Abstract
Shannon wavelets are studied together with their differential properties (known as connection coefficients). It is shown that the Shannon sampling theorem can be considered in a more general approach suitable for analyzing functions ranging in multifrequency bands. This generalization coincides with the Shannon wavelet reconstruction of functions. The differential properties of Shannon wavelets are also studied through the connection coefficients. It is shown that Shannon wavelets are -functions and their any order derivatives can be analytically defined by some kind of a finite hypergeometric series. These coefficients make it possible to define the wavelet reconstruction of the derivatives of the -functions.
1. Introduction
Wavelets [1] are localized functions which are a very useful tool in many different applications: signal analysis, data compression, operator analysis, and PDE solving (see, e.g., [2] and references therein). The main feature of wavelets is their natural splitting of objects into different scale components [1, 3] according to the multiscale resolution analysis. For the functions, that is, functions with decay to infinity, wavelets give the best approximation. When the function is localized in space, that is, the bottom length of the function is within a short interval (function with a compact support), such as pulses, any other reconstruction, but wavelets, leads towards undesirable problems such as the Gibbs phenomenon when the approximation is made in the Fourier basis. In this paper, it is shown that Shannon wavelets are the most expedient basis for the analysis of impulse functions (pulses) [4]. The approximation can be simply performed and the reconstruction by Shannon wavelets range in multifrequency bands. Comparing with the Shannon sampling theorem where the frequency band is only one, the reconstruction by Shannon wavelets can be done for functions ranging in different frequency bands. Shannon sampling theorem [5] plays a fundamental role in signal analysis and, in particular, for the reconstruction of a signal from a digital sampling. Under suitable hypotheses (on a given signal function) a few sets of values (samples) and a preliminary chosen basis (made by the sinc function) enable us to completely reconstruct the continuous signal. This reconstruction is alike the reconstruction of a function as a series expansion (such as polynomial, i.e., Taylor series, or trigonometric functions, i.e., Fourier series), but for the first time the reconstruction (in the sampling theorem) makes use of the sinc function, that is a localized function with decay to zero. Together with the Shannon sampling theorem (and reconstruction), also the wavelets series become very popular, as well as the bases with compact support. It has been recognized that on the sinc functions one can settle the family of Shannon wavelets. The main properties of these wavelets will be shown and discussed. Moreover, the connection coefficients [6–9] (also called refinable integrals) will be computed by giving some finite formulas for any order derivatives (see also some preliminary results in [2, 10–12]). These coefficients enable us to define any order derivatives of the Shannon scaling and wavelet basis and it is shown that also the derivatives are orthogonal.
2. Shannon Wavelets
Sinc function or Shannon scaling function is the starting point for the definition of the Shannon wavelet family [11]. It can be shown that the Shannon wavelets coincide with the real part of the harmonic wavelets [2, 10, 13, 14], which are the band-limited complex functionswith . Harmonic wavelets form an orthonormal basis and give rise to a multiresolution analysis [1–3, 14, 15]. In the frequency domain, they are very well localized and defined on compact support intervals, but they have a very slow decay in the space variable. However, in dealing with real problems it is more expedient to make use of real basis. By focussing on the real part of the harmonic family, we can take advantage of the basic properties of harmonic wavelets together with a more direct physical interpretation of the basis.
Let us take, as scaling function , the sinc function (Figure 1) and for the dilated and translated instancesThe parameters give, respectively, a compression (dilation) of the basic function (2.2) and a translation along the -axis. The family of translated instances is an orthonormal basis for the banded frequency functions [5] (Shannon theorem). For this reason, they can be used to define the Shannon multiresolution analysis as follows. The scaling functions do not represent a basis, in a functional space, therefore we need to define a family of functions (based on scaling) which are a basis; they are called the wavelet functions and the corresponding analysis the multiresolution analysis.
Letbe the Fourier transform of the function andits inverse transform. The Fourier transform of (2.2) gives uswithAnalogously for the dilated and translated instances of scaling function it isFrom the given scaling function, it is possible to define the corresponding wavelet function [1, 15] according to the following.
Theorem 2.1. The Shannon wavelet, in the Fourier domain, is
Proof. It can be easily shown that the scaling function (2.6)
fulfills the conditionwhich characterizes the
multiresolution analysis [1] withThus the corresponding wavelet
function can be defined as [1, 15]
With we havethen analogously with we obtainso that (2.9) follows.
For the whole family of dilated-translated instances, it isThe Shannon wavelet function in the real domain can be obtained from (2.9) by the inverse Fourier transform (Figure 1) and by the space shift and compression we have the whole family of dilated and translated instances:
By summarizing (2.3) and (2.17), the Shannon wavelet theory is based on the following functions [11]:in the space domain, and collecting (2.8) and (2.15), we have in the frequency domain
The inner product is defined aswhich, according to the Parseval equality, can be expressed also aswhere the bar stands for the complex conjugate.
With respect to the inner product (2.20), we can show the following theorem [11].
Theorem 2.2. Shannon wavelets are orthonormal functions in the sense that with being the Kroenecker symbols.
