724 BELL SYSTEM TECHNICAL JOURNAL 



It is of engineering interest to determine the noise produced by a very 

 large impulse, exceeding greatly the ampHtude of the signal carrier. When 

 such a large impulse arrives, it causes a sudden jump, or discontinuity, in 

 the phase of the carrier. The excursion of the instantaneous frequency 

 corresponding to the phase jump is indefinitely large and the problem ac- 

 cordingly cannot be solved satisfactorily by means of the usual assumption 

 of quasi-stationary frequency. The problem of large impulsive interference 

 provides, therefore, the principal justification for the more exact method of 

 analysis here employed. In the paragraph following, the problem is re- 

 stated in terms providing a suitable basis for mathematical analysis. 



We assume, as before, that the Hmiter is deUvering to the frequency de- 

 tector a steady carrier current of constant amphtude h and frequency /o . 

 At time / = a brief disturbance occurs specified by the statement that 

 the instantaneous frequency, 0'(/), of the current suddenly executes an 

 impulse of moment 9. That is: ^'(0 is zero at all times except at / = 0, 

 when it goes to infinity and back to zero again in such a way that the area 

 of the impulse so formed is 0. The carrier current ampHtude then remains 

 constant but the phase, ^(/), of the carrier takes a sudden jump of radians 

 at / = 0. What is the voltage output of the frequency detector? 



The general formula (28) gives directly the envelope function of the volt- 

 age across the impedance (26) for a phase function d{t). In this formula we 

 have now to put d{i) = before time / = and 0, after / = 0. We do this by 

 dividing the interval of integration into two parts, (— «» , 0) and (0, t)\ thus 



a{t) -f ib(t) = M^-^'"°+^^' U e^'"°+^^^ dr -f e'® f e^''^'^'^' dr^ 



= hk ^'- ^^''"^'"'" + -% M (^ - ^^"^'l:' + -" . (30) 



Let the radian frequency interval by which the applied frequency co© 

 is set oflF from the resonant frequency coc be 



A = coo — tOc (31) 



as indicated on the curves of Fig. 2. Whena/wc is small compared to unity, 

 as it is in practical circuits, jS is very nearly equal to Wc . (See the formulas 

 following equations (26).) Then 



iwo + 7 = ^wo + a + iwc = a — iA -f- 2^coo 



and 



ia>o + 7 = *wo + a — iwc = a -h iA. (32) 



