3-3] 



DETECTION PROBABILITY FOR A PULSE RADAR 



155 



2= 0.10 



0.02 



0.2 



1.6 



0.6 0.8 1.0 ^ 1.2 

 NORMALIZED RANGE, K4 (R/Rg) 



Fig. 3-8 Average Detection Probability as a Function of Normalized Range. 



When n = I, the probability density function departs from normality to 

 the greatest degree, and the width of the transition region in Fig. 3-8 is 

 largest. The approximation will also be poorest when the false alarm 

 number is small. Curves are plotted in Fig. 3-8 for the two values r{ = 10^ 

 and 17 = 10*. These are considered small values for this parameter. With 

 the false alarm number rj equal to 10^, the approximation is already quite 

 good for values of Pd greater than 20 per cent. For larger values of r? or n, 

 the approximation becomes very good.^*. 



It is of interest to compare the average detection probability on a 

 fluctuating target with that obtained on a constant target of the same size. 

 The detection probability on a nonfluctuating target was previously 

 determined in the case of Ro = 20.4 n.mi., n = 10, and rj = 10^. For the 

 fluctuating case, we first determine the value of K from Fig. 3-7 for these 

 parameters. This value is found to be 4.5 db or a factor of 2.82. The fourth 

 root of K is then 1.3. In Fig. 3-8, the normalized range for Pd = 0.9 is 

 found to be 0.58. The actual range giving an average 90 per cent probability 

 of detection is thus 



i^More detailed development of these ideas can be found in P. Swerling, "Probability of 

 Detection for Fluctuating Targets," Research Memorandum 1217, The RAND Corporation, 

 Santa Monica, Calif. (17 March 1954). 



