Linear analog electronic filters |
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In electronics and signal processing, a **Gaussian filter** is a filter whose impulse response is a Gaussian function (or an approximation to it, since a true Gaussian response is physically unrealizable as it has infinite support). Gaussian filters have the properties of having 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. It is considered the ideal time domain filter, just as the sinc is the ideal frequency domain filter.^{ [1] } These properties are important in areas such as oscilloscopes ^{ [2] } and digital telecommunication systems.^{ [3] }

Mathematically, a Gaussian filter modifies the input signal by convolution with a Gaussian function; this transformation is also known as the Weierstrass transform.

The one-dimensional Gaussian filter has an impulse response given by

and the frequency response is given by the Fourier transform

with the ordinary frequency. These equations can also be expressed with the standard deviation as parameter

and the frequency response is given by

By writing as a function of with the two equations for and as a function of with the two equations for it can be shown that the product of the standard deviation and the standard deviation in the frequency domain is given by

- ,

where the standard deviations are expressed in their physical units, e.g. in the case of time and frequency in seconds and hertz, respectively.

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

^{ [4] }^{ [5] }^{ [6] }

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.

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The Gaussian function is for and would theoretically require an infinite window length. However, since it decays rapidly, it is often reasonable to truncate the filter window and implement the filter directly for narrow windows, in effect by using a simple rectangular window function. In other cases, the truncation may introduce significant errors. Better results can be achieved by instead using a different window function; see scale space implementation for details.

Filtering involves convolution. The filter function is said to be the kernel of an integral transform. The Gaussian kernel is continuous. Most commonly, the discrete equivalent is the sampled Gaussian kernel that is produced by sampling points from the continuous Gaussian. An alternate method is to use the discrete Gaussian kernel ^{ [7] } which has superior characteristics for some purposes. Unlike the sampled Gaussian kernel, the discrete Gaussian kernel is the solution to the discrete diffusion equation.

Since the Fourier transform of the Gaussian function yields a Gaussian function, the signal (preferably after being divided into overlapping windowed blocks) can be transformed with a Fast Fourier transform, multiplied with a Gaussian function and transformed 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, the Gaussian can be approximated by several runs of a very simple filter such as the moving average. The simple moving average corresponds to convolution with the constant B-spline (a rectangular pulse), and, for example, four iterations of a moving average yields a cubic B-spline as filter window which approximates the Gaussian quite well. A moving average is quite cheap to compute, so levels can be cascaded quite easily.

In the discrete case the standard deviations are related by

where the standard deviations are expressed in number of samples and *N* is the total number of samples. Borrowing the terms from statistics, the standard deviation of a filter can be interpreted as a measure of its size. The cut-off frequency of a Gaussian filter might be defined by the standard deviation in the frequency domain yielding

where all quantities are expressed in their physical units. If is measured in samples the cut-off frequency (in physical units) can be calculated with

where is the sample rate. The response value of the Gaussian filter at this cut-off frequency equals exp(-0.5)≈0.607.

However, it is more common to define the cut-off frequency as the half power point: where the filter response is reduced to 0.5 (-3 dB) in the power spectrum, or 1/√2 ≈ 0.707 in the amplitude spectrum (see e.g. Butterworth filter). For an arbitrary cut-off value 1/*c* for the response of the filter the cut-off frequency is given by

For *c*=2 the constant before the standard deviation in the frequency domain in the last equation equals approximately 1.1774, which is half the Full Width at Half Maximum (FWHM) (see Gaussian function). For *c*=√2 this constant equals approximately 0.8326. These values are quite close to 1.

A simple moving average corresponds to a uniform probability distribution and thus its filter width of size has standard deviation . Thus the application of successive moving averages with sizes yield a standard deviation of

(Note that standard deviations do not sum up, but variances do.)

