Spectral Transformations - Biomedical Signal Analysis: Contemporary Methods and Applications - page 47

8

ω

, d m=3

x x x x x x x x x x x x x

'

3,n

m

4

ω

, d m=2

x x x xx xxxx

'

2,n

2

ω

, d m=1 x x x x x

1,n '

x x

ω , d

' m=0

x

0,n

0

n

0 0.5 1.0 1.5 2.0

Figure 2.7
Dyadic sampling grid for the DWT.

pieces (or projections) give finer and finer details of f ( t ). For audio
signals, these scales are essentially octaves. They represent higher and
higher frequencies. For images and all other signals, the simultaneous
appearance of multiple scales is known as multiresolution .
Mallat and Meyer's method [165] for signal decomposition based on
orthonormal wavelets with compact carrier will be reviewed here. We
will establish a link between these wavelet families and the hierarchic
filter banks. In the last part of this section, we will show that the FIR
PR-QMF hold the regularization property, and produce orthonormal
wavelet bases.

Multiscale-Analysis Spaces

Multiscale signal analysis provides the key to the link between wavelets
and pyramidal dyadic trees. A wavelet family is used to decompose a
signal into scaled and translated copies of a basic function. As stated
before, the wavelet family consists of scaling and wavelet functions.
Scaling functions ϕ ( t ) alone are adequate to code a signal completely,
but a decomposition based on both scaling and wavelet functions is most
ecient.