110 BELL SYSTEM TECHNICAL JOURNAL 



where the subscripts stand for "total low" frequency, "direct current." 

 and "low frequency," respectively. We have 



Idc = average I d = 7 a (4.1-3) 



Mean Square Iff = average (FiC — Idc)' = la — Idc 



Probably the simplest method of obtaining Idc is to square the given ex- 

 pression for V and pick out the terms independent of time. Thus if 



V = P cos pt + Q cos qt + TV (4.1-4) 

 we have 



i^c = « (y + I" + '^) (4-1-5) 



I(/ may also be obtained by picking out the low frequency terms. How- 

 ever, here we wish to use the square law device, and the linear rectifier in the 

 next section, to illustrate a general method of deahng with the statistical 

 properties of the output of a non-linear device when the input voltage is 

 restricted to a relatively narrow band. 



If none of the low frequency spectrum is removed by filters, 



Id = a~ ' (4.1-6) 



where R is the envelope of V . The probability density and the statistical 

 properties of 1 1( may be derived from this relation when the distribution 

 function of R is known. Before discussing these properties we shall 

 establish (4.1-6). 



Equation (4.1-6) is a special case of a more general result established 

 in Section 4.3. However, its truth may be seen by taking the example 



V = P cos pt + Q cos qt + Vn (4.1-4) 



where /p = p/liv and/g = (//27r lie within, or close to, the band of the noise 

 voltage Vn . 



By using formulas of the type 



cos pt = cos [{p — co„i)/ + o:j] 



= cos {p — U,n)t cos Wmt — S\\\ (p — COmjt SUl (ji^t 



*^ When part of the low -frequency spectrum is removed, the problem becomes much 

 more difficult. Idc may be obtained as above, but to get /L it is necessary to first deter- 

 mine the power spectrum of I (Section 4.5) and then integrate over the appropriate por- 

 tion of it. Concerning the distribution of /// , our present knowledge tells us only that it 

 lies between the one given by (4.1-6) and the normal law which it approaches when only 

 a narrow portion of the low frequency spectrum is passed by the audio frequency filter 

 (Section 4.3). 



