3-4] 



EFFECT OF SCANNING ON DETECTION PROBABILITY 



161 



The Number of Pulses per Scan. In the system model adopted in 

 Paragraph 3-3, to develop an analytical method for calculating the proba- 

 bility of detection, it was assumed that n equal-amplitude pulses were 

 received on a scan over a target. With a complex scan and realistic beam 

 shapes, the pulses received are not all of the same amplitude; neither is it 

 clear just what n should be in many situations. For instance, with the 

 Palmer scan illustrated in Fig. 3-11 b, the pulses received on a single scan 

 over a target may be grouped into several separate packets by the cyclical 

 motion imposed on the basic scan. The grouping and number of pulses in 

 the individual packets can then change with the location of the target in the 

 scan pattern. 



In order to analyze situations of this nature correctly and in detail, 

 extensive analytical investigations are often required. More commonly, 

 it is quite adequate to make reasonable approximations which will allow 

 the methods developed in Paragraph 3-3 to be applied. This is what we 

 shall do here. 



We consider first the problem of estimating the effect of the antenna 

 beamshape in a linear scan over a target. We suppose that the antenna 

 pattern has a Gaussian shape similar to that defined in Equation 3-45: 



Two-way power pattern of antenna '-^ ^-eVo.ise^ (3-45) 



where Q = angular position of the antenna 



= antenna beamwidth (half-power, one-way). 



We wish to approximate this antenna pattern by a uniform pattern so 

 that the results of the preceding paragraph are applicable. This type of 

 approximation is indicated in Fig. 3-12. In making this approximation, the 



Fig. 3-12 Rectangular Approximation to Gaussian Beam Shape (Equal Area 

 Approximation) . 



total integrated power will be maintained constant. That is, the integral of 

 the uniform approximation will be made equal to the integral of the antenna 

 power pattern between the effective limits of integration. This will result 

 in an equivalent loss in signal power for pulses in the uniform pattern in 

 comparison with pulses in the center of the more realistic pattern. This loss 

 is referred to as the scan loss. Following current practice, we suppose that 



