DISCRETE WAVELET TRANSFORMS IN WALSH ANALYSIS

where ε 3 =exp(2πi/3). Lang (see [30]) considered the discrete transform O(2, 2) associated with the
vector b (0) =(1,a,0,b), where 0 <a≤ 1anda 2 + b 2 = 1. For this transform, nonzero coefficients
in (3) and (4) are determined by the formulas
c (0)
0 = c (1)
1 = 1+a + b
2 √ , c (0)
1 = −c (1)
0 = 1+a − b
2
2 √ ,
2
c (0)
2 = c (1)
3 = 1 − a − b
2 √ , c (0)
3 = −c (1)
2 = 1 − a + b
2
2 √ .
2
When processing a particular signal, a selected discrete wavelet transform is applied iteratively:
after the first step, the vectors d (1)
j−1
, ..., d(p−1)
j−1
remain in the memory, and the vector a j−1 passes
into the approximating vector a j−2 and detailing vectors d (1)
j−2
, ..., d(p−1)
j−2
, etc. At each step, the
dimensions of the vectors decrease p times. As a result, after j 0 steps, we obtain the following vectors:
a j−j0 , d (1)
j−1
, ..., d(p−1)
j−1
; ...; d(1)
j−j 0
, ..., d (p−1)
j−j 0
.
The vector a j is reconstructed from these vectors by the inverse transform.
If a multiple-scale analysis (MSA) of {V j } in L 2 (R + ) is constructed by using the vector b (0) (see [9,
15]), then, similarly to [33, Theorem 7.7], the formulas (3) and (4) allow one to quickly pass from one
level of approximating an arbitrary function f ∈ L 2 (R + ) by subspaces V j to another 2 . The freedom of
choosing the vector b (0) allows one to adapt the transform O(p, n) to the signal processed according
to the root-mean-square, entropy, or other criteria (see examples in [17, 19] and [33, Sec. 11.4.1]).
3. Coding of the generalized Weierstrass function. It was proved in a recent paper [36] that
for encoding some fractal signals, the discrete wavelet transform O(p, n) has advantages over the
discrete Haar transform and the zone coding method from [25, Sec. 11.3]. We illustrate this result by
the following example of coding values of the generalized Weierstrass function
W α,β (x) =
∞∑
α k e βkπix , 0 <α≤ 1, β ≥ 1 α .
k=1
As four original signals, we choose the values of the function W α,β (x) at the 243 nodes of the uniform
partition of the interval [0, 1) for the pairs of indices α and β listed in Table 1.
Let T N ) be the matrix composed of the numbers w(N)
l,k
= w k (l/N), 0 ≤ l, k ≤ N − 1.
As usually, we denote by TN ∗ the complex conjugate matrix T N. Then the direct and inverse discrete
multiplicative transforms defined in the space C N have the form
=(w (N)
l,k
̂x = N −1 T ∗ N x, x = T N ̂x, x ∈ C N .
The zone coding method Z(p) used in [25, Sec. 11.3] consists of applying the direct discrete multiplicative
transform to the vector x and subsequent setting to zero the last p n p n−1 components of the
vector ̂x, i.e., ̂x(p n−1 ), ̂x(p n−1 +1), ..., ̂x(p n − 1). Then, after applying the inverse discrete multiplicative
transform, we obtain the vector ˜x. The recovery error is estimated as δ = ‖x − ˜x‖ 2 / √ N.When
performing computational experiments, orthogonal discrete wavelet transforms O(3, 1) and O(3, 2)
were compared in [36] not only with the method of zone coding Z(3), but also and with the discrete
Haar transform H(3) defined by the coefficients (5). Results of calculations are presented in Table 1.
2 Recall that closed subspaces V j ⊂ L 2 (R +), j ∈ Z, formanMSAifV j ⊂ V j+1, ⋂ V j = {0}, and ⋃ V j = L 2 (R +). A
jth-level approximation of a function f ∈ L 2 (R +) is the orthogonal projection of this function onto V j.
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