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A procedure for determining the current parameters of the oscillator is based on the calculation of the wavelet coefficients of the signal system using fast discrete wavelet transform and application of the differentiat ion formulas wavelet expansions.

Any function f ( t ) quadratically summable R = ( − ∞ , ∞ ) in the space can be expanded at a given level of resolution j = j 0 in the wavelet series [

f ( t ) = ∑ k s j 0 , k φ j 0 , k ( t ) + ∑ j ≥ j 0 ∑ k d j , k ψ j , k ( t ) , (1)

where

φ j , k ( t ) = 2 j / 2 φ ( 2 j t − k ) , ψ j , k ( t ) = 2 j / 2 ψ ( 2 j t − k ) . (2)

As introduced φ ( t ) and ψ ( t ) we will take Daubechies [

φ ( t ) = 2 ⋅ ∑ k = 0 2 M − 1 h k φ ( 2 t − k ) , ψ ( t ) = 2 ⋅ ∑ k = 0 2 M − 1 g k φ ( 2 t − k ) , (3)

where M: positive integer.

Within the framework of multiresolution analysis [

For Daubechies h k wavelets are real and

s j , k = 〈 f , φ j , k 〉 , d j , k = 〈 f , ψ j , k 〉 , (4)

where for any f 1 ( t ) and f 2 ( t ) ( L 2 ( R ) . Here its means that L 2 ( R ) space of quadratically summable functions on R

〈 f 1 , f 2 〉 = ∫ R f 1 ( t ) f 2 ( t ) d t . (5)

Using formulas fast wavelet transform

s j , n = ∑ k = 0 2 M − 1 h k ⋅ s j − 1 , 2 n + k , d j , n = ∑ k = 0 2 M − 1 g k ⋅ s j − 1 , 2 n + k . (6)

When time signal f ∈ L 2 ( R ) 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 j = 0 . Therefore, the analysis begins with the calculation

s 0 , k : = 〈 f , φ o , k 〉 = ∫ R f ( t ) φ ( t − k ) d t . (7)

These values can be calculated, for example by means of numerical integration. In the case where f the initially set is a discrete array { f ( k ) } , k ∈ Z ( Z : set of integers), just suppose

s 0 , k : = f ( k ) , k ∈ Z . (8)

Applying wavelet transform (7), we can now calculate all the factors s j , n , d j , n .

Denote the matrix elements of the operator in terms of T S S ( n ) ( j , k ; j , k ′ ) , T S D ( n ) ( j , k ; j , k ′ ) , T D S ( n ) ( j , k ; j , k ′ ) , T D D ( n ) ( j , k ; j , k ′ ) . Here

T S S ( n ) ( j , k ; j , k ′ ) = ∫ R φ j , k ( t ) ( T ( n ) φ j , k ′ ( t ) ) d t , (9)

replacement of the lower indices S → D ( D → S ) in the left side of (9) corresponds to the substitution φ → ψ ( ψ → φ ) of the integral sign on the right side. We denote further r j , l ; j , l ′ ( n ) = T S S ( n ) ( j , l ; j , l ′ ) , r k ( n ) = T S S ( n ) ( 0 , 0 ; 0 , k ) . It’s obvious that r j , l ; j , l ′ ( k ) = r j , k / k = l − l ′ ( k ) , where r j , k ( k ) = r j , 0 ; j , k ( k ) .

Elements r j , k ( n ) of the same level are related

r j , k ( n ) = 2 ∑ i = 0 L ∑ m = 0 L h i ⋅ h m ⋅ r j , 2 k − i + m ( n ) , ( L = 2 M − 1 ) , (10)

and the elements adjacent levels-ratio

r j + 1 , l ; j + 1 , l ′ ( n ) = 2 n ⋅ r j , l ; j , l ′ ( n ) . (11)

Conditions normalization coefficients r 0 , k ( n ) is determined as [

∑ k = 1 L k ⋅ r 0 , k ( n ) = n ! . (12)

If j = 0 (10) for the value of r 0 , k ( n ) (the r k ( n ) ) we obtain the system of equations

r k ( n ) = 2 ∑ i = 0 L ∑ m = 0 L h i ⋅ h m ⋅ r 2 k − i + m ( n ) , ( k = 0 , 1 , ⋯ , L − 2 ) . (13)

In the domain of the L wavelet coefficients r k ( n ) in length, they have the symmetry property

r k ( n ) = ( − 1 ) k ⋅ r − k ( n ) ( k = 1 , 2 , ⋯ ) . (14)

Given a L ′ ≥ n ( L ′ = L − 1 ) system of linear algebraic Equations (12) - (14) has a unique exact solution u = ( r 0 ( n ) , r 1 ( n ) , ⋯ , r L − 2 ( n ) ) . The matrix elements of the differentiation operator recurrently expressed in terms of matrix elements of the op-

erator of differentiation T ( n − 1 ) = d n − 1 d t n − 1 [

Solving the system of linear algebraic equations (12) - (14) n = 1 for all matrix elements r 0 , l ; 0 , l ′ ( k ) = r k ( 1 ) | k = l ′ − l can be recovered from the elements.

