What is the Nyquist sampling theorem and how does it apply to bandpass sampling of RF signals?

The Nyquist sampling theorem states that a signal with maximum frequency f_max must be sampled at a rate f_s ≥ 2×f_max to avoid aliasing. However: for RF signals, which are bandpass (occupying a narrow bandwidth around a center frequency), the sampling rate can be much lower than 2×f_carrier through bandpass sampling (also called subsampling or undersampling): (1) Baseband sampling (standard Nyquist): a signal from DC to f_max requires f_s ≥ 2×f_max. For a 1 GHz signal: f_s ≥ 2 GHz. This is an extremely high sampling rate for the ADC. (2) Bandpass sampling: for a signal centered at f_c with bandwidth BW: the signal occupies f_c - BW/2 to f_c + BW/2. The minimum sampling rate: f_s ≥ 2×BW (not 2×f_c). This is because the bandpass signal has only BW Hz of information content, regardless of the carrier frequency. Example: a 5G NR signal at f_c = 3.5 GHz with BW = 100 MHz. Baseband Nyquist: f_s ≥ 7 GHz (very fast ADC). Bandpass Nyquist: f_s ≥ 200 MHz (much more practical). (3) Valid sampling rates: not every f_s ≥ 2×BW works. The sampling rate must be chosen so that the aliased signal falls within a Nyquist zone without overlapping with other aliases. The valid sampling rates satisfy: f_s_valid = 2×f_upper / n, where f_upper = f_c + BW/2, and n is a positive integer, subject to: f_s ≥ 2×BW. For f_c = 3.5 GHz, BW = 100 MHz: f_upper = 3.55 GHz. n = 1: f_s = 7.1 GHz (direct sampling). n = 2: f_s = 3.55 GHz. n = 3: f_s = 2.367 GHz. n = 10: f_s = 710 MHz. n = 17: f_s = 417.6 MHz. n = 35: f_s = 202.9 MHz (near the minimum 200 MHz). Each valid n places the aliased signal at a different IF frequency; the designer chooses the n that results in a convenient f_s and IF. (4) Anti-aliasing: a bandpass filter before the ADC must pass only the desired signal band (f_c - BW/2 to f_c + BW/2) and reject all other frequencies. Without this filter: noise and interferers outside the signal band alias into the same Nyquist zone, degrading the SNR. The filter must have sharp roll-off to prevent near-band noise from aliasing.