Abstract:
The invention concerns a procedure for the estimation of unknown parameters (Δω, Δφ, ε,g a   sync ,g b   code ) of a received CDMA-signal (r desc (v)) which is transmitted by means of a transmission channel (11), in which the CDMA-signal has experienced changes to the parameters (Δω,Δφ,ε,g b   sync ,g b   code ) with the following steps: (a) formation of a cost function (L), which is dependent on the estimated values (Δ{tilde over (ω)},Δ{tilde over (φ)},{tilde over (ε)}, . . . ) of combined unknown parameters (Δω,Δφ,ε, . . . ); (b) partial differentiation of the cost function in respect to the said estimate values (Δ{tilde over (ω)},Δ{tilde over (φ)},{tilde over (ε)}, . . . ) of the unknown parameters (Δω,Δφ,ε, . . . ); (c) formation of a matrix-vector-equationfrom the presupposition that all partial differentials of the cost function are zero and thus a minimum of the cost function exists, and (d) computation of at least some of the matrix elements of the matrix elements of the matrix-vector-equation with the use of the Fast-Hadamard-Transformation.

Description:
BACKGROUND OF THE INVENTION 
   The invention concerns a procedure for the estimation of the parameters of a CDMA (Code Division Multiple Access) Signal and also concerns a corresponding computer program. The parameters to be estimated are, for instance, the time-shift, the frequency-shift and the phase-shift, to which the CDMA signal is subjected in the transmission signal and the gain factors. 
   As to the present state of the technology, one can refer to DE 43 02 679 A1, wherein a procedure for instantaneous frequency detection for a complex base-band is disclosed. This known procedure does not, however, adapt itself to the simultaneous determination of the time-shift and the phase-shift and further, this known procedure requires, in the case of broadband signals, a high investment in implementation. 
   All references cited herein are incorporated herein by reference in their entireties. 
   BRIEF SUMMARY OF THE INVENTION 
   Thus the invention has the purpose of creating a procedure for the estimation of parameters of a CDMA signal and a corresponding computer program, which calls for a small numerical complexity and a small cost in time and equipment, i.e., a small computational time. 
   The basis of the invention is, that by means of the employment of the Fast Hadamard-Transformation for computation of the coefficients, the numerical complexity can be substantially reduced. 

   
     BRIEF DESCRIPTION OF SEVERAL VIEWS OF THE DRAWINGS 
     An embodiment of the invention given below is described in more detail with reference to the drawings, wherein: 
       FIG. 1  is a block circuit diagram of a sender-model based on the invention procedure, 
       FIG. 2  is the code tree of a OVSF (Orthogonal Variable Spreading Factor) usable in the invented procedure (Spread Codes), 
       FIG. 3  is the block circuit diagram of a model based on the invented procedure of the transmission channel, 
       FIG. 4  is a schematic presentation to exhibit the structure of a coefficient matrix required by the numerical solution, and 
       FIG. 5  is a signal-flow-graph, which, with the Fast Hadamard-Transformation employed by the invented procedure, in natural form. 
   

   DETAILED DESCRIPTION OF THE INVENTION 
   In the following, the invented procedure is more closely described with the aid of an example embodiment. In the case of the following mathematical presentation, the following formula symbols are used: 
   
     
       
             
             
           
         
             
                 
             
           
           
             
               ε 
               Time shift 
             
             
               {circumflex over (ε)}, {tilde over (ε)} 
               Estimated value of the time shift 
             
             
               Δω 
               Frequency shift 
             
             
               Δ{circumflex over (ω)}, Δ{tilde over (ω)} 
               Estimated value of the frequency shift 
             
             
               Δφ 
               Phase shift 
             
             
               Δ{circumflex over (φ)}, Δ{tilde over (φ)} 
               Estimated value of the phase shift 
             
             
               ν 
               Time Index on the chip surface 
             
             
               c b  (ν) 
               Normed capacity, unscrambled chip signal 
             
             
                 
               of the b-ten code channel. 
             
             
               g b   code   
               Gain factor 
             
             
               g a   sync   
               Gain factor of the a-ten Synchronization Channel 
             
             
               J 
               Square root of minus one 
             
             
               l 
               Time index on symbol plane 
             
             
               n(ν) 
               Additive disturbance 
             
             
               r desc  (ν) 
               Unscrambled Measurement Signal 
             
             
               r b  (l) 
               Capacity normalized, undisturbed symbol of the 
             
             
                 
               b-ten code channel, which uses the b-ten 
             
             
                 
               spread code 
             
             
               REAL{. . .} 
               Real Part Operator 
             
             
               S desc  (ν) 
               Unscrambled reference signal 
             
             
               sync a  (ν) 
               Capacity normalized, unscrambled chipsignal 
             
             
                 
               of the a-ten synchronization channel. 
             
             
               SF b   
               Spreadfactor of the b-ten code channel 
             
             
               w b  (ν) 
               Spreadcode of the b-ten code channel 
             
             
               x(ν) 
               Chipsignal, which if employed for the Fast 
             
             
                 
               Hadamard-Transformation. 
             
             
               x b  (l) 
               Symbol signal, as a result of the Q-ten stage of the 
             
             
                 
               Fast Hadamard-Transformation. The symbols were 
             
             
                 
               spread with a spread code of the code class Q 
             
             
                 
               and the code number b. 
             
