Digital Communications OverviewDigital Communications OverviewSimplest Abstraction: Point to Point Binary Data Communications

Physical Layer, first in OSI multiple layersKb bits is present at source � Desired to be delivered at sink

Vector1: KI b ×

Course N

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s, Sharif,

Channel adds: delay, phase, noise, interference, …Noise assumed to be additive at the input of the receiver (demodulator)Receiver Extracts the information vector estimation

220

sec/ ratebit or rateon Transmissi

)( ofset support largest

)( signals analog 2 toMap

iesprobabilitprior or priori a }{sent be likely toequally Not

12,...,0by Enumeratedwords2

bitsT

KW

txT

tx

iM

p

bb

ip

i

K

i

KK

b

bb

==

−==

π

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…Two main domain:

1) Baseband : CDs, Disk Drives, Computer peripherals, Ethernet, …2) Passband : Carrier Modulated Data Communications Modems, Cell phone,…

Unified Framework using complex envelope representation (LP equivalent)Baseband case has zero imaginary part

Modulation:

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, Modulation:

Transformation from information bits to complex envelope

Digital modulation is very similar to analog modulation m(t) ~ I‘I’ assumed to be random (random information bits) but m(t) assumed deterministic

What is sent to the channel is always analog� Digital to analog is a main part221

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…Channel:Simplest form: propagation delay and propagation loss

Demodulation: Transformation from channel output into ‘estimation’ of information bits

p

cpc

j

pzpzpcp

tfjfj

pzppcpc

etXLtRf

eetXLtXLtR

ϕ

πτπ

ττπϕ

ττ

−

−

−==

−=−=

)()(2

})(Re{)()(22

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, Transformation from channel output into ‘estimation’ of information bits

Modulation and Demodulation Schemes � Assessing the Fidelity of System

What is sent to the channel is always analog� Digital to analog is a main part

222

Course N

otes, Sim

ulatio

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ommunicatio

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s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…Performance of Communication Systems3 main metrics: Fidelity, Complexity, Bandwidth Efficiency

� Fidelity metric typically measures how often data transmission errors are made given the level of transmitted power.

Error rate as a function of SNR: might be bit error, packet error, frame error …

At a fixed power, decreasing the transmission rate (lower BW) increases fidelity

Course N

otes, Sim

ulatio

n of C

ommunicatio

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s, Sharif,

At a fixed power, decreasing the transmission rate (lower BW) increases fidelity

In practice a measure of SNR is used which is not a function of BW� Eb/N0

223

i

i

i

b

p

i

i

i

b

p

z

b

p

bzb

EK

Ldttx

K

L

dttXEK

LKdttREE

bKbK

∑∫∑

∫∫−

=

∞+

∞−

−

=

∞+

∞−

∞+

∞−

==

==

12

0

2

212

0

2

2

2

2

|)(|

}|)(|{/}|)(|{

ππ

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…� Complexity metric is almost always translated directly into cost

Depends on many marketing and management decisions

� Bandwidth Efficiency metric measures how much bandwidth a modulation uses to implement the communication.

A measure of how well a system uses the bandwidth resources

Spectral characteristics of the underlying signals are needed

pTT

bb

TB

M

B

W 2logzbits/sec/H

BWon Transmissi

RateBit : Efficiency Spectral ===η

Course N

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ulatio

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ommunicatio

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s, Sharif,

Spectral characteristics of the underlying signals are needed

1) To calculate the spectral efficiency 2)To define the BW the radio needs

D can be interpreted as spectral density of the transmitted energy per bit

BT is the bandwidth of D 224

∫

∑∑∞+

∞−

−

=

−

=

=

==

=

dffDLE

bitHzJoulesfXK

fSK

KfSEfD

z

bK

i

bK

zz

Xpb

i

i

i

b

x

i

i

b

bXX

)(

//|)(|1

)(1

}/)({)( :bit per SpectrumEnergy Average

2

212

0

12

0

ππ

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…� Other aspects, depending on the system, structure, …

Wireless: Power efficiency, cost, weight, …

Performance Limits of Digital Communications System

Clude Shannon: Channel Capacity for Error free Transmission

AWGN:

Course N

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s, Sharif,

Linear increase in bandwidth efficiency needs exponential increase in SNR

In most communication system < 15 bits/s/Hz usually much less

225

)1(log

)1(log

2

2

SNR

SNRBCW

b

Tb

+<

+=<

η

)/1(log)1(log

//

022

0

NESNR

BNWEPPSNR

bbb

TbbNs

ηη +=+<

==

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…Performance Limits of Digital Communications System …

Solving the equation for different values of Eb/N0

Below the curve is achievable

Results:

)/1(log 02 NEbbb ηη +=

Course N

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s, Sharif,

Results:

1) When BW is the most restricted

Use highest possible Eb/N0

Many systems Eb/N0 >10dB

2) When Eb is the most restricted

Lower the BW efficiency

Even less than 1 bit/s/Hz in deep space

3) Min (Eb/N0) Still having reliable communication ln2= -1.59dB 226

0→bη

Course N

otes, Sim

ulatio

n of C

ommunicatio

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s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…Signal Space Representation

Orthogonal basis to represent each signal as a vector

Linear Modulation

Complex baseband Transmitted waveform

dtttxttsststs

MNtttsts

j

S

ijiij

N

j

jiji

NM

)()()(|)()()(

)}(),...({)}(),...,({

*1

0

1010

ϕϕϕ

ϕϕ

∫∑ >==<=

≤−

=

−−

Course N

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s, Sharif,

Examples: 1) Baseband Line Codes

227

TBaudRateRateSymboltppulsefixedbsymboldTransmitte

emRfSfPfST

AfS

mRbbEmbEnTtpbAtx

n

m

mfTj

ccX

nmnn

n

n

/1/)(:}{:

)()(,|)(|)()(

)()()()()(

222

*

=

==

==−=

∑

∑∞

−∞=

−

+

π

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…Linear Modulation…

Examples: 2) Baseband Line Codes, Manchester Code, twice BW usages, RFID, Ethernet

Better synchronization properties when long runs of one or zero happen

Other Examples: 3) Nonlinear with memory line codes� Miller code

1)(0)( fortsforts ±±

Course N

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s, Sharif,

Sign changes

when run length 2 happens

Example 4) RZ code

228

1)(0)( 10 fortsforts ±±

Course N

otes, Sim

ulatio

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ommunicatio

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s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…Linear Modulation …

Examples: 5) Pass-band linear modulations, Orthogonal, Bi-orthogonal modulations

PSK : |bn| constant QAM : Im(bn ) and Re(bn ) change PAM : QAM with real bn

Design Choices:

p(t), 1/T, M, Constellation,

)()Im()()()Re()( nTtpbAtxnTtpbAtxn

nq

n

nd −=−= ∑∑

Course N

otes, Sim

ulatio

n of C

ommunicatio

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s, Sharif, p(t), 1/T, M, Constellation,

Mapping from bits to symbols,

Power

16 QAM Example:

100 MB/s, 0.99 Power containment

Rectangular Pulse ~ sinc2

1/T= (100 Mb/s )/ 4 bit/symbol= 25 Msymbol/s

B=10.2 to have the 0.99 of the energy

Normalized BW