Estimation of hidden frequencies for 2D stationary process (Q2759341)

This paper deals with the following second-order stationary random field model: NEWLINE\[NEWLINEy(m,n)=\sum_{k=1}^{p}\{C_{k}\cos(m\lambda_{k}+n \mu_{k})+D_{k}\sin(m\lambda_{k}+n \mu_{k})\}+x(m,n),\;m,n=0,\pm 1,\pm 2,\ldots,NEWLINE\]NEWLINE where \(\{C_{k}, D_{k}\}\) is a set of uncorrelated random variables and uncorrelated with \(\{x(m,n)\}\), \(Var(C_{k})=Var(D_{k})\), and \(\{x(m,n);\;(m,n)\in {\mathbb Z}^2\}\) is a stationary random field with an absolutely continuous spectral distribution. The problem is to estimate the number of the frequencies \(p\) and the unknown frequencies \((\lambda_{k},\mu_{k})\), given observations \(y(m,n),\;m,n=1,\ldots,N\).NEWLINENEWLINENEWLINEThe authors propose a method for estimating \(p\) and the hidden frequencies that is based on the properties of the periodogram. It is shown that the periodogram of the regular random field \(x(m,n)\) has a uniform upper bound of \(O(\ln(N^2))\), where \(N^2\) is sample size. The behaviour of the periodogram is studied and consistent estimators of \(p\) and \((\lambda_{k},\mu_{k})\) are constructed.