# What is Allan Variance?

What is Allan Variance? What is it used to measure in oscillators?

#### Editorial Team - everything RF

May 11, 2023

Allan Variance is a statistical tool used to measure the stability and accuracy of clock sources and other time-varying systems. It is named after David W. Allan, who developed this technique in the 1960s as a way to analyze the stability of atomic clocks. Today, Allan variance is widely used in fields like telecommunications, navigation, and metrology, to evaluate the performance of clocks, oscillators, and other devices that rely on accurate timekeeping.

Allan Variance is expressed mathematically as

The Allan variance is a widely used tool for characterizing the stability of oscillators, which are devices that produce periodic signals with a well-defined frequency and phase. Oscillators are used in many applications, including clocks, communication systems, and navigation systems. The stability of an oscillator is an important parameter that affects the accuracy and reliability of these applications.

The Allan variance of an oscillator is a measure of the short-term stability of its frequency. It is computed by measuring the frequency difference between two adjacent periods of the oscillator and comparing the variance of these differences at different averaging intervals.

For example, suppose an oscillator that is supposed to oscillate once per second, but that actually varies slightly from this ideal rate due to noise and other sources of error. To measure the Allan variance of this oscillator, pairs of successive readings from the oscillator are taken, and the difference between each pair is computed. Then the variance of these differences is computed, and this variance is plotted as a function of the averaging interval used to compute it.

The resulting plot is known as the Allan deviation curve, and it typically has a characteristic shape that reveals important information about the stability and accuracy of the oscillator. For short averaging intervals, the Allan deviation is dominated by the short-term noise in the oscillator signal, which causes the variance to increase rapidly as the averaging interval decreases. As the averaging interval increases, however, the noise begins to average out, and the Allan deviation reaches a minimum value, known as the Allan variance floor. This floor represents the fundamental limit on the accuracy and stability of the oscillator and is determined by its underlying physics.

For short averaging intervals, the Allan deviation curve reflects the short-term noise and jitter in the oscillator signal, which can be caused by factors such as thermal fluctuations, power supply noise, and electromagnetic interference. As the averaging interval increases, the Allan deviation curve transitions to a region where the noise is averaged out, and the oscillator's long-term frequency stability is revealed. At longer averaging intervals, the Allan deviation curve typically reaches a minimum value known as the Allan variance floor, which represents the fundamental limit of the oscillator's stability.

The Allan variance of an oscillator can be affected by many factors, including the type of oscillator circuit, the quality of the components used, and the operating conditions. For example, an oscillator that is subject to temperature variations may have a higher short-term noise and a higher Allan variance than an oscillator that is kept at a constant temperature. Similarly, an oscillator that is powered by a noisy power supply may have a higher short-term noise and a higher Allan variance than an oscillator that is powered by a clean power supply.

One of the key advantages of the Allan variance method is that it can reveal different types of noise and instability in the oscillator signal. For example, the Allan deviation curve can show whether the oscillator has white noise (random fluctuations with a flat power spectrum), flicker noise (fluctuations that decrease with increasing frequency), or other types of noise. This information can be useful in designing and optimizing oscillator circuits for specific applications.

There are several important applications of Allan variance in the real world. In telecommunications, for example, Allan variance is used to measure the frequency stability of oscillators used in cell phone towers and other wireless infrastructure. In navigation, it is used to evaluate the performance of GPS and other satellite-based positioning systems. In metrology, it is used to compare the accuracy of different types of clocks and to establish traceability to international standards. Allan variance can also be used to evaluate the effectiveness of filters and amplifiers in reducing noise and improving signal quality. It can also be used to compare the performance of different RF systems or to establish standards for performance.

Hence, Allan variance is a powerful tool for measuring the stability and accuracy of oscillators and other time-varying systems. By providing a quantitative measure of the noise and error in these systems, it helps engineers and scientists to optimize their performance and to ensure that they meet the stringent requirements of modern technology.