Gaussian filter: Information from Answers.com

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Linear analog electronic filters
Network synthesis filters Butterworth filter Chebyshev filter Elliptic (Cauer) filter Bessel filter Gaussian filter Optimum "L" (Legendre) filter Linkwitz-Riley filter Image impedance filters Constant k filter m-derived filter General image filters Zobel network (constant R) filter Lattice filter (all-pass) Bridged T delay equaliser (all-pass) Composite image filter mm'-type filter Simple filters RL filter RC filter RLC filter LC filter
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Shape of a typical Gaussian filter

In electronics and signal processing, a Gaussian filter is a filter whose impulse response is a Gaussian function. Gaussian filters are designed to give no overshoot to a step function input while minimizing the rise and fall time. This behavior is closely connected to the fact that the Gaussian filter has the minimum possible group delay. Mathematically, a Gaussian filter modifies the input signal by convolution with a Gaussian function; this transformation is also known as the Weierstrass transform.

1 Definition
2 Digital implementation
3 Communications Applications
4 See also
5 References

Definition

The one dimensional Gaussian filter has an impulse response given by

$g(x)= \sqrt{\frac{a}{\pi}}\cdot e^{-a \cdot x^2}$

or with the standard deviation as parameter

$g(x) = \frac{1}{\sqrt{2\cdot\pi}\cdot\sigma}\cdot e^{-\frac{x^2}{2\sigma^2}}.$

In two dimensions, it is the product of two such Gaussians, one per direction:

$g(x,y) = \frac{1}{2\pi \sigma^2} e^{-\frac{x^2 + y^2}{2 \sigma^2}}$ ^[1]^[2]

where x is the distance from the origin in the horizontal axis, y is the distance from the origin in the vertical axis, and σ is the standard deviation of the Gaussian distribution.

When applied in two dimensions, this formula produces a surface whose contours are concentric circles with a Gaussian distribution from the center point. Values from this distribution are used to build a convolution matrix which is applied to the original image. Each pixel's new value is set to a weighted average of that pixel's neighborhood. The original pixel's value receives the heaviest weight (having the highest Gaussian value) and neighboring pixels receive smaller weights as their distance to the original pixel increases.

Digital implementation

This article may require cleanup to meet Wikipedia's quality standards. Please improve this article if you can. (November 2008)

Since the Gaussian function decays rapidly, it is often reasonable to truncate the filter window and implement the filter directly for narrow windows; in other cases this introduces significant errors, and one should instead use a different window function; see scale space implementation for details.
The Gaussian kernel is continuous; most commonly the discrete analog is the sampled Gaussian kernel (sampling points from the continuous Gaussian), but the discrete Gaussian kernel is a better analog and has superior characteristics.
Since the Fourier transform of the Gaussian function yields a Gaussian function, again, you can apply the Fast Fourier transform to the signal (preferably divided into overlapping windowed blocks), multiply with a Gaussian function and transform back. This is the standard procedure of applying an arbitrary finite impulse response filter, with the only difference that the Fourier transform of the filter window is explicitly known.
Due to the central limit theorem you can approximate the Gaussian by several runs of a very simple filter like the moving average. The simple moving average corresponds to convolution with the constant B-spline, and e.g. four iterations of a moving average yields a cubic B-spline as filter window which approximates the Gaussian quite well. You can interpret the standard deviation of a filter window as a measure of its size. For standard deviation $σ$ and sample rate $f$ you obtain the frequency $\frac{f}{\sigma}$ which can be considered the cut-off frequency. A simple moving average corresponds to a uniform probability distribution and thus its filter window with size $n$ has standard deviation $\sqrt{\frac{n^2-1}{12}}$ . Thus $m$ moving averages with sizes $n_1,\dots,n_m$ yield a standard deviation of $\sqrt{\frac{n_1^2+\dots+n_m^2-m}{12}}$ . (Note that standard deviations do not sum up, but variances do.)

Communications Applications

Please help improve this article by expanding it. Further information might be found on the talk page. (March 2009)

It is used in GSM since it applies GMSK modulation
the Gaussian filter is also used in GFSK.

References

^ Shapiro, L. G. & Stockman, G. C: "Computer Vision", page 137, 150. Prentence Hall, 2001
^ Mark S. Nixon and Alberto S. Aguado. Feature Extraction and Image Processing. Academic Press, 2008, p. 88.

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Gaussian filter

Contents

Definition

Digital implementation

Communications Applications

See also

References