Deconvolution problems and superresolution in Hilbert-transform spectroscopy based on a.c. Josephson effect

The analysis of the numerical aspects of Hilbert transform spectroscopy based on the a.c. Josephson effect is presented. The resolving power of Hilbert transform spectroscopy is determined by such factors as the linewidth of the Josephson oscillations (intrinsic or natural resolution limit) and the limitation of the measurement interval (extrinsic or technical resolution limit) like in any spectroscopic technique based on some integral transformations. The deconvolution problem in Hilbert transform spectroscopy is posed and its solution is considered using the approach of the 1st kind integral equation for the spectrum of the incident radiation constructed from the input data of the Hilbert transform spectroscopy—the ‘hilbertogram’. The program package RECOVERY based on the maximum likelihood method is used for this purpose. This method allows to attain the maximum possible resolution enhancement in output result for a given signal-to-noise ratio in the input experimental data. The samples of numerical simulations and the spectrum of frequency-modulated BWO radiation measured by means of the Josephson junction made from high-T_c superconductor are presented. It is shown also that the integral equation approach allows to recover the sought spectrum beyond the intrinsic resolution limit and to achieve the superresolution.