Shannon Interpolation Research Articles

EXPERIENCES WITH THE BRILLINGER SPECTRAL ESTIMATOR APPLIED TO SIMULATED IRREGULARLY OBSERVED PROCESSES

Abstract. Shannon interpolation is used to assign values from a readily simulated discrete time process to the times of a point process, simulated by Ogata's thinning technique. The result is a set of unequally spaced samples from a hypothetical continuous time process with spectrum equal to that of the discrete time process for frequencies |ω| ≤π/Δ and identically equal to zero for |ω| > π/Δ, where Δ is the discrete time step. The spectra are theoretically known both for the sampled process and for the sampling point process. We calculate Brillinger spectral estimates for examples of a process with autoregressive spectrum, sampled at the times of a Hawkes Self Exciting Point Process. The success of the Brillinger estimator is demonstrated but it is shown to have an inherently high variance. An approximate confidence interval is discussed.

Upgrading a slow-scan interferometer for Fourier spectroscopic measurements up to its nyquist frequency limit

No readily available general theory of unequal sampling exists as yet to recover Fourier spectra from unevenly-spaced interferogram data. Application of the so-called finite-sum Shannon interpolation was found to be an appropriate software method to retain uniformly-spaced interferogram data, immediately transformable by the widely used Cooley Tukey fast Fourier algorithm. A rudimentary interpolation was chosen by the properly developed band-matrix formalism to save computation time and to find a good compromise for suitable spectral recovery. The method requires only control of sample spacings with the help of an auxiliary laser interferometer which can be easily implemented in the existing step-scan interferometer. The upgraded interferometer has been used for measurements of impurity contaminations (C, O) in Si and measurements of layered dopant structures in Si wafers.

Utilisation d'une ligne de microphotodiodes en spectrographie Fabry-Perot

A self-scanned photodiode array has been used to detect the image from a Fabry-Perot spectrograph, the recording being a cut along a diameter of the interference pattern. A microcomputer plus a 12-bit analogto- digital converter are employed to store numerically the data as a central logic system and to eliminate the fixed pattern noise of the array. From the spatial photosensibility distribution, we can restore the profile of the light distribution at the sampled points. Using the Shannon interpolation method on these ideal samples, we can thus pinpoint precisely the maxima of the fringes and linearize the spectra. Theoretical and experimental comparisons with photographic plates pointed out an improvement in the SNR for the photodiode system and open up interesting and useful applications to Fabry-Perot spectrography.