PATENT DOCUMENT

Abstract:
A transmitter is described which has a determination unit for determining a plurality of symbols ( 301 ) such that each symbol ( 301 ) comprises a tail component ( 302 ) and the plurality is determined in such a way that the tail components ( 302 ) are substantially equal and a tone generation unit for arranging the plurality of symbols ( 301 ) to form a plurality of long preambles ( 300 ) such that the plurality of long preambles ( 300 ) are tone-interleaved.

Full Description:
FIELD OF THE INVENTION 
       [0001]    The invention relates to a transmitter, a method for generating a plurality of long preambles and a communication device. 
       BACKGROUND OF THE INVENTION 
       [0002]    With the introduction of multiple antennas in the realization of MIMO (Multiple Input Multiple Output) systems, the transmission capacity of OFDM (orthogonal frequency division multiplexing) systems is largely expanded as the number of communication channels in space is multiplied. This benefit is derived, however, at the expense of a more complex receiver design compared with the SISO (single input single output) solution. A direct consequence of the MIMO structure that contributes to the added complexity is the need for identification of more spatial channels. More sophisticated channel estimation schemes and more pilot symbols are therefore needed. 
         [0003]    Several challenging issues are posed when considering the design of pilots for channel estimation. The most pressing of all is associated with the provision of adequate information for identifying all channel responses between every transmit and receive antenna pair. More preambles certainly allow for more accurate channel estimation and simpler design, but are too expensive to engage due to a reduction in throughput. An excessive rise in overhead can undermine the whole idea of enhancing data throughput in MIMO transmission altogether. 
         [0004]      FIG. 1  shows a long preamble  100  according to prior art. The long preamble  100  is used for the single antenna case and comprises three OFDM symbols  102  of FFT (fast Fourier transform) size N FFT =8 and each OFDM symbol comprises a cyclic prefix (guard interval)  101  of the length N CP =3 modulated symbols. 
         [0005]    In this conventional design, one cyclic prefix  101  is provided for each OFDM symbol  102  in the long preamble  100  to protect it against inter-symbol interference (ISI). 
         [0006]    An object of the invention is to provide a method for generating long preambles such that the transmission overhead necessary for long preamble transmission can be reduced compared to prior art. 
         [0007]    The object is achieved by a transmitter, a method for generating a plurality of long preambles and a communication device with the features according to the independent claims. 
       SUMMARY OF THE INVENTION 
       [0008]    A transmitter is provided comprising a determination unit for determining a plurality of symbols such that each symbol comprises a tail component and the plurality is determined in such a way that the tail components are substantially equal. The transmitter further comprises a tone generation unit for arranging the plurality of symbols to form a plurality of long preambles such that the plurality of long preambles is tone-interleaved. 
         [0009]    Further, a method for generating a plurality of long preambles and a communication device according to the transmitter described above are provided. 
     
    
     
       SHORT DESCRIPTION OF THE FIGURES 
         [0010]      FIG. 1  shows a long preamble according to prior art. 
           [0011]      FIG. 2  shows a transmitter according to an embodiment of the invention. 
           [0012]      FIG. 3  shows three long preambles according to an embodiment of the invention. 
           [0013]      FIG. 4  illustrates the frequency domain power in the transmission of long preambles according to an embodiment of the invention. 
           [0014]      FIG. 5  illustrates the time domain values according to an embodiment of the invention. 
           [0015]      FIG. 6  illustrates the time frequency domain values according to an embodiment of the invention. 
       
    
    
