In the paper we consider the partial sum process $\sum_{k=1}^{[nt]}X_k^{(n)}$, where $\{X_k^{(n)}=\sum_{j=0}^{\infty} a_{j}^{(n)}ξ_{k-j}(b(n)), \ k\in \bz\},\ n\ge 1,$ is a series of linear processes with tapered filter $a_{j}^{(n)}=a_{j}\ind{[0\le j\le ł(n)]}$ and heavy-tailed tapered innovations $ξ_{j}(b(n), \ j\in \bz$. Both tapering parameters $b(n)$ and $ł(n)$ grow to $\infty$ as $n\to \infty$. The limit behavior of the partial sum process depends on the growth of these two tapering parameters and dependence properties of a linear process with non-tapered filter $a_i, \ i\ge 0$ and non-tapered innovations. We consider the case where $b(n)$ grows relatively slow (soft tapering), and all three cases of growth of $ł(n)$ (strong, weak, and moderate tapering). In these cases the limit processes (in the sense of convergence of finite dimensional distributions) are Gaussian.