JPL's Wireless Communication Reference Website Chapter: Cellular Telephone Networks Section: Outage probability

Outage Probability in Rayleigh-Fading Channels

• Path loss
• Shadowing
• Rayleigh fading

Definition of some 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

Laplace Images

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

• the area-mean C/I ratio (AMCI). This can be computed from the reuse pattern and the path loss law. In the case of "40 log d",

    AMCI := SQR(3 * C);
  

  with C the cluster size.
• the receiver threshold z, i.e., the minimum required C/I ratio

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

• Here we use the standard deviation of the shadowing expressed in natural units. That is, for 6 dB of shadowing, one must define

  sigma         := 6 * 0.23026;
  

• the instruction

  Prob_outage := 1 - integral1;
  

  subtracts two numbers of about equal size, so it is likely to cause large errors. A way to avoid this can be as follows :

  BEGIN
         Integral1 := 0;
         FOR I := 1 TO 20 DO
         BEGIN
              s           := z / AMCI * exp(-SQRT(2) * sigma * HER[1,I]);
              integrand1  := HER[2,I]  * ( 1 - exp (6* ln (  SU(s,sigma) )   );
              Integral1   := integral1   +  integrand1;
         END;
         integral1 := integral1 * rootpiinv; 
  END; {FUNCTION}
  

  In this case, any error in the numerical integration method occurs in both terms, so they cancel.

Exercise

JPL's Wireless Communication Reference Website © 1993, 1995.