A digital signal can be derived from the original time-varying continuous analog signal by creating a sampled sequence of quantized values. It is intuitively obvious that the accuracy and resolution of this quantized signal is based on the number of samples taken per unit time. Harry Nyquist published an early version, later formalized by Claude Shannon, of what became known as the Nyquist-Shannon theorem, which states that if a system evenly samples an analog signal at a rate that exceeds the highest signal frequency with a factor of at least two, the original analog signal can be perfectly recovered from the discrete version produced by sampling.

Perfect? Yes, that’s what the theorem says. To comply with the conditions of the theorem, we must recognize that the “highest frequency” consists of all kinds of harmonics, which can be quite extensive. They can be filtered, but this limits the applicability of the theorem with respect to the bandwidth of the original signal.

However, perhaps surprisingly, there is something to be said about the deliberate sampling of an analog signal, i. using a sampling rate of less than twice the maximum frequency component of the signal. Sampling is also called band sampling, harmonic sampling, or super-Nyquist sampling.

The benefits of sampling stem from the pseudonym effect. Suppose a signal F_{in} samples are taken at speed, F_{s}, less than twice the maximum frequency. Then the alias signal appears on F_{s} – Ф_{in}. For example, suppose we sample a 70-MHz signal with a 100 MSPS sampling rate. The component with the alias will appear at 30 MHz (100 – 70). The key is that we need to know in advance that the signal we see is a pseudonym. We can then recover the actual frequency using F_{s} – Ф_{in} connection.

The use of subsampling uses the ADC as a mixer or step-down converter. Sampling is useful for demodulating or retrieving information from carrier signals in radios. Think again about a 70 MHz signal and assume that there is a 20-MHz signal bandwidth. If the sampling rate F_{s} is 100 MSPS, the component with the alias will appear between 20 MHz to 40 MHz (30 ± 10 MHz).

Sampling uses the concept of what are called Nyquist zones. Nyquist zones subdivide the spectrum into areas evenly spaced at intervals from F_{s}/ 2. Each Nyquist zone contains a copy of the spectrum of the desired signal or its mirror image. Odd Nyquist zones contain exact copies of the signal spectrum.

Think again about the 70-MHz signal, but accept the 56-MSPS sampling rate. The 70-MHz signal in this case is located in the third zone of Nyquist and is connected back to the first zone of Nyquist, centered on 14 MHz. A digital step-down converter (DDC) can recover this digital frequency mixing signal, which converts it to a baseband and additional digital filtering.

Radio designers sometimes use this sampling technique as a way to eliminate the analog frequency conversion stage. The nickname of the signal is not spectrally inverted in this case, as it is located in zone Nyquist three, zone with odd number. However, spectral inversion can be reversed and reversed by FPGA or other means after receiving data from the ADC. However, spectral inversion worsens some demodulating processes. Spectrally, inversion is not problematic, as long as the spectrum is relatively flat in the band. There may also be problems if the whole signal spreads to two Nyquist zones. Here, the component with the alias in the first Nyquist zone cannot be extracted by any kind of digital downward conversion.

The main advantages of sampling include the fact that the ADC can consume less power than a version running at higher frequencies. And slower ADCs are also cheaper. In addition, it is easier to capture signals with slower ADCs, as setup and hold times are quieter than those at higher speeds. Conversely, undersampling systems use narrowband filters, which can be difficult to implement depending on the bandwidth and alias frequencies of interest.