basebandcom11-3

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Dr. Uri Mahlab 1


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Dr. Uri Mahlab 2

Informationsource

Transmitter Channel Receiver Decision

Communication system


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Dr. Uri Mahlab 3

Information

source

Pulse

generator

Trans

filter

channel

(X(t (XT(t

Timing

Receiver

filter

Clock

recovery

network

A/D

+

Channel noise

n(t)

Output

Block diagram of an Binary/M-ary signalingscheme

+

HT(f)

HR(f)

Y(t)

Hc(f)


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Dr. Uri Mahlab 4

Information

source

Pulse

generator

Trans

filter

(X(t (XT(t

Timing

Block diagram Description

HT

(f)

{dk}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}

Tb

Tb

)t(pg

Tb

)t(pg

Tb

For bk=1

For bk=0

;"0"a

;"1"aa

k

k

k

dif

dif


5/56

Dr. Uri Mahlab 5

Information

source

Pulse

generator

Trans

filter

(X(t (XT(t

Timing

Block diagram Description (Continue - 1)

HT

(f)

{dk}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}

Tb

Tb

)t(pg

Tb

For bk=1

For bk=0

Transmitter

filter

Tb

)t(pg

Tb


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Dr. Uri Mahlab 6

Information

source

Pulse

generator

Trans

filter

(X(t (XT(t

Timing


HT

(f)

{dk}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}

Tb

k

bgk kTtpatX )()(

Tb

100110

2Tb

3Tb 4Tb

5Tb

t6Tb


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Dr. Uri Mahlab 7

Information

source Pulsegenerator Transfilter

(X(t (XT(t

Timing


HT(f)

{dk}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}

Tb

kbgk kTtpatX )()(

Tb

100110

2Tb

3Tb 4Tb

5Tb

t6Tb

Tb 2Tb

3Tb 4Tb

5Tb

t6Tb


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Dr. Uri Mahlab 8

Informationsource

Pulsegenerator Transfilter

(X(t

Timing


HT(f)

Tb 2Tb

3Tb 4Tb

5Tb

t6Tb

Tb 2Tb

3Tb 4Tb

5Tb

t6Tb

Channel noise n(t)

+

t

Receiverfilter

HR(f)


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Dr. Uri Mahlab 10

Information

sourcePulse

generator

Transfilter

channel

(X(T (Xt(T

Timing

Receiverfilter

Clockrecoverynetwork

A/D

+

Channel noisen(t)

Output


+

HT(f)

HR(f)

Y(t)

Hc(f)

kbdrk tnkTttpAtY )()()( 0


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Dr. Uri Mahlab 11

Block diagram Description

Tb 2Tb

3Tb 4Tb

5Tb

t

6Tb

Tb 2Tb

3Tb 4Tb

5Tb

t6Tb

t

1 0 0 0 1 0

1 0 0 1 1 0

t


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Dr. Uri Mahlab 12

Information

sourcePulse

generator

Transfilter

channel

(X(t (XT(t

Timing

Receiverfilter

Clockrecoverynetwork

A/D

+

Channel noisen(t)

Output


+

HT(f)

HR(f)

Y(t)

Hc(f)


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Dr. Uri Mahlab 13

Transfilter

channel

Pg(t)Receiver

filter

Explanation of Pr(t)

HT(f) HR(f)Hc(f)

k

bdrk tnkTttpAtY )()()( 0

Pr(t)

HT(f) Hc(f) HR(f)Pg(f) Pr(f)

1)0(pr


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Dr. Uri Mahlab 14

The output of the pulse generator X(t),is given by

Tpa bgk

kkttX

Pg(t) is the basic pulse whose amplitude ak depends on.the kth input bit


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Dr. Uri Mahlab 15

For tm =mTb+td and td is the total time delay in thesystem, we get.

t

t

t

tmY

t1

t2 t3tm

The input to the A/D converter is

kbdrk tnkTttpAtY )()()( 0


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Dr. Uri Mahlab 16

tnTpAAt m0brmK

kmmkmY

t

tmY

t1

t2 t3tm

The output of the A/D converter at the sampling time

tm

=mTb

+td

k


Tb 2Tb

3Tb 4Tb

5Tb

t6Tb


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Dr. Uri Mahlab 17

tnTpAAt m0brmK

kmmkmY

t

tmY

t1

t2 t3tm

ISI - Inter SymbolInterference


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Dr. Uri Mahlab 18

Transfilter channel

Pg(t)Receiver

filter

Explanation of ISI

HT(f) HR(f)Hc(f)Pr(t)

