DISCRETE WAVELET TRANSFORMS IN WALSH ANALYSIS

DOI 10.1007/s10958-021-05476-2
Journal of Mathematical Sciences, Vol. 257, No. 1, August, 2021
DISCRETE WAVELET TRANSFORMS IN WALSH ANALYSIS
Yu. A. Farkov UDC 517.518, 621.391
Abstract. A review of discrete wavelet transforms defined through Walsh functions and used for image
processing, compression of fractal signals, analysis of financial time series, and analysis of geophysical
data is presented. Relationships of the discrete transforms considered with wavelet bases recently
constructed and frames on the Cantor and Vilenkin groups are noted.
Keywords and phrases: Walsh function, Haar system, Weierstrass function, wavelet, frame, zerodimensional
group, discrete transform, image processing, signal coding, analysis of geophysical data.
AMS Subject Classification: 42C40, 65T60
1. Introduction. It was noted in the introduction to the monograph [1] that the interpretation
of Walsh functions as characters of the Cantor dyadic group was proposed by I. M. Gelfand. A wide
class of locally compact Abelian groups (called Vilenkin groups in modern literature) containing the
Cantor group as a special case was defined in [43]. Independently, Walsh functions as characters of
the Cantor dyadic group were studied by Fain (see [23]). Basic information from the theory of Walsh
series and transforms is presented in [25, 37]. Orthogonal wavelets with compact supports on the
Cantor group and the corresponding wavelets on the positive half-line were defined in [30, 35]. As was
noted in [14], approximative properties of orthogonal wavelets with compact support on the Cantor
group are significantly different from properties of Daubeshies wavelets. In addition, Lang wavelets
(see [30]) on a fixed segment of the positive half-line can have arbitrarily high dyadic smoothness, and
for classical wavelets, the smoothness increases with increasing of the length of the support (see [7,
Sec. 7.1], [34, Sec. 7.3], and [35]). The main methods of constructing orthogonal, biorthogonal, periodic,
and nonstationary wavelets on Vilenkin groups were discussed in recent papers [14–16] (see also the
references therein). The additive group of the field of p-adic numbers Q p and the Vilenkin group G p
are locally compact zero-dimensional groups (see [1, 27]). Constructions of wavelets on the field Q p
and their applications both in the theory of functions and in mathematical physics were analyzed
in [28, 39]. Recently, it was proved (see [8]) that in L 2 (Q p ) any orthogonal wavelet basis consisting of
functions with bounded spectrum is a modification of the Haar system.
It is well known that the fast discrete wavelet transform splits signals into low-frequency (approximating)
and high-frequency (detailing) components followed by partial sampling; the inverse transform
performs the signal recovery (see, e.g., [33, Sec. 7.3]). In this paper, we present a review of discrete
wavelet transforms defined by using generalized Walsh polynomials and used for image processing
(see [19]), compressing fractal signals (see [17, 36]), analysis of financial time series (see [36]), and
analysis of geophysical data (see [32, 36, 38]).
Let p be an integer, p ≥ 2, and let R + =[0, +∞). As usual, by Z, Z + ,andN we denote the sets
of integers, nonnegative integers, and natural numbers, respectively. We denote by 〈s〉 p the remainder
when dividing an integer s by p, andby[x] the integer part of x. Foreachx ∈ R + we set
x j = 〈[p j x]〉 p , x −j = 〈[p 1−j x]〉 p , j ∈ N.
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie
Obzory, Vol. 160, Proceedings of the International Conference on Mathematical Modelling in Applied Sciences
ICMMAS’17, Saint Petersburg, July 24–28, 2017, 2019.
1072–3374/21/2571–0127 c○ 2021 Springer Science+Business Media, LLC 127

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]).
