# Patent

## Title
Method of determining an acquisition indicator bit in a communication system

## Abstract

A method of determining an acquisition indicator bit AIs at a receiver in a communication system which includes receiving multiplexed acquisition indicator bits y=B×AI+n, where B is the signature matrix known at both base station and UE, AI is all the acquisition indicator bits and n represents noise. Next, an estimated value of the acquisition indicator bit AÎMMSE (s) as a function of




  
    ∑
    j
    
      
    
  
  ⁢
  
    
  
  ⁢
  
    
      
        B
        T
      
      ⁡
      
        (
        
          s
          ,
          j
        
        )
      
    
    ×
    
      y
      ⁡
      
        (
        j
        )
      
    
  




is calculated, where BT(s,.) is the s-th row vector of the transposed matrix BT for acquisition indicator bit AIs. Finally, the acquisition indicator bit AIs is set as follows:

AIs=−1, if AÎMMSE(s)<R

AIs=0, if R≦AÎMMSE(s)<U

AIs=1, if AÎMMSE(s)≧U,

where R and U are decision thresholds.

## Background
CROSS REFERENCE
This is a continuation-in-part of U.S. patent application Ser. No. 10/190,579 filed Jul. 9, 2002, now U.S. Pat. No. 6,944,205 which is incorporated herein by reference.


BACKGROUND AND SUMMARY OF THE INVENTION
The present invention relates generally to communication systems of coded data and, more specifically, to an improvement in determining an acquisition indicator bit at a receiver in a communication system.
Although the present invention will be described with respect to 3rd generation wideband code division multiple access (3G WCDMA) system, the same method can be used for determining an acquisition bit AIs at a receiver in other communication systems. General and specific references will also be made to the 3G WCDMA standard 3GPP TS 25.211, Physical channels and mapping of transport channels onto physical channels (FDD) (Release 4) and 3GPP TS 25.213, “Spreading and modulation (FDD)” (Release 4).
In 3rd generation CDMA systems, physical connections between UE (User Equipment) and base stations are established through the Physical Random Access Channel (PRACH) and Acquisition Indicator Channel (AICH). The UE transmits PRACH signal that carries the RACH preamble and message to a base station, to request a connection. When the base station recognizes a PRACH preamble, it responds with AICH to UE to indicate if the connection request is granted. An example of the AICH from 3GPP TS 25.211 is shown in FIG. 1.
Upon the AICH reception, the problem is to estimate the AIs based on the received real valued symbols a0, a1, . . . , a31 which are transmitted through AICH, where AIs ε{−1,0,1} and s=0,1, . . . , 15. An AIs of 0 indicates that the signature s is not a member of the set of available signatures. An AIs of 1 indicates a positive acknowledgement, and an AIs of −1 indicates a negative acknowledgement. At the base station, AIs is code multiplexed with others, {AIn: n≠s}, by the following formula:




  
    
      
        
          
            a
            j
          
          =
          
            
              ∑
              
                s
                =
                0
              
              15
            
            ⁢
            
              
            
            ⁢
            
              
                AI
                s
              
              ⁢
              
                b
                
                  s
                  ,
                  j
                
              
            
          
        
        ,
        
          j
          =
          0
        
        ,
        1
        ,
        
          
        
        ⁢
        ...
        ⁢
        
          
        
        ,
        31
        ,
      
    
    
      
        (
        1
        )
      
    
  




where {bs,j: j=0,1, . . . 31} are given in Table 21 in 3GPP TS 25.211 and shown below in the example as B

The present invention is a method of determining an acquisition indicator bit AIs at a receiver in a communication system. The method includes receiving multiplexed acquisition indicator bits y=B×AI+n, where B is the signature matrix known at both base station and UE, AI is the set of all the acquisition indicator bits and n represents noise, for example, AWGN (additive white Gaussian noise). Next, an estimated value of the acquisition indicator bit




  A
  ⁢
  
    
  
  ⁢
  
    
      
        I
        ^
      
      MMSE
    
    ⁡
    
      (
      s
      )
    
  
  ⁢
  
    
  
  ⁢
  as
  ⁢
  
    
  
  ⁢
  a
  ⁢
  
    
  
  ⁢
  function
  ⁢
  
    
  
  ⁢
  of
  ⁢
  
    
  
  ⁢
  
    
      ∑
      j
      
        
      
    
    ⁢
    
      
    
    ⁢
    
      
        
          B
          T
        
        ⁡
        
          (
          
            s
            ,
            j
          
          )
        
      
      ×
      
        y
        ⁡
        
          (
          j
          )
        
      
    
  




is calculated, where BT (s,.) is the s-th row vector of the transposed matrix BT for acquisition indicator bit AIs. Finally, the acquisition indicator bit AIs is set as follows:

AIs=−1, if AÎMMSE (s)<R

AIs═0, if R≦AÎMMSE (s)<U

AIs=1, if AÎMMSE (S)≧U,

where R and U are decision thresholds.

