Spectral Transformations - Biomedical Signal Analysis: Contemporary Methods and Applications - page 45

2 2 dt

|| f || | f ( t ) | (2.35)

A, B > 0 are the frame bounds. A tight, exact frame that has A = B =1
represents an orthonormal basis for L 2 ( R ). A notable characteristic of
orthonormal wavelets

{ ψ mn ( t ) } is

ψ mn ( t ) ψ m n ( t ) dt = 1 ,m = m ,n = n

(2.36)

0 , else

In addition they areorthonormal in both indices. This means that for the
same scale m they are orthonormal both in time and across the scales.
For the scaling functions the orthonormal condition holds only for a
given scale

ϕ mn ( t ) ϕ ml ( t ) dt = δ n−l

(2.37)

The scaling function can be visualized as a low-pass filter. While scaling
functions alone can code a signal to any desired degree of accuracy,
eciency can be gained by using the wavelet functions. Any signal
f

L 2 ( R )atthescale m can be approximated by its projections on
the scale space.
The similarity between ordinary convolution and the analysis equa-
tions suggests that the scaling function coecients and the wavelet func-
tion coecients may be viewed as impulse responses of filters, as shown
in Figure 2.6. The convolution of f ( t )with ψ m ( t )isgivenby

∈

y m ( t )= f ( τ ) ψ m ( τ − t ) dτ (2.38)

where

ψ m ( t )=2 −m/2 ψ (2 −m t )

(2.39)

Sampling y m ( t )at n 2 m yields

y m ( n 2 m )=2 −m/2 f ( τ ) ψ (2 −m τ

− n ) dτ = d m,n (2.40)