5-9] AN APPLICATION TO THE ANALYSIS OF AN MTI SYSTEM 271 



of ground objects, the scanning of the antenna, and the motion of the 

 platform on which the antenna is mounted. The clutter power is denoted 

 hj C = x^ = y, and the autocorrelation function of the x and y processes 

 is denoted by Cp = XyX2 = yxji- 



During a pulse, the echo received from a moving target is assumed to be 

 a sinusoidal signal with a doppler shift: 



Target echo = a cos (ojc + co^)/. (-5-98) 



The peak signal power is denoted by 6" = a} jl. 



We shall assume that the signal plus clutter is rectified by a square-law 

 second detector to give the following video signal during a pulse: 



Video = V = \a cos (coc + co^)/ + -v cos (xsct + y sin cor/]". (5-99) 



The video frequencies are, of course, limited by the video bandwidth. 

 Squaring this expression and retaining only the low-frequency components 

 which will be passed by the video amplifier gives 



Video = y = \{a} -|- a;- + .V^ + lax cos cod/ — lay sin cod/). (5-100) 



The cancellation unit acts to generate the difference of video signals 

 separated by a repetition period. Denoting the residue from the cancel- 

 lation unit by r(/), we have 



Residue signal = r{t) = v{i) — v{t — T) = V\ — v-i 



= ^{xi^ - X2'' + yi^ - y2^ + 2axi cos co^/i (5-101) 



— 2^X2 cos 0)^/2 — 2ayi sin cod/i + 2ay2 sin cod/2). 



In order to evaluate the effect of the cancellation unit in reducing the 

 clutter, it is convenient to define a video signal-to-clutter ratio. This ratio is 

 defined as the difference between the video power with a signal v^j^^ 

 and the video power with clutter only y^ divided by this latter quantity. 



Video signal-to-clutter ratio -= {S/C)v = (y|+c — vl)/ v^- (5-102) 



Similarly, a signal-to-clutter ratio is defined for the residue signal output 

 of the cancellation unit: 



Residue signal-to-clutter ratio = {S/N)r = (r|+c — ^c)/ ^c' (5-103) 



With these definitions, a gain factor may be determined as the quotient of 

 these two ratios: 



System gain factor = G = ^^^ (5-104) 



In order to evaluate this gain factor in terms of the system parameters, 

 the average values of the squares of the video and residue signals must be 

 calculated. This is somewhat complicated because of the large number of 