The constants R and U may be equal absolute values, for example, −0.5 and 0.5, respectively. The estimated value AÎMMSE (s) is calculated by




  
    A
    ⁢
    
      
    
    ⁢
    
      
        
          I
          ^
        
        MMSE
      
      ⁡
      
        (
        s
        )
      
    
  
  =
  
    
      1
      
        (
        
          32
          +
          
            
              σ
              ^
            
            2
          
        
        )
      
    
    ⁢
    
      
        ∑
        
          j
          =
          0
        
        31
      
      ⁢
      
        
      
      ⁢
      
        
          
            B
            T
          
          ⁡
          
            (
            
              s
              ,
              j
            
            )
          
        
        ×
        
          
            y
            ⁡
            
              (
              j
              )
            
          
          .
        
      
    
  




The method pre-stores a matrix B or BT and is performed in software.

These and other aspects of the present invention will become apparent from the following detailed description of the invention, when considered in conjunction with accompanying drawings.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a diagram of the structure of the acquisition indicator channel AICH of the 3G WCDMA system defined in 3GPP TS 25.211.
FIG. 2 is a flow chart of a method of determining an acquisition indicator bit AIs incorporating the principles of the present invention.



DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
The method of the present invention, which is performed at the receiver of a communication system, is illustrated in FIG. 2. The input signal y is received at 10. An estimated acquisition indicator (AÎML(s)) is calculated from the input signal y. The resulting estimated acquisition indicator corresponding to a signature index s, AÎML(s) is then used for determining an acquisition indicator bit AIs having one of the values −1,0,1. The decided acquisition indicator bit is outputted at 16. The details of the method are described as follows:
In matrix form, equation (1) can be expressed as




  
    
      
        
          
            
              
                a
                =
                
                  B
                  ×
                  AI
                
              
              ,
            
          
        
        
          
            where
          
        
        
          
            
              
                a
                =
                
                  
                    [
                    
                      
                        a
                        0
                      
                      ,
                      
                        a
                        1
                      
                      ,
                      ...
                      ⁢
                      
                        
                      
                      ,
                      
                        a
                        31
                      
                    
                    ]
                  
                  T
                
              
              ,
            
          
        
        
          
            
              
                B
                =
                
                  [
                  
                    
                      
                        
                          b
                          
                            0
                            ,
                            0
                          
                        
                      
                      
                        
                          b
                          
                            1
                            ,
                            0
                          
                        
                      
                      
                        
                          b
                          
                            2
                            ,
                            0
                          
                        
                      
                      
                        ⋯
                      
                      
                        
                          b
                          
                            15
                            ,
                            0
                          
                        
                      
                    
                    
                      
                        
                          b
                          
                            0
                            ,
                            1
                          
                        
                      
                      
                        
                          b
                          
                            1
                            ,
                            1
                          
                        
                      
                      
                        
                          b
                          
                            2
                            ,
                            1
                          
                        
                      
                      
                        ⋯
                      
                      
                        
                          b
                          
                            15
                            ,
                            1
                          
                        
                      
                    
                    
                      
                        ·
                      
                      
                        ·
                      
                      
                        ·
                      
                      
                        ·
                      
                      
                        ·
                      
                    
                    
                      
                        ·
                      
                      
                        ·
                      
                      
                        ·
                      
                      
                        ·
                      
                      
                        ·
                      
                    
                    
                      
                        
                          b
                          
                            0
                            ,
                            31
                          
                        
                      
                      
                        
                          b
                          
                            1
                            ,
                            31
                          
                        
                      
                      
                        
                          b
                          
                            2
                            ,
                            31
                          
                        
                      
                      
                        ⋯
                      
                      
                        
                          b
                          
                            15
                            ,
                            31
                          
                        
                      
                    
                  
                  ]
                
              
              ,
            
          
        
        
          
            and
          
        
        
          
            
              AI
              =
              
                
                  
                    [
                    
                      
                        AI
                        0
                      
                      ,
                      
                        AI
                        1
                      
                      ,
                      ...
                      ⁢
                      
                        
                      
                      ,
                      
                        AI
                        15
                      
                    
                    ]
                  
                  T
                
                .
              
