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Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

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Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
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Aliasing01:18

Aliasing

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Accurate signal sampling and reconstruction are crucial in various signal-processing applications. A time-domain signal's spectrum can be revealed using its Fourier transform. When this signal is sampled at a specific frequency, it results in multiple scaled replicas of the original spectrum in the frequency domain. The spacing of these replicas is determined by the sampling frequency.
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Properties of Fourier series I01:20

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The Fourier series is a powerful tool in signal processing and communications, allowing periodic signals to be expressed as sums of sine and cosine functions. A foundational property of the Fourier series is linearity. If we consider two periodic signals, their linear combination results in a new signal whose Fourier coefficients are simply the corresponding linear combinations of the original signals' coefficients. This property is crucial in applications like frequency modulation (FM) radio,...
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Properties of Laplace Transform-I01:15

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The Laplace transform is a powerful mathematical tool used to convert functions from the time domain into the frequency domain, greatly simplifying the analysis and solution of linear time-invariant systems. This transformation is facilitated by several universal properties: Linearity, Time-Scaling, Time-Shifting, and Frequency Shifting.
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Frequency Response of a Circuit01:20

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Inductive circuits present intriguing challenges in electrical engineering, particularly during the transition from the time domain to the frequency domain. This transformation involves converting inductors into impedances and utilizing phasor representation.
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Frequency-Domain Interpretation of PD Control01:24

Frequency-Domain Interpretation of PD Control

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Proportional-Derivative (PD) controllers are widely used in fan control systems to improve stability and performance. A fan control system can be effectively represented using a Bode plot to illustrate the impact of a PD controller through its transfer function. The Bode plot visually conveys how PD control modifies the fan's response across various frequencies, providing a frequency domain interpretation of the controller's behavior.
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Updated: Mar 20, 2026

Measurement of Scattering Nonlinearities from a Single Plasmonic Nanoparticle
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Quantitative analysis of a frequency-domain nonlinearity indicator.

Brent O Reichman1, Kent L Gee1, Tracianne B Neilsen1

  • 1Department of Physics and Astronomy, Brigham Young University, Provo, Utah 84602, USA.

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Summary

A new indicator quantifies acoustic nonlinearity, revealing how nonlinear effects, absorption, and spreading impact sound waves. This helps predict harmonic changes before wave distortion, crucial for understanding sound propagation in various media.

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Area of Science:

  • Acoustics
  • Nonlinear acoustics
  • Wave propagation

Background:

  • Understanding acoustic nonlinearity is crucial for predicting wave behavior.
  • Existing methods struggle to quantitatively separate nonlinear effects from absorption and geometric spreading.
  • The generalized Burgers equation provides a framework for analyzing nonlinear wave phenomena.

Purpose of the Study:

  • To develop a quantitative, frequency-domain indicator for acoustic nonlinearity.
  • To analyze the interplay of nonlinearity, absorption, and geometric spreading on the pressure spectrum.
  • To investigate the spatial rate of change of the pressure spectrum level using a normalized quadspectrum.

Main Methods:

  • Derivation of a nonlinearity indicator from the generalized Burgers equation.
  • Calculation of nonlinear effects using the pressure-squared-pressure quadspectrum.
  • Analysis of plane wave propagation of initial sinusoids in inviscid and thermoviscous media.

Main Results:

  • The normalized quadspectrum (Q/S) quantifies the spatial rate of change of the pressure spectrum level.
  • Harmonic growth and decay are captured, with significant changes preceding sawtooth formation.
  • In inviscid media, the spatial rate of change becomes uniform across harmonics at large distances.
  • In thermoviscous media, nonlinear gains are offset by absorption, causing a greater negative rate of change for higher harmonics.

Conclusions:

  • The developed indicator provides a quantitative understanding of frequency-domain nonlinearity.
  • The Q/S parameter effectively predicts changes in acoustic wave spectra due to nonlinear propagation.
  • This work offers insights into sound propagation, particularly the competing effects of nonlinearity and dissipation.