3-4] EFFECT OF SCANNING ON DETECTION PROBABILITY 157 



R, = R^- (k - l)AR. (3-36) 



We shall limit our discussion to a consideration of fluctuating targets 

 which can be handled rather generally thanks to the simplicity of the 

 expression for the average probability of detection (Equation 3-31). At 

 each look, the average probability of detection is given by Equation 3-37 

 below. 



PdiRi.) = ^-^'(«^/«o)\ (3.37) 



The cumulative probability that a target is detected at the range Rk or 

 before is denoted by Pc(Rk) and is given by the well-known expression for 

 the probability of at least one success in a sequence of k trials: 



Pc{Rk) = 1 - n[l - P.iRd]. (3-38) 



An additional refinement needs to be introduced. Equation 3-39 implicitly 

 assumes that the last look occurred at Rk-. Actually, the last look may occur 

 anywhere between Rk and Rk-i = Rk -\- AR with equal probability. That 

 is, there will be a random phase between the antenna scan and the relative 

 motion of the target. To take this effect into account, an average value of 

 the cumulative probability of detection must be computed: 



\rJo 



Fortunately, the calculations shown in Equations 3-38 and 3-39 do not 

 have to be carried out every time the cumulative probability of detection is 

 desired. With properly normalized variables, a universally applicable series 

 of detection curves can be derived. 



In order to do this, a normalized range denoted by p is defined: 



p = K''*(R/Ro). (3-40) 



The normalized range decrement is defined similarly: 



Ap = K''\AR/Ro). (3-41) 



With these definitions, the average single-glimpse probability of detection 

 takes the following form: 



Pa = e-"' (3-42) 



With this form for the single-glimpse probability of detection, universal 

 curves of the average cumulative probability of detection have been 

 calculated on the basis of Equations 3-38 and 3-39. These curves are plotted 

 in Fig. 3-9. 



From the appearance of these curves, it would seem desirable to make 

 the normalized decrement Ap as small as possible in order to obtain the 



