VKEDCO: Vladimir Kulyukin's Education Coop: CDF(2

Monday, December 14, 2015

Three Scales of SCDF(S, 2, 2, L, 3) and NCDF(S, 2, 2, L, 3) and Formulas for Individual Elements at Each Scale: Conceptual Note 4 on Ripples in Mathematics

Introduction

This is my conceptual note 4 on Chapters 3 and 4 in "Ripples in Mathematics" by A. Jensen & A. la Cour-Harbo. This note continues to discuss the notation SCDF(S, 2, 2, L, J) and NCDF(S, 2, 2, L, J) introduced in conceptual note 3. Recall that in SCDF(S, 2, 2, L, J) the first letter stands for standard whereas in NCDF(S, 2, 2, L, J) the first letter stands for normalized. Jensen and la Cour-Harbo give the normalization formulas in equations 3.40 and 3.41 on p. 23, where the signal samples, i.e., the elements of S, are normalized by sqrt(2) and the elements of D are normalized by 1/sqrt(2). In this note, I will derive the formulas for the three iterations for SCDF(S, 2, 2, L, 3) and NCDF(S, 2, 2, L, 3) and derive the formulas for the individual elements for L = 1, 2, and 3. These manual derivations have helped me verify my implementation of CDF(2, 2) on shorter signals.

Three Scales of SCDF(S,2,2,L,J) and NCDF(S, 2, 2, L, J)

Figures 1 and 2 show how three scales of SCDF(S, 2, 2, L, 3) and NCDF(S, 2, 2, L, 3) are computed, respectively. In Figure 2, if the images for NCDF(S, 2, 2, L, 3) appear blurry, even when enhanced, each element of S, at each of the three scales, is multiplied by sqrt(2) and each element of D, at each scale of the three scales, is divided by sqrt(2), per the normalization formulas on p. 23 in "Ripples in Mathematics."


Figure 1. Three Scales of SCDF(2, 2, L, 3).

Figure 2. Three Scales of NCDF(2, 2, L, 3)

Formulas for Individual Elements at Each Scale of SCDF(S, 2, 2, L, 3)

Figure 3 - 7 define formulas for the individual elements of S and D at each of the three scales of SCDF(S, 2, 2, L, 3).

Figure 3. Formulas for the S elements at L = 1

Figure 4. Formulas for the D elements at L = 1

Figure 5. Formulas for the S elements at L = 2

Figure 6. Formulas for the D elements at L = 2

Figure 7. Formulas for the S and D elements at L = 3

Formulas for Individual Elements at Each Scale of NCDF(S, 2, 2, L, 3)

Figures 8 - 12 state the formulas for the individual elements of S and D at each scale of NCDF(S, 2, 2, L, 3).

Figure 8. Formulas for the S elements at L = 1

Figure 9. Formulas for the D elements at L = 1

Figure 10. Formulas for the S elements at L = 2

Figure 11. Formulas for the D elements at L = 2

Figure 12. Formulas for the S and D elements at L = 3

Monday, November 30, 2015

Individual Elements of Standard & Normalized CDF(2, 2): Conceptual Note 3 on Ripples in Mathematics

Vladimir Kulyukin

Introduction

This is my conceptual note 3 on Chapters 3 and 4 in "Ripples in Mathematics" by A. Jensen & A. la Cour-Harbo. This note continues to discuss the notation for CDF(2, 2) I introduced in conceptual note 2. I changed CDF(2, 2) to CDF(S, 2, 2, L, J) where S denotes the signal, which is a sequence of samples whose length is an integral power of 2. Specifically, the length of S is 2^J. L is the current scale or iteration. It starts at 0, which denotes the original, unmodified signal, and goes up to J. In this note, I will separate standard CDF(2, 2) and normalized CDF(2, 2). The normalization formulas are given in Ch. 3 in equations 3.40 and 3.41 on p. 23, where the signal samples, i.e., the elements of S, are normalized by sqrt(2) and the elements of D are normalized by 1/sqrt(2). I will also state the formulas for computing individuals elements of S and D at a given scale.

Changing CDF(S, 2,2, L, J) to SCDF(S,2,2,L,J) and NCDF(S, 2, 2, L, J)

Let us call CDF(S, 2, 2, L, J) standard CDF(2, 2) and change the notation to SCDF(S, 2, 2, L, J) where the first letter in the abbreviation stands for standard. The meaning of the other symbols in SCDF(S, 2, 2, L, J) is the same as in CDF(S, 2, 2, L, J). Let us also introduce the notation NCDF(S, 2, 2, L, J) to denote normalized CDF(2, 2). Figures 1 and 2 show how two scales of SCDF(S, 2, 2, L, J) and NCDF(S, 2, 2, L, J) are computed, respectively. I will still keep the notation CDF(S, 2, 2, L, J) to denote either SCDF(S, 2, 2, L, J) or NCDF(S, 2, 2, L, J) when the context permits safe omission of normalization.

Figure 1. Two scales of SCDF(S, 2, 2, L, J)

Figure 2. Two scales of NCDF(S, 2, 2, L, J)

Formulas for Individual Elements at Two Scales of CDF(S, 2, 2, 2, 2)

Figures 3 and 4 give formulas for computing individual elements of CDF(S, 2, 2, 2, 2). Specifically, Figure 3 shows how to compute individual elements of CDF(S, 2, 2, 1, 2) and Figure 4 shows how to compute individual elements of CDF(S, 2, 2, 2, 2). If normalization is needed, then each element should be normalized, as shown in Figure 2.