3-3] DETECTION PROBABILITY FOR A PULSE RADAR 147 



the adoption of a suitable approximation to P s+n{u). Such an approxima- 

 tion can be based on the assumption that n(S /N) ^ I. In this case, the 

 distribution of u is very nearly normal. This is so because when S /N ^ 1, 

 V itself tends to be normally distributed (see Equation 5-80), while with 

 n y> I, the distribution of the sum u tends to normality by the central limit 

 theorem.^ 



The mean and standard deviation of the video voltage v can be found 

 from Equation 5-81 or Fig. 5-12. 



The mean and standard deviation of the sum signal u will be larger by «, the 

 number of components in the sum, and by the square root of w, respectively. 



Integrator output, d-c voltage = « = 2n{N + S) (3-20) 



Integrator output, rms a-c voltage = o-„ = 2^1nN(N -i- 2S). (3-21) 



The probability density function of the integrator output for noise alone 

 can be established by standard statistical procedures to have the following 

 form. 



''"^"^ = Wuhlji^)"' ^~""'- P-22) 



For the statistics-minded, we may note that each variable Vk is given by the 

 sum of squares of two independent normal variables Xk andjy/c- Thus, u will 

 be the sum of the squares of 2n normal variates, and Pn{u) will be the 

 probability density function of a chi-squared distribution with 2n degrees 

 of freedom.^ 



The Decision Element. The threshold type of decision element 

 assumed for this system corresponds closely to the detection operations 

 which would be performed by an automatic system such as might be 

 employed in the terminal seeker of a guided missile. In many important 

 cases, however, the human operator is the decision element. It is postulated 

 that the human operator does something very similar to the threshold type 

 of decision element. The functions of the human operator, though, would 

 probably deviate somewhat from those performed by an ideal decision 

 threshold. For instance, the threshold of human operators appears to vary 



8J. L. Lawson and G. E. Uhlenbeck, Threshold Signals, pp. 46-52, McGraw-Hill Book Co., 

 Inc., New York, 1950. 



^See P. G. Hoel, Introduction to Mathematical Statistics, pp. 134-136, John Wiley & Sons, Inc., 

 New York. 



