.NODE

Appendix E. erf( )

Appendix E erf()

NOTE: In the mathematical literature you will see many tabulations of the function erf ( ), sometimes with definitions different from what is presented here. Although the definitions may be transformed from one to another fairly easily, it is sometimes confusing when you need one function and have only a table for another. Table E.1 shows how to convert tabulated data (or software functions) provided under an alternate definition into my format. The functions defined here as erf ( ) and erfc ( ) may be computed from any of the other forms:

Equation E.1

graphics/xeequ01.gif

Equation E.2

graphics/xeequ02.gif

Equation E.3

graphics/xeequ03.gif

Equation E.4

graphics/xeequ04.gif

Equation E.5

graphics/xeequ05.gif

Equation E.6

graphics/xeequ06.gif

Equation E.7

graphics/xeequ07.gif

Equation E.8

graphics/xeequ08.gif

 

Table E.1. Alternate Definitions for the Error Function

Function name

Definition

Range for a > 0

Comment

My error function [136] , [137] erf ( a )

graphics/748equ01.gif

[0,1]

I like the minimal simplicity of this definition.

My complementary error function erfc ( a )

graphics/748equ02.gif

[0,1]

Used in probability analysis and communication theory to emphasize connection with Gaussian probability density function.

Error function [138] (alt. definition) erf 2 ( a )

graphics/748equ03.gif

[ ½,1]

Used in probability analysis and communication theory to emphasize connection with Gaussian probability density function.

Yet another error function [139] erf 3 ( a )

graphics/748equ04.gif

[0, ½]

Variant; same as ( erf 2 ( a ) “ ½).

Complementary error function [140] , [141] Q ( a )

graphics/748equ05.gif

[ ½,0]

Variant; same as (1 “ erf 2 ( a )).

MathCad built-in function Pnorm ( a , m , s )

graphics/748equ06.gif

[ ½,1]

For m = 0 and s = 1, same as erf 2 ( x ); NOTE: In versions of MathCad earlier than 2001 do not use the built-in function erf ( ), as it is a totally unrelated function. Use Pnorm instead.

[136] John M. Wozencraft, Irwin Mark Jacobs, Principles of Communication Engineering , John Wiley & Sons, 1965, ISBN 0 471 96240 6

[137] John A. Aseltine, Transform Method in Linear System Analysis , McGraw-Hill, 1958, U.S. Lib. Congress cat. no. 58-8038

[138] Harry Van Trees, Detection, Estimation, and Modulation Theory: Part I , John Wiley and Sons, 1968 ISBN 471 89955 0

[139] Athanasios Papoulis, Probability, Random Variables, and Stochastic Processes , McGraw-Hill, 1965, ISBN 07-048448-1

[140] John M. Wozencraft, Irwin Mark Jacobs, Principles of Communication Engineering , John Wiley & Sons, 1965, ISBN 0 471 96240 6

[141] Bernard Sklar, Digital Communications , Prentice Hall, 1988, ISBN 0-13-211939-0

Fundamentals

Transmission Line Parameters

Performance Regions

Frequency-Domain Modeling

Pcb (printed-circuit board) Traces

Differential Signaling

Generic Building-Cabling Standards

100-Ohm Balanced Twisted-Pair Cabling

150-Ohm STP-A Cabling

Coaxial Cabling

Fiber-Optic Cabling

Clock Distribution

Time-Domain Simulation Tools and Methods

Points to Remember

Appendix A. Building a Signal Integrity Department

Appendix B. Calculation of Loss Slope

Appendix C. Two-Port Analysis

Appendix D. Accuracy of Pi Model

Appendix E. erf( )

Notes

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High-Speed Signal Propagation[c] Advanced Black Magic
High-Speed Signal Propagation[c] Advanced Black Magic
ISBN: 013084408X
EAN: N/A
Year: 2005
Pages: 163
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