Page 49 - 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




            2                        3                        3                         3
            1.8
                                     2                        2                         2
            1.6
                                     1                        1                         1
            1.4
            1.2                      0                        0                         0
           Amplitude  0.8 1         0.4  0  200  400  600  800 Approximation  1400  1600  1800  2000  1  0  200  400  600  800 Approximation  1400  1600  1800  2000  0.4  0  200  400  600  800 Approximation  1400  1600  1800  2000
                                                                           1200
                                                                         1000
                                                                                                     1200
                                                                                                  1000
                                                1000
                                                  1200
            0.6                     0.2                       0.5                      0.2
            0.4                      0                        0                         0
            0.2                     -0.2                      -0.5                     -0.2
            0                       -0.4                      -1                       -0.4
             0  500  1000  1500  2000  2500  3000  3500  4000  0  200  400  600  800  1000  1200  1400  1600  1800  2000  0  200  400  600  800  1000  1200  1400  1600  1800  2000  0  200  400  600  800  1000  1200  1400  1600  1800  2000
                     Original Signal           Detail                    Detail                   Detail
                      (a)                      (b)                       (c)                      (d)
                                                              3
            3                        3                                                  3
                                                              2
            2                        2                                                  2
            1                        1                        1                         1
            0                        0                        0                         0
            0  200  400  600  800  1000  1200  1400  1600  1800  2000  0  200  400  600  800  1000  1200  1400  1600  1800  2000  0  200  400  600  800  1000  1200  1400  1600  1800  2000  0  200  400  600  800  1000  1200  1400  1600  1800  2000
                     Approximation            Approximation            Approximation             Approximation
                                                              0.4
           0.4                       1                                                  1
                                                              0.2
           0.2                      0.5                                                0.5
            0                        0                        0                         0
           -0.2                     -0.5                      -0.2                     -0.5
           -0.4                      -1                       -0.4                      -1
            0  200  400  600  800  1000  1200  1400  1600  1800  2000  0  200  400  600  800  1000  1200  1400  1600  1800  2000  0  200  400  600  800  1000  1200  1400  1600  1800  2000  0  200  400  600  800  1000  1200  1400  1600  1800  2000
                      Detail                   Detail                    Detail                   Detail
                      (e)                      (f)                       (g)                      (h)
          Fig. 4 – Comparison of MSICA with different wavelet transforms in decomposing a Piece-Regular signal. (a) Original Signal. Approxi-
          mation and detail by (b) Daubechies 3 wavelet; (c) Haar wavelet; (d) Biorthogonal 2.2 wavelet; (e) Coiflets 4 wavelet; (f) Fejer-Korovkin
          4 wavelet; (g) discrete Meyer wavelet; and (h) our proposed MSICA.
             2                                                 4.   MSICA FOR SIGNAL DENOISING
            1.8
                                                               We discuss now how MSICA can be beneficial in sig-
                                                               nal denoising. Specifically, we compare MSICA with
            1.6
            1.4                                                the other wavelet transforms and show how MSICA can
                                                               suppress the noise via a simple wavelet thresholding.
            1.2
            Amplitude  1                                       We also show that, in the case of multi-channel trans-
                                                               mission, MSICA outperforms the other wavelet trans-
                                                               forms and is able to extract and filter out the noise of
            0.8
                                                               the noisier communication channel by exploiting chan-
            0.6
                                                               nel diversity.
            0.4
                                                               Let us assume that the original signal is passed through
            0.2
                                                               an AWGN channel, the noisy output signal is then,
             0
              0    500  1000  1500  2000  2500  3000  3500  4000
                             Reconstructed Signal                        (  ) =    (  ) +    (  ) ,    = 1, … ,   ,  (26)
          Fig. 5 – Reconstructed signal by MSICA using the approximation  where   (  ) is the original signal and   (  ) is the AWGN
          and detail in Fig. 4(h).
                                                               with zero mean and variance of    . The goal of sig-
                                                                                              2
                                                                                                
          reconstruct the original signal we need to multiply the  nal denoising is to remove the noise and obtain an es-
          inverse of the separation matrix (B ) by the vector of  timate ̂  (  ) of   (  ) that minimizes the Mean Squared
                                        −1
          approximation and detail y so as to obtain the obser-  Error (MSE), defined as,
          vation vector x and reconstruct the original signal   (  )
          from x, i.e.,                                                             1               2
                                                                         MSE ( ̂  ) =  ∑ ( ̂  (  ) −   (  )) .  (27)
                                                                                     
                                                                                       =1
                            (  )
                                        −1
                       [  1    ] = x = B y,           (24)     Note that the model presented in (26) is not general
                            (  )
                                                               since the noise may be non-additive, and the relation
                          2
                                                               between the noisy observed signal and the original signal
               (  ) = (   (  ) ↑ 2) + (   (  ) ↑ 2) ∗   (   + 1),  (25)  may be stochastic. Nevertheless, (26) is a widely used
                                 2
                      1
                                                               model in many practical situations as it serves well as
          where    (  ) ↑ 2,    = 1, 2, denotes the upsampling of  a motivating example to give a good sense as to what
                   
             (  ) by a factor of 2. Fig. 5 shows the reconstructed sig-  happens in more realistic channels.
             
          nal using (24) and (25). This figure shows that MSICA  Let us emphasize that there are many existing ap-
          can successfully reconstruct the original signal from the  proaches in the literature to perform signal denoising,
          approximation and detail obtained in the decomposition  which can be roughly divided into two main categories:
          procedure.                                           1) denoising in the original signal domain (e.g., in time
                                             © International Telecommunication Union, 2020                    29
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