146 THE CALCULATION OF RADAR DETECTION PROBABILITY 



The video signal and noise samples are statistical variables. Their 

 probability density functions are determined in Paragraph 5-7. We denote 

 the video voltage out of the square law detector by v. This voltage is equal 

 to the square of the video envelope r in Equations 5-78 and 5-79 in Para- 

 graph 5-7. Making the transformations y = r^ ^ = a} 11^ and A^ = cr^ in 

 these equations provides the probability density functions of the video 

 voltage for signal plus noise and noise alone. 



Probability density video voltage signal plus noise = 



P.M^) = :^ exp 1^2^ - ^j hiyj'h^Sjm (3-13) 



Probability density video voltage noise alone = 



P.(.)=2^exp[^j. (3-14) 



The video voltage when no signal is present is thus represented by a series 

 of independent samples at intervals of r = 1 jW which are chosen from a 

 statistical population with the probability density of Equation 3-14. When 

 the signal is present, the sample is chosen from a population with the 

 probability density of Equation 3-13. 



An interpretation of these expressions may be given as follows. For a 

 given value of noise power A^ and a given value of signal power S the 

 probability that the video voltage will have a value between v and v -]- dv 

 may be expressed as Ps+N{v)dv. 



Next we examine the effects of integration. We denote the sum signal 

 at the pulse integrator output by u. 



« - ^1 + ^2 + ^3+ ••• + Vn (3-15) 



The components of the sum Vk are independent because they are separated 

 in time by the repetition period while the correlation time of the video 

 voltage is approximately the pulse length r, which is on the order of micro- 

 seconds. 



Probability density functions giving the distribution of the signal plus 

 noise and noise alone of the sum signal out of the integrator are required 

 in order to determine whether a decision threshold will be exceeded. These 

 probability density functions are denoted by 



Probability density integrator output, signal plus noise = Ps-\-n{h) (3-16) 



Probability density integrator output, noise alone = PNi'i)- (3-17) 



The probability density function of the integrator output when a signal 

 is present is quite complicated, and we will not attempt a detailed study of 

 this function here.'' Some calculations are greatly simplified, however, by 



''For such a study see J. I. Marcum, // Statistical Theory of Target Detection by Pulsed Radar, 

 RM-754; and /I Statistical Theory of Target Detection by Pulsed Radar: Mathematical Appendix, 

 RM-753, The RAND Corp., Santa Monica, Calif. 



