**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**.

Figure 3. Individual elements of CDF(S, 2, 2, 1, 2) |

Figure 4. Individual elements of CDF(S, 2, 2, 2, 2) |

**SCDF(S, 2, 2, 2, 2) Example**

**Figures 5**and

**6**give two scales of SCDF(

*S*, 2, 2, 1, 2) where the signal

*S*= [1, 2, 3, 4].

Figure 5. SCDF(S, 2, 2, 1, 2) |

Figure 6. SCDF(S, 2, 2, 2, 2) |

**NCDF(S, 2, 2, 2, 2) Example**

**Figures 7**and

**8**give two scales of NCDF(

*S*, 2, 2, 1, 2) where the signal

*S*= [1, 2, 3, 4].

Figure 7. NCDF(S, 2, 2, 1, 2) |

Figure 8. NCDF(S, 2, 2, 2, 2) |