# A Note on Some New Generalized Wavelets

Authors: A. Zothansanga; Nikhil Khanna; S. K. Kaushik; Dilip Kumar
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we define new real wavelets based on the Hartley kernel and Boas transforms. These wavelets have possible application in analyzing both the symmetries of an asymmetric real signal. We give various results to obtain their higher vanishing moments. Finally, we give a sufficient condition under which Hartley-Boas-Like wavelets associated with Riesz projector forms a convolution filter with transfer function vanishing for the positive frequencies.


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## Body

## 1. Introduction

Initially, wavelets arrived as the possible resolution to the dilemma occurred due to the inherent drawbacks in Fourier transforms and short time Fourier transforms, and epitomized an embarkment from the Fourier analysis. But, actually they are a natural addendum to them. The major shortcoming of Fourier transform (FT) lies in the fact that it annunciates meticulously the information of each frequency a signal embraces, but is reticent about when these were actually emanated. This makes the FT abominably susceptible to errors. On the other hand, wavelet acts as a protraction of Fourier analysis whose working is based on the principle of turning the data or information in the form of coefficients that can be maneuvered, saved, communicated, examined, or availed to reconstruct the original information or signal. For more details on wavelets, see [1–5].The fundamental technique in both of them is similar. A signalfx in the standard Fourier analysis is contemporaneously examined by even and odd (in quadrature) functions, which is represented by(1)fx=d.c. term+cosine terms+sine terms,whereas it may be represented by(2)fx=ϕterm+ψterms,in the case of standard wavelet analysis using the multiresolution analysis (MRA), where “ϕ term” elucidate the analysis of fx with a scaling function ϕx, and the “ψ terms” exhibit those ones procured from scaled variants of a mother wavelet function ψx.The coefficients of the analyzing functions, i.e., sines and cosines in the case of Fourier analysis or wavelets in the case of wavelet analysis, help us to determine the extent to which these functions are reformed in order to reconstruct a signal. One may note that it is possible to formulate a signal by summing up wavelets of varied sizes at several different positions, in the same manner as with sines and cosines. The unique feature of wavelet analysis is that wavelets can change their frequency by means of translation and dilation. This is the reason that the wavelets become reconciled to the numerous sections of a signal, by virtue of a narrow window to explore at brief, higher frequency sections, and a wide window to view smaller frequency sections of a signal. This process is known as multiresolution analysis (MRA) [6], where one can obtain comprehensive overview of the signal by studying it at low resolution and to get an overall picture and progressively more finer details by examining it with enhanced resolution. This notion of MRA was introduced by Mallat [7] and Meyer [8]. Although, wavelets emerge highly improbable to have the radical influence upon pure mathematicians which Fourier analysis has shown, but wavelets are well matched to a broad spectrum of applications. After carefully analyzing equations (1) and (2), one can observe that the “ϕ term,” in wavelet analysis using MRA perform the same function as the “d.c. term” of the Fourier series; the wavelet coefficients, “ψ terms,” can be considered as the harmonic components of the Fourier series which are also considered as the scaled variants of the Fourier kernel.On that account, one may find it normal to identify every wavelet having even symmetry with another one possessing odd symmetry, and other way round. It would be interesting to generate the in quadrature version of a symmetric (or an anti-symmetric) wavelet by means of Boas transform, an integral transform related to the Hilbert transform, which emerged as a result of the analysis of the class of functions with Fourier transforms vanishing on a finite interval. In view of this, the decomposition of a signal owing to generalized version of MRA is based on the given formula:(3)fx=ϕterm+ψterms+orthogonal ofψterms.Thereupon, we introduce generalized wavelets which apparently have Hartley transform kernel like structure. In other words, we replace the cosine and sine kernel byψ and the Boas transform of ψ, respectively. Soares et al. [9] exercised the similar idea to define new wavelets using Hilbert transforms (HT).Paley and Wiener [10] examined functions in L2ℝ whose Fourier transform vanishes outside −α,α which brought about the Paley–Wiener class Bα2 consisting of functions band limited to −α,α. Due to which Boas [11] later studied the other class of functions, i.e., Bℝ−−α,α with members f such that f^γ=0 in −α,α. This study led to the introduction of new transform called as the Boas transform (BT). The BT of a function f∈L2ℝ is defined by(4)fBx=1πp.v.∫0∞fx+z−fx−zz2sinzdz=1πp.v.∫−∞∞fx+zz2sinzdz,for any x for which the integral exists. The BT and the HT share a close similarity which can be seen in the following relation:(5)fBx=fℌx−fℌ∗gx,where(6)gx=2π1/21−cosxx2.One may note that if the HT of functionf, i.e., fℌx lies in L1ℝ, then f^Bγ=−isgnγf^γ1−g^γ, where(7)sgnγ=γγ,ifγ≠0,0,ifγ=0,and g^γ is the FT of g⋅, given by(8)g^γ=0,ifγ>1,1−γ,ifγ≤1.In 1960, Goldberg [12] investigated BT widely and provided some substantial results and properties of BT. Later, inversion of BT was studied in [13, 14]. For complete details on Boas transforms, one may see [15]. For f∈L2ℝ, the inversion formula is given by(9)fx=−1π∫−∞∞fBx+zrzdz,where(10)rz=coszz+∫01sinzuudu.The BT of wavelets were introduced and studied in [4, 16]. As a consequence, we can now explore the link between the BT and the FT of wavelets, but the characterization of wavelets ψx for which ψ^γ=0 a.e. on −1,1 drafted by the BT of wavelets ψBx was the original impetus behind this whole idea. In [16], Khanna et al. explored the application of the continuous wavelet transforms in pursuance of executing signal filtering processes. Some recent additions to the work on BT are the notions of fractional Boas transform (FRBT) operation [17], parameter (p, q)-Boas transform of a signal in linear canonical transform domain [18], and the generalized wavelets based on Fourier kernel [19].The remainder of the paper is organized as follows. In Section2, we introduce new real wavelets based on Hartley kernel and BT. We give sufficient conditions on K-wavelets and obtain their higher vanishing moments. We also derive the generalized moment formula for the Hartley-Boas-Like wavelets by imposing sufficient condition on the wavelets. Finally, we give one possible application of these new wavelets.


