Appendix 3

Prev TOC

Filters with gaussian magnitude approximation

In pulse communication systems there is a demand for filters whose impulse responses have the following properties:

  • No ringing and overshoot;
  • Symmetry about the time for which the response is a maximum.

A filter that satisfies the above conditions is called a Gaussian filter. The three most common filter types with widely available design tables and curves which approach the ideal Gaussian filter are:

  1. The gaussian magnitude filter;
  2. The maximally flat group delay filter;
  3. The equiriple group-delay filters.

Although, the delay performance of a gaussian filter is worse than the Bessel approximation, a gaussian filter has a better step response. By definition, a gaussian function has the form:


where T is the mean value and s the standard deviation. If the impulse response of a filter is of this form, then the filter will be said to be gaussian. This impulse response has no overshoot. If we denote w02=2/s2 , then the ideal gaussian magnitude shape derived from the Fourier transform of g(t) can be written as:


The frequency w0 is a normalizing frequency and it can be related to the -3dB point as:


The magnitude of the gaussian transfer and the group-delay response are shown below.

Fig.A.3.1: Ideal gaussian transfer and the group delay

When w=w0, the value of the magnitude is e=2.71828 and the relative attenuation is 1Np or 8.68dB. It can be shown that a gaussian magnitude shape is unrealizable but approximations of this transfer can be obtained by using the following series expansion:


An nth-order approximation consists of the first 2n powers in the series. The attenuation of the gaussian function approximated up to n=6 is shown in fig. A.3.2.

Fig.A.3.2: Attenuation performance of gaussian approximations

The approximation of the Gaussian function with a finite number of network elements can be made better by increasing the number of network elements. However, it is possible to approximate the gaussian function up to a certain level. Gaussian-to-6dB and gaussian-to-12 dB approximations will approximate the transfer up to -6dB point and -12dB point respectively. To show the difference between those approximations, the frequency transfer of a 5th order gaussian, gaussian-to-6dB and gaussian-to-12dB transfer is depicted in fig. A.3.3. With these gaussian approximations it is possible to achieve a higher stopband attenuation with the same number of network elements at the expenses of a small decrease of performance in the time domain response and group delay.

Fig.A.3.3: Gaussian, gaussian-to-6dB and gaussian-to-12dB transfer

The corresponding group-delay and step response of the approximations shown above are illustrated in fig.A.3.4. This explains the degradation of the group delay and correspondingly, the degradation of the step response when approximating the gaussian transfer.

Fig.A.3.4: Group delay and step response of gaussian-to-6(12) dB


Featured Video
Mechanical Engineer I for Air Techniques, Inc at Melville, NY
Proposal Support Coordinator for Keystone Aerial Surveys at Philadelphia, PA
Upcoming Events
The Rise of Mechatronics at Dassault Systèmes San Diego 5005 Wateridge Vista Drive San Diego CA - Sep 12, 2017
The Rise of Mechatronics at Buca di Beppo - Pasadena 80 West Green Street Pasadena CA - Sep 13, 2017
The Rise of Mechatronics at Dassault Systèmes Santa Clara 3979 Freedom Circle, Ste 750 Santa Clara CA - Sep 14, 2017
The 30th Annual Integrated Process Excellence Symposium & Training at Wyndham Grand Bonnet Creek Resort Orlando FL - Sep 18 - 20, 2017
Kenesto: 30 day trial
SolidCAM: Cutting Webinar

Internet Business Systems © 2017 Internet Business Systems, Inc.
25 North 14th Steet, Suite 710, San Jose, CA 95112
+1 (408) 882-6554 — Contact Us, or visit our other sites:
AECCafe - Architectural Design and Engineering EDACafe - Electronic Design Automation GISCafe - Geographical Information Services TechJobsCafe - Technical Jobs and Resumes ShareCG - Share Computer Graphic (CG) Animation, 3D Art and 3D Models
  Privacy Policy Advertise