908 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



IV DERIVATION OF TIMING WAVE FROM PULSE TRAIN 



4.0 General 



The retiming wave R{t) must have a fixed relation to the received 

 pulses, with certain tolerable fixed and random deviations to be con- 

 sidered later. Such a timing wave can be derived from the pulse train 

 with the aid of a sufficiently narrow band-pass filter, the simplest form 

 of which is a resonant circuit consisting of a coil and capacitor in series 

 or in parallel. 



A train of rectangular "on-ofT" pulses is shown in Fig. 7 as it would 

 appear at the output of a regenerative repeater and at the input of the 

 next repeater, (dotted) with uniform intervals T between sampling 

 points. 



As indicated in Fig. 7, the pulse train can be regarded as being made up 

 of two components. One of these is an infinite sequence of pulses of one 

 polarity, the other an infinite sequence of randomly positive and negative 

 polarity. 



It will be recognized that the first of the above components at the out- 

 put has a fundamental frequency equal to the pulse repetition fre- 

 quency, / = l/T", and the forced response of a resonant circuit to this 

 component will be the pulse repetition frequency, regardless of any im- 

 perfections in tuning. In order that this frequency be present in the re- 

 ceived pulse train, it is necessary that the spectrum of the received pulses 

 extend beyond the pulse repetition frequency, so that there will be a 

 ripple in a long sequence of received pulses of one polarity, as indicated 

 in the illustration. 



The second random component of the pulse train will have a fre- 

 quency spectrum that is nearl}^ uniform over the band of the tuned 

 circuit, and which will vary in amplitude depending on the composition 

 of the pulse train. The response of the tuned circuit to this component 

 is thus rather complex, and must be treated on an approximate statistical 

 basis. It will consist of an almost periodic wave with random amplitude 

 and phase modulation, and with mean frequencj^ equal to the resonant 

 frequency. 



Owing to the presence of the second component, there will be a 

 variation with time in the amplitude and phase of the response of a mis- 

 tuned resonant circuit, and resultant deviations in timing. The re- 

 generated pulses will thus not be uniformly spaced, but will in general 

 have random deviations from the desired exact positions. Such deviations 

 can be created by superposing on a train with uniform spacing a random 

 dipulse train, as indicated in Fig. 7. The resonant circuit response to this 



