Page 48 - ITU Journal, Future and evolving technologies - Volume 1 (2020), Issue 1, Inaugural issue
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ITU Journal on Future and Evolving Technologies, Volume 1 (2020), Issue 1




                                                               Since D is a diagonal matrix, we conclude that multi-
                                                               plying Q by the observation vector x results in uncor-
                                                                         
                                                               related outputs. Hence, the whitening matrix W can be
                                                               considered to be equal to Q .
                                                                                          
                                                               Now, we need to find the elements of matrix W so as to
                                                               determine finally the   (  ) and ℎ(  ) filters. Since   (  )
                                                               is a WSS signal, we have,
           Fig. 3 – Whitening process in ICA as a filter-bank structure.
                                                                              2
                                                                                          2
                                                                                                        2
                                                                     (0) =    {   (2  )} =    {   (2   + 1)} =    ,
                                                                     
                                                                                                          
          Using (8) and (10), we can write,                              (1) =    {   (2  )    (2   − 1)} =      .  (19)
                                                                                                   2
                                                                                                     
                                                                         
                       (  ) =       (  ) +       (  ) ,  (11)  Hence, with regard to (8), we can recast C as,
                                       12 2
                             11 1
                                                                                                       
                     1
                       (  ) =       (  ) +       (  ) .                               2             1    
                     2
                             21 1
                                       22 2
                                                                                        2
                                                                             
                                                                                      
                                                                                                2
          Also,    (  ) and    (  ) can be written as,           C =    {xx } = [                  2  ] =    [      1  ], (20)
                                                                                   2
                                                                                                  
                                                                    
                                                                                     
                1
                         2
                                                               where the eigenvalues and eigenvectors of C are,
                      (  ) =   (2  ) = [  (  ) ∗   (  )] ↓ 2,                                           
                    1
                     (  ) =   (2   − 1) = [  (  ) ∗   (   − 1)] ↓ 2,           2              2
                   2
                                                      (12)                  =    (1 +   )     =    (1 −   ) ,
                                                                                         2
                                                                                                
                                                                                 
                                                                          1
          where ∗ and ↓ 2 denote convolution and downsampling            q = [   √ 1 2  ]  q = [  − √ 1 2  ] .  (21)
          by a factor of 2, respectively, and   (  ) is the unit impulse   1     √ 1 2   2      √ 1 2
          signal. Using (11) and (12), we can rewrite    (  ) and
                                                  1
             (  ) as,                                          Hence, the whitening matrix W can be presented as,
          2
              (  ) =      (2  ) +       (2   − 1)                  W = Q = [ q     q ] = [    √ 1 2  √ 1 2  ] .  (22)
                                                                                         
                                                                            
                    11
                               12
            1
                 =    11  [  (  ) ∗   (  )] ↓ 2 +    12  [  (  ) ∗   (   − 1)] ↓ 2  1  2     − √ 1 2  √ 1 2
                 = [  (  ) ∗ (     (  ) +      (   − 1))] ↓ 2,
                                     12
                            11
              (  ) =      (2  ) +       (2   − 1)              Comparing (14) and (22),   (  ) and ℎ(  ) can be written
                    21
                               22
            2
                 =    21  [  (  ) ∗   (  )] ↓ 2 +    22  [  (  ) ∗   (   − 1)] ↓ 2  as,
                                                                                  1
                                                                                           1
                 = [  (  ) ∗ (     (  ) +      (   − 1))] ↓ 2.               (  ) = √   (  ) + √   (   − 1),  (23)
                            21
                                     22
                                                                                   2
                                                                                           2
                                                      (13)                ℎ(  ) = − √   (  ) + √   (   − 1).
                                                                                            1
                                                                                   1
          Note that, if we consider the low-pass filter   (  ) and                  2       2
          high-pass filter ℎ(  ) as,                           From (23), it is clear that   (  ) and ℎ(  ) are a low and
                                                               high-pass filter, respectively, as we wanted to demon-
                      (  ) =      (  ) +      (   − 1),  (14)  strate. Hence, we can conclude that the whitening pro-
                            11
                                     12
                    ℎ(  ) =      (  ) +      (   − 1),         cess in the ICA (presented as a filter-bank structure in
                                      22
                            21
                                                               Fig. 3) decomposes the observation signals into uncorre-
          (13) can be rewritten in a simpler form as,          lated approximation and detail. Moreover, the rotation
                                                               process in the ICA makes sure that the approximation
                                                                                                      1
                          (  ) = [  (  ) ∗   (  )] ↓ 2,  (15)  and the detail are statistically independent . Hence,
                        1
                          (  ) = [  (  ) ∗ ℎ(  )] ↓ 2.         if the even and odd samples of a one-dimensional sig-
                        2
                                                               nal are considered as the observations of the ICA, we
          From (15), we observe that the whitening process can be  ensure the approximation and detail to be statistically
          modeled as a filter-bank structure, as shown in Fig. 3.  independent. As we will see in the next section, the sta-
          Now, we need to show that   (  ) and ℎ(  ) are indeed  tistical independency between the approximation and
          low-pass and high-pass filters. In order to do so, we  detail can be very beneficial in signal denoising, espe-
          consider the covariance matrix of x as follows,      cially when the even and odd samples are transmitted
                                                               through different (noisy) channels.
                                   
                                             
                       C =   {xx } = QDQ ,            (16)     Fig. 4 showcases a signal decomposition by different
                          
                                                               wavelet transform and MSICA, where a Piece-Regular
          where Q is an orthogonal matrix of eigenvectors and D  signal is decomposed into the approximation and detail.
          is a diagonal matrix of eigenvalues. Interestingly, the  As shown in Fig. 4(h), like all the other wavelet trans-
          covariance matrix of z = Q x can be written as,      forms, MSICA is also able to decompose the original
                                   
                                                               signal into approximation and detail, where the approx-
           C =    {zz } =    {Q xx Q} = Q    {xx } Q. (17)     imation and detail coefficients contain the low and high-
                                    
                                            
                                                  
                       
                                
              
                                                               frequency components, respectively. As said earlier, to
          Given (16) and (17), we can write,
                                                               1 Based on our simulations, the separation matrix was always close
                                       
                                
                       C = Q Q D Q Q = D.             (18)      to (22), which means that the rotation matrix was always close
                                   ⏟
                            ⏟
                          
                              I      I                          to the identity matrix.
          28                                 © International Telecommunication Union, 2020
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