Abstract:
A permuted sequences combination uses a frame structure in which two sync words, each comprising M complex symbols, are appended at the frame start. One benefit is the reduction of the large variance of the timing estimation error in the conventional correlation method. In at least one embodiment, the first sync word, {right arrow over (s)} 1 , is a predetermined constant amplitude zero autocorrelation (CAZAC) sequence. The second sync word, {right arrow over (s)} 2 , is a permutation of the first such that the combination of the two received sync signal vectors perform sliding window processing where the peak occurs at the correct frame start. The permuted sequences combination can be used in both AWGN channel and multi-path environments.

Description:
CROSS-REFERENCE TO RELATED PATENT APPLICATIONS 
     The present application claims priority to U.S. Provisional Application Ser. No. 60/874,141 entitled “Method and System for Frame Synchronization Using Correlation Between Permuted Sequences” filed on Dec. 11, 2006, and incorporated herein by reference in its entirety. 
    
    
     FIELD 
     The present application relates to the filed of time estimation in communication systems. More specifically, it relates to frame synchronization using correlation between permuted sequences. 
     BACKGROUND 
     Frame synchronization is important in digital communications. Heretofore, various approaches have been proposed to conduct frame synchronization. Specifically in orthogonal frequency division multiplexing (OFDM) systems, a conventional correlation method appends cyclic periodic training sequences at the start of the data frame at the transmitter and, at the receiver, the periodicity of the signals within an observation window is explored to obtain frame synchronization. In IEEE 802.11a and IEEE 802.16g standards, for example, the preambles contain two (or more) identical sync words of M symbols. Only M complex multiplications are needed for each timing estimate. However, when the timing offset is less than M samples (M is the period of the training sequence), the frame synchronization metric is not narrow around its peak that corresponds to the correct timing start. When the signal-to-noise ratio (SNR) is not high enough, the probability of coarse synchronization error (timing estimate errors within the length of the training sequence) is high. 
     A number of techniques have been proposed to sharpen the timing metric and thus reduce the variance of the timing estimation error. For example, a different sync words have been used to minimize coarse timing synchronization errors, specifically in OFDM. In a double correlation technique for MPSK systems, the nonlinear Bessel function is approximated by a second order power series, resulted in estimate which approaches the maximum likelihood (ML) solution of the frame start in the presence of the frequency offset. The complexity of this method, however, is high, requiring about M 2  complex multiplications for each timing estimate. Other frame synchronization techniques have been proposed. However, prior systems and methods fail to show or suggest a permuted sequences combination for frame synchronization such as described in the present application. 
     SUMMARY 
     According to one aspect described in the present application, a first sync word {right arrow over (s)} 1  is a predetermined constant amplitude zero autocorrelation (CAZAC) sequence, while the second {right arrow over (s)} 2 , which is the permutation of the first sync word, is obtained in such a way that the permutation matrix γ is a sparse matrix of order M having ones only at the (i,k)th component if 
                     (     i   -   1     )     ⁢   p     +   q     ⁢     =     mod   ⁢           ⁢   M       ⁢     k   -   1       ,         
where p and q are integers less than M. According to the specific permutation used in the design of {right arrow over (s)} 2 , the combination of two received signal vectors creates sliding window processing, resulting in a peak that occurs at the correct start of a frame. The permuted sequences combination described can perform well in AWGN channels and multi-path channels.
 
