Some statistical properties of random noise

A considerable volume of knowledge is now available on random fluctuations (noise) as regards the behaviour in amplitude. Familiar names in this field are those of Uhlenbeck and Ornstein( 7 ), Fürth( 1 ) and Rice ( 4 ), although very many others have made valuable contributions. A particular class of problem, of considerable practical importance, exists when the frequency spectrum is limited to a relatively narrow range. The resulting noise has then the character of a more or less regular oscillation modulated randomly in amplitude and phase. In this case, if we write the fluctuation in the form (where R ( t ) and θ( t ) are variables changing slowly in comparison with sinω 0 t ), it is clear that the magnitude of the envelope R ( t ) and the phase θ( t ) are now the significant quantities. Rice ( 4 ), among others, has made a study of the statistical properties of R , deriving in particular the correlation function R ( t ) R ( t + τ) in terms of the characteristics of the (power) spectrum ω( f ). Fürth and the writer ( 2 ) have extended this work and carried out a collateral experimental investigation.