d.comm Pass Band

8/2/2019 d.comm Pass Band

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1 Dr. Uri Mahlab


2/88

INTRODUCTION

In order to transmit digital information over *

bandpass channels, we have to transferthe information to a carrier wave of

.appropriate frequency

We will study some of the most commonly *

used digital modulation techniques wherein

the digital information modifies the amplitude

the phase, or the frequency of the carrier in

.discrete steps

2 Dr. Uri Mahlab


3/88

transmittingThe modulation waveforms for

binary information over bandpass channels:

ASK

FSK

PSK

DSB

3 Dr. Uri Mahlab


4/88

OPTIMUM RECEIVER FOR BINARY

DIGITAL MODULATION SCHEMS:

The function of a receiver in a binary communication *system is to distinguish between two transmitted signals

.S1(t) and S2(t) in the presence of noise

The performance of the receiver is usually measured *

in terms of the probability of error and the receiver

is said to be optimum if it yields the minimum

.probability of error

In this section, we will derive the structure of an optimum *

receiver that can be used for demodulating binary

.ASK,PSK,and FSK signals4 Dr. Uri Mahlab


5/88

Description of binary ASK,PSK, and

FSK schemes:

-Bandpass binary data transmission system

ModulatorChannel

(Hc(f

Demodulator

(receiver)

{bk}

Binary

data

Input

{bk}

Transmit

carrier Clock pulses Noise(n(t Clock pulses

Local carrier

Binary data output(Z(t +

+

(V(t

+

5 Dr. Uri Mahlab


6/88

*Explanation:The input of the system is a binary bit sequence {bk} with a *

.bit rate r b and bit duration Tb

The output of the modulator during the Kth bit interval *

.depends on the Kth input bit bk

The modulator output Z(t) during the Kth bit interval is *

a shifted version of one of two basic waveforms S1(t) or S2(t) and

:Z(t) is a random process defined by

bb kTtTkfor )1(:

1bif])1([

0bif])1([)(

k2

k1

b

b

Tkts

TktstZ

.1

6 Dr. Uri Mahlab


7/88

The waveforms S1(t) and S2(t) have a duration *

of Tb and have finite energy,that is,S1(t) and S2(t) =0

],0[ bTtif and

b

b

T

T

dttsE

dttsE

0

2

22

0

2

11

)]([

)]([Energy

:Term

7 Dr. Uri Mahlab


8/88

:The received signal + noise

dbdb

db

db

tkTttTk

tntTkt

tntTkt

tV

)1(

)(])1([s

or

)(])1([s

)(

2

1

8 Dr. Uri Mahlab


9/88

Choice of signaling waveforms for various types of digital*

modulation schemes

for0(t)=2(t),S1S

2];,0[ ccb fTt

.The frequency of the carrier fc is assumed to be a multiple of rb

Type of

modulation

ASK

PSK

FSK

bTtTS 0);(1 bTtts 0);(2

)sinor(

cos

twA

twA

c

c

)sin(

cos

twAor

twA

c

c

0

)sin(

cos

twA

twA

c

c

}])sin{([(

})cos{(

twwAor

twwA

dc

dc

}])sin{(or[

})cos{(

twwA

twwA

dc

dc

9 Dr. Uri Mahlab


10/88

Receiver structure:

Threshold

device or A/D

converter

(V0(t

Filter

(H(f output

Sample every

Tb seconds

)()()( tntztv

10 Dr. Uri Mahlab


11/88

:{Probability of Error-{Pe*

The measure of performance used for comparing *

probability of error!!!digital modulation schemes is the

The receiver makes errors in the decoding process *!!! due to the noise present at its input

The receiver parameters as H(f) and threshold setting are *

minimize the probability of error!!!chosen to

11 Dr. Uri Mahlab


12/88

*can be written asbThe output of the filter at t=kT:

)()()( 000 bbb kTnkTskTV

12 Dr. Uri Mahlab


13/88

b:The signal component in the output at t=kT

bkT

bb dkThZkTs )()()(0

termsISI)()()1

dkThZ b

kT

Tk

b

b

h( ) is the impulse response of the receiver filter*

ISI=0*

b

b

kT

Tk

bb dkThZkTs)1(

0 )()()(

13 Dr. Uri Mahlab


14/88

and making*1Substituting Z(t) from equation

change of the variable, the signal component

:will look like that

b

b

T

bb

T

bb

b

kTsdThs

kTsdThskTs

0

k012

0

k011

0

1bwhen)()()(

0bwhen)()()()(

14 Dr. Uri Mahlab


15/88

:The noise component n0(kTb) is given by *

bkT

bbdkThnkTn )()()(

0

he output noise n0(t) is a stationary zero mean Gaussian random process

:The variance of n0(t) is*

dffHfGtnEN n22

00 )()()}({

:The probability density function of n0(t) is*

n

NNnfn ;

2

n-exp

2

1)(

0

2

0

0

15


16/88

The probability that the kth bit is incorrectly decoded*

:is given by

}1|)({2

1

}0|)({2

1

})(Vand1

)(Vand0{

00

00

00

00

kb

kb

bk

bke

bTkTVP

bTkTVP

TkTbor

TkTbPP.2

16 Dr. Uri Mahlab


17/88

:The conditional pdf of V0 given bk = 0 is given by*

0

0

2

020

0

01\

0

0

2010

0

00\

-,

2

)(V-exp

2

1)(

-,2

)(V-exp

2

1)(

0

0

V

N

s

N

Vf

VN

s

NVf

k

k

bV

bV

:It is similarly when bk is 1*

.3

17 Dr. Uri Mahlab


18/88

Combining equation 2 and 3 , we obtain an*

:expression for the probability of error- Pe as

0

0

0

0

2

020

0

0

0

2

010

0

2

)(V-exp

2

1

2

1

2

)(V-exp

2

1

2

1

T

T

e

dVN

S

N

dVN

S

NP

.4

18 Dr. Uri Mahlab


19/88

:Conditional pdf of V0 given bk

:The optimum value of the threshold T0* is*

2

0201*0

SST

)( 0

00

v

kv bf )(

k0

01b vvf

19 Dr. Uri Mahlab


20/88

Substituting the value of T*0 for T0 in equation 4*

we can rewrite the expression for the probability

:of error as

00102

0102

2/)(

2

2/)(

0

0

2

010

0

2exp

21

2

)(exp

2

1

Nss

sse

dZZ

dV

N

sV

NP

20 Dr. Uri Mahlab


21/88


22/88


23/88

If we let P(t) =S2(t)-S1(t), then the numerator of the*

:quantity to be maximized is

bT

bb

bbb

dThPdThP

TPTSTS

0

00102

)()()()(

)()()(

Since P(t)=0 for t


24/88

:Hence can be written as*22

2

2

)()(

)2exp()()(

dffGfH

dffTjfPfH

n

b

(*)

We can maximize by applying Schwarzs*

:inequality which has the form

dffX

dffX

dffXfX2

2

2

2

1

21

)(

)(

)()(

(**)

2

24 Dr. Uri Mahlab

A l i S h i li E i (**) i h


25/88

Applying Schwarzs inequality to Equation(**) with-

)(

)2exp()()(

)()()(

2

1

fG

fTjfPfX

fGfHfX

n

b

n

and

We see that H(f), which maximizes ,is given by-

)()2exp()()(

*

fGfTjfPKfH

n

b

!!! Where K is an arbitrary constant

(***)

