U. Glaeser

For the discrete wavelet expansion, called DWT, a two-dimensional set of coefficients
in such a way that
© 2002 by CRC Press LLC
akl
is constructed
(27.5)
where the function ψ(
t),
called wavelet (small wave), generates the expansion set ψ(2
t − l),
which is an orthogonal
basis. The wavelet ψ(
t)
is an oscillating and quickly decaying function, which has its energy sufficiently
well localized in time and in frequency. Several different wavelet classes have already been proposed [5].
Introducing another basic function ϕ(), t called the scaling function,
a multiresolution signal representation,
starting from some resolution k,
can be formulated:
Two fundamental self-similarity equations have to be fulfilled:
k
(27.6)
(27.7a)
(27.7b)
where h0(
n)
and h1(
n)
are impulse responses of two discrete-time complementary filters—a lowpass filter
and a highpass filter, respectively. For a finite even length N,
responses h0(
n)
and h1(
n)
are related to each
other by
(27.8)
If resolution k is large enough, κ = k can only be taken into account in expansion (27.6). In other words,
we can assume that
Thus, from Eqs. (27.7a) and (27.7b) we conclude that
modeled as
k
∞
∑
∞
∑
∞
∑
x() t
2 k/2 aklψ 2 k =
( t– l)
k=−∞ l=−∞
x() t 2 k/2 bkl ϕ 2 k ( t– l)
2 κ/2 aκlψ 2 k =
+
( t– l)
l=−∞
∑
κ=k l=−∞
(27.9)
x(
t)
is a signal of resolution k + 1 and can be
(27.10)
Assuming that functions ϕ (2 t − l) and ψ (2 k t − l) in expansion (27.9) form an orthonormal basis,
after some manipulations, we conclude that
(27.11a)
(27.11b)
If the signal x(t) is of finite duration, the sums in expressions (27.9) and (27.10) are finite. The sets {b (k+1)n}
and {b kl, a kl} form alternative parametric descriptions for the signal x(t). Although both sets have the same
∞
∞
∑ ∑
ϕ() t = 2h0( n)ϕ(
2t– n)
n
∑
ψ() t = 2h1( n)ϕ(
2t– n)
n
h1( n)
( – 1)
n = h0( N – 1 – n),
n = 0,…,N – 1
∞
∑
x() t 2 k/2 bklϕ 2 k ( t– l)
2 k/2 aklψ 2 k =
+ ( t– l)
l=−∞
∞
∑
x() t 2 k+1
=
n=−∞
∞
∑
l=−∞
( )/2 b( k+1 )nϕ2 k+1 t– n
∑
bkl = h0( n – 2l)bk+1
n
∑
( )
( )n
akl =
h1( n – 2l)bk+1
n
( )n