37 nyquist criterion for zero isi

8/13/2019 37 Nyquist Criterion for Zero ISI

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EE 370 Chap. VII: Principles of Digital Data Trans. ver. 1.0. Lect. 37

Nyquist Criterion for ZeroISI

N!"ist propose# a con#ition for p"lsesp$t% to have &ero'I(I )hen trans*itte# thro"gh achannel )ith s"fficient +an#)i#th to allo) the spectr"* of all the trans*itte# signal to

pass. N!"ist propose# that a &ero'I(I p"lsep$t% *"st satisf the con#ition

1 0$ %

0 , - , 3 ,b b b

tp t

t T T T

== = K

p"lse that satisfies the a+ove con#ition at *"ltiples of the +it perio# Tb )ill res"lt in

&ero'I(I if the )hole spectr"* of that signal is receive#. The reason for )hich these &ero'I(I p"lses $also calle# N!"ist'criterion p"lses% ca"se no I(I is that each of these p"lses at

the sa*pling perio#s is either e!"al to 1 at the center of p"lse an# &ero the points other

p"lses are centere#.

In fact, there are *an p"lses that satisf these con#itions. /or ea*ple, an s!"are p"lse

that occ"rs in the ti*e perio# 'Tb to Tbor an part of it $it *"st +e &ero at 'Tb an# Tb%)ill satisf the a+ove con#ition. lso, an triang"lar )avefor* $f"nction% )ith a )i#ththat is less than -Tb)ill also satisf the con#ition. sinc f"nction that has &eros at t Tb,-Tb, 3Tb, 2 )ill also satisf this con#ition. The pro+le* )ith the sinc f"nction is that iteten#s over a ver long perio# of ti*e res"lting in a lot of processing to generate it. The

s!"are p"lse re!"ire# a lot of +an#)i#th to +e trans*itte#. The triang"lar p"lse isrestricte# in ti*e +"t has relativel large +an#)i#th.

There is a set of p"lses no)n as raise#'cosine p"lses that satisf the N!"ist criterionan# re!"ire slightl larger +an#)i#th than )hat a sinc p"lse $)hich re!"ires the *ini*"*

+an#)i#th ever% re!"ires.

The spectr"* of these p"lses is given +

( ){ }4 -11 sin

- - -

$ % 0-

1-

b bx

x

bx

bx

P

+ < +

,

)here bis the fre!"enc of +its in ra#4s $b -4Tb%, an# xis calle# the ecess

+an#)i#th an# it #efines ho) *"ch +an#)i#th )o"l# +e re!"ire# a+ove the *ini*"*

+an#)i#th that is re!"ire# )hen "sing a sinc p"lse. The ecess +an#)i#th x for this

tpe of p"lses is restricte# +et)een

Notes are provi#e# + : Dr. 5a6ih +"l (a"#. 8inor *o#ifications + Dr. 8"!ai+el


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0-

bx

.

(etching the spectr"* of these p"lses )e get

5e can easil verif that )hen x 0, the a+ove spectr"* +eco*es a rect f"nction, an#

therefore the p"lse p$t% +eco*es the "s"al sinc f"nction. /or x b4-, the spectr"* is

si*ilar to a sinc f"nction +"t #ecas $#rops to &ero% *"ch faster than the sinc $it eten#s

over - or 3 +it perio#s on each si#e%. The epense for having a p"lse that is short in ti*e isthat it re!"ires a larger +an#)i#th than the sinc f"nction $t)ice as *"ch for x b4-%.

(etch of the p"lses an# their spectr"* for the t)o etre*e cases of x b4- an# x 0

are sho)n +elo).

5e can #efine a factor r calle# the roll'off factor to +e

-Ecess 9an#)i#th

8ini*"* 9an#)i#th 4 -

x x

b b

r

= = = .



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The roll'off factor r specifies the ratio of etra +an#)i#th re!"ire# for these p"lsesco*pare# to the *ini*"* +an#)i#th re!"ire# + the sinc f"nction.


37 nyquist criterion for zero isi

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