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Unformatted text preview: nd clarity, we consider a first-order system which has only a single
mode, e At. Let t he impulse response of this system b et h (t) = A e At
j (t) = (2.72) T his result shows t hat a n i nput pulse spreads o ut (disperses) as it passes through
a system. S ince Th is also t he s ystem's time constant or rise time, t he a mount of
spread in t he p ulse is equal to t he t ime constant (or rise time) of t he system. Time Constant and Rate o f Information Transmission In pulse communications systems where information is conveyed through pulse
amplitudes, t he r ate of information transmission is p roportional to t he r ate of pulse
transmission. W e shall demonstrate t hat t o avoid t he d estruction of information
caused by dispersion of pulses during their transmission through t he channel (transmission m edium), t he r ate of information transmission should not exceed t he b andwidth of t he c ommunications channel.
Since a n i nput pulse spreads o ut by Th seconds, t he consecutive pulses should
be spaced Th s econds a part in order to avoid interference between pulses. Thus, t he
r ate of pulse transmission should not exceed 11Th pulses/second. B ut 11Th = Fe,
t he channel's b andwidth, so t hat we can t ransmit pulses through a communications
channel a t a r ate of Fe pulses/second and still avoid significant interference between
t he pulses. T he r ate of information transmission is therefore proportional t o t he
channel's b andwidth (or to t he reciprocal of its time constant).t. t T his discussion (Secs. 2.7-2 through 2.7-6) shows t hat t he system time constant
determines much of a system's b ehavior-its filtering characteristics, rise time, pulse
dispersion, a nd so on. In t urn, t he t ime constant is d etermined by t he s ystem's·
c haracteristic roots. Clearly t he c haracteristic roots a nd t heir relative amounts in
the impulse response h (t) d etermine t he b ehavior of a system. (2.73) a nd let t he i nput be I n general, t he transmission of a pulse through a system causes pulse dispersion (or spreading). Therefore, t he o utput pulse is generally wider t han t he i nput
pulse. This s ystem behavior can have serious consequences in communication systems where information is t ransmitted by pulse amplitudes. Dispersion (or spreading) causes interference or overlap with neighboring pulses, thereby distorting pulse
amplitudes a nd i ntroducing errors in t he received information.
Earlier we saw t hat if a n i nput j (t) is a pulse of width T f, t hen Ty, t he w idth
of t he o utput y (t), is 2 .7-6 159 2.7 Intuitive Insights into S ystem Behavior T heoretically, a c hannel of b andwidth Fe c an t ransmit u p t o 2Fe p ulse amplitudes p er second
correctly. 3 O ur d erivation here, being very simple a nd q ualitative, yields only half t he t heoretical
limit. I n p ractice i t is difficult t o a ttain t he u pper t heoretical limit; transmission r ates o f Fe pulses
p er second are m ore common. e (A-€)t T he s ystem response y (t) is t hen given by y (t) = Ae At * e (A-€)t From t he convolution table we o btain = Ae At (l-e-€t)
_ __ (2.74) E Now, a s E ---> 0, b oth t he n umerator a nd t he d enominator of t he t erm in the parentheses approach zero. Applying...
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