Morlet et al., introduced the wavelet transform at the beginning of the 1980s to evaluate the seismic data. There are two categories of wavelet transform; continuous-time wavelet transforms (CWTs) and the discrete wavelet transform (DWTs).
The continuous-time wavelet transform is also called the integral wavelet transforms (IWTs). It can be used in most of the data analysis applications where it concern with the affine invariant time-frequency representation (Alfred Mertins, 1999).
The most famous version is the discrete wavelet transform (DWT). Since this transform is the computationally very efficient, it has the excellent signal compaction properties for many classes of real-world signals. Therefore, discrete wavelet transform can be applied in most of the technical fields such as numerical integration, image compression, pattern recognition and de-noising (Alfred Mertins, 1999).
CONTINUOUS-TIME WAVELET TRANSFORM
The wavelet transform of a continuous-time signal x (t) is defined as
Thus, the wavelet transform is computed as the inner product of x (t) and translated and scaled versions of a single function Ïˆ (t), the so-called wavelet. If is considered to be a bandpass impulse response, then the wavelet analysis can be understood as a bandpass analysis. The center frequency and the bandwidth of the bandpass are influenced by varying the scaling parameter 'a'. The variation of 'b' simply means a translation in time, so that for a fixed 'a' the transform of equation (3.1.1) can be seen as a convolution of with the time-reversed and scaled wavelet:
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The prefactor is introduced in order to ensure that all scaled functions
Ïˆ () with a Ð„ IR have the same energy. For the wavelet transform the condition that must be met in order to ensure perfect reconstruction is
where denotes the Fourier transform of the wavelet. This condition is known as the admissibility condition for the wavelet Ïˆ (t).
Obviously, in order to satisfy the equation (3.1.3) the wavelet must satisfy
Moreover, must decrease rapidly for and for . That is, must be a bandpass impulse response. Since a bandpass impulse response looks like a small wave, the transform is named wavelet transform. The theory of wavelet transforms contracts with the general properties of the wavelets and wavelet transform. It identifies a framework within one can propose wavelets to taste and wishes.
Fig 1. Comparison of the analysis kernels of the short-time Fourier transform and the wavelet transform for high and low analysis frequencies.
THE DISCRETE WAVELET TRANSFORM
In other practical applications and particularly in the application described in this report the signal of interest is sampled. The discrete wavelets are not time-discrete, only the translation- and the scale step are distinct. Basically implementing the wavelet filter bank as a digital filter bank spontaneously seems to do the job.
But intuitively is not good sufficient. If a wavelet spectrum is added to the scaling function spectrum, a new scaling function will be obtained with a spectrum twice as wide as the first one.
The result of this addition is that the first scaling function can be stated in terms of the second, because all the information is limited in the second scaling function. This can be expressed formally in the so-called multiresolution formulation or two-scale relation:
The two-scale relation states that the scaling function at a certain scale can be expressed in terms of translated scaling functions at the next smaller scale. Do not make confused here: smaller scale represents more detail.
The first scaling function restored a set of wavelets and then the wavelets can also be described in this set in terms of translated scaling functions at the next scale. The wavelet at level j can be write specially:
which is the two-scale relation between the scaling function and the wavelet.
Since the signal Æ’(t) could be articulated in terms of dilated and explained wavelets up to a scale j-1, this leads to the result that Æ’(t) can also be expressed in terms of dilated and translated scaling functions at a scale j:
To be reliable the case of discrete scaling functions since only discrete dilations and translations are allowed.
