## Tuesday, December 15, 2015

### SCDF(S, 2, 2, L, 3): Example 3: S = [1, 1, 1, 1, 4, 4, 4, 4]

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 = [1, 1, 1, 1, 4, 4, 4, 4]. 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.

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 = [1, 1, 1, 1, 4, 4, 4, 4].
 Figure 1. Formulas for individual elements of S at scale L = 1
 Figure 2. Formulas for individual elements of D at scale L = 1
 Figure 3. Computing elements of S and D at scale L = 1

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 S = [1, 1, 1, 1, 4, 4, 4, 4] at scale L = 2.
 Figure 4. Formulas for individual elements of S and D at scale L = 2
 Figure 5. Computing elements of S and D at scale L = 2

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 sampleS = [1, 1, 1, 1, 4, 4, 4, 4] 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

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 =  S = [1, 1, 1, 1, 4, 4, 4, 4] .

 Figure 8. Summary of SCDF(S, 2, 2, 3, 3) for three scales on S = [1, 1, 1, 1, 4, 4, 4, 4]