RESPONSE OF RECTIFIER TO SIGNAL AND NOISE 101 



IT. Spectrum of Output 



A much more powerful method of attack on this problem is obtained by 

 the use of multiple Fourier series. In this section we shall use Fourier 

 analysis to obtain not only the direct-current output of the rectifier, but 

 also the spectral distribution of the sinusoidal components in the output of 

 the rectifier. We represent the input spectrum by 



N 



E = Fo cos Pot 4- E ^« cos pj (13) 



n=l 



This representation is more general than that given by (4) in that a frequency 

 spectrum as well as an amplitude distribution is defined; it may be shown 

 that the probability density for the sum of N sinusoidal waves with incom- 

 mensurable frequencies approaches (4) when N is large. The first term 

 represents the sinusoidal signal; the mean power which would be dissipated 

 by this signal in unit resistance is 



Ws = Pill. (14) 



The noise is represented by a large number A^ of sinusoidal components with 

 incommensurable frequencies (or commensurable frequencies with random 

 phase angles) distributed along the frequency range /i to ji in such a way 

 that the mean noise power in band width A/ is: 



«'(/)A/ =1 E P\- vLjP\j)ll (15) 



Here v is the number of components per unit band width and /*(/) represents 

 the amplitude of a component in the neighborhood of frequency /". Note 

 also that the mean total noise input power, Wn , is given by 



I^„ = jf w{j) df=^l P\f) df (16) 



The linear rectifier is specified by the current- voltage relationship (8), 

 which is equivalent to 



It Jc z^ 



where C is an infinite contour going from — «; to + co vvith an indentation 

 below the pole at the origin. We may expand / in the multiple Fourier 

 series 



1 Bennett and Rice, "Note on Methods of Computing Modulation Products," Phil. 

 Mag., Sept. 1934. The present application represents an extension to N variables of the 

 theory there given for two. 