2

25 Dr. Uri Mahlab


26/88

Substituting equation (***) in(*) , we obtain-

:the maximum value of as2

dffG

fP

n )(

)(2

max2

:And the minimum probability of error is given by-

22exp

21 max

2

2max/

QdZZPe

26 Dr. Uri Mahlab


27/88

:Matched Filter Receiver*

If the channel noise is white, that is, Gn(f)= /2 ,then the transfer -

:function of the optimum receiver is given by

)2exp()()( * bfTjfPfH

From Equation (***) with the arbitrary constant K set equal to /2-

:The impulse response of the optimum filter is

dfjftjfTfPth b )2exp()]2exp()([)(

*

27 Dr. Uri Mahlab


28/88

Recognizing the fact that the inverse Fourier *

of P*(f) is P(-t) and that exp(-2 jfTb) represent

:a delay of Tb we obtain h(t) as

)()( tTpth b :Since p(t)=S1(t)-S2(t) , we have*

)()()( 12 tTStTSth bb

The impulse response h(t) is matched to the signal *

:S1(t) and S2(t) and for this reason the filter is called

MATCHED FILTER28 Dr. Uri Mahlab

I l f th M t h d Filt *


29/88

:Impulse response of the Matched Filter *

(S2(t

(S1(t 2 \Tb

2 \Tb

1

0

0

1-

2

0

Tb

t

t

t

t

t

(a)

(b)

(c)

2 \Tb(P(t)=S2(t)-S1(t

(P(-tTb- 0

2

(d)

2 \Tb0

Tb

(h(Tb-t)=p(t

2

(e)

(h(t)=p(Tb-t

29 Dr. Uri Mahlab

C l ti R i *


30/88

:Correlation Receiver*

bT

bb dThVTV )()()(0

The output of the receiver at t=Tb*

Where V( ) is the noisy input to the receiver

Substituting and noting *: that we can rewrite the preceding expression as

)()()( 12 bb TSTSh

)T(0,for0)( b h

b b

b

T T

T

b

dSVdSV

dSSVTV

0 0

12

0120

)()()()(

)]()()[()(

(# #)

30 Dr. Uri Mahlab


31/88

Equation(# #) suggested that the optimum receiver can be implemented *

as shown in Figure 1 .This form of the receiver is called

A Correlation Receiver

Threshold

device

(A\D)

integrator

integrator

-

+Sample

every Tb

seconds

bT

0

bT

0

)(1 tS

)(2 tS

)()(

)()(

)(

2

1

tntS

or

tntS

tV

Figure 1

31 Dr. Uri Mahlab

In actual practice the receiver shown in Figure 1 is actually *


32/88

In actual practice, the receiver shown in Figure 1 is actually *

.implemented as shown in Figure 2

In this implementation, the integrator has to be reset at the

- (end of each signaling interval in order to ovoid (I.S.I

!!! Inter symbol interference

:Integrate and dump correlation receiver

Filterto

limit

noise

power

Threshold

device

(A/D)R(Signal z(t

+

(n(t

+

White

Gaussiannoise

High gainamplifier)()( 21 tStS

Closed every Tb seconds

c

Figure 2

The bandwidth of the filter preceding the integrator is assumed *

!!! to be wide enough to pass z(t) without distortion

32


33/88

Example: A band pass data transmission schemeuses a PSK signaling scheme with

sec2.0T,Tt0,cos)(

/10,Tt0,cos)(

bb1

b2

mtwAtS

TwtwAtS

c

bcc

The carrier amplitude at the receiver input is 1 mvolt and

the psd of the A.W.G.N at input is watt/Hz. Assumethat an ideal correlation receiver is used. Calculate the

.average bit error rate of the receiver

11

10

33 Dr. Uri Mahlab

S l ti


34/88

:Solution

34 Dr. Uri Mahlab

:Solution Continue


35/88

=Probability of error = Pe *

:Solution Continue

35 Dr. Uri Mahlab

* Binary ASK signaling


36/88

* Binary ASK signaling

schemes:

1bif])1([1)T-(k

0bif])1([

)(k2

b

k1

b

b

b

TktskTt

Tkts

tz

The binary ASK waveform can be described as

Where andtAtS ccos)(2 0)(1 ts

We can represent

:Z(t) as

)cos)(()( tAtDtZ c

36 Dr. Uri Mahlab


37/88

Where D(t) is a lowpass pulse waveform consisting of

.rectangular pulses

:The model for D(t) is

k

bk Tktgbtd 1or0b],)1([)( k

elswhere0

Tt01)(

btg

)()( TtdtD

37 Dr. Uri Mahlab

Th l d i i i b


38/88

:The power spectral density is given by

)()([4

)(2

cDcDzffGffG

AfG

The autocorrelation function and the power spectral density

:is given by

b

bD

b

bb

b

DD

Tf

fTffG

T

TT

T

R

22

2sin

)(

4

1)(

for0

for44

1

)(

38 Dr. Uri Mahlab


39/88

:The psd of Z(t) is given by

)

2

2

22

2

2

()(sin

)(

)(sin

)()((16

)(

cb

cB

cb

cb

cz

ffTffT

ffT

ffT

ffffA

fG

39 Dr. Uri Mahlab


40/88

If we use a pulse waveform D(t) in which the individual pulses

g(t) have the shape

elsewere0

Tt0)2cos(12)(

b tra

tgb

40 Dr. Uri Mahlab

C h t ASK


41/88

Coherent ASKWe start with

The signal components of the receiver output at the

:of a signaling interval are

0)(andcos)(12

tstAts c

b

b

T

bb

T

b

TA

dttststskT

dttststskTs

0

2

122O2

0

12101

2)]()()[()(S

and

0)]()()[()(

41 Dr. Uri Mahlab

:The optimum threshold setting in the receiver is


42/88

:The optimum threshold setting in the receiver is

b

bb TAkTskTs

T42

)()( 20201*0

:The probability of error can be computed aseP

max2

1

22

22

max

42exp

2

1

be

b

TAQdz

zp

TA

42 Dr. Uri Mahlab


43/88

:The average signal power at the receiver input is given by

4

2A

sav We can express the probability of error in terms of the

:average signal power

bave

TSQp

The probability of error is sometimes expressed in *

: terms of the average signal energy per bit , as

bavav TsE )(

av

eEQP

43 Dr. Uri Mahlab


44/88

N h t ASK R i


45/88

Noncoharent ASK Receiver

filterbandpasstheof

outputat thenoisetheisn(t)when

0Aand1bbitdtransmitte

kthwhen theAwheresin)(

cos)(cos

)(cos)(

:haveoutput wefilterAt the

kk

k

Attn

ttntA

tntAtY

cs

ccck

ck

45


46/88

:The pdf is

0r,2

exp)(

0r,2

exp)(

0

22

0

0

0

1|

0

2

0

0|

N

Ar

N

ArI

N

rrf

N

r

N

rrf

k

k

bR

bR

B

TTBN

N

2

filter.bandpasstheofoutputat thepowernoise

0

0

2

0

0 ))cos(exp(

2

1)( duuxXI

46 Dr. Uri Mahlab


47/88

pdfs of the envelope of the noise and the signal

:pulse noise

47 Dr. Uri Mahlab

:The probability of error is given by


48/88

2

2exp

)(

ionapproximattheUsing

22

)(exp

2

1

and

8exp

2exp

where

2

1

2

1

)1b|error(2

1)0b|error(

2

1

2

2

00

2

0

1

2

0

2

0

2

0

0

10

kk

x

x

xQ

N

AQdr

N

Ar

N

p

N

Adr

N

r

N

rp

pp

ppp

A

e

A

e

ee

e


48 Dr. Uri Mahlab


49/88

0

2

0

2

0

2

2

0

0

2

2

01

1

Aif8

exp2

1

8exp

2

41

2

1

Hence,

8exp

24

toreducecanwex,largefor

NN

A

N

A

A

Np

NA

ANp

p

e

e

e

49 Dr. Uri Mahlab


50/88

BINERY PSK SIGNALING

SCHEMES:The waveforms are *

0bforcos)(

1bforcos)(

k2

k1

tAts

tAts

c

c