     According to another aspect described in the present application, a method for synchronizing a communication signal with a frame structure and each frame of the frame structure includes at least one symbol, includes receiving a first synchronization word in a first sequence of a communication signal. The first synchronization word provides a reference symbol. The method further includes receiving a second synchronization word in a second sequence of the communication signal where the second synchronization word is a permutation of the first synchronization word. 
     According to yet another aspect described in the present application, an apparatus synchronizes a communication signal having a frame structure using correlations between permuted sequences. The apparatus includes a modulator and a transmitter. The modulator provides a first synchronization word in a first sequence of a communication signal and a second synchronization word in a second sequence of the communication signal. The second synchronization word is a permutation of the first synchronization word. The transmitter transmits the communication signal including the first and second synchronization words. 
     According to yet still another aspect described in the present application, a method of frame synchronization using permuted sequences includes receiving a communication signal at a receiver where the communication signal including a frame structure with a first frame having a first communication word and a second frame having a second communication word. The method further includes de-permuting the first communication word from the received communication signal and performing a correlation of the first communication word and the second communication word. 
     These and other features, aspects and advantages of the present invention will become apparent from the following description, appended claims, and the accompanying exemplary embodiments shown in the drawings, which are briefly described below. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a graphical representation of an example frame structure. 
         FIG. 2(   a ) is a graph showing the average timing metric over 1000 symbols with a conventional QPSK OFDM scheme. 
         FIG. 2(   b ) is a graph showing the average timing metric over 1000 symbols with a conventional QPSK scheme. 
         FIG. 2(   c ) is a graph showing the average timing metric over 1000 symbols with a QPSK OFDM permuted sequences scheme. 
         FIG. 2(   d ) is a graph showing the average timing metric over 1000 symbols with a QPSK single carrier permuted sequences scheme. 
         FIG. 2(   e ) is a graph showing the average timing metric over 1000 symbols with a QPSK OFDM symmetric sequences scheme. 
         FIG. 2(   f ) is a graph showing the average timing metric over 1000 symbols with a QPSK single carrier symmetric sequences scheme. 
         FIG. 3(   a ) is a graphical representation of a conventional sync pattern. 
         FIG. 3(   b ) is a graphical representation of a symmetric sync pattern. 
         FIG. 4(   a ) is a graph comparing the absolute sample error of a permuted combination where q=0, p=2, a symmetric correlation, and a conventional correlation scheme (s&amp;c) at different SNRs for AWGN (additive white Gaussian noise) channels in a QPSK on OFDM system. 
         FIG. 4(   b ) is a graph comparing the absolute sample error of the permuted combination where q=0, p=2, a symmetric correlation, and a conventional correlation scheme (s&amp;c) at different SNRs for AWGN channels in a QRSK single carrier system. 
         FIG. 5(   a ) is a graph comparing the absolute sample error of a permuted combination where q=0, p=2, a symmetric correlation, and the conventional correlation scheme (s&amp;c) at different SNRs for fading channels in QPSK on OFDM. 
         FIG. 5(   b ) is a graph comparing the absolute sample error of a permuted combination where q=0, p=2, a symmetric correlation, and the conventional correlation scheme (s&amp;c) at different SNRs for fading channels in a QRSK single carrier system. 
     
    
    
     DETAILED DESCRIPTION 
     Exemplary embodiments are described below with reference to the accompanying drawings. It should be understood that the following description is intended to describe exemplary embodiments of the invention, and not to limit the invention. 
     Frame synchronization techniques can be divided into two groups: a data-aided and a non-data aided, or blind synchronization. In general, the latter technique may require complicated computation and its performance is usually worse than the former. The former technique may require an extra small percent of bandwidth, however, it is much simpler to implement and hence is widely accepted in industry. In this paper, we will focus on the former: data-aided frame synchronization schemes, specifically, for those whose frame structures have two successive sync words. 
       FIG. 1  illustrates a frame structure  10  for a frame used in data-aided frame synchronization. While the case of modulation using the AWGN (additive white Gaussian noise) channel with only timing offset is used for illustration purposes, other modulation schemes can also be used. Considering the frame structure shown in  FIG. 1 , the first 2M symbols are the sync pattern. (2M≦N). The transmitted signal, 
                     x   ⁡     (   n   )       =     {               s   1     ⁡     (   n   )       ,           0   ≤     n   ⁢           ⁢   mod   ⁢           ⁢   N     &lt;   M                   s   2     ⁡     (   n   )       ,           M   ≤     n   ⁢           ⁢   mod   ⁢           ⁢   N     &lt;     2   ⁢   M                 d   ⁡     (   n   )               2   ⁢   M     ≤     n   ⁢           ⁢   mod   ⁢           ⁢   N     &lt;   N                     (   1   )               
where s 1 (n) and s 2 (n) has power σ s   2 ·σ s   2  is the transmitted signal power.
 
     In the AWGN case where only timing offset, τ, is an unknown random parameter, the received signal can be written as
 
 r ( n )= x ( n −τ)+ w ( n )  (2)
 
With the two identical sync words {right arrow over (s)} 1  and {right arrow over (s)} 2  used to detect the frame start, the timing metric for the start position, {circumflex over (τ)} is given by
 
                     τ   ^     =         arg   ⁢           ⁢   max       i   =     [       -   M     ,     N   -   M   +   1       ]         ⁢              r   →     i   H     ⁢       r   →       i   +   M                        (   3   )               
where {right arrow over (r)} i  is the received signal vector of length M starting from the ith sample; (·) H  represents the Hermitian operator. The above estimator in (3) depends on only the correlation term related to the sync pattern as compared to other solutions which may also include the auto-correlation term of the received signal in the timing metric.
 
