Gaussian filter

In electronics and signal processing, mainly in digital 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 would have infinite impulse response). 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. A Gaussian filter will have the best combination of suppression of high frequencies while also minimizing spatial spread, being the critical point of the uncertainty principle. These properties are important in areas such as oscilloscopes and digital telecommunication systems. 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: 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. 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.