A gaussian kernel requires values, e.g. for a of 3 it needs a kernel of length 17. A running mean filter of 5 points will have a sigma of . Running it three times will give a of 2.42. It remains to be seen where the advantage is over using a gaussian rather than a poor approximation.

When applied in two dimensions, this formula produces a Gaussian surface that has a maximum at the origin, whose contours are concentric circles with the origin as center. A two dimensional convolution matrix is precomputed from the formula and convolved with two dimensional data. Each element in the resultant matrix new value is set to a weighted average of that elements neighborhood. The focal element receives the heaviest weight (having the highest Gaussian value) and neighboring elements receive smaller weights as their distance to the focal element increases. In Image processing, each element in the matrix represents a pixel attribute such as brightness or a color intensity, and the overall effect is called Gaussian blur.

The Gaussian filter is non-causal which means the filter window is symmetric about the origin in the time-domain. This makes the Gaussian filter physically unrealizable. This is usually of no consequence for applications where the filter bandwidth is much larger than the signal. In real-time systems, a delay is incurred because incoming samples need to fill the filter window before the filter can be applied to the signal. While no amount of delay can make a theoretical Gaussian filter causal (because the Gaussian function is non-zero everywhere), the Gaussian function converges to zero so rapidly that a causal approximation can achieve any required tolerance with a modest delay, even to the accuracy of floating point representation.

- GSM since it applies GMSK modulation
- the Gaussian filter is also used in GFSK.
- Canny Edge Detector used in image processing.

In probability theory, a **normal****distribution** is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is

A **wavelet** is a wave-like oscillation with an amplitude that begins at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one recorded by a seismograph or heart monitor. Generally, wavelets are intentionally crafted to have specific properties that make them useful for signal processing.

In probability and statistics, **Student's t-distribution** is any member of a family of continuous probability distributions that arise when estimating the mean of a normally-distributed population in situations where the sample size is small and the population's standard deviation is unknown. It was developed by English statistician William Sealy Gosset under the pseudonym "Student".

In signal processing and statistics, a **window function** is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the interval, usually near a maximum in the middle, and usually tapering away from the middle. Mathematically, when another function or waveform/data-sequence is "multiplied" by a window function, the product is also zero-valued outside the interval: all that is left is the part where they overlap, the "view through the window". Equivalently, and in actual practice, the segment of data within the window is first isolated, and then only that data is multiplied by the window function values. Thus, tapering, not segmentation, is the main purpose of window functions.

In mathematics, a **Gaussian function**, often simply referred to as a **Gaussian**, is a function of the form

In probability theory and statistics, the **Rayleigh distribution** is a continuous probability distribution for nonnegative-valued random variables. It is essentially a chi distribution with two degrees of freedom.

In signal processing, a **finite impulse response** (**FIR**) **filter** is a filter whose impulse response is of *finite* duration, because it settles to zero in finite time. This is in contrast to infinite impulse response (IIR) filters, which may have internal feedback and may continue to respond indefinitely.

In mathematics and in signal processing, the **Hilbert transform** is a specific linear operator that takes a function, *u*(*t*) of a real variable and produces another function of a real variable H(*u*)(*t*). This linear operator is given by convolution with the function . The Hilbert transform has a particularly simple representation in the frequency domain: It imparts a phase shift of ±90° to every frequency component of a function, the sign of the shift depending on the sign of the frequency. The Hilbert transform is important in signal processing, where it is a component of the analytic representation of a real-valued signal *u*(*t*). The Hilbert transform was first introduced by David Hilbert in this setting, to solve a special case of the Riemann–Hilbert problem for analytic functions.

The **Voigt profile** is a probability distribution given by a convolution of a Cauchy-Lorentz distribution and a Gaussian distribution. It is often used in analyzing data from spectroscopy or diffraction.

**Gaussian noise**, named after Carl Friedrich Gauss, is statistical noise having a probability density function (PDF) equal to that of the normal distribution, which is also known as the Gaussian distribution. In other words, the values that the noise can take on are Gaussian-distributed.