     DETAILED DESCRIPTION  
       [0016]    Illustratively, the symbols are determined in such a way that they share a common tail and long preambles are generated such that they are tone-interleaved. Due to the common tail, no cyclic prefix of the symbols in the long preambles is necessary. Therefore, the long preambles can be significantly shortened compared to prior art. In particular, in case of a MIMO (multiple input multiple output) system, where relatively many long preamble symbols are necessary for channel estimation, a substantial reduction in transmission overhead can be achieved without compromise in performance. The common tail may be seen as an apparent cyclic prefix which is absorbed into each long preamble. The tone-interleaving property allows for simple channel estimation at the receiver. 
         [0017]    In one embodiment, an iterative method is employed to meet the requirement of unit preamble amplitude in the frequency domain and orthogonality in space-time-for optimal channel estimation performance. A template vector is constructed in one embodiment as a basis to facilitate the design of long preambles transmitted at different time and space regardless of the number of transmit and receive antennas in the corresponding communication system. 
         [0018]    Embodiments of the invention emerge from the dependent claims. The embodiments which are described in the context of the transmitter are analogously valid for the method for generating a plurality of long preambles and the communication device. 
         [0019]    The symbols are in one embodiment modulation symbols. For example, the symbols are OFDM symbols. 
         [0020]    The symbols can be determined based on a decomposition of a plurality of frequencies into passband-frequencies and stopband-frequencies. 
         [0021]    In one embodiment, the symbols are determined based on a decomposition of a Fourier transform matrix in at least two matrices according to the decomposition of the frequencies. 
         [0022]    The symbols are for example determined based on a determination of the nullspace of one of the matrices, into which the Fourier transform matrix is decomposed. 
         [0023]    The transmitter is for example adapted according to a MIMO system. 
         [0024]    The communication device may be adapted as a base station. The communication device may also be adapted as a mobile communication device. 
         [0025]    Illustrative embodiments of the invention are explained below with reference to the drawings. 
         [0026]      FIG. 2  shows a transmitter  200  according to an embodiment of the invention. 
         [0027]    The transmitter  200  has the structure according to a MIMO (multiple input multiple output) system. The transmitter  200  comprises three transmit antennas  201  which is each used to transmit one of a plurality of data streams  202 , wherein the data streams  202  are supplied to the transmitter  200  by some data sources (not shown). Alternatively, one single data stream  202  may be supplied to the transmitter  200  which is mapped to three different data streams by the transmitter  200 . 
         [0028]    The transmitter  200  comprises an OFDM (orthogonal frequency division multiplexing) unit  203  which modulates subcarriers sent by the transmit antennas  201  according to OFDM modulation such that the data corresponding to the data streams  202  are transmitted. 
         [0029]    The transmitter  200  further comprises a preamble generator that is adapted to generate long preambles that are prepended to blocks of data before the blocks of data are sent. 
         [0030]      FIG. 3  shows three long preambles  300  according to an embodiment of the invention. 
         [0031]    Each of the long preambles.  300  is sent by one transmit antenna  201  of the transmitter  200 . The long preambles  300  are followed by some data to be sent (corresponding to the respective data stream  202 ) and there may also be sent short preambles before the long preambles  300  are sent. 
         [0032]    As can be seen from  FIG. 3 , the long preambles  300  do not comprise an explicit cyclic prefix (guard interval) between the OFDM symbols  301  that are contained in the long preambles  300 . The values of the OFDM symbols  301  are tuned in such a way that an identical tail  302  is shared by all OFDM symbols  301  corresponding to the same long preamble  300 . The tail  302  serves the function of an official cyclic prefix. 
         [0033]    In the long preamble  300  shown top in  FIG. 3 , the values {6,7,8} can be seen as both part (tail) of each OFDM  301  symbol of length  8  as well as the cyclic prefix of the following OFDM symbol  301 . This idea is-applied to every transmit antenna  201 . 
         [0034]    In the following, it is described how the values of the long preambles  300  are calculated. 
         [0035]    Denoting the s-th  0 FDM symbol in the a-th transmit antenna in the time domain by the vector t a,s , its frequency domain equivalent can be obtained through the Discrete Fourier Transform (DFT) matrix F by 
         [0000]      f a,s =Ft a,s    (1) 
         [0036]    Rearranging the elements in f a,s  into the bandpass subfrequencies denoted by e n,s  and the bandstop sub-frequencies denoted by O n,s , and observing that t a,s  is composed of its head component h a,s  and its tail component c a  in the form 
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         [0000]    comprises the corresponding components of the DFT matrix F. 
         [0037]    Since the tail values are common to all OFDM symbols, the subscript in the tail c is free from s. For a general MIMO system with N Tx  transmit antennas and N sym  OFDM symbols in each long preamble, (3) can be repeatedly used to write 
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         [0038]    In this embodiment, the following criteria are taken into account: 
         [0039]    C1) The long preamble power should be evenly distributed to each sub-frequency in the passband for optimal channel estimation performance (see [2]), every element in the vector e n,s  is therefore a complex exponent. 
         [0040]    C2) The long preambles should be tone-interleaved in the frequency domain, so different sub-frequencies at different transmit antennas in the passband are loaded at different times. For the simplest case, each sub-frequency at a particular antenna is loaded only once throughout the long preamble transmission. This implies that the combined set of loaded sub-frequencies pertaining to the vectors e a,1 , e a,2 , . . . , e a , N sym , equals to the set of passband sub-frequencies for each antenna a=1, 2, . . . , N Tx  to allow all sub-channels to be estimated from the long preamble information. 
         [0041]    For example, in a system with N FFT =16, length of cyclic prefix N CP =4 and N Tx =N sym =3, where the passband covers sub-frequencies {1,2,3,5,6,10,11,12,13,14,15} and the stopband {0,7,8,9}, 
         [0000]      | f   1,1   |=|f   2,2   |=|f   3,3 |=[0100100000100100] T , 
         [0000]      | f   1,2   |=|f   2,3   |=|f   3,1 |=[0010010000010010] T , 
         [0000]      | f   1,3   |=|f   2,1   |=|f   3,2 |=[0001001000001001] T    
         [0000]    can be chosen. This is illustrated in  FIG. 4 . 
         [0042]      FIG. 4  illustrates the frequency domain power in the transmission of long preambles according to an embodiment of the invention. 
         [0043]    The nine diagrams  401  to  409  shown in  FIG. 4  illustrate the frequency domain loading at different sub-frequencies, transmit antennas and times for tone-interleaving long preambles. 
         [0044]    It can be seen that each sub-frequency at every antenna is loaded only once in the whole period of 3 OFDM symbol duration to enable efficient channel estimation. The union of all loaded sub-frequencies at each antenna over all symbols (time) is therefore the set of passband sub-frequencies {1,2,3,5,6,10,11,12,13,14,15}. 
         [0045]    C3) No sub-frequency should be simultaneously loaded at two different antennas in the same OFDM symbol interval to avoid contention and interference, which causes the signals transmitted via different antennas to add up and become indistinguishable at the receiver. Referring to the example above, in transmitting the first OFDM symbol of the long preamble, antenna  1  loads sub-frequencies {1,4,10,13}, antenna  2  {2,5,11,14} and antenna  3  {3,6,12,15}, and thus, no violation of the condition. 
         [0046]    The above can be consolidated to the requirement that if the long preamble values at each of the sub-frequencies in the passband is recorded to form a N sym ×N Tx  matrix, this time-space matrix has to be orthogonal and the amplitude of each element needs to be either unity or zero. Referring again to the same example, the time-space matrix for sub-frequencies 
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         [0000]    where * represents some arbitrary real values. All of these matrices are orthogonal and formed by elements with amplitudes of either 1 or 0. 
         [0047]    The problem now is to solve for h a,s , e a,s  and c s , for a=1, 2, . . . , N Tx , s=1, 2, . . . , N sym  subject to the orthogonality constraint. 
         [0048]    Equations (4) and (5) can be expressed in compact form 
         [0000]      e a =A a x a    (6) 
         [0000]      O=B a x a    (7) 
         [0000]    for antenna a. If every sub-frequency in the passband is loaded only once throughout the duration of the N sym  OFDM symbols in antenna a, the orthogonality condition is easily met by applying cyclic shift of the tone assignment in antenna a to other antennas. As such, only one antenna a needs to be considered. The solutions to the linear equation (7) are given by 
         [0000]      x a =B a , ⊥α  (8) 
         [0000]    wherein α is arbitrary and B a , ⊥ is the null space of B a  (i.e. the columns of B a , ⊥ form a basis of the null space of B a ). B a , ⊥ can be obtained by singular value decomposition (SVD) of the matrix B a . Substituting (8) into (6) and defining M=A a B a , ⊥ yields 
         [0000]      e a =M a α  (9) 
         [0000]    to which there is no closed-form solution. In fact, due to the nonlinear nature of the equation, it has possibly no solution. Therefore it is more tractable, instead, to find, given M a  and that all entries in e a  are complex exponents, the vector {circumflex over (α)} that minimizes the cost function 
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                                   _ 
                                 
                               
                             
                              
                           
                           2 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               [ 
                               
                                 
                                   α 
                                   _ 
                                 
                                 - 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                             M 
                                             _ 
                                           
                                           H 
                                         
                                          
                                         
                                           M 
                                           _ 
                                         
                                       
                                       ) 
                                     
                                     
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                     
                                       M 
                                       _ 
                                     
                                     H 
                                   
                                    
                                   
                                     e 
                                     _ 
                                   
                                 
                               
                               ] 
                             
                             H 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   M 
                                   _ 
                                 
                                 H 
                               
                                
                               
                                 M 
                                 _ 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             [ 
                             
                               
                                 α 
                                 _ 
                               
                               - 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           M 
                                           _ 
                                         
                                         H 
                                       
                                        
                                       
                                         M 
                                         _ 
                                       
                                     
                                     ) 
                                   
                                   
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     M 
                                     _ 
                                   
                                   H 
                                 
                                  
                                 
                                   e 
                                   _ 
                                 
                               
                             
                             ] 
                           
                           + 
                           
                             
                               
                                 e 
                                 _ 
                               
                               H 
                             
                              
                             
                               e 
                               _ 
                             
                           
                           - 
                           
                             
                               
                                 e 
                                 _ 
                               
                               H 
                             
                              
                             
                               
                                 
                                   M 
                                   _ 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         M 
                                         _ 
                                       
                                       H 
                                     
                                      
                                     
                                       M 
                                       _ 
                                     
                                   
                                   ) 
                                 
                               
                               
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 M 
                                 _ 
                               
                               H 
                             
                              
                             
                               
                                 e 
                                 _ 
                               
                               . 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The subscript a has been omitted in the above for ease of notation. Clearly, the minimum value of J 1 (α) can be found by setting {circumflex over (α)}=(M H M) −1 M H e and maximizing 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             J 
                             2 
                           
                            
                           
                             ( 
                             
                               e 
                               _ 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               e 
                               _ 
                             
                             H 
                           
                            
                           
                             
                               
                                 M 
                                 _ 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       M 
                                       _ 
                                     
                                     H 
                                   
                                    
                                   
                                     M 
                                     _ 
                                   
                                 
                                 ) 
                               
                             
                             
                               - 
                               1 
                             
                           
                            
                           
                             
                               M 
                               _ 
                             
                             H 
                           
                            
                           
                             e 
                             _ 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               
                                 
                                   e 
                                   _ 
                                 
                                 H 
                               
                                
                               
                                 e 
                                 _ 
                               
                             
                             - 
                             
                               
                                 
                                   e 
                                   _ 
                                 
                                 H 
                               
                                
                               
                                 U 
                                 _ 
                               
                             
                           
                           ⊥ 
                           
                             
                               
                                 U 
                                 _ 
                               
                               ⊥ 
                               H 
                             
                              
                             
                               e 
                               _ 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where a singular value decomposition 
         [0000]    
       
         
           
             
               
                 
                   
                     M 
                     _ 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               U 
                               _ 
                             
