144 THE CALCULATION OF RADAR DETECTION PROBABILITY 



The decision element in the radar system is assumed to be simply a thresh- 

 old or bias. When the integrated video voltage exceeds this threshold, 

 detection is said to have occurred. When this voltage fails to exceed the 

 threshold, no detection occurs. The bias may be exceeded for one of two 

 reasons. (1) The integrated video signal-plus-noise may exceed the bias; 

 in this case, "target detection" takes place. (2) The integrated noise alone 

 may exceed the bias; in this case, a"false alarm" takes place. Fig. 5-16 shows 

 how a decision threshold or bias is used to distinguish between the dis- 

 tribution of signal plus noise and noise alone. The selection of the threshold 

 level thus will represent a compromise between the desire for maximum 

 sensitivity to integrated signal plus noise and the system penalties incurred 

 by false alarms. 



Method of Analysis. Using the radar system model already described 

 and the mathematical theory presented in Chapter 5, we will trace the 

 progress of noise and signal plus noise through the elements of the receiver. 

 The objectives of this analysis are to derive the target detection and false-alarm 

 probabilities as functions of S jN ratio, threshold level, and the amount of 

 integration. 



The analysis will be performed for both constant and fluctuating radar 

 targets to determine the. probability of detection on a single scan.^ 



Finally, the concept of single-scan probability of detection will be 

 employed in Paragraph 3-4 to develop the multiple-scan probability of 

 detection for a moving target. This quantity — also called the cmnulative 

 probability of detection — is the one most directly related to system perform- 

 ance in the tactical-use environment.^ For example, the detection ranges 

 specified for the examples in Chapter 2 were expressed in terms of the 

 cumulative probability of detection. 



From the standpoint of clear exposition, it is rather unfortunate that 

 a true understanding of the radar detection problem is wrapped in com- 

 plexities of statistical theory which do not convey to the practicing designer 

 a real feel for the problem. The author has attempted to alleviate this 

 problem by confining some of the more detailed mathematical derivation 

 to Chapter 5; the analysis that follows herein applies some of the results 

 of these derivations as they pertain to the assumed model. 



Signal Analysis. As previously mentioned, the input to the square- 

 law detector consists of noise with a power spectrum equal to the power 

 transfer function of the amplifier and — when a signal is present — a signal 



''This quantity is often called the "blip-scan" ratio or the "single glimpse" detection prob- 

 ability. 



5See Paragraph 2-12 and Fig. 2-19. 