If in this equation, a scale can be set up to j-1 (!), it is also have to insert wavelets in order to keep the same detail level. The signal Æ’(t) can be expressed as follows
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If the scaling function and the wavelets are orthonormal or a tight frame, then the coefficients and are found by taking the inner products
If and are replaced in the inner products by suitably scaled and translated versions of equations (3.2.1) and (3.2.2) and manipulate a bit, the inner product can also be written as an integration, can get at the important result:
These above two equations state that the wavelet- and scaling function coefficients on a certain scale can be established by calculating a weighted sum of the scaling function coefficients from the previous scale. Now recall from the section on the scaling function that the scaling function coefficients approached from a low-pass filter and recollect from the section on subband coding how a filter bank is iterated by frequently splitting the low-pass spectrum into a low-pass and a high-pass part. The filter bank iteration started with the signal spectrum, so if the signal spectrum can be assumed as the output of a low-pass filter at the previous (imaginary) scale, then the sampled signal as the scaling function coefficients from the previous (imaginary) level. In other words, the sampled signal Æ’ (k) is simply equal to at the largest scale but there is more. As the signal processing theory a discrete weighted sum like the ones in equations (3.2.6) and (3.2.7) is the same as a digital filter and since the coefficients can be obtained that is come from the low-pass part of the splitted signal spectrum, the weighting factors h(k) in equation (3.2.6) must form a low-pass filter. And since the coefficients come from the high-pass part of the splitted signal spectrum, the weighting factors g(k) must form a high-pass filter. This means that the equations (3.2.6) and (3.2.7) together form one stage of an iterated digital filter bank and refers to the coefficients h(k) as the scaling filter and the coefficients g(k) as the wavelet filter.
By now the wavelet transform has to represent as an iterated digital filter bank is potential and later discrete wavelet transform or DWT will be used.
Because of it is useful property of the equations (3.2.6) and (3.2.7), the subsampling property.
The effect of this is that only every other is used in the convolution, with the result that the output data rate is equal to the input data rate. Even though this is not a new idea, it has always been exploited in subband coding schemes; it is kind of pleasant to see it pop up here as piece of the deal.
The subsampling property had come up at the end of the section on the scaling function, of how to select the width of the scaling function spectrum. The filter bank of the number of samples is iterated for the next stage. It will be clear that this is where the iteration definitely has to terminate and this determines the width of the spectrum of the scaling function. Normally the iteration will stop at the point where the number of samples has become smaller than the length of the scaling filter or the wavelet filter, whichever is the longest, so the length of the longest filter determines the width of the spectrum of the scaling function.
Fig 2. Implementation of equations (3.2.6) and (3.2.7) as one stage of an iterated filter bank.
One criticism of wavelet analysis is the arbitrary choice of the wavelet function. In choosing the wavelet function, there are several factors which should be considered (Christopher Torrence and Gilbert P.Compo, 1998).
Orthogonal or nonorthogonal
In orthogonal wavelet analysis, the number of convolutions at each scale is proportional to the width of the wavelet basis at that scale. This produces a wavelet spectrum that contains discrete'blocks' of wavelet power and is useful for signal processing as it gives the most compact representation of the signal. Unfortunately for time series analysis, an aperiodic shift in the time series produces a different wavelet spectrum. Conversely, a nonorthogonal analysis is highly redundant at large scales, where the wavelet spectrum at adjacent times is highly correlated. The nonorthogonal transform is efficiently useful for time series analysis. Where smooth, continuous variations in wavelet amplitude are estimated.
Complex or real
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a complex wavelet function will return information about both amplitude and phase and is better adapted for capturing oscillatory behavior. A real wavelet function returns only a single component and can be used to isolate peaks or discontinuities.
For concreteness, the width of a wavelet function is defined here as the e-folding time of the wavelet amplitude. The resolution of a wavelet function is determined by the balance between the width in real space and the width in fourier space. A narrow (in time) function will have good time resolution but poor frequency resolution, while a broad functiuon will have poor time resolution, yet good frequency resolution.
The wavelet function should reflect the type of features present in the time series. For time series with sharp jumps or steps, one would choose a boxcar-like function such as the Harr, while for smoothly varying time series one would choose a smooth function such as a damped cosine. If one is primarily interested in wavelet power spectra, then the choice of wavelet function is not critical, and one function will give the same qualitative results as another. ]5
Several families of wavelets that have proven to be especially useful are included in this toolbox. What follows is an introduction to some wavelet families.
Other Real Wavelets
It is now recognised as the first known wavelet. The Haar is the simplest wavelet. The Harr is not the continuous wavelet so it is not differentiable. This property can, however, be an advantage for the analysis of signals with sudden transitions, such as monitoring of tool failure in machine.
The Haar wavelet
The Haar wavelet's mother wavelet function Ïˆ(t) can be described as
and its scaling function Ï†(t) can be described as
The Haar transform is the simplest of the wavelet transforms This transform cross-multiplies a function against the Haar wavelet with various shifts and stretches, like the Fourier transform cross-multiplies a function against a sine wave with two phases and many stretches.
