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2013/02/11 15:08:09 -0600
The first summary is given in four tables of the basic relationships and equations, primarily developed in Chapter: The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients , for the scaling function φ(t), scaling coefficients h(n), and their Fourier transforms Φ(ω) and H(ω) for the multiplier M=2 or two-band multiresolution system. The various assumptions and conditions are omitted in order to see the “big picture" and to see the effects of increasing constraints.
Case | Condition | φ ( t ) | Φ ( ω ) | Signal Space |
1 | Multiresolution |
![]() |
![]() | distribution |
2 | Partition of 1 | ∑ φ ( t – n ) = 1 | Φ ( 2 π k ) = δ ( k ) | distribution |
3 | Orthogonal |
![]() | ∑ | Φ ( ω + 2 π k ) | 2 = 1 | L2 |
5 | SF Smoothness |
![]() | poly ∈Vj | |
6 | SF Moments |
![]() | Coiflets |
Case | Condition | h ( n ) | H ( ω ) | Eigenval.{T} |
1 | Existence |
![]() |
![]() | |
2 | Fundamental | ∑ h ( 2 n ) = ∑ h ( 2 n + 1 ) | H ( π ) = 0 | EV = 1 |
3 | QMF |
![]() | | H ( ω ) | 2 + | H ( ω + π ) | 2 = 2 | EV ≤ 1 |
4 | Orthogonal |
![]() | | H ( ω ) | 2 + | H ( ω + π ) | 2 = 2 | one EV =1 |
L2 Basis | and H(ω)≠0,|ω|≤π/3 | others <1 | ||
6 | Coiflets | ∑ nkh ( n ) = 0 |
Case | Condition | ψ ( t ) | Ψ ( ω ) | Signal Space |
1 | MRA |
![]() |
![]() | distribution |
3 | Orthogonal |
![]() | L2 | |
3 | Orthogonal |
![]() | L2 | |
5 | W Moments |
![]() | poly not∈Wj |
Case | Condition | h1 ( n ) | H1 ( ω ) | Eigenval.{T} |
2 | Fundamental | ∑ h1 ( n ) = 0 | H1 ( 0 ) = 0 | |
3 | Orthogonal | h1 ( n ) = ( – 1 ) nh ( 1 – n ) | | H1 ( ω ) | = | H ( ω + π ) | | |
3 | Orthogonal | ∑ h1 ( n ) h1 ( 2 m – n ) = δ ( m ) |
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H1
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