DISCRETE WAVELET TRANSFORMS IN WALSH ANALYSIS

These numbers are digits of the expansion of the number x in base p:
x = ∑ x j p −j−1 + ∑ x j p −j
j<0
j>0
(for p-periodically rational x, the decomposition with a finite number of nonzero terms is obtained).
It is clear that
∞∑
[x] = x −j p j−1
j=1
andthereexistsanumberk = k(x) such that x −j =0forallj > k. By definition, the equality
z = x ⊕ y means that
z = ∑ 〈x j + y j 〉 p p −j−1 + ∑ 〈x j + y j 〉 p p −j
j<0
j>0
and, respectively, z = x ⊖ y, ifz ⊕ y = x.
Assuming that ε p =exp(2πi/p), on the interval [0, 1) we define the function
{
1, x ∈ [0, 1/p),
w 1 (x) =
ε l p, x ∈ [ lp −1 , (l +1)p −1) ,l∈{1,...,p− 1},
and continue it to the half-line R + with period 1. The Walsh generalized system {w l : l ∈ Z + } is
determined by the formula
w 0 (x) ≡ 1, w l (x) =
k∏ (
w1 (p j−1 x) ) ν j
, l ∈ N, x ∈ R + ,
where ν j are the digits of the p-ary decomposition of the number l:
l =
j=1
k∑
ν j p j−1 , ν j ∈{0, 1 ...,p− 1}, ν k ≠0, k = k(l).
j=1
The following notation will be used below.
γ(l) :=
k∑
ν j .
It is well known that the system {w l : l ∈ Z + } is an orthonormal basis in L 2 [0, 1]. In the case where
p = 2, the operations ⊕ and ⊖ coincide and the functions w l (x) are classical Walsh functions. Discrete
wavelet transforms with more than two input channels are used in multidimensional multirate signal
processing systems (see [41]). In the next section, using the functions w l (x), we define an orthogonal
discrete wavelet transform whose number of input channels coincides with the number p. Witha
special choice of parameters, we obtained from it the discrete transforms introduced by Lang in [30].
2. Orthogonal discrete wavelet transform. For an arbitrary natural n, wesetN = p n and
N 1 = p n−1 . According to [12], for any complex vector b =(b 0 ,b 1 ,...,b N−1 ) satisfying the condition
j=1
|b l | 2 + |b l+N1 | 2 + ···+ |b l+(p−1)N1 | 2 =1, 0 ≤ l ≤ N 1 − 1, (1)
we construct an orthonormal wavelet basis in the space of N-periodic complex sequences with the
standard scalar product. The condition (1) also appears in algorithms for constructing orthogonal
wavelet bases on the Vilenkin group G p (see [14, 15, 18]).
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