NEGATIVE FEEDBACK 433 



(30), it can be reasoned that a differentiation process is necessary for 

 the recovery of the signal itself. 



Differentiation of the argument of the cosine term in (31) yields the 

 instantaneous frequency (rate of change of phase with respect to time) 

 of the received wave given by (30). Now it can be shown that with a 

 strictly linear frequency detector, the low-frequency output is propor- 

 tional to the response of the conversion system at the instantaneous 

 frequency. The recovered signal is, therefore, proportional to the 

 variable part of the instantaneous frequency and hence to the time 

 derivative of the variable phase term in the original wave. 



In the case of non-linearity in the characteristic of the frequency 

 detector the output can be expressed, to a sufficiently close degree of 

 approximation, as a power series in the derivative of the phase term 

 d{t). Hence the output of the receiver can be written in the form 



<j{t) = aAB Z hn 



[IM^ 



(35) 



= aAB E hnlpiS - kp2(x'y (36) 



n 



where the coefficients hn are based upon the transfer admittance 

 characteristic of the receiver. 



What is now desired is the relationship between a and 6". This can 

 be expressed in the general form 



a = aAB L c„[pi5]". (37) 



n 



Equations (36) and (37) can now be equated. Replacing a in the 

 right-hand side of (36) by the series (37) we shall have 



cipiS + CiipiSy + csipiSy + • • • 



= b£piS - kaABp2(c^prS + c,(pisy + Mpisy +•■•)] 



+ b^LpiS - kaABp,{c,p,S + C2{piSy + cipiSy +•••)? (38) 



-f bsipiS - kaABp2(c,p,s + C2(pisy -f- czipisy +•••)? 



After expanding, coefficients of like powers of piS can be equated. 

 Then solving for the first three orders of Cn we find 



^ ^1 ^ bi , . 



^' 1 -f abikABp2 1 - fx^ ^ ^ 



- = (T^ <■''' 



_ hz _ laABkp-jhi^ ,.. . 



'' (1 - fji^r (1 - M^)^ ^^'^ 



