jxu

Error probability of digital signaling

15, Jan. 2001

jxu

CONTENTS

■ Error probability of digital signalingPairwise error probabilityBounds on error probabilityBit and symbol error probabilities

■ Error probability of digital signaling for an AWGNchannel

BPSK, M-PSK, M-PAM, M-QAM...

■ Error probability of digital signaling in the fadingchannel

error probability in a flat fading channelerror probability over frequency selective fading channel

jxu

Vector space representation of received signals

■ In a flat fading channel with AWGN, the received complex envelopecan be defined as:

)(~)(~)()(~ tntstgtr i +=

(t) is M complex low-pass waveforms

{ (t)} (k: from 0 to M-1)

g(t): a complex Gaussian random variable.

: zero mean complex AWGN with apower spectral density (psd) of N0 watts/Hz.

)(~ tn

iS~

kS~

).~,...,~,~(~)~,...,~,~(~

)~,...,~,~(~

110

110

110

−

−

−

===

N

Niiii

N

nnnn

ssss

rrrr

nsgr i~~~ +=

The vector can be expressed as:

)(~)(~)(~ tntsgtr i +=If fmT<<1

jxu

Probability of error■ the set M signal vectors■ The decision regions■

{ } 1

0

~ −

=

M

mmS

{ }mmsgrsgrrR mmm ≠∀−≤−= ˆ,~~~~:~ 2

ˆ

2

rdsgrpsePmR mm

~)~~(1)~( ∫−= ∑−

=

=1

0

)~

(1

)(M

mmSeP

MeP

jxu

Upper bounds and lower boundson error probability

■ Upper bound can be obtained by computing theminimum squared Euclidean distance between anytwo-signal points

the pair wise error probability ,

■ lower bound on the error probability

)4

~()1()(

0

2min

2

N

dQMeP

α−≤

)4

~(

2)(

0

2min

2

N

dQ

MeP

α≥

2

,

2min

~~min~

mnmn

ssd −=

)4

~()

~,

~(

0

2min

2

N

dQSSP kj

α≤

jxu

Pairwise error probability

decisionboundary

jSg~

kSg~

It can be defined for each pair of signal vectors in the signalconstellation

22 ~~~

kijk SSd −=the squared Euclidean distance

)4

~

()~

()~

()~

,~

(0

22

N

dQSePSePSSP

jk

kjkj

α===

So the pairwise error probabilitybetween the message vectors

andkS

~jS

~

jxu

Bit and symbol error probabilities

■ The symbol error probability is PM and the bit error probability is Pb

■ Gray codes :symbol errors correspond to single bit errors where k= log2M.

MbM PP

M

P≤≤

2log

k

PP M

b ≈

jxu

Error probability of BPSK

■ two signal waveforms are s1(t)=g(t) and s2(t)= -g(t).

■ When noise n is present, The received signal from thedemodulator is r = s1+n

bb EsEs −== 21 ,

02

02 /)(

0

2/)(

0

1

1)(__

1)( NErNEr bb e

NsrpANDe

Nsrp +−−− ==

ππ

Fig. Conditional pdfsof two signals

)2

()()(0

0

11 N

EQdrsrpseP b== ∫ ∞−

)2

()(2

1)(

2

1)(

021 N

EQsePsePeP b=+=

jxu

Error probability of M-PSK

■ The probability of M-ary symbol error, is just the probability thatthe received angle outsider the region [-�/M, � /M]

∫−−=M

MsM dpP/

/)(1)(

π

πθθγ

jxu

Error probability of M-PAM

■ the Gray coded 8PAM system signal constellation

=

0

222

N

EQP h

i

α

=

0

22

N

EQP h

o

α

−=+−=

0

2

0

2)

11(2

22

N

EQ

MP

MP

M

MP h

im

α

jxu

Error probability of M-QAM

■ Rectangular QAM signal constellations most frequently usedin practice

■ two -PAM systems in quadraturee.g.16-QAM system can be treated as two independent Gray coded 4-

PAM systems in quadrature, each operating with half the power of the16-QAM system

The symbol error probabilityfor each-PAM system is

the probability of error symbolreception in the M-QAM systemis

−

−=21

6)

11(2)( s

sM MQ

MP

γγ

2)1(1)(MsM PP −−=γ

jxu

Error probability in a flat fading channel(1/2)

■ When the channel experiences fading, the error probability mustbe averaged over the fading statistics.

■ The bit error probability of a fading channel is given by

and the average symbol error probability is the symbol errorprobability over the

■ If the channel is Rayleigh faded.

γγγ dpPP eb )()(0∫∞

=

)exp(1

)(00 γ

γγ

γ −=p

[ ] ,2

0

22

00 N

EaE

N

E bb σγ ==is the mean value of the SNR.

γγγ dpPP MM )()(0∫∞

=)(γp

jxu

Error probability in a flat fading channel(2/2)

■ Rayleigh fading converts an exponential dependency of thebit error probability on the bit energy-to-noise ratio into aninverse linear one.

Bit error probability for M-QAM onan AWGN channel and a Rayleighfading channel with AWGN

Bit error probability forBPSK and QPSK on anAWGN channel and aRayleigh fading channelwith AWGN

jxu

Doppler spread impacts

■ For OFDM, if the channel is time-varying, inter-channel interference(ICI) is introduced due to a loss ofsub-channel orthogonality

■ At low , additive noise dominates the

performance so that the extra noise due toICI has little effect. However, ICI dominates

the performance and causes an error floorat large .

Figure: BER for 16-QAM OFDM on aRayleigh fading channel with various Dopplerfrequencies

bγ~

bγ~

jxu

Error probability over frequency selectivefading channel

■ The frequency selective channel will lead to ISI in the receiver,which will cause signal distortion. The result is irreducible biterror probability floor, which means increasing the signalenergy does not have any effect on the bit error probability.

■ This irreducible error floor can also due to the random FMcaused by the time-varying Doppler spread, which in turn is dueto the motion of the mobile.

■ The computer simulations are the main tool used for analyzingfrequency selective fading effects

jxu

Reference

[1] Gordon L. Stuber, "Principles of MobileCommunication (Second Edition)", Kluwer AcademicPublishers

[2] John G. Proakis, “Digital communications (ThirdEdition)”, by McGraw-Hill, Inc

[3] L.Ahlin & J. Zander, Principle of WirelessCommunications , Studentlitteratur, 1998