A sequence of special functions in Hardy space are constructed from Cauchy kernel on unit disk. Applying projection operator of the sequence of functions leads to an analytic sampling approximation to f, any given function in. That is, f can be approximated by its analytic samples in . Under a mild condition, f is approximated exponentially by its analytic samples. By the analytic sampling approximation, a signal in $\mathcal{H}^{2}({\mathbb T})$ can be approximately decomposed into components of positive instantaneous frequency. Using circular Hilbert transform, we apply the approximation scheme in $\mathcal{H}^{s}({\mathbb T}^{s})$ to L() such that a signal in L() can be approximated by its analytic samples on . A numerical experiment is carried out to illustrate our results.