Nyquist Theorem — The Foundation of Sampling Theory

The Nyquist-Shannon Sampling Theorem states:
To perfectly reconstruct a continuous-time signal from discrete samples, the sampling frequency (fs) must be at least twice the highest frequency component (fmax) in the signal:

    fs ≥ 2 x fmax

The threshold fs/2 is called the Nyquist frequency. Sampling below this rate causes loss of signal information and aliasing. Aliasing is where high-frequency signals fold into lower frequencies and distort the result, this topic is covered in more detail later in the tutorial.

Why It Matters for ADCs

When measuring a signal with an analog-to-digital converter (ADC) Nyquist Sampling Theorem tells us the frequency range we can hope to capture with that ADC based on its sampling rate. It also tells us what frequency cutoff we should set for our low pass filter stage before the ADC input. This common low pass filter stage is often referred to as an anti-aliasing filter since it prevents aliasing or sampling of signals above the Nyquist limit. Without this filter, aliasing can corrupt measurements and lead to incorrect interpretations of the signal.

While the Nyquist theorem provides the theoretical minimum frequency (fs/2), real-world ADCs are not ideal. Factors such as finite analog bandwidth, aperture jitter, incomplete settling, and sampling distortion mean that, in practice, the sampling rate must be higher than 2× the signal bandwidth to ensure accurate and clean conversion. For this reason, ADC designers and end users typically sample at 2.5× to 5× the highest input frequency to provide sufficient margin for signal integrity and spectral purity.

Why It Matters for DACs

The highest frequency a digital-to-analog converter or DAC can ideally generate without introducing spectral artifacts is limited by the Nyquist frequency, which is half the sampling rate $f_s/2$fs/2 of the DAC. For example, a DAC operating at 100 MSPS has a Nyquist frequency of 50 MHz, meaning the highest frequency signal it can theoretically output without spectral images overlapping is 50 MHz.

However just like ADCs, real DACs are not ideal. They exhibit imperfections such as sample-and-hold effects, finite analog bandwidth, glitch energy, digital-to-analog settling limitations, and output noise. These non-idealities degrade signal fidelity as output frequency increases — especially near the Nyquist limit. As a result, the practical upper limit for generating high-purity waveforms is often well below the Nyquist frequency. In most applications, designers of DAC-based systems output signals with frequency content no higher than 40–80% of the fs/2. 

To maintain signal quality, DAC designers often use a reconstruction filter is used to suppress image frequencies. These image frequencies are an artifact of the DAC sampling rate (fs) and are higher frequency signal content then the desired output signal (often appear as stair steps in the output signal). These reconstruction filters ensure that only the desired signal frequency content is passed to downstream circuitry.

Summary table of DAC output frequencies

 

Nyquist Zones and Aliasing Explained

Nyquist zones break the frequency spectrum into bands of width fs/2. Each zone determines how a signal is “seen” by the ADC during sampling:

• Zone 1: 0 to fs/2 – Signal is sampled correctly, no aliasing
• Zone 2: fs/2 to fs – Signal aliases back into Zone 1
• Zone 3: fs to 3fs/2 – Signal aliases and folds again into Zone 1
• And so on for successive zones

This zone-based view is key to understanding aliasing and under-sampling. Zones above 1 are typically frequencies we want to filter out with an anti-aliasing filter. But there are some ADC applications where capturing zones higher than 1 is desirable. This is typically referred to as undersampling. 

Aliasing and Nyquist Zones Example

The chart below shows how signals alias into Zone 1 depending on their true frequency:

From the chart above we can see that if you sample a 60 MHz signal at 100 MSPS, it lies in Zone 2, and you will observe that 60MHz signal incorrectly as:
    f_alias = |fs - f_signal| = |100 MHz - 60 MHz| = 40 MHz

If we sample a 120MHz signal at the same 100MSPS rate we will measure it as only 20MHz!. This is why we need an anti-aliasing filter to prevent incorrect signal measurements of frequencies outside our ADC's Nyquist Zones. 

Using an ADC to undersample a signal is a niche area where aliasing is desirable and planned for. Undersampling allows a low sampling rate ADC to be used to measure higher frequency signals above its Nyquist frequency. This has the advantage of reducing the cost of a design since the higher the ADC's sample rate the more expensive parts are and the more design time it takes to implement the design correctly. Using an ADC for undersampling only works when you have knowledge of that signal's frequency spectrum. It works best for repetitive signals or signals that vary within a known Nyquist zone. If a signal can vary through multiple Nyquist zones unpredictably then undersampling will not work. It also will miss sudden unexpected changes or glitches in a signal cycle since it is not sampling every continuous cycle of a signal.

Recommended Video from Texas Instruments Precision Lab Series

We are big fans of TI's Precision Lab Series and thought we would share this video from the series that covers a lot of what was discussed in the tutorial through the lens of an ADC designer. It lot only covers the need for anti-aliasing filters but also covers the charge bucket filter which is sometimes mislabeled as an anti-aliasing filter. Enjoy!

Summary

Nyquist Limit: Always sample at ≥ 2× highest signal frequency
Aliasing: Happens when signals in Zone 2+ fold into Zone 1
Under-Sampling: Useful for repetitive RF signals with narrow bandwidth
Use Filters: Always apply anti-aliasing filters (ADC) or reconstruction filters (DAC)