Baseband Transmission

in Noise (Proakis: chapter 8.3, 10.5)

EEE3012 – Spring 2020

Di i l C i i SDigital Communication System

i SBasic Setup

S[ ]x k

XT( )s t ( )y t ( )v t

mV( )H f

thresh.[ ]x̂ k

X

( )n t ( ) NbT

( )n t ( )2

on

NS f =

the LP filter suppresses noise, but (ideally) not signal

( )

/ 2N

( )S f ( )H fis this the optimum

filter?/ 2oN

( )S f ( )H f

cont … / 2oN

Not necessarily! the optimum filter should minimize the error rate

it may pay to distort the signal if more noise can be rejected

i di h O i ilFinding the Optimum Filter

Focus on the “no ISI” case Signals are sent every Tb seconds Signals are sent every Tb seconds s1(t) is sent for a “0”

(t) i t f “1” s2(t) is sent for a “1” each pulse lasts for Tb seconds

Example: FSK( ) ù( )( ) ( )

1

2

cos0

cosc

bc

s t A tt T

s t A t

w

w w

ù= ú £ £ú= +D úûû

[ ]x k ( )s t ( )y t ( )v t V( )H f

[ ]x̂ k

cont … S[ ]

XT

( )n t ( ) oNS f =

( ) ( ) ( )mV

thresh.

bT

[ ]

( ) ( )2nS f =

The signals are thusg

( )( ) ( )1 k k k bs t t n t t t t T

y tì - + £ £ +ïï=í( )

( ) ( )2 k k k b

y ts t t n t t t t T

íï - + £ £ +ïî

sampling at givesk bt t T= +

( ) [ ] 0T N kì +ï [ ] 0E N( ) [ ]( ) [ ]

01

02

0

1b k

kb k

s T N x kV

s T N x k

+ =ïï=íï + =ïî

[ ]

( )22 2

0

2k

k

oN N

E NN H f dfs s

¥

-¥

=

= = ò2 ¥ò

cont …

the Vk are Gaussian random variables

Ziemer

depends on Pr[0] and Pr[1]

if Pr[0]=Pr[1], then

( ) ( )1k s T s Té ù= +ë û( ) ( )01 022opt b bk s T s Té ù= +ë û

cont … ( ) ( )101 022opt b bk s T s Té ù= +ë û

with this formulation

[ ] ( )P |"0" | "0"f d¥

ò[ ] ( )Pr error|"0" | "0"kVk

f v dv= ò( )01 bk s T

Qæ ö- ÷ç ( ) ( )02 01b bs T s T

Qæ ö- ÷ç ÷( )01 b

N

Qs

÷ç ÷= ç ÷ç ÷çè ø

( ) ( )02 01

2b b

N

Qs

÷ç ÷= ç ÷ç ÷çè ø

[ ]P |"1"[ ]Pr error|"1"=

the error probability is minimized by maximizing( ) ( )02 01b bs T s T

z-( ) ( )02 01b b

N

zs

=

( ) ( )02 01b bs T s Tz

-

cont …( ) ( )02 01b b

N

zs

=

( ) ( ) ( ) let ( ) ( ) ( )2 1g t s t s t= -

( ) ( )2

2 g Tg T( ) ( ) ( )0 02 01g t s t s t= - ( ) ( )02 02 2

k k

bb

N N

g Tg Tz

s s= =

now,( ) ( ) ( )1g t H f G f- é ù( ) ( ) ( )0g t H f G fé ù= ë û

2¥

ò ( ) ( )

( )

2

2

2

bj fTH f G f e df

N f df

p

z

¥

-¥

¥=ò

ò ( )2

2oN H f df

-¥ò

S h liSchwarz’s Inequality

This inequality states that2 2 2

( ) ( ) ( ) ( )2 2 2

*X f Y f df X f df Y f df¥ ¥ ¥

-¥ -¥ -¥£ò ò ò

with equality if( ) ( )X f kY f=( ) ( )f f

cont …( ) ( )*X f Y f df

¥

ò Proof:

( ) ( )2

X f Y f df¥

ò

( ) ( )

( )2

let X f Y f df

Y f dfa -¥

¥

-¥

-=

òò

( ) ( )

( ) ( ) ( )2

* *

X f Y f df

X f df X f Y f df

a

a

-¥

¥ ¥

+

+

ò

ò ò

¥

a is arbitrary

( ) ( ) ( )

( ) ( ) ( )2

2*

+

X f df X f Y f df

X f Y f df Y f df

a

a a

-¥ -¥

¥ ¥

= +

+

ò ò

ò ò( ) ( ) ( ) + X f Y f df Y f dfa a-¥ -¥

+ò òthese will now cancel

( ) ( ) ( )( ) ( )

2*

2 2 X f Y f dfX f Y f df X f dfa

¥

¥ ¥ -¥+

òò ò( ) ( ) ( )

( )2

X f Y f df X f dfY f df

a ¥-¥ -¥

-¥

+ = -ò òò

( ) ( ) ( ) ( )2 2 2

*¥ ¥ ¥

ò ò òcont …2

( ) ( ) ( ) ( )*X f Y f df X f df Y f df-¥ -¥ -¥

£ò ò ò

( ) ( ) ( )( ) ( )

2*

2 2

2

X f Y f dfX f Y f df X f dfa

¥

¥ ¥ -¥

¥+ = -ò

ò òò

( ) ( ) ( )( )

2Y f df

¥-¥ -¥

-¥

ò òò

always +ve

( ) ( )( )

2*

2X f Y f dfX f df

¥

¥-¥£

òò

( )( )

