272 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 



terms resulting from the squaring of Equations 5-100 and 5-101, and the 

 details will not be given here. The following average values originally- 

 determined in Paragraph 5-6 in connection with a discussion of a square-law 

 device are used in these calculations: 



X = y = x^ = y^= xy- — x-y — 



x^ = y^ = c, X1X2 = yiy2 = Cp{T) (5-105) 



x^ = y^ = 3C2 = xi^xs^ = y.'^y^'' = C^[l + 2p2(T)l. 



The following results were determined for the video and residue signal to 

 clutter ratios: 



{S/C\ = 2{S/C) [1 + i {S/C)] 



1 - p{T) coscodT] 



{S/C)r = 2(S/C) 

 The gain in signal to clutter ratio will simply be 



1 - pHD I (^-'0') 



System gam factor = G = ^ _^ a)(S/C) 1 _ 2(7-) (5-108) 



This expression essentially summarizes the ability of a noncoherent MTI 

 system to reduce clutter. Various interesting observations might be made 

 from a study of this factor. For instance, the depth of the blind speed nulls 

 at harmonics of the repetition frequency can be determined as a function of 

 the normalized autocorrelation function of the clutter at the repetition 

 period. The average gain over all doppler frequencies can also be found as 

 a function of the same parameters. These details will not be explored here. 

 The primary purpose of the example has been served by the derivation of 

 Equation 5-108, which showed how a performance equation could be arrived 

 at by a straightforward application of the techniques for signal and noise 

 analysis previously developed. 



5-10 AN APPLICATION TO THE ANALYSIS OF A 

 MATCHED FILTER RADAR 



In this paragraph, we shall consider how the mathematical techniques 

 which have been developed can be applied to the derivation of optimum 

 radar systems. Besides providing a good illustration of the application 

 of these techniques, this example will also provide an insight into the 

 important basic factors which affect system performance and set theoretical 

 performance boundaries which a practical system may approach but not 

 surpass. 



We shall be primarily concerned with the detection performance of radar 

 systems. A fundamental problem in detecting a radar target is to distin- 



