3. Derivatives of the smoothing function

According to definition (1), the smoothing function coincides with the function at points t=t₀. However, this in not the case for the derivatives, i.e.

Obviously, for the s-th derivative of a general parameter X,

Equation (31) allows to estimate accuracy of the s-th derivative :

Particularly, if t=t₀ and polynomial basic functions (6),

and thus

Much more complicated is the determination of the derivatives by the argument t₀. Most important are first and second derivatives, especially at the extrema. Let's determine the vectors h at a moment . For small , one may expand vectors into series restricting maximum order to 2:

And similarly for h_k and Hereafter the last index s=0,1,2 corresponds to a coefficient at For the coefficient C₀:

The coefficients of the power series (35) for parameter X may be written as , as all differences (t-t₀) become . However, for it is more suitable to use the expressions

For further study of the polynomial fits, we will measure times in units of , practically using dimensionless units . Introducing the sums one may easily obtain

for "unweighted" parabolic fits, and, for the weights (5) the matrices are equal to

Other matrices , , may be consequently determined by using above mentioned expressions.

To determine extremum of the smoothing function, one has to solve an equation

After determining the root t₀ for the given signal values x_k, one has to obtain an accuracy estimate of it. Assuming small variations of the moment of the extremum caused by small , one may write a linearized equation

or

where the first and second derivatives of the smoothing function are evaluated at argument t₀. Assuming again , one may obtain

As an illustration of the derived above expressions, we show in Fig. 2 (click here) the dependence of h[C_*,k] on k for 19-point "wp" approximation.

  
Figure 2: Projective 19-point vectors h[C_*,k] for polynomial "wp" fits

Statistical properties of the test functions used for the period determination by using the moments of "characteristic events" are studied earlier (Andronov 1987, 1991).