            
          
        
      
    
    
      
        (
        2
        )
      
    
  




The received soft bits from a rake receiver can be modeled as

y=a+n 

    
    
        or

y=B×AI+n,  (3)

where n denotes noise, for example, the AWGN.

    
    


The maximum likelihood estimator for the unknown acquisition indicator, AÎML, in equation (3) is well known (see S. M. Kay, Fundamentals of Statistical Signal Processing (Estimation Theory), Prentice Hall) and is given by

AÎML=(BT×B)−1×BT×y.  (4)

Due to orthogonality (see Table 21 in 3GPP TS 25.211),

BT×B=32×I16×16,  (5)

where I16×16 is the identity matrix with dimension 16.

Thus, equation (4) can be reduced to




  
    
      
        
          A
          ⁢
          
            
          
          ⁢
          
            
              I
              ^
            
            ML
          
        
        =
        
          
            1
            32
          
          ×
          
            B
            T
          
          ×
          
            y
            .
          
        
      
    
    
      
        (
        6
        )
      
    
  




It is assumed that the table index, s, for AICH signature patterns is equivalent to the one for the RACH preamble signatures from Table 3 in 3GPP TS 25.213. In other words, the index s is known at the user equipment UE so that only the portion corresponding to s of equation (6) is of interest.
Let Ts(y) be a statistic corresponding to an index s, given y




  
    
      
        
          
            
              T
              s
            
            ⁡
            
              (
              y
              )
            
          
          =
          
            
              A
              ⁢
              
                
              
              ⁢
              
                
                  
                    I
                    ^
                  
                  ML
                
                ⁡
                
                  (
                  s
                  )
                
              
            
            =
            
              
                1
                32
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    0
                  
                  31
                
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      B
                      T
                    
                    ⁡
                    
                      (
                      
                        s
                        ,
                        j
                      
                      )
                    
                  
                  ×
                  
                    y
                    ⁡
                    
                      (
                      j
                      )
                    
                  
                
              
            
          
        
        ,
      
    
    
      
        (
        7
        )
      
    
  




where BT(s,.) denotes the s-th row vector of the matrix BT.

Assuming uniform prior for the following hypotheses,

H−1: AIs=−1,

H0: AIs=0

H1: AIs=1, s=0,1, . . . , 15,

the 3-arry MAP decision rule (see S. M. Kay, Fundamentals of Statistical Signal Processing (Detection Theory), Prentice Hall) is to decide

AIs=−1, if Ts(y)<R

AIs=0, if R≦Ts(y)<U

AIs=1, if Ts(y)≧U,

where U and R are constants. The {Hi, i=−1,0,1} denote the hypotheses that the acquisition indicator bit is i. Assuming perfect channel, the constants R and U may be equal absolute values, for example, −0.5 and 0.5, respectively.

To improve the estimation, an MMSE (Minimum Mean Square Error) estimator that is given by the following may be used

AÎMMSE=(BT×B+{circumflex over (σ)}2 I)−1×BT×y,  (8)

where {circumflex over (σ)}2 represents the estimated noise variance at the receiver. The MMSE estimator is the solution that takes into account the background noise.

With (8), (4) can be reduced to




  
    
      
        
          A
          ⁢
          
            
          
          ⁢
          
            
              I
              ^
            
            MMSE
          
        
        =
        
          
            1
            
              (
              
                32
                +
                
                  
                    σ
                    ^
                  
                  2
                
              
              )
            
          
          ×
          
            B
            T
          
          ×
          
            y
            .
          
