MATHEMATICAL ANALYSIS OF RANDOM NOISE 123 



and for i? < it is 

 Ao{R) = \B\, R <\B\ 



\B\ \B\ \B\ 1 / , , (4.3-16) 



A,(R) = +LeJ + L^i arc sin L„-' + - \/R' - B% \B\ < R 



2 TT K IT 



where tlie arc sines lie between and ir/l. Ao{R) and its first derivative 

 with respect to R are continuous. 



From (4.3-15), the d.c. output current is, for i^ > 0, 



he = f T-f + - arc sin | + - VR- - bA p(R) dR (4.3-15) 



J H ]_ 2 IT K IT J 



where p(R) is the probability density of the envelope of the input V, e.g., 

 p(R) is of the form (3.7-10) for noise alone, and of the form (3.10-11) for 

 noise plus a sine wave. Similarly, the rms value of the low frequency 

 current If/ , excluding d.c, may be computed from 



ilf = fa - lie 



where, if 5 > 0, 



T]l= f T-f + - arc sin I + - VW^=^^\ p(R) dR (4.3-16) 



J B [_ 2 TT K T J 



If T' consists of a sine wave of amplitude P plus noise T',v , so it may be 

 represented as (4.1-13), and if P » rms V^ , the distribution of R is 

 approximately normal. If, in addition, P — B ^ rms F.v > 0, (4.3-15), 

 (4.3-16), and (3.10-19) lead to the approximations 



^-l + ^ + ^I+P (4.3-17) 



2 T 2irP 



Ti P- - B 



^f ~ V¥^^ h 



The second expression for Idc assumes P » B. When B = Q, these re- 

 duce to the first terms of (4.2-5) and (4.2-7). By using a different 

 method Middleton has obtained a more precise form of this result. 



Incidentally, for a given applied voltage, /dc(+) for a positive bias | B \ 

 is related to /dc( — ) for a negative bias — | 5 | by 



/dc(-) = \B\ -\- /do(+) (4.3-18) 



Also r.m.s. 7^/(+) is equal to r.m.s. Iff{ — ). Equation (4.3-18) follows 

 from a physical argument based on the areas underneath a curve of I for 



