MATHEMATICAL ANALYSIS OF RANDOM NOISE 

 The probability density function in (3.6-1) is 

 p(I,0,t) = (2Tr"'\M\-"'exp 



73 



L 2 \M\ 



(Mnl' + M33r' + 2Mi3/r) 



] 



(3.6-4) 



and when this is put in (3.6-1) and the integration with respect to f per- 

 formed we get 



dl dl 



s,-3/2 r 



(3.6-5) 



for the probabihty of a maximum occurring in the rectangle dl dt. As is 

 mentioned just below expression (3.6-1), the expected number of maxima 

 in the interval /i , /2 may be obtained by integrating (3.6-1) from h to t^ 

 after replacing x by /, and / from — oo to + °° after replacing y by /. "When 

 we use (3.6-4) it is easier to integrate with respect to / first. The expected 

 number is then 



Mn 



(4) 



d^ 



= (/2-/x)'^Vr=^^r^,i 



27r 27r L-i^oJ 



Hence the expected number of maxima per second is 



27rL-^d 



/ A(/) df 

 / Mf) df 



(3.6-6) 



For a band pass filter, the expected number of maxima per second is 



11/2 



uii-fVi 



(3.6-7) 



where fb and fa are the cut-off frequencies. Putting /„ = so as to get a 

 low pass filter, 



W 



/ft - = .775/6 



(3.6-8) 