                              
                             
                               U 
                               _ 
                             
                           
                           ⊥ 
                         
                         ] 
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 S 
                                 _ 
                               
                             
                           
                           
                             
                               
                                 0 
                                 _ 
                               
                             
                           
                         
                         ] 
                       
                     
                      
                     
                       
                         V 
                         _ 
                       
                       H 
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0000]    has been carried out. Comparing (10) and (11), it is evident that to have J 1 (α)=0 implies e H U ⊥ U ⊥   H e=0 or 
         [0000]        J   3 (θ)= c   T   G   r   c+s   T   G   r +2 s   T   G   m   c =0   (13) 
         [0000]    where 
         [0000]        c+js =cos θ+ j  sinθ= e    (14) 
         [0000]        G   r   +jG   m   =U   ⊥   U   ⊥   H    (15) 
         [0000]    and the subscript of θ in c and s has been dropped for simplicity in notation. The solution can be estimated using Newton-Raphson&#39;s iterative method upon expanding (13) into its first order Taylor series 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       J 
                       3 
                     
                      
                     
                       ( 
                       
                         θ 
                         _ 
                       
                       ) 
                     
                   
                   ≈ 
                   
                     
                       
                         J 
                         3 
                       
                        
                       
                         ( 
                         
                           
                             θ 
                             _ 
                           
                           0 
                         
                         ) 
                       
                     
                     + 
                     
                       
                         
                           [ 
                           
                             
                               
                                  
                                 
                                   
                                     J 
                                     3 
                                   
                                    
                                   
                                     ( 
                                     
                                       θ 
                                       _ 
                                     
                                     ) 
                                   
                                 
                               
                               
                                  
                                 
                                   θ 
                                   _ 
                                 
                               
                             
                              
                             
                                
                               
                                 
                                   θ 
                                   _ 
                                 
                                 = 
                                 
                                   
                                     θ 
                                     _ 
                                   
                                   0 
                                 
                               
                             
                           
                           ] 
                         
                         T 
                       
                        
                       Δ 
                        
                       
                           
                       
                        
                       
                         θ 
                         _ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0000]    to yield the minimum norm solution of 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         θ 
                         _ 
                       
                       
                         n 
                         + 
                         1 
                       
                     
                     = 
                     
                       
                         
                           θ 
                           _ 
                         
                         n 
                       
                       - 
                       
                         
                           
                             J 
                             3 
                           
                            
                           
                             ( 
                             
                               
                                 θ 
                                 _ 
                               
                               n 
                             
                             ) 
                           
                         
                         
                           
                              
                             
                               
                                 
                                    
                                   
                                     
                                       J 
                                       3 
                                     
                                      
                                     
                                       ( 
                                       
                                         θ 
                                         _ 
                                       
                                       ) 
                                     
                                   
                                 
                                 
                                    
                                   
                                     θ 
                                     _ 
                                   
                                 
                               
                                
                               
                                  
                                 
                                   
                                     θ 
                                     _ 
                                   
                                   = 
                                   
                                     
                                       θ 
                                       _ 
                                     
                                     n 
                                   
                                 
                               
                             
                              
                           
                           2 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 where 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                        
                       
                         
                           J 
                           3 
                         
                          
                         
                           ( 
                           
                             θ 
                             _ 
                           
                           ) 
                         
                       
                     
                     
                        
                       
                         θ 
                         _ 
                       
                     
                   
                   = 
                   
                     2 
                      
                     
                       ( 
                       
                         
                           
                             C 
                             _ 
                           
                            
                           
                             
                               G 
                               _ 
                             
                             r 
                           
                            
                           
                             s 
                             _ 
                           
                         
                         - 
                         
                           
                             S 
                             _ 
                           
                            
                           
                             
                               G 
                               _ 
                             
                             r 
                           
                            
                           
                             c 
                             _ 
                           
                         
                         + 
                         
                           
                             S 
                             _ 
                           
                            
                           
                             
                               G 
                               _ 
                             
                             m 
                           
                            
                           
                             s 
                             _ 
                           
                         
                         + 
                         
                           
                             C 
                             _ 
                           
                            
                           
                             
                               G 
                               _ 
                             
                             m 
                           
                            
                           
                             c 
                             _ 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and C and S are diagonal matrices with their main diagonals being c and s respectively, and subscripts of θ again omitted. 
         [0049]    The simple example introduced above is now used to illustrate the procedures involved in attaining the proposed tone-interleaving long preambles. Accordingly, it is assumed that N FFT =16, N CP =4, N Tx =N sym =3 with the passband over the sub-frequencies {1,2,3,5,6,10,11,12,13,14,15} and the stopband {0,7,8,9}. The following steps are carried out: 
         [0050]    S1) For a tone-interleaving design, 
         [0000]    
       
         
           
             
               
                 
                   F 
                   _ 
                 
                 
                   a 
                   , 
                   1 
                 
                 
                   [ 
                   11 
                   ] 
                 
               
               = 
               
                 
                   F 
                   _ 
                 
                  
                 
                   { 
                   
                     1 
                     , 
                     4 
                     , 
                     10 
                     , 
                     13 
                   
                   } 
                 
               
             
             , 
             
               { 
               
                 0 
                 , 
                 1 
                 , 
                 … 
                  
                 
                     
                 
                 , 
                 11 
               
               } 
             
           
         
       
       
         
           
             
               
                 
                   F 
                   _ 
                 
                 
                   a 
                   , 
                   2 
                 
                 
                   [ 
                   11 
                   ] 
                 
               
               = 
               
                 
                   F 
                   _ 
                 
                  
                 
                   { 
                   
                     2 
                     , 
                     5 
                     , 
                     11 
                     , 
                     14 
                   
                   } 
                 
               
             
             , 
             
               { 
               
                 0 
                 , 
                 1 
                 , 
                 … 
                  
                 
                     
                 
                 , 
                 11 
               
               } 
             
           
         
       
       
         
           
             
               
                 
                   F 
                   _ 
                 
                 
                   a 
                   , 
                   3 
                 
                 
                   [ 
                   11 
                   ] 
                 
               
               = 
               
                 
                   F 
                   _ 
                 
                  
                 
                   { 
                   
                     3 
                     , 
                     6 
                     , 
                     12 
                     , 
                     15 
                   
                   } 
                 
               
             
             , 
             
               { 
               
                 0 
                 , 
                 1 
                 , 
                 … 
                  
                 
                     
                 
                 , 
                 11 
               
               } 
             
           
         
       
       
         
           
             
               
                 
                   F 
                   _ 
                 
                 
                   a 
                   , 
                 
                 
                   [ 
                   21 
                   ] 
                 
               
               = 
               
                 
                   F 
                   _ 
                 
                  
                 
                   { 
                   
                     0 
                     , 
                     2 
                     , 
                     3 
                     , 
                     5 
                     , 
                     6 
                     , 
                     7 
                     , 
                     8 
                     , 
                     9 
                     , 
                     11 
                     , 
                     12 
                     , 
                     14 
                     , 
                     15 
                   
                   } 
                 
               
             
             , 
             
               { 
               
                 0 
                 , 
                 1 
                 , 
                 … 
                  
                 
                     
                 
                 , 
                 11 
               
               } 
             
           
         
       
       
         
           
             
               
                 
                   F 
                   _ 
                 
                 
                   a 
                   , 
                   2 
                 
                 
                   [ 
                   21 
                   ] 
                 
               
               = 
               
                 
                   F 
                   _ 
                 
                  
                 
                   { 
                   
                     0 
                     , 
                     1 
                     , 
                     3 
                     , 
                     4 
                     , 
                     6 
                     , 
                     7 
                     , 
                     8 
                     , 
                     9 
                     , 
                     10 
                     , 
                     12 
                     , 
                     13 
                     , 
                     15 
                   