             
                 
             
           
        
       
     
       
       x(v) Chipsignal, which if employed for the Fast Hadamard Transformation. 
       Xb(l) Symbol signal, as a result of the Q-ten stage of the Fast Hadamard-Transformation. The symbols were spread with a spread code of the code class Q and the code number b. 
     
  
   In the following is described an estimation procedure for the approximation of unknown parameters which exhibit a small degree of complexity. This procedure is, in the case of the mobile function, that is to say, is operable in accord with the standards 3 GPP and CDMA2000, or generally by all mobile radio systems which employ “Orthogonal Variable Spreading Factor Codes” or “Wash-codes” as a spreading sequence. In  FIG. 1 , the block circuit diagram of the model of the sender  1  is based on the invented procedure. The symbols r b (l) of different code channels are separated by means of orthogonal spreading codes w b (v). The symbols r b (l) and the spreading codes w b (v) are spread upon the multiplier  2   o  to  2   N2 . Each code channel can possess a different gain factor g b   code  which is fed to a multiplier  3   o  to  3   N2 . As synchronization channels, unscrambled synchronization-chip-signals sync a (v) were sent which possess the gain factors g a   sync  which are fed to the multipliers  4   o  to  4   N1 . The codes sync a (v) of the synchronization channels are not orthogonal to the spreading codes w b (v). The unscrambled reference signal s desc (V) is the sum of the signals of all N 2  code channels and the signals of all N 1  synchronization channels and are created by the addition units  5  and  6 . 
   The spread code, used in the embodiment example, as these are presented in  FIG. 2 , are “Orthogonal Variable Spreading Factor Codes” (OVSF) and can have their origins from different code classes. The code-tree is, for instance, described in more detail in T. Ojampera, R. Prasad, “Wideband CDMA for Third Generation Mobile Communications”, Artech House, ISBN 0-89006-735-x, 1998, pages 111–113. 
   In the WCDMA-System in accord with 3GPP in general, the summation signal from the code channels is unscrambled by an unscrambling code. The synchronization channels are not scrambled. This fact is given consideration in the employed sender model, since the model describes the generation of a unscrambled sender signal s desc (v). In consideration of this, the synchronization channels send unscrambled code sequences sync a (v). 
   The model of the transmission channel  11 , as shown schematically in  FIG. 3 , takes into consideration an additive disturbance n(v), a normalized time-shift on the chip period Δω and a phase-shift Δφ which bias the scrambled reference signal, and repeats itself in the measurement signal:
 
 r   desc ( v )= s   desc ( v +ε)· e   +jΔω(v+ε)   ·e   +jΔφ   +n ( v )  (1)
 
   In the block circuit drawing are provided, on this account, two multipliers  7  and  8 , a time delay element  9  and an addition device  10 . 
   For the in-common-estimation of all unknown parameters, that is, the timeshift ε, the frequency-shift Δω, the phase-shift Δφ and the gain factors g a   sync  and g b   code  of the synchronization or code channel, a maximum-likelihood-approximation procedure is employed, which uses the following cost function: 
                     L   1     ⁡     (       Δ   ⁢           ⁢     ω   ~       ,     Δ   ⁢           ⁢     ϕ   ~       ,     ɛ   ~     ,       g   ~     a   sync     ,       g   ~     b   code       )       =       ∑     v   =   0       N   -   1       ⁢     |           r   desc     ⁡     (     v   -     ɛ   ~       )       ·     ⅇ       -   jΔ     ⁢           ⁢       ω   ~     ·   v         ·     ⅇ       -   jΔ     ⁢           ⁢     ϕ   ~           -       ∑     a   =   0         N   1     -   1       ⁢         g   ~     a   sync     ·       sync   a     ⁡     (   v   )           -       ∑     b   =   0         N   2     -   1       ⁢         g   ~     b   code     ·       c   b     ⁡     (   v   )             ⁢     |   2                 (   2   )               
wherein sync a (v) denotes the complex value, unscrambled, capacity normalized, undeformed chip-signal of the a-ten synchronization channel, also c b (v) stands for the complex valued, unscrambled, capacity normalized, chip-signal of the b-ten code channelg a   sync  of the gain factor of the a-ten synchronization channel and g b   code  represents the gain factor of the b-ten code channel.
 
   For the minimizing of the cost function, this is linearized, in which process a series development of the first order of the exponential function, as well as the measuring signal is used: 
   
     
       
         
           
             
               
                 
                   L 
                   ⁡ 
                   
                     ( 
                     
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ω 
                           ~ 
                         
                       
                       , 
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ϕ 
                           ~ 
                         
                       
                       , 
                       
                         ɛ 
                         ~ 
                       
                       , 
                       
                         
                           g 
                           ~ 
                         
                         a 
                         sync 
                       
                       , 
                       
                         
                           g 
                           ~ 
                         
                         b 
                         code 
                       
                     
                     ) 
                   
                 
                 = 
                 
                   
                     ∑ 
                     
                       v 
                       = 
                       0 
                     
                     
                       N 
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     | 
                     
                       
                         
                           r 
                           desc 
                         
                         ⁡ 
                         
                           ( 
                           ν 
                           ) 
                         
                       
                       - 
                       
                         j 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               r 
                               desc 
                             
                             ⁡ 
                             
                               ( 
                               ν 
                               ) 
                             
                           
                           · 
                           Δ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ω 
                             ~ 
                           
                           · 
                           ν 
                         
                       
                       - 
                       
                           
                       
                       ⁢ 
                       
                         
 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         j 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               r 
                               desc 
                             