The Haar transform is came from the Haar matrix. An example of a 4x4 Haar matrix is as follows.
The Haar transform can be use of as a sampling process in which rows of the transform matrix operate as samples of finer and finer resolution.
Ingrid Daubechies, one of the brightest stars in the world of wavelet research, created what are called compactly supported orthonormal wavelets -- therefore making discrete wavelet analysis practicable.
A name of the Daubechies family wavelets is written dbN, where N is the order, and db the "surname" of the wavelet. The db1 wavelet is the same as Haar wavelet. Here is the wavelet functions psi of the next nine members of the family:
The biorthogonal wavelets exhibit the property of linear phase. This means that biothogonal wavelets are needed in the signal and image reconstruction. By means of two wavelets, one for decomposition (on the left side) and the other for reconstruction (on the right side) in its place of the same single one, interesting properties are derived.
This wavelet function has 2N moments same as 0 and the scaling function has 2N-1 moments equal to 0. The Daubechies and Coiflets functions have length 6N-1.
The symlets are practically symmetrical wavelets proposed by Daubechies as modifications to the db family. The properties of the two wavelet families are related. Here are the wavelet functions psis.
This Morlet wavelet has no scaling function, but it is explicit.
This wavelet has no scaling function and is consequent from a function that is proportional to the second derivative function of the Gaussian probability density function.
The Meyer wavelet and scaling function are already defined in the frequency domain.
The admissibility and the regularity conditions are the most important properties of wavelets (Alfred Mertins, 1999 and C. Valens, 1999). It can be shown that square integrable functions Ïˆ(t)satisfying the admissibility condition,
Can be used to first analyze and then reconstruct a signal without loss of information. In equation (3.5.1), Î¨(Ï‰) stands for the Fourier transform of Ïˆ(t). The admissibility condition implies that the Fourier transform of Ïˆ(t) vanishes at the zero frequency, i.e.
This means that wavelets must have a band-pass like spectrum. This is a very important to build a proficient wavelet transform.
A zero at the zero frequency also means that the average value of the wavelet in the time domain must be zero,
and therefore it must be oscillatory. In other words, Ïˆ(t)must be a wave.
As can be seen from the equation (3.1.1) the wavelet transform of a one-dimensional function is two-dimensional; a two-dimensional function for wavelet transform is four-dimensional. The time-bandwidth result of the wavelet transform is the square of the input signal and for most practical applications this is not an advantageous property. As a result one imposes some further conditions on the wavelet functions in order to make the wavelet transform quickly decrease with decreasing scale s. These are the regularity conditions and they define that the wavelet function should have some smoothness and awareness in both time and frequency domains. Regularity is a rather complex concept and so it is need to make clear using the concept of vanishing moments.
If the wavelet transform of equation (3.1.1) expands into the Tylor series at t=0 until order n, it will be
Here stands for the pth derivative of f and O (n+1) means the rest of the expansion. Now, if the moments of the wavelet are defined by Mp,
then the equation rewrite the equation (3.5.4) can be alter into the finite development
From the admissibility condition of the 0th moment so that Ïˆ(t)=0. If other moments to be completed up to Mn zero as well, then the wavelet transform coefficients will decay as fast as for a smooth signal Æ’(t). This is identified in literature as the vanishing moments or approximation order. If a wavelet has N vanishing moments, after that the approximation order of the wavelet transform is N. The moments do not have to be precisely zero; a small value is often good adequate. In fact, experimental research defines that the number of vanishing moments needed to depend closely on the application.
A BANDPASS FILTER
According to the redundancy removed, there are two hurdles needed to take before transforming the wavelet into a normal form. But it is still need to carry on by trying to reduce the number of wavelets needed in the wavelet transform and save the difficulty of the problem analytical solutions for the last part.
Even with discrete wavelets there still have an infinite number of scalings and translations to manipulate the wavelet transform. The simplest way to tackle this problem is basically not to use an infinite number of discrete wavelets. The translations of the wavelets are limited by the duration of the signal under examination so that a higher boundary is needed for the wavelets.