        
      
    
    
      
        (
        9
        )
      
    
  




Let Ts(y) be a statistic corresponding to an index s, given y





  
    
      
        
          
            
              T
              s
            
            ⁡
            
              (
              y
              )
            
          
          =
          
            
              A
              ⁢
              
                
              
              ⁢
              
                
                  
                    I
                    ^
                  
                  MMSE
                
                ⁡
                
                  (
                  s
                  )
                
              
            
            =
            
              
                1
                
                  (
                  
                    32
                    +
                    
                      
                        σ
                        ^
                      
                      2
                    
                  
                  )
                
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    0
                  
                  31
                
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      B
                      T
                    
                    ⁡
                    
                      (
                      
                        s
                        ,
                        j
                      
                      )
                    
                  
                  ×
                  
                    y
                    ⁡
                    
                      (
                      j
                      )
                    
                  
                
              
            
          
        
        ,
      
    
    
      
        (
        10
        )
      
    
  




As an illustrative example, let AI0=−1, AI1=−1, AI2=1, AI3=−1, AI4=−1, AI5=−1, AI6=−1, AI7=−1, AI8=−1, AI9=−1, AI10=0, AI11=−1, AI12=−1, AI13=−1, AI14=−1, and AI15=0.
Since AI is defined by

AI=[AI0 AI1 AI2 AI3 AI4 AI5 AI6 AI7 AI8 AI9 AI10 AI11 AI12 AI13 AI14 AI15]^T, 

where T denote the transpose, that is





  
    
      
        AI
        =
        
          [
          
            -
            1
          
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          -
          1
        
      
    
  
  
    
      
        
          
        
        ⁢
        1
      
    
  
  
    
      
        
          
        
        ⁢
        
          -
          1
        
      
    
  
  
    
      
        
          
        
        ⁢
        1
      
    
  
  
    
      
        
          
        
        ⁢
        
          -
          1
        
      
    
  
  
    
      
        -
        1
      
    
  
  
    
      
        -
        1
      
    
  
  
    
      
        -
        1
      
    
  
  
    
      
        -
        1
      
    
  
  
    
      0
    
  
  
    
      
        -
        1
      
    
  
  
    
      
        -
        1
      
    
  
  
    
      
        -
        1
      
    
  
  
    
      
        -
        1
      
    
  
  
    
      
        
          
            
          
          ⁢
          0
          ]
        
        .
      
    
  




The matrix B (of size 32×16) in (2) is given by





  
    
      
        B
        =
        
        ⁢
        
          [
          
            1
            ,
          
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
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            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          -
          1
        
      
    
  
  
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          -
          1
        
      
    
  
  
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        1
      
    
  
  
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        1
      
    
  
  
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          -
          1
        
      
    
  
  
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          -
          1
        
      
    
  
  
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          -
          1
        
      
    
  
  
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          -
          1
        
      
    
  
  
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        1
      
    
  
  
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          1
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
        ⁢
        
          
            -
            1
          
          ,
        
      
    
    
      
        
          
          ⁢
          1
          ]
        
        .
      
    
  




For instance, b0,0=1 and b2,1=−1.

The transposed B is then





  
    
      
        
          B
          T
        
        =
        
          [
           
          ⁢
          
            1
            ,
            
              
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    
                      
                    
                    ⁢
                    
                      1
                      ,
                    
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
                
                  
                    1
                    ,
                  
                
              
            
          
        
      
    
  
  
    
      
        
        ⁢
        
          
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
          
        
      
    
  
  
    
      
        
        ⁢
        
          
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
          
        
      
    
  
  
    
      
        
        ⁢
        
          
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
          
        
      
    
  
  
    
      
        
        ⁢
        
          
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
          
        
      
    
  
  
    
      
        
        ⁢
        
          
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
          
        
      
    
  
  
    
      
        
        ⁢
        
          
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
          
        
      
    
  
  
    
      
        
        ⁢
        
          
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
          
        
      
    
  
  
    
      
        
        ⁢
        
          
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
          
        
      
    
  
  
    
      
        
        ⁢
        
          
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
          
        
      
    
  
  
    
      
        
        ⁢
        
          
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
          
        
      
    
  
  
    
      
         
        ⁢
        
          1
          ,
          
            
          
          ⁢
          1
          ,
          
            
          
          ⁢
          
            -
            1
          
          ,
          
            
          
          ⁢
          
            -
            1
          
          ,
          
            
          
          ⁢
          
            -
            1
          
          ,
          
            
          
          ⁢
          
            -
            1
          
          ,
          
            
          
          ⁢
          1
          ,
          
            
          
          ⁢
          1
          ,
          
            
          
          ⁢
          1
          ,
          
            
          
          ⁢
          1
          ,
          
            
          
          ⁢
          
            -
            1
          
          ,
          
            
          
          ⁢
          
            -
            1
          
          ,
          
            
          
          ⁢
          
            -
            1
          
          ,
          
            
          
          ⁢
          
            -
            1
          
          ,
          
            
          
          ⁢
          1
          ,
          
            
          