Proof. which is zero for . For it isand, according to (2.7), by the change of variable For (and ), it is trivially For , it isand analogously .
Moreover, we have the following theorem [11].
Theorem 2.3. The translated instances of the Shannon scaling functions , at the level , are orthogonal in the sense that being .
Proof. It isWhen ,
we haveSince ,
there follows thatthat is,When ,
let's say ,
we havethat is,
When ,
the last expression is always different from zero, in fact (since )that is,and only if .
Therefore, in order to have the orthogonality it must
be
,
so thatand, in particular, when ,
As a consequence of this proof we have thatThe scalar product of the (Shannon) scaling functions with the corresponding wavelets is characterized by the following [11].
Theorem 2.4. The translated instances of the Shannon scaling functions , at the level , are orthogonal to the Shannon wavelets in the sense that being .
Proof. It iswhich is zero for (since, according to (2.7), the compact support
of the characteristic functions do not intersect).
On the contrary, it can be easily seen that, for ,
it isand this product, in general,
does not vanish.
3. Reconstruction of a Function by Shannon Wavelets
Let be a function such that for any value of the parameters , it isand the Paley-Wiener space, that is, the space of band-limited functions such that,For the representation with respect to the basis (2.18), it is . According to the sampling theorem (see, e.g., [5]) we have the the following.
Theorem 3.1 (Shannon). If and the series uniformly converges to , and
Proof. In order to compute the values of the coefficients,
we have to evaluate the series in correspondence of the integer:having taken into account (2.39).
The convergence follows from the hypotheses on .
In particular, the importance of the band-limited frequency can be easily seen
by applying the Fourier transform to (3.3):so thatIn other words, if the function
is band limited (i.e., with compact support in the frequency domain), it can be
completely reconstructed by a discrete Fourier series. The Fourier coefficients
are the values of the function sampled at the integers.
As a generalization of the Paley-Wiener space, and in order to generalize the Shannon theorem, we define the space of functions such that the integralsexist and are finite. According to (2.20) and (2.21), it is in the Fourier domain that Let us prove the following.
Theorem 3.2 (Shannon generalization). If and the series converges to , with and given by (3.8) and (3.9). In particular, when it is
Proof.
The representation (3.10) follows from the
orthogonality of the scaling and Shannon wavelets (Theorems
2.2,
2.3,
2.4). The
coefficients, which exist and are finite, are given by (3.8). The convergence
of the series is a consequence of the wavelet axioms.
It should be noticed thatso that for a band-limited
frequency signal, that is, for a signal whose frequency belongs to the first
band ,
this theorem reduces to the Shannon. But, more in general, one has to deal with
a signal whose frequency range in different bands, even if
practically banded, since it is .
In this case, we have some nontrivial contributions to the series coefficients
from all the bands, ranging from :In the frequency domain, (3.10)
givesthat is,There follows that the Fourier
transform is made by the composition of coefficients at different frequency
bands. When ,
for all ,
we obtain the Shannon sampling theorem as a special case.
Of course, if we limit the dilation factor , for a truncated series, we have the approximation of , given by
By rearranging the many terms of the series with respect to the different scales, for a fixed we have
where represent the component of the function at the scale (i.e., in the band ), and results from a multiscale approximation or better from the multiband reconstruction.
3.1. Examples
Let us first compute the approximate wavelet representation of the even function
The bottom length (i.e., the main part) of the function is concentrated in the interval . With a low scale we can have a good approximation (Figures 2, 4) of the function even with a small number of translation. In fact, with the absolute value of the approximation error is less than (see Figure 4). The higher number of the translation parameter improves the approximation of the function on its “tails,” in the sense that by increasing the number of translation parameters the oscillation on “tails” is reduced. We can see that with the approximation error is reduced up to . Moreover, the approximation error tends to zero with .
(a)
(b)
(c)
(d)
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The multiscale representation is given byso that at the higher scales there are the higher frequency oscillations (see Figure 2). It should be also noticed that the lower scale approximations represent the major content of the amplitude. In other words, gives a good representation of (3.18) in the origin, while , with its higher oscillations, makes a good approximation of the tails of (3.18). Therefore, if we are interested in the evolution of the peak in the origin, we can restrict ourselves to the analysis of the lower scales. If we are interested in the evolution either of the tails or the high frequency, we must take into consideration the higher scales (in our case ).
If we compare the Shannon wavelet reconstruction with the Fourier integral approach, in the Fourier method the following hold.
(1) It is impossible to have a series expansion except for the periodic functions.(2) It is impossible to focus, as it is done with the Shannon series, on the contribution of each basis to the function. In other words, the projection of on each term of the Fourier basis is not evident. There follows that it is impossible to decompose the profile with the components at different scales.(3) The integral transform performs an integral over the whole real axis for a function which is substantially zero (over ), except in the “small” interval .