2X f df

Y f df¥ -¥

-¥

£òò

( ) ( ) ( ) ( )2 2 2

*X f Y f df X f df Y f df¥ ¥ ¥

-¥ -¥ -¥£ò ò ò

( ) ( )equality when X f kY f=

( ) ( ) ( ) ( )2 2 2

*¥ ¥ ¥

ò ò òcont … ( ) ( ) ( ) ( )*X f Y f df X f df Y f df-¥ -¥ -¥

£ò ò ò

Return to our maximization2¥

ò ( ) ( )

( )

2

2

2

bj fTH f G f e df

N

p

z

¥

-¥

¥=ò

ò let

( )2

2oN H f df

-¥ò let

( ) ( ) ( ) ( )*; bj fTH f X f G f e Y fp2= =

( ) ( )( )

2 2

222

2 2G f df H f dfG f df

N Nz

¥ ¥

¥-¥ -¥

¥£ =ò ò

òò ( )

( )2

o oN NH f df¥ -¥

-¥

òò

( )22 2 G f dfz

¥£ òcont … ( )2

o

G f dfN

z-¥

£ ò

The maximum z occurs under equality

( ) ( )X f kY f( ) ( )X f kY f=

( ) ( ) 2* bj fTH f kG f e p-=

in the time domain,,

( ) ( )*g t G f- «

( ) ( ) ( ) ( )2 1b b bh t g T t s T t s T t= - = - - -

cont … ( ) ( ) ( )2 1b bh t s T t s T t= - - -

Ziemer

this filter is said to be matched to s1(t)

the matched filter assumed no ISI and symbol width pulses

for raised cosine pulse shapes square root of the response at the transmitter square root of the response at the transmitter square root at the receiver

l iImplementation

Implement matched filters with correlators

( )bs T t-

0 bt T< <

( ) ( ) ( )y t s t n t= + ( )v t

t T

( )bv T

( ) ( ) ( )v t y t h t= * ( ) ( )bT

y s T t dt t t= - +ò

0 bt< <bt T=

( ) ( ) ( )v t y t h t= * ( ) ( )0 by s T t dt t t= +ò

( ) ( )0

( )bT

b s bs s T t d N R T t Nt t t= - + + = - +ò( ) ( ) ( )

0(0)

bT

b sv T y s d R Nt t t= = +ò0ò

a correlator

bT

òcont … ( ) ( ) ( )0

bT

bv T y s dt t t= ò

if we have rectangular pulses

( ) ( ) ( ) ( ); bs t u t h t u T t= = -

b bT T

ò ò( ) ( ) ( ) ( )0 0

b bT T

bv T y u d y dt t t t t= =ò òj t i t t d d th i tjust integrate and dump the input

b biliError Probability

Recall that

[ ]( ) ( )02 01P b bs T s T

Q Q zæ ö é ù- ÷ç ÷ ê úç[ ]( ) ( )02 01Pr error

2 2b b

N

Q Q zs

ç ÷= = ê úç ÷ç ÷ç ê úè ø ë û

( )1/ 2

2max

2

o

G f dfN

z¥

-¥

é ùê ú= ê úë û

ò ( ) ( ) ( )2 1G f S f S f= -

using Parseval’s theorem

oë û

( ) ( ) ( )1/ 2 1/ 2

22max 2 1

2 2g t dt s t s t dtN N

z¥ ¥

¥ ¥

é ù é ùé ùê ú ê ú= = -ë ûê ú ê úë û ë û

ò òo oN N-¥ -¥ê ú ê úë û ë ûò ò

( ) ( )1/ 2

22 dz¥é ùé ùê úòcont … ( ) ( )max 2 1

2

o

s t s t dtN

z-¥

é ùê ú= -ë ûê úë ûò

expanding gives

1/ 22 é ù( ) ( ) ( ) ( )1/ 2

2 2max 2 1 1 2

2 2o

s t dt s t dt s t s t dtN

z¥ ¥ ¥

-¥ -¥ -¥

é ù= + -ê úê úë ûò ò ò

2E 1Esignal correlation

1 ( ) ( )12 1 21 2

1 s t s t dtE E

r¥

-¥ò

we want as negative as possible

best performance when r12 = -1

lExample

Square full-width pulses, polar signaling the matched filter is ( )h t

A( )

( ) ( ) ( ) ( )1/ 2

2 22 1 1 2

2 2s t dt s t dt s t s t dtz¥ ¥ ¥é ù

= + -ê úò ò ò

bT

( ) ( ) ( ) ( )2 1 1 22o

s t dt s t dt s t s t dtN

z-¥ -¥ -¥

+ê úê úë ûò ò ò

b bAT E= b bAT E= 2 2b bAT E- =-22 2

b bAT E b b b b

[ ] 21 2Pr error 4 bb

EQ Q E Qz é ù é ùé ù ê ú ê ú= = =ê ú ê ú ê úê ú[ ]

2 2 bo o

Q Q QN Nê ú ê úê úë û ë û ë û

“SNR”

Optimum Linear Receiver

Problem with zero-forcing equalizer a solution to have zero ISI causes noise enhancement !

Minimum-mean square error (MMSE) equalizer a more balanced solution to reduce the effects of both

noise and ISI in a combined manner !

1 é ù21 | ( ) | (MSE criterion)2

where ( ) ( ) ( ) for ( ) ( ) ( )

b nJ E y nT a

y t x t c t x t a q t kT w t

é ù= -ë û

= Ä = +å design the receive filter to minimize the MSE J.

where ( ) ( ) ( ) for ( ) ( ) ( )k bky t x t c t x t a q t kT w t= Ä = - +å

g

cont …

Optimum linear receiver consisting of the cascade connection of matched filter and transversal equalizer.

*( )( )( ) / 2

Q fC fS f N

=+( ) / 2q oS f N+

cont …

Block diagram of adaptive equalizer.- training mode- decision-directed mode