Statistical estimation of the parameters of a moving source from array data
For the model \(X_ 1(t)=S(t)+e_ 1(t)\); \(X_ 2(t)=S(a_ 0+b_ 0t)+e_ 2(t)\), with known 'shifted signal', given by \(S(a+bt)\), a,b parameters and e(t) noise, the problem is to estimate the variable time delay, i.e. (a,b). One determines \((\hat a,\hat b)\) which maximizes the criterion: \[ Q_ T(a,b)=\sum_{| s| <T/2}\psi (\omega_ s)\underline d^ T_{X_ 2}(\omega_ s)\underline d^ T_{X_ 1}(- \omega_ s/b)\exp (-i\lambda a) \] where \(\underline d^ T_ s(\omega)\) is the Fourier transform of the signal S(t) at frequency \(\omega\), \(\psi\) is a weighting (empirical) function, \(\omega_ s=2\pi s/T\), \(s=0,\pm 1,... \). Under regularity conditions, it is proved that the derived estimate is consistent and asymptotically normal. Estimates of the spectra are also obtained, and applications to seismology are discussed in detail.
scientific article; zbMATH DE number 3976161