                   } 
                 
               
             
             , 
             
               { 
               
                 0 
                 , 
                 1 
                 , 
                 … 
                  
                 
                     
                 
                 , 
                 11 
               
               } 
             
           
         
       
       
         
           
             
               
                 
                   F 
                   _ 
                 
                 
                   a 
                   , 
                   3 
                 
                 
                   [ 
                   21 
                   ] 
                 
               
               = 
               
                 
                   F 
                   _ 
                 
                  
                 
                   { 
                   
                     0 
                     , 
                     1 
                     , 
                     2 
                     , 
                     4 
                     , 
                     5 
                     , 
                     7 
                     , 
                     8 
                     , 
                     9 
                     , 
                     10 
                     , 
                     11 
                     , 
                     13 
                     , 
                     14 
                   
                   } 
                 
               
             
             , 
             
               { 
               
                 0 
                 , 
                 1 
                 , 
                 … 
                  
                 
                     
                 
                 , 
                 11 
               
               } 
             
           
         
       
       
         
           
             
               
                 
                   F 
                   _ 
                 
                 
                   a 
                   , 
                   1 
                 
                 
                   [ 
                   12 
                   ] 
                 
               
               = 
               
                 
                   F 
                   _ 
                 
                  
                 
                   { 
                   
                     1 
                     , 
                     4 
                     , 
                     10 
                     , 
                     13 
                   
                   } 
                 
               
             
             , 
             
               { 
               
                 12 
                 , 
                 13 
                 , 
                 … 
                  
                 
                     
                 
                 , 
                 15 
               
               } 
             
           
         
       
       
         
           
             
               
                 
                   F 
                   _ 
                 
                 
                   a 
                   , 
                   2 
                 
                 
                   [ 
                   12 
                   ] 
                 
               
               = 
               
                 
                   F 
                   _ 
                 
                  
                 
                   { 
                   
                     1 
                     , 
                     4 
                     , 
                     10 
                     , 
                     13 
                   
                   } 
                 
               
             
             , 
             
               { 
               
                 12 
                 , 
                 13 
                 , 
                 … 
                  
                 
                     
                 
                 , 
                 15 
               
               } 
             
           
         
       
       
         
           
             
               
                 
                   F 
                   _ 
                 
                 
                   a 
                   , 
                   3 
                 
                 
                   [ 
                   12 
                   ] 
                 
               
               = 
               
                 
                   F 
                   _ 
                 
                  
                 
                   { 
                   
                     1 
                     , 
                     4 
                     , 
                     10 
                     , 
                     13 
                   
                   } 
                 
               
             
             , 
             
               { 
               
                 12 
                 , 
                 13 
                 , 
                 … 
                  
                 
                     
                 
                 , 
                 15 
               
               } 
             
           
         
       
       
         
           
             
               
                 
                   F 
                   _ 
                 
                 
                   a 
                   , 
                   1 
                 
                 
                   [ 
                   22 
                   ] 
                 
               
               = 
               
                 
                   F 
                   _ 
                 
                  
                 
                   { 
                   
                     0 
                     , 
                     2 
                     , 
                     3 
                     , 
                     5 
                     , 
                     6 
                     , 
                     7 
                     , 
                     8 
                     , 
                     9 
                     , 
                     11 
                     , 
                     12 
                     , 
                     14 
                     , 
                     15 
                   
                   } 
                 
               
             
             , 
             
               { 
               
                 12 
                 , 
                 13 
                 , 
                 … 
                  
                 
                     
                 
                 , 
                 15 
               
               } 
             
           
         
       
       
         
           
             
               
                 
                   F 
                   _ 
                 
                 
                   a 
                   , 
                   2 
                 
                 
                   [ 
                   22 
                   ] 
                 
               
               = 
               
                 
                   F 
                   _ 
                 
                  
                 
                   { 
                   
                     0 
                     , 
                     1 
                     , 
                     3 
                     , 
                     4 
                     , 
                     6 
                     , 
                     7 
                     , 
                     8 
                     , 
                     9 
                     , 
                     10 
                     , 
                     12 
                     , 
                     13 
                     , 
                     15 
                   
                   } 
                 
               
             
             , 
             
               { 
               
                 12 
                 , 
                 13 
                 , 
                 … 
                  
                 
                     
                 
                 , 
                 15 
               
               } 
             
           
         
       
       
         
           
             
               
                 
                   F 
                   _ 
                 
                 
                   a 
                   , 
                   3 
                 
                 
                   [ 
                   22 
                   ] 
                 
               
               = 
               
                 
                   F 
                   _ 
                 
                  
                 
                   { 
                   
                     0 
                     , 
                     1 
                     , 
                     2 
                     , 
                     4 
                     , 
                     5 
                     , 
                     7 
                     , 
                     8 
                     , 
                     9 
                     , 
                     10 
                     , 
                     11 
                     , 
                     13 
                     , 
                     14 
                   
                   } 
                 
               
             
             , 
             
               { 
               
                 12 
                 , 
                 13 
                 , 
                 … 
                  
                 
                     
                 
                 , 
                 15 
               
               } 
             
           
         
       
     
         [0000]    is set where F rows , columns (on the right hand side of the above equations) denotes the sub-matrix generated by the set of rows and columns (as specified by the subscripts) of the DFT matrix F. 
         [0051]    S2) The matrices A a  ε           
         [0000]      12×40  and B a  ε             36×40  in (6) and (7) are then formed according to (4) and (5).
 
         [0052]    S3) A singular value decomposition of B a , which is of rank 36, is then performed. The null space matrix B a , ⊥ ε           
         [0000]      40×4  can be easily derived by extracting the 4 columns in the right singular matrix of B a  that correspond to the 4 zero singular values. 
         [0053]    S4) The matrix M=A a B a , ⊥ ε           
         [0000]      12×4  is then computed to be 
         [0000]    
       
         
           
             M 
             = 
             
               [ 
               
                 
                   
                     
                       0.1314 
                       - 
                       
                         0.0656 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       0.0498 
                       + 
                       
                         0.0640 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       
                         - 
                         0.0820 
                       
                       + 
                       
                         0.2557 
                          
                         j 
                       
                     
                   
                   
                     
                       0.4933 
                       + 
                       0.2236 
                     
                   
                 
                 
                   
                     
                       0.2629 
                       - 
                       
                         0.2779 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       0.1534 
                       - 
                       
                         0.3681 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       
                         - 
                         0.3530 
                       
                       - 
                       
                         0.2282 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       0.1734 
                       - 
                       
                         0.0768 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       
                         - 
                         0.3232 
                       
                       - 
                       
                         0.1455 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       0.2639 
                       - 
                       
                         0.0042 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       
                         - 
                         0.1283 
                       
                       - 
                       
                         0.0616 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       0.0598 
                       + 
                       
                         0.0469 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       0.0101 
                       + 
                       
                         0.1900 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       
                         - 
                         0.0842 
                       
                       + 
                       
                         0.3566 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       
                         - 
                         0.4363 
                       
                       - 
                       
                         0.1382 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       0.1316 
                       + 
                       
                         0.2945 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       
                         - 
                         0.362 
                       
                       - 
                       
                         0.0994 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       
                         - 
                         0.2704 
                       
                       + 
                       
                         0.1212 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       
                         - 
                         0.2168 
                       
                       + 
                       
                         0.1434 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       0.0571 
                       + 
                       
                         0.4973 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       0.2859 
                       - 
                       
                         0.0424 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       0.2388 
                       + 
                       
                         0.0233 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       
                         - 
                         0.0252 
                       
                       - 
                       
                         0.3567 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       0.1752 
                       + 
                       
                         0.2056 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       
                         - 
                         0.0840 
                       
                       - 
                       
                         0.4155 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       0.2432 
                       + 
                       
                         0.2530 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       0.0386 
                       + 
                       
                         0.0040 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       
                         - 
                         0.1925 
                       
                       + 
                       
                         0.0210 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       
                         - 
                         0.3013 
                       
                       - 
                       
                         0.0169 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       
                         - 
                         0.2694 
                       
                       + 
                       
                         0.2126 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       
                         - 
                         0.0367 
                       
                       - 
                       
                         0.4315 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       
                         - 
                         0.1925 
                       
                       + 
                       
                         0.0210 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       0.1290 
                       - 
                       
                         0.2757 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       
                         - 
                         0.3740 
                       
                       - 
                       
                         0.2405 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       
                         - 
                         0.2904 
                       
                       - 
                       
                         0.0006 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       0.0848 
                       - 
                       
                         0.1165 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       0.2518 
                       + 
                       
                         0.0465 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       
                         - 
                         0.0582 
                       
                       + 
                       
                         0.1704 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       0.1981 
                       - 
                       
                         0.1552 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       
                         - 
                         0.2905 
                       
                       - 
                       
                         0.1578 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       0.3446 
                       - 
                       
                         0.2602 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       0.0283 
                       + 
                       
                         0.4322 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       
                         - 
                         0.0882 
                       
                       + 
                       
                         0.1392 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       
                         - 
                         0.1142 
                       
                       + 
                       
                         0.2943 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       
                         - 
                         0.0759 
                       
                       - 
                       
                         0.3421 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       
                         - 
                         0.2172 
                       
                       + 
                       
                         0.0169 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       0.3478 
                       - 
                       
                         0.1786 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       0.0531 
                       - 
                       
                         0.2601 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
       The null space U ⊥ ε   
       [0054]      12×8  of M is similarly obtained from the 8 columns of the right singular matrix of M associated with the 8 zero singular values. 
         [0055]    S5) G r  ε           
         [0000]      12×12  and G m  ε             12×12  are calculated in accordance with (15). A random vector θ 0  ε             12×1  is fit into (13), (17) and (18) to produce θ n  after n iterations.
 
         [0056]    S6) The vector θ is then created using (14). With reference to (4) and (6), the time domain preamble can be obtained from {circumflex over (x)} a =(A a ) + e a  where the superscript+denotes pseudo-inversion, and therefore 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         t 
                         _ 
                       
                       
                         a 
                         , 
                         1 
                       
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       x 
                                       _ 
                                     
                                     ^ 
                                   
                                   a 
                                 
                                 ] 
                               
                               
                                 1 
                                 , 
                                 2 
                                 , 
                                 … 
                                  
                                 
                                     
                                 
                                 , 
                                 12 
                               
                             
                           
                         
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       x 
                                       _ 
                                     
                                     ^ 
                                   
                                   a 
                                 
                                 ] 
                               
                               
                                 37 
                                 , 
                                 38 
                                 , 
                                 39 
                                 , 
                                 40 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         t 
                         _ 
                       
                       
                         a 
                         , 
                         2 
                       
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       x 
                                       _ 
                                     
                                     ^ 
                                   
                                   a 
                                 
                                 ] 
                               
                               
                                 13 
                                 , 
                                 14 
                                 , 
                                 … 
                                  
                                 
                                     
                                 
                                 , 
                                 24 
                               
                             
                           
                         
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       x 
                                       _ 
                                     
                                     ^ 
                                   
                                   a 
                                 
                                 ] 
                               
                               
                                 37 
                                 , 
                                 38 
                                 , 
                                 39 
                                 , 
                                 40 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         t 
                         _ 
                       
                       
                         a 
                         , 
                         3 
                       
                     
                     = 
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   [ 
                                   
                                     
                                       
                                         x 
                                         _ 
                                       
                                       ^ 
                                     
                                     a 
                                   
                                   ] 
                                 
                                 
                                   25 
                                   , 
                                   26 
                                   , 
                                   … 
                                    
                                   
                                       
                                   
                                   , 
                                   36 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   [ 
                                   
                                     
                                       
                                         x 
                                         _ 
                                       
                                       ^ 
                                     
                                     a 
                                   
                                   ] 
                                 
                                 
                                   37 
                                   , 
                                   38 
                                   , 
                                   39 
                                   , 
                                   40 
                                 
                               
                             
                           
                         
                         ] 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
         [0057]    The resulting time domain preamble values are 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               t 
                               _ 
                             
                             
                               a 
                               , 
                               1 
                             
                           
                           = 
                           
                             [ 
                             
                               
                                 
                                   
                                     0.2763 
                                     - 
                                     
                                       0.6893 
                                        
                                       
                                           
                                       
                                        
                                       j 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       - 
                                       0.0888 
                                     
                                     - 
                                     
                                       0.0868 
                                        
                                       
                                           
                                       
                                        
                                       j 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       - 
                                       0.1507 
                                     
                                     - 
                                     
                                       0.2166 
                                        
                                       j 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       - 
                                       0.3636 
                                     
                                     - 
                                     
                                       0.4049 
                                        
                                       
                                           
                                       
                                        
                                       j 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     0.5349 
                                     + 
                                     
                                       0.1983 
                                        
                                       
                                           
                                       
                                        
                                       j 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     0.7882 
                                     - 
                                     
                                       0.0394 
                                        
                                       
                                           
                                       
                                        
                                       j 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       - 
                                       0.1296 
                                     
                                     - 
                                     
                                       0.0342 
                                        
                                       
                                           
                                       
                                        
                                       j 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     0.0847 
                                     - 
                                     
                                       0.0864 
                                        
                                       
                                           
                                       
                                        
                                       j 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     0.1093 
                                     - 
                                     
                                       0.0820 
                                        
                                       
                                           
                                       
                                        
                                       j 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     0.0330 
                                     + 
                                     
                                       0.8921 
                                        
                                       
                                           
                                       
                                        
                                       j 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     0.2270 
                                     + 
                                     
                                       0.4489 
                                        
                                       
                                           
                                       
                                        
                                       j 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       - 
                                       0.2883 
                                     
                                     - 
                                     
                                       0.3459 
                                        
                                       
                                           
                                       
                                        
                                       j 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       - 
                                       0.0723 
                                     
                                     + 
                                     
                                       0.0313 
                                        
                                       
                                           
                                       
                                        
                                       j 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       - 
                                       0.1907 
                                     
                                     + 
                                     
                                       0.0824 
                                        
                                       
                                           
                                       
                                        
                                       j 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       - 
                                       0.7950 
                                     
                                     + 
                                     
                                       0.3435 
                                        
                                       
                                           
                                       
                                        
                                       j 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     0.0256 
                                     - 
                                     
                                       0.0111 
                                        
                                       
                                           
                                       
                                        
                                       j 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                         , 
                       
                     
                   
                   
                     
                       
                         
                           
                             t 
                             _ 
                           
                           
                             a 
                             , 
                             3 
                           
                         
                         = 
                         
                           [ 
                           
                             
                               
                                 
                                   0.6914 
                                   + 
                                   
                                     0.2712 
                                      
                                     
                                         
                                     
                                      
                                     j 
                                   
                                 
                               
                             
                             
                               
                                 
                                   0.0024 
                                   + 
                                   
                                     0.1242 
                                      
                                     
                                         
                                     
                                      
                                     j 
                                   
                                 
                               
                             
                             
                               
                                 
                                   0.0545 
                                   + 
                                   
                                     0.2581 
                                      
                                     
                                         
                                     
                                      
                                     j 
                                   
                                 
                               
                             
                             
                               
                                 
                                   0.0457 
                                   + 
                                   
                                     0.5423 
                                      
                                     
                                         
                                     
                                      
                                     j 
                                   
                                 
                               
                             
                             
                               
                                 