As a second example, let us consider the approximate wavelet representation of the odd function
The bottom length (i.e., the main part) of the function is concentrated in the interval . Also in this case, for a localized function, with a low scale we can have a good approximation (Figures 3, 4) of the function even with a small number of translation. However, in this case, the error can be reduced (around the origin) by adding some translated instances, but it remains nearly constant far from the origin. In fact, with the absolute value of the approximation error is less than ( in the origin, Figure 4). The higher number of the translation parameter improves the approximation of the function on its “tails,” in the sense that by increasing the number of translation parameters the oscillation on “tails” is reduced and becomes constant (around ). But we can see that with the approximation error in the origin is reduced up to .
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4. Reconstruction of the Derivatives
Let and let be a differentiable function with sufficiently high. The reconstruction of a function given by (3.10) enables us to compute also its derivatives in terms of the wavelet decompositionso that, according to (3.10), the derivatives of are known when the derivativesare given.
By a direct computation, we can easily evaluate the first and second derivatives of the scaling functionrespectively. However, on this way, higher-order derivatives cannot be easily expressed. Indeed, according to (3.10), we have to compute the wavelet decomposition of the derivatives: with being the connection coefficients [6–9, 11] (or refinable integrals).
Their computation can be easily performed in the Fourier domain, thanks to equality (2.21). In fact, in the Fourier domain the -order derivatives of the (scaling) wavelet functions areand according to (2.19),
Taking into account (2.21), we can easily compute the connection coefficients in the frequency domainwith the derivatives given by (4.8).
For the explicit computation, we need some preliminary theorems (for a sketch of the proof see also [11]).
Theorem 4.1. For given , , it is where
Proof. When , (4.10) trivially follows. When , by a partial integration we get the iterative formulafrom where by the explicit computation of iterative terms and rearranging the many terms, (4.10) holds.
The following corollary follows. From Theorem 4.1, after a substitution , we have the following corollary.
Corollary 4.2. For given , , it is
In particular, taking into account that we have the following corollary.
Corollary 4.3. For given , and , it is
More in general, the following corollary holds.
Corollary 4.4. For given , and it is
As a particular case, the following corollaries hold.
Corollary 4.5. For given , and it is
Corollary 4.6. For given , it is
Thus we can show that the following theorem holds.
Theorem 4.7. The any order connection
coefficients (4.5)1of the scaling functions are
or, shortly,
Proof. From (4.9), (4.8), (4.7), (2.21), (2.19), it isthat is,The last integral, according to (4.15) (with ), gives (4.20).
Thus we have at the lower-order derivatives
Analogously for the connection coefficients (4.5)2, we have the following theorem.
Theorem 4.8. The any order connection coefficients (4.5)2 of the Shannon wavelets (2.18)2 are respectively, for , and .
Proof. From (4.9), (4.8), (4.7), (2.21), (2.19), it isfrom where, according to the definition (2.7), it isand (for )By taking into account (4.16), (4.24) is proven.
Theorem 4.9. The connection coefficients are recursively given by the matrix at the lowest scale level:
Moreover, it is
Let us now prove that the mixed connection coefficients (4.6) are zero. It is enough to show the following theorem.
Theorem 4.10. The mixed coefficients (4.6)1 of the Shannon wavelets are
Proof. From (4.9), (4.8), (4.7), (2.21), (2.19), it isfrom where, sincethe theorem is proven.
As a consequence, we have that the -order derivatives of the Shannon scaling and wavelets areIn other words, the following theorem holds.
Theorem 4.11. The derivatives of the Shannon scaling function are orthogonal to the derivatives of the Shannon wavelets
Proof. It follows directly from (4.33) and the orthogonality of the Shannon functions according to Theorem 2.4.
4.1. First- and Second-Order Connection Coefficients
For the first and second derivatives of the Shannon wavelets, we have (see [11])with (4.24) respectively.
A disadvantage in (4.33) is that derivatives are expressed as infinite sum. However, since the wavelets are mainly localized in a short range interval, a good approximation can be obtained with a very few terms of the series. The main advantage of (4.33) is that the derivatives are expressed in terms of the wavelet basis.
Analogously, we obtain for the first and second derivative of the scaling functionwith (4.20) The coefficients of derivatives are real values as can be shown by a direct computation If we consider a dyadic discretization of the -axis such thatthat is,there results
Thus (4.33) at dyadic points becomes For instance, (see the above tables) in ,
Analogously, it isfrom where, in , it is
Outside the dyadic points, the approximation is quite good even with low values of the parameters . For instance, we have (Figure 5) the approximation
5. Conclusion
In this paper, the theory of Shannon wavelets has been analyzed showing the main properties of these functions sharply localized in frequency. The reconstruction formula for the functions has been given not only for the function but also for its derivatives. The derivative of the Shannon wavelets has been computed by a finite formula (both for the scaling and for the wavelet) for any order derivative. Indeed, to achieve this task, it was enough to compute connection coefficients, that is, the wavelet coefficients of the basis derivatives. These coefficients were obtained as a finite series (for any order derivatives). In Latto's method [6, 8, 9], instead, these coefficients were obtained only (for the Daubechies wavelets) by using the inclusion axiom but in approximated form and only for the first two order derivatives. The knowledge of the derivatives of the basis enables us to approximate a function and its derivatives and it is an expedient tool for the projection of differential operators in the numerical computation of the solution of both partial and ordinary differential equations [2, 3, 10, 13].