MATHEMATICAL AX A LYSIS OF RANDOM NOISE 87 



The asymptotic expression for puiy) may be obtained from the integral 

 (3.8-9) for pit, R). Indeed, replacing the variables of integration x, y in 

 (3.8-9) by 



x' = X 



y' = X + y, 



integrating a portion of the y' integral by parts, and assuming b < I 

 (a' > 1, by Schwarz's inequality, so that 6 < 1 always) leads to 



'--(09;-e;-') 



when R is large. 



If, instead of an ideal band pass filter, we assume that w(f) is given by 



""'^^^ ^ ^V^ e-'^-^'"^''^''^ /„ » a (3.8-16) 



we find that 



h= 1 

 hi = 4:ir'a~ 

 b[ = 167r -ia 

 a- = 3,b = 

 An = ill + 1) 



Some rough work indicates that the sum of the series in (3.8-12) is near 

 3.97. This gives the expected number of maxima of the envelope as 



N = 2.52(7 (3.8-17) 



per second. 



The pass band is determined by a. It appears difficult to compare this 

 with an ideal band pass filter. If we use the fact that the filter given by 



.a)=».exp[-,(^— ^J_ 



passes the same average amount of power as does an ideal band pass filter 

 whose pass band is fb — fa , we have 



fb — fa = (T^/2ir 



and the expression for N becomes 1.006 (fb — fa)- 



3.9 Energy Fluctuation 

 Some information regarding the statistical behavior of the random vari- 

 able 



rll+T 



E = / /'(/) dt (3.9-1) 