                             ⁡ 
                             
                               ( 
                               ν 
                               ) 
                             
                           
                           · 
                           Δ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ϕ 
                           ~ 
                         
                       
                       - 
                       
                         
                           
                             r 
                             desc 
                             ′ 
                           
                           ⁡ 
                           
                             ( 
                             ν 
                             ) 
                           
                         
                         · 
                         
                           ɛ 
                           ~ 
                         
                       
                       - 
                       
                         
                           ∑ 
                           
                             a 
                             = 
                             0 
                           
                           
                             
                               N 
                               1 
                             
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             
                               g 
                               ~ 
                             
                             a 
                             sync 
                           
                           · 
                           
                             
                               sync 
                               a 
                             
                             ⁡ 
                             
                               ( 
                               ν 
                               ) 
                             
                           
                         
                       
                       - 
                       
                         
                           ∑ 
                           
                             b 
                             = 
                             0 
                           
                           
                             
                               N 
                               2 
                             
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             
                               g 
                               ~ 
                             
                             b 
                             code 
                           
                           · 
                           
                             
                               c 
                               b 
                             
                             ⁡ 
                             
                               ( 
                               ν 
                               ) 
                             
                           
                         
                       
                     
                     ⁢ 
                     
                       | 
                       2 
                     
                   
                 
               
             
             
               
                 ( 
                 3 
                 ) 
               
             
           
         
       
     
   
   The cross terms between the unknown parameters are neglected, so that the minimizing of the cost function with a linear equation can be undertaken. This is reliable, as long as the unknown parameters are small, which, if necessary, can be attained by several reiterations. This means that the here presented method can be applied only for the more refined approximating. 
   For the computation of the partial derivatives of the linearized cost function in accord with the unknown parameters, the following formulations are employed: An unknown parameter x is a real value number, the constants c and d are complex numbers and a cost function employed as a squared amount:
 
 L=|c·x+d|   2 =( c·x+d )·( c·x+d )*=| c|   2   ·x   2   +c*·d·x+c·d*·x+|d|   2   (4)
 
Now, the partial differential may be computed:
 
                     ∂   L       ∂   x       =       2   ·          c        2     ·   x     +       2   ·   REAL     ⁢       {     c   ·     d   *       }     .                 (   5   )               
With equation 5, the partial derivative with respect to the frequency shift to:
 
                     ∂   L         ∂   Δ     ⁢           ⁢     ω   ^         =         2   ⁢       ∑     v   =   0       N   -   1       ⁢                  r   desc     ⁡     (   v   )            2     ·     v   2     ·   Δ     ⁢           ⁢     ω   ^           +     2   ⁢       ∑     v   =   0       N   -   1       ⁢     REAL   ⁢     {       -   j     ·       r   desc     ⁡     (   v   )       ·   v   ·       a   0   *     ⁡     (   v   )         }             =   0             (   6   )               
with the following
 
                       a   0     ⁡     (   v   )       =         r   desc     ⁡     (   v   )       -     j   ⁢           ⁢         r   desc     ⁡     (   v   )       ·   Δ     ⁢           ⁢     ϕ   ^       -         r   desc   ′     ⁡     (   v   )       ·     ɛ   ^       -       ∑     a   =   0         N   1     -   1       ⁢         g   ^     a   sync     ·       sync   a     ⁡     (   v   )           -       ∑     b   =   0         N   2     -   1       ⁢         g   ^     b   code     ·       c   b     ⁡     (   v   )               ,           (   7   )               
the partial derivative with respect to the phase shift, to
 
   
     
       
         
           
             
               
                 
                   
                     ∂ 
                     L 
                   
                   
                     
                       ∂ 
                       Δ 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ϕ 
                       ^ 
                     
                   
                 
                 = 
                 
                   
                     
                       2 
                       ⁢ 
                       
                         
                           ∑ 
                           
                             v 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             
                               
                                  
                                 
                                   
                                     r 
                                     desc 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     v 
                                     ) 
                                   
                                 
                                  
                               
                               2 
                             
                             · 
                             Δ 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ϕ 
                             ^ 
                           
                         
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         
                           ∑ 
                           
                             v 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           REAL 
                           ⁢ 
                           
                             { 
                             
                               
                                 - 
                                 j 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   
                                     r 
                                     desc 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     v 
                                     ) 
                                   
                                 
                                 · 
                                 
                                   
                                     a 
                                     1 
                                     * 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     v 
                                     ) 
                                   
                                 
                               
                             
                             } 
                           
                         
                       
                     
                   
                   = 
                   0 
                 
               
             
             
               
                 ( 
                 8 
                 ) 
               
             
           
         
       
     
   
   with 
                       a   1     ⁡     (   v   )       =         r   desc     ⁡     (   v   )       -     j   ⁢           ⁢         r   desc     ⁡     (   v   )       ·   Δ     ⁢           ⁢       ω   ^     ·   v       -         r   ′     ⁡     (   v   )       ·     ɛ   ^       -       ∑     a   =   0         N   1     -   1       ⁢         g   ^     a   sync     ·       sync   a     ⁡     (   v   )           -       ∑     b   =   0         N   2     -   1       ⁢         g   ^     b   code     ·       c   b   ′     ⁡     (   v   )               ,           (   9   )               
the partial derivative with respect to the time shift, to
 
   
     
       
         
           
             
               
                 
