MATHEMATICAL ANALYSIS OF TLiNDOM NOISE 83 



The double integral in (3.8-9) becomes 

 _a=.2 /t f> (2sr Y nib"" T(m + ^Wn - m + 2) 



n — 0^2^ 



2 e 



(M) 



n=0 



where Aq = 1 and 



- (*)(!) ••• (>n - h) 



^^^ Y: ^'^^^^ "T (^' - '" + I)*'"' < « (3.8-10) 



m=0 WZ 



^„ ~ {n + 1)(1 - b)-'" - ^ (1 - by"\ n large 



The term corresponding to m = in (3.8-10) is » + 1. 

 We thus obtain 



Pit, R) = -— -, '—J^ E -7 ^ An 



\2 ^V 



(3.8-11) 



-a222 t1/2 





4'\/7r ^0 n=0 



(M) 



We are interested in the expected number, .V, of maxima per second. 

 From the similar work for /, it follows that N is the coefficient of dt when 

 (3.8-1) is integrated with respect to R from to 20 . Thus from (3.8-7) and 



dR = VWibfdz = (2boBy"b7'^'dz 



= [26o(a- - \)f"dz 



we find 



N = [ pit, R) dR 

 Jo 



^ ja' - If (b^Y f ^ (I + 4) A^ 

 i2ayi- \Trbo/ h (n . l\ a- 



(3.8-12) 



Equations (3.8-11) and (3.8-12) have been derived on the assumption 

 that ivii) is symmetrical about /„, , i.e. the band pass filter attenuation is 