128

We denote by G(p, n) thesetofallvectorsb ∈ C N satisfying the condition (1). Using any method
of unitary matrix expansion 1 (see, e.g., [34, Sec. 2.6] and [14, p. 24]), for an arbitrary fixed vector
b (0) =(b (0)
0 ,b(0) 1 ,...,b(0) N−1
)ofG(p, n), we find the numbers b(s)
k
,0≤ k ≤ N − 1, 1 ≤ s ≤ p − 1, such
that the matrices
⎛
b (0)
l
b (0)
l+N 1
... b (0)
l+(p−1)N 1
M l =
b (1)
l
b (1)
l+N 1
... b (1)
l+(p−1)N
⎜
1
⎝.............................
⎟
⎠ , 0 ≤ l ≤ N 1 − 1,
b (p−1)
l
b (p−1)
l+N 1
... b (p−1)
l+(p−1)N 1
⎞
are unitary. Next, using the fast Vilenkin–Chrestenson transform, we calculate the coefficients
andthensetb (s)
k
k
= 1 N−1
∑
b (s)
j
w j (k/N), 0 ≤ k ≤ N − 1, 0 ≤ s ≤ p − 1, (2)
N 1
c (s)
j=0
= c (s)
k
=0fork ≥ N.
The orthogonal discrete wavelet transform O(p, n) associated with the vector b (0) ∈ G(p, n), is
defined by formulas
a j−1,k = ∑
l⊖pk a j,l, d (1)
j−1,k = ∑
l⊖pk a j,l, ..., d (p−1)
j−1,k = ∑
l⊖pk a j,l, (3)
l∈Z +
c (0)
l∈Z +
c (1)
l∈Z +
c (p−1)
where the coefficients c (0)
k
of the low-frequency filter and the coefficients c (1)
k
, ..., c(p−1)
k
of the highfrequency
filters are defined by the formula (2) (see [7, Sec. 5.6] and [33, Sec. 7.3.2]). The dimension of
the vector a j with components a j,l is assumed to be a multiple of the number of input channels p. The
transform (3) turns the vector a j into an approximating vector a j−1 and the detalizing vectors d (1)
j−1 ,
..., d (p−1)
j−1
whose dimensions are p times smaller than the dimension of the vector a j . The inversion
formula for this transform is as follows:
a j,l = ∑
l⊖pk a j−1,k + c (1)
l⊖pk d(1) j−1,k
+ ···+ c(p−1)
l⊖pk d(p−1) j−1,k . (4)
l∈Z +
c (0)
The discrete transform O(2, 1) associated with the vector b (0) =(1/ √ 2, 1/ √ 2) coincides with the
classical Haar transform. In this case, the formulas (3) have the form
a j−1,k = a j,2k + a j,2k+1
√
2
, d j−1,k = a j,2k − a j,2k+1
√
2
.
For arbitrary p and n, the discrete Haar transform corresponds to the values
b (0)
0
= b (0)
1
= ···= b (0)
p−1 = √ 1 , b (0)
p
k
=0,k ≥ p;
then
c (s)
k
= √ 1 exp 2πiks , 0 ≤ k, s ≤ p − 1.
p p
In particular, for p =3wehave
c (0)
0 = c (0)
1 = c (0)
2 = c (1)
0 = c (2)
0 =
√ √ √
3
3 , c(1) 1 = c (2) 3
2 =
3 ε 3, c (1)
2 = c (2) 3
1 =
3 ε2 3, (5)
1 For a complete description of the set of all unitary extensions of a matrix by its first row, see [18, Theorem 9]; for
methods of matrix expansion in constructing multi-dimensional frames, see [29].
129

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.
130

Table 1. Mean-square errors of compression of the values of the function W α,β
obtained for the zone coding method, the discrete Haar transform, and
orthogonal wavelet transform.