:The binary PSK waveform Z(t) can be described by *

)cos)(()( tAtDtZ c

.D(t) - random binary waveform *

50 Dr. Uri Mahlab


51/88

:The power spectral density of PSK signal is

b

bD

cDcDZ

TffTfG

Where

ffGffGA

fG

22

2

2

sin)(

,

)]()([4

)(

51 Dr. Uri Mahlab


52/88

Coherent PSK:The signal components of the receiver output are

b

b

b

b

kT

Tk

bb

kT

Tk

bb

TAdttststskTs

TAdttststskTs

)1(

212202

)1(

212101

)]()()[()(

)]()()[()(

52 Dr. Uri Mahlab



53/88


bav

av

av

be

T

bc

e

TA

E

A

E

s

TAQp

TAdttA

QP

b

2

and

2s

arescheme

PSKfor thebitperenergysignal

theendpowersignalaverageThe

or

4)cos2(

2

where

2

2

2

av

2

0

222

max

max

53 Dr. Uri Mahlab


54/88

av

bave

E

Q

Tsp

2

2

:errorofyprobabilittheexpresscanwe

54 Dr. Uri Mahlab

DIFFERENTIALLY COHERENT *


55/88

DELAY

LOGIC

NETWORK

LEVEL

SHIFT

bT

BINERY

SEQUENCE

1oro

dk

1kd

1

tA ccos

tA Ccos

Z(t)

DIFFERENTIALLY COHERENT

:PSK

DPSK modulator

55 Dr. Uri Mahlab


56/88

DPSK demodulator

Filter to

limit noise

power

Delay

Lowpass

filter orintegrator

Threshold

device

(A/D)

Z(t)

)(tn

bT

kb

bkTatsample

56 Dr. Uri Mahlab


57/88

Differential encoding & decoding

Input

Seque-nce

1 1 0 1 0 0 0 1 1

Encodedsequence 1 1 1 0 0 1 0 1 1 1

Transmit

Phase 0 0 0 pi pi 0 pi 0 0 0

PhaseCompari-son

output+ + - + - - - + +

Output

Bit

sequence1 1 0 1 0 0 0 1 1

57 Dr. Uri Mahlab


58/88

* BINARY FSK SIGNALING

SCHEMES ::The waveforms of FSK signaling

1bfor)cos()(

0bfor)cos()(

k2

k1

ttAtS

ttAtS

dC

dc

:Mathematically it can be represented as

')'(cos)( dttDtAtZ dc

0bfor1

1bfor1)(

k

ktD

58 Dr. Uri Mahlab

P t l d it f FSK i l


59/88

Power spectral density of FSK signals

Power spectral density of a binary FSK signal

with bd rf 2

59

2

2

ee

dd

wf

wf

Dr. Uri Mahlab


60/88

Coherent FSK:The local carrier signal required is

)cos()cos()()( 12 ttAttAtsts dcdc

The input to the A/D converter at sampling timewhere)(or)(is 0201 bbb kTskTskTt

b

b

T

b

T

b

dttststskTs

dttststskTs

0

12101

0

12202

)]()()[()(

)]()()[()(

60 Dr. Uri Mahlab

Th b bilit f f th l ti i i


61/88

The probability of error for the correlation receiver is

:given by

)cos()(

and)cos()(when

)]()([2

where

2

1

2

0

2

12

2

max

max

ttAts

ttAts

dttsts

QP

dc

dc

T

e

b

61 Dr. Uri Mahlab

.Which are usually encountered in practical system


62/88

y p y

:We now have

bd

bdb

TTTA

22sin12

22

max

62dbc wTw cw,1

:When

Dr. Uri Mahlab


63/88

Noncoherent FSK

0r,2

exp)(

and

0r,2

exp)(

:isfilterbottomtheof)(Renvelopetheofpdftheinterval,

signalingkththeduringmittedbeen transhas)cos()(thatAssuming

2

0

2

2

0

22|

1

0

22

1

0

10

0

11)(|

1

1

12

11

N

r

N

rrf

nAr

NArI

Nrrf

kT

tAts

sR

tsR

b

dc

63 Dr. Uri Mahlab

N h d d l t f bi FSK


64/88

Noncoharenr demodulator of binary FSK

ENVELOPE

DETECTOR

ENVELOPE

DETECTOR

THRESHOLD

DEVICE

(A/D)

dc ff

filter

Bandpass

dc ff

filter

bandpass

+

-

)(2 bkTR

)(1 bkTR

0*0 T

Z(t)+n(t)