Pg(f) Pr(f)Transfilter

channelReceiver

filter

HT(f) HR(f)Hc(f)

t

f

FourierTransform

BandPassFilter

f

FourierTransform

t


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Dr. Uri Mahlab 20

-The pulse generator output is a pulse waveform

k

bgk kTtpatX )()(

a

aa

p

k

g 1)0(

If kth input bit is 1

if kth input bit is 0

k


-The A/D converter input Y(t)


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Dr. Uri Mahlab 21

Datarate

Errorrate

Transmittedpower

Noisepower

NoiseSpectraldensity

Systemcomplexity

Type of Important Parameters Involved In The DesignOf a PAM System


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Dr. Uri Mahlab 22

5.2 BASEBAND BINARY PAM SYSTEMS

Pulse shapespg(t)

pr(t) HR(f) HT(t)

Design of a basebandbinary PAM system

- minimize the combined effects of inter symbolinterference and noise in order to achieve minimumprobability of error for given data rate.


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Dr. Uri Mahlab 23

0nfor00nfor1)nT(p br

5.2.1 Baseband pulse shaping

The ISI can be eliminated by proper choiceof received pulse shape pr (t).

Does not Uniquely Specify Pr(t) for all

values of t.


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Dr. Uri Mahlab 24

0nfor0

0nfor1)nT(pThen

T2/1fforT)T

kf(Pif

br

k

bb

b

r

k

T2/)1k2(

T2/)1k2(

rr

rr

b

b

df)ft2jexp()f(p)t(p

df)ft2jexp()f(p)t(p

Theorem

Proof

To meet the constraint, Fourier Transform Pr(f) of Pr(t), shouldsatisfy a simple condition given by the following theorem


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Dr. Uri Mahlab 25

b

b

b

b

T2/1

T2/1k b

brbr

k

T2/1

T2/1

b

b

rbr

df)fnT2jexp())T

k

f(p()nT(p

'df)nT'f2jexp()T

k'f(p)nT(p

k

Tk

Tk

brbr

b

b

dftfnTjfpnTp

2/)12(

2/)12(

)2exp()()(

b

b

T2/1

T2/1

bbbrn

)nsin(df)fnT2jexp(T)nT(p

Which verify that the Pr(t) with a transform Pr(f)Satisfy ZERO ISI


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Dr. Uri Mahlab 26

The condition for removal of ISI given in the theorem is calledNyquist (Pulse Shaping) Criterion

mk

m0brkmm )t(n)T)km((PAA)t(Y

0nfor0

0nfor1)nT(p br

Tb 2Tb-Tb-2Tb

1

n

)nsin()nT(p br


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Dr. Uri Mahlab 27

The Theorem gives a condition for the removal of ISI using a Pr(f) witha bandwidth larger then rb/2/.

ISI cant be removed if the bandwidth of Pr(f) is less then rb/2.

HT(f) Hc(f) HR(f)Pg(f) Pr(f)

Tb 2Tb

3Tb 4Tb

5Tb

t6Tb


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Dr. Uri Mahlab 28

Rate of decayof pr(t)

Shaping filters

pr(t)

Particular choice of Pr(t) for a

given application

The smallest values near Tb, 2Tb, In such that timing error (Jitter)

will not Cause large ISI

Shape of Pr(f) determines the easewith which shaping filters can be

realized.


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Dr. Uri Mahlab 29

A Pr(f) with a smooth roll - off characteristics is preferableover one with arbitrarily sharp cut off characteristics.

Pr(f) Pr(f)


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Dr. Uri Mahlab 30

In practical systems where the bandwidth available fortransmitting data at a rate of rb bits\sec is between rb\2 to rbHz, a class of pr(t) with a raised cosine frequency

characteristicis most commonly used.A raise Cosine Frequency spectrum consist of a flat amplitude portion and a roll offportion that has a sinusoidal form.

tr

trsin

)t4(1

t2cos)t(P

)f(PFT

2

rf

2

r

2/rf,0

),2

rf(

4cosT

2/rf,T

)f(P

b

b

2r

r1

bb

b

b2

b

bb

r


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Dr. Uri Mahlab 31

raised cosine frequency characteristic


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Dr. Uri Mahlab 32

The BW occupied by the pulse spectrum is B=rb/2+.The minimum value of B is rb/2 and the maximum value is rb.

