Spectral Transformations - Biomedical Signal Analysis: Contemporary Methods and Applications - page 56

scalings and translations of a mother wavelet. We will show an ecient
signal coding that uses scaling and wavelet functions at two successive
scales. In other words, we give a recursive algorithm which supports the
computation of wavelet coe cients of a function f ( t )

∈ L 2 ( R ).

Assume we have a signal or a sequence of data { c 0 ( n ) | n ∈ Z } ,and

c 0 ( n )isthe n th scaling coecient for a given function f ( t ):

c 0,n = <f,ϕ 0n >

for each n Z . This assumption makes the recursive algorithm work.

The decomposition and reconstruction algorithm is given by theorem
2.2 [278].

∈

Theorem 2.2:
Let

be a multiscale analysis with associated scaling function
ϕ ( t ) and scaling filter h 0 ( n ). The wavelet filter h 1 ( n )isdefinedby
equation (2.52), and the wavelet function is defined by equation (2.53).
Given a function f ( t )

{ V k }

L 2 ( R ), define for n

∈ ∈ Z

c 0,n = <f,ϕ 0n > (2.68)

and for every m ∈ N and n ∈ Z ,

c m,n = <f,ϕ mn > and d m,n = <f,ψ mn > (2.69)

Then the decomposition algorithm is given by

c m+1,n = √ 2
k

d m+1,n = √ 2
k

c m,k h 0 ( k − 2 n ) d m,k h 1 ( k − 2 n )

(2.70)

and the reconstruction algorithm is given by

c m,n = √ 2
k

2 k )+ √ 2
k

c m+1,n h 0 ( n − d m+1,n h 1 ( n − 2 k ) (2.71)

From equation (2.70) we obtain for m = 1 at resolution 1/2 the
wavelet d 1,n and the scaling coecients c 1,n :