•−•  ••−•    −•−•  •−  ••−•  •
RF Cafe Morse Code >Hear It<

Job Board

About RF Cafe™

Sitemap

RF Cafe Software

RF Cascade Worbook
 RF Cascade Workbook 2005 - RF Cafe
Calculator Workbook
RF Workbench
Smith Chart™ for Visio
Smith Chart™ for Excel
RF & EE Symbols Word
RF Stencils for Visio

Your RF Cafe
Progenitor & Webmaster

Click here to read about RF CafeKirt Blattenberger
BSEE
KB3UON
EIEIO

Carpe Diem!
(Seize the Day!)

5th MOB:
My USAF radar shop

Airplanes and Rockets:
My personal hobby website

Equine Kingdom:
My daughter Sally's horse riding website

Error Function erf(x) & Complimentary Error Function erfc(x)

Note: The error function and complimentary function used in communications are not exactly the same as the ones typically used in statistics. The relationship between the two is given at the bottom of the page.

In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape which occurs in probability, statistics, materials science, and partial differential equations.

"In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape which occurs in probability, statistics, materials science, and partial differential equations." - Wikipedia

The Gaussian probability density function with mean = 0 and variance =1 is

   Probability density function equation - RF Cafe

The error function erf(x) is defined as:

   Error function equation for communications - RF Cafe

 

Note that erf(0) = 0.5, and that erf(∞)=1.

 

The complimentary error function erfc(x) is defined as:

   Complimentary error function equation for communications - RF Cafe

 

The following graph illustrates the region of the normal curve that is being integrated.

   Complimentary error function graph - RF Cafe

 

For large values of x (>3), the complimentary error function can be approximated by:

   Approximation complimentary error function equation for communications - RF Cafe

 

The error in the approximation is about -2% for x=3, and -1% for x=4, and gets progressively better with larger values of x.

Approximations

RF Cafe visitor Ilya L. provided an approximation for the error function and complimentary error function that was published by Sergei Winitzki titled, "A handy approximation for the error function and its inverse." February 6, 2008.

Here are the main results:

Error function approximation:

 

      ,Error function approximation - RF Cafe where  RF Cafe - Error function approximation constant "a"

 

Complimentary error function:

      Complimentary error function approximation

 

 

NOTE: I used to have an alternative approximation formula for the complimentary error function for large values of x, but decided to remove it since the source for it is not generally available to the public. It can be found as equation #13, on page 641, of IEEE Transactions on Communications volume COM-27, No. 3, dated March 1979. A subscription to the IEEE service is required to access the article.

Try Using SEARCH to Find What You Need.  >10,000 Pages Indexed on RF Cafe !

Copyright 1996 - 2016
Webmaster:  Kirt Blattenberger, BSEE - KB3UON
Family Websites:  Airplanes and Rockets | Equine Kingdom

All trademarks, copyrights, patents, and other rights of ownership to images
and text used on the RF Cafe website are hereby acknowledged.