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

Abstracts

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.

vanishing moments; additional constraints; transient detection

Este artigo apresenta uma melhoria para a formulação de Sherlock e Monro para a parametrização wavelet para a obtenção das restrições que garantem três momentos nulos. A fim de testar a formulação apresentada, um exemplo de detecção de transitórios em sinais é apresentada.

Momentos nulos; restrições adicionais; detecção de transitórios

1 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.

2 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

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

which ensures at least one vanishing moment.

According to ⁽⁷⁾, to ensure two vanishing moments it is necessary that . This provides, ⁽⁹⁾,

Equation (2.4) has a real solution if the angles α_i satisfy the following condition

3 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

Lemma 3.1.Considering (2.2), the following equalities are true

Proof. Proof by induction:

In the case of N = 1, in (3.3) and (3.4), respectively, there are obtained the following values

Furthermore, it is necessary to demonstrate for N>1 that if

then

a) Demonstrate that the validity of (3.5) and (3.6) implies the validity of (3.7):

b) Demonstrate that the validity of (3.5) and (3.6) implies the validity of (3.8):

□

Lemma 3.2.The equations in (2.2) imply

Proof. For N = 1 the verification of the validity of (3.9) and (3.10) is immediate.

For N > 1 show that if

then

and if

then

Demonstrate that the validity of (3.11) implies the validity of (3.12):

From Lemma 3.1 it follows that

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

where

From equation (2.3), equation (3.16) has the following implications:

Leading to some properties:

Using such properties, equations (3.15-3.16) can be written as:

where

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

and following the reasoning

Equation (3.20) ensures the third vanishing moment, and it has a real solution if the angles α_i satisfy the condition

4 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, α_{p = 1} = {-17. 38º, 16. 83º, -45. 10º, 90. 65}. In order to ensure two vanishing moments complies (2.4) must be used resulting in α_{p = 2} = {-17. 38º, 16. 83º, 3. 12º, 42. 43º}. In order to ensure at least three va nis hing moments apply (3.20) which leads to α_{p = 3} = {-17. 38º, -47. 23º, 93. 44º, 16. 17º}. 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.

Figure 1:
On the left: α_{p = 1}, α_{p = 2} and α_{p = 3}. On the right: the sampling frequency of each wavelet.

Let f(t) be the signal shown in Figure 2,

Figure 2:
Signal f(t).

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 Figures 3, 4 and 5. Each figure shows the decomposition of the first and second wavelet levels, respectively.

Figure 3:
Analysis of f(t) with α_{p = 1} in the first and second level of decomposition.

Figure 4:
Analysis of f(t) with α_{p = 2} in the first and second level of decomposition.

Figure 5:
Analysis of f(t) with α_{p = 3} in the first and second level of decomposition.

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⁾.

5 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.

ACKNOWLEDGMENTS

The authors would like to thank the Brazilian agencies FAPESP (grant 2011/17610-0) and CNPq (research fellowships and PhD grant 160545/2013-7).

• ¹
  [1] A. Daamouche & F. Melgani. Swarm Intelligence Approach to Wavelet Design for Hyperspectral Image Classification, Geoscience and Remote Sensing Letters, IEEE, 6(4), pp. 825-829, Oct. 2009.
• ²
  [2] I. Daubechies. "Ten Lectures on Wavelets". Philadelphia, PA: SIAM, (1992).
• ³
  [3] M. A. Q Duarte, R. K. H. Kawakami & H. M. Paiva. Bi-objective optimization in a wavelet identification procedure for fault detection in dynamic systems, Industrial Electronics and Applications (ICIEA), 2013 8th IEEE Conference on, pp. 1319-1324, June 2013.
• ⁴
  [4] T. Froese, S. Hadjiloucas, R. K. H. Galvão, V. M. Becerra & C. J. Coelho. Comparison of extrasystolic ECG signal classifiers using discrete wavelet transforms, Pattern Recognit. Lett. , 27, pp. 393-407, Apr. 2006.
• ⁵
  [5], R. K. H. Galvão G. E. José, H. A. D. Filho, M. C. U. Araujo, E. C. Silva, H. M. Paiva, T. C. B. Saldanha & E. S. O. N. Souza. Optimal wavelet filter construction using X and Y data, Chemometrics and Intelligent Laboratory Systems, 70, pp. 1-10, Jan. 2004.
• ⁶
  [6], S. Hadjiloucas N. Jannah, F. Hwang &.R. K. H. Galvão On the application of optimal wavelet filter banks for ECG signal classification, Journal of Physics: Conference Series, 490 (2014).
• ⁷
  [7] S. Mallat. "A Wavelet Tour of Signal Processing". San Diego, EUA: Academic Press, (1998).
• ⁸
  [8] H. M. Paiva&. R. K. H. Galvão Optimized orthonormal wavelet filters with improved frequency separation. Digital Signal Processing, 22(4), pp. 622-627, July 2012.
• ⁹
  [9], H. M. Paiva M. N. Martins, R. K. H. Galvao & J. Paiva. On the space of orthonormal wavelets: additional constraints to ensure two vanishing moments, Signal Processing Letters, IEEE, 16(2), pp. 101-104, Feb. 2009.
• ¹⁰
  [10] J. P. L. M. Paiva, C. A. Kelencz, H. M. Paiva, R. K. H. Galvão& M. Magini. Adaptive wavelet EMG compression based on local optimization of filter banks. Physiological Measurement, 29 (2008), 843-856.
• ¹¹
  [11] B. G. Sherlock & D. M. Monro. On the space of orthonormal wavelets. Signal Processing, IEEE Transactions on, 46(6), pp. 1716-1720, June 1998.
• ¹²
  [12] J. C. Uzinski. Vanishing moments and wavelet regularity in the fault detection of signals (in Portuguese), (M. Sc. thesis) Ilha Solteira, SP, Brazil, (2013).
• ¹³
  [13], J. C. Uzinski H. M. Paiva, F. Villarreal, M. A. Q. Duarte &.R. K. H. Galvão Additional constraints to ensure three vanishing moments for orthonormal wavelet filter banks, Congresso de matemática aplicada e computacional - Centro-Oeste, Cuiabá - MT, July 2013.
• ¹⁴
  [14] P. P. Vaidyanathan. "Multirate Systems and Filter Banks". Englewood Cliffs, NJ: Prentice-Hall, 1993.
• ¹⁵
  [15] H. Zou & A. H. Tewfik. Parametrization of compactly supported orthonormal wavelets. Signal Processing, IEEE Transactions on, 41(3), pp. 1428-1431, Mar. 1993.

Publication Dates

• Publication in this collection
  Apr 2015

History

• Received
  26 Nov 2013
• Accepted
  04 Feb 2015