Coiflet

Theory

Some theories about Coiflet :

Theorem 1

For a wavelet system { ϕ , ϕ ~ , ψ , ψ ~ , h , h ~ , g , g ~ }, the following three equations are equivalent:

M ψ ~ ( 0 , l ] = 0 for  l  =0,1,...,L-1 ∑ n ( − 1 ) n n l h [ n ] = 0 for  l  =0,1,...,L-1 H ( l ) ( π ) = 0 for  l  =0,1,...,L-1
and similar equivalence holds between ψ and h ~

Theorem 2

For a wavelet system { ϕ , ϕ ~ , ψ , ψ ~ , h , h ~ , g , g ~ }, the following six equations are equivalent:

M ϕ ~ ( t 0 , l ] = δ [ l ] for  l  =0,1,...,L-1 M ϕ ~ ( 0 , l ] = t 0 l for  l  =0,1,...,L-1 ϕ ^ ( l ) ( 0 ) = ( − j t 0 ) t for  l  =0,1,...,L-1 ∑ n ( n − t 0 ) l h [ n ] = δ [ l ] for  l  =0,1,...,L-1 ∑ n n l h [ n ] = t 0 l for  l  =0,1,...,L-1 H ( l ) ( 0 ) = ( − j t 0 ) t for  l  =0,1,...,L-1
and similar equivalence holds between ψ ~ and h ~

Theorem 3

For a biorthogonal wavelet system { ϕ , ψ , ϕ ~ , ψ ~ }, if either ψ ~ or ψ possesses a degree L of vanishing moments, then the following two equations are equivalent:
M ψ ~ ( t 0 , l ] = δ [ l ] for  l  =0,1,..., L ¯ − 1 M ψ ( t 0 , l ] = δ [ l ] for  l  =0,1,...,  L ¯ − 1
for any L ¯ such that L ¯ ≪ L

Coiflet coefficients

Both the scaling function (low-pass filter) and the wavelet function (High-Pass Filter) must be normalised by a factor 1 / 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.

Matlab function

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.