278 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 



almost all radar receivers whose inputs consist of a series of pulses incor- 

 porate such a device in one form or another. 



The effect of the pulse integrator on the noise can also be determined 

 from Equation 5-125. The noise output will be 



/oo „_i 



n(T)J2 P(r - kT)dT. (5-12S) 



The noise power is determined by squaring no{to) and finding its average 

 value : 



/°° /• "^ n-l n-l 

 / w(ri)w(r2)S ^p{ri — kT)p(T2 — mT)dTidT2. 



(5-129) 



Since n(t) was assumed to have a uniform spectrum with density D, the 

 average value of the product «(ri)«(T2) is an impulse function with weight 

 D: 



/oo /■ oo „_!„_! 



/ D8{ti — T2)2Z ^P(tT- ~ kT)p{T2—mT)dTldT2 



= dI E jipiT2 - kT)p{T2 - 7nT)dT, (5-130) 



j -co 



= £>/ Y.pKr2- kT)dr2 

 7-- 



r oo 



In evaluating the integral of the double sum, we made use of the fact that 

 when the pulse functions in the integrand do not coincide {k 7^ m), their 

 product is zero: 



-my 



{T)dT. (5-131) 



It is apparent that the effect of the pulse integrator is to increase the signal- 

 to-noise ratio for a single pulse by the factor n. This could, of course, be 

 inferred at the outset from Equation 5-121, since the signal power is 

 directly proportional to n. 



The shape of the matched filter response in this case is of some interest. 

 Denoting the spectrum of an individual pulse by P(co), the spectrum of the 

 pulse train will be 



n-l 



Pulse train spectrum = S{i^) = P(co) X) ^~''*"^ (5-132) 



\ sm coT/2 / 



