## 3 Short Remarks on Haar Wavelets  
Wavelets are mathematical functions that are based on time frequency. They analyze physical situations with the signal containing discontinuities and sharp spikes better than traditional Fourier Methods. Wavelets are very often applied in mathematics, quantum physics and electrical engineering. Intersection between these subjects in the last decade caused increasing wavelet applications such as image compression, turbulence, human vision and radar. Currently, Digital Signal Processors implement sampled signals. Only approximated values of the signal projection coefficients can be often calculated by selecting a particular signal approximation. A great number of wavelet functions that have different features exist, and, a short time ago, polynomial structures started to be applied for creating wavelets. In [13], a wavelet having a form of a cubic spline is presented in pre-filtering applications. Haar wavelets have a wide application field in industry. Representation of discrete signals using Haar wavelets is very compact due to the use of microprocessors (see [14]). A brief overview on the Haar wavelets function \(\psi_{(d,jn)}(t)=\psi_{j}(2^{d}t-n)\) is considered with a support of size \(2^{-d}\) of the Nyquist frequency. Furthermore, \(d\) is a scale parameter, \(j\) is a phase parameter and \(n\) is a time translation parameter. Here, the _pyramidal_ packet is represented by the indices \((d,j,n)\), \(d\) is the level of the tree (scaling parameter), \(j\) is the frequency cell (oscillation parameter) and \(n\) the time cell (localization parameter). Basically, the Haar basis has the following two properties:  
* the \(\psi_{(d,j,n)}^{h}(t)\) are orthonormal,
* any \(\mathcal{L}^{2}(\Re)\) function \(f(t)\) can be approximated, up to arbitrarily low precision, by a finite linear combination of the \(\psi_{(d,j,n)}^{h}(t)\). In particular, to be more precise, coefficients \(w_{(d,j,n)}\) (the weight coefficients) are calculated as follows:  
\[w_{(d,j,n)}=\int_{I}f(t)\psi_{(d,j,n)}^{h}(t)d(t).\]  
For \(j=0\), it is possible to define the following coefficients:  
\[s_{(d,0,n)}=\int_{I}f(t)\psi_{(d,0,n)}^{h}(t)d(t),\]  
where \(f(t)\) is the required signal, \(I\) is the considered time interval and \(\psi_{(d,0,n)}^{h}(t)\) is the well-known _mother function_. To conclude:  
\[f(t)=\sum_{n}s_{(d,n)}\psi_{(d,0,n)}^{h}(t)+\sum_{j}\sum_{n}w_{(d,j,n)}\psi_{( d,j,n)}^{h},\]  
where \(s_{(d,n)}=w_{(d,0,n)}\). It is shown that there are just two independent parameters and that parameter \(d\) characterizes the _level of the tree_. The wavelet packets under consideration are derived from the Haar mother wavelet. A set of Haar functions are presented in Figure 4. The localisation of the Haar functions through the tree is accomplished by means of the parameter tuple, \((d,j,n)\), as shown in Figure 4. Moreover, there is a relation of parameter \(j\) to the frequency shift, a relation of parameter \(n\) to the time shift, and the supplementary parameter \(\mathsf{d}\) is necessary to address the level of the tree directly depending on the number of the examined samples. It means that if eight samples are to be considered, the level of the tree will be three. Wavelet functions are constructed like a tree, so \(d=1\) is the highest degree of improvement concerning the time. Figure 4 demonstrates four wavelet functions at \(d=1\) and the four wavelet functions for \(d=2\). The appropriate coefficients, \(w(d,j,n)\) representing the wavelet functions, for a tree consisting of \(d=3\) is shown in the right part of Figure 5. \(w(1,0,0.3)\) indicates the coefficients of the first level on the extreme left with time shifts 0 through 3. One of the motivations to choose Haar wavelets is that they are the most structurally simple wavelets to be applied. This means that, with a window of four samples, we can already achieve very good results.