A Note on Some New Generalized Wavelets
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.
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 signal in the standard Fourier analysis is contemporaneously examined by even and odd (in quadrature) functions, which is represented bywhereas it may be represented byin the case of standard wavelet analysis using the multiresolution analysis (MRA), where “ term” elucidate the analysis of with a scaling function , and the “ terms” exhibit those ones procured from scaled variants of a mother wavelet function .
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) , 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  and Meyer . 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 “ term,” in wavelet analysis using MRA perform the same function as the “d.c. term” of the Fourier series; the wavelet coefficients, “ 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:
Thereupon, we introduce generalized wavelets which apparently have Hartley transform kernel like structure. In other words, we replace the cosine and sine kernel by and the Boas transform of , respectively. Soares et al.  exercised the similar idea to define new wavelets using Hilbert transforms (HT).
Paley and Wiener  examined functions in whose Fourier transform vanishes outside which brought about the Paley–Wiener class consisting of functions band limited to . Due to which Boas  later studied the other class of functions, i.e., with members such that in . This study led to the introduction of new transform called as the Boas transform (BT). The BT of a function is defined byfor any for which the integral exists. The BT and the HT share a close similarity which can be seen in the following relation:where
One may note that if the HT of function , i.e., lies in , then , whereand is the FT of , given by
In 1960, Goldberg  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 . For , the inversion formula is given bywhere
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 for which a.e. on drafted by the BT of wavelets was the original impetus behind this whole idea. In , 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 , parameter (p, q)-Boas transform of a signal in linear canonical transform domain , and the generalized wavelets based on Fourier kernel .
The remainder of the paper is organized as follows. In Section 2, we introduce new real wavelets based on Hartley kernel and BT. We give sufficient conditions on -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 , 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 that and . In view of this, the Hartley kernel can be expressed as . This drives us to define Hartley-Boas-Like wavelets as . The motivation behind this generalization is very clear and is to endow the characterization of those wavelets whose Fourier transform vanishes a.e. on . 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 that is a wavelet under the given sufficient condition.
Proposition 1. Let be a wavelet such that and . Then, is again a wavelet.
Proof. Since , it follows that is a bounded function. So, there exists a positive real number such that , for all . This givesAlso, we haveIt is clear that has energy 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.
The th moment of a function is defined by . If for . Then, is said to have 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 number of vanishing moments cannot analyze polynomial functions of degree . 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  was the one who first formulated compactly supported orthonormal wavelets with a fixed degree of smoothness. It was found that for , the low-pass filter is given by with being -periodic function. This can be observed using the following given result.
Theorem 1. (see ). Let be a function such that , and that for . If is an orthonormal system in , where . Then, , for all .
Further, one cannot assume the existence of compactly supported orthonormal wavelet in . This can be seen in the next result which is a corollary of the previous result.
Theorem 2. Let be a compactly supported function. Then, cannot be an orthonormal system in .
One may recall that a function has fast decay with decay rate , if , for all , where is a constant. For more details, see . Next, we give the definition of -wavelet of order .
Definition 1. Let be a wavelet such that . Then, is said to be a -wavelet of order if , for , where .
We note that the moment formula for the Hilbert transform of function is given by
The above given formula holds true if .
Next, we give the sufficient condition on -wavelet of order in order to obtain the higher vanishing moments of Hartley-Boas-Like wavelets.
Theorem 3. Let be a -wavelet of order such that(i) has fast decay with decay exponent ,(ii), and .If forms an orthonormal system in , then , for all .
Proof. Note thatwhere denotes the Fourier transform. On account of , , , we haveFurther, given the fact that , and , it follows that for . By means of equations (13), equation (15) reduces toIn view of Theorem 1 and since , for , it follows that for .
In the following result, a sufficient condition is given aiming to obtain higher vanishing moments of wavelets.
Theorem 4. (see ). Let be such that for some , . If is an orthogonal system on , then
Next result is the generalization of the Theorem 4. Here, we formulate another sufficient condition on to procure higher vanishing moments for Hartley-Boas-Like wavelets.
Theorem 5. Let be a -wavelet of order such that and , for some . If is an orthogonal system on , then
Proof. We haveSince for , following the similar steps as done in the proof of Theorem 3, it follows that
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. Let be a wavelet such that and , for some . Further, let be a function with . If , then
Proof. We haveThis gives
In this section, we give an application of Hartley-Boas-Like wavelets in the area of electrical engineering.
A continual filter is an operator which is defined by , where and is called as Riesz projector, where is the identity operator and is the Hilbert transform, respectively. It should be noted that acts as a convolution filter if for any , we have , with is known as weight function and is known as the transfer function. For more details, one may see .
In the following result, we give a sufficient condition under which behaves as a convolution filter with transfer function vanishing for the positive frequencies.
Theorem 7. Let be a wavelet such that . Then, forms a convolution filter with transfer function given byand the Fourier transform of the convolution filter vanishes for all positive frequencies.
Proof. We haveThus, is a convolution filter.
The transfer function is given byFurther, the Fourier transform of the convolution filter is given by
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 between -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.
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Each author equally contributed to this paper and read and approved the final manuscript.
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