                                   0.2222 
                                   - 
                                   
                                     0.5255 
                                      
                                     
                                         
                                     
                                      
                                     j 
                                   
                                 
                               
                             
                             
                               
                                 
                                   0.5689 
                                   - 
                                   
                                     0.5470 
                                      
                                     
                                         
                                     
                                      
                                     j 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     - 
                                     0.0639 
                                   
                                   + 
                                   
                                     0.1177 
                                      
                                     
                                         
                                     
                                      
                                     j 
                                   
                                 
                               
                             
                             
                               
                                 
                                   0.1209 
                                   - 
                                   
                                     0.0025 
                                      
                                     
                                         
                                     
                                      
                                     j 
                                   
                                 
                               
                             
                             
                               
                                 
                                   0.1346 
                                   - 
                                   
                                     0.0233 
                                      
                                     
                                         
                                     
                                      
                                     j 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     - 
                                     0.6270 
                                   
                                   - 
                                   
                                     0.6354 
                                      
                                     
                                         
                                     
                                      
                                     j 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     - 
                                     0.1713 
                                   
                                   - 
                                   
                                     0.4730 
                                      
                                     
                                         
                                     
                                      
                                     j 
                                   
                                 
                               
                             
                             
                               
                                 
                                   0.0542 
                                   + 
                                   
                                     0.4470 
                                      
                                     
                                         
                                     
                                      
                                     j 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     - 
                                     0.0723 
                                   
                                   + 
                                   
                                     0.0313 
                                      
                                     
                                         
                                     
                                      
                                     j 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     - 
                                     0.1907 
                                   
                                   + 
                                   
                                     0.0824 
                                      
                                     
                                         
                                     
                                      
                                     j 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     - 
                                     0.7950 
                                   
                                   + 
                                   
                                     0.3435 
                                      
                                     
                                         
                                     
                                      
                                     j 
                                   
                                 
                               
                             
                             
                               
                                 
                                   0.0256 
                                   - 
                                   
                                     0.0111 
                                      
                                     
                                         
                                     
                                      
                                     j 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
                  
                 
                   
                     t 
                     _ 
                   
                   
                     a 
                      
                     .2 
                   
                 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         0.6947 
                         - 
                         
                           0.3002 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         0.1021 
                         - 
                         
                           0.0441 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         0.2404 
                         - 
                         
                           0.1039 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         0.4874 
                         - 
                         
                           0.1039 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           0.5328 
                         
                         + 
                         
                           0.2302 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           0.7391 
                         
                         + 
                         
                           0.3194 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         0.0852 
                         - 
                         
                           0.0368 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           0.0996 
                         
                         + 
                         
                           0.0431 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           0.0896 
                         
                         + 
                         
                           0.0387 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         0.8277 
                         - 
                         
                           0.3577 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         0.4694 
                         - 
                         
                           0.2029 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           0.4134 
                         
                         + 
                         
                           0.1786 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           0.0723 
                         
                         + 
                         
                           0.0313 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           0.1907 
                         
                         + 
                         
                           0.0824 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           0.7950 
                         
                         + 
                         
                           0.3435 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         0.0256 
                         - 
                         
                           0.0111 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
           
         
       
     
         [0058]    Tone interleaving is accomplished by setting 
         [0000]      t 1,1 =t 2,2 =t 3,3 =t a,1 , 
         [0000]      t 1,2 =t 2,3 =t 3,1 =t a,2 , 
         [0000]      t 1,3 =t 2,1 =t 3,2 =t a,3    
         [0000]    which are mapped to their frequency domain values via the DFT matrix F as described in (1). 
         [0059]      FIG. 5  illustrates the time domain values according to an embodiment of the invention. 
         [0060]    The time domain values are shown in the diagrams  501  to  509  for all antennas according to the example above. The tails (shown bold in  FIG. 3 ) formed by the last four samples are identical for each antenna. 
         [0061]      FIG. 6  illustrates the time frequency domain values according to an embodiment of the invention. 
         [0062]    The frequency domain values are shown in the diagrams  601  to  609  for all antennas according to the example above. 
         [0063]    There are situations where the procedure described above is inadequate for satisfactory tone-interleaving designs. This occurs when the number of transmit antennas is too large to allow sufficient degrees of freedom in (9) to solve (13). As a result, the long preamble tones will fail to acquire unit amplitudes in the frequency domain. In such cases, orthogonality in space-time for every sub-frequency can be achieved by introducing special orthogonal matrices as an alternative to the tone-interleaving strategy. The idea is to produce common tail long preambles from a small set of tone-interleaving sequences by linear combination using orthogonal matrices. 
         [0064]    To illustrate the concept, the design example above is considered. Careful observations shows that the interleaving tones have amplitudes deviating slightly from unity, by a maximum of about 1,6%. Instead of designing N Tx =3 sets of interleaving tones, n Tx =2&lt;N Tx  sets can be designed by the same procedure to arrive at 
         [0000]    
       
         
           
             
               
                 t 
                 
                   a 
                   , 
                   1 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         0.6187 
                         + 
                         
                           0.2740 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         0.4016 
                         - 
                         
                           0.2019 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         0.1971 
                         + 
                         
                           0.2353 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         0.3187 
                         + 
                         
                           1.2004 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         0.1768 
                         + 
                         
                           0.2978 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         0.0916 
                         - 
                         
                           0.7011 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         0.2225 
                         - 
                         
                           0.0926 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           0.1547 
                         
                         + 
                         
                           0.0382 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           0.6187 
                         
                         - 
                         
                           0.2740 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           0.4016 
                         
                         + 
                         
                           0.2019 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           0.1971 
                         
                         - 
                         
                           0.2353 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         0.3187 
                         - 
                         
                           1.2004 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           0.1768 
                         
                         - 
                         
                           0.2978 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         0.0916 
                         + 
                         
                           0.7011 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           0.2225 
                         
                         + 
                         
                           0.0926 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         0.1547 
                         - 
                         
                           0.0382 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               
                 t 
                 
                   a 
                   , 
                   2 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         0.4234 
                         - 
                         
                           0.6445 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         0.1488 
                         - 
                         
                           0.9199 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           0.0241 
                         
                         + 
                         
                           0.8497 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           0.2119 
                         
                         + 
                         
                           0.2570 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           0.1768 
                         
                         - 
                         
                           0.2978 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           0.0916 
                         
                         + 
                         
                           0.7011 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           0.2225 
                         
                         + 
                         
                           0.0926 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         0.1547 
                         - 
                         
                           0.0382 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         0.4234 
                         - 
                         
                           0.6445 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         0.1488 
                         - 
                         
                           0.9199 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           0.0241 
                         
                         + 
                         
                           0.8497 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           0.2119 
                         
                         + 
                         
                           0.2570 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           0.1768 
                         
                         - 
                         
                           0.2978 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         0.0916 
                         + 
                         
                           0.7011 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           0.2225 
                         
                         + 
                         
                           0.0926 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                   
                     
                       
                         0.1547 
                         - 
                         
                           0.0382 
                            
                           
                               
                           
                            
