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Bandpass Sampling

Let's say we have a continous, real-valued, and band-limited signal with the following spectrum:

Hz
Hz

Now we sample the time-domain signal of this spectrum, with a certain sampling frequency.

Hz
This will repeat our signal's spectrum and cause ambiguities. We can no longer tell whether a certain frequency is actually present in our continuous signal or if it is from content a multiple of the sampling frequency away. The spectrum of the sampled signal will look like this:

Typically, it is required that the sampling frequency for such a signal is at least twice its maximum frequency to avoid the aliased spectral content to overlap with our original content. In our case, this would be a frequency of at least ??? Hz. You can confirm this with the graph above. For this kind of signal (bandpass signal), we would first bring it down to baseband by means of mixing. Then, we could sample it in baseband at a much lower rate and be happy.

There is an alternative to this known as bandpass-sampling, which performs both downconversion and sampling in a single step. The idea is to intentionally undersample the signal at a frequency below the nyquist rate. Because of the nature of bandpass-signals, we have some space below the minimum frequency, which we can use to store some aliased versions without detrimental effects to the spectral shape. However, the sampling frequency used for this needs to be considered carefully.

A signal with a center frequency of Hz and a bandwidth of Hz

In practice, we need to make sure that there is no other spectral content out of the band of interest to begin with. Let's look at what happens when we have some content (like noise, or simply another transmitter) around our band of interest.

TODO! working on it :)

References