ABSTRACT
This chapter begins by deriving the sampling theorem for nonperiodic band-limited functions. It aims to address several issues raised earlier concerning the sampling theorem, and shows how the theorem can help with the task of constructing and computing the Fourier transform from the discrete-time samples. The almost band-limited property allows to apply the sampling theorem when the bandwidth is properly chosen; the almost time-limited property allows us to use a nite interpolating formula when the number of terms is properly chosen. The interpolating formula is very expensive to compute, because it involves trigonometric function in every term. The chapter shows that the interplay between the Fourier series and the Fourier transform is critical in developing the sampling theorem.