T S S ( 1 ) ( j , l ; j , l ′ ) = r j , l ; j , l ′ ( 1 ) ( j ≥ 1 ) using the recurrence relation (11). The remaining matrix elements are n = 1 defined as follows. By implementing the equivalent of (10) in the n = 1 expression

T S S ( 1 ) ( j , l ; j , l ′ ) = 2 ∑ i = 0 L ∑ m = 0 L h i ⋅ h m ⋅ r j , 2 l + i ; j , 2 l ′ + m (15)

replacement indexes S → D and respective substitute h → g , obtain for all j ≥ 0 presentation

T S D ( 1 ) ( j , l ; j , l ′ ) = 2 ∑ i = 0 L ∑ m = 0 L h i ⋅ g m ⋅ r j , 2 l + i ; j , 2 l ′ + m ( 1 ) , T D S ( 1 ) ( j , l ; j , l ′ ) = 2 ∑ i = 0 L ∑ m = 0 L g i ⋅ h m ⋅ r j , 2 l + i ; j , 2 l ′ + m ( 1 ) , T D D ( 1 ) ( j , l ; j , l ′ ) = 2 ∑ i = 0 L ∑ m = 0 L g i ⋅ g m ⋅ r j , 2 l + i ; j , 2 l ′ + m ( 1 ) . (16)

Let f ( t ) represented as g ( t ) = T ( n ) f ( t ) a series of wavelet coefficients ( s f j , k ; d f j , k ) and ( s g j , k ; d g j , k ) . Then

s g j , k = ∑ k ′ = 0 L ( T S S ( n ) ( j , k ; j , k ′ ) ⋅ s f j , k ′ + T S D ( n ) ( j , k ; j , k ′ ) ⋅ d f j , k ′ ) , d g j , k = ∑ k ′ = 0 L ( T D S ( n ) ( j , k ; j , k ′ ) ⋅ s f j , k ′ + T D D ( n ) ( j , k ; j , k ′ ) ⋅ d f j , k ′ ) . (17)

The matrix elements are n = 2 connected to the matrix elements n = 1 by relations with

T S S ( 2 ) ( j , k ; j , k 2 ) = ∑ k 1 = 0 L ( T S S ( 1 ) ( j , k ; j , k 1 ) ⋅ T S S ( 1 ) ( j , k ; j , k 2 ) + T S S ( 1 ) ( j , k ; j , k 1 ) ⋅ T D S ( 1 ) ( j , k ; j , k 2 ) ) , T S D ( 2 ) ( j , k ; j , k 2 ) = ∑ k ′ = 0 L ( T S S ( 1 ) ( j , k ; j , k 1 ) ⋅ T S D ( 1 ) ( j , k ; j , k 2 ) + T S D ( 1 ) ( j , k ; j , k 1 ) ⋅ T D D ( 1 ) ( j , k ; j , k 2 ) ) , T D S ( 2 ) ( j , k ; j , k 2 ) = ∑ k ′ = 0 L ( T D S ( 1 ) ( j , k ; j , k 1 ) ⋅ T S S ( 1 ) ( j , k ; j , k 2 ) + T D D ( 1 ) ( j , k ; j , k 1 ) ⋅ T D S ( 1 ) ( j , k ; j , k 2 ) ) , T D D ( 2 ) ( j , k ; j , k 2 ) = ∑ k ′ = 0 L ( T D S ( 1 ) ( j , k ; j , k 1 ) ⋅ T S D ( 1 ) ( j , k ; j , k 2 ) + T D D ( 1 ) ( j , k ; j , k 1 ) ⋅ T D D ( 1 ) ( j , k ; j , k 2 ) ) . (18)

Replacing in the equation oscillator Van der Pol x ( t ) , d x / d t and d 2 x / d t 2 expansions in the wavelet basis with the carrier. Applying the formula of differentiation (12) and taking t the points k = 0 , 1 , ⋯ , L , we get in the parameter λ system of linear algebraic equations type

A Λ = b (19)

where A = ( a 0 , a 1 , ⋯ , a L ) T - the vector of one element λ , b = ( b 0 , b 1 , ⋯ , b L ) T T: 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 Λ that will best meet all the equations of the system (19), i.e., minimize the residual (the difference between the vector A Λ and the vector), i.e. | A Λ − b | → min . Such a solution can be obtained by solving function MATHCAD system.

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 x ( t ) 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 [

Abdullayev, N.T., Dyshin, O.A. and Dadashova, D.A. (2017) Identification of the Oscillator Parameters of the Van der Pol Oscillator Using Wavelet Analysis. Open Access Library Journal, 4: e3718. https://doi.org/10.4236/oalib.1103718