Extraction of Walsh Harmonics by Linear Combinations of Dyadic Shifts
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Journal of Mathematical Sciences, Vol. 249, No. 6, September, 2020
EXTRACTION OF WALSH HARMONICS BY LINEAR COMBINATIONS OF DYADIC SHIFTS M. S. Bespalov ∗ A. G. and N. G. Stoletov Vladimir State University 87, Gor’kogo St., Vladimir 600000, Russia [email protected]
K. M. Malkova A. G. and N. G. Stoletov Vladimir State University 87, Gor’kogo St., Vladimir 600000, Russia [email protected]
UDC 517.984.5+621.391
We solve two problems of extracting any term from a signal in the form of a finite sum of the Fourier series with respect to the discrete Walsh functions (in the first case) and the Walsh functions (in the second case) by a linear combination of group shifts of the original signal. We propose a vector version of the discrete time- and frequencythinning wavelet Haar bases, which widely used in encoding and decoding algorithms for the discrete Haar transform. Bibliography: 10 titles.
The following problem was proposed by V. I. Danchenko: Find parameters of the phaseamplitude operator m μk f (t − αk ) k=1
extracting a given term (harmonic) from the polynomial f (t) =
n
ak ϕk (t),
k=1
where {ϕk (t)} is an orthogonal system. In the case of the trigonometric system {sin(kt + αk )}, the problem is solved by using the finite moment method and the Carath´eodory theorem (cf. [1]). In the case of the Walsh system {wk (t)}, the problem was studied in [2]. In this paper, we obtain a simple formula for extracting harmonics from the Walsh polynomial. An analogous formula for the Haar polynomials does not exist (except for the first two harmonics which coincide with the Walsh harmonics). ∗
To whom the correspondence should be addressed.
Translated from Problemy Matematicheskogo Analiza 103, 2020, pp. 21-30. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2496-0838
838
1
Preliminaries
For integers k, s ∈ N ∪ {0} we introduce the operation k⊕s=
n
|kj − sj |2j−1 ,
n
k=
j=1
kj 2j−1 ,
s=
j=1
n
sj 2j−1 ,
kj , sj ∈ {0, 1},
j=1
which can be also defined for t, ω ∈ [0, 1) ∞ |tj − ωj | , t⊕ω = 2j j=1
∞ tj t= , 2j
ω=
j=1
∞ ωj j=1
2j
,
tj , ωj ∈ {0, 1}.
We introduce the mth rank intervals Δkm = [k/2m , (k + 1)/2m ). In the nondiscrete case, we automatically pass from [0, 1) to [0, 1]∗ by replacing the half-interval with the interval Δkm in the topology of [0, 1]∗ . We introduce the delta-signal δ = (0 1) and its circular shift δ → = (0 1) for the standard basis. Denote S = (1 1) and A = (1 − 1) for the Haar basis in the space R2 . The discrete 1 1 sends the standard basis to the Haar basis. Haar transform with the matrix H = 1 −1 The inverse discrete Haar transform is realized by the operator 12 H calculating half-sums and half-differences. −1 An orthogonal system {fk }N k=0 of elements in an N -dimensional Euclidean space X is a basis for X. The expansion of x in this basis has the form x=
N −1 k=0
ak fk , rk
(1.1)
where the coefficients ak = x, fk and corrections rk = fk , fk are calculated in the discrete inner product N −1 x(k)y(k), x, y ∈ X = RN . x, y =
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