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Related Concept Videos

Convolution: Math, Graphics, and Discrete Signals01:24

Convolution: Math, Graphics, and Discrete Signals

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.
To simplify the convolution integral, it is assumed that both the input signal and impulse response are zero for negative time values. The graphical convolution process...
Convolution Properties II01:17

Convolution Properties II

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.
The area property asserts that the area under the...
Convolution Properties I01:20

Convolution Properties I

Convolution computations can be simplified by utilizing their inherent properties.
The commutative property reveals that the input and the impulse response of an LTI (Linear Time-Invariant) system can be interchanged without affecting the output:
Collisions in Multiple Dimensions: Introduction01:05

Collisions in Multiple Dimensions: Introduction

It is far more common for collisions to occur in two dimensions; that is, the initial velocity vectors are neither parallel nor antiparallel to each other. Let's see what complications arise from this. The first idea is that momentum is a vector. Like all vectors, it can be expressed as a sum of perpendicular components (usually, though not always, an x-component and a y-component, and a z-component if necessary). Thus, when the statement of conservation of momentum is written for a problem,...
Properties of DTFT II01:24

Properties of DTFT II

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.
The frequency differentiation property is illustrated by considering a DTFT pair and differentiating both sides with respect to ω. Multiplying by j...
Deconvolution01:20

Deconvolution

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.
Deconvolution involves several mathematical techniques to derive the impulse response. One common approach is polynomial division. In this method, the input and output sequences are treated as coefficients of...

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Related Experiment Video

Updated: May 29, 2026

Blood Flow Imaging with Ultrafast Doppler
05:57

Blood Flow Imaging with Ultrafast Doppler

Published on: October 14, 2020

High-speed multidimensional convolution.

C E Kim1, M G Strintzis

  • 1Department of Electrical Engineering, University of Pittsburgh, Pittsburgh, PA 15261.

IEEE Transactions on Pattern Analysis and Machine Intelligence
|August 27, 2011
PubMed
Summary
This summary is machine-generated.

This study introduces fast algorithms for multidimensional convolutions using overlap-and-add and overlap-and-save methods. The overlap-and-save technique offers superior speed and storage efficiency for complex, high-dimensional computations.

Related Experiment Videos

Last Updated: May 29, 2026

Blood Flow Imaging with Ultrafast Doppler
05:57

Blood Flow Imaging with Ultrafast Doppler

Published on: October 14, 2020

Area of Science:

  • Digital Signal Processing
  • Computational Mathematics

Background:

  • Multidimensional convolutions are computationally intensive operations.
  • Efficient algorithms are crucial for processing large datasets in various scientific fields.

Purpose of the Study:

  • To develop and analyze fast computation algorithms for 2D and general multidimensional convolutions.
  • To compare the performance of overlap-and-add and overlap-and-save techniques in multidimensional settings.

Main Methods:

  • Detailed description of overlap-and-add and overlap-and-save techniques.
  • Decomposition of total convolution into partial convolutions executable via fast cyclic convolution algorithms.
  • Assessment of computational burden with radix-2 Fast Fourier Transform (FFT) algorithms.

Main Results:

  • Both overlap-and-add and overlap-and-save allow tradeoffs between speed and storage.
  • The overlap-and-save method demonstrates a significant advantage in speed and storage for multidimensional convolutions compared to overlap-and-add.
  • Computational complexity is analyzed in conjunction with radix-2 FFT.

Conclusions:

  • The overlap-and-save method is recommended for multidimensional convolution due to its efficiency.
  • Fast algorithms, particularly overlap-and-save with FFT, can significantly reduce computational load.
  • The findings provide practical guidance for optimizing high-dimensional signal processing tasks.