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Compactly Supported One-Cyclic Wavelets Derived from Beta Distributions H

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Compactly Supported One-Cyclic Wavelets Derived from Beta Distributions H 1 Compactly Supported One-cyclic Wavelets Derived from Beta Distributions H. M. de Oliveira, G. A. A. de Araujo´ Resumo—New continuous wavelets of compact support are n t−b introduced, which are related to the beta distribution. They can 1 t − b n 1 @ P ( a ) n( ) = (−1) : (4) be built from probability distributions using ”blur”derivatives. pjaj a pjaj @tn These new wavelets have just one cycle, so they are termed unicycle wavelets. They can be viewed as a soft variety of Haar Now wavelets whose shape is fine-tuned by two parameters a and b. Close expressions for beta wavelets and scale functions as well as @nP ( t−b ) 1 t − b their spectra are derived. Their importance is due to the Central a n (n) n = (−1) n P ( ); (5) Limit Theorem applied for compactly supported signals. 1 @b a a Index Terms—One cycle-wavelets, continuous wavelet, blur so that derivative, beta distribution, central limit theory, compactly 1 Z +1 @nP ( t−b ) CWT (a; b) = f(t) · a dt: supported wavelets. p n (6) jaj −∞ @b If the order of the integral and derivative can be commuted, I. PRELIMINARIES AND BACKGROUND it follows that Avelets are strongly connected with probability dis- tributions. Recently, a new insight into wavelets was W 1 @n Z +1 t − b presented, which applies Max Born reading for the wave- CWT (a; b) = f(t) · P ( )dt: p n (7) function [1] in such a way that an information theory focus jaj @b −∞ a has been achieved [2]. Many continuous wavelets are derived Defining the LPFed signal as the ”blur”signal from a probability density (e.g. Sombrero). This approach also sets up a link among probability densities, wavelets and Z +1 1 t − b Z +1 “blur derivatives” [3]. To begin with, let P (:) be a probability fe(a; b) = f(t)· P ( )dt = f(t)·Pa;b(t)dt; 1 p a density, P 2 C , the space of complex signals f : R ! C −∞ jaj −∞ infinitely differentiable. (8) a If an interesting interpretation can be made: set a scale and dn−1P (t) take the average (smoothed) version of the original signal - lim = 0 (1) t!1 dtn−1 the blur version fe(a; b). The “blur derivative” then @n fe(a; b) (9) @bn dnP (t) (t) = (−1)n (2) th dtn is the n derivative regarding the shift b of the blur signal at the scale a. The blur derivative coincide with the wavelet is a wavelet engendered by P (:). Given a mother wavelet transform CWT (a; b) at the corresponding scale. Details arXiv:1502.02166v1 [math.CA] 7 Feb 2015 that holds the admissibility condition [4,5] then the continuous (high-frequency) are provided by the derivative of the low- wavelet transform is defined by pass (blur) version of the original signal. Z +1 1 t − b CWT (a; b) = f(t) · p ( )dt; (3) A. Revisiting Central Limit Theorems −∞ jaj a There are essentially three kinds of central limit theorems: 8a 2 − f0g b 2 R , R. for unbounded distributions, for causal distributions and for Continuous wavelets have often unbounded support, such as compactly supported distributions [9]. The random variable Morlet, Meyer, Mathieu, de Oliveira wavelets [6-8]. In the case corresponding to the sum of N independent and identically where the wavelet was generated from a probability density, distributed (i.i.d.) variables converges to: a Gaussian distribu- one has tion, a Chi-square distribution or a Beta distribution (see Table 1This work was partially supported by the Brazilian National Council I). The Gaussian pulse always has been playing a very central for Scientific and Technological Development (CNPq) under research grant role in Engineering and it is associated with Morlet’s wavelet, #306180. Authors are with Department of Electronics and Systems, Fede- which is known to be of unbounded support. This is the only ral University of Pernambuco, CTG-UFPE C.P 7800, 50.711-970, Recife, Brazil, +55-81-2126-8210 fax: +55-81-2126-8215 (e-mail: [email protected], wavelet that meets the lower bound of Gabor’s uncertainty [email protected]). inequality [10]. The Gabor concept of logon naturally leads to 2 the Gaussian waveforms as an efficient signalling in the time- 2 2 − σ ! −jm! frequency plan. Nevertheless, in the cases where a constraint P (!) ∼ e 2 ; (16) in signal duration is imposed, it can