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Related Concept Videos

Basic Operations on Signals01:22

Basic Operations on Signals

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Basic signal operations include time reversal, time scaling, time shifting, and amplitude transformations. These operations are fundamental in signal processing and analysis.
Time Reversal mirrors a continuous-time signal about the vertical axis at t=0. This is achieved by substituting t with −t. For example, if a signal x(t) is considered, the time-reversed signal is x(−t). This operation can be graphically represented, showing the mirrored signal.
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Sampling Continuous Time Signal01:11

Sampling Continuous Time Signal

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In signal processing, a continuous-time signal can be sampled using an impulse-train sampling technique, followed by the zero-order hold method. Impulse-train sampling involves the use of a periodic impulse train, which consists of a series of delta functions spaced at regular intervals determined by the sampling period. When a continuous-time signal is multiplied by this impulse train, it generates impulses with amplitudes corresponding to the signal's values at the sampling points.
In the...
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Basic Continuous Time Signals01:22

Basic Continuous Time Signals

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Basic continuous-time signals include the unit step function, unit impulse function, and unit ramp function, collectively referred to as singularity functions. Singularity functions are characterized by discontinuities or discontinuous derivatives.
The unit step function, denoted u(t), is zero for negative time values and one for positive time values, exhibiting a discontinuity at t=0. This function often represents abrupt changes, such as the step voltage introduced when turning a car's...
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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...
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Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

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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.
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Basic Discrete Time Signals01:16

Basic Discrete Time Signals

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The unit step sequence is defined as 1 for zero and positive values of the integer n. This sequence can be graphically displayed using a set of eight sample points, showing a step function starting from n=0 and remaining constant thereafter.
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Related Experiment Video

Updated: Dec 30, 2025

A Method for Tracking the Time Evolution of Steady-State Evoked Potentials
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Stationary time-vertex signal processing.

Andreas Loukas1, Nathanaël Perraudin2

  • 11Laboratoire de Traitement des Signaux 2, École Polytechnique Fédérale Lausanne, Lausanne, 1015 Switzerland.

EURASIP Journal on Advances in Signal Processing
|January 28, 2020
PubMed
Summary

This study introduces joint stationarity for high-dimensional graph-dependent processes, improving covariance estimation and MMSE recovery. This method enhances accuracy even with approximate graph knowledge or non-strict stationarity.

Keywords:
Graph signal processingHarmonic analysisMultivariate time-vertex processesPSD estimationStationarity

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Area of Science:

  • Statistics
  • Time Series Analysis
  • Graph Signal Processing

Background:

  • High-dimensional multivariate processes often exhibit complex structures dependent on graph topologies.
  • Existing methods struggle with accurate covariance estimation and efficient recovery for such processes.

Purpose of the Study:

  • To introduce a novel definition of stationarity, termed joint stationarity, for graph-dependent processes.
  • To demonstrate the benefits of joint stationarity in reducing estimation variance and computational complexity.
  • To enable reliable covariance structure learning and efficient MMSE recovery.

Main Methods:

  • Definition of time-vertex wide-sense stationarity (joint stationarity) extending beyond product graphs.
  • Theoretical analysis of covariance structure learning from single realizations.
  • Development of algorithms for MMSE recovery (interpolation, denoising) with near-linear computational time.

Main Results:

  • Joint stationarity allows reliable covariance structure learning from a single process realization.
  • MMSE recovery problems are solved in nearly linear computational time relative to edges and timesteps.
  • Experiments show accuracy improvements in recovering high-dimensional processes on graphs.

Conclusions:

  • Joint stationarity offers significant advantages for analyzing and recovering high-dimensional graph-dependent processes.
  • The method is robust to approximate graph knowledge and deviations from strict stationarity.
  • This framework advances the field of graph signal processing and time series analysis.