104 BELL SYSTEM TECHNICAL JOURNAL 



The amplitude of the typical difference product between the signal and 

 the rih. noise component is 



Asn — 2 ^100.--010---0 



J,{I\z)J,{l\z)J,{P^z)-' ■Ji{P,.z)-- -MPnz) W 



T J 



dz 



Z" 



Using the same process as before, we replace J\{Pnz) by 



UPnz) =^-^6-"'^''" ^ (32) 



and obtain in the limit as N becomes indefmitely large 



TT Jo 2 



aP 



^2 



Relations between the i/''i function and liessel functions are discussed in 

 Appendix II. 



The shape of the spectrum of the beats between Pi) and the noise input 

 evidently consists of the superjjosition of the noise spectra above and below 

 po , so that if we write n'snif) A/ for the mean energy from this source in that 

 part of the filter output lying in the band of width A/ at/, 



■Wsnimi = ^^ [(Atnf + (^7n)'1 (34) 



Asn — l^f »nJp„=po+2ir/ (35) 



Asn ^^ [Asn]pn=:po—27rf W^) 



P„ = ^^ (37) 



"■•"^^ = W„ ^ Hiwj + '\2W..)] (3S) 



X W/o+/) +w(r„- f)] 



The total noise from this source in the output of a particular filter of transfer 

 admittance Y(f) is obtained by integrating Wsn{j)Y{f)df throughout the band 

 of the filter. In the particular case in which the original band of noise is 



