124 BELL SYSTEM TECHNICAL JOURNAL 



the two cases. Both of the above relations follow from formulas given by 

 Middle ton when T' is the sum of a sine wave plus noise. They may also be 

 derived from (4.3-15) and (4.3-16). 



4.4 Output Power Spectrum 



The remainder of Part IV will be concerned with methods of solving the 

 following problem: Given a non-linear device and an input voltage con- 

 sisting of noise alone or of a signal plus noise. WTiat is the power spectrum 

 of the output? 



In some ways the answer to this problem gives us less information than 

 the methods discussed in the first three sections. For example, beyond 

 giving the rms value, it tells us very httle about the probabiUty density of 

 the current corresponding to a given frequency band of the output. On 

 the other hand, this rms value may be found (by integrating the power 

 spectrum) for any band we choose to study. The methods described earlier 

 depended on the input being confined to a relatively narrow band and gave 

 information regarding only the entire band corresponding to a given har- 

 monic (0th, 1st, 2nd, etc.) of the input. There was no way to study the 

 output when part of a band was eliminated by filters except by obtaining 

 the power spectrum of some function of the envelope. 



At present there appear to be two general methods available for the 

 determination of the output power spectrum each with its own advantages 

 and disadvantages. First there is the direct method which has been used 

 by W. R. Bennett*, F. C. Williams**, J. R. Ragazzini"^ and others. The 

 noise is represented as the sum of a finite number of sinusoidal components. 

 The typical modulation product is computed and the output power spectrum 

 is obtained by considering the density and amplitude of these products. 

 The chief advantage of this method lies in its close relation to the known 

 theory of modulation in non-linear circuits. Generally, the lower order 

 modulation products are the only ones which contribute significantly to the 

 output power and when they are known, the problem is well along towards 

 solution. The main disadvantage is the labor of counting the modulation 

 products falling in a given interval. However, Bennett has developed a 

 method for doing this.^^ 



The fundamental idea of the second method is to obtain the correlation 

 function for the output current. From this the output power spectrum may 

 be obtained by Fourier's transform. The correlation function method and 

 its variations are of more recent origin than the direct method. They have 



* Cited in Section 4.0. Also much of this writer's work on interference in broad band 

 communication systems may be carried over to noise theory without any change in the 

 methods used. 



** Cited in Section 4.1. 



«Proc. I.R.E. Vol. 30, pp. 277-288 (June 1942), "The Effect of Fluctuation Voltages 

 on the Linear Detector." 



"^.S.TJ., Vol. 19 (1940), pp. 587-610, Appendix B. 