0

2

4exp

2

1

N

APe

64 Dr. Uri Mahlab

P b bilit f f bi di it l d l ti *


65/88

Probability of error for binary digital modulation *

:schemes

65 Dr. Uri Mahlab


66/88

M-ARY SIGNALING

SCHEMESARY coherent PSK-:MThe M possible signalsthat would be transmitted

:during each signaling interval of duration Ts are

sTt0,1,...1,0,2

cos)(

Mk

M

ktAtS ck

:The digital M-ary PSK waveform can be represented

k

kcs tkTtgAtZ )cos()()(

66 Dr. Uri Mahlab


67/88

k k

skcskc kTtgtAkTtgtAtZ )()(sinsin)()(coscos)(

:In four-phase PSK (QPSK), the waveform are

S

c

c

c

c

Tt

tAtS

tAtS

tAtS

tAtS

0allfor

sin)(

cos)(

sin)(

cos)(

4

3

2

1

67 Dr. Uri Mahlab


68/88

If th t S th t itt d i l


69/88

If we assume that S 1 was the transmitted signal

:during the signaling interval (0,Ts),then we have

0

2

0

01

4cos2

)4

cos()cos()(

LTA

dttAtATS

s

T

ccs

s

0

2

0

02

4cos

2

A

4cos)cos()(

LT

dttAtATS

s

T

ccs

s

69 Dr. Uri Mahlab

QPSK i h


70/88

Z(t)

)(tn

)45cos( tA c

)45cos( tA c

ST

0

ST

0

)(01 SkTV

)(02 SkTV

QPSK receiver scheme

70 Dr. Uri Mahlab

:The outputs of the correlators at time t=TS are


71/88

:The outputs of the correlators at time t TS are

S

s

T

cs

T

cs

ss

sss

sss

dttAtnTn

dttAtnTn

TnTn

TnTSTV

TnTSTV

0

0

02

0

0

01

0201

020202

010101

)45cos()()(

)45cos()()(

bydefinedvariablesrandomGaussianmeanzeroare)(&)(where

)()()(

)()()(

71 Dr. Uri Mahlab


72/88

Probability of error of

QPSK:

2

2

0

0

002

0011

2NLQ

))((

))((

ecs

s

sec

PTAQ

LTnP

LTnPP

72 Dr. Uri Mahlab

correctlyreceivedissignaledtransmitty that theprobabilitThe-Pc


73/88

sin2

4Mfor

2221

:issystemfor the

)1)(1(

ygyp

22

2

1

21

M

TAQP

TAQPPP

P

PPP

se

s

ecce

e

ececc

c

73 Dr. Uri Mahlab

Phasor diagram for M-ary PSK ; M=8


74/88

Phasor diagram for M-ary PSK ; M=8

74 Dr. Uri Mahlab

The average power requirement of a binary PSK


75/88

The average power requirement of a binary PSK

:scheme are given by

sin

1)()(

Z&smallveryisIf

sin

1

Z)(

)(

2

21

22

2

2

1

M

SS

ZP

M

Z

S

S

bav

Mav

e

bav

Mav

75 Dr. Uri Mahlab

* COMPARISION OF POWER-BANDWIDTH


76/88

* COMPARISION OF POWER-BANDWIDTH

:FOR M-ARY PSK4

10

eP

Value

of M bM

Bandwidth

Bandwidth

)(

)(

bav

mav

S

S

)(

)(

4

8

16

32

0.5

0.333

0.25

0.2

0.34 dB

3.91 dB

8.52 dB

13.52 dB

76 Dr. Uri Mahlab

* M ary for four phase


77/88

* M-ary for four-phase

Differential PSK:RECEIVER FOR FOUR PHASE DIFFERENTIAL PSK

Integrate

and dump

filter

STDelay

ST

Delay

shiftphase

090

Integrate

and dump

filter

)(01 tV

)(02 tV

)(tn

Z(t)

77 Dr. Uri Mahlab

:The probability of error in M-ary differential PS


78/88

:The probability of error in M-ary differential PS

M

TAQP Se2

sin22 22

:The differential PSK waveform is

)cos()()( kk

cS tkTtgAtZ

78 Dr. Uri Mahlab

:Transmitter for differential PSK*


79/88

:Transmitter for differential PSK*

Serial to

parallel

converter

Diff

phase

mod.

