1 Dr. Uri Mahlab

INTRODUCTION

In order to transmit digital information over* bandpass channels, we have to transfer

the 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

The modulation waveforms for transmitting :binary information over

bandpass channels

ASK

FSK

PSK

DSB

3 Dr. Uri Mahlab

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 signals 4 Dr. Uri Mahlab

Description of binary ASK,PSK, and :FSK schemes

-Bandpass binary data transmission system

ModulatorChannel

)Hc(fDemodulator

)receiver(

{ bk}

Binarydata

Input

{bk}

Transmitcarrier

Clock pulsesNoise

)n(t Clock pulses

Local carrier

Binary data output)Z(t

+

+

)V(t

+ 7ּ

5 Dr. Uri Mahlab

: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(:

1b if ])1([

0b if ])1([)(

k2

k1

b

b

Tkts

TktstZ

.1

6 Dr. Uri Mahlab

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

222

0

211

)]([

)]([Energy:Term

7 Dr. Uri Mahlab

: The received signal + noise

dbdb

db

db

tkTttTk

tntTkt

tntTkt

tV

)1(

)(])1([s

or

)(])1([s

)(

2

1

8 Dr. Uri Mahlab

Choice of signaling waveforms for various types of digital*modulation schemes S1(t),S2(t)=0 for

2

];,0[ ccb fTt

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

Type ofmodulation

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

: Receiver structure

Thresholddevice or A/D

converter

) V0(t

Filter)H(f output

Sample everyTb seconds

)()()( tntztv

10 Dr. Uri Mahlab

:}Probability of Error-{Pe*

The measure of performance used for comparing* !!! digital modulation schemes is the probability of error

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 * !!!chosen to minimize the probability of error

11 Dr. Uri Mahlab

: The output of the filter at t=kTb can be written as*

)()()( 000 bbb kTnkTskTV

12 Dr. Uri Mahlab

: The signal component in the output at t=kTb

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

Substituting Z(t) from equation 1 and making*change of the variable, the signal component

:will look like that

b

b

T

bb

T

bb

b

kTsdThs

kTsdThs

kTs

0

k012

0

k011

0

1b when )()()(

0b when )()()(

)(

14 Dr. Uri Mahlab

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

bkT

bb dkThnkTn )()()(0

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

:The variance of n0(t) is*

dffHfGtnEN n

2200 )()(){(}

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

n

NNnfn ;

2

n-exp

2

1)(

0

2

00

15

The probability that the kth bit is incorrectly decoded*:is given by

{1|)(}2

1

{0|)(}2

1

{)(V and 1

)(V and 0}

00

00

00

00

kb

kb

bk

bke

bTkTVP

bTkTVP

TkTbor

TkTbPP.2

16 Dr. Uri Mahlab

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

00

2020

0

01\

00

2010

0

00\

- , 2

)(V-exp

2

1)(

- , 2

)(V-exp

2

1)(

0

0

VN

s

NVf

VN

s

NVf

k

k

bV

bV

:It is similarly when bk is 1*

.3

17 Dr. Uri Mahlab

Combining equation 2 and 3 , we obtain an*:expression for the probability of error- Pe as

0

0

00

2020

0

00

2010

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

:Conditional pdf of V0 given bk

:The optimum value of the threshold T0* is*

20201*

0

SST

)( 0

00 v

kv bf )(k

0

01b v

vf

19 Dr. Uri Mahlab

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/)(

00

2010

0

2exp

2

1

2

)(exp

2

1

Nss

ss

e

dZZ

dVN

sV

NP

20 Dr. Uri Mahlab

The optimum filter is the filter that maximizes*the ratio or the square of the ratio

)maximizing eliminates the requirement S01<S02(

0

0102 )()(

N

TSTS bb

2

2

21 Dr. Uri Mahlab

:Transfer Function of the Optimum Filter* The probability of error is minimized by an*

appropriate choice of h(t) which maximizes

Where

0

201022 )]()([

N

TsTs bb

bT

bbb dThssTsTs0

120102 )()]()([)()(

And dffHfGN n

2

0 )()(

2

22 Dr. Uri Mahlab

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<0 and h( )=0 for <0*:the Fourier transform of P0 is

dffTjfHfPTP

fHfPfP

bb )2exp()()()(

)()()(

0

0

23 Dr. Uri Mahlab

:Hence can be written as* 22

2

2

)()(

)2exp()()(

dffGfH

dffTjfPfH

n

b (*)

We can maximize by applying Schwarz’s*:inequality which has the form

dffX

dffX

dffXfX2

2

2

2

1

21

)(

)(

)()(

(**)

2

24 Dr. Uri Mahlab

Applying Schwarz’s 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()()(

*

fG

fTjfPKfH

n

b

!!!Where K is an arbitrary constant

(***)

2

25 Dr. Uri Mahlab

Substituting equation (***) in(*) , we obtain-:the maximum value of as

2

dffG

fP

n )(

)(2

max2

:And the minimum probability of error is given by-

22exp

2

1 max2

2max/

QdZZ

Pe

26 Dr. Uri Mahlab

: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

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

:Impulse response of the Matched Filter*

)S2(t

)S1(t2\ Tb

2\ Tb

1

0

0

1-

2

0Tb

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

: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,for 0)( b h

b b

b

T T

T

b

dSVdSV

dSSVTV

0 0

12

0

120

)()()()(

)]()()[()(

(# #)

30 Dr. Uri Mahlab

Equation(# #) suggested that the optimum receiver can be implemented* as shown in Figure 1 .This form of the receiver is called

A Correlation Receiver

Thresholddevice

)A\D(

integrator

integrator

- +

Sampleevery 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* . 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

limitnoisepower

Thresholddevice

)A/D(R)Signal z(t

+

)n(t

+

WhiteGaussian

noise

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

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

sec2.0T ,Tt0 ,cos)(

/10 ,Tt0 ,cos)(

b b1

b2

mtwAtS

TwtwAtS

c

bcc

The carrier amplitude at the receiver input is 1 mvolt andthe 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

1110

33 Dr. Uri Mahlab

:Solution

34 Dr. Uri Mahlab

=Probability of error = Pe*

:Solution Continue

35 Dr. Uri Mahlab

* Binary ASK signaling schemes:

1b if ])1([

1)T-(k

0b if ])1([

)(

k2

b

k1

b

b

b

Tkts

kTt

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 c36 Dr. Uri Mahlab

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

: The model for D(t) is

k

bk Tktgbtd 1or 0b ],)1([)( k

elswhere 0

Tt0 1)( btg

)()( TtdtD

37 Dr. Uri Mahlab

: The power spectral density is given by

)()([4

)(2

cDcDz ffGffGA

fG

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)(

for 0

for 44

1

)(

38 Dr. Uri Mahlab

: The psd of Z(t) is given by

)

2

2

22

2

2

(

)(sin

)(

)(sin

)()((16

)(

cb

cB

cb

cb

cz

ffT

ffT

ffT

ffT

ffffA

fG

39 Dr. Uri Mahlab

If we use a pulse waveform D(t) in which the individual pulsesg(t) have the shape

elsewere 0

Tt0 )2cos(12)( b tra

tg b

40 Dr. Uri Mahlab

Coherent ASKWe start with The signal components of the receiver output at the

: of a signaling interval are

0)( and cos)( 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

bbb T

AkTskTsT

42

)()( 20201*

0

: The probability of error can be computed as eP

max2

1

22

22max

42exp

2

1

be

b

TAQdz

zp

TA

42 Dr. Uri Mahlab

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

4

2Asav

We can express the probability of error in terms of the: average signal power

bav

e

TSQp

The probability of error is sometimes expressed in* : terms of the average signal energy per bit , as

bavav TsE )(

av

e

EQP

43 Dr. Uri Mahlab

Noncoherent ASK: The input to the receiver is*

0b when )(

1b when )(cos)(

k

k

tn

tntAtV

i

ic

white.and Gaussian,

mean, zero be toassumed is which

inputreceiver at the noise the)( tni

44 Dr. Uri Mahlab

Noncoharent ASK Receiver

filter bandpass theof

output at the noise theis n(t)when

0A and 1bbit dtransmitte

kth when theA where

sin)(

cos)(cos

)(cos)(

:have output wefilter At the

kk

k

A

ttn

ttntA

tntAtY

cs

ccck

ck

45

:The pdf is

0r ,2

exp)(

0r ,2

exp)(

0

22

00

01|

0

2

00|

N

Ar

N

ArI

N

rrf

N

r

N

rrf

k

k

bR

bR

BT T

BN

N

2

filter. bandpass

theofoutput at thepower noise

0

0

2

0

0 ))cos(exp(2

1)( duuxXI

46 Dr. Uri Mahlab

pdf’s of the envelope of the noise and the signal* : pulse noise

47 Dr. Uri Mahlab

2

2exp

)(

ionapproximat theUsing

22

)(exp

2

1

and

8exp

2exp

where2

1

2

1

)1b|error(2

1)0b|error(

2

1

2

2

00

2

0

1

20

2

0

2

00

10

kk

x

x

xQ

N

AQdr

N

Ar

Np

N

Adr

N

r

N

rp

pp

ppp

A

e

Ae

ee

e

: The probability of error is given by

48 Dr. Uri Mahlab

02

0

2

0

2

20

0

2

20

1

1

A if 8

exp2

1

8exp

2

41

2

1

Hence,

8exp

2

4

to reducecan we x,largefor

NN

A

N

A

A

Np

N

A

A

Np

p

e

e

e

49 Dr. Uri Mahlab

BINERY PSK SIGNALING SCHEMES

: The waveforms are*

0bfor cos)(

1bfor cos)(

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

: The power spectral density of PSK signal is

b

bD

cDcDZ

Tf

fTfG

Where

ffGffGA

fG

22

2

2

sin)(

,

)]()([4

)(

51 Dr. Uri Mahlab

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

: The probability of error is given by

bav

av

av

be

T

bc

e

TA

E

A

E

s

TAQp

TAdttA

QP

b

2

and2

s

are scheme

PSK for the bit per energy signal

theend power signal average The

or

4)cos2(

2

where

2

2

2

av

2

0

222

max

max

53 Dr. Uri Mahlab

av

bave

EQ

Tsp

2

2

:error ofy probabilit theexpresscan we

54 Dr. Uri Mahlab

DELAY

LOGICNETWORK

LEVELSHIFT

bT

BINERYSEQUENCE

1or o

dk

1kd

1

tA ccos

tA Ccos

Z(t)

DIFFERENTIALLY COHERENT* : PSK

DPSK modulator

55 Dr. Uri Mahlab

DPSK demodulator

Filter tolimit noise

power

Delay

Lowpassfilter or

integrator

Thresholddevice

(A/D)

Z(t)

)(tn

bT

kb̂

bkTat

sample

56 Dr. Uri Mahlab

Differential encoding & decoding

InputSeque-nce

1 1 0 1 0 0 0 1 1Encodedsequence 1 1 1 0 0 1 0 1 1 1TransmitPhase 0 0 0 pi pi 0 pi 0 0 0PhaseCompari-sonoutput

+ + - + - - - + +OutputBitsequence 1 1 0 1 0 0 0 1 1

57 Dr. Uri Mahlab

* 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

0bfor 1

1bfor 1)(

k

ktD

58 Dr. Uri Mahlab

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

Coherent FSK: The local carrier signal required is

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

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

b

b

T

b

T

b

dttststskTs

dttststskTs

0

12101

0

12202

)]()()[()(

)]()()[()(

60 Dr. Uri Mahlab

The probability of error for the correlation receiver is : given by

)cos()(

and )cos()(

when

)]()([2

where

2

1

2

0

212

2max

max

ttAts

ttAts

dttsts

QP

dc

dc

T

e

b

61 Dr. Uri Mahlab

. Which are usually encountered in practical system

: We now have

bd

bdb

T

TTA

2

2sin1

2 22max

62dbc wTw c w, 1

:When

Dr. Uri Mahlab

Noncoherent FSK

0r ,2

exp)(

and

0r ,2

exp)(

:isfilter bottom theof )(R envelope theof pdf theinterval,

signalingkth theduring mittedbeen trans has )cos()( that Assuming

20

22

0

22|

10

221

0

10

0

11)(|

1

1

12

11

N

r

N

rrf

n

Ar

N

ArI

N

rrf

kT

tAts

sR

tsR

b

dc

63 Dr. Uri Mahlab

Noncoharenr demodulator of binary FSK

ENVELOPEDETECTOR

ENVELOPEDETECTOR

THRESHOLDDEVICE

(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

Probability of error for binary digital modulation* :schemes

65 Dr. Uri Mahlab

M-ARY SIGNALING SCHEMES

: M-ARY coherent PSK

The M possible signals that 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

k k

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

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

S

c

c

c

c

Tt

tAtS

tAtS

tAtS

tAtS

0 allfor

sin)(

cos)(

sin)(

cos)(

4

3

2

1

67 Dr. Uri Mahlab

Phasor diagram for QPSK

)45cos( and )45cos( 00 tAtA cc That are derived from a coherent local carrier

reference tA ccos

68

If we assume that S 1 was the transmitted signal: during the signaling interval (0,Ts),then we have

0

2

0

01

4cos

2

)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

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

S

s

T

cs

T

cs

ss

sss

sss

dttAtnTn

dttAtnTn

TnTn

TnTSTV

TnTSTV

0

002

0

001

0201

020202

010101

)45cos()()(

)45cos()()(

by defined variablesrandomGaussian mean zero are )( & )( where

)()()(

)()()(

71 Dr. Uri Mahlab

Probability of error of QPSK:

2

2

0

0

002

0011

2N

LQ

))((

))((

ecs

s

sec

PTA

Q

LTnP

LTnPP

72 Dr. Uri Mahlab

sin2

4Mfor

2221

:is system for the

)1)(1(

correctly received is signal ed transmitty that theprobabilit The

22

2

1

21

M

TAQP

TAQPPP

P

PPP

- P

se

secce

e

ececc

c

73 Dr. Uri Mahlab

Phasor diagram for M-ary PSK ; M=8

74 Dr. Uri Mahlab

The average power requirement of a binary PSK : scheme are given by

sin

1

)(

)(

Z& small very is If

sin

1

Z)(

)(

2

21

222

21

MS

S

ZP

M

Z

S

S

bav

Mav

e

bav

Mav

75 Dr. Uri Mahlab

*COMPARISION OF POWER-BANDWIDTH: FOR M-ARY PSK

410eP

Valueof M b

M

Bandwidth

Bandwidth

)(

)(

bav

mav

S

S

)(

)(

48

1632

0.50.333

0.250.2

0.34 dB3.91 dB8.52 dB

13.52 dB

76 Dr. Uri Mahlab

* M-ary for four-phase Differential PSK:

RECEIVER FOR FOUR PHASE DIFFERENTIAL PSK

Integrateand dump

filter

ST

Delay

ST

Delay

shift

phase

090

Integrateand dump

filter

)(01 tV

)(02 tV

)(tn

Z(t)

77 Dr. Uri Mahlab

: The probability of error in M-ary differential PSK

M

TAQP S

e 2sin22 2

2

: The differential PSK waveform is

)cos()()( kk

cS tkTtgAtZ

78 Dr. Uri Mahlab

: Transmitter for differential PSK*

Serial toparallel

converter

Diffphasemod.

Envelopemodulator BPF

(Z(t

3

4

2400br

Data

Binary

Clocksignal

2400 Hz

4

1200

M

rs

Hzfc 1800

600 Hz

79 Dr. Uri Mahlab

* M-ary Wideband FSK Schemas: Let us consider an FSK scheme witch have the

: following properties

ST

0

2

s

FOR 0

FOR 2)()(

elsewhere 0

Tt0 cos)(

ji

jiTA

tStS

and

tAtS

S

ji

ii

80 Dr. Uri Mahlab

:Orthogonal Wideband FSK receiver

MAXIMUMSELECTOR

ST

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

component noise The-)(

outputfilter th -j theofcomponent signal The-)(

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

42

0

sTAN

: The probability of correct decoding as

-

11|andsent

112

1113121

)({|,...,}

{ |,...,,}

11

1

11dyyfyYyYP

sentSYYYYYYpP

SYs

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* : is given by

M

iiYMyYSYY yfyyf

iM2

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

84 Dr. Uri Mahlab

s

s

SY

Y

SY

My

Y

SY

y y M

iiiYc

ii

iY

TA

S

TA

N

yN

Sy

Nyf

yN

y

Nyf

dyyfdyyf

dyyfdyyfP

yN

y

Nyf

i

i

2

22

and

, 2

)(exp

2

1)(

, 2

exp2

1)(

where

)()(

)()(...

and

, 2

exp2

1)(

2

01

2

0

10

2011

0

1|

0

2

0

-

11|

1

11|2

1

0

2

0

11

11

1

11

1 1

where

85 Dr. Uri Mahlab

Probability of error for M-ary orthogonal* : signaling scheme

86 Dr. Uri Mahlab

The probability that the receiver incorrectly* decoded the incoming signal S1(t) is

Pe1 = 1-Pe1

The probability that the receiver makes * an error in decoding is

Pe = Pe1

We assume that , and We can see that increasing values of M lead to smaller power requirements and also to more complex transmitting receiving equipment .

2M )inteegr positive a ( log2 ssb rMrr

87 Dr. Uri Mahlab

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

7.0r /S if 0

7.0/S if 1

bav

av

b

e

r

P

The maximum errorless rb at W data can be transmittedusing an M- ary orthogonal FSK signaling scheme

eSS

r avavb 2log

7.0

The bandwidth of the signal set as M 88 Dr. Uri MahlabDr. Uri Mahlab