A reader asked me to discuss undersampling, so here goes. First, though, some background. Say you need to measure signals from 4-5 MHz. To give yourself a 10 percent margin, you set a Nyquist limit of 5.5 MHz (Fn) and thus establish a sample frequency (Fs) of 11 Msamples/sec. Unfortunately, without an anti-alias filter, signals at frequencies above the Nyquist limit alias into your sampled data.
Suppose your signal source also produces energy at 15.3 MHz. That signal aliases to 15.3-11 MHz, or 4.3 MHz, right in the midst of your 4-5-MHz measurement bandwidth. Signals between the Nyquist limit and your sampling frequency can alias, too. A signal at 6.3 MHz will alias to 6.3 MHz-11 MHz, or -4.7 MHz. The sign simply indicates phase, so the -4.7 MHz "signal" appears at 4.7 MHz, within the bandwidth you plan to measure. Figure 1 shows how signals above Fs/2 alias into a measurement range. These unwanted signals become indistinguishable from your signals of interest.
Although aliasing can cause problems, you can purposely alias and measure signals, a technique called undersampling. Recall the Nyquist theorem specifies a bandwidth for measurements, not absolute frequencies. Take the US FM-radio spectrum between 88 and 108 MHz, which has a 20-MHz bandwidth. In theory, you can alias this bandwidth and measure it with Fs>43.2 Msamples/sec.
To use undersampling effectively, first, use a band-pass filter to pass to a digitizer only the signals in the desired bandwidth. Second, choose a sampling rate that aliases the bandwidth to frequencies on the positive side of the aliased-frequency axis. Third, do not let parts of your bandwidth alias to separate portions of the aliased-frequency span between -Fs/2 and Fs/2. (See the Web version of this column for a diagram.) To allow for small timing errors in your sampling system, I recommend you choose a bandwidth slightly wider than the actual band of interest. For the FM radio spectrum, try an 85-110-MHz bandwidth.
How do you determine the sampling frequency? Angelo Ricotta at the Institute of Atmospheric Sciences and Climate at the Italian National Research Council provides a thorough analysis of undersampling and two equations that produce sampling ranges for a given bandwidth. Assume Fl = 85 MHz and Fh = 110 MHz: first, find a set of integers, 0, 1 ... n, where n & (Fl / (Fh- Fl)). Then, find the range of Fs for each n: (2Fh / (n + 1)) & Fs& (2Fl/n). For the 25-MHz wide radio band, undersampling rates (in Msamples/sec) include, 55 & Fs & 56.6 (n=3); 73.3 &Fs& 85 (n=2); 110 & Fs& 170 (n=1); and 220 & Fs (n=0). Caution: Undersampling requires that your ADC must operate properly with signals at the top of the bandwidth.
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