Envelope

modulatorBPF

(Z(t

3

4

2400br

Data

Binary

Clock

signal

2400 Hz

4

1200

M

rsHzfc 1800

600 Hz

79 Dr. Uri Mahlab

* M Wid b d FSK


80/88

* M-ary Wideband FSK

Schemas:Let us consider an FSK scheme witch have the: following properties

ST

0

2

s

FOR0

FOR2)()(

elsewhere0Tt0cos)(

ji

jiTAtStS

and

tAtS

S

ji

ii

80 Dr. Uri Mahlab

:Orthogonal Wideband FSK receiver


81/88

g

MAXIMUM

SELECTORST

0

ST

0

ST

0

Z(t)

)(tn

noise

gausian

)(1 tS

)(2 tS

)(tSM

.

.

.

.

)(1 tY

)(2 tY

)(tYM

81 Dr. Uri Mahlab

:The filter outputs are


82/88

p

componentnoiseThe-)(

outputfilterth-jtheofcomponentsignalThe-)(

where

)()(

)()()()(

M1,2,....,j,)]()()[()(

0

0

0 0

1

0

1

S

sj

sj

sjsj

T T

jj

T

jsj

Tn

TS

TnTS

dttntSdttStS

dttStntSTY

S S

s

82 Dr. Uri Mahlab

:N0 is given by


83/88

4

20

sTAN

:The probability of correct decoding as

-

11|andsent

112

1113121

)(}|,...,{

}|,...,,{

11

1

11dyyfyYyYP

sentSYYYYYYpP

SY

s

yYM

Mc

:In the preceding step we made use of the identity

dyyfyYyXPYXP Y )()|()(

83 Dr. Uri Mahlab

The joint pdf of Y2 Y3 YM *


84/88

The joint pdf of Y2 ,Y3 ,,YM

:is given by

M

i

iYMyYSYY yfyyf iM2

2:|...2 )(),...,(111

84 Dr. Uri Mahlab

ii

iY yy

yf ,exp1

)(2

where


85/88

s

s

SY

Y

SY

My

Y

SY

y y M

iiiYc

iiY

TA

S

TA

N

yN

Sy

Nyf

yN

y

Nyf

dyyfdyyf

dyyfdyyfP

yNN

yf

i

i

2

22

and

,2

)(exp

2

1)(

,2

exp2

1)(

where

)()(

)()(...

and

,2

p2

)(

2

01

2

0

1

0

2

011

0

1|

0

2

0

-

11|

1

11|2

1

00

11

11

1

11

1 1

85 Dr. Uri Mahlab

Probability of error for M-ary orthogonal *

i i


86/88

: signaling scheme

86 Dr. Uri Mahlab

The probability that the receiver incorrectly *

d d d h i i i l S ( ) i


87/88

decoded the incoming signal S1(t) is

Pe1 = 1-Pe1The probability that the receiver makes *

an error in decoding is

Pe = Pe1

assume that , and

can see that increasing values of M lead to smaller poweruirements and also to more complex transmitting

eiving equipment.

2M )inteegrpositivea(log 2 ssb rMrr

87 Dr. Uri Mahlab

In the limiting case as M the probability of error Pe satisfies


88/88

g p y

7.0r/Sif0

7.0/Sif1

bav

av

b

e

r

P

The maximum errorless rb at W data can be transmitted

using an M- ary orthogonal FSK signaling scheme

e

SS

ravav

b 2log7.0

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