In image processing, a **Gabor filter**, named after Dennis Gabor, is a linear filter used for texture analysis, which essentially means that it analyzes whether there is any specific frequency content in the image in specific directions in a localized region around the point or region of analysis. Frequency and orientation representations of Gabor filters are claimed by many contemporary vision scientists to be similar to those of the human visual system. They have been found to be particularly appropriate for texture representation and discrimination. In the spatial domain, a 2D Gabor filter is a Gaussian kernel function modulated by a sinusoidal plane wave.

In image processing, a **Gaussian blur** is the result of blurring an image by a Gaussian function.

In probability theory, the **Rice distribution** or **Rician distribution** is the probability distribution of the magnitude of a circularly-symmetric bivariate normal random variable, possibly with non-zero mean (noncentral). It was named after Stephen O. Rice.

In probability theory, calculation of the **sum of normally distributed random variables** is an instance of the arithmetic of random variables, which can be quite complex based on the probability distributions of the random variables involved and their relationships.

In imaging science, **difference of Gaussians** (**DoG**) is a feature enhancement algorithm that involves the subtraction of one Gaussian blurred version of an original image from another, less blurred version of the original. In the simple case of grayscale images, the blurred images are obtained by convolving the original grayscale images with Gaussian kernels having differing width. Blurring an image using a Gaussian kernel suppresses only high-frequency spatial information. Subtracting one image from the other preserves spatial information that lies between the range of frequencies that are preserved in the two blurred images. Thus, the DoG is a spatial band-pass filter that attenuates frequencies in the original grayscale image that are far from the band center.

The **Gabor transform**, named after Dennis Gabor, is a special case of the short-time Fourier transform. It is used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. The function to be transformed is first multiplied by a Gaussian function, which can be regarded as a window function, and the resulting function is then transformed with a Fourier transform to derive the time-frequency analysis. The window function means that the signal near the time being analyzed will have higher weight. The Gabor transform of a signal x(t) is defined by this formula:

In probability theory, the **inverse Gaussian distribution** is a two-parameter family of continuous probability distributions with support on (0,∞).

In the areas of computer vision, image analysis and signal processing, the notion of scale-space representation is used for processing measurement data at multiple scales, and specifically enhance or suppress image features over different ranges of scale. A special type of scale-space representation is provided by the Gaussian scale space, where the image data in *N* dimensions is subjected to smoothing by Gaussian convolution. Most of the theory for Gaussian scale space deals with continuous images, whereas one when implementing this theory will have to face the fact that most measurement data are discrete. Hence, the theoretical problem arises concerning how to discretize the continuous theory while either preserving or well approximating the desirable theoretical properties that lead to the choice of the Gaussian kernel. This article describes basic appproaches for this that have been developed in the literature.

In mathematics, the **Weierstrass transform** of a function *f* : **R** → **R**, named after Karl Weierstrass, is a "smoothed" version of *f*(*x*) obtained by averaging the values of f, weighted with a Gaussian centered at *x*.

In probability theory, the **rectified Gaussian distribution** is a modification of the Gaussian distribution when its negative elements are reset to 0. It is essentially a mixture of a discrete distribution and a continuous distribution as a result of censoring.

- ↑ Filtering in the Time and Frequency Domains by Herman J. Blinchikoff, Anatol I. Zverev
- ↑ http://www.radiomuseum.org/forumdata/users/4767/file/Tektronix_VerticalAmplifierCircuits_Part1.pdf
- ↑ https://kh6htv.files.wordpress.com/2015/11/an-07a-risetime-filters.pdf
- ↑ R.A. Haddad and A.N. Akansu, "A Class of Fast Gaussian Binomial Filters for Speech and Image Processing," IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 39, pp 723-727, March 1991.
- ↑ 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. - ↑ Lindeberg, T., "Scale-space for discrete signals," PAMI(12), No. 3, March 1990, pp. 234-254.

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