     Assuming that the noise at the receiver is AWGN and the data symbols are randomly distributed with mean zero, then E└|{right arrow over (r)} i   H {right arrow over (r)} i+M |┘ for i=μ is the expectation of the sliding correlation of the two successively received vectors as a function of the frame start with timing estimate error μ, given by 0&lt;μ=|{circumflex over (τ)}−τ|≦M. As such, E└|{right arrow over (r)} i   H {right arrow over (r)} i+M |   i=0   ┘ represents the expectation of the sliding correlation of the two successively received vectors without a timing error. The difference of these two can be rewritten as 
     
       
         
           
             
               
                 
                   
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     The expectation of the difference of the sliding correlation of the two successive received sync words only depends on 
                      ∑     i   =   1     μ     ⁢         s   1   *     ⁡     (   i   )       ⁢       s   2     ⁡     (   i   )                ,         
which is when {right arrow over (s)} 1 ={right arrow over (s)} 2 , equals to μσ s   2 . Using a length shift register sequence (m-sequence) of length 63 (M=63) as the sync word, the average value of the timing metric (3) over 1000 symbols of the conventional correlation methods in both QPSK (quadrature phase shift keying) on OFDM (orthogonal frequency division multiplexing) systems and QPSK (single carrier) systems have been plotted in  FIG. 2   a  and  FIG. 2   b  when SNR=10 dB. The expectation of the timing metric is not narrow around the correct timing offset and hence the timing estimation error has large variance, resulting in large coarse synchronization errors.
 
     From equation (4), it can be seen that the quality of a sync word can depend on several parameters. To maximize the difference of the value of the timing metric between μ=0 and 0&lt;μ≦M, a linear frame synchronization or permuted sequences combination method can be used. In such a method, {right arrow over (s)} 2  is a permuted (interleaved) version of {right arrow over (s)} 1 . Correspondingly, at the receiver, de-permute {right arrow over (s)} 2  is de-permute and the correlation of the two sync words is taken. 
     Denote             as an permuted sequences combination operator where
 
{right arrow over (s)} 1              {right arrow over (s)}   2 ={right arrow over (s)} 1   H γ{right arrow over (s)} 2   (7)
 
γ is a permutation matrix given by
 
γ=└ e   γ     0      . . . e   γ     M−1   ┘ T   (8)
 
e i  is the ith row of the identity matrix. The subscript [γ 0  . . . γ M−1 ] is given by
 
[γ 0  . . . γ M−1 ] T =γ[0  . . . M− 1] T   (9)
 
Note that when the sync word {right arrow over (s)} 1  and {right arrow over (s)} 2  are composed of real symbols, we can drop the conjugate operator in (7).

     When γ=I, the identity matrix; 
                               s   →     1     ⊗       s   →     2       =         s   →     1   H     ⁢       s   →     2                   =         [         s   1     ⁡     (   0   )       ⁢           ⁢   …   ⁢           ⁢       s   1     ⁡     (     M   -   1     )         ]     *     ⁡     [             s   2     ⁡     (   0   )               ⋮               s   2     ⁡     (     M   -   1     )             ]                     (   10   )               
Also, assuming that the sync words {right arrow over (s)} 1  and the permuted {right arrow over (s)} 2  have fixed signal energy Mσ s   2 , then in order to have the largest combination, the two sync words vectors should have the same components as the case in the conventional method. Support for this proposition is described below with reference to Equations (25) through (29). Specifically, the largest combination Mσ s   2  occurs at μ=0 if the sync word {right arrow over (s)} 2  is given by
 
{right arrow over (s)} 2 =γ T {right arrow over (s)} 1   (11)
 
where γ T  is the transpose of the permutation matrix and also the inverse permutation matrix. In fact, if the second sync word {right arrow over (s)} 2  is designed according to (11), then from (7), assuming the noise is small compared with the signal, the expectation of the sliding correlation of the two successively received vectors without a timing error is
 
                       E   ⁢     ⌊                r   →     i     ⊗       r   →       i   +   M                i   =   0       ⌋       =                s   →     1     ⊗       s   →     2            =           s   →     1   H     ⁢   γ   ⁢           ⁢       s   →     2       =           s   →     1   H     ⁢     γγ   T     ⁢       s   →     1       =         s   →     1   H     ⁢       s   →     1               ⁢     
     ⁢   since           (   12   )                 γγ   T     =   I           (   13   )               
Consequently, assuming that M is large and μ is small in comparison to M then ignoring the second term on the RHS of (4), (4) is reduced to
 
