Mathematical Analysis of Random Noise

T H I S p a p e r deals with t h e m a t h e m a t i c a l analysis of noise o b t a i n e d by passing r a n d o m noise t h r o u g h p h y s i c a l devices. T h e r a n d o m noise considered is t h a t which arises f r o m s h o t effect in v a c u u m t u b e s or f r o m t h e r m a l a g i t a t i o n of electrons in resistors. O u r m a i n i n t e r e s t is in the s t a tistical p r o p e r t i e s of such noise a n d we leave to one side m a n y p h y s i c a l results of which N y q u i s t ' s law m a y be given as an example. 1 A b o u t half of t h e work given here is believed to be new, t h e b u l k of t h e new results a p p e a r i n g i n P a r t s I I I a n d I V . I n order t o p r o v i d e a s u i t a b l e i n t r o d u c t i o n to these results a n d also to bring o u t their relation to the work of others, this p a p e r is w r i t t e n as an exposition of t h e s u b j e c t i n d i c a t e d in the title. W h e n a b r o a d b a n d of r a n d o m noise is applied to some physical device, such as an electrical n e t w o r k , t h e statistical p r o p e r t i e s of the o u t p u t are o f t e n of i n t e r e s t . For example, w h e n t h e noise is d u e to s h o t effect, its m e a n a n d s t a n d a r d deviations are given b y C a m p b e l l ' s t h e o r e m ( P a r t I ) w h e n t h e p h y s i c a l device is linear. Additional i n f o r m a t i o n of this s o r t is given by t h e (auto) correlation f u n c t i o n which is a rough m e a s u r e of the d e p e n d e n c e of values of the o u t p u t s e p a r a t e d by a fixed t i m e i n t e r v a l .