SCDF(S, 2, 2, L, 3): Example 2: S = [8, 7, 6, 5, 4, 3, 2, 1]

Vladimir Kulyukin

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Problem
  

This post is a detailed example of computing individual elements of SCDF(S, 2, 2, L, 3) at each of the three scales on S = [8, 7, 6, 5, 4, 3, 2, 1]. SCDF(S, 2, 2, L, J) notation is defined here. This notation is based on the notation introduced in Chapters 3 and 4 in "Ripples in Mathematics" by A. Jensen & A. la Cour-Harbo.

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SCDF(S, 2, 2, 1, 3): Scale L = 1
  
Figures 1 and 2 show the formulas for the elements of S and D at scale L = 1. Figure 3 shows an example of computing the first scale of the signal S = [8, 7, 6, 5, 4, 3, 2, 1].  

Figure 1. Formulas for elements of S at scale L = 1

Figure 2. Formulas for elements of D at scale L = 1

Figure 3. Computing elements of S and D at scale L = 1

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SCDF(S, 2, 2, 2, 3): Scale L = 2
  
Figure 4 shows the individual formulas for S and D at scale L = 2. Figure 5 shows the computation of individual elements of S and D of the sample [8, 7, 6, 5, 4, 3, 2, 1] at scale L = 2.

Figure 4. Formulas for elements of S and D at scale L = 2

Figure 5. Computing elements of S and D at scale L = 2

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SCDF(S, 2, 2, 3, 3): Scale L = 3
  
Figure 6 shows the individual formulas for S and D at scale L = 3. Figure 7 shows the computation of individual elements of S and D of the sample [8, 7, 6, 5, 4, 3, 2, 1] at scale L = 3.

Figure 6. Formulas for individual elements of S and D at scale L = 3

Figure 7. Computing elements of S and D at scale L = 3

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Summary of SCDF(S, 2, 2, 3, 3) at All Three Scales
  
Figure 8 summarizes the three scales of SCDF(S, 2, 2, 3, 3), where S =  [8, 7, 6, 5, 4, 3, 2, 1] .

Figure 8. Computing three scales of SCDF(S, 2, 2, 3, 3) for S =  [8, 7, 6, 5, 4, 3, 2, 1]

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