Analysis of mean square estimator errors of basic frequency for periodically non-stationary random processes⋆

Abstract

Functional for estimation of the basic frequency for the mean and covariance functions of periodically non-stationary random processes (PNRPs), grounded on the reduced LS functional, analyzed. It is obtained that for the case of a Gaussian PNRP, the maximum points of the proposed functional are unbiased and consistent estimators of the basic frequency if the covariance function decays with time lag. Such analysis is provided using solutions of the nonlinear equations with appropriate conditions for maximum existence. The solutions are obtained using the small parameter method.

Keywords

• eol>periodically non-stationary random process
• basic frequency estimator
• quasi-optimal functional

1. Introduction

k Z

k N where is the basic frequency, P is the period, and jointly stationary random processes. be represented in the form of a Fourier series in a following way:

The moment functions of processes

determines the periodic changes in the mean and covariance , functions ( ), which can (1) are

m t   kZmke ik 0t m 0   mkc cosk 0t mks sink 0t  k N , R t ,   R k  e ik 0t R 0     R kc  cosk 0t R ks  sink 0t  k Z k N , where and . The Fourier coefficients of the series in (2) and (3) are defined by the modulating processes , and expressed as

Discovering the hidden periodicities using the model of signal as PNRP has been considered in a number of works [3–17]. The statistical properties of the estimator of the non-stationarity period were not analyzed in these investigations, but were briefly characterized in [18, 19].

As shown in [ 2 ], the biases and the variances of the LS estimators for the mean and covariance functions and the component estimators formed on the basis of their cyclic statistics quickly converge as the realization length draws. 2. Mean square functional It was proved in [20, 21] that the LS estimation of the basic frequency for the PNRP mean function can be reduced to finding the maximum point of the following squared functional: R k    R l k ,l  e il 0

l Z Fˆ1    1 Tmˆ 2  ,t dt 2T T , , (2) (3) and (4) (5) (6) (9) where

L mˆ  ,t  mˆ 0     mˆkc  cosk t mˆks  sink t  k 1 , and in (6) is the number of chosen harmonics, is the realization length.

We substitute into the functional in (5) the component estimator for the mean function [ 1, 2 ] rather than its LS estimator.

From the time averaging of (5), we have:

Fˆ1   mˆ 02  1 L mˆkc  2  mˆks  2  

2k 1 

L   mˆkc  mˆls   mˆks  mˆlc  J 0 k l  T,   k ,l 1 k l

L   mˆkc  mˆlc   mˆks  mˆls  J 0 k l  T, 

k ,l 1 , increases. They are the higher order smallness in comparison to first two terms in equation (9). So, we have:

And denote:

Qˆ1    1 L mˆkc  2  mˆks  2  2k 1 

.  mˆkc  dmˆkc   mˆks  dmˆks    L 

  0 k 1  d  d  

.

C kn     C k  1 S  kn     S k  

1 P d  2 P d  2  t  ,  t  , n     M k

M k   Dt mˆ t 2 , 1 N kn    

N k   Dt mˆ t 2 .

1 c k l   d lC kn    

  d l   0 ,

l   d lS kn       s k   d l 

 0 ,  l n     mkl   d M k   d l  0 , nk l   d lN kn    

  d l  0 , we can decompose the left-hand side of the equation (11) into a Taylor series in the neighborhood of the point . Then for a first approximation, we will have: (10) (11) (12) (13) (14) or

Let us consider the properties of the maximum point of the functional in (10). This point can be found as a solution to the following nonlinear equation:

When to introduce in (10) the normalized deterministic and fluctuation parts as follows:   1

L  c k0c k1 s k0s k1  p T k 1 ,   1 1 L c k0mk1 s k0nk1 s k1nk0 c k1mk0  

    2       p T k 1  2c k1mk1 c k0mk  c k2mk0 s k0nk2 s k2nk0  , where k 1 

L  p T    c k1 2  s k1 2 c k0c k2 s k0s k2  Gaussian PNRP is an asymptotically unbiased and consistent estimator for the basic frequency of the mean function, and to a first approximation, its bias and variance are defined by the formulae:  R kc l u  R kc l u  cosk 0u  cosl 0u  mksmls R ks l u sink 0u  sinl 0u   R ks l u sink 0u  sinl 0u   R kc l u  R kc l u  cosk 0u  cosl 0u   2mkcmls R lck u sink 0u  sinl 0u  R ks l u sink 0u  sinl 0u    R ks l u  R lss u  cosk 0u  cosl 0u du  o T 3  ,

3. Conclusion

(15) (16) It was shown that LS estimation of the basic frequency for mean and covariance functions of the PNRP can be reduced to searching for the maximum points of the simplified functional, which are defined by the sums of the powers of the harmonics of the test frequency and its multiples. The values of the maximum points of functional are close to the time-averaged powers of the time changes of related moment function. It was proved that the estimator, obtained with such functional for a Gaussian PNRP is asymptotically unbiased and consistent if the variance function tends to zero with time lag increasing. Formulae for the estimator variances were derived to a first approximation.

Declaration on Generative AI The authors have not employed any Generative AI tools.

References

[1] S.M. Kay , Modern Spectral Estimation: Theory and Application , Prentice Hall, New Jersey, 1988 .

[2] I. Javorskyj , R. Yuzefovych , O. Dzeryn , M.   Varyvoda , Analysis of Deterministic Components of Biperiodically Correlated Random Signals, 10th International Conference on Advanced Computer Information Technologies, ACIT 2020 , 65 - 68 .