138 BELL SYSTEM TECHNICAL JOURNAL 



mensionality — in order that S{u)) be inversely proportional to co in agree- 

 ment with experiment. In this case the only allowed value oinis — 1. 



(c) From (b) we have the interesting result that S{(ji) is independent of D 

 when it is inversely proportional to oj. This means that very slowly diffusing 

 substances can contribute as much to contact noise as rapidly diffusing 

 substances. This result can be derived on quite dimensional grounds and is 

 not dependent upon the special assumptions underlying our treatment. 



(d) A system comprising a high resistance layer modulated by the three- 

 dimensional diffusion of particles or heat gives 5(co) « co~' ^. See Case a.(i) 

 in Section 5. 



(e) In a system composed of a localized contact disturbed by a diffusing 

 surface layer (See Case a.(ii), Section 5), the self-covariance Ri{t)Ri{l -\- u) 

 is inversely proportional to A + Du where A may be considered the contact 

 area. We have S{w) oc — log a -\- const, for co « Z)/A and 5'(co) a ijT'' 

 for w » D/^. 



(f) In a system involving the contact between relatively large areas of 

 rough surfaces covered with diffusing surface layers (Cases b. and c, Section 

 5), we have been successful in obtaining S{<ji) «■ co~\ and also in obtaining 

 a reasonable dependence upon the average resistance. 



APPENDIX I 



Spectral Density and the Self-Covariance 



Here we consider in detail the spectral density, the self-covariance, and 

 the relation between these two quantities, first for the case of a single random 

 variable. The treatment is subsequently extended to the case of a set of 

 random variables which necessitates the consideration of the spectral density 

 matrix and the covariance matrix. 



Let y(/)be a real random variable whose time average vanishes, y{l) = 0. 

 Now the m.s. value of y can be defined 



3^ = Lim ^ f y\l, r) dl (I-l) 



where y(t, r) ^ y{t) in the interval —-</<- and vanishes outside this 

 interval. Evidently y{t, t) can be expressed by the Fourier integral 



.+00 



where 



.+{T/2) 



y{l, t) = [ z{o,, Tje'-" do, (1-2) 



''-00 



z{w, t) = -— / y{l)e^'"^ dl, 



l-K J-{tI2) 