## 2. Hartley-Boas-Like Wavelets

In [20], Khanna et al. introduced wavelets associated with Riesz projectors which have applications in analysis of an asymmetric real signal. It is observed that it is necessary to execute a complex wavelet analysis in order to study even and odd parts of an asymmetric real signal by means of an even wavelet and its odd counterpart (or HT). But this task can also be done using a real wavelet based on the Hartley kernel and BT.We observe thatsinxB=cosx and cosxB=−sinx. In view of this, the Hartley kernel can be expressed as casx=cosx−cosxB=sinx+sinxB. This drives us to define Hartley-Boas-Like wavelets as ψBℋx=ψx±ψBx. The motivation behind this generalization is very clear and is to endow the characterization of those wavelets whose Fourier transform vanishes a.e. on −1,1. Further, the inadequacy of wavelets to analyze both the symmetries of an asymmetric real signal can be invigorated by employing such kind of real wavelet.In the following result, we prove thatψBℋx is a wavelet under the given sufficient condition.Proposition 1.
Letψ be a wavelet such that ψ,ψ^∈L1ℝ and ψ^0=0. Then, ψBℋx is again a wavelet.Proof.
Sinceψ^∈L1ℝ, it follows that ψ is a bounded function. So, there exists a positive real number M such that ψx≤M, for all x∈ℝ. This gives(11)EψBℋ=ψBℋx22≤ψ22+ψB22+2MψB1<∞.
Also, we have(12)CψBℋ=∫ℝψ^Bℋγ2γdγ=∫ℝψ^γ∓isgnγψ^γ1−g^γ2γdγ=∫γ>11±iψ^γ2γdγ+∫γ≤11∓iγψ^γ2γdγ≤4∫ℝψ^γ2γdγ<+∞.
It is clear thatψBℋ has energy EψBℋ<+∞ and it also satisfies the admissibility condition.One of the significant role played by wavelets is the effective reconstruction of signals which requires the higher number of vanishing moments.Themth moment of a function fx is defined by Jmfx=∫ℝxmfxdx. If Jmfx=0 for m=0,1,2,…,p−1. Then, f is said to have p vanishing moments. For more details, see [20, 21]. The number of vanishing moments actually identifies what kind of signals the wavelet can analyze. A wavelet with p number of vanishing moments cannot analyze polynomial functions of degree p. With large number of vanishing moments, the wavelet becomes much smoother. This reduces the number of wavelet coefficients and therefore relatively smaller number of coefficients can store the whole signal information. This will eventually help in compression of the signal in addition to examine signals with singularities and discontinuities. Thus, it would be interesting to choose the number of vanishing moments which varies from application to application.Daubechies [22] was the one who first formulated compactly supported orthonormal wavelets with a fixed degree of smoothness. It was found that for ϕ,ψ∈Cpℝ, the low-pass filter m0 is given by m0γ=1+e−iγ/2p+1ℒγ with ℒ∈Cpℝ being 2π-periodic function. This can be observed using the following given result.Theorem 1.
(see [6]). Let ψ∈Cpℝ be a function such that ψx≤M1+x−p−1−μfor someμ>0, and that ψm∈L∞ℝ for m=1,2,…,p. If ψj,kj,k∈ℤ is an orthonormal system in L2ℝ, where ψj,kx=2j/2ψ2jx−k. Then, Jmψx=0, for all m=0,1,…,p.Further, one cannot assume the existence of compactly supported orthonormal wavelet inC∞ℝ. This can be seen in the next result which is a corollary of the previous result.Theorem 2.