                           j 
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
         [0065]    Since the n Tx =2 sets have a common tail (the last 4 samples) in the time domain, if they are linearly combined by any arbitrary ratio k 1  and k 2 , the resulting signal will always end up with a fixed tail so long as k 1 +k 2  is fixed. For instance, if the first set is multiplied by 1 and the second set is multiplied by e j2π/3  or the first set by e j2π/3  and the second by 1, k 1 +k 2 =1+e j2π/3  in either case and the resulting time domain waveforms share a common tail. 
         [0066]    Both waveforms do now occupy all the passband sub-frequencies instead of being tone-interleaving. This can be understood when the portions of the two tone-interleaving sets are mixed in according to the linear combination. Since the linear combining ratios 1 and e j2π/3  are both of magnitude 1, multiplying the 2 sets by these factors does not change their amplitude in frequency. As the 2 sets are still interleaved in frequency after scaling, adding them together will result in a waveform loaded at all passband frequencies. 
         [0067]    Following this principle, sets of long preamble sequences can be easily created that meet the common time domain tail and frequency domain mask requirements by linearly combining a small tone-interleaving set in complex exponent ratios for each antenna at different times. If it is ensured that the sequences chosen by different antennas are orthogonal in space-time for channel estimation, then the design is complete. Suppose there are n Tx  tone-interleaving sets, and set i is multiplied with the linear combining coefficient e i   θa,s  for antenna a (space) and at long preamble symbol (time) s, then it is sufficient for the orthogonality condition to hold by having 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         
                           n 
                           Tx 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           E 
                           _ 
                         
                         i 
                       
                     
                     = 
                     
                       
                         
                           1 
                            
                           
                               
                           
                            
                           c 
                         
                         _ 
                       
                       H 
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       
                         
                           E 
                           _ 
                         
                         i 
                         H 
                       
                        
                       
                         
                           E 
                           _ 
                         
                         i 
                       
                     
                     = 
                     
                       I 
                       _ 
                     
                   
                   , 
                   
                     
                       ∀ 
                       i 
                     
                     = 
                     1 
                   
                   , 
                   2 
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     n 
                     Tx 
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where [E i ] a,s =e i   θa,s , a=1, 2, . . . , n Tx , s=1, 2, . . . , N sym , 1 is a column vector of ones and c ε         
 
 N   Tx   ×1  is an arbitrary complex vector. The first equation guarantees that the linear combination ratios add to a constant over all symbols to realize a common tail for each antenna in the time domain. The second equation ensures orthogonality across space-time.
 
         [0068]    Referring to the example, if 
         [0000]    
       
         
           
             
               
                 E 
                 _ 
               
               1 
             
             = 
             
               
                 1 
                 
                   3 
                 
               
                
               
                 [ 
                 
                   
                     
                       
                          
                         
                           j 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             π 
                             / 
                             3 
                           
                         
                       
                     
                     
                       1 
                     
                     
                       1 
                     
                   
                   
                     
                       1 
                     
                     
                       
                          
                         
                           j 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             π 
                             / 
                             3 
                           
                         
                       
                     
                     
                       1 
                     
                   
                   
                     
                       1 
                     
                     
                       1 
                     
                     
                       
                          
                         
                           j 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             π 
                             / 
                             3 
                           
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
       
         
           
             
               
                 E 
                 _ 
               
               2 
             
             = 
             
               
                 1 
                 
                   3 
                 
               
                
               
                 [ 
                 
                   
                     
                       1 
                     
                     
                       
                          
                         
                           j 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             π 
                             / 
                             3 
                           
                         
                       
                     
                     
                       
                          
                         
                           j 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             π 
                             / 
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                          
                         
                           j 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             π 
                             / 
                             3 
                           
                         
                       
                     
                     
                       1 
                     
                     
                       
                          
                         
                           j 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             π 
                             / 
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                          
                         
                           j 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             π 
                             / 
                             3 
                           
                         
                       
                     
                     
                       
                          
                         
                           j 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             π 
                             / 
                             3 
                           
                         
                       
                     
                     
                       1 
                     
                   
                 
                 ] 
               
             
           
         
       
     
         [0000]    is set, the conditions in (21) are satisfied with 
         [0000]    
       
         
           
             
               
                 E 
                 1 
               
               + 
               
                 E 
                 2 
               
             
             = 
             
               
                 
                   3 
                   + 
                   
                     3 
                   
                 
                 6 
               
                
               
                 1 
                 _ 
               
                
               
                 
                   
                     1 
                     _ 
                   
                   H 
                 
                 . 
               
             
           
         
       
     
         [0069]    The complete long preamble sequence at the 3 antennas are obtained as E 1              
         [0000]    t a,1 +E 2            
 
t a,2  where          
 
represents the Kronecker product. The common tails are shared at different times (symbols). Space time orthogonality at each sub-frequency can be easily verified from the frequency domain values
 
         [0000]    
       
         
           
             
               
                 f 
                 _ 
               
               
                 1 
                 , 
                 1 
               
             
             = 
             
               
                 
                   f 
                   _ 
                 
                 
                   2 
                   , 
                   2 
                 
               
               = 
               
                 
                   
                     f 
                     _ 
                   
                   
                     3 
                     , 
                     3 
                   
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           0.0000 
                           + 
                           
                             0.0000 
                              
                             
                                 
                             
                              
                             j 
                           
                         
                       
                     
                     
                       
                         
                           
                             - 
                             0.3047 
                           
                           + 
                           
                             0.4904 
                              
                             
                                 
                             
                              
                             j 
                           
                         
                       
                     
                     
                       
                         
                           0.2451 
                           - 
                           
                             0.5227 
                              
                             
                                 
                             
                              
                             j 
                           
                         
                       
                     
                     
                       
                         
                           0.4849 
                           + 
                           
                             0.3134 
                              
                             
                                 
                             
                              
                             j 
                           
                         
                       
                     
                     
                       
                         
                           0.0161 
                           - 
                           
                             0.5771 
                              
                             
                                 
                             
                              
                             j 
                           
                         
                       
                     
                     
                       
                         
                           
                             - 
                             0.5554 
                           
                           - 
                           
                             0.1579 
                              
                             
                                 
                             
                              
                             j 
                           
                         
                       
                     
                     
                       
                         
                           
                             - 
                             0.4398 
                           
                           + 
                           
                             0.3741 
                              
                             
                                 
                             
                              
                             j 
                           
                         
                       
                     
                     
                       
                         
                           
                             - 
                             0.0000 
                           
                           - 
                           
                             0.0000 
                              
                             
                                 
                             
                              
                             j 
                           
                         
                       
                     
                     
                       
                         
                           
                             - 
                             0.0000 
                           
                           + 
                           
                             0.0000 
                              
                             
                                 
                             
                              
                             j 
                           
                         
                       
                     
                     
                       
                         
                           
                             - 
                             0.0000 
                           
                           + 
                           
                             0.0000 
                              
                             
                                 
                             
                              
                             j 
                           
                         
                       
                     
                     
                       
                         
                           0.5386 
                           + 
                           
                             0.2080 
                              
                             
                                 
                             
                              
                             j 
                           
                         
                       
                     
                     
                       
                         
                           
                             - 
                             0.5730 
                           
                           - 
                           
                             0.0711 
                              
                             
                                 
                             
                              
                             j 
                           
                         
                       
                     
                     
                       
                         
                           0.2687 
                           - 
                           
                             0.5110 
                              
                             
                                 
                             
                              
                             j 
                           
                         
                       
                     
                     
                       
                         
                           0.2337 
                           + 
                           
                             0.5279 
                              
                             
                                 
                             
                              
                             j 
                           
                         
                       
                     
                     
                       
                         
                           0.3492 
                           - 
                           
                             0.4597 
                              
                             
                                 
                             
                              
                             j 
                           
                         
                       
                     
                     
                       
                         
                           
                             - 
                             0.5480 
                           
                           - 
                           
                             0.1817 
                              
                             
                                 
                             
                              
                             j 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
       
         
           
             
               
                 f 
                 _ 
               
               
                 p 
                 , 
                 
                   q 
                    
                   
                      
                     
                       p 
                       , 
                       
                         q 
                         = 
                         1 
                       
                       , 
                       2 
                       , 
                       
                         3 
                         ; 
                         
                           p 
                           ≠ 
                           q 
                         
                       
                     
                   
                 
               
             
             = 
             
               [ 
               
                 
                   
