Undersampling Shifts Signal Bandwidths

In Specifying & Creating Data-Acquisition Systems, Part 3, I noted unwanted high-frequency signals could alias or "fold" into the signals you plan to measure. Anti-alias filters can remove those unwanted signals. In some cases, though, you can take advantage of aliasing and use it with a technique called under-sampling or bandwidth sampling.

So far, my data-acquisition examples assumed signals of interest exist between DC and an upper frequency; the measurement bandwidth. Sampling at a rate more than twice this bandwidth lets you reconstruct the original signals. The Nyquist-Shannon theorem does not require the bandwidth start at DC, though. Say you have a signal at 55MHz that carries "information" that causes the signal to vary between 51MHz and 59MHz. You might think an ADC must sample at a rate faster than 59 ∙ 10⁶ x 2, or more than 118 Msamples/sec. But the signal has only an 8MHz bandwidth, so you can sample at a lower rate once you purposely alias the signals.

The diagram illustrates how aliasing works for a given sample rate, fs, over a frequency spectrum. Think of the spectrum as drawn on a long piece of clear plastic you can fold along the [n-(1/2)]∙fs and n∙fs lines. A look through the folded plastic shows the final spectrum of the data an ADC would digitize. It's a mess! Also, a frequency inversion occurs for the information shown in section 2, between fs/2 and fs. Signals between n∙fs and [n+1-(1/2)]∙fs experience no such inversion.

To alias only the signals between fs and (3/2)∙fs, for example, use a bandpass filter to eliminate all signals outside this bandwidth of interest. Then choose a sample rate that lets you "fold" the frequencies between fs and (3/2)∙fs down to between DC to fs/2. So how do you determine the proper sampling rate?

I'll skip the intermediate math, which you can find in the reference listed below. The sample frequency must meet two conditions that require you know the bandwidth (BW) of the signals you want to measure and the center frequency (fc) of this bandwidth. First:

(2∙ fc - BW)/n ≥ fs ≥ (2∙ fc + BW)/(n + 1)

Where n equals integer values 1, 2, 3, and so on.

And second: fs > 2∙BW

I created an Excel spreadsheet (see references, below) to calculate and compare values in the first equation for the 55MHz signal with an 8MHz bandwidth. The spreadsheet also determined which values met the condition fs > 2∙BW.

References: Lyons, Richard G., "Understanding Digital Signal Processing," 2 ed., Pearson Education, Inc., 2004. ISBN: 978-0-13-108989-7. Download the Excel 2003 spreadsheet. (Values in megahertz.)

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