Hilbert Transformation and Properties of Solar Cycles in Envelope−Instantaneous Frequency Variables

Abstract

When analyzing a narrowband signal, the Hilbert transform is often used, which makes it possible to describe the process through slowly changing functions: the envelope (amplitude) and, weakly dependent on time, the characteristic signal frequency—the “instantaneous” frequency. Based on the smoothness of these characteristics, one can evaluate the process and compare it at different periods. This approach was used to analyze the spectral components of a series of average monthly Wolf numbers. This description of the main and second harmonics, supplemented by the properties of the long-period component, gives a fairly complete picture of the entire series of monthly averages. The work examines the correspondence of the characteristics of reliable data, with this approach, to the accepted description in terms of the parameters of cycles (maximum of the cycle, duration of the cycle, and its growth branches) and constructs an “envelope” of the maxima of the cycles. The time dynamics of the instantaneous frequencies of the fundamental and second harmonics of the entire series are also presented, and significant differences in their behavior are noted in the intervals corresponding to the reconstructed and reliable parts.