                     
                       0.0000 
                       + 
                       
                         0.0000 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       0.5770 
                       + 
                       
                         0.0187 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       0.3302 
                       + 
                       
                         0.4736 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       0.0290 
                       - 
                       
                         0.5766 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       0.4918 
                       + 
                       
                         0.3025 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       0.1410 
                       + 
                       
                         0.5599 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       
                         - 
                         0.1041 
                       
                       - 
                       
                         0.5679 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       0.0000 
                       - 
                       
                         0.0000 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       0.0000 
                       - 
                       
                         0.0000 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       0.0000 
                       + 
                       
                         0.0000 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       
                         - 
                         0.4495 
                       
                       + 
                       
                         0.3624 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       0.2249 
                       + 
                       
                         0.5318 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       0.3082 
                       + 
                       
                         0.4882 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       0.3403 
                       - 
                       
                         0.4664 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       0.2235 
                       + 
                       
                         0.5323 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       0.1167 
                       + 
                       
                         0.5654 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
         [0070]    For instance, sub-frequency with index 1 is loaded in space-time with the values 
         [0000]    
       
         
           
             
               [ 
               
                 
                   
                     
                       
                         - 
                         0.3047 
                       
                       + 
                       
                         0.4904 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       0.5770 
                       + 
                       
                         0.0187 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       0.5770 
                       + 
                       
                         0.0187 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       0.5770 
                       + 
                       
                         0.0187 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       
                         - 
                         0.3047 
                       
                       + 
                       
                         0.4904 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       0.5770 
                       + 
                       
                         0.0187 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
                 
                   
                     
                       0.5770 
                       + 
                       
                         0.0187 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       0.5770 
                       + 
                       
                         0.0187 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                   
                     
                       
                         - 
                         0.3047 
                       
                       + 
                       
                         0.4904 
                          
                         
                             
                         
                          
                         j 
                       
                     
                   
                 
               
               ] 
             
             = 
             
               
                 ( 
                 
                   
                     - 
                     0.3047 
                   
                   + 
                   
                     0.4904 
                      
                     
                         
                     
                      
                     j 
                   
                 
                 ) 
               
                
               
                 [ 
                 
                   
                     
                       1 
                     
                     
                       
                          
                         
                           j 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             π 
                             / 
                             3 
                           
                         
                       
                     
                     
                       
                          
                         
                           j 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             π 
                             / 
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                          
                         
                           j 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             π 
                             / 
                             3 
                           
                         
                       
                     
                     
                       1 
                     
                     
                       
                          
                         
                           j 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             π 
                             / 
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                          
                         
                           j 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             π 
                             / 
                             3 
                           
                         
                       
                     
                     
                       
                          
                         
                           j 
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             π 
                             / 
                             3 
                           
                         
                       
                     
                     
                       1 
                     
                   
                 
                 ] 
               
             
           
         
       
     
         [0000]    which is an orthogonal matrix as expected. 
         [0071]    The procedure is summarized below:
   1. Given a number of antennas N Tx , design tone-interleaving sets for n Tx  antennas, where n Tx &lt;N Tx .   2. Design mixing matrices E i ε             
 
         N     Tx     ×N     Tx   , i= 1 ,  2 , . . . , n Tx  that satisfy (21) and (22) wherein every element in E i  is a complex exponent. 
       [0074]    In the extreme case when the degrees of freedom are still inadequate to solve (13), extra freedom can be released by lifting the constraint in (21) through zero tail adoption. This works by assigning any arbitrary orthogonal matrix E 1  and E 2 =−E 1  to two tone-interleaving sets such that c=0 in (21). 
         [0075]    Since the two mixing matrices work to subtract the common tails of the tone-interleaving sets in the time domain, the resulting tail is always zero. The advantage of this design is that the tone-interleaving sets can be readily re-used when the number of transmit antennas in the system configuration. changes by simply employing another orthogonal matrix E 1 . 
         [0076]    Illustrating again with the same example, by applying FFT as the mixing matrix 
         [0000]    
       
         
           
             
               
                 E 
                 _ 
               
               1 
             
             = 
             
               
                 - 
                 
                   
                     E 
                     _ 
                   
                   2 
                 
               
               = 
               
                 
                   1 
                   
                     3 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         1 
                       
                       
                         1 
                       
                     
                     
                       
                         1 
                       
                       
                         
                            
                           
                             j 
                              
                             
                                 
                             
                              
                             2 
                              
                             
                                 
                             
                              
                             
                               π 
                               / 
                               3 
                             
                           
                         
                       
                       
                         
                            
                           
                             j 
                              
                             
                                 
                             
                              
                             4 
                              
                             
                                 
                             
                              
                             
                               π 
                               / 
                               3 
                             
                           
                         
                       
                     
                     
                       
                         1 
                       
                       
                         
                            
                           
                             j 
                              
                             
                                 
                             
                              
                             4 
                              
                             
                                 
                             
                              
                             
                               π 
                               / 
                               3 
                             
                           
                         
                       
                       
                         
                            
                           
                             j 
                              
                             
                                 
                             
                              
                             2 
                              
                             
                                 
                             
                              
                             
                               π 
                               / 
                               3 
                             
                           
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
         [0000]    a zero tail non-tone-interleaving long preamble sequence can be generated. If the system is reduced to N Tx =2, a simple change to a size 2 FFT matrix to 
         [0000]    
       
         
           
             
               
                 E 
                 _ 
               
               1 
             
             = 
             
               
                 - 
                 
                   
                     E 
                     _ 
                   
                   2 
                 
               
               = 
               
                 
                   1 
                   
                     2 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         1 
                       
                     
                     
                       
                         1 
                       
                       
                         
                           - 
                           1 
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
         [0000]    leads to a zero-tail preamble design with N FFT =16, N Tx =N sym =2. 
         [0077]    In the following, an alternative for solving equation (9) is given. 
         [0078]    Assuming that e a  is known, the minimum squared error solution to (9) can be written as 
         [0000]      {circumflex over (α)}=M a   + e a    (22) 
         [0079]    Substituting {circumflex over (α)} for α in (9) results in the iterative equation 
         [0000]        e   a   [n+ 1 ]=M   a   M   a   +   e   a   [n].    
         [0080]    However, since the operation on the right hand side does not warrant an exponential vector required on the left, the iteration has to be modified as 
         [0000]    
       
         
           
             
               
                 x 
                 _ 
               
               
                 a 
                  
                 
                   [ 
                   
                     n 
                     + 
                     1 
                   
                   ] 
                 
               
             
             = 
             
               
                 
                   
                     M 
                     _ 
                   
                   a 
                 
                  
                 
                   
                     M 
                     _ 
                   
                   a 
                   † 
                 
                  
                 
                   
                     
                       
                         e 
                         _ 
                       
                       
                         a 
                          
                         
                           [ 
                           n 
                           ] 
                         
                       
                     
                      
                     
                       
 
                     
                     [ 
                     
                       
                         e 
                         _ 
                       
                       
                         a 
                          
                         
                           [ 
                           
                             n 
                             + 
                             1 
                           
                           ] 
                         
                       
                     
                     ] 
                   
                   i 
                 
               
               = 
               
                 
                   
                     [ 
                     
                       
                         x 
                         _ 
                       
                       
                         a 
                          
                         
                           [ 
                           
                             n 
                             + 
                             1 
                           
                           ] 
                         
                       
                     
                     ] 
                   
                   i 
                 
                 
                    
                   
                     
                       [ 
                       
                         
                           x 
                           _ 
                         
                         
                           a 
                            
                           
                             [ 
                             
                               n 
                               + 
                               1 
                             
                             ] 
                           
                         
                       
                       ] 
                     
                     i 
                   
                    
                 
               
             
           
         
       
     
         [0000]    where [] i  denotes the i-th element of the vector in the argument.