Function Z(3) H(3) O(3, 1) O(3, 2)
W 0.6;9 0.3774 0.2835 0.2068 0.1310
W 0.8;6 0.6660 0.5393 0.4826 0.4629
W 0.8;9 0.6572 0.4599 0.3448 0.2934
W 0.9;4 1.3786 0.9706 0.8763 0.8269
4. Biorthogonal discrete wavelet transform. To generalize the discrete transform (3) to the
biorthogonal case, we take two vectors b (0) =(b (0)
0 ,b(0) 1 ,...,b(0) N−1 )and˜b (0) =(˜b (0) (0) (0)
0 ,˜b 1 ,...,˜b
N−1 )such
that
b (0) ˜b(0)
l l + b (0)
l+N 1˜b(0)
l+N 1
+ ···+ b (0)
l+(p−1)N 1˜b(0)
l+(p−1)N 1
=1, 0 ≤ l ≤ N 1 − 1,
and the unitarity of the matrix M l is replaced by the condition M l ˜Ml = I, whereI is the identity
matrix, and ˜M l is defined in the same way as M l replacing the first row by the vector ˜b (0) .Afterthis,
similarly to (2), the coefficients c (s)
k
, ˜c(s) k
,0≤ k ≤ N − 1, 0 ≤ s ≤ p − 1, are defined by the biorthogonal
discrete wavelet transform
a j−1,k = ∑
c (0)
l⊖pk a j,l,
l∈Z +
d (1)
j−1,k = ∑
c (1)
l⊖pk a j,l,
l∈Z +
..., d (p−1)
j−1,k = ∑
c (p−1)
l⊖pk a j,l.
l∈Z +
(6)
The inversion formula is as follows:
a j,l = ∑
˜c (0)
l⊖pk a j−1,k + ˜c (1)
l⊖pk d(1) j−1,k
+ ···+ ˜c(p−1)
l⊖pk d(p−1) j−1,k .
l∈Z + (7)
Algorithms for constructing matrices for which the condition M l ˜Ml = I is satisfied are available
in [34, Sec. 2.6] and [22]. The corresponding wavelet biorthogonal systems on the Vilenkin group G p are
constructed in [20]. While in Fourier analysis, filters with a finite impulse response of a discrete wavelet
transforms are trigonometric polynomials, the transform filters (6) and (7) are Walsh polynomials
m s (ω) =
N−1
∑
k=0
c (s)
N−1
k w ∑
k(ω), ˜m s (ω) =
k=0
˜c (s)
k w k(ω), s =0, 1,...,p− 1.
As in the orthogonal case, the transforms (6) and (7) allow one to quickly pass from one level of
approximation of a function f ∈ L 2 (G p ) to another (approximations are performed by using two MSA
{V j } and {Ṽj} defined by polynomials m 0 (ω) and ˜m 0 (ω), respectively; see [10]). In the case p =2,
the discrete transforms (6) and (7) were used in [19] for image processing. We also note that by
the orthogonal and biorthogonal discrete wavelet transforms defined above, similarly to well-known
constructions (see, for example, [33, 42]), one can construct wavelet packages and separable filter sets.
5. Nonstationary discrete wavelet transform. For each j ∈ Z + , we choose complex numbers
b (0)
j+1,k , k =0, 1,...,pj+1 − 1suchthat
∑p−1
∣
∣b (0)
j+1,νp+l
l=0
∣ 2 =1, ν =0, 1,...,p j − 1, (8)
131

and introduce the sequence of filters
with the coefficients
p j+1 −1
j+1 (ω) := ∑
c (0)
j+1,k w k(ω), j ∈ Z + , ω ∈ R + , (9)
m (0)
k=0
p j+1
c (0)
j+1,k = ∑−1
p−j−1
l=0
b (0)
j+1,l w k((j +1)/l). (10)
Applying the inversion formula of the discrete Vilenkin—Krestenson transform (see [25, Sec. 11]), from
(9) and (10) we obtain
b (0)
j+1,k = m(0) j+1 ((j +1)/k), k =0, 1,...,pj+1 − 1, j ∈ Z + .