Larger values ofimply that morebandwidth is required for agiven bit rate, however it lead for faster decaying pulses, whichmeans that synchronization will be less critical and will notcauselarge ISI.

=rb/2 leads to a pulse shape with two convenient properties.The half amplitude pulse width is equal to Tb, and there are zerocrossings at t=3/2Tb, 5/2Tb. In addition to the zero crossing

at Tb, 2Tb, 3Tb,...

Summary


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Dr. Uri Mahlab 33

5.2.2

Optimum transmitting and receiving

filters

pulse shaping noise immunity

HT ,HR

The transmitting and receiving filters are chosen to providea proper


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Dr. Uri Mahlab 34

)22exp()()()()( drcRTg ftjfPKfHfHfp

-One of design constraints that we have for selecting the filtersis the relationship between the Fourier transform of pr(t) andpg(t).

In order to design optimum filter Ht(f) & Hr(f), we will assume that Pr(f),Hc(f) and Pg(f) are known.

Where td, is the time delay Kc normalizing constant.

Portion of a baseband PAM system


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Dr. Uri Mahlab 35

If we choose Pr(t) {Pr(f)} to produce Zero ISI we are leftonly to be concerned with noise immunity, that is will choose

HT(f) Hc(f) HR(f)

Pg(f) Pr(f)

effectsnoiseofminimum )f(Hand)f(H RT


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Dr. Uri Mahlab 36

Noise Immunity

Problem definition:

For a given :Data Rate - rb

Transmission power - STNoise power Spectral Density - Gn(f)Channel transfer function - Hc(f)Raised cosine pulse - Pr(f)

Choose

effectsnoiseofminimum )f(Hand)f(H RT


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Dr. Uri Mahlab 37

Error probability Calculations

At the m-th sampling time the input to the A/D is:

mk

m0brkmm )t(n)T)km((PAA)t(Y

We decide:

0)t(Y"0"0)t(Y"1"

m

m

ifif

"1"sentTosentwas"1"

"0"sentTosentwas"0"

obPr0)t(YobPr

obPr0)t(YobPrP

m

merror


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Dr. Uri Mahlab 38

"0")t(A)t(Y

"1")t(A)t(Y

mm

mm

ifn

ifn

0

0

A=aKc

5.0obProbPr "1"sentTo"0"sentTo

A)t(nobPrA)t(nobPr2

1P m0m0error

The noise is assumed to be zero mean Gaussian at the receiver inputthen the output should also be Zero mean Gaussian with variance No

given by:

df)f(H)f(GN2

Rn0


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Dr. Uri Mahlab 39

0202 N2/)An

0

N2/n

0

eN2

1e

N2

1

0 A

b

N2/)z

0

error dzeN2

1bnobPrP 0

2

y(tm)

b


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Dr. Uri Mahlab 40

02m02m N2/)A)t(y

0

N2/)A)t(y

0

eN2

1e

N2

1

-A A 0)t(YobPrP merror 0)t(YobPr m

y(tm)

0y(tm)


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Dr. Uri Mahlab 41

02m02m N2/)A)t(y

0

N2/)A)t(y

0

eN2

1e

N2

1

-A A

y(tm)

VreceivedVTransmit


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Dr. Uri Mahlab 42

u

2

0N/A

2

e

A

N/xz0

2

0

Ax

0

2

0

e

dz2/zexp2

1uQ

N

AQdz2/zexp

2

1P

dxN2/xexpN2

1

dxN2/xexp

N2

12/1P

0

0

Q(u)


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Dr. Uri Mahlab 43

u

2

0N/A

2

e

dz2/zexp2

1uQ

N

AQdz2/zexp

2

1P

0

U

Q(u)

u

u

2dz2/zexp

2

1uQ

dz=


44/56

Dr. Uri Mahlab 44

RatioNoisetoSignalN

A

0

Perror decreases as0N/A increase

Hence we need to maximize the signalto noise Ratio

Thus for maximum noise immunity the filter transfer functions HT(f)

and HR(f) must be xhosen to maximize the SNR

O ti filt d i l l ti


45/56

Dr. Uri Mahlab 45

Optimum filters design calculations

We will express the SNR in terms of HT(f) and HR(f)

We will start with the signal:

k

bgk kTtpatX )()(

b

2

g

2

2

k

b

g

XT

)f(paaE

T

)f(p)f(G

The psd of the transmitted signal is given by::

)f(G)f(H)f(G X2

TX

And the average transmitted power ST is


46/56

Dr. Uri Mahlab 46

df)f(H)f(PTK

AS

df)f(H)f(PT

aS

2

T

2

g

b

2

c

2

T

aKAaKA

2

T

2

g

b

2

T

c

kck

df)f(H)f(P

TKSA2

T

2

g

b

2

cT2

The average output noise power of n0(t) is given by:

df)f(H)f(GN2

Rno


47/56

Dr. Uri Mahlab 47

The SNR we need to maximize is

)f(H)f(H)f(H)f(Pwhere

df)f(H)f(H

)f(Pdf)f(H)f(G

TSNA

TRcr

2

Rc

2

r2

Rn

bT

o

2

Or we need to minimize

22

Rc

2

r2

Rn mindf)f(H)f(H

)f(Pdf)f(H)f(Gmin


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Dr. Uri Mahlab 48

Using Schwartzs inequality

2

22

df)f(W)f(Vdf)f(Wdf)f(V

The minimum of the left side equaity is reached when

V(f)=const*W(f)If we choose :

)f(H)f(H

)f(P)f(W

)f(G)f(H)f(V

cR

r

2/1

nR


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Dr. Uri Mahlab 49

whenminimizedis2

constantpositivearbitraryanK

)f(P)f(HK

)f(G)f(PK

)f(H

)f(G)f(H)f(PK)f(H

2

gc

2/1

nr

2

c2

T

2/1

nc

r2R

The filter should have alinear phase response in a total time delay of td


50/56

Dr. Uri Mahlab 50

Finally we obtain the maximum value of the SNR to be:

maxo

2

error

2

c

2/1

nr

bT

maxo

2

N

AQP

df

)f(H

)f(G)f(P

TS

N

A


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Dr. Uri Mahlab 51

For AWGN with

and

pg(f) is chosen such that it does not change much over thebandwidth of interest we get.

)f(H

)f(pK)f(H

)f(H

)f(pK)f(H

c

r

2

2

T

c

r

1

2

R

Rectangular pulse can be used at the input of HT(f).

elsewhere0

T;2/tfor1)t(p

b

g

2/)f(Gn


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Dr. Uri Mahlab 52

5.2.3 Design procedure and Example

The steps involved in the design procedure.

Example:Design a binary baseband PAM system totransmit data at a a bit rate of 3600 bits/sec with a biterrorprobability less than .10 4

The channel response is given by:

elewhere

fforfHc

_0

240010)(

2

The noise spectral density is HzwattfGn /10)(14


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Dr. Uri Mahlab 53

Solution:

HzwattfG

HzB

p

bitsr

n

e

b

/10)(

2400

10

sec/3600

4

4

If we choose a braised cosine pulse spectrum with6006/ br

24001200),1200(

2400cos

2400,0

3600

1

1200,

3600

1

)( 2

ff

f

f

fpr

W h ( )


54/56

Dr. Uri Mahlab 54

We choose a pg(t)

973.0)2400(,)0(

)sin()(

)10)(28.0(10/;,0

1200,1)(

4

gg

g

b

g

pp

f

ffp

Telsewhere

ttp

2/1

2/1

1

)()(

)()(

fPfH

fpKfH

rR

rT

We choose

)()()()()(

)10)(3600( 31

fpfHfHfHfp

K

rRcTg

Pl f P (f) H (f) H (f) H (f) d P (f)


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Dr. Uri Mahlab 55

Plots of Pg(f),Hc(f),HT(f),HR(f),and Pr(f).

To maintain a4

10P


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D U i M hl b 56

To maintain a 10eP

2

4

142

2/1

max

0

2

max0

2

max0

2

4

max0

2

max0

2

)(10

10)06.14)(3600(

)(

)()()(

1

06.14)/(

75.3)/(

10)/((

)/(

dffPdffH

fGfP

N

A

TS

NA

NA

NAQ

NA

r

c

nr

b

T

For Pr

(f) with raised cosine shape

1)( dffPr

And hence dBmST 23)10)(3600)(06.14(10

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