Gaussian filter

Gaussian filters are frequency filters, which have no overshoot in the step response and having the same maximum flank steepness in the transition region. A special feature of this filter has both the transfer function and the impulse response of the path of a Gaussian bell-shaped curve, as shown in the figure on the right, which is also the name of this type of filter is derived.

Application of this filter are in digital modulation method and in the field of image processing.

Transfer function

The magnitude of the transfer function is given in the Gaussian filter by

The constant

The impulse response of the Gaussian filter is

Applications

Digital Signal Processing

Gaussian filters have a constant group delay in the stop - and passband and no overshoot in the step response. Application of this filter is primarily for pulse shaping with applications in digital signal processing.

The pulse shaping takes place at the digital modulation methods such as Gaussian Minimum Shift Keying ( GMSK ), since thereby the individual, generally rectangular transmission symbols can be converted into pulses of the Gaussian bell-shaped curve with a smaller bandwidth than the original rectangular transmission symbols. Thus a higher spectral efficiency of the modulation method is connected.

In mobile radio systems such as GSM Gaussian filters are used in the GMSK modulation on the radio interface for the transmission of digital voice and control information.

Other applications are in modulation techniques, such as Chirp Spread Spectrum, wherein the discontinuous frequency change is smoothed using a Gaussian filter in successive chirps.

Image processing

In image processing, Gaussian filters are used for smoothing or blurring of the image content. It can thus be reduced picture noise: Smaller structures are lost, coarser structures remain, however obtained. Spectral smoothing is a low-pass filter the same.

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

For practical realizations in the context of digital image processing, the discrete impulse response is often used as a two dimensional matrix.

Alternatively, in the literature in the description of the Gaussian filter is used instead of the constant equal to the variance in the term of the impulse response - which expresses the mathematical proximity of the impulse response of a Gaussian filter to the function of the normal distribution. Wherein one dimension is the impulse response of:

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

By utilizing the separability, the computing time can be greatly reduced.

Example of a 3 × 3 filter: