Signals, Systems, and Signal Processing

This chapter discusses many interesting properties of bandlimited signals. The subspace of bandlimited signals is introduced. It is shown that uniformly shifted versions of an appropriately chosen sinc function constitute an orthogonal basis for this subspace. It is also shown that the integral and the energy of a bandlimited signal can be obtained exactly from samples if the sampling rate is high enough. For non-bandlimited functions, such a result is only approximately true, with the approximation getting better as the sampling rate increases. A number of less obvious consequences of these results are also presented. Thus, well-known mathematical identities are derived using sampling theory. For example, the Madhava–Leibniz formula for the approximation of π can be derived like this. When samples of a bandlimited signal are contaminated with noise, the reconstructed signal is also noisy. This noise depends on the reconstruction filter, which in general is not unique. Excess bandwidth in this filter increases the noise, and this is quantitatively analyzed. An interesting connection between bandlimited signals and analytic functions (entire functions) is then presented. This has many implications, one being that bandlimited signals are infinitely smooth.