^{1}

^{*}

^{2}

^{*}

^{1}

^{*}

This study deduces a general inversion of continuous wavelet transform (CWT) with timescale being real rather than positive. In conventional CWT inversion, wavelet’s dual is assumed to be a reconstruction wavelet or a localized function. This study finds that wavelet’s dual can be a harmonic which is not local. This finding leads to new CWT inversion formulas. It also justifies the concept of normal wavelet transform which is useful in time-frequency analysis and time-frequency filtering. This study also proves a law for CWT inversion: either wavelet or its dual must integrate to zero.

Continuous wavelet transform (CWT) [

CWT inversion has found few applications. The reason lies in that wavelet’s dual is assumed to be a reconstruction wavelet in conventional CWT inversion [

There has been a general inversion for linear time-frequency transform [

Deconvolution Theorem. For a time signal

can be inverted by

if

where

and Fourier transform of kernel’s dual

and constant

The name of this theorem comes from the fact that it gives the general way to inverting frequency-indexed convolutions. It is noted that the linear time-frequency transform (1) is a set of frequency-indexed convolutions. According to the deconvolution theorem, we will give a general inversion of CWT with timescale being real. This inversion gives an explicit definition of wavelet’s dual. The inversion implies a law: either wavelet or wavelet’s dual must integrate to zero. Also according to the inversion, we find that wavelet’s dual can be a harmonic besides a wavelet. Thus new CWT inversion formulas are obtained. The new formulas suggest the concept of normal wavelet transform, which is useful in time-frequency analysis and time-frequency filtering.

For a time signal

where ^{1}-norm rather than L^{2}-norm. It has been proved that only L^{1}-norm CWT spectrum is unbiased in detecting frequency [

CWT (6) is composed of timescale-indexed convolutions. It can be regarded as a special time-frequency transform. Then, there is a CWT inversion corollary.

Inversion Corollary. For a time signal

if

satisfies

Proof. CWT (6) is a special time-frequency transform (1) with

and

Letting

Then

According to the deconvolution theorem, this inversion corollary is proved.

This inversion corollary gives a general way to inverting CWT. Relation (8) and (9) establish an explicit definition for wavelet’s dual. This means that a function

Observing the relation (8), one can find that it is necessary for the wavelet and its dual of CWT (6) to satisfy

To make (14) true, there must be that either

or

Thus, there is a CWT inversion law.

Inversion Law. In CWT inversion, either wavelet or its dual must integrate to zero.

This law applies to any CWT inversion and can never be violated. Such law breaks the traditional zero-integration requirement on wavelet. The zero-integration requirement on wavelet is made in the case that the dual of the wavelet is exactly the wavelet itself. Such case is very special. As shown by the inversion corollary, a CWT with its wavelet being unevenly undulant is still possible to be inverted.

Making a CWT inversion is equivalent to finding a dual of wavelet. In tradition, wavelet’s dual is assumed to be a reconstruction wavelet or a localized function [

where

This means, for a time signal

Particularly, if

Inversion (20) has been found by Liu and Hsu 2012. It plays a main role in the concept of normal wavelet transform [

In inversion (19), there is requirement for the wavelet is that

This requirement can be easily to meet by letting

where

For time signal

where window

1)

and

2)

where “

If applying a normal wavelet transform to a harmonic

It is easy to observe that

1)

2)

Relations (27) and (28) assure that the normal wavelet transform is accurate and useful in time-frequency analysis. At first, Relation (27) means that the normal wavelet transform can exactly (i.e. without bias) detect the immediate (i.e. local) frequency of a harmonic. Secondly, relation (28) means that the normal wavelet transform can exactly detect the immediate amplitude and phase of a harmonic. It is important to note that, relation (27) does not hold if the CWT (6) is not defined in L^{1}-norm. Different from the S-transform [

According to (20), the normal wavelet transform can be inverted simply by

because

This inversion suggests that

where S is some time-frequency area and

We here provide a numeric example of time-frequency analysis and time-frequency filtering by using the normal wavelet transform. A test time signal

The normal wavelet transform spectrum is obtained (

According to deconvolution theorem, this study explicates the way to inverting continuous wavelet transform (CWT) and the definition of wavelet’s dual. We prove that, in CWT inversion, either wavelet or its dual must integrate to zero. This study shows that wavelet’s dual can be a harmonic, which leads to new CWT inversion for-

mulas. One of the formulas justifies the concept of normal wavelet transform, which is useful in time-frequency analysis and time-frequency filtering.

We thank the Editor and the referee for their comments. This study is supported by NSFC 41074050 and by 2011YQ120045 of Ministry of Science and Technology of the People’s Republic of China.

LintaoLiu,XiaoqingSu,GuochengWang, (2015) On Inversion of Continuous Wavelet Transform. Open Journal of Statistics,05,714-720. doi: 10.4236/ojs.2015.57071