3-3] 



DETECTION PROBABILITY FOR A PULSE RADAR 



153 



P^{ 



'' - wS 



exp ( - 



■2n{N+S) 



xy2)dx. 



2ylnN{N+2S) 



In Fig. 3-6, Pd{S) is plotted as a function of ^. For small values of S, PdiS) 

 is very small, while for large S, Pd(S) is approximately unity. A transition 



S=b/2n -N 



SIGNAL POWER 



Fig. 3-6 Probability of Detection as a Function of the Signal Amplitude. 



occurs when S = b jln — N. The width of the transition region is inversely 

 proportional to n. When n is large, then, it is reasonable to approximate 

 Pd{S) as zero for S < b jln — A^and unity for S larger than this transition 

 point. We shall subsequently indicate that this is a fair approximation even 

 when n = \. With this approximation, the integral in Equation 3-26 is 

 easily evaluated. 



- - P - - ibllnN - 1) 



P. = (1/6')/ exp (- SIS)ds = exp ^,„ '■ (3-28) 



Jb/2n-N o/l\ 



It is convenient for. some developments to work directly with the range 

 to the target instead of the average signal to noise ratio. The expression 

 in Equation 3-10 giving the signal to noise ratio as the fourth power of the 

 ratio of an ideal range to the actual range provides this relation: 



S/N = (Ro/Ry (3-29) 



In addition, a factor K(n,r]) is defined 



K = (b/2nN - 1). (3-30) 



With this notation, the average detection probability can be represented in 

 the following very simple form. 



Pd = ^-if(«/-Ro)\ (3_31) 



The K factor can be evaluated from the data in Fig. 3-3 giving the relation 

 between the relative bias, the number of pulses integrated, and the false- 

 alarm number. The results of such an evaluation are shown in Fig. 3-7, 

 where the K factor is plotted versus the number of pulses integrated for 

 representative values of the false-alarm number. 



