# Nyquist Sampling

Example of two sine waves that cannot be differentiated because they are not sampled at the Nyquist rate

Suppose you have a signal, and you want to sample it just often enough to be able to uniquely characterize it. This situation crops up everywhere in this digital world, from digital music to pixels on a screen to, of course, radio astronomy. And it turns out, for the general case (i.e. arbitrary signals that you don’t have other information about, such as sparseness), this well-posed question has a definitive answer: you must sample at a rate, fs that is more than twice the maximum frequency, fmax, in the spectrum of the signal. This is called the Nyquist rate.

$f_ s > 2f_{max} \,\!$

If you sample your signal at, or above, the Nyquist rate (also known as “Nyquist sampling” the signal), you can reconstruct the signal. If you don’t, you’ll have to deal with aliasing, which is described below.

### 1 Aliasing

Aliasing occurs when you fail to Nyquist sample a signal. As a general rule, for a sample rate of fs, you will not be able to differentiate between any sine waves with frequencies $N\frac{f_ s}{2}\pm \Delta f$, where N is any integer. All signal power for these frequencies will pile up (“alias”) at $\frac{f_ s}{2}-\Delta f$. As long as, for each Δf, you only have a signal at one of the aliasing frequencies, you haven’t irrevocably destroyed your signal; you have all the measurements you need to reconstruct your signal, if you know which of the frequencies that alias to fracfs2 − Δf were occupied by your signal. In these cases, aliasing on purpose can be useful, as described below in the discussion of Nyquist Zones.

However, if your signal has power at, say, both $\frac{f_ s}{2}+\Delta f$ and $\frac{f_ s}{2}-\Delta f$, then aliasing is a real problem. Because you no longer have the information to distinguish them, these formerly distinct signal frequencies now appear as a single frequency, with no way to tell how much power came from which component.

#### Nyquist Zones

For a sampling frequency fsamp, various zones exist where, if a signal is band limited to that zone, the full orignal spectrum can be recovered.

As described above, aliasing isn’t always bad. A chosen sample frequency, fs, defines zones (as shown in the figure above) where a band-limited signal in a higher Nyquist zone will alias down into the first Nyquist zone. For even-numbered Nyquist zones, the spectrum will appear in reversed order, and for odd-numbered zones, it appears in original order. This can mean, for example, that a 100-MHz band ranging from 700-800 MHz need only be sampled at 100 MHz to characterize it. As long as you know which Nyquist zone was occupied, you have all the measurements you need to reconstruct the original signal perfectly.

One difficulty in purposely aliasing signals in this manner is that the boundaries of the Nyquist zones are fundamentally tied to the sample rate, which also dictates the bandwidth that a signal can have within a Nyquist zone. Thus, in the case above, a 100-MHz band from 700-800 MHz can use aliasing effectively, but one from 720-820 MHz cannot.