A Two Channel System Approximation for Bandlimited Functions

Abstract

The approximation of stable linear time-invariant (LTI) systems is studied for the Paley–Wiener space $ \mathcal {PW}_{\pi }^{1}$ of bandlimited functions with absolutely integrable Fourier transform. For pointwise sampling, it is known that there exist stable LTI systems and functions such that the approximation process diverges, regardless of the oversampling factor. Recently, it was shown that the divergence can be overcome by using more general measurement functionals that are based on a complete orthonormal system. However, this approach requires the approximation process to have an increased bandwidth. In this paper, a two channel approximation process is presented that is uniformly convergent for all stable LTI systems and all functions in $ \mathcal {PW}_{\pi }^{1}$ . An advantage of the two channel approach compared with the one channel approach is the reduction of the approximation bandwidth, which can be exactly the same as the input function bandwidth.