Letψ∈L2ℝ∩C∞ℝ be a compactly supported function. Then, ψj,kj,k∈ℤ cannot be an orthonormal system in L2ℝ.One may recall that a functionf has fast decay with decay rate d∈ℕ, if fx≤Md/1+xd, for all x∈ℝ, where Md is a constant. For more details, see [23]. Next, we give the definition of K-wavelet of order p.Definition 1.
Letψ∈L1ℝ be a wavelet such that ψ1,ψ^∈L1ℝ. Then, ψ is said to be a K-wavelet of order p if ∫ℝxmKxdx=0, for m=0,1,…,p, where Kx=∫−111−1/γe−2πiγxψ1^−γdγ.We note that the moment formula for the Hilbert transform of functionf is given by(13)xmfxℌ=xmfℌx−1π∑i=0m−1xi∫ℝqm−1−ifqdq,m≥0.The above given formula holds true ifxmfx∈Lpℝ,1<p<∞.Next, we give the sufficient condition onK-wavelet of order p in order to obtain the higher vanishing moments of Hartley-Boas-Like wavelets.Theorem 3.
Letψ∈Cpℝ be a K-wavelet of order p such that(i)
ψ has fast decay with decay exponent d∈ℕ,(ii)
ψm∈L∞ℝ, m=1,2,…,p and xpψx∈L2ℝ.
Ifψj,kj,k∈ℤ forms an orthonormal system in L2ℝ, then ∫ℝxmψBℋxdx=0, for all m=0,1,2,…,minp,d−2.Proof.
Note that(14)∫ℝxmψBℋxdx=∫ℝxmψxdx±∫ℝxmψBxdx=∫ℝxmψxdx±∫ℝxmψℌxdx∓∫ℝxmψℌ∗gxdx=∫ℝxmψxdx±∫ℝxmψℌxdx∓∫ℝxm∫ℝT−xψℌ−tgtdtdx=∫ℝxmψxdx±∫ℝxmψℌxdx∓∫ℝxm∫ℝℱT−xψℌ−γg^γdγdx=∫ℝxmψxdx±∫ℝxmψℌxdx±i∫ℝxm∫−11sgn−γℱT−xψ−γ1−γdγdx,where ℱ⋅ denotes the Fourier transform. On account of ψ, ψ1, ψ^∈L1ℝ, we have(15)∫ℝxmψBℋxdx=∫ℝxmψxdx±∫ℝxmψℌxdx∓12π∫ℝxm∫−111−1γe−2πiγxψ1^−γdγdx=∫ℝxmψxdx±∫ℝxmψℌxdx∓12π∫ℝxmKxdx.
Further, given the fact thatψ, ψ^∈L1ℝ and xpψx∈L2ℝ, it follows that xmψx∈L2ℝ for m=0,1,…,p. By means of equations (13), equation (15) reduces to(16)∫ℝxmψBℋxdx=∫ℜxmψxdx±∫ℝxmψxℌdx±1π∫ℝ∑r=0m−1xr∫ℝqm−1−rψqdqdx.
In view of Theorem1 and since xmψx∈L2ℝ, for m=0,1,…,p, it follows that ∫ℝxmψBℋxdx=0 for m=0,1,2,…,p.In the following result, a sufficient condition is given aiming to obtain higher vanishing moments of wavelets.Theorem 4.
(see [21]). Let ψx be such that for some p∈ℕ, xpψx,γp+1ψ^γ∈L1ℝ. If ψj,kj,k∈ℤ is an orthogonal system on ℝ, then(17)∫ℝxmψxdx=0,form=1,2,…,p.Next result is the generalization of the Theorem4. Here, we formulate another sufficient condition on K−wavelet to procure higher vanishing moments for Hartley-Boas-Like wavelets.Theorem 5.
Letψ be a K-wavelet of order p such that xpψx∈L1ℝ∩L2ℝ and γp+1ψ^γ∈L1ℝ, for some p∈ℕ. If ψj,kj,k∈ℤ is an orthogonal system on ℝ, then(18)∫ℝxmψBℋxdx=0,form=0,1,…,p.Proof.
We have(19)∫ℝxmψBℋxdx=∫ℝxmψxdx±∫ℝxmψxℌdx±1π∫ℝ∑r=0m−1xr∫ℝqm−1−rψqdqdx.
Sincexmψx∈L2ℝ for m=0,1,…,p, following the similar steps as done in the proof of Theorem 3, it follows that(20)∫ℝxmψBℋxdx,form=0,1,2,…,p.In the following result, using the relationship between the two wavelets, the generalized moment formula for the Hartley-Boas-Like wavelets is given.Theorem 6.
Letψ be a wavelet such that ψ,ψ^∈L1ℝ,ψ^0=0 and xpψx∈L1ℝ∩L2ℝ, for some p∈ℕ. Further, let ζ∈L1ℝ be a function with ζ,ζ1,ζ1^∈L1ℝ. If ψ^γ=g^γ+γζ^γ, then(21)xpψx−ζxBℋ=xpψBℋx−xpζBℋx∓1π∑r=0p−1xr∫ℝqp−1−rψq−ζqdq∓xpψx−ζxℌ∗gx.Proof.
We have(22)xpψxBℋ=xpψx±xpψxB=xpψx±xpψℌx∓1π∑r=0p−1xr∫ℝqp−1−rψqdq∓xpψxℌ∗gx=xpψBℋx±xpζℌx∓xpζBx∓1π∑r=0p−1xr∫ℝqp−1−rψqdq∓xpψxℌ∗gx=xpψBℋx±xpζℌx−xpζBℋx+xpζxBℋ∓xpζxℌ−xpζxℌ∗gx∓1π∑r=0p−1xr∫ℝqp−1−rψqdq∓xpψxℌ∗gx=xpψBℋx−xpζBℋx+xpζxBℋ∓1π∑r=0p−1xr∫ℝqp−1−rψq−ζqdq∓xpψx−ζxℌ∗gx.
This gives(23)xpψx−ζxBℋ=xpψxBℋ−xpζxBℋ=xpψBℋx−xpζBℋx∓1π∑r=0p−1xr∫ℝqp−1−rψq−ζqdq∓xpψx−ζxℌ∗gx.


