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Deconvolution

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...

Convolution Properties II

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...

Downsampling

Downsampling

When considering a sampled sequence with zero values between sampling instants, one can replace it by taking every N-th value of the sequence. At these integer multiples of N, the original and sampled sequences coincide. This process, known as decimation, involves extracting every N-th sample from a sequence, thereby creating a more efficient sequence.
The Fourier transform of the decimated sequence reveals a combination of scaled and shifted versions of the original spectrum. This...

Convolution Properties I

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:

Convolution: Math, Graphics, and Discrete Signals

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...

Reconstruction of Signal using Interpolation

Reconstruction of Signal using Interpolation

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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Related Experiment Video

Updated: Mar 17, 2026

Whole-cell Super-Resolution Imaging via DNA-PAINT on a Spinning Disk Confocal with Optical Photon Reassignment

Whole-cell Super-Resolution Imaging via DNA-PAINT on a Spinning Disk Confocal with Optical Photon Reassignment

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Deconvolution: It Fans Back, Out, and Ahead [Retrospectroscope].

Sergio A Gonzalez, Max E Valentinuzzi, Pedro D Arini

|July 15, 2016

Summary

This summary is machine-generated.

Convolution

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Area of Science:

Mathematics
Signal Processing

Background:

The historical development of convolution is complex.
Its origins trace back to the mid-18th century.

Purpose of the Study:

To provide a detailed historical account of convolution.
To highlight the evolution of its modern applications.

Main Methods:

Historical review of mathematical concepts.
Analysis of scientific literature.

Main Results:

Convolution's roots are in the 18th century.
Modern applications are recent, within the last 50-60 years.

Conclusions:

Convolution has a long but evolving history.
Its contemporary significance is a recent development.