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8/18/2019 Lec7 Equalizer
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BEIT- 5th Semester 1
Equalization
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BEIT- 5th Semester 2
Agenda
Nyquist criterion for zero ISI
Zero forcing equalizer
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BEIT- 5th Semester 3
Intersymbol Interference (ISI)Spreading of a pulse beyond Tb will cause
interference with neighboring pulses
ISI!y"uist #riterion for $ero ISI%
& 1 for t & '
p(t)& ' for t & ±nTb
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BEIT- 5th Semester *
Rb-Rb
P()
-!"Rb -#"Rb -$"Rb % $"Rb #"Rb
$
&(t)
$ for t ' %
&(t) ' sinc(πR bt) '
% for t ' ±nb
Pulse satisfying Nyquist riterion
8/18/2019 Lec7 Equalizer
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BEIT% 5th Semester
Raised osine function+
The characteristic is also
called full-cosine-roll *off characteristic
,ecays as 1t3
The time domain representation
is (ta.ing in/erse 0ourier Transform)%
)(sin41
)cos()(
22 t Rc
t R
t R Rt p b
b
bb
%
$
+P(ω
)+
ω, ' %
ω, ' ωb "#
ω
tb
#b-b
-
#b
&(t)
Ref. /.P.0at1i2 1. 34Ref. /.P.0at1i2 1. 34
)4
()4
(cos
)4
()2
cos1(2
1)(
2
bb
bb
Rrect
R
Rrect
R P
=
ω, ' ωb "5
5
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BEIT- 5th Semester
61ic1 one is really required7
Eliminate or minimize ISI with neighboring
pulses at all instance of time t.
Eliminate or minimize ISI with neighboring
pulses at their respective sampling instances
when decision is made.
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BEIT- 5th Semester
Zero 8orcing Equalizer
&r (t)&%(t)
#b
(N9#)b(N-#)b
-#b b (N9$)b-b (N-$)b
In&ut :ut&ut
Nb
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BEIT- 5th Semester
Zero 8orcing Equalizer
&r (t)
#-!#-!41 #!
#! -1
∑
N
N n
br N nT t pC t p )()(
0;f
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BEIT- 5th Semester
Set of #N9$ simultaneous
equations in #N9$ <ariables
0ilter output p'(t) is pr (t 6 !Tb)% delayed /ersion of pr (t)
The samples of p'(t) at t & .Tb are%
!y"uist criterion re"uires%
∑
N
N nbr nb T nk pckT p])[()(
0
0
1][
0 k p
8or = ' %
8or = '±
$2±
#2 ...2±
N
it1 = '±
$2±
#2 ...2±
N
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BEIT- 5th Semester 1'
Set of #N9$ simultaneous
equations in #N9$ <ariables
=
N
N
N
N
c
c
c
c
c
c
c
1
1
0
1
1
rrr
rrr
rrr
rrr
rrr
rrr
rrr
...
...
[0]p ... 1]-[2Np [2N]p
[1]p ... 2]-[2Np 1]-[2Np
...
1][-Np ... [N]p 1][Np
[-N]p ... 1]-[Np [N]p1]-[-Np ... 2]-[Np 1]-[Np
1][-2Np ... [0]p [1]p
[-2N]p ... [-1]p [0]p
0
0
...
0
10
...
0
0
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BEIT- 5th Semester 11
E,am&le>
&r (%) ' $?
&r ($) ' -%.!? &r (#) ' %.$
&r (-$) ' -%.#? &r (-#) ' %.%@
8ind -$2 %2 $.
esign t1e t1ree-ta& gain equalizer
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BEIT- 5th Semester 12
E,am&le>
&r (%) ' $?
&r ($) ' -%.!? &r (#) ' %.$
&r (-$) ' -%.#? &r (-#) ' %.%@
=
1
0
1
1 0.3- 0.1
0.2- 1 0.3-
0.05 0.2- 1
0
1
0
c
c
c
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BEIT- 5th Semester 13
E,am&le>
&r (%) ' $?
&r ($) ' -%.!? &r (#) ' %.$
&r (-$) ' -%.#? &r (-#) ' %.%@
=
1
0
1
1 0.3- 0.1
0.2- 1 0.3-
0.05 0.2- 1
0
1
0
c
c
c
c-$ ' %.#$%2 c% ' $.$!2 c$ ' %.!$B
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BEIT- 5th Semester 1*
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Sync1ronization
BEIT- 5th Semester 15
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BEIT- 5th Semester 1
Sync1ronization ec1nique
#loc.s at transmitter and recei/er tic. together
Synchroni$ation of Symbols
0rame
7eneral methods 8ne master cloc. for T9 and :9
Transmitting separate synchroni$ation signal (pilot)
Self synchroni$ation
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BEIT- 5th Semester 1
Sync1ronization in a Recei<er
Regenerator 0P8
/it
Syncloc=
8rame
Sync
:ut&ut
message
8rame
indicator
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BEIT- 5th Semester 1
:&en-loo& /it Sync1ronizer
Square 0a
e<ice /P8P1ase
AdCusto
cloc=
y(t) y#(t)
y(t) ' cos(#π
r bt 9 )
y#(t)
cos(#πr bt)
cos(#π
r bt 9 )
r b
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BEIT- 5th Semester 1
losed-loo& /it Sync1ronizer
Zero rossing
etector Integrator 0P8
Doltage-controlled
loc=
b "#
z(t)
c(t)
y(t) <(t)
c(t)
z(t)
c(t)
y(t)
t
t
t
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BEIT- 5th Semester 2'
b "#
y(t)
z(t)
c(t)
z(t)c(t)
<(t)
t
t
t
t
t
b
b "5
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BEIT- 5th Semester 21
atc1ed 8ilter
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BEIT- 5th Semester 22
atc1ed 8ilter
Signal to noise ratio &
7(f) 0ourier Transform of the .nown signal g(t)
;(f) 0re"uency response of the filter
g(t)
61ite noise
(t)
,(t) y(t)
)]([
|)(|2
20
t n E
T g =η
y()
($)
0inear time in<ariant
filter of im&ulse
res&onse 1(t)
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BEIT- 5th Semester 23
<a9imum noise with white noise w(t) ha/ing
power spectral density !'2
#orrespondingly ;(f) assumes its optimum /alue
;opt(f)%
Ta.ing in/erse 0ourier transform we get%
Since for a real signal g(t) we ha/e 7=(f) & 7(-f)%
we may write
∫ ∞
∞−
= df f G N
2
0
max |)(|2
η
)2exp()()( * fT j f kG f H opt
π −=
∫ ∞
∞−
−−= df t T f j f Gk t hopt )](2exp[)()( *
π
∫ ∞
∞−
−−−= df t T f j f Gk t hopt )](2exp[)()( π
∫ ∞
∞−
−−= df t T f j f Gk )](2exp[)( π )( t T kg −=
(#)
(!)
(5)
(@)
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BEIT- 5th Semester 2*
atc1ed 8ilter for Rectangular Pulse
7(t) input
rectangular pulse
0or a special
case% matchedfilter may be
reali$ed with an
integrate-and-
dump-circuit
%
A
g(t)
t
%
=A#
g(t)
t
%
A
g(t)
t
Energy ' A#
(a)
(b)
(c)
8igures>
a) Rectangular &ulse2
b) atc1ed filter out&ut
c) Integrator out&ut
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BEIT- 5th Semester