Allan's-variance.md

+Allan variance is a statistical method for measuring the stability and accuracy of a time and frequency signal over time. It was first introduced by David W. Allan in 1966, and it is widely used to analyze the performance of oscillators, clocks, and other time-keeping devices.
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+Allan variance is a measure of the frequency stability of a signal over time, and it is calculated by dividing the data into overlapping segments, and then calculating the variance of the frequency or phase differences between each segment. It is represented by the symbol, "σy²(τ)" where τ is the averaging time.
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+The Allan variance is usually plotted as a function of the averaging time, and the resulting plot is called the Allan deviation. The Allan deviation plot allows for the analysis of the stability of the signal over different time scales.
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+There are three main regions that can be identified in an Allan deviation plot:
+-   Short-term stability: This region corresponds to averaging times of less than 1 second. In this region, the Allan deviation is dominated by random noise and is inversely proportional to the square root of the averaging time.
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+-   Medium-term stability: This region corresponds to averaging times between 1 second and 1 day. In this region, the Allan deviation is dominated by the stability of the oscillator or clock.
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+-   Long-term stability: This region corresponds to averaging times greater than 1 day. In this region, the Allan deviation is dominated by other factors such as aging, temperature effects, and environmental disturbances.
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+Allan variance is useful for measuring the stability and accuracy of a time and frequency signal over different time scales. It allows for the identification of different sources of error and instability, such as short-term noise, medium-term oscillator stability, and long-term environmental effects. Additionally, Allan variance can be used to compare the performance of different oscillators or clocks, and to determine the optimal averaging time for a specific application.
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