                     E   ⁢     ⌊                  r   →     i   H     ⁢       r   →       i   +   M                i   =   μ       -                r   →     i   H     ⁢       r   →       i   +   M                i   =   0         ⌋       =                  s   →     1   H     ⁢     T   μ     ⁢     T   μ   H     ⁢       s   →     2       -       ∑     i   =   1     μ     ⁢         s   1   *     ⁡     (   i   )       ⁢       s   2     ⁡     (   i   )                  -              s   →     1   H     ⁢       s   →     2                      (   14   )               
Now a good timing metric should have a delta like timing metric, which means the first term on the RHS of (14) has the following property given by
 
                         s   →     1   H     ⁢       T   μ     ⊗     T   μ   H       ⁢     γ   Y     ⁢       s   →     1       =     {             M   ⁢           ⁢     σ   s   2       ,           μ   =   0               0   ,           M   ≥        μ        &gt;   0                     (   15   )               
By the definition of, the left hand side (LHS) of (15) can be rewritten as
 
{right arrow over (s)} 1   H T μ             T μ   H γ T {right arrow over (s)} 1 ={right arrow over (s)} 1   H T μ   T γT μ γ T {right arrow over (s)} 1   (16)
 
And when μ=0, T μ =I, it follows that
 
{right arrow over (s)} 1   H T μ   T           T μ γ T {right arrow over (s)} 1 ={right arrow over (s)} 1   H {right arrow over (s)} 1   (17)
 
Note that the RHS of (17) equals to the modular square of the sync word {right arrow over (s)} 1 .

     It is notable that there are some special sequences (for example, CAZAC sequence, m-sequence) with a delta like auto-correlation satisfying the property 
                         s   →     1   H     ⁢     T   μ     ⁢       s   →     1       =     {             M   ⁢           ⁢     σ   s   2       ,           μ   =   0                   0   ⁢           ⁢   or     -   1     ,           M   ≥        μ        &gt;   0                     (   18   )               
By employing such sequences, it is possible to design only the permutation to make the LHS of (15) circulant auto-correlations of the sync word {right arrow over (s)} 1 . Thus, using such sequences, the permutation matrix γ is chosen to satisfy
 
T μ   T γT μ γ T =T ν , ν≠0  (19)
 
ν is an integer.
 
     For this, a specific permutation matrix of order M is introduced whose (i,k)th component is given by 
                     γ   ⁡     (     i   ,   k     )       =     {           0   ,                 (     i   -   1     )     ⁢   p     +   q     ⁢     ≠     mod   ⁢           ⁢   M       ⁢     k   -   1                 1   ,                 (     i   -   1     )     ⁢   p     +   q     ⁢     =     mod   ⁢           ⁢   M       ⁢     k   -   1                       (   20   )                 gcd   ⁡     (     p   ,   M     )       =   1           (   21   )               
i.e. p is relatively prime to M·gcd(·) represents the greatest common divisor. q is a given integer that can be selected from [0 . . . M−1]. For example, when M=5, p=3, q=0, the permutation matrix defined by (20) can be given by
 
                   γ   =     [         1       0       0       0       0           0       0       0       1       0           0       1       0       0       0           0       0       0       0       1           0       0       1       0       0         ]             (   22   )               
Combining (6) and (19):
 
γT μ γ T =T μ+ν , ν≠0  (23)
 
     As shown in the description below accompanying Equations (30) through (41), when using the permutation matrix γ given by (20), and in order to satisfy (23), (p−1) should be relatively prime to M, where p is the multiplicative inverse of p in the finite filed of 
     
       
         
           
             
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     Simulations were conducted to verify the above results. Using, for example, a m-sequence of length 63 as the sync word, the permutation matrix is given by (20) when q=0, p=2. The average value of the timing metric of the permuted sequences combination methods in QPSK on OFDM systems and QPSK (single carrier) systems have been plotted when SNR=10 dB. As it is shown in  FIG. 2   c  and  FIG. 2   d , the expectation of the timing metric is a delta like function around the correct timing position. 
     One can show that, when p=M−1 and q=M−1, the second sync word is the reverse version of the first sync word {right arrow over (s)} 1 , and hence form a symmetric sync pattern with the first sync word. Taking the example in the above, when p=4, q=4, the permutation matrix can be given by 
     
       
         
           
             
               
                 
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     The above permutation matrix can give rise to a symmetric permutation. This scheme can be referred to as a symmetric correlation method. Indeed, symmetric correlation demonstrates the symmetry between the two received sync words ( FIG. 3(   b )) instead of the similarity of the two sync words ( FIG. 3(   a )). Using the triangle as the sync word, in  FIG. 3(   a ), the two sync words still have a large part of similarity when 0&lt;μ≦M−1. While in  FIG. 3(   b ), the two sync words do not maintain such similarity even when there is a small timing shift. As one embodiment of permuted sequences combination method, the symmetric correlation method is also robust to the frequency offset. While in the permuted sequences combination method, the estimator of timing offset is also a function of the frequency offset. 
     In simulation with the symmetric correlation method, for m-sequence of length 63, (M=63) as shown in  FIG. 2(   e ) and  FIG. 2(   f ), the expectation of the timing metric is similar to the permuted sequences combination method given by  FIG. 2(   c ) and  FIG. 2(   d ) for QPSK on OFDM and single carrier respectively. 
       FIG. 4  and  FIG. 5  illustrate simulation results showing the performance of the permuted sequences combination method “p-c” (with q=0, p=2), and the symmetric correlation method “s-c” in comparison with the conventional correlation method “c-c”. Also shown is a performance comparison to a method proposed by Schmidl and Cox (s&amp;c) (based on the combination of the cross and auto-correlation). In these simulations, the timing offset used was 49. In  FIG. 4(   a ) and  FIG. 4(   b ), an AWGN channel is applied with QPSK on OFDM (FFT size was 64, CP=16) and QPSK on single carrier. In  FIG. 5(   a ) and  FIG. 5(   b ), Rayleigh fading channels are applied in which there were three channel taps inside CP (the variance of the three channel taps are 1, 0.16, and 0.02 respectively), again with QPSK on OFDM and QPSK on single carrier respectively. 
     From the results, it can be seen that with AWGN, the permuted sequences combination method and the symmetric correlation method have similar performance. Both methods outperform the conventional correlation method, especially at SNRs greater than −2 dB. For the fading channels above, the permuted sequence combination and symmetric correlation methods still outperform the conventional method. Performance may depend on the power of the taps of the fading multipath. 
     As mentioned above, sync words {right arrow over (s)} 1  and {right arrow over (s)} 2  have the same components. The sync words {right arrow over (s)} 1  and {right arrow over (s)} 2  have constant amplitudes and signal power Mσ s   2 . That is, 
                            s   →     1          =              s   →     2          =       M   ⁢           ⁢     σ   s   2       =         ∑     i   =   1       M   -   1       ⁢              s   1     ⁡     (   i   )            2       =       ∑     i   =   1       M   -   1       ⁢              s   2     ⁡     (   i   )            2                     (   25   )               
According to (7), the permuted sequences combination of {right arrow over (s)} 1  and {right arrow over (s)} 2  is given by
 
                         s   →     1     ⊗       s   →     2       =       ∑     i   =   0       M   -   1       ⁢         s   1   *     ⁡     (   i   )       ⁢       s   2     ⁡     (     γ   i     )                   (   26   )               
combining (25) and (26), we have
 
                                  s   →     1          =       ⁢            s   →     2                      =       ⁢     2   ⁢   M   ⁢           ⁢     σ   s   2                   =       ⁢         ∑     i   =   1       M   -   1       ⁢              s   1     ⁡     (   i   )            2       +       ∑     i   =   1       M   -   1       ⁢              s   2     ⁡     (   i   )            2       -     2   ⁢       ∑     i   =   0       M   -   1       ⁢         s   1   *     ⁡     (   i   )       ⁢       s   2     ⁡     (     γ   i     )             +     2   ⁢         s   →     1     ⊗       s   →     2                         (   27   )               
Note that the LHS of (27) is a constant, in order to maximize the last term on the RHS of (27), the first term should be minimized, that is the component of and should be the same. Specifically, the largest permuted sequences combination occurs when
 
s 1 (i)=s 2 (γ i )  (28)
 
Or, writing in matrix notation,
 
{right arrow over (s)} 1 =γ{right arrow over (s)} 2   (29)
 
     As mentioned above, for γT μ γ T  to equal T μ+ν , ν≠0, (p−1) must be relatively prime to M. Since that T μ+ν  is a circulant matrix, it can be fully diagonalized by Fast Fourier Transform (FFT) and Inverse Fast Fourier Transform (IFFT) matrix. Accordingly, for γT μ γ T =T μ+ν  it follows that:
 
 Fγγ   H   FT   μ   F   H   Fγ   T   F   H   =FT   μ+ν   F   H , ν≠0  (30)
 
Then using the diagnalization property of circulant matrix, (30) can be rewritten as
 
Fγγ H Λ μ Fγ T F H =Λ μ+ν , ν≠0  (31)
 
     F is the FFT matrix given by 
                     F   =     [         1       1       …       1           1       ω       …         ω     M   -   1               ⋮       ⋮       ⋮       ⋮           1         ω     M   -   1           …         ω       (     M   -   1     )     2             ]       ⁢     
     ⁢   where           (   32   )               ω   =     exp   ⁡     (       -   j     ⁢           ⁢   2   ⁢     π   /   M       )               (   33   )               
Λ i  is a diagonal matrix whose diagonal elements are given by
 
Λ i ( k,k )=ω i(k−1)   ,k= 1 , . . . , M   (34)
 
     From (20) and (32), we can get for M=5, p=3, q=1, after mathematical manipulation, Fγγ H  is given by 
                       F   ⁢           ⁢     γγ   H       =     [         1       0       0       0       0           0       0         ϖ     -   2           0       0           0       0       0       0         ϖ     -   4               0         ϖ     -   1           0       0       0           0       0       0         ϖ     -   3           0         ]       ⁢     
     ⁢   where           (   35   )                 ϖ   =     exp   ⁡     (       -   j     ⁢           ⁢   2   ⁢           ⁢     π   /   5       )         ⁢     
     ⁢     Generally   ,             (   36   )                 F   ⁢           ⁢     γγ   H       =     {               ω     -     q   ⁡     (     k   -   1     )           ,               (     k   -   1     )     ⁢   p     ⁢     =     mod   ⁢           ⁢   M       ⁢     i   -   1                 0   ,               (     k   -   1     )     ⁢   p     ⁢     ≠     mod   ⁢           ⁢   M       ⁢     i   -   1             ⁢     
     ⁢   Also   ⁢           ⁢   since               (   37   )                 F   ⁢           ⁢     γ   T     ⁢     F   H       =       (     F   ⁢           ⁢     γγ   H       )     H             (   38   )               
then substituting together (35) into the LHS of (31), after mathematical manipulation, it follows that
 
     
       
         
           
             
               
                 
                   
                     
                       F 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         γγ 
                         H 
                       
                       ⁢ 
                       
                         Λ 
                         μ 
                       
                       ⁢ 
                       F 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         γ 
                         T 
                       
                       ⁢ 
                       
                         F 
                         H 
                       
                     
                     = 
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               ϖ 
                               2 
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               ϖ 
                               4 
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               ϖ 
                               6 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               ϖ 
                               8 
                             
                           
                         
                       
                       ] 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     Generally 
                     , 
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       γγ 
                       H 
                     
                     ⁢ 
                     
                       Λ 
                       μ 
                     
                     ⁢ 
                     F 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       γ 
                       T 
                     
                     ⁢ 
                     
                       F 
                       H 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           … 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             ω 
                             
                               p 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               μ 
                             
                           
                         
                         
                           … 
                         
                         
                           0 
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           … 
                         
                         
                           
                             ω 
                             
                               
                                 ( 
                                 
                                   M 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               ⁢ 
                               p 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               μ 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
     where p is the multiplicative inverse of p in the finite field of M. 
     The RHS of (40) is a diagonal matrix whose diagonal elements or (k,k)th component can be expressed as ω (M−1)pμ . Combining (31), (34) and (40), it is possible to satisfy γT μ γ T =T μ+ν , ν≠0, 
                       (     p   -   1     )     ⁢   μ     ⁢     ≠     mod   ⁢           ⁢   M       ⁢   0           (   41   )               
That is, (p−1) is relatively prime to M, where
 
     
       
         
           
             pp 
             ⁢ 
             
               = 
               
                 mod 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 M 
               
             
             ⁢ 
             1. 
           
         
       
     
     The permuted sequences combination embodiments described herein sharpens the timing metric compared with the conventional correlation method. Analysis and simulation show that the permuted sequences combination method outperforms the previous frame synchronization schemes in AWGN channels and fading channels. 
     The foregoing description of exemplary embodiments has been presented for purposes of illustration and description. It is not intended to be exhaustive or to limit the present invention to the precise form disclosed, and modifications and variations are possible in light of the above teachings or may be acquired from practice of the present invention. The embodiments were chosen and described in order to explain the principles of the present invention and its practical application to enable one skilled in the art to utilize the present invention in various embodiments and with various modifications as are suited to the particular use contemplated.