134 BELL SYSTEM TECHNICAL JOURNAL 



and the value of li'o taken there is the same as here and is i/'o//3. The value 

 of Giif) given there leads to the approximation, for low frequencies: 



Woij) 



-iTT V h- fa) 



7n/'o4/3 



(4.7-9) 



when < / < /b - /„ , and to W^f) -= ior fb - fa < f < fa ■ By setting 

 P equal to zero in the curve given in Fig. 8 for Wdf) corresponding to the 

 square law detector, we see that the low frequency portion of the power 

 spectrum is triangular in shape and is zero at / = /3. Thus, looking at 

 (4.7-9), we see that to a first approximation the shape of the output power 

 spectrum is the same for a linear detector as for a square law detector when 

 the input consists of a relatively narrow band of noise. 



An approximate rms value of the low frequency output current may be 

 obtained by integrating (4.7-9) 



rms low freq. current = ~y^ X rms applied voltage (4.7-10) 



It is seen that this is half of the direct current. It must be kept in mind 

 that (4.7-10) is an approximation because we have neglected \pr and higher 

 powers. The true value may be obtained from (4.2-8). It is seen that the 

 coefficient (Stt)"^'^ = 0.200 should be replaced by 



K-i)"=«- 



= 0.209 



Wcif) for other types of band pass filters may be obtained by using the 

 corresponding G's given in appendix 4C. It turns out that (4.7-10) holds 

 for all three types of filters. This is a special case of Middleton's theorem, 

 mentioned several times before, that the total power in any modulation 

 product (it will be shown later in Section 4.9 that the term i/'" in (4.7-6) 

 corresponds to the n order modulation products) depends only on the 

 total input power of the applied noise, not on its spectral distribution. 



4.8 The Characteristic Function Method 



As mentioned in the preceding parts, especially in connection with equa- 

 tion (1.4-3), the ch. f. of a random variable x is the average value of exp 



