156 THE CALCULATION OF RADAR DETECTION PROBABILITY 



/^9o% = 0.58 X 20.4/1.3 = 9.1 n.mi. (3-34) 



This range is substantially less than the range (14.7 n.mi.) which gives 

 Pd = 0.9 in the case of a nonfluctuating target. From Fig. 3-8, the nor- 

 malized range giving Pd = 0.1 is 1.23, which yields an actual range corre- 

 sponding to an average detection probability of 10 per cent of 19.3 n.mi. 

 This value corresponds to 17.5 n.mi. in the nonfluctuating case. Thus, 

 while the fluctuations degrade the performance at high probabilities, they 

 enhance the performance on small and distant targets. The complete curve 

 of detection probability versus range in this case is plotted in Fig. 3-5. 



3-4 THE EFFECT OF SCANNING AND THE CUMULATIVE 

 PROBABILITY OF DETECTION 



Because the beamwidth of a high-gain antenna is normally much smaller 

 than the search area within which a target might appear, the beam must 

 be made to scan over the area. For AEW or ground-mapping systems 

 where the beam is narrow in only one dimension, this motion is generally 

 very simple, either a wigwag or a complete rotation. For systems where 

 the beam is narrow in both azimuth and elevation, the motion of the beam 

 can become quite complex. 



The efi^ect of scanning is to provide multiple looks at the target, giving 

 multiple chances for detection. In this case, it is the cumulative probability 

 of detection which is most significantly related to the tactical use of the 

 system. Complex scans can produce a nonuniform coverage of the scan 

 area, with holes in the pattern and undesired modulation of the received 

 pulse packet. 



Multiple-Scan Probability of Detection. In a typical detection 

 situation, the radar will periodically scan the target and there will be a 

 number of looks at the target when a detection can be made. Moreover, 

 since the target will normally move during the scan time, the average 

 signal-to-noise ratio and thus the average single glimpse probability will 

 vary from scan to scan. This situation is conveniently described by the 

 cumulative probability of detection. When the target is closing on the 

 radar, the cumulative probability of detection at a given range is defined as 

 the probability that the target is detected on or before reaching that range. 



We shall assume that the radar closes on the target at a constant rate 

 — Ry and the scan time t^c is also constant. Thus, the range interval which 

 is closed during a scan is given by 



Range decrement = A/^ = —Rtsc (3-35) 



If the first look occurs at the range i?i, then the ^'-th look will occur at the 

 range 



