BAND WIDTH AXD TRANS.UISSION PERFORMANCE 561 



and we have assumed linear phase in computing pulses 2 to 7. A good 

 approximation to the Gaussian filter can be obtained both as to phase and 

 amplitude with a number of tuned circuits in tandem, coupled through 

 buffers. A fair approximation can also be obtained by combining a 3- or 

 4-section maximally flat filter" with a tuned circuit through a buffer. 



Rectangular or near-rectangular shaping filters produce pulses with over- 

 shoot as shown by pulses 8 to 1 1. The filter corresponding to pulses 8 and 9 

 is assumed to have rectangular shape and linear phase. Filters of this sort 

 have no simple approximation in practice and are included for comparison 

 with filters 10 and 11 which are made up of simple maximally flat networks. 

 In pulses 9, 10 and 11 the "unshaped pulse" is assumed to be very narrow 

 and of ampHtude sufficient to yield pulses of the heights shown. 



Let us now regard these pulses as received pulses and compare them in 

 respect to shape for use in various kinds of pulse systems. 



PPM. In PPM the pulses may occupy any time position in the assigned 

 inter\'al and so the tails of pulses 8 to 11 may "crosstalk" into time assigned 

 to an adjacent channel. To allow guard tune for the train of tails or to 

 design for satisfactory operation in the presence of the tails is uneconomical 

 of frequency space. It follows that pulses which are more definitely bounded 

 in time such as those obtained with Gaussian filters are more desirable and 

 likely to be more economical of frequency space in general despite their 

 wider spectrum. 



In PPM where the trailing (or leading) edge of a pulse is used to convey 

 the information a flat top pulse such as pulse 2 is no better than one in which 

 the flat portion is absent and the two transitions brought together.^* The 

 latter pulse would, in fact, be superior since more time would then be avail- 

 able for additional channels or for greater swing. 



We are thus led to conclude that one of the pulses in the 4 to 6 group is 

 the preferred shape for PPM. We chose pulse 4 in our illustrative calcula- 

 tions and defined bandwidth as 2/To , but pulses 5 or 6 would have given 

 substantially the same results. 



PAM. In PAM the pulses occur at standardized, regular times so that 

 if pulse 9 were used the accompanying tails, which disappear completely 

 at instants To, 2To , etc., from the pulse peak, need not theoretically pro- 

 duce crosstalk between channels if the channels are spaced To and the pulse 

 ampHtudes are measured instantaneously at the time the nuUs occur. As a 

 practical matter both the precise pulse shape and the instantaneous measure- 



" W. W. Mumford, "Maximallv-Flat Filters in Wave Guide," Bell Sys. Tech. //., Vol. 27, 

 October, 1948, pp. 684-713. ' . 



38 Such a pulse would look like pulse 4 if the latter were shrunk to occupy 0.6 of the time 

 shown in the plot. The spectrum would accordingly be that of pulse 4 expanded by the 

 factor 1.7 but would not include more significant band width than is necessary to form pulse 

 2 as shown. This deduction follows from the fact that the rise time of pulse 2 is the same 

 with or without the flat top. 



