DISTORTION IN NOISE MODULATED FM SIGNAL 887 



If the transducer is perfect except for an echo, its response to a unit 

 impulse 8(t) is 



g(t) = 8(t) + r8(t - T) (5.1) 



where r and T are the amplitude and the delay of the echo. The results 

 obtained using (5.1) agree, as they should, with the results obtained in 

 Reference 4. Of course, r must be assumed small compared to unity in 

 order that condition (1.5) may hold. 



When the power spectrum of the signal is equal to a constant Po over 

 the band (/„ , /&) and zero elsewhere we have for phase and frequency 

 modulation, respectively, 



PM: w^{j) = Po, fa<f <h, 



FM: w,{j) = Po/(27r/)^ fa<f <fi>. 



When fa = the autocorrelation functions are 



(5.2) 



(5.3) 



PM: lAr = PoMsmv)/v, 



FM: yPo - ypr = A[-l + cos v + vSi{v)], 



V = 27rf,T, A = Po.U2Th)-' = (<T/f,)\ 



The mean square values of the signals are Pofb (radians)" for PM and 

 Pofb (radians/sec)^ for FM. If, for FM, a is the rms frequency deviation 

 in cps (so that the "peak" deviation is, say, 4o- cps) then (27ro-) = Pofb • 

 The difference i/'o — \pT is used in the FM case to avoid difficulty at/ = 0. 

 It ^\^ll be noticed that our formulas are such that the \J/'s may be re- 

 placed by {\l/ — \l/o)'s without altering the values of the various ex- 

 ponents, etc. In microwave systems the quantity A is often small in 

 comparison with unity. 



As an example of the use of the second order modulation approxima- 

 tion (4.10) consider the case where the attenuation, a, is zero and the 

 phase shift ^ = a2(f — fp) /2 radians, a2 being small. Then, since G ^ 

 1 — a, ^ ^ —j3, we have Gu ^ I and 



Bu ^ -[13 for f = fp + 2i] 



= — OoM /2. 



When we take the FM case of (5.2) and substitute in the approxima- 

 tion (4.10), the interchannel interference power spectrum is found to be 



{2Tnty (27r)2(/ - ^0- (5.5) 



= (27r)-''(a2Po/2)'(2/b - /). 



5! si/-/, 