be expected that beta 2 waveforms will be the most efficient signalling in the time- 1 − (t−m) p(t) ∼ p e 2σ2 (17) frequency plan. The concept of wavelet entropy was recently 2πσ2 introduced and Morlet wavelet also revealed to be a special wavelet [2,11]. Among all wavelets of compact support, it can According to Gnedenko and Kolmogorov, if all marginal be expected that the one linked to the beta distribution could probability densities have bounded support, then the corres- also play a valuable practical and theoretical role. ponding theorem is [9]: Tabela I Theorem 2: Central Limit Theorem for distributions of DIFFERENT VERSIONS OF THE CENTRAL LIMIT THEOREM: compact support. Let fpi(t)g be distributions such that UNBOUNDED DISTRIBUTIONS, CAUSAL DISTRIBUTIONS AND Suppf(pi(t))g = (ai; bi)(8i). Let COMPACTLY SUPPORTED PROBABILITY DISTRIBUTIONS. Marginal Distribution Central Limit Distribution as N ! 1 N −(t−m)2 X 1 a = ai < +1; (18) G(tjm; σ2) = · e 2σ2 Unbounded Support sqrt2πσ2 α −t β i=1 Causal Distribution χ2(tjm; σ2) = t ·β βα+1Γ(α+1) N α β X Compact Support beta(tjα, β) = K · t · (1 − t) ; 0 < t < 1 b = bi < +1: (19) i=1 Let pi(t) be a probability density of the random variable It is assumed without loss of generality that a = 0 and ti,where i = 1; 2; 3::N i.e. pi(t) > 0 , (8t) and b = 1. The random variable defined by (11) holds as N ! 1, Z +1 ( k · tα(1 − t)β; 0 ≤ t ≤ 1 pi(t)dt = 1: (10) p(t) ∼ (20) −∞ 0; otherwise If pi(t) $ Pi(!) , then Pi(0) = 1 and (8!)jPi(!)j ≤ 1 . where Suppose that all variables are independent. The density p(t) of the random variable corresponding to the sum m(m − m2 − σ2) α = 2 ; (21) N σ X t = t (11) and i (1 − m)(α + 1) i=1 β = : (22) m is given by the iterate convolution [12] In spite of the fact that a general theory of deriving wavelets p(t) = p1(t) ∗ p2(t) ∗ p3(t) ∗ :::pN (t): (12) from probability distributions is well known, the particular jΘi(!) application discussed in this paper search for discovering a If pi(t) $ Pi(!) = jPi(!)je , i = 1; 2; 3::N and p(t) $ P (!) = jP (!)jejΘ(!) , then link between the most noteworthy compact support distribution and wavelets. N N Y X II. β-WAVELETS: NEW COMPACTLY SUPPORTED jP (!)j = jP (!)j; Θ(!) = Θ (!): (13) i i WAVELETS i=1 i=1 The beta distribution is a continuous probability distribution The mean and the variance of a given random variable t i defined over the interval 0 ≤ t ≤ 1 [13]. It is characterized are, respectively by a couple of parameters, namely α and β, according to: Z +1 mi = τ · pi(τ)dτ; (14) 1 P (t) = tα−1 · (1 − t)β−1; 1 ≤ α; β ≤ +1: (23) −∞ B(α; β) Z +1 2 2 The normalizing factor is B(α; β) = Γ(α)·Γ(β) , where Γ(·) σi = (τ − mi) · pi(τ)dτ: (15) Γ(α+β) −∞ is the generalized factorial function of Euler and B(·; ·) is the The following theorems can be proved [9]. Beta function [13]. Theorem 1: Central Limit Theorem for distributions of un- The following parameters can be computed: bounded support Supp(P ) = [0; 1] (24) If the distributions fpi(t)g are not a lattice (a Dirac comb) α 3 mean = α+β (25) and E(ti ) < 1, and mode = α−1 (26) lim σ2 = +1 α+β−2 N!1 2 αβ variance = σ = (α+β)2(α+β+1) (27) PN then t = i=1 ti holds, as N ! 1, characteristicfunction = M(α; α + β; jν); (28) 3 where M(·; ·; ·) is the Kummer confluent hypergeometric function [14], [15]. The N th moment of P (·) can be found r h −1 p β p i using Supp( ) = p α + β + 1; α + β + 1 βα α = [a; b] (37) Z 1 N B(α + N; β) moment(N) = t · pi(t)dt = 0 B(α; β) s B(α + β; N) α + β + 1 = : (29) lengthSupp( ) = T (α; β) = (α + β) : (38) B(α; N) αβ The derivative of the beta distribution can easily be found. The parameter R = bjaj = βα is referred to as ”cyclic balance”, and is defined as the ratio between the lengths of the α − 1 β − 1 causal and non-causal piece of the wavelet. It can be easily P 0(t) = − P (t): (30) t 1 − t shown that the instant of transition tzerocross from the first to A random variable transform can be made by an affine the second half cycle is given by transform in order to generate a new distribution with zero- s (α − β) α + β + 1 mean and unity variance [12], which implies a non-normalized tzerocross = : (39) 1 (α + β − 2) αβ support T = σ = T (α; β). Let a new random variable be defined by T · (t − m). This Although scale and wavelets can be found for any α; β > 1, variable has zero-mean and unity variance.
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