Consider a symmetric continuous time α stable process observed with an additive constant error. The objective of this paper is to give a non-parametric estimator of this error by using discrete observations. As the time of process is continuous and the observations are discrete, we encountered the aliasing phenomenon. Our process sample is taken in a way to circumvent the difficulty related to aliasing and we smoothed the periodogram by using Jackson Kernel. The rate of convergence of this estimator is studied when the spectral density is zero at origin. Few long memory processes are taken here as examples. We have applied our estimator to the concrete case of modeling noise of a bird captured under stress.