Additional Constraints to Ensure Three Vanishing Moments for Orthonormal Wavelet Filter Banks and Transient Detection

. This article presents an improvement to the formulation of Sherlock and Monro for the wavelet parameterization for the obtainment of the restrictions which ensure three vanishing moments. In order to test the formulation presented, a transient signal detection is presented.


INTRODUCTION
Sherlock and Monro [11] started the study of the angular parameterization of orthonormal filter banks, adapting the work of [14] on the factorization of paraunitary matrices and parameterizing the space of orthonormal wavelets by a set of angular parameters.
Initially the formulation had a weak point, there were no restrictions to ensure a number of vanishing moments greater than one. Additional constraints to ensure at least two vanishing moments were obtained by [9]. This article is an extension of [13] presenting an improvement to the formulation of [11]. In order to ensure a third vanishing moment for wavelet filter banks, additional constraints are presented to the work of [9] and [11]. An application of the formulation with three vanishing moments for transient detection is presented in this paper.
There are several works using the formulation proposed in [11] for several applications without the extensions to ensure more than one vanishing moment, for example, [1,4,5,10]. However, after obtaining the constraints that ensure at least two vanishing moments, in [9], other papers have presented the use of that formulation and this restriction to various applications, some examples are, [3,8,6]. The restriction achieved in this paper ensures at least three vanishing moments and was already applied to fault detection in [12], presenting promising results.
The remainder of the paper is organized as follows: in Section 2, formulas for assuring at least one and two vanishing moments are presented, in Section 3, the proposed formulation for a third vanishing moment is presented, Section 4 for presents an application to the proposed formulation, finally conclusions are presented in Section 5.

FIRST AND SECOND VANISHING MOMENTS
According to [11], let be the transfer functions of the lowpass and highpass filters, respectively, for an orthonormal filter bank with length-2N , where and h (1) If the filter bank is to characterize a wavelet transform, the regularity condition G (N) (z)| z=1 = 0 must be satisfied [2,7]. Which, according to [9], leads to According to [7], to ensure two vanishing moments it is necessary that dG (N) (z) dz z=1 = 0. This provides, [9], Equation (2.4) has a real solution if the angles α i satisfy the following condition (2.5)

THE THIRD VANISHING MOMENT
In order to obtain a third vanishing moment, [13], it is necessary that Replacing (2.1) in the second derivative of G (N) (z) when z = 1 and writing conveniently, it becomes (3.2) Lemma 3.1. Considering (2.2), the following equalities are true Furthermore, it is necessary to demonstrate for N > 1 that if a) Demonstrate that the validity of (3.5) and (3.6) implies the validity of (3.7): Demonstrate that the validity of (3.5) and (3.6) implies the validity of (3.8): Proof. For N = 1 the verification of the validity of (3.9) and (3.10) is immediate. 12) and if (3.14) Demonstrate that the validity of (3.11) implies the validity of (3.12): Demonstrate that the validity of (3.13) implies the validity of (3.14):

From Lemma 3.1 it follows that
From lemmas 3.1 and 3.2 it follows that (3.2) can be written as Leading to some properties: Using such properties, equations (3.15-3.16) can be written as: Applying (3.1) in (3.17) Then (3.18) should be written in terms of α N−2 , firstly rewriting (3.18), Decomposing the second parcel of (3.19) gives (3.21)

TRANSIENT DETECTION IN A SIGNAL
Consider a orthonormal filter bank with length-8 (N = 4), which initial configuration characterizes a wavelet with at least one vanishing moment,  Figure 1 shows the functions ψ(t ): α p=1 , α p=2 and α p=3 , respectively. Each wavelet has the sampling frequency which is denoted by ω s = 2π/ T s.
Let f (t ) be the signal shown in Figure 2,  This signal has two transients (discontinuities), when t = 377s and t = 995s, and it was analyzed using α p=1 , α p=2 and α p=3 , according to   Comparing Figures 3, 4 and 5 it is noticed that in spite of the good identification of transients using α p=2 , the analysis with α p=3 also provides a good result. But for α p=1 , the detection is not quite clear. The amplitude of the detail coefficients, besides the transients that appear in the first decomposition level with α p=2 , decrease in the case of α p=3 . The presence of high frequency coefficients indicates that the transients are slightly more highlighted when the analysis is done using α p=3 .
There are other formulations to work with wavelet filter banks, for example, [15], but the formulation of Sherlock and Monro stands for the mathematical and computational simplicities. However, initially there were no constraints to ensure a number of vanishing moments greater than one. An extension of this formulation introducing restrictions to ensure two vanishing moments was done by [9]. In [13] the constraints to ensure at least three vanishing moments were presented.
Several papers on applications using this formulation before the extension for three vanishing moments have been published, some examples are, such as pattern recognition [4], linear estimation [5], and signal compression [10].

CONCLUSIONS
This paper presented the constraints that ensure three vanishing moments and also demonstrations and calculations for the obtaining. It also presents a brief application of the formulation for transient detection in signals, a comparative way between wavelets with different regularities.
In this paper, an application example was used to test the three different wavelets of Sherlock and Monro. Through this example it was noticed that those wavelets are efficient in transient detection, specially when regularity is of at least two vanishing moments. In the case that the parameterization satisfies at least three vanishing moments it was obtained a good identification of transients and better compression or the regular parts of the signal. This fact supports the idea that the more regular is the wavelet the better is the compression of the regular parts of the signal to be decomposed.