Sampling a Bandpass Signal
A bandpass signal with a spectrum of centered at with a bandwidth has an upper frequency . A bandpass signal sampled at a rate of can be recovered by a bandpass filter; that is, aliasing is not present.[more]
However, the uniform sampling theorem for bandpass signals states that a bandpass signal sampled at a sample rate can be recovered if the following condition is satisfied:
where . Significantly, the lower limit of the sampling rate can be less than .
The uniform sampling theorem for bandpass signals can be visualized by knowing that the spectrum of a sampled signal is a periodic version of with a period of .
In this example, and , resulting in , and the signal can be recovered by a bandpass filter for a range of sample rates of and .
In this Demonstration, the periodic version of is displayed along with the required recovery filter. Use the slider to change the sample rate. The output of the recovery filter is also shown, and a flag indicates if aliasing is present. The impact of aliasing on the spectrum at the output of the recovery filter is highlighted. This Demonstration confirms that the spectrum at the output of the recovery filter is for the range of sample rates between 3 and 3.5 and when ; aliasing is present for all other sample rates.[less]
 F. T. Ulaby and A. E. Yagel, Signals and Systems: Theory and Applications, Ann Arbor, MI: Michigan Publishing, 2018. https://ss2.eecs.umich.edu.
 R. E. Ziemer and W. H. Tranter, Principles of Communications: Systems, Modulation and Noise, Boston: Houghton Mifflin, 1976.