DISCRETE WAVELET TRANSFORM

Given a digital signal Fo(k), where k ranges from 0 to N-1 the Discrete Wavelet Transform (DWT) of Fo consists of two FIR (finite impulse response) filters, a low-pass and a high-pass one:

F-1,h = ∑k h^(k-2n) Fo(k)
F-1,g = ∑k g^(k-2n) Fo(k)
where h^(k) are the coefficients of the low-pass filter and g^(k) are those of the high-pass one. These filters must satisfy certain conditions described below.

 

Now we get to the relationship with the wavelet. We associate to the digital signal Fo a function (denoted with the same name, and distinguished by using x as argument) belonging to Vo:

Fo(x) = ∑ Fo(k) so(x-k)

Vo = V-1 + W-1 (semidirect decomposition) and V-1 has basis s-1,k(x)=so(x/2 -k) (scaling function). W-1 has basis w-1,k(x) (wavelet function).

Let s^j,k denote the dual basis to sj,k. The projectionis of Fo on V-1 and W-1 are

F-1,h(x) = ∑ s-1,n(x) ∑k Fo(k) h^(k-2n)
F-1,g(x) = ∑ w-1,n(x) ∑k Fo(k) g^(k-2n)
where
h^(k-2n) = < s^-1,n | so,k >
  = Int 1/sqrt(2) s^*(x/2) s(x+2n-k)
g^(k-2n) = < w^-1,n | so,k >
  = Int 1/sqrt(2) w^*(x/2) s(x+2n-k)


Marco Corvi - Page hosted by geocities.com.