154 



THE CALCULATION OF RADAR DETECTION PROBABILITY 



2 5 10 20 50 100 200 500 1000 



NO. OF PULSES INTEGRATED,n 



Fig. 3-7 The Factor K{n,r]) as a Function of n. 



It is of interest to note that we can infer from the slopes of the curves in 

 this figure the trade-off of signal-to-noise ratio with n, the number of pulses 

 integrated. From Fig. 3-7, a typical slope is about — 6 db for a factor of 10 

 in the number of pulses integrated. This is equivalent to a variation of K 

 with n of the following form: 



A^ ^ «-«-6. (3-32) 



Because the average probability of detection is a function of the ratio 

 K/(S/N), a variation in K is equivalent to an inverse variation in the 

 average signal to noise ratio. The trade-off between signal to noise ratio 

 and n, then, is simply 



S/N^ 



(3-33) 



Because of the rather gross approximation which had to be made to 

 obtain Equation 3-31, there is a question about its range of validity. A 

 reasonable validation of this equation is obtained by comparing it with some 

 examples of exact calculations and observing the error. This is done in 

 Fig. 3-8 where the average detection probability as given by Equation 3-31 

 has been plotted as a function of the normalized range K^'*{R/Ro). It is 

 approximately a straight line on the normal probability coordinates used 

 in that figure. Also plotted in Fig. 3-8 are the exact values of Pd for n = 1. 

 This is the case when the approximations made introduce the greatest error. 