                   
                     ∂ 
                     L 
                   
                   
                     ∂ 
                     
                       ɛ 
                       ^ 
                     
                   
                 
                 = 
                 
                   
                     
                       2 
                       ⁢ 
                       
                         
                           ∑ 
                           
                             v 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             
                               
                                  
                                 
                                   
                                     r 
                                     desc 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     v 
                                     ) 
                                   
                                 
                                  
                               
                               2 
                             
                             · 
                             Δ 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             v 
                             ^ 
                           
                         
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         
                           ∑ 
                           
                             v 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           REAL 
                           ⁢ 
                           
                             { 
                             
                               
                                 - 
                                 
                                   
                                     r 
                                     desc 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     v 
                                     ) 
                                   
                                 
                               
                               · 
                               
                                 
                                   a 
                                   2 
                                   * 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   v 
                                   ) 
                                 
                               
                             
                             } 
                           
                         
                       
                     
                   
                   = 
                   0 
                 
               
             
             
               
                 ( 
                 10 
                 ) 
               
             
           
         
       
     
   
   with 
                       a   2     ⁡     (   v   )       =         r   desc     ⁡     (   v   )       -     j   ⁢           ⁢         r   desc     ⁡     (   v   )       ·   Δ     ⁢           ⁢       ω   ^     ·   v       -     j   ⁢           ⁢         r   desc   ′     ⁡     (   v   )       ·   Δ     ⁢           ⁢     ϕ   ^       -       ∑     a   =   0         N   1     -   1       ⁢         g   ^     a   sync     ·       sync   a     ⁡     (   v   )           -       ∑     b   =   0         N   2     -   1       ⁢           g   ^     b   code     ·     c   b       ⁢     (   v   )             ,           (   11   )               
the partial derivative, with respect to the gain factors of the synchronization channels, to
 
   
     
       
         
           
             
               
                 
                   
                     ∂ 
                     L 
                   
                   
                     ∂ 
                     
                       
                         g 
                         ^ 
                       
                       μ 
                       sync 
                     
                   
                 
                 = 
                 
                   
                     
                       2 
                       ⁢ 
                       
                         
                           ∑ 
                           
                             v 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             
                                
                               
                                 
                                   sync 
                                   μ 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   v 
                                   ) 
                                 
                               
                                
                             
                             2 
                           
                           · 
                           
                             
                               g 
                               ^ 
                             
                             μ 
                             sync 
                           
                         
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         
                           ∑ 
                           
                             v 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           REAL 
                           ⁢ 
                           
                             { 
                             
                               
                                 - 
                                 
                                   
                                     sync 
                                     μ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     v 
                                     ) 
                                   
                                 
                               
                               · 
                               
                                 
                                   a 
                                   3 
                                   * 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   v 
                                   ) 
                                 
                               
                             
                             } 
                           
                         
                       
                     
                   
                   = 
                   0 
                 
               
             
             
               
                 ( 
                 12 
                 ) 
               
             
           
         
       
     
   
   with 
                     a   3     ⁡     (   v   )       =         r   desc     ⁡     (   v   )       -     j   ⁢           ⁢         r   desc     ⁡     (   v   )       ·   Δ     ⁢           ⁢       ω   ^     ·   v       -     j   ⁢           ⁢         r   desc     ⁡     (   v   )       ·   Δ     ⁢           ⁢     ϕ   ^       -         r   desc   ′     ⁡     (   v   )       ·     ɛ   ^       -       ∑     a   =   0         N   1     -   1       ⁢         g   ^     a   sync     ·       sync   a     ⁡     (   v   )           -       ∑     b   =   0         N   2     -   1       ⁢         g   ^     b   code     ·       c   b     ⁡     (   v   )                     (   13   )               
and the partial derivatives with respect to the gain factors of the code channels to
 
   
     
       
         
           
             
               
                 
                   
                     ∂ 
                     L 
                   
                   
                     ∂ 
                     
                       
                         g 
                         ^ 
                       
                       μ 
                       code 
                     
                   
                 
                 = 
                 
                   
                     
                       2 
                       ⁢ 
                       
                         
                           ∑ 
                           
                             v 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             
                                
                               
                                 
                                   c 
                                   μ 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   v 
                                   ) 
                                 
                               
                                
                             
                             2 
                           
                           · 
                           
                             
                               g 
                               ^ 
                             
                             μ 
                             code 
                           
                         
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         
                           ∑ 
                           
                             v 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           REAL 
                           ⁢ 
                           
                             { 
                             
                               
                                 - 
                                 
                                   
                                     c 
                                     μ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     v 
                                     ) 
                                   
                                 
                               
                               · 
                               
                                 
                                   a 
                                   4 
                                   * 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   v 
                                   ) 
                                 
                               
                             
                             } 
                           
                         
                       
                     
                   
                   = 
                   0 
                 
               
             
             
               
                 ( 
                 14 
                 ) 
               
             
           
         
       
     
   
   with 
   
     
       
         
           
             
               
                 
                   
                     a 
                     4 
                   
                   ⁡ 
                   
                     ( 
                     v 
                     ) 
                   
                 
                 = 
                 
                   
                     
                       r 
                       desc 
                     
                     ⁡ 
                     
                       ( 
                       v 
                       ) 
                     
                   
                   = 
                   
                     
                       j 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           
                             r 
                             desc 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                         · 
                         Δ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ω 
                           ^ 
                         
