(5-88) 



€ COS ip. 



5-8] APPLICATION TO EVALUATION OF ANGLE TRACKING NOISE 267 

 Error signal = [ SiN r "^ ^^^ ^°^ ("«/ + ip)\{\IK) cos w,t 



= ( S/^ \ 



VI + S/NJ 



In this expression the wavy bar indicates the time average, which eliminates 

 the fluctuating terms. The factor cos <p indicates that the error derived is 

 the projection of the total error on the axis represented by one of the angle 

 tracking loops. Complete directional control of the antenna requires it to be 

 controlled in two directions, normally azimuth and elevation. The error 

 signal for the other loop is obtained from a demodulator with a reference 

 sin cos/. 



This error represents the input to the antenna controller which moves the 

 antenna in order to null the error and track the target. In order to arrive at 

 a definite result in this example, we shall assume that the antenna controller 

 is composed of a single integrator, although in a practical system the 

 dynamic response of the angle tracking loop might be quite complicated. 

 With this assumption, the response of the whole loop becomes the same as 

 that of a low-pass RC filter, and the power transfer function has the 

 following familiar form. 



Angle tracking loop power transfer function = ^ ^^^ (5-89) 



where K = gain around the tracking loop = bandwidth (rad/sec) 



As noted above, the gain K will be attenuated by a factor depending on the 

 signal-to-noise ratio. Thus we shall express K as the product of this factor 

 and a design bandwidth /3 achieved at high signal-to-noise ratios: 



Our primary interest in this example is to determine the response of the 

 loop to internally generated noise. It will turn out that the spectrum of 

 the equivalent noise input to the loop is very much broader than ^ and 

 relatively flat in the low-frequency region. If we denote the power density 

 of this input noise spectrum by D in angular units squared per rad/sec, 

 the variance of the tracking noise will be given by 



Mean square trackmgnoise = :^j_^ ^^^^-p^, = ^ = [j^^-sJnKJ )' 



(5-91) 



The next problem is to determine the magnitude D of the input-power 

 density spectrum. 



