Signal Processing and Linear Systems-B.P.Lathi copy

I n contrast for t he low frequency sinusoid t he

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

Ask a homework question - tutors are online