                         · 
                         v 
                       
                     
                     - 
                     
                       j 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           
                             r 
                             desc 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                         · 
                         Δ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ϕ 
                         ^ 
                       
                     
                     - 
                     
                       
                         
                           r 
                           desc 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           v 
                           ) 
                         
                       
                       · 
                       
                         ɛ 
                         ^ 
                       
                     
                     - 
                     
                       
                         ∑ 
                         
                           a 
                           = 
                           0 
                         
                         
                           
                             N 
                             1 
                           
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           
                             g 
                             ^ 
                           
                           a 
                           sync 
                         
                         · 
                         
                           
                             sync 
                             a 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                       
                     
                     - 
                     
                       
                         ∑ 
                         
                           b 
                           = 
                           0 
                         
                         
                           
                             N 
                             2 
                           
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           
                             g 
                             ^ 
                           
                           b 
                           code 
                         
                         · 
                         
                           
                             c 
                             b 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 15 
                 ) 
               
             
           
         
       
     
   
   The equations (12, 13) and the equations (14, 15) are valid for all synchronization channels or for all code channels. The equations (6, 7), (8, 9), (10, 11), (12, 13), (14, 15) can be summarized in a matrix-vector statement: 
   
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             A 
                             
                               0 
                               , 
                               0 
                             
                           
                         
                         
                           
                               
                           
                         
                         
                           
                             A 
                             
                               0 
                               , 
                               
                                 3 
                                 ⁢ 
                                 a 
                               
                             
                           
                         
                         
                           
                             A 
                             
                               0 
                               , 
                               
                                 4 
                                 ⁢ 
                                 b 
                               
                             
                           
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           ⋰ 
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           
                             A 
                             
                               
                                 3 
                                 ⁢ 
                                 a 
                               
                               , 
                               0 
                             
                           
                         
                         
                           
                               
                           
                         
                         
                           
                             A 
                             
                               
                                 3 
                                 ⁢ 
                                 a 
                               
                               , 
                               
                                 3 
                                 ⁢ 
                                 a 
                               
                             
                           
                         
                         
                           
                             A 
                             
                               
                                 3 
                                 ⁢ 
                                 a 
                               
                               , 
                               
                                 4 
                                 ⁢ 
                                 b 
                               
                             
                           
                         
                       
                       
                         
                           
                             A 
                             
                               
                                 4 
                                 ⁢ 
                                 b 
                               
                               , 
                               0 
                             
                           
                         
                         
                           
                               
                           
                         
                         
                           
                             A 
                             
                               
                                 4 
                                 ⁢ 
                                 b 
                               
                               , 
                               
                                 3 
                                 ⁢ 
                                 a 
                               
                             
                           
                         
                         
                           
                             A 
                             
                               
                                 4 
                                 ⁢ 
                                 b 
                               
                               , 
                               
                                 4 
                                 ⁢ 
                                 b 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                   · 
                   
                     [ 
                     
                       
                         
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ω 
                               ^ 
                             
                           
                         
                       
                       
                         
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ϕ 
                               ^ 
                             
                           
                         
                       
                       
                         
                           
                             ɛ 
                             ^ 
                           
                         
                       
                       
                         
                           
                             
                               g 
                               ^ 
                             
                             a 
                             sync 
                           
                         
                       
                       
                         
                           
                             
                               g 
                               ^ 
                             
                             b 
                             code 
                           
                         
                       
                     
                     ] 
                   
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           b 
                           0 
                         
                       
                     
                     
                       
                         ⋮ 
                       
                     
                     
                       
                         
                           b 
                           
                             3 
                             ⁢ 
                             a 
                           
                         
                       
                     
                     
                       
                         
                           b 
                           
                             4 
                             ⁢ 
                             b 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
             
               
                 ( 
                 16 
                 ) 
               
             
           
         
       
     
   
   whereby, the coefficients of the first row come to:
 
b a=0   (17)
 
   
     
       
         
           
             
               
                 
                   A 
                   
                     0 
                     , 
                     0 
                   
                 
                 = 
                 
                   
                     ∑ 
                     v 
                   
                   ⁢ 
                   
                     
                       
                          
                         
                           
                             r 
                             desc 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                          
                       
                       2 
                     
                     · 
                     
                       v 
                       2 
                     
                   
                 
               
             
             
               
                 ( 
                 18 
                 ) 
               
             
           
         
       
     
   
   
     
       
         
           
             
               
                 
                   A 
                   
                     0 
                     , 
                     1 
                   
                 
                 = 
                 
                   
                     ∑ 
                     v 
                   
                   ⁢ 
                   
                     
                       
                          
                         
                           
                             r 
                             desc 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                          
                       
                       2 
                     
                     · 
                     v 
                   
                 
               
             
             
               
                 ( 
                 19 
                 ) 
               
             
           
           
             
               
                 
                   A 
                   
                     0 
                     , 
                     2 
                   
                 
                 = 
                 
                   
                     ∑ 
                     v 
                   
                   ⁢ 
                   
                     REAL 
                     ⁢ 
                     
                       { 
                       
                         j 
                         · 
                         
                           
                             r 
                             desc 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                         · 
                         
                           
                             r 
                             desc 
                             
                               ′ 
                               * 
                             
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                         · 
                         v 
                       
                       } 
                     
                   
                 
               
             
             
               
                 ( 
                 20 
                 ) 
               
             
           
           
             
               
                 
                   A 
                   
                     0 
                     , 
                     
                       3 
                       ⁢ 
                       a 
                     
                   
                 
