Gaussian filter

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Magnitude frequency response of a Gaussian filter with normalized frequency and a bandwidth of 1.
Impulse response of a Gaussian filter

Gaussian filters are frequency filters which have no overshoot in the step response and at the same time have maximum edge steepness in the transition area. As a special feature of this filter, both the transfer function and the impulse response have the course of a Gaussian bell curve , as shown in the figures, from which the name of this filter type is derived.

This filter is used in digital modulation processes and in image processing .

Transfer function

The magnitude of the transfer function is given by Gaussian filters

with the constant

.

The impulse response of a Gaussian filter is

.

This shows that the Gaussian filter represents an idealization because it is non- causal : Half of the impulse response (curve at t <0) has already appeared at the filter output when the triggering signal, the impulse, is at the filter input , occurs at t = 0.

Applications

digital signal processing

A square pulse, shown dotted in blue, is converted into the signal curve shown in red by the pulse shaping of a Gaussian filter.

Gaussian filters have a constant group delay in the stop and pass band and no overshoot in the step response . This filter is primarily used for pulse shaping with applications in digital signal processing .

The pulse shaping takes place in digital modulation methods such as Gaussian Minimum Shift Keying (GMSK), since the individual, mostly rectangular transmission symbols can be converted into pulses of the Gaussian bell curve with a lower bandwidth requirement than the original rectangular transmission symbols. This is associated with a higher spectral efficiency of the modulation method.

In mobile radio systems such as GSM , Gaussian filters are used as part of GMSK modulation on the radio interface to transmit digital voice and control information.

Further applications are in modulation techniques such as the chirp spread spectrum , in which the discontinuous frequency change in chronologically successive chirps is smoothed by Gaussian filters.

Image processing

Halftone image smoothed with a Gaussian filter

In image processing , Gaussian filters are used to smooth or soften the image content. This can reduce the image noise: Smaller structures are lost, while coarser structures are retained. Spectrally, the smoothing is equivalent to a low-pass filter .

Since an image has two dimensions, the impulse response must be expanded to two dimensions for image processing. The impulse response has the two arguments and according to the spatial directions:

.

For practical implementations in the context of digital image processing, the discrete impulse response is mostly used in the form of a two-dimensional matrix .

Alternatively, in the literature when describing Gaussian filters, instead of the constants , the variance in the expression of the impulse response is used equivalently - which expresses the mathematical proximity of the impulse response of a Gaussian filter to the function of the normal distribution . For one dimension the impulse response is:

The impulse response in two dimensions results from the product of the two directions in x and y :

The computing time can be significantly reduced by utilizing the separability .

Example of a 3 × 3 filter:

literature

  • Karl Dirk Kammeyer, Volker Kühn: MATLAB in communications engineering . 1st edition. J. Schlembach Fachverlag, 2001, ISBN 3-935340-05-2 .

Web links

Individual evidence

  1. ^ F. Dellsperger: Passive Filters. Bern University of Applied Sciences, University for Technology and Computer Science HTI, Department of Electrical and Communication Technology, 2012, p. 25 , accessed on July 17, 2017 .
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