Undersampling of Modulated Signals

In the case of a higher frequency carrier onto which lower frequency information is modulated, it is necessary to translate the information content down to a frequency band low enough for the A-to-D converter (ADC) to use. Modern (as of 2006) ADCs easily operate at sample rates beyond 100 MHz while delivering 12 to 14 bits or more of resolution. That is a phenomenal achievement considering that a mere 15 years ago radar design teams I worked on were struggling to do 10-12 bits at 20 MHz using custom-designed circuits. If you had an ADC running at 216 MHz, it would be possible to directly sample the FM radio band signal (88 - 108 MHz) and demodulate the information in software. We're not quite there yet, so it is still necessary to use and RF mixer to downconvert the RF to a baseband frequency that can be used. AM radio (520 – 1,610 kHz) could, though, be directly sampled at RF.

There is another option called undersampling (sometimes called bandpass sampling, or super-Nyquist sampling) whereby the aliasing phenomenon is exploited to enable the ADC to sample the signal using a rate that intentionally aliases the modulated carrier (f_CARRIER) into the operating range of the ADC. Determining the necessary sampling frequency (f_SAMPLE), is a fairly simple 2-step process that uses the following two formulas:

(eq. 1)                (eq. 2)

As an example, consider a 20 MHz signal modulated on a 2 GHz carrier. Per the Nyquist sampling theorem, a clock rate of at least 4.04 GHz  (2 x 2.02 MHz) would be needed to directly sample this signal. Now, working through the above equations, we first solve equation 1 for the minimum sampling frequency:

Next, plug that value into equation 2 to calculate Z, and round the answer down to the nearest integer value:

Finally, use the final value of Z to calculate the Undersampling frequency: