636 BELL SYSTEM TECHNICAL JOURNAL 



The following result is derived in Appendix 6. 



Theorem 15: Let the average power of two ensembles be Ni and iV2 and 

 let their entropy powers be Ni and Ni . Then the entropy power of the 

 sum, Nz , is bounded by 



Ni + N2< Nz<Ni-{- N2. 



White Gaussian noise has the peculiar property that it can absorb any 

 other noise or signal ensemble which may be added to it with a resultant 

 entropy power approximately equal to the sum of the white noise power and 

 the signal power (measured from the average signal value, which is normally 

 zero), provided the signal power is small, in a certain sense, compared to 

 the noise. 



Consider the function space associated with these ensembles having n 

 dimensions. The white noise corresponds to a spherical Gaussian distribu- 

 tion in this space. The signal ensemble corresponds to another probability 

 distribution, not necessarily Gaussian or spherical. Let the second moments 

 of this distribution about its center of gravity be aij . That is, if 

 p{xi J • • ' y Xn) is the density distribution function 



aij =/•••/ P(^i ~ ^i)(^3 — «j) dXi, " ' , dxn 



where the ai are the coordinates of the center of gravity. Now aij is a posi- 

 tive definite quadratic form, and we can rotate our coordinate system to 

 align it with the principal directions of this form, aij is then reduced to 

 diagonal fonn ba . We require that each bu be small compared to N, the 

 squared radius of the spherical distribution. 



In this case the convolution of the noise and signal produce a Gaussian 

 distribution whose corresponding quadratic form is 



N -t- bu . 



The entropy power of this distribution is 



[n(iV-FMr 



or approximately 



= [(NT + iibiiiNr-']'^'' 



^ N + -i:bii. 



n 

 The last term is the signal power, while the first is the noise power. 