## 3. Application

In this section, we give an application of Hartley-Boas-Like wavelets in the area of electrical engineering.A continual filter is an operatorτ:L2ℝ⟶L2ℝ which is defined by τfx=P−fx, where f∈L2ℝ and P−=1/2ℑ−iℌ is called as Riesz projector, where ℑ is the identity operator and ℌ is the Hilbert transform, respectively. It should be noted that τ acts as a convolution filter if for any x, we have τx=θ∗x, with θ∈L1∩L2ℝ is known as weight function and θ^γ is known as the transfer function. For more details, one may see [20].In the following result, we give a sufficient condition under whichP−ψBℋx behaves as a convolution filter with transfer function vanishing for the positive frequencies.Theorem 7.
Letψ be a wavelet such that ψ,ψ^∈L1ℝandψ^0=0. Then, P−ψBℋx forms a convolution filter with transfer function given by(24)1±i,ifγ<−1,1∓iγ,if−1≤γ<0,12,ifγ=0,0,ifγ>0,and the Fourier transform of the convolution filter vanishes for all positive frequencies.Proof.
We have(25)τψBℋx=P−ψBℋx=ψBℋx−iψBℋxℌ=ψBℋx−i1πp.v.1x∗ψBℋx=P−δ∗ψBℋx=P−δ∗ψx±P−δ∗ψℌx∓P−δ∗ψℌ∗gx=P−δ∗ψx±P−δ∗ψℌx∓P−δ∗ψℌ∗gx=P−δ∗ψx±P−δ∗ψBx=P−δBℋ∗ψx.
Thus,P−ψBℋx is a convolution filter.
The transfer function is given by(26)FP−δBℋγ=121−sgnγ1∓isgnγ1−g^γ=1±i,ifγ<−1,1∓iγ,if−1≤γ<0,12,ifγ=0,0,ifγ>0.
Further, the Fourier transform of the convolution filter is given by(27)FP−ψBℋγ=1±iψ^γ,ifγ<−1,1∓iγψ^γ,if−1≤γ<0,ψ^γ2,ifγ=0,0,ifγ>0.


## 4. Conclusion

New generalized real wavelets based on the Hartley kernel and Boas transforms are introduced which gives an alternate way to study even and odd parts of an asymmetric real signal. The relationship betweenK-wavelets and higher order vanishing moments of Hartley-Boas-Like wavelets is also studied. A sufficient condition on the wavelets is introduced in order to generate the generalized moment formula for the Hartley-Boas-Like wavelets. Towards the end, one possible application of these new wavelets in the form of convolution filter is given.



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