Lec7 Equalizer

8/18/2019 Lec7 Equalizer

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BEIT- 5th Semester 1

Equalization




Agenda

Nyquist criterion for zero ISI

Zero forcing equalizer




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



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



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



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.



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



BEIT- 5th Semester

Zero 8orcing Equalizer

&r (t)

#-!#-!41 #!

#! -1

∑

N

N n

br N nT t pC t p )()(

0;f



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



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




E,am&le>

&r (%) ' $?

&r ($) ' -%.!? &r (#) ' %.$

&r (-$) ' -%.#? &r (-#) ' %.%@

8ind -$2 %2 $.

esign t1e t1ree-ta& gain equalizer




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




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



BEIT- 5th Semester 1*



Sync1ronization





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




Sync1ronization in a Recei<er

Regenerator 0P8

/it

Syncloc=

8rame

Sync

:ut&ut

message

8rame

indicator




:&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




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



BEIT- 5th Semester 2'

b "#

y(t)

z(t)

c(t)

z(t)c(t)

<(t)

t

t

t

t

t

b

b "5




atc1ed 8ilter




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)




<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)

(@)



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



BEIT- 5th Semester

Documents

Lec7 Equalizer