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Related Concept Videos

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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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In the study of discrete-time signal processing, understanding the properties of the Discrete-Time Fourier Transform (DTFT) is crucial for analyzing and manipulating signals in the frequency domain. Several properties, including frequency differentiation, convolution, accumulation, and Parseval's relation, offer powerful tools for signal analysis.
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Properties of the z-Transform II01:16

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The property of Accumulation in signal processing is derived by analyzing the accumulated sum of a discrete-time signal and using the time-shifting property to determine its z-transform. This principle reveals that the z-transform of the summed signal is related to the z-transform of the original signal by a multiplicative factor.
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In signal processing, Discrete-Time Fourier Transforms (DTFTs) play a critical role in analyzing discrete-time signals in the frequency domain. Various properties of the DTFTs such as linearity, time-shifting, frequency-shifting, time reversal, conjugation, and time scaling help understand and manipulate these signals for different applications.
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Massera's theorem on arbitrary discrete time domains.

Martin Bohner1, Jaqueline G Mesquita2, Sabrina Streipert3

  • 1Department of Mathematics and Statistics, Missouri S&T, 400 W. 12th St., Rolla, MO 65409, USA.

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This study generalizes Massera's theorems for discrete systems, proving that u-bounded solutions in linear systems and under specific conditions for nonlinear equations lead to periodic solutions.

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boundednessisolated time scaleslinear dynamic equationsnonlinear dynamic equationsperiodicity

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Area of Science:

  • Mathematics
  • Dynamical Systems Theory
  • Differential Equations

Background:

  • Massera's theorems provide conditions for periodic solutions in differential equations.
  • Extending these theorems to discrete domains is crucial for broader applications.
  • Existing definitions for linear and nonlinear equations lack generality for discrete systems.

Purpose of the Study:

  • To generalize Massera's theorems for arbitrary discrete domains.
  • To introduce new definitions for linear and nonlinear equations applicable to discrete systems.
  • To establish conditions for asymptotic and necessary periodic solutions.

Main Methods:

  • Development of novel definitions for linear and nonlinear equations in discrete settings.
  • Application of these definitions to analyze u-bounded solutions.
  • Formulation and proof of generalized Massera's theorems.

Main Results:

  • Sufficient conditions are identified for scalar nonlinear equations where u-bounded solutions approach periodicity.
  • It is proven that for linear systems, a u-bounded solution guarantees a periodic solution.
  • The findings are illustrated with examples demonstrating practical relevance.

Conclusions:

  • The generalized theorems offer a robust framework for analyzing periodic solutions in discrete dynamical systems.
  • The new definitions enhance the applicability of Massera's theorems to a wider range of problems.
  • The results contribute to a deeper understanding of solution behavior in discrete systems.