Next, we define the unitary matrices
⎛
b (0)
j+1,νp
b (0)
j+1,νp+1
... b (0) ⎞
j+1,νp+(p−1)
M ν,j =
b (1)
j+1,νp
b (1)
j+1,νp+1
... b (1)
j+1,νp+(p−1)
⎜
⎝.....................................
⎟
⎠ ,
b (p−1)
j+1,νp
b (p−1)
j+1,νp+1
... b (p−1)
j+1,νp+(p−1)
where ν =0, 1,...,p j − 1. The first rows in M ν,j are formed by numbers from (8), and the remaining
row are defined as in [34, Sec. 2.6] or [14, p. 24]. Now, just like in (10), we set
p j+1
c (s)
j+1,k = ∑−1
p−j−1
l=0
b (s)
j+1,l w k((j +1)/l), s =1,...,p− 1,
and define the nonstationary discrete wavelet transform:
a j,k = ∑
j+1,l⊖pk a j+1,l, d (1)
j,k = ∑
j+1,l⊖pk a j+1,l, ..., d (p−1)
j,k
= ∑
l∈Z +
c (0)
l∈Z +
c (1)
l∈Z +
c (p−1)
j+1,l⊖pk a j+1,l. (11)
Its inversion formula is as follows:
a j+1,l = ∑
c (0)
j+1,l⊖pk a j,k + c (1)
j+1,l⊖pk d(1) j,k
+ ···+ c(p−1)
j+1,l⊖pk d(p−1) j,k
. (12)
l∈Z +
Numerical experiments performed in [21] demonstrate the possibility of using discrete transforms (11)
and (12) for compressing fractal signals. In these experiments, at each approximation level, the transform
coefficients (11) and (12) were chosen so that the projection of the signal onto the corresponding
approximating space was maximum (see [38]).
Some generalizations and modifications of the discrete wavelet transforms discussed above can be
obtained by extending the sets of admissible parameter values as a result of the transition from wavelets
to frames by analogy with the constructions in [2, 6]. In this connection, we note that according to [14,
18], a rigid frame in L 2 (G p ) can be constructed along any vector b =(b 0 ,b 1 ,...,b N−1 ) satisfying the
condition
∣ ∣
b 0 =1, ∣b l 2 ∣ ∣
+ b l+N1 2 ∣ ∣
+ ···+ b l+(p−1)N1 2 ≤ 1, 0 ≤ l ≤ N1 − 1. (13)
In addition, instead of above-mentioned constructions of wavelets and frames on the group G p ,
one can use their generalizations for the Vilenkin group G P associated with the sequence P =
132

{p 1 ,p 2 ,...,p j ,...}, p j ∈ N, p j ≥ 2. For example, for frame constructions similarly to (13) for each
j ∈ Z + , the complex numbers b (0)
j+1,k , k =0, 1,...,n j+1 − 1aretakensuchthat
where n 0 =1,n j =
with the coefficients
b (0)
j+1,0 =1,
p j+1 −1
∑
l=0
∣ (0) b
j+1,νp j+1 +l
∣ 2 ≤ 1, ν =0, 1,...,n j − 1,
j∏
p l . Then the low-frequency filters are determined by the formula
l=1
n j+1 −1
j+1 (ω) = ∑
c (0)
j+1,k W k(ω)
m (0)
k=0
n j+1 −1
c (0)
j+1,k = ∑
n−1 j+1
b (0)
j+1,l W k((j +1)/l),
l=0
where W k (ω) are generalized Walsh functions associated with the sequence P. High-frequency filters
are defined in a similar way (see Sec. 2 for the case where p j = p for all j). Note that the algorithm
for constructing nonstationary orthogonal wavelets on the group G P is described in [16], which also
contains the corresponding generalization of the Haar system.
