5-9] AN APPLICATION TO THE ANALYSIS OF AN MTI SYSTEM 269 



that is, the noise spectrum may be assumed to be uniform without appre- 

 ciable error: 



[1 + 2(S /N)]T 

 Power density of demodulator output noise = D = 7- — ; — LT\i,^ A^(cos). 



(1 + o/i\)^kl 



(5-95) 

 A further simplification can often be made when the ratio of the scanning 

 to the repetition frequencies is small. In this case, the factor N(cos) is 

 approximately unity. For example, when the ratio of these frequencies is 

 1 : 10, the value of A^(co,) is 0.97. 



Substituting the power density D given in Equation 5-95 into the relation 

 already derived for the mean square tracking noise (Equation 5-91) and 

 assuming that N(olIs) is unity gives the following expression for the tracking 

 noise variance: 



Mean square tracking noise 



(S/N)[l + 2(.V/A^)] / 7^\ 



(1 + s/Nr \4ky' ^^"^^^ 



This expression represents the end product of our analysis of the effect of 

 internally generated noise on the performance of a conically scanned angle 

 tracking loop. It \& interesting that the tracking noise from this source has 

 a maximum at a signal-to-noise ratio of 1.35 db. The decrease in tracking 

 noise at small signal-to-noise ratios is due to the loss in loop gain and 

 consequent narrowing of the loop bandwidth. When this begins to happen 

 in a practical system, dynamic tracking lags usually cause an early loss of 

 the target. We also note that the rms tracking noise is directly proportional 

 to the square root of the repetition period and inversely proportional to the 

 modulation constant of the antenna. This constant, expressed in per cent 

 modulation per unit error, is itself inversely proportional to the antenna 

 beamwidth. 



The analysis in this example was intended to illustrate the sort of 

 considerations which are appropriate to a study of noise in a radar tracking 

 loop which incorporates a variety of components — some of them nonlinear 

 or time dependent. Similar analyses can be made of other types of tracking 

 systems such as monopulse tracking loops, range tracking loops, and 

 frequency tracking loops. The effects of externally generated random 

 disturbances such as glint or amplitude noise will also be handled in a 

 similar fashion. 



5-9 AN APPLICATION TO THE ANALYSIS OF AN MTI 

 SYSTEM 



In this paragraph we shall make some observations on the performance 

 of a radar system which provides moving target indication (MTI). This 

 analysis will supply another example to illustrate the use of the mathe- 

 matical techniques which have been developed. The MTI system which we 



