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:

(A.3.1)

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:

(A.3.2)

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

(A.3.3)

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:

(A.3.4)

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


EMA:

Featured Video
Editorial
Peggy AycinenaWhat Would Joe Do?
by Peggy Aycinena
Retail Therapy: Jump starting Black Friday
Peggy AycinenaIP Showcase
by Peggy Aycinena
REUSE 2016: Addressing the Four Freedoms
More Editorial  
Jobs
Principal Circuit Design Engineer for Rambus at Sunnyvale, CA
AE-APPS SUPPORT/TMM for EDA Careers at San Jose-SOCAL-AZ, CA
FAE FIELD APPLICATIONS SAN DIEGO for EDA Careers at San Diego, CA
Manager, Field Applications Engineering for Real Intent at Sunnyvale, CA
Development Engineer-WEB SKILLS +++ for EDA Careers at North Valley, CA
ACCOUNT MANAGER MUNICH GERMANY EU for EDA Careers at MUNICH, Germany
Upcoming Events
2016 IEEE International Electron Devices Meeting at Hilton San Francisco Union Square 333 O’Farrell Street San Francisco CA - Dec 3 - 7, 2016
Zuken Innovation World 2017, April 24 - 26, 2017, Hilton Head Marriott Resort & Spa in Hilton Head Island, SC at Hilton Head Marriott Resort & Spa Hilton Head Island NC - Apr 24 - 26, 2017
CST Webinar Series



Internet Business Systems © 2016 Internet Business Systems, Inc.
595 Millich Dr., Suite 216, Campbell, CA 95008
+1 (408)-337-6870 — Contact Us, or visit our other sites:
AECCafe - Architectural Design and Engineering TechJobsCafe - Technical Jobs and Resumes GISCafe - Geographical Information Services  MCADCafe - Mechanical Design and Engineering ShareCG - Share Computer Graphic (CG) Animation, 3D Art and 3D Models
  Privacy Policy