          ⁢
          1
          ,
          
            
          
          ⁢
          
            -
            1
          
          ,
          
            
          
          ⁢
          
            -
            1
          
          ,
          
            
          
          ⁢
          1
          ,
          
            
          
          ⁢
          1
          ,
          
            
          
          ⁢
          1
          ,
          
            
          
          ⁢
          1
          ,
          
            
          
          ⁢
          
            -
            1
          
          ,
          
            
          
          ⁢
          
            -
            1
          
          ,
          
            
          
          ⁢
          
            -
            1
          
          ,
          
            
          
          ⁢
          
            -
            1
          
          ,
          
            
          
          ⁢
          1
          ,
          
            
          
          ⁢
          1
          ,
          
            
          
          ⁢
          1
          ,
          
            
          
          ⁢
          1
          ,
          
            
          
          ⁢
          
            -
            1
          
          ,
          
            
          
          ⁢
          
            -
            1
          
          ,
        
      
    
  
  
    
      
        
        ⁢
        
          
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
          
        
      
    
  
  
    
      
        
        ⁢
        
          
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
          
        
      
    
  
  
    
      
        
        ⁢
        
          
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
          
        
      
    
  
  
    
      
        
        ⁢
        
          
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                
                  -
                  1
                
                ,
              
            
            
              
                1
                ,
              
            
            
              
                1
                ]
              
            
          
        
      
    
  




From equation (3), a=B×AI.

Note that the size of y, a, B. and AI are 32×1, 32×1, 32×16, and 16×1, respectively.

The calculated a is then





  
    
      
        a
        =
        
        ⁢
        
          [
          
            -
            12
          
        
      
    
  
  
    
      
        
        ⁢
        
          -
          12
        
      
    
  
  
    
      
        
        ⁢
        2
      
    
  
  
    
      
        
        ⁢
        2
      
    
  
  
    
      
        
        ⁢
        
          -
          4
        
      
    
  
  
    
      
        
        ⁢
        
          -
          4
        
      
    
  
  
    
      
        
        ⁢
        
          -
          2
        
      
    
  
  
    
      
        
        ⁢
        
          -
          2
        
      
    
  
  
    
      
        
        ⁢
        2
      
    
  
  
    
      
        
        ⁢
        2
      
    
  
  
    
      
        
        ⁢
        4
      
    
  
  
    
      
        
        ⁢
        4
      
    
  
  
    
      
        
        ⁢
        
          -
          2
        
      
    
  
  
    
      
        
        ⁢
        
          -
          2
        
      
    
  
  
    
      
        
        ⁢
        
          -
          4
        
      
    
  
  
    
      
        
        ⁢
        
          -
          4
        
      
    
  
  
    
      
        
        ⁢
        0
      
    
  
  
    
      
        
        ⁢
        0
      
    
  
  
    
      
        
        ⁢
        2
      
    
  
  
    
      
        
        ⁢
        2
      
    
  
  
    
      
        
        ⁢
        0
      
    
  
  
    
      
        
        ⁢
        0
      
    
  
  
    
      
        
        ⁢
        
          -
          2
        
      
    
  
  
    
      
        
        ⁢
        
          -
          2
        
      
    
  
  
    
      
        
        ⁢
        2
      
    
  
  
    
      
        
        ⁢
        2
      
    
  
  
    
      
        
        ⁢
        0
      
    
  
  
    
      
        
        ⁢
        0
      
    
  
  
    
      
        
        ⁢
        
          -
          2
        
      
    
  
  
    
      
        
        ⁢
        
          -
          2
        
      
    
  
  
    
      
        
        ⁢
        0
      
    
  
  
    
      
        
          
          ⁢
          0
          ]
        
        .
      
    
  




From equation (3), the noise is assumed as the additive white Gaussian. Suppose the noise values are given by




  
    
      
        n
        =
        
        ⁢
        
          [
          0.1703
        
      
    
  
  
    
      
        
        ⁢
        0.3561
      
    
  
  
    
      
        
        ⁢
        0.2907
      
    
  
  
    
      
        
        ⁢
        0.2298
      
    
  
  
    
      
        
        ⁢
        0.0160
      
    
  
  
    
      
        
        ⁢
        0.2692
      
    
  
  
    
      
        
        ⁢
        0.2262
      
    
  
  
    
      
        
        ⁢
        
          -
          0.1017
        
      
    
  
  
    
      
        
        ⁢
        
          -
          0.1501
        
      
    
  
  
    
      
        
        ⁢
        
          -
          0.1177
        
      
    
  
  
    
      
        
        ⁢
        
          -
          0.5866
        
      
    
  
  
    
      
        
        ⁢
        
          -
          0.0930
        
      
    
  
  
    
      
        
        ⁢
        0.0471
      
    
  
  
    
      
        
        ⁢
        0.1252
      
    
  
  
    
      
        
        ⁢
        0.5740
      
    
  
  
    
      
        
        ⁢
        
          -
          0.1396
        
      
    
  
  
    
      
        
        ⁢
        0.2478
      
    
  
  
    
      
        
        ⁢
        0.3177
      
    
  
  
    
      
        
        ⁢
        0.3741
      
    
  
  
    
      
        
        ⁢
        
          -
          0.3945
        
      
    
  
  
    
      
        
        ⁢
        0.0843
      
    
  
  
    
      
        
        ⁢
        0.0946
      
    
  
  
    
      
        
        ⁢
        
          -
          0.4007
        
      
    
  
  
    
      
        
        ⁢
        
          -
          0.2951
        
      
    
  
  
    
      
        
        ⁢
        0.4304
      
    
  
  
    
      
        
        ⁢
        
          -
          0.0523
        
      
    
  
  
    
      
        
        ⁢
        0.1550
      
    
  
  
    
      
        
        ⁢
        0.0350
      
    
  
  
    
      
        
        ⁢
        
          -
          0.2527
        
      
    
  
  
    
      
        
        ⁢
        
          -
          0.2225
        
      
    
  
  
    
      
        
        ⁢
        0.1764
      
    
  
  
    
      
        
          
          ⁢
          
            -
            0.3777
          
          ]
        
        .
      
    
  




Then from equation (3), the observed samples, y, is




  
    
      
        y
        =
        
        ⁢
        
          [
          
            -
            11.8297
          
        
      
    
  
  
    
      
        
        ⁢
        
          -
          11.6439
        
      
    
  
  
    
      
        
        ⁢
        2.2907
      
    
  
  
    
      
        
        ⁢
        2.2298
      
    
  
  
    
      
        
        ⁢
        
          -
          3.9840
        
      
    
  
  
    
      
        
        ⁢
        
          -
          3.7308
        
      
    
  
  
    
      
        
        ⁢
        
          -
          1.7738
        
      
    
  
  
    
      
        
        ⁢
        
          -
          2.1017
        
      
    
  
  
    
      
        
        ⁢
        1.8499
      
    
  
  
    
      
        
        ⁢
        1.8823
      
    
  
  
    
      
        
        ⁢
        3.4134
      
    
  
  
    
      
        
        ⁢
        3.9070
      
    
  
  
    
      
        
        ⁢
        
          -
          1.9529
        
      
    
  
  
    
      
        
        ⁢
        
          -
          1.8748
        
      
    
  
  
    
      
        
        ⁢
        
          -
          3.4260
        
      
    
  
  
    
      
        
        ⁢
        
          -
          4.1396
        
      
    
  
  
    
      
        
        ⁢
        0.2478
      
    
  
  
    
      
        
        ⁢
        0.3177
      
    
  
  
    
      
        
        ⁢
        2.3741
      
    
  
  
    
      
        
        ⁢
        1.6055
      
    
  
  
    
      
        
        ⁢
        0.0843
      
    
  
  
    
      
        
        ⁢
        0.0946
      
    
  
  
    
      
        
        ⁢
        
          -
          2.4007
        
      
    
  
  
    
      
        
        ⁢
        
          -
          2.2951
        
      
    
  
  
    
      
        
        ⁢
        2.4304
      
    
  
  
    
      
        
        ⁢
        1.9477
      
    
  
  
    
      
        
        ⁢
        0.1550
      
    
  
  
    
      
        
        ⁢
        0.0350
      
    
  
  
    
      
        
        ⁢
        
          -
          2.2527
        
      
    
  
  
    
      
        
        ⁢
        
          -
          2.2225
        
      
    
  
  
    
      
        
        ⁢
        0.1764
      
    
  
  
    
      
        
          
          ⁢
          
            -
            0.3777
          
          ]
        
        .
      