                 = 
                 
                   
                     ∑ 
                     v 
                   
                   ⁢ 
                   
                     REAL 
                     ⁢ 
                     
                       { 
                       
                         j 
                         · 
                         
                           
                             r 
                             desc 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                         · 
                         v 
                         · 
                         
                           
                             sync 
                             a 
                             * 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 ( 
                 21 
                 ) 
               
             
           
           
             
               
                 
                   
                     A 
                     
                       0 
                       , 
                       
                         4 
                         ⁢ 
                         b 
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       v 
                     
                     ⁢ 
                     
                       REAL 
                       ⁢ 
                       
                         { 
                         
                           j 
                           · 
                           
                             
                               r 
                               desc 
                             
                             ⁡ 
                             
                               ( 
                               v 
                               ) 
                             
                           
                           · 
                           v 
                           · 
                           
                             
                               c 
                               b 
                               * 
                             
                             ⁡ 
                             
                               ( 
                               v 
                               ) 
                             
                           
                         
                         } 
                       
                     
                   
                 
                 , 
               
             
             
               
                 ( 
                 22 
                 ) 
               
             
           
         
       
     
   
   The coefficients of the second line show:
 
b 1 =0   (23)
 
   
     
       
         
           
             
               
                 
                   A 
                   
                     1 
                     , 
                     0 
                   
                 
                 = 
                 
                   
                     ∑ 
                     v 
                   
                   ⁢ 
                   
                     
                       
                          
                         
                           
                             r 
                             desc 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                          
                       
                       2 
                     
                     · 
                     v 
                   
                 
               
             
             
               
                 ( 
                 24 
                 ) 
               
             
           
           
             
               
                 
                   A 
                   
                     1 
                     , 
                     1 
                   
                 
                 = 
                 
                   
                     ∑ 
                     v 
                   
                   ⁢ 
                   
                     
                        
                       
                         
                           r 
                           desc 
                         
                         ⁡ 
                         
                           ( 
                           v 
                           ) 
                         
                       
                        
                     
                     2 
                   
                 
               
             
             
               
                 ( 
                 25 
                 ) 
               
             
           
           
             
               
                 
                   A 
                   
                     1 
                     , 
                     2 
                   
                 
                 = 
                 
                   
                     ∑ 
                     v 
                   
                   ⁢ 
                   
                     REAL 
                     ⁢ 
                     
                       { 
                       
                         j 
                         · 
                         
                           
                             r 
                             desc 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                         · 
                         
                           
                             r 
                             
                               ′ 
                               * 
                             
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 ( 
                 26 
                 ) 
               
             
           
           
             
               
                 
                   
                     A 
                     
                       1 
                       , 
                       
                         3 
                         ⁢ 
                         a 
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       v 
                     
                     ⁢ 
                     
                       REAL 
                       ⁢ 
                       
                         { 
                         
                           j 
                           · 
                           
                             
                               r 
                               desc 
                             
                             ⁡ 
                             
                               ( 
                               v 
                               ) 
                             
                           
                           · 
                           
                             
                               sync 
                               a 
                               * 
                             
                             ⁡ 
                             
                               ( 
                               v 
                               ) 
                             
                           
                         
                         } 
                       
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     
                       A 
                       
                         1 
                         , 
                         
                           4 
                           ⁢ 
                           b 
                         
                       
                     
                     = 
                     
                       
                         ∑ 
                         v 
                       
                       ⁢ 
                       
                         REAL 
                         ⁢ 
                         
                           { 
                           
                             j 
                             · 
                             
                               
                                 r 
                                 desc 
                               
                               ⁡ 
                               
                                 ( 
                                 v 
                                 ) 
                               
                             
                             · 
                             
                               
                                 c 
                                 b 
                                 * 
                               
                               ⁡ 
                               
                                 ( 
                                 v 
                                 ) 
                               
                             
                           
                           } 
                         
                       
                     
                   
                   , 
                 
               
             
             
               
                 ( 
                 27 
                 ) 
               
             
           
         
       
     
   
   The coefficients of the third row show: 
   
     
       
         
           
             
               
                 
                   b 
                   2 
                 
                 = 
                 
                   
                     ∑ 
                     v 
                   
                   ⁢ 
                   
                     REAL 
                     ⁢ 
                     
                       { 
                       
                         
                           
                             r 
                             desc 
                             ′ 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                         · 
                         
                           
                             r 
                             desc 
                             * 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 ( 
                 29 
                 ) 
               
             
           
           
             
               
                 
                   A 
                   
                     2 
                     , 
                     0 
                   
                 
                 = 
                 
                   
                     ∑ 
                     v 
                   
                   ⁢ 
                   
                     REAL 
                     ⁢ 
                     
                       { 
                       
                         
                           - 
                           j 
                         
                         · 
                         
                           
                             r 
                             desc 
                             ′ 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                         · 
                         
                           
                             r 
                             desc 
                             * 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                         · 
                         v 
                       
                       } 
                     
                   
                 
               
             
             
               
                 ( 
                 30 
                 ) 
               
             
           
           
             
               
                 
                   A 
                   
                     2 
                     , 
                     1 
                   
                 
                 = 
                 
                   
                     ∑ 
                     v 
                   
                   ⁢ 
                   
                     REAL 
                     ⁢ 
                     
                       { 
                       
                         
                           - 
                           j 
                         
                         · 
                         
                           
                             r 
                             desc 
                             ′ 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                         · 
                         
