152 THE CALCULATION OF RADAR DETECTION PROBABILITY 



signal is constant. The energy reflected from an aircraft in flight is not 

 generally constant. Such an object is a complex reflector of electromagnetic 

 waves. As it moves in flight, it vibrates and turns relative to the radar 

 system, and various parts of the aircraft reflect signals with more or less 

 random amplitudes and with slightly diflPerent doppler shifts. As a result, 

 the signal reflected from the aircraft fluctuates and can be represented as a 

 noise process. ^^ The power in the signal will be distributed similarly to the 

 square of the envelope of narrow-band noise with a probability density 

 function of the same form as that in Equation 3-14. This distribution is 

 often called a Rayleigh distribution. The probability density function of 

 the signal power S will thus have the following form : 



Probability density of signal power from fluctuating target = 



1 



P(S) = -^e-^'s. (3-25) 



The factor S in this expression represents the average signal power. 



Because the rate of turn of aircraft is relatively slow, the spectrum of the 

 fluctuations in S is normally less than about 3 cps in width,^^ and S is 

 reasonably well correlated over intervals of less than about 50 msec. We 

 shall suppose that the observation time is less than this and assume that 

 S is constant during a look, but independent from look to look. 



With the signal-to-noise ratio fluctuating from scan to scan, the proba- 

 bility of detection will also fluctuate, and an average value of Pd must be 

 calculated by weighting the various values of Pd by their probability of 

 occurrence. Thus the average probability of det ction is of the form 



Average probability of detection for a fluctuating target = 



Pd= Pd(u,S)PiS)^S. (3-26) 



This integral can be evaluated approximately by making use of the 

 previously noted Gaussian approximation to the distribution of integrated 

 video signal plus noise. Using the values given in Equations 3-20 and 3-21 

 for the mean and variance of the integrated video, the approximate proba- 

 bility of detection is 



or, with an appropriate change of variable, (3-27) 



l^A detailed discussion of the fluctuations in apparent size of aircraft is given in Paragraph 

 4-8. 



i^See Fig. 4-24 for an example. 