    
  




The y is the noisy version of the received acquisition indicator bits carried over AICH.
Assume that the signature index s is 2, then by equation (3), the AI2 can be estimated as




  
    
      
        
          A
          ⁢
          
            
              
                I
                ^
              
              ML
            
            ⁡
            
              (
              2
              )
            
          
        
        =
        
        ⁢
        
          
            
              1
              32
            
            ⁢
            
              
                ∑
                
                  j
                  =
                  0
                
                31
              
              ⁢
              
                
                  
                    B
                    T
                  
                  ⁡
                  
                    (
                    
                      2
                      ,
                      j
                    
                    )
                  
                
                ×
                
                  y
                  ⁡
                  
                    (
                    j
                    )
                  
                
              
            
          
          =
          
            
              1
              /
              32
            
            *
            
              (
              
                
                  1
                  *
                  
                    -
                  
                  ⁢
                  11.8297
                
                +
              
            
          
        
      
    
  
  
    
      
        
        ⁢
        
          
            1
            *
            
              -
            
            ⁢
            11.6439
          
          +
          
            1
            *
            2.2907
          
          +
          
            ...
            ⁢
            
              
            
            ⁢
            
              -
            
            ⁢
            1
            *
            0.1764
          
          +
        
      
    
  
  
    
      
        
          
          ⁢
          
            
              -
            
            ⁢
            1
            *
            
              -
            
            ⁢
            0.3777
          
          )
        
        =
        
          1.0434
          .
        
      
    
  




This is the multiplication of two vectors, the 3rd row of BT and y. Since AÎML (2) is greater than 0.5 using U=0.5, the decision can be made as AÎ(2)=1, which is matched with the initial setting, AI2=1.

Although the present invention has been described and illustrated in detail, it is to be clearly understood that the same is by way of illustration and example only, and is not to be taken by way of limitation. The scope of the present invention are to be limited only by the terms of the appended claims.

## Claims

1. A method of determining an acquisition indicator bit AIs at a receiver in a communication system, the method comprising:
receiving multiplexed acquisition bits y=B×AI+n, where B is the signature matrix known at both base station and the receiver, AI is all the acquisition indicator bits and n represents noise;
calculating an estimated value of the AI bit of interest AÎMMSE (s) as a function of





  
    
      ∑
      j
    
    ⁢
    
      
        
          B
          T
        
        ⁡
        
          (
          
            s
            ,
            j
          
          )
        
      
      ×
      
        y
        ⁡
        
          (
          j
          )
        
      
    
  
  ,




where BT (s,.) is the s-th row vector of the transposed matrix BT for acquisition indicator bit AIs; and
setting the acquisition indicator bit AIs as follows:


AIs=−1, if AÎMMSE(s)<R



AIs=0, if R≦AÎMMSE(s)<U



AIs=1, if AÎMMSE(s)≧U



where R and U are decision thresholds,
wherein the estimated value AÎMMSE(S) is calculated by





  
    
      A
      ⁢
      
        
          
            I
            ^
          
          MMSE
        
        ⁡
        
          (
          s
          )
        
      
    
    =
    
      
        1
        
          (
          
            J
            +
            
              
                σ
                ^
              
              2
            
          
          )
        
      
      ⁢
      
        
          ∑
          0
          
            J
            -
            1
          
        
        ⁢
        
          
            
              B
              T
            
            ⁡
            
              (
              
                s
                ,
                j
              
              )
            
          
          ×
          
            y
            ⁡
            
              (
              j
              )
            
          
        
      
    
  
  ,



wherein {circumflex over (σ)}2 represents estimated noise variance at the receiver, and
J is an integer greater than or equal to 0.

2. The method according to claim 1, wherein R and U are equal absolute values.

3. The method according to claim 1, wherein R and U are −0.5 and 0.5, respectively.

4. The method according to claim 1, wherein the estimated value AÎMMSE(s) is calculated for J=32, thereby resulting in the equation:




  
    A
    ⁢
    
      
        
          I
          ^
        
        MMSE
      
      ⁡
      
        (
        s
        )
      
    
  
  =
  
    
      1
      
        (
        
          32
          +
          
            
              σ
              ^
            
            2
          
        
        )
      
    
    ⁢
    
      
        ∑
        0
        31
      
      ⁢
      
        
          
            B
            T
          
          ⁡
          
            (
            
              s
              ,
              j
            
            )
          
        
        ×
        
          
            y
            ⁡
            
              (
              j
              )
            
          
          .

5. The method of any of claim 1, wherein the method pre-stores a matrix B or BT.

6. The method of any of claim 1, wherein the method is performed in software.