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Related Concept Videos

Variance01:15

Variance

The deviations show how spread out the data are about the mean. A positive deviation occurs when the data value exceeds the mean, whereas a negative deviation occurs when the data value is less than the mean. If the deviations are added, the sum is always zero. So one cannot simply add the deviations to get the data spread. By squaring the deviations, the numbers are made positive; thus, their sum will also be positive.The standard deviation measures the spread in the same units as the data.
Pole and System Stability01:24

Pole and System Stability

The transfer function is a fundamental concept representing the ratio of two polynomials. The numerator and denominator encapsulate the system's dynamics. The zeros and poles of this transfer function are critical in determining the system's behavior and stability.
Simple poles are unique roots of the denominator polynomial. Each simple pole corresponds to a distinct solution to the system's characteristic equation, typically resulting in exponential decay terms in the system's response.
Stability01:28

Stability

The time response of a linear time-invariant (LTI) system can be divided into transient and steady-state responses. The transient response represents the system's initial reaction to a change in input and diminishes to zero over time. In contrast, the steady-state response is the behavior that persists after the transient effects have faded.
The stability of an LTI system is determined by the roots of its characteristic equation, known as poles. A system is stable if it produces a bounded...
BIBO stability of continuous and discrete -time systems01:24

BIBO stability of continuous and discrete -time systems

System stability is a fundamental concept in signal processing, often assessed using convolution. For a system to be considered bounded-input bounded-output (BIBO) stable, any bounded input signal must produce a bounded output signal. A bounded input signal is one where the modulus does not exceed a certain constant at any point in time.
To determine the BIBO stability, the convolution integral is utilized when a bounded continuous-time input is applied to a Linear Time-Invariant (LTI) system.
Standard Deviation01:10

Standard Deviation

The most commonly used measure of variation is the standard deviation. It is a numerical value measuring how far data values are from their mean. The standard deviation value is small when the data are concentrated close to the mean, exhibiting slight variation or spread. The standard deviation value is never negative, it is either positive or zero. The standard deviation is larger when the data values are more spread out from the mean, which means the data values are exhibiting more...
Stability of Equilibrium Configuration01:23

Stability of Equilibrium Configuration

Understanding the stability of equilibrium configurations is a fundamental part of mechanical engineering. In any system, there are three distinct types of equilibrium: stable, neutral, and unstable.
A stable equilibrium occurs when a system tends to return to its original position when given a small displacement, and the potential energy is at its minimum. An example of a stable equilibrium is when a cantilever beam is fixed at one end and a weight is attached to the other end. If the weight...

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Stability variances: a filter approach.

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Summary

This study presents a novel frequency-domain method for calculating Allan variance and its variants. This approach offers improved statistical properties for discrete-time measurements compared to traditional time-domain methods.

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Area of Science:

  • Metrology and Measurement Science
  • Signal Processing
  • Time Series Analysis

Background:

  • The Allan variance is a standard tool for characterizing frequency stability in oscillators and sensors.
  • Existing time-domain methods for Allan variance calculation have limitations with discrete-time data.
  • The relationship between Allan variance and power spectral density (PSD) is established for continuous-time signals.

Purpose of the Study:

  • To analyze the Allan variance estimator as a combination of discrete-time linear filters.
  • To develop a new frequency-domain method for computing Allan variance and its variants.
  • To establish a valid relationship between Allan variance and PSD for discrete-time signals.

Main Methods:

  • Analysis of Allan variance variants (overlapping, modified, Hadamard, overlapping Hadamard) as discrete-time linear filters.
  • Development of frequency-domain equations for Allan variance estimation.
  • Demonstration of equivalence between frequency-domain and time-domain data periodization.
  • Derivation of a new equation relating Allan variance to the PSD of discrete-time signals.

Main Results:

  • The proposed frequency-domain method is equivalent to periodizing data in the time domain.
  • Frequency-domain variance estimators exhibit superior statistical properties compared to classical time-domain estimators.
  • A new, accurate equation is provided for relating Allan variance to the PSD of discrete-time signals.
  • The previously known equation for continuous-time signals is shown to be invalid for discrete-time measurements.

Conclusions:

  • The novel frequency-domain approach provides a statistically superior method for Allan variance estimation with discrete-time data.
  • This work corrects and extends the relationship between Allan variance and PSD for practical, real-world measurements.
  • The developed frequency-domain method enables efficient and accurate computation of Allan variance and its variants.