                           
                             r 
                             desc 
                             * 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 ( 
                 31 
                 ) 
               
             
           
           
             
               
                 
                   A 
                   
                     2 
                     , 
                     2 
                   
                 
                 = 
                 
                   
                     ∑ 
                     v 
                   
                   ⁢ 
                   
                     
                        
                       
                         
                           r 
                           desc 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           v 
                           ) 
                         
                       
                        
                     
                     2 
                   
                 
               
             
             
               
                 ( 
                 32 
                 ) 
               
             
           
           
             
               
                 
                   A 
                   
                     2 
                     , 
                     
                       3 
                       ⁢ 
                       a 
                     
                   
                 
                 = 
                 
                   
                     ∑ 
                     v 
                   
                   ⁢ 
                   
                     REAL 
                     ⁢ 
                     
                       { 
                       
                         
                           
                             r 
                             desc 
                             ′ 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                         · 
                         
                           
                             sync 
                             a 
                             * 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 ( 
                 33 
                 ) 
               
             
           
           
             
               
                 
                   
                     A 
                     
                       2 
                       , 
                       
                         4 
                         ⁢ 
                         b 
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       v 
                     
                     ⁢ 
                     
                       REAL 
                       ⁢ 
                       
                         { 
                         
                           
                             
                               r 
                               desc 
                               ′ 
                             
                             ⁡ 
                             
                               ( 
                               v 
                               ) 
                             
                           
                           · 
                           
                             
                               c 
                               b 
                               * 
                             
                             ⁡ 
                             
                               ( 
                               v 
                               ) 
                             
                           
                         
                         } 
                       
                     
                   
                 
                 , 
               
             
             
               
                 ( 
                 34 
                 ) 
               
             
           
         
       
     
   
   The coefficients of the fourth row show, 
   
     
       
         
           
             
               
                 
                   b 
                   
                     3 
                     ⁢ 
                     a 
                   
                 
                 = 
                 
                   
                     ∑ 
                     v 
                   
                   ⁢ 
                   
                     REAL 
                     ⁢ 
                     
                       { 
                       
                         
                           
                             sync 
                             a 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                         · 
                         
                           
                             r 
                             desc 
                             * 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 ( 
                 35 
                 ) 
               
             
           
           
             
               
                 
                   A 
                   
                     
                       3 
                       ⁢ 
                       a 
                     
                     , 
                     0 
                   
                 
                 = 
                 
                   
                     ∑ 
                     v 
                   
                   ⁢ 
                   
                     REAL 
                     ⁢ 
                     
                       { 
                       
                         
                           - 
                           
                             
                               sync 
                               a 
                             
                             ⁡ 
                             
                               ( 
                               v 
                               ) 
                             
                           
                         
                         · 
                         j 
                         · 
                         
                           
                             r 
                             desc 
                             * 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                         · 
                         v 
                       
                       } 
                     
                   
                 
               
             
             
               
                 ( 
                 36 
                 ) 
               
             
           
           
             
               
                 
                   A 
                   
                     
                       3 
                       ⁢ 
                       a 
                     
                     , 
                     1 
                   
                 
                 = 
                 
                   
                     ∑ 
                     v 
                   
                   ⁢ 
                   
                     REAL 
                     ⁢ 
                     
                       { 
                       
                         
                           - 
                           
                             
                               sync 
                               a 
                             
                             ⁡ 
                             
                               ( 
                               v 
                               ) 
                             
                           
                         
                         · 
                         j 
                         · 
                         
                           
                             r 
                             desc 
                             * 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 ( 
                 37 
                 ) 
               
             
           
           
             
               
                 
                   A 
                   
                     
                       3 
                       ⁢ 
                       a 
                     
                     , 
                     2 
                   
                 
                 = 
                 
                   
                     ∑ 
                     v 
                   
                   ⁢ 
                   
                     REAL 
                     ⁢ 
                     
                       { 
                       
                         
                           
                             sync 
                             a 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                         · 
                         
                           
                             r 
                             desc 
                             
                               ′ 
                               * 
                             
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 ( 
                 38 
                 ) 
               
             
           
           
             
               
                 
                   A 
                   
                     
                       3 
                       ⁢ 
                       a 
                     
                     , 
                     
                       3 
                       ⁢ 
                       μ 
                     
                   
                 
                 = 
                 
                   
                     ∑ 
                     v 
                   
                   ⁢ 
                   
                     REAL 
                     ⁢ 
                     
                       { 
                       
                         
                           
                             sync 
                             a 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                         · 
                         
                           
                             sync 
                             μ 
                             * 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 ( 
                 39 
                 ) 
               
             
           
           
             
               
                 
                   A 
                   
                     
                       3 
                       ⁢ 
                       a 
                     
                     , 
                     
                       4 
                       ⁢ 
                       b 
                     
                   
                 
                 = 
                 
                   
                     ∑ 
                     v 
                   
                   ⁢ 
                   
                     REAL 
                     ⁢ 
                     
                       { 
                       
                         
                           
                             sync 
                             a 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                         · 
                         
                           
                             c 
                             b 
                             * 
                           
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 ( 
                 40 
                 ) 
               
             
           
         
       
