Coiflet

Some theorems about Coiflets:{{cite web|title=COIFLET-TYPE WAVELETS: THEORY, DESIGN, AND APPLICATIONS|url=https://seagrant.mit.edu/ESRDC_library/Wei_Dong_PhD_Thesis.pdf|access-date=2015-01-22|archive-url=https://web.archive.org/web/20160305125125/https://seagrant.mit.edu/ESRDC_library/Wei_Dong_PhD_Thesis.pdf|archive-date=2016-03-05|url-status=dead}}

For a wavelet system $\backslash \{\backslash phi,\backslash tilde\{\backslash phi\},\backslash psi,\backslash tilde\{\backslash psi\},h,\backslash tilde\{h\},g,\backslash tilde\{g\}\backslash \}$, the following three

equations are equivalent:

: $$

\begin{array}{lcl}

\mathcal{M_\tilde{\psi}}(0,l] = 0 & \text{for }l =0,1,\ldots,L-1 \\

\sum_n (-1)^n n^l h[n]=0 & \text{for }l =0,1,\ldots,L-1 \\

H^{(l)}(\pi)=0 & \text{for }l=0,1,\ldots,L-1

\end{array}

and similar equivalence holds between $\backslash psi$ and $\backslash tilde\{h\}$

For a wavelet system $\backslash \{\backslash phi,\backslash tilde\{\backslash phi\},\backslash psi,\backslash tilde\{\backslash psi\},h,\backslash tilde\{h\},g,\backslash tilde\{g\}\backslash \}$, the following six equations

are equivalent:

: $$

\begin{array}{lcl}

\mathcal{M_\tilde{\phi}}(t_0,l] = \delta[l] & \text{for } l=0,1,\ldots,L-1 \\

\mathcal{M_\tilde{\phi}}(0,l] = t_0^l & \text{for } l=0,1,\ldots,L-1 \\

\hat{\phi}^(l)(0)=(-jt_0)^t& \text{for }l=0,1,\ldots,L-1 \\

\sum_n (n-t_0)^l h[n]= \delta[l] & \text{for } l=0,1,\ldots,L-1 \\

\sum_n n^l h[n]=t_0^l & \text{for } l=0,1,\ldots,L-1 \\

H^{(l)}(0)=(-jt_0)^t & \text{for } l=0,1,\ldots,L-1 \\

\end{array}

and similar equivalence holds between $\backslash tilde\{\backslash psi\}$ and $\backslash tilde\{h\}$

For a biorthogonal wavelet system $\backslash \{\backslash phi,\backslash psi,\backslash tilde\{\backslash phi\},\backslash tilde\{\backslash psi\}\backslash \}$, if either $\backslash tilde\{\backslash psi\}$ or $\backslash psi$

possesses a degree L of vanishing moments, then the following two equations are equivalent:

: $$

\begin{array}{lcl}

\mathcal{M_\tilde{\psi}}(t_0,l] = \delta[l] & \text{for } l=0,1,\ldots, \bar{L}-1 \\

\mathcal{M_\psi}(t_0,l] = \delta[l] & \text{for }l=0,1,\ldots, \bar{L}-1 \\

\end{array}

for any $\backslash bar\{L\}$ such that $\backslash bar\{L\}\; \backslash ll\; L$

Both the scaling function (low-pass filter) and the wavelet function (high-pass filter) must be normalised by a factor $1/\backslash sqrt\{2\}$. Below are the coefficients for the scaling functions for C6–30. The wavelet coefficients are derived by reversing the order of the scaling function coefficients and then reversing the sign of every second one (i.e. C6 wavelet = {−0.022140543057, 0.102859456942, 0.544281086116, −1.205718913884, 0.477859456942, 0.102859456942}).

Mathematically, this looks like $B\_k\; =\; (-1)^k\; C\_\{N\; -\; 1\; -\; k\}$, where k is the coefficient index, B is a wavelet coefficient, and C a scaling function coefficient. N is the wavelet index, i.e. 6 for C6.

F = coifwavf(W) returns the scaling filter associated with the Coiflet wavelet specified by the string W where W = "coifN". Possible values for N are 1, 2, 3, 4, or 5.{{cite web|title=coifwavf|url=http://www.mathworks.com/help/wavelet/ref/coifwavf.html|website=www.mathworks.com/|accessdate=22 January 2015}}

Category:Orthogonal wavelets

Category:Wavelets