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Related Concept Videos

Parseval's Theorem01:18

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Parseval's theorem is a fundamental concept in signal processing and harmonic analysis. It asserts that for a periodic function, the average power of the signal over one period equals the sum of the squared magnitudes of all its complex Fourier coefficients. This theorem, named after Marc-Antoine Parseval, provides a powerful tool for analyzing the energy distribution in signals.
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Related Experiment Video

Updated: Jul 17, 2026

Measurement of X-ray Beam Coherence along Multiple Directions Using 2-D Checkerboard Phase Grating
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Double Spectral theorem and Two Type Magnitude Coherence Functions.

Y Liu1, Z Xie, D M Xie

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PubMed
Summary

New magnitude coherence functions (MCFs) precisely capture signal coherence. Derived from the double spectral theorem (DST), these functions overcome limitations of magnitude-squared coherence functions (MSCFs) by preserving phase information.

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Area of Science:

  • Signal Processing
  • Time Series Analysis
  • Spectral Analysis

Background:

  • Frequency domain analysis is crucial for understanding signal coherence.
  • Magnitude-squared coherence functions (MSCFs) are widely used but can omit vital phase information.
  • The double spectral theorem (DST) provides a foundational framework for frequency domain coherence theories.

Purpose of the Study:

  • To introduce novel magnitude coherence functions (MCFs) that retain phase information.
  • To address the limitations of existing MSCFs in representing signal coherence.
  • To theoretically derive and validate new coherence measures based on the DST.

Main Methods:

  • Mathematical derivation of two new MCFs: same type magnitude coherence function (SMCF) and difference magnitude coherence function (DMCF).
  • Utilizing the double spectral theorem (DST) as the core theoretical basis.
  • Deriving theoretical conditions for MCFs equaling 1, -1, or 0 based on DST.

Main Results:

  • Successful mathematical derivation of SMCF and DMCF from the DST.
  • Theoretical establishment of conditions under which SMCF and DMCF equal 1, -1, or 0.
  • Demonstration that SMCF and DMCF can precisely extract coherence components between two signals.

Conclusions:

  • SMCF and DMCF offer a more complete representation of signal coherence than MSCFs.
  • These new functions provide exact extraction of coherence information by component.
  • The DST is a powerful tool for developing advanced coherence analysis methods.