JPL's Wireless Communication Reference Website

Chapter: Cellular Telephone Networks
Section: Outage probability


Outage Probability in Rayleigh-Fading Channels

This pages discusses a Pascal program that computes the capture and outage probability in cellular mobile radio channel with

Definition of some constants

First of all we define the following constants:
Var
  HER          : array[1 .. 2, 1 .. 20] of real;
  N            : Integer  
  rootpiinv, ln10: real

BEGIN 
  rootpiinv   :=  1/(sqrt(3.141592)); 
  N           :=  20;
  ln10        := ln(10);

HER[1, 1]   :=  0.2453407083;
HER[1, 2]   :=  0.7374737285;
HER[1, 3]   :=  1.2340762153;
HER[1, 4]   :=  1.7385377121;
HER[1, 5]   :=  2.25497400203;
HER[1, 6]   :=  2.78880605844;
HER[1, 7]   :=  3.34785456736;
HER[1, 8]   :=  3.94476404017;
HER[1, 9]   :=  4.603682449510;
HER[1,10]   :=  5.387480890013;


HER[2, 1]   :=  4.6224366960E-1;
HER[2, 2]   :=  2.8667550536E-1;
HER[2, 3]   :=  1.0901720602E-1;
HER[2, 4]   :=  2.4810520887E-2;
HER[2, 5]   :=  3.2437733422E-3;
HER[2, 6]   :=  2.2833863601E-4;
HER[2, 7]   :=  7.8025564785E-6;
HER[2, 8]   :=  1.0860693707E-7;
HER[2, 9]   :=  4.3993409922E-10;
HER[2,10]   :=  2.2293936455E-13;

For I := 1 to 10 do
BEGIN
    HER[1,10+I]  := -1 * HER[1,I];
    HER[2,10+I]  :=  HER[2,I];
END;

END
Here the variables HER[1,*] and HER[2,*] are used to efficiently compute integrals from minus infinity to plus infinity over f(x) exp{-x2}, with f(x) some function which behaves sufficiently "nice". HER[1,*] are sample points (we suggest to take take 20 samples), and HER[2,*] are weight factors. The method we use is called the Hermitian quadrature method; HER[1,*] are zeros of a Hermite polynomial. An example of its use is given below.

Laplace Images

In the evaluation of outage probabilities, one needs to find the Laplace image of the pdf of the received interference power. This can be computed as follows:

FUNCTION SU(VAR s :REAL; VAR sigma : REAL) : REAL;
{-----------------------------------------------------}
{     Computation of Laplace image                    }
{                                                     }
{    s:      variable of Laplace Image Function       }
{    sigma: amount of shadowing                       }  
{-----------------------------------------------------}
VAR
  m       : REAL;
  aux     : REAL;
  sqrt2sigma : real;

BEGIN
  SQRT2SIGMA := sigma * SQRT(2);
  aux := 0;
  FOR I := 1 to 20 DO
    BEGIN
      aux := aux + her[2,I] / (1 + s * exp(SQRT2SIGMA * HER[1,I]));
    END;
    SU   :=  aux * rootpiinv;
END;  { Function SU}

Outage Probability

We use the above image function, plus

FUNCTION  Prob_outage(var AMCI: REAL; 
                   var sigma :real) : REAL;
{-----------------------------------------------------}
{          CALCULATES OUTAGE            }
{-----------------------------------------------------}
var
    integrand1   : REAL;
    integral1    : REAL;
    s           : REAL;
    I            : INTEGER;
    aux          : REAL;
  
BEGIN
       Integral1 := 0;
       FOR I := 1 TO 20 DO
         BEGIN
            s           := z / AMCI * exp(-SQRT(2) * sigma * HER[1,I]);
            integrand1  :=   HER[2,I]  * exp (6* ln (  SU(s,sigma)   );
                     {take sixth power because of 6 co-channel interferers)
            Integral1   := integral1   +  integrand1;
         END;
          integral1 := integral1 * rootpiinv; 
          Prob_outage := 1 - integral1;
   END; {FUNCTION}

Hints and Warnings

Exercise

Write a Pascal or C program to compute outage probabilities.



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