6. Periodic discrete wavelet transforms. As was noted in Sec. 2, the orthonormal wavelet basis
in the space of N-periodic complex sequences was constructed in [12] in an arbitrary complex vector
b =(b 0 ,b 1 ,...,b N−1 ) satisfying the condition (1). Another periodic discrete wavelet transform can be
obtained in the standard way by using the Lang wavelet periodization procedure (see [30]), similarly to
classical wavelets on the real line R (see, e.g., [33, Sec. 7.5.2]). In this section, we describe a construction
of a periodic discrete wavelet transform defined by using Dirichlet–Walsh-type kernels (see [10, 13,
17]).
For a given N = p n ,wesetx n,k = k/N, 0≤ k ≤ N − 1, and choose a vector b =(b 0 ,b 1 ,...,b N−1 )
whose components b k are nonzero. The functions Φ b N (x) andϕ n,k(x) are defined by the formulas
Consider the spaces
N−1
Φ b N (x) :=N ∑
−1
k=0
b k w k (x), ϕ n,k (x) :=Φ b N (x ⊖ x n,k), x ∈ R + .
V n := span { 1, w 1 (x), ..., w N−1 (x) } ,
W n (s) := span { w sN (x), w sN+1 (x), ..., w (s+1)N−1 (x) } , s =1,...,p− 1.
Recall that the scalar product in the space of 1-periodic functions on R + is determined by the formula
〈f,g〉 =
∫ 1
0
f(x)g(x) dx.
We see that the orthogonal direct sum of the spaces V n , W (1)
n
other words, if W n := W (1)
n
⊕···⊕W (p−1)
n
{ϕ n,k } N−1
k=0 is a basis of the space V n, and similar bases {ψ (s)
n,k }N−1 k=0
spaces W (s)
n , s =1,...,p− 1.
,coincideswithV n+1 ;in
,thenV n ⊕W n = V n+1 . It was proved in [13] that the system
were constructed for each of the
,...,W (p−1)
n
133

We choose a number α ∈ (0, 1) and consider the case where the components of the vector b are
given by the equalities
{
α, if k =0ork = N − 1,
b k =
1, otherwise.
Then for Φ a N
(x), the following equality holds:
NΦ a N (x) =D N (x) − (1 − α)(1 + w N−1 (x)),
where
D N (x) :=
N−1
∑
j=0
w j (x) =
{
N, x ∈ [0,N −1 ),
0, x ∈ [N −1 , 1).
The last equality expresses the well-known property of Dirichlet–Walsh-type kernels (see, e.g., [25,
Sec. 1.5]).
For any functions f n ∈ V n and g n ∈ W n , the following equalities are valid:
where
f n (x) =
N−1
∑
k=0
g (s)
n
p−1
∑
C n,k ϕ n,k (x), g n (x) =
N−1
(x) = ∑
In these formulas, the sequences of coefficients
C n = {C n,k },
D (s)
n
k=0
D (s)
n,k ψ(s) n,k (x).
s=0
g n
(s) (x),
= {D (s)
n,k
}, 1 ≤ s ≤ p − 1,
are uniquely determined by f n and g n , respectively. As was shown in [13], an arbitrary function
f n+1 ∈ V n+1 can be decomposed into the direct sum of the functions f n ∈ V n and g n
(s) ∈ W n
(s) ,and
also f n+1 can be reconstructed from this decomposition by using the following discrete transforms.
1. The direct periodic discrete wavelet transform
∑p−1
N−1
∑
C n,ν = A (n)
pk+j,ν C n+1,pk+j, D (s)
where
j=0 k=0
p−1
n,ν = ∑
j=0
B (n)
pν+j,s C n+1,pν+j, 0 ≤ ν ≤ N − 1, (14)
1
⎧⎪ ⎨
A (n)
pk+j,ν = p + 1 − α
αpN , ν = k,
B (n)
⎪ ⎩ ε γ(ν)−γ(k) 1 − α
p
αpN , ν ≠ k, pk+j,s = p−1 ε −js
p .
Recall that we denote by γ(k) the sum of digits in p-ary decomposition of the number k.
2. The inverse periodic discrete wavelet transform
where
134
C n+1,pk+j =
N−1
∑
ν=0
Q (n)
pk+j,ν C ∑
n,ν +
Q (n)
⎧
⎪⎨ 1 − ε γ(ν)
pk+j,ν = p
⎪ ⎩ − ε γ(ν)
p
p−1
ε js
p D (s)
n,k
s=1
, 0 ≤ k ≤ N − 1, 0 ≤ j ≤ p − 1, (15)
(1 − α) γ n+1,pk+j
, k = ν,
N
(1 − α) γ n+1,pk+j
, k ≠ ν.
N

Note that the decomposition and reconstruction algorithms based on the formulas (14) and (15)
are much simpler (both in structure and in the amount of arithmetic operations) compared with the
corresponding algorithms for the case of trigonometric wavelets constructed in [5].
7. Applications to the analysis of financial time series. Main results on applications of wavelet
transforms to the analysis of financial time series are presented in the books [24, 26]. In a recent paper
[31], shares of 16 large Russian companies from the beginning of 2010 to the end of 2016 were
analyzed. Increments of time series of stock values have a strongly marked chaotic nature and have a
large amplitude of individual interference, against which a weak common signal can be distinguished
only on the basis of its correlation in different scalar components of a multidimensional time series.
Classical methods of analysis based on correlations between adjacent samples are poorly effective in
processing financial time series, because from the point of view of the correlation theory of random processes,
stock price increments formally have all the signs of white noise (in particular, “flat spectrum”
and “delta-shaped” autocorrelation function). Due to this, in [31], the passage from original signals to
sequences of their nonlinear properties calculated in time fragments of small length was performed. As
these properties, the entropy of wavelet coefficients over a Daubechies basis, multifractality indicators,
and the autoregressive measure of signal nonstationarity were used. Measures of synchronous behavior
of the properties of time series in a sliding time window by using the method of principal components,
moduli values of all pairwise correlation coefficients, and a multiple spectral measure of coherence are
constructed, which is a generalization of the quadratic coherence spectrum between two signals. Using
the method proposed, two time intervals of synchronization of the Russian stock market were identified:
from December 2013 to March 2014 and from October 2014 to January 2016. Computational
experiments showed that the same result is obtained if the entropy of wavelet coefficients is calculated
using discrete transforms defined in Secs. 2 and 5.
Various proximity measures based on wavelet coefficients for comparison time series were studied
in [3, 4]. In [36]; this technique was used for determining measures of proximity of financial time
series, which are the rates of various currencies against the US dollar. It was shown that not only
the discrete Daubechies transform is applicable for solving this problem, but also the discrete splash
transform O(2, 2). Measures of proximity between the time series corresponding to the Swiss franc
and 28 other time series were calculated. The exchange rate data were taken from the website Federal
Reserve Statistical Release 3 for the period from February 26, 1991, to December 31, 1998. The number
of samples for this period is 2048. With the help of the applied methodology, it is possible to identify
several currencies, the rates of which are closest to the Swiss franc, with respect to the following
proximity measures:
∣ ∣
∣ ∣
˜d
(x)
∑
D 1 (x, y) =− ln
∣
j,k
˜d (x)
j,k
(y) ˜d
j,k
∣ , D ∑
2(x, y) =1−
∣
(y)
where
j,k
and ˜d
j,k
are normalized detailing coefficients of the series x and y, respectively. Namely,
in both cases, the first six places are occupied by the exchange rates of the following countries: The
Netherlands, Germany, Austria, Belgium, France, and Denmark.
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3 See www.federalreserve.gov/releases/h10/hist/default1999.htm
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Yu. A. Farkov
Russian Presidential Academy of National Economy and Public Administration, Moscow, Russia
E-mail: farkov@list.ru, farkov-ya@ranepa.ru
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