Based on dual framelets, we construct the sampling approximation for the whole Sobolev space H-s(R) where s > 1/2. In particular, the sampling system has adjustable shift parameters. By the B-spline sampling system, we construct the approximation of Hilbert transform of any function of Hs(R). Combining the approximation of the function Hs(R) and that of its Hilbert transform, we establish a reconstruction method for the analytic signal. Particularly, the reconstruction series converges exponentially with respect to the scale level. Moreover, the numerical singularity emerging in computation of Hilbert transform can be removed by adjusting the shift parameters. That is, the method of reconstruction of analytic signal is numerically and L-2-stable. Several numerical experiments are carried out to check the efficiency of our reconstruction method.