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ITU Journal on Future and Evolving Technologies, Volume 1 (2020), Issue 1




          ture of statistically independent signals, each character-  2.  BLIND SOURCE SEPARATION
          izing signal information at a different scale. Having this  In BSS, a set of mixtures of different source signals is
          perspective in multi-scale decomposition can be benefi-  available and the goal is to separate the source signals
          cial in signal denoising for two reasons. First, since most  when we have no information about the mixing system
          of the signal information in one scale is not included in  or the source signals (hence the name “blind”) [26, 27].
          the other scales, such decomposition provides the ad-
          vantage of noise suppression at the individual scales in     1  -         -    1   -             -    1
          order to trade off noise suppression for signal-quality     2  -          -    2   -             -    2
          preservation. Second, since the noise signal is statisti-  �     A       �       �      B       �
          cally independent from the original signal, by decompos-
          ing the noisy signal into statistically independent scales,         -     -        -             -     
          the noise is expected to be separated in finer scales.     Mixing System         Separating System
          Our Contribution: Given this motivation and per-     Fig. 1 – Mixing and separating systems in Blind Source Separa-
          spective, we propose a new method for Multi-Scale de-  tion (BSS).
          composition exploiting Independent Component Anal-   As in Fig. 1, the mixing and separating systems can be
          ysis (ICA), called MSICA, in which the original digital  represented as,
          signal is decomposed into approximation and detail with
          statistically independent components. Specifically, we                 x(  ) = As(  ),
          extract two correlated signals from the original signal                y(  ) = Bx(  ),             (1)
          and apply a linear transformation to the extracted sig-
          nals so as to decompose the original signal into multiple  where s(  ) = [   (  ), … ,    (  )] is the vector of sources
                                                                                            
                                                                             1
                                                                                       
          scales. Since we need a suitable transform to decompose  that are mixed by the mixing matrix A and create the
          the original signal into statistically independent compo-  observations vector x(  ) = [   (  ), … ,    (  )] . Let A
                                                                                                         
                                                                                                    
                                                                                          1
          nents, we consider our problem as a Blind Source Sepa-  be a square matrix (   ×   ) of full column rank, which
          ration (BSS) problem in which the extracted signals are  means that the number of sources is equal to the num-
          considered as the observations of the source separation  ber of observations and that the sources are linearly in-
          problem. To relate this problem with the concept of  dependent. The goal of BSS is to find the separating
          multi-scale decomposition, we introduce an equivalent  matrix B such that y(  ) = [   (  ), … ,    (  )] is an es-
                                                                                                         
                                                                                                    
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          filter-bank structure for the proposed method, which is  timation of the sources.
          similar to the structure introduced in [14] for the DWT  A method to solve the BSS problem is via ICA, which
          implementation. We also propose a method for multi-  exploits the assumption of source independence and es-
          channel transmission in which MSICA outperforms com-  timates B such that the outputs    ’s be statistically in-
                                                                                               
          mon wavelet transforms in denoising of the received sig-  dependent [28]. As studied in [28, 29], the ICA can be
          nal.  We show that if the even and odd samples of    performed by two steps: 1) whitening (or decorrelating)
          the original signal are transmitted through two Addi-  and 2) rotation. To illustrate the ICA model, we con-
          tive White Gaussian Noise (AWGN) channels, MSICA     sider two independent components,    ,    = 1, 2, with a
                                                                                                  
          is able to extract and filter out the noise from the noisier  uniform distribution,
          channel. This key property of MSICA—which exploits
          channel diversity and generalizes to the case in which              (   ) = {  1 if |   | ≤ 0.5,   (2)
                                                                                             
          more than two channels are available—can be used to                        0   otherwise,
          increase the transmission efficiency in noisy communi-
          cation channels, although the marginal return dimin-  where the joint density of    and    is uniform on a
                                                                                         1
                                                                                                2
          ishes as the number of channels increases. It should be  square, as illustrated in Fig. 2(a). This follows from
                                                               the definition that the joint density of two independent
          noted that, although single-channel ICA has been used  variables is the product of their marginal densities. Let
          in previous works [21, 22, 23, 24, 25] (including the spa-  us now mix these two independent components by the
          tial case of using even and odd samples), in this work  following mixing matrix A,
          single-channel ICA has been studied as a technique for
          signal decomposition into statistically independent ap-                     1  2       
          proximation and detail and its performance in denoising          [     1  ] = [  2  1  ] [     1  ] ,  (3)
          has been compared with other wavelet transforms.                    2     ⏟⏟⏟⏟⏟       2
          Article Organization: In Section 2, we provide some                           A
          background on BSS and ICA. In Section 3, we propose  where the mixed variables    and    have a uniform
                                                                                         1
                                                                                                2
          our ICA-based transform for multi-scale decomposition.  distribution on a parallelogram, as shown in Fig. 2(b).
          In Section 4, we examine the performance of MSICA in  Note that    and    are not independent anymore. To
                                                                          1
                                                                                2
          signal denoising and show how to increase transmission  show this consider whether it is possible to predict the
          efficiency when multiple (noisy) channels are available.  value of one of them, say    , from the value of the other;
                                                                                      2
          Finally, in Section 5, we draw the main conclusions and  it is clear that if    attains one of its maximum or min-
                                                                               1
          wrap up the article by discussing future work.       imum values, then this completely determines the value


          26                                 © International Telecommunication Union, 2020
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