This work focuses on the symmetric alpha stable processes with continuous time frequently used in modeling the signal with indefinitely growing variance when the spectral measure is mixed: sum of a continuous meseare and discrete measure. The objective of this paper is to estimate the spectral density of the continuous part from discrete observations of the signal. For that, we propose a method based on a sample of the signal at a periodic instant. The Jackson polynomial kernel is used for construct a periodogram. We smooth this periodogram by two spectral windows taking into account the width of the interval where the spectral density is nonzero. This technique allows to circumvent the phenomenon of aliasing often encountered in the estimation from the discrete observations of a process with a continuous time.