Perfectly Bandlimited Characteristic Functions

We have seen when discussing the sampling theorem in Section 2.2 that the conditions of the theorem are exactly met only if the signal being sampled is perfectly bandlimited. This is rarely the case, since perfect bandlimitedness implies that the signal cannot be time–limited. Such a signal can be easily defined mathematically, but measured signals are always time–limited, so the condition of the sampling theorem can be met only approximately. While the sinc function wave is theoretically bandlimited, its time truncated versions are not, so the sampling theorem can be applied only approximately. However, sampling theory proved to be very powerful despite its approximate applicability. The situation is similar with the quantizing theorems. Bandlimitedness of the CF would imply that the PDF is not amplitude–limited. Since measured signals are always amplitude–limited, the quantization theorems can be applied only approximately. Similarly to the sampling theorem, this does not prevent the quantizing theorems from being very powerful in many applications. Most distributions that are applied in practice, like the Gaussian, exponential or chi–squared are not bandlimited. This fact does not prevent the application of the quantizing theorems if the quantum step size is significantly smaller than the standard deviation. Nevertheless, it is of interest to investigate the question whether there are distributions whose characteristic functions are perfectly bandlimited, similarly to the sinc function. In the following paragraphs we will discuss some examples of distributions whose CFs are perfectly bandlimited.