In the equation (3.5.2), the wavelet has a band-pass same as spectrum. From Fourier theory, compression in time is equivalent to stretching the spectrum and shifting it upwards:
The time compression of the wavelet by a factor of 2 will stretch the frequency spectrum of the wavelet by a factor of 2 and also shift all frequency components up by a factor of 2. By using this insight, the finite spectrum of the signal can be covered with the spectra of dilated wavelets in the same way as covering the signal in the time domain with translated wavelets. To get an excellent coverage of the signal spectrum the stretched wavelet spectra should handle each other, as well as they were standing hand in hand. This is shown in the Figure. 3. This can be approved by correctly designing the wavelets.
Fig 3. Touching wavelet spectra resulting from scaling of the mother wavelet in the time domain.
Summarizing, if one wavelet can be seen as a band-pass filter, after that a series of dilated wavelets can be seen as a band-pass filter bank. If the ratio between the center regularity of a wavelet spectrum and the width of this spectrum, it can be recognized that it is the same for all wavelets. This ratio is usually referred to as the fidelity factor Q of a filter and in the case of wavelets one speaks consequently of a constant-Q filter bank.
INTERMEZZO: A CONSTRAINT
When the signal has infinite energy it will not be possible to cover its frequency spectrum and its time period with wavelets. Usually this constraint is formally stated as
and it is equivalent to stating that the L2-norm of the signal Æ’(t) should be finite.
THE SCALING FUNCTION
The scaling method was introduced by Mallat. The low-pass nature of the scaling function spectrum it is sometimes referring to as the averaging filter.
Scaling function spectrum () (cork)
Wavelet spectra (Ïˆ)
Fig 4. How an infinite set of wavelets is replaced by one scaling function.
If the scaling function can be seen as being just a signal with a low-pass spectrum, then it can be decomposed in wavelet components and expresses
Since the scaling function is selected as its spectrum neatly fitted in the space left open by the wavelets, the expression of equation (3.8.1) uses an infinite number of wavelets up to a certain scale j (shown in Fig. 4). Therefore when a signal is analyzed that is used in the combination of the scaling and wavelets functions, the scaling function deals with the spectrum or else all the wavelets are covered up to the scale j while the rests is done by the wavelets. The number of wavelets from the infinite number can be limited by using this way. By using the scaling function, the problem of infinite number of wavelets can be avoided and the lower bound can be set for the wavelets. Therefore, the scaling function can be used rather than using the wavelets. In the signal representation, any information cannot be lost because it is possible to reconstruct the original signal. But in the wavelet-analysis view, the important scale information is removed. Therefore the width of scaling function spectrum is an important parameter in the wavelet transformation design. If there has shorter the spectrum, the more wavelet coefficients will be obtained and the more scale information. The low-pass spectrum of the scaling function can state some sort of admissibility condition as
which shows that the 0th moment of the scaling function cannot vanish.
In conclusion, if any the wavelet can be seen as a band-pass filter and the scaling function as a low-pass filter, then the series of wavelets with the scaling function also can be seen as the filter bank.
If the wavelet transform has to be considered as a filter bank, then the wavelet transform can be considered as a signal through this filter bank. The wavelet and scaling function transform coefficients are the outputs of the different filter stages. The subband coding means analyzing the signal by passing it through a filter bank.
There have many ways to build the filter bank that is needed in the subband coding such as building many band-pass filter to split the spectrum. Building the filter bank in the subband has the advantage and disadvantage. As the advantage, the width of every band freely can be chosen just as if the spectrum of the signal which has to analyze is covered in the places where it might be interesting. But every filter has to be designed separately so it is the time consuming process.
The way to split the signal spectrum is that the signal spectrum is split into two parts; a low-pass and a high-pass. The high-pass part contains the smallest details and the low-pass part contains some details so it can be split again and again until the numbers of bands have been created. In this way an iterated filter bank has been created. The number of bands is limited by the amount of data or computation power available. The process of splitting the spectrum is shown in the Figure. 5. This process also has disadvantage and advantage. The advantage is that only two filters are needed to design and the disadvantage is that the signal spectrum coverage is fixed. In the beginning of splitting section, this is the same as applying a wavelet transform to the signal. If the wavelet transform is implemented as an iterated filter bank, the wavelets are not needed to specify explicitly.
Fig 5. Spliting the signal spectrum with an iterated filter bank.