10.07.2015   SonSolar

Frame wavelet function

For a function to be a mother it must be oscillatory, have a fast toward zero and integrate to zero. These condition are said to satisfy the Grossmann-Morlet admissibility condition for every function

he L2(R) (the set of square inferable function):

-dm

Where H(m) is the Fourier transformation of h(t). The constant ch is the admissibility constant of the function h(t). The requirement that (3) is finite allows for inversion of the wavelet transformation given by:

(3)

Where the daughter wavelets are generated from a signal mother wavelet function h(t) by dilatation a (a>0) and transformation b:

ha, b =

h(~r)

(4)

Wj{a, b)= J/(t)h£b(t)dt=( fba, b)

(2)

Ch =

07

To constitute a frame, the discrete wavelet transformation generated by sampling the wavelet parameters (time/scale) on a grid or lattice, not necessarily equally spaced, must satisfy the admissibility condition; and, lattice points must be sufficiency close satisfy basic information — theoretic need. The family of basis function {ha, b(t)} with a, b є Z is said to be a frame if it satisfies the property that there exists two frame bounds A>0 and B<<x Where A, B are independent of f(t) such that of all the functions

he L2(R), the sum of square moduli of Wj(a, b) must be between the two positives bounds:

2

(5)

a, b

Recently, a class of frame wavelets has been introduced for its simplicity and availability that suits adaptive processes, particularly in a neural network architecture [15].

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