the channel through which x 3  and x 4  pass is inferior to the channel through which x 1  and x 2  pass, superior performance cannot be obtained due to error propagation. 
     In this case, after estimating x 1  and x 2 , x 3  and x 4  are estimated. In general, when Equation 28 is satisfied, x 1  and x 2  are first estimated. When Equation 28 is not satisfied, x 3  and x 4  are first estimated. 
     
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         1 
                       
                       
                         n 
                         r 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                              
                             
                               h 
                               
                                 k 
                                 , 
                                 1 
                               
                             
                              
                           
                           2 
                         
                         + 
                         
                           
                              
                             
                               h 
                               
                                 k 
                                 , 
                                 2 
                               
                             
                              
                           
                           2 
                         
                       
                       ) 
                     
                   
                   ≥ 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         1 
                       
                       
                         n 
                         r 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                              
                             
                               h 
                               
                                 k 
                                 , 
                                 3 
                               
                             
                              
                           
                           2 
                         
                         + 
                         
                           
                              
                             
                               h 
                               
                                 k 
                                 , 
                                 4 
                               
                             
                              
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     28 
                   
                   ) 
                 
               
             
           
         
       
     
     The detection method will now be described in detail with reference to  FIG. 2 . 
       FIG. 2  is a flowchart illustrating a detection method according to an exemplary embodiment of the present invention. 
     As shown in  FIG. 2 , if receiving signals from n r  receiving antennas (Step S 100 ), it is determined which channel is superior from among a plurality of channels through which the signals are received (Step S 110 ). In the exemplary embodiment of the present invention, since the detection of the signals from four antennas is exemplified, it is determined whether a channel from transmitting antennas  3  and  4  is superior to a channel from transmitting antennas  1  and  2 . 
     When it is determined that the channel from the transmitting antennas  3  and  4  is superior to the channel from the transmitting antennas  1  and  2 , Cholesky decomposition is performed in order to calculate F (in Equation 6, F A  and F B ) of the ML metric that is generated for the purpose of performing sphere decoding (Step S 120 ). In this case, since the Cholesky decomposition is already well-known, a detailed description thereof will be omitted in the exemplary embodiment of the present invention. 
     If the ML metric is calculated by performing the Cholesky decomposition, x 3  and c 4  are detected by using two dependent sphere decoders (Step S 130 ). Since the method of detecting x 3  and c 4  by using the sphere decoders is already well-known, a detailed description thereof will be omitted in the exemplary embodiment of the present invention. 
     If x 3  and c 4  are detected, x 1  and x 2  are detected by a single symbol detection method using Equation 17 (Step S 140 ). When detecting x 1  and x 2 , an Alamouti decoding scheme is used. 
     Steps S 120  to S 140  are performed when the channel from the transmitting antennas  3  and  4  is superior to the channel from the transmitting antennas  1  and  2 . Meanwhile, when the channel from the transmitting antennas  1  and  2  is superior to the channel from the transmitting antennas  3  and  4 , after Cholesky decomposition is performed (Step S 160 ), x 1  and x 2  are detected by using a sphere decoder (Step S 170 ) and x 3  and x 4  are then detected by using the single symbol detection method (Step S 180 ). 
     As described above, after x 1 , x 2 , x 3  and x 4  are all detected, the radius of a sphere decoder is updated (Step S 150 ). The detection method is repeatedly performed until only one constellation point exists in the updated radius of the sphere decoder. 
     Referring to  FIGS. 3 to 6 , the running times and the CPU clocks when using the 4-QAM method and the 16-QAM method according to the exemplary embodiment of the present invention are compared with the running time and the CPU clock when using the method according to the related art. 
       FIGS. 3 and 4  are exemplary views illustrating the comparison result between a running time and the number of CPU clocks when using a 4-QAM method according to an exemplary embodiment of the present invention and a running time and the number of CPU clocks when using a method according to the related art. 
     In this case, the scheme, which is proposed by Z. Yang, C. Liu, and J. He, is called a “best known scheme”.  FIG. 3  shows a comparison result between running times when using a detection method according to an exemplary embodiment of the present invention and when using the detection method according to the best known scheme. As shown in  FIG. 3 , it can be understood that the running time when using the detection method according to the exemplary embodiment of the present invention is shorter than the running time when using the detection method according to the best known scheme. In particular, an improvement width is large in a region having a low SNR (signal-to-noise ratio). 
     As shown in  FIG. 4 , it can be understood that the number of CPU clocks when using the 4-QAM method according to the exemplary embodiment of the present invention is smaller than the number of CPU clocks when using the method according to the best known scheme. In this case, the number of CPU clocks is in proportion to complexity of a system. When the number of CPU clocks increases, the complexity of the system increases. 
       FIGS. 5 and 6  are exemplary views illustrating a comparison result between a running time and the number of CPU clocks when using a 16-QAM method according to an exemplary embodiment of the present invention and a running time and the number of CPU clocks when using a method according to the related art. 
     As shown in  FIGS. 5 and 6 , it can be understood that the detection method according to the exemplary embodiment of the present invention using the 16-QAM method is superior to the detection method according to the related art in terms of the running time and the number of CPU clocks. This point is the same as that of the 4-QAM method. 
     In this case, a program that realizes a function corresponding to the structure according to the exemplary embodiment of the present invention and a recording medium that has the program recorded therein are within the range of the present invention. 
     While this invention has been described in connection with what is presently considered to be practical exemplary embodiments, it is to be understood that the invention is not limited to the disclosed embodiments, but, on the contrary, is intended to cover various modifications and equivalent arrangements included within the spirit and scope of the appended claims. 
     According to the above-described exemplary embodiment, when detecting the space-time code B for the four transmitting antennas, as compared with the case of using the existing sphere decoding scheme, it is possible to provide the same ML performance with low complexity.