PROPERTIES OF SINE WAVE PLUS NOISE 



153 



Table — Continued 



APPENDIX II 

 Second Moments Associated with Ic and Is 

 The in-phase and quadrature components of the noise current In 



M 



Ie(t) = ^Cn COS [(cOn " q)t " (Pn] 

 n=l 



M 



Is(t) = ^Cn sin [(co„ — q)t — <p„] 



(3.3) 



are closely related to the envelope R and phase angle 6 of the total current, 

 this relationship being being shown by the equations (3.4) and (3.5). /«(/) 

 and Is{t) and their time derivatives may be regarded as random variables. 

 In much of our work we have to deal with the probability distribution of 

 these random variables. By virtue of the representation (3.3) and the 

 central limit theorem^^ this distribution is normal in the several variables. 

 The coefficients in the quadratic form occurring in the exponent are deter- 



" Section 2.10 of Reference A. 



