5-11] DETERMINATION OF SIGNAL'S TIME OF ARRIVAL 289 



Because the video signal plus noise is gated, the noise power and thus the 

 noise power density will be smaller than the value during a pulse by the 

 duty ratio d. Taking these factors and considerations into account, the 

 effective power density is determined from Equation 5-87 to be 



Power density at zero frequency of gated video noise 



^ {lcj''Yl{SlN)hd,SINy>\. (5-149) 



The d-c component of the video corresponds to the signal which is used 

 to locate the target. From Equation 5-148 it will be noted that this 

 component is proportional to the received signal power. Thus the video 

 signal is modulated by a scan modulation function which indicates how 

 the received power fluctuates during a scan. The first term in Equation 

 5-148 gives the square of the d-c level during a pulse for large signal-to- 

 noise ratios. To obtain the d-c level of our gated signal, we must multiply 

 the signal level during a pulse by the duty ratio d. The resulting video 

 signal has the following approximate form: 



Gated video signal = (2o-^)(*S'/A^)'^(scan power modulation), S/N^ 1. 



(5-150) 



We assume that the antenna pattern of the system has a Gaussian shape 

 and that the same antenna is used for both transmission and reception. 

 The beamwidth of the pattern 6 is defined as the angle between the half- 

 power points for one-way transmission. The gain of the two-way power 

 pattern would thus be down by a factor of 4 at these points. The antenna 

 angle is denoted by ^. We assume that the scan modulation is generated 

 by a constant velocity scan at the rate i/'. Supposing that the signal max- 

 imum occurs at the time / = 0, the scan modulation has the form 



Scan power modulation = exp (— \p^/0.1SQ^) 



= exp (- ipyyO.lSQ') (5-151) 



Because we have assumed a square-law second detector, the scan modula- 

 tion of the video voltage will be proportional to this scan power modulation. 

 We are now in a position to apply the result of Equation 5-145, giving the 

 rms error in time of arrival, which in turn yields the rms angular error after 

 multiplication by the scan rate. We first compute the signal-to-noise ratio 

 in the output of the video filter matched to the scan modulation. The 

 energy in the signal is given by the integral of its square: 



Signal energy at low frequencies 



= (2cr2)2(^/7V)V2 / exp ( - V'VVO.0902) dt 



= 0.53{2a'y(S/Nyd\e/i^). (5-152) 



