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Convolution: Math, Graphics, and Discrete Signals01:24

Convolution: Math, Graphics, and Discrete Signals

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In any LTI (Linear Time-Invariant) system, the convolution of two signals is denoted using a convolution operator, assuming all initial conditions are zero. The convolution integral can be divided into two parts: the zero-input or natural response and the zero-state or forced response, with t0 indicating the initial time.
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Convolution Properties II01:17

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The important convolution properties include width, area, differentiation, and integration properties.
The width property indicates that if the durations of input signals are T1 and T2, then the width of the output response equals the sum of both durations, irrespective of the shapes of the two functions. For instance, convolving two rectangular pulses with durations of 2 seconds and 1 second results in a function with a width of 3 seconds.
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Convolution Properties I01:20

Convolution Properties I

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Convolution computations can be simplified by utilizing their inherent properties.
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Deconvolution

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Deconvolution, also known as inverse filtering, is the process of extracting the impulse response from known input and output signals. This technique is vital in scenarios where the system's characteristics are unknown, and they must be inferred from the observable signals.
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Properties of DTFT II01:24

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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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Related Experiment Video

Updated: Dec 28, 2025

Closed-loop Neuro-robotic Experiments to Test Computational Properties of Neuronal Networks
11:18

Closed-loop Neuro-robotic Experiments to Test Computational Properties of Neuronal Networks

Published on: March 2, 2015

10.7K

Understanding Convolutional Neural Networks With Information Theory: An Initial Exploration.

Shujian Yu, Kristoffer Wickstrom, Robert Jenssen

    IEEE Transactions on Neural Networks and Learning Systems
    |February 20, 2020
    PubMed
    Summary

    This study introduces a new method to measure information flow in convolutional neural networks (CNNs) using Rényi

    Related Experiment Videos

    Last Updated: Dec 28, 2025

    Closed-loop Neuro-robotic Experiments to Test Computational Properties of Neuronal Networks
    11:18

    Closed-loop Neuro-robotic Experiments to Test Computational Properties of Neuronal Networks

    Published on: March 2, 2015

    10.7K

    Area of Science:

    • Information theory
    • Machine learning
    • Deep learning

    Background:

    • A novel functional estimator for Rényi's α-entropy was recently proposed.
    • Its utility and applications are not widely known to practitioners.

    Purpose of the Study:

    • To demonstrate the straightforward measurement of information flow in convolutional neural networks (CNNs) using the novel estimator.
    • To introduce the partial information decomposition (PID) framework for analyzing convolutional layer representations.
    • To develop quantities for analyzing synergy and redundancy in CNNs.

    Main Methods:

    • Utilized a novel functional estimator based on the normalized eigenspectrum of a Hermitian matrix in a reproducing kernel Hilbert space (RKHS).
    • Applied the estimator to measure information flow in CNNs without approximation.
    • Integrated the partial information decomposition (PID) framework.

    Main Results:

    • Enabled straightforward, non-approximate measurement of information flow in realistic CNNs.
    • Developed three quantities to analyze synergy and redundancy in convolutional layer representations.
    • Validated two fundamental data processing inequalities.
    • Revealed inner properties of CNN training.

    Conclusions:

    • The novel estimator provides a practical tool for analyzing information flow in CNNs.
    • The PID framework offers new insights into the internal workings of convolutional layers.
    • The findings contribute to understanding information processing within deep learning models.