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Related Concept Videos

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The simplest mechanical waves are associated with simple harmonic motion and repeat themselves for several cycles. These simple harmonic waves can be modeled using a combination of sine and cosine functions. Consider a simplified surface water wave that moves across the water's surface. Unlike complex ocean waves, in surface water waves, water moves vertically, oscillating up and down, whereas the disturbance of the wave moves horizontally through the medium. If a seagull is floating on the...
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Related Experiment Video

Updated: Mar 30, 2026

Using Wavelet Entropy to Demonstrate how Mindfulness Practice Increases Coordination between Irregular Cerebral and Cardiac Activities
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A Wavelet Perspective on the Allan Variance.

Donald B Percival

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    |November 4, 2015
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    Summary
    This summary is machine-generated.

    This study reveals how wavelet variance theory, particularly with Haar wavelets, deepens understanding of Allan variance for time and frequency standards. It enables accurate confidence intervals without assuming noise types, even with gappy or contaminated data.

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    Area of Science:

    • Signal Processing
    • Time and Frequency Standards
    • Geophysics

    Background:

    • Allan variance is crucial for characterizing high-performance time and frequency standards.
    • Wavelets and discrete wavelet transform (DWT) emerged in geophysical and signal processing literature.
    • A connection between Allan variance and Haar wavelet transform was previously noted.

    Purpose of the Study:

    • To review wavelet variance theory, focusing on Haar wavelets and their link to Allan variance.
    • To explore wavelet variance estimation for constructing Allan variance confidence intervals without a priori noise assumptions.
    • To discuss specialized wavelet variance estimators for gappy or contaminated data and their adaptation for Allan variance.

    Main Methods:

    • Review of basic wavelet variance theory, emphasizing Haar-based approaches.
    • Interpretation of Allan variance estimators using the maximal overlap discrete wavelet transform (MODWT).
    • Adaptation of specialized wavelet variance estimators for gappy and contaminated data.

    Main Results:

    • Wavelet variance using Haar wavelets is directly related to Allan variance.
    • Wavelet variance estimation provides a method for confidence intervals of Allan variance.
    • Estimators for gappy/contaminated data can be adapted for Allan variance.

    Conclusions:

    • Wavelet variance theory offers a deeper understanding and improved estimation for Allan variance.
    • This approach bypasses the need for pre-specifying noise models for confidence intervals.
    • Generalizations of Allan variance are possible using wavelets beyond the Haar.