Department of Pharmaceutical Sciences (DiFarma), University of Salerno, Via Ponte Don Melillo, 84084 Fisciano (SA), Italy
Department of Mathematics and Computer Science, University of Salerno, Via Ponte Don Melillo, 84084 Fisciano (SA), Italy
Abstract
The discrete harmonic wavelet transform has been reviewed
and applied towards given functions. The absolute error of reconstruction
of the functions has been computed.
1. Introduction
The discrete
harmonic wavelet transform was
developed by Newland in 1993 [1, 2]. Similar to the ordinary
discrete wavelet transform, the classical harmonic wavelet transform can also
perform multiresolution analysis of a function. In addition, it has a fast
algorithm based on fast Fourier transform for numerical implementation. A
distinct advantage of harmonic wavelets is that they are disjoint in frequency
domain (see Figure 1) and the Fourier transform of the successive levels
decreases in propagation of their
bandwidth (1.1).
(1.1)Calculating its inverse Fourier
transform, we obtain
(1.2)where 
and 
.
This function represents a class of pulsed functions due to its compact support in the
space domain.
Figure 1: Values of the Fourier transform of
harmonic wavelets of different levels.
2. Discretisation of a Real Function
The goal of the
wavelet transform is to decompose any arbitrary given function 
into an infinite summation of wavelets at
different scales according to the expansion
(2.1)or in the alternative form
[3]
(2.2)
The first sum is a smooth approximation of 
,
where the wavelets for 
have been rolled together into scaling
functions. The second sum is an addition of the details of 
at a specific level of resolution.
For complex wavelet coefficients, we have to define
two amplitude coefficients
(2.3)and the corresponding pair of
complex coefficients for the terms of scaling function,
(2.4)If 
is real, then 
is the complex conjugate of 
,
that is, 
,
but to allow the general case, when 
is complex, we will consider 
and 
as two different amplitudes. Then the
expansion formulas (2.1) and
(2.2) become [2]
(2.5)
Our primary purpose is to compute the coefficients 
and 
of this expansion.
An important condition for the function is
that
(2.6)
Let us consider a real-valued function 
,
represented by its discrete sequence
(2.7)where 
.
Recalling the definition of the discrete Fourier transform, the corresponding
Fourier coefficients are
(2.8)Note that
(2.9)where the asterisk stands for
the complex conjugate; 
and 
are always real numbers.
Furthermore, we will consider the coefficient 
,
defined by the first formula in (2.3). Firstly, we will substitute 
in terms of its Fourier transform
(1.1)
(2.10)into the first formula of (2.3),
and we obtain the following integral
(2.11)where we have reversed the order
of integration. The second integral over 
represents the Fourier transform of 
multiplied by 
,
and (2.11) becomes
(2.12)
To derive a discrete algorithm of decomposition of the
function, we must replace the operation of integration by summation, and (2.12)
becomes
(2.13)This identity represents the
inverse discrete Fourier transform for the sequence of frequency coefficients 
.
Analogous transformation towards the computation of 
will lead us to the following [2]:
(2.14)
Computation of the amplitudes 
and 
in the reviewed algorithm involves special
approach, and 
and 
[2].
Also, it is easy to show from (2.13) that if 
,
then 
and
(2.15)
Summarizing the stated above, the sequence of
operations for computation of wavelet amplitude coefficients is as
follows:
(i)
represent the given function 
by a discrete sequence 
,
where 
;
(ii)
compute the set of frequency coefficients by
fast Fourier transform 
,
where 
;
(iii)
the inverse fast Fourier transform of the
octave blocks 
generates the amplitudes of the harmonic
wavelet expansion of the function 
.
It is important to mention that this algorithm works
for only the functions which satisfy the following conditions.
(i)
The discrete transform covers the unit internal
of 
.
(ii)
The analysed function is periodic in 
with period 
.
The algorithm
was applied to the given functions which satisfy the mentioned conditions.
3. Implementation of Newland's Algorithm towards a Given Function
Let us review functions which satisfy the stated
conditions. For example, it is 
and 
.
Following the algorithm, we discretise the interval 
into 
equally spaced nods, and obtain discrete set
of values of functions
(3.1)
The fast Fourier transform (2.8) of the obtained
discrete sequence gives us the set Fourier coefficients 
.
Recalling that 
,
and 
we can easily find these three coefficients. Another part of coefficients from 
to 
is obtained by computation of the inverse fast
s Fourier transform
(2.13) of coefficients from 
to 
.
To reconstruct the function from its wavelet
coefficients, we followed the reverse algorithm of decomposition, that is: the
fast Fourier transform of the wavelet coefficients 
represents the discrete Fourier transform of
the reconstructed function 
.
Then, taking into account the shifting property (2.9), we can find 
as inverse fast Fourier transform of 
.
The results of decomposition and reconstruction of
functions 
and 
are presented in Figures 2 and
3.
Figure 2: Arbitrary given function: (a)
,
(b)
(dashed line), and its reconstructed clone
(solid line) from wavelet coefficients for
.
Figure 3: Arbitrary given function: (a)
,
(b)
(dashed line), and its reconstructed clone
(solid line) from wavelet coefficients for
.
One can notice that the plots of the reconstructed
functions are defined within the interval from 
to 
.
The difference between the algorithm and its corresponding computer code consists
in that we put
in the code instead of 
,
and so forth . Therefore, the reconstruction of the function begins from point 
to 1, and not from 
to 
.
To show the efficiency of the algorithm, it is worth
to estimate the absolute error of the reconstructed function in the discrete
nods. It is well known that the absolute error is given by
(3.2)where 
is the value of the reconstructed point. The
dependence of absolute error of the reconstruction of the function from 
is represented in Figure 5 and for two partial
cases, when 
and 
can be found in Figure 4. As we can see, small
numbers of the level of decomposition 
give a very good approximation, when we
reconstruct the function.
Figure 4: Absolute error of the reconstruction of
for
(solid line) and
(dashed line).
Figure 5:
Absolute error of the reconstruction
of
after regression analysis.
4. Discussion of Results and Conclusion
Wavelets are
considered as a new powerful tool for time-frequency analysis of nonlinear
phenomena. In our paper, we discussed the harmonic wavelet transform and
applied its algorithm towards decomposition and reconstruction of functions
with a unit period. This algorithm might be useful for the wavelet solution of
partial differential equations, when it is reduced to a system of ordinary
differential equations [4, 5]. The algorithm of the decomposition consists
of fast Fourier transform of the given discredited
vector function, in
which approximation error is
proportional to 
and the corresponding approximation was
obtained in our simulations (see Figure 5). It means that the
increase of the length of 
leads us to a slow, but steady
increase of the approximation error. The line of
the dependence of the error from 
was obtained by implementing the method of
least squares [6].
Note that the line of the plot takes discrete values due to the fact that 
takes only integer values of 
.
The only disadvantage of harmonic wavelets is that its
decay rate is relatively low (proportional to 
), therefore, its localisation is not precise.
However, we have this disadvantage for the restricted Fourier transform of a
harmonic wavelet of a specific level.
The application of harmonic wavelets towards
particular problems is still new. The subject is developing very fast, however,
there are still many questions remain unanswered. For example, what is the best
choice of wavelet to use for a particular problem? How far does the harmonic
wavelet transform computational simplicity compensate its slow decay rate in
the 
-domain? How it can be used for the solution
of integrodifferential equations, and many others. This work is in progress.
Acknowledgment
The work of Aleksey Kudreyko has been supported by Istituto Nazionale di Alta
Matematica Francesco Severi (Rome-IT) under Scholarship U 2007/000394, 02/07/2007.
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