1. Decomposition of a Function into a Wavelet Series
1.1. The Definition of the Scale Function and the Basis Wavelet
Any function quadratically summable in the space can be expanded at a given level of resolution in the wavelet series  
As introduced and we will take Daubechies  scaling (scale) wavelet basis function and they are defined by the equations:
where M: positive integer.
Within the framework of multiresolution analysis  functions and serve as a high-frequency and low-frequency filters, RHR, respectively . The general properties of the scaling functions and wavelet coefficients are uniquely determined. Example of calculation for Daubechies wavelet is given in  .
For Daubechies wavelets are real and
where for any and ( . Here its means that space of quadratically summable functions on
Using formulas fast wavelet transform
1.2. Dependence of Signal Scale on Its Resolution Level
When time signal analysis should choose, the thinnest scale to the subsequent signal synthesis receives it in its original form. We can assume that such a scale is associated with the level of permissions . Therefore, the analysis begins with the calculation
These values can be calculated, for example by means of numerical integration. In the case where the initially set is a discrete array , ( : set of integers), just suppose
Applying wavelet transform (7), we can now calculate all the factors .
Denote the matrix elements of the operator in terms of , , , . Here
replacement of the lower indices in the left side of (9) corresponds to the substitution of the integral sign on the right side. We denote further , . It’s obvious that , where .
Elements of the same level are related
and the elements adjacent levels-ratio
Conditions normalization coefficients is determined as 
If (10) for the value of (the ) we obtain the system of equations
In the domain of the wavelet coefficients in length, they have the symmetry property
2. Solution of a System of Linear Algebraic Equations
2.1. Operator Matrix Elements of Differentiation the Signal
Given a system of linear algebraic Equations (12) - (14) has a unique exact solution . The matrix elements of the differentiation operator recurrently expressed in terms of matrix elements of the op-
erator of differentiation  .
Solving the system of linear algebraic equations (12) - (14) for all matrix elements can be recovered from the elements.
using the recurrence relation (11). The remaining matrix elements are defined as follows. By implementing the equivalent of (10) in the expression
replacement indexes and respective substitute , obtain for all presentation
2.2. Representation of Time Functions in the Form of Series with Wavelet Coefficients
Let represented as a series of wavelet coefficients and . Then
The matrix elements are connected to the matrix elements by relations with
Replacing in the equation oscillator Van der Pol and expansions in the wavelet basis with the carrier. Applying the formula of differentiation (12) and taking the points , we get in the parameter system of linear algebraic equations type
where - the vector of one element , a sign of transposition.
Overdetermined system (19) does not have an exact solution, so instead of an exact solution is organized search of the vector that will best meet all the equations of the system (19), i.e., minimize the residual (the difference between the vector and the vector), i.e. . Such a solution can be obtained by solving function MATHCAD system.
3. Advantages of Determining the Parameters of an Oscillator by Solving a System of Equations in the Space of Wavelet Coefficients
Thus, the calculation of the original signal wavelet coefficients using fast wavelet transform and application of the formulas of differentiation of the discrete wavelet decomposition reduces the problem of determining the oscillator parameters directly to solving a system of linear algebraic equations directly in the space of wavelet coefficients, which significantly increases the speed of calculations in comparison with method of direct and inverse wavelet transforms. This approach has undoubted advantages compared with the numerical signal by differentiating the approximate mathematical formulas. In addition, if a normally distributed random process is added to the analyzed signal, the formal application of numerical differentiation formulas based on the Newton interpolation polynomials leads to large errors in the evaluation of the derivatives, as shown in  . Based wavelet approach provides at least an order of magnitude less than the magnitude of error, and this error can be reduced even further by a suitable choice of wavelet basis. The undoubted advantage of wavelet transform is the possibility of eliminating the incorrectness of the operation of numerical differentiation of noisy time series by transitioning into the space of wavelet coefficients.