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Related Concept Videos

Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

109
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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Linear Approximation in Time Domain01:21

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Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
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Reconstruction of Signal using Interpolation01:10

Reconstruction of Signal using Interpolation

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Signal processing techniques are essential for accurately converting continuous signals to digital formats and vice versa. When a continuous signal is sampled with a period T, the resulting sampled signal exhibits replicas of the original spectrum in the frequency domain, spaced at intervals equal to the sampling frequency. To handle this sampled signal, a zero-order hold method can be applied, which creates a piecewise constant signal by retaining each sample's value until the next...
233
Aliasing01:18

Aliasing

159
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.
If the sampling frequency is below the Nyquist rate, these replicas overlap, preventing the original...
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Bandpass Sampling01:17

Bandpass Sampling

203
In signal processing, bandpass sampling is an effective technique for sampling signals that have most of their energy concentrated within a narrow frequency band. This type of signal is known as a bandpass signal. The key principle of bandpass sampling involves sampling the signal at a rate that is greater than twice the signal's bandwidth to prevent aliasing.
A bandpass signal has a spectrum with a lower frequency limit, denoted as ω1, and an upper frequency limit, denoted as ω2....
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Routh-Hurwitz Criterion II01:19

Routh-Hurwitz Criterion II

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In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.
The first scenario occurs when a singular zero appears in the first column of the Routh table. This situation creates a division by zero issues. To resolve this, a small positive or negative number, denoted as epsilon (∈), is substituted for the zero. The stability analysis proceeds by assuming a sign for ∈. If ∈ is positive, any sign change in the first...
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Numerical computation of the equilibrium-reduced density matrix for strongly coupled open quantum systems.

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A spectrum adaptive kernel polynomial method.

Tyler Chen1

  • 1Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012, USA and Department of Computer Science and Engineering, Tandon School of Engineering, New York University, 370 Jay Street, New York, New York 11201, USA.

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We introduce a spectrum adaptive kernel polynomial method (KPM) that avoids costly pre-computation. This novel approach uses the Lanczos algorithm, simplifying spectral density approximation and offering practical benefits.

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Area of Science:

  • Numerical Analysis
  • Computational Physics
  • Applied Mathematics

Background:

  • Kernel polynomial method (KPM) is vital for spectral density approximation.
  • Traditional KPM requires pre-computation for parameter estimation, increasing costs.
  • Existing methods for parameter estimation add computational overhead.

Purpose of the Study:

  • To develop a spectrum adaptive KPM that eliminates the need for prior parameter estimation.
  • To integrate the Lanczos algorithm for efficient and adaptive spectral density approximation.
  • To demonstrate the practical and pedagogical advantages of decoupling computation from approximation.

Main Methods:

  • Implementation of a spectrum adaptive KPM utilizing the Lanczos algorithm without reorthogonalization.
  • Theoretical analysis to validate the use of the Lanczos algorithm in finite precision arithmetic.
  • Numerical examples to illustrate the method's effectiveness and benefits.

Main Results:

  • The proposed method allows KPM parameter selection after the main computation.
  • The Lanczos algorithm is shown to be suitable for this adaptive KPM, even with finite precision.
  • Decoupling computation from approximation offers significant practical advantages.

Conclusions:

  • The spectrum adaptive KPM provides a more efficient alternative to traditional methods.
  • The approach simplifies the KPM workflow by deferring parameter selection.
  • This method enhances both the practicality and pedagogical value of spectral density approximation.