     
   
   and the coefficients of the fifth row are 
                   b     4   ⁢   b       =       ∑   v     ⁢     REAL   ⁢     {         c   b     ⁡     (   v   )       ·       r   desc   *     ⁡     (   v   )         }                 (   41   )                 A       4   ⁢   b     ,   0       =       ∑   v     ⁢     REAL   ⁢     {       -       c   b     ⁡     (   v   )         ·   j   ·       r   desc   *     ⁡     (   v   )       ·   v     }                 (   42   )                 A       4   ⁢   b     ,   1       =       ∑   v     ⁢     REAL   ⁢     {       -       c   b     ⁡     (   v   )         ·   j   ·       r   desc   *     ⁡     (   v   )         }                 (   43   )                 A       4   ⁢   b     ,   2       =       ∑   v     ⁢     REAL   ⁢     {         c   b     ⁡     (   v   )       ·       r   desc     ′   *       ⁡     (   v   )         }                 (   44   )                 A       4   ⁢   b     ,     3   ⁢   a         =       ∑   v     ⁢     REAL   ⁢     {         c   b     ⁡     (   v   )       ·       sync   a   *     ⁡     (   v   )         }                 (   45   )                 A       4   ⁢   b     ,     4   ⁢   μ         =       ∑   v     ⁢     REAL   ⁢     {         c   b     ⁡     (   v   )       ·       c   μ   *     ⁡     (   v   )         }                 (   46   )               
to the codes of the synchronization channels. Because of the orthogonal characteristic of the code channels, the coefficients A 4b,4μ  for b=μ equal zero. The structure of the matrix A is presented in  FIG. 4 .
 
For the computation of the coefficients:
     A 0 4b , A 1 4b , A 2 4b , A 3a,4b , A 4b,0 , A 4b,1 , A 4b,2 , A 4b,3a  and b 4b  
 
correlation products of the form:
   
                   R     x   ,   c       =       ∑   v     ⁢     REAL   ⁢     {       x   ⁡     (   v   )       ·       [       c   b     ⁡     (   v   )       ]     *       }                 (   47   )               
must be computed, whereby the signal x(v) can be one of the following: x(v)=r(v), c(v)=r′(v) or x(v)=sync(v). The direct calculation of this correlation would have a high numerical complexity.
 
   The algorithms for the estimation of all unknown parameters can be implemented with a reduced numerical complexity, in case the gain factors of a plurality of code channels must be estimated. In this case, the Fast Hadamard-Transformation for the computation of the coefficients A 0,4b , A 1,4b , A 2,4b , A 3a,4b , A 3a,4b , A 4b,0 , A 4b,1 , A 4b,2 , A 4b3a , and b 4b  can be efficiently employed. 
   The capacity normalized, unscrambled, undistorted chip signal c b (l·SF b +v)−r b (l)·w b (v) (48) of a code/channel emerges from the spreading of the symbol r b (l) of the code channel with its spreading code w b (v). The magnitude of SF b  presents the spreading factor of the code channel. 
   The equation (47) and the equation (48) can be brought together in the expression: 
   
     
       
         
           
             
               
                 
                   R 
                   
                     x 
                     , 
                     c 
                   
                 
                 = 
                 
                   
                     
                       ∑ 
                       l 
                     
                     ⁢ 
                     
                       REAL 
                       ⁢ 
                       
                         { 
                         
                           
                             
                               [ 
                               
                                 
                                   r 
                                   b 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   l 
                                   ) 
                                 
                               
                               ] 
                             
                             * 
                           
                           · 
                           
                             
                               ∑ 
                               v 
                             
                             ⁢ 
                             
                               
                                 x 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       l 
                                       · 
                                       
                                         SF 
                                         b 
                                       
                                     
                                     + 
                                     v 
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 
                                   w 
                                   b 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   v 
                                   ) 
                                 
                               
                             
                           
                         
                         } 
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       l 
                     
                     ⁢ 
                     
                       REAL 
                       ⁢ 
                       
                         { 
                         
                           
                             
                               [ 
                               
                                 
                                   r 
                                   b 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   l 
                                   ) 
                                 
                               
                               ] 
                             
                             * 
                           
                           · 
                           
                             
                               x 
                               b 
                             
                             ⁡ 
                             
                               ( 
                               l 
                               ) 
                             
                           
                         
                         } 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 49 
                 ) 
               
             
           
         
       
     
   
   The inner sum from equation (49) can now be computed efficiently for all codes in a code class with the Fast Hadamard-Transformation, so that the cross-correlation-coefficients now need only to be computed on the symbol plane. 
   In  FIG. 5  is presented the signal flow sheet of a Fast Hadamard-Transformation of the natural form of the length four. The chip-signal x(v) transformed in the first stage of the transformation in the code class CC=1. The results in the first stage of the transformation, x 0 (1+0),x 1 (1+0),x 0 (1+1) and x 1 (1+1), represent the inner sum of the equation 49 for the code channels, which the spreading codes from the code class CC=1 employ. In the second stage of the transformation, one obtains the results of the inner summation of equation (49) for the code channels, which the spreading code uses from the code class CC−2. 
   The numerical complexity lessens, because, first, the Fast Hadamard-Transformation possesses a complexity of M·log M in comparison to the complexity of the direct computation with equation (49) of M 2 . Further, in the computation of the inner summation of equation (49) only two real value signals must be considered, and it need not, as is the case with the direct computation from equation (47) be carried out by computations with complex valued signals. 
   While the invention has been described in detail and with reference to specific examples thereof, it will be apparent to one skilled in the art that various changes and modifications can be made therein without departing from the spirit and scope thereof.