# How to use undersampling

**Aliasing**

To avoid aliasing, we know from the Nyquist criterion that we must digitize a signal at a sampling rate of at least twice the bandwidth of the signal. For 'baseband' signals with frequency components starting at DC and extending up to some maximum frequency, this means the sampling rate must be at least twice this maximum frequency. In this case, the bandwidth and the maximum frequency are the same.

**Bandpass Signals**

Let's consider 'bandpass' signals like the IF (intermediate frequency) output from a standard communications receiver with a 70 MHz center frequency and a 5 MHz bandwidth, for example.

Does this mean that we can digitize this 70 MHz bandpass signal with an A/D operating at twice the bandwidth, or just above 10 MHz? If so, this can be a big benefit compared to baseband sampling, where we would otherwise need to use an A/D converter with a sampling rate of about 150 MHz. This is the basic mission of 'undersampling'.

**Undersampling**

In order to apply undersampling successfully, a careful frequency plan must be developed. One tried and true technique is the 'fan-fold' paper method. Start with a small stack of semitransparent fan-fold computer printer paper, or the imaginary equivalent. Holding the paper with the folds in the vertical direction, plot the frequency axis from left to right along the bottom edge with the inward creases at multiples of the A/D sampling frequency, Fs, and the outward creases at odd multiples of Fs/2, as shown in *Figure 1*.

*Figure 1. Fan-fold paper showing the spectrum of a RF input signal.*

The vertical axis is used to plot the spectral amplitude of the signal, such as wideband RF signal shown. In order to see what happens after sampling, simply collapse the stack of fanfold paper, hold it up to a light and look through the stack. You'll see the spectra from all sheets superimposed on top of each other, which represents the exact frequency content in the A/D output samples. As shown in *Figure 2*, signals on all of the sheets above Fs/2 are effectively "folded" down into sheet 1 between 0 Hz and Fs/2.

*Figure 2. Looking through the collapsed stack reveals the resulting spectrum.*

For the signals on every odd numbered sheet, the effect is a frequency translation by a multiple of Fs. For the signals on even numbered sheets, there is a reversal of the frequency axis on that sheet, followed by a translation by an odd multiple of Fs/2. Again, this is much easier to follow by visualizing the fan-fold model.