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.
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