3-5] DETECTION PROBABILITY FOR A PULSED DOPPLER RADAR 165 



It is not completely correct to substitute the average values of the 

 detection probability in each channel into Equation 2-48 when considering 

 a fluctuating target because the signal will fluctuate similarly in the two 

 channels. Instead, we should use the procedure previously used in Equation 

 3-28 for finding the approximate average value of a single channel to deter- 

 mine the average of the two-channel expression given in Equation 3-48. 



Equation 3-28 was derived on the basis of an approximation to the curve 

 in Fig. 3-6 that Pa was zero out to the value 6" = KN and unity for higher 

 values of S. With this approximation, Pi and P2 become 



^'^""^ X.SyKN/a"- (3-49) 



prn .^2vi~o, ^<AW/(i ~aY 



Adopting these approximate expressions, it is apparent that the product 

 P1P2 is equal to Pi when a < 1 /2, and to P2 when a > 1 /2. This observation 

 materially simplifies Equation 3-48, since only one term is retained: 



The average value of Pd2 in each case will be of the exponential form first 

 given in Equation 3-31: 



- _ exp - K{R/R,)V/i.\ - ar-\ = {P,iy"'-'\ cc<h 



^''- ~ exp - K{R/R,Y{\/a^-) = (Prf)l/«^ a > f ^^'^'^ 



The minimum value is attained when a = \, and the received pulse is 

 divided equally between the two channels: 



min Prf2 - P/. (3-52) 



Of more interest is the average value, which can be used as a smooth 

 replacement for oscillatory curves similar to that in Fig. 3-14 in many cases. 

 The function in Equation 3-51 could be integrated by numerical means. 

 It is more expedient, though, and a good approximation to simply use the 

 average of 1 /a^, which = 2 in Equation 3-51. Thus, 



ave P,2 - P/. (3-53) 



On the average, then, the effect of straddling can be interpreted as a 3-db 

 loss in signal-to-noise ratio. _ 



It should also be noted that Pd2 is of the same general form as Pa itself. 

 Thus, if the effects of eclipsing can be neglected, the methods developed 

 in Paragraph 3-4 for determining the cumulative probability of detection 

 and the normalized curves in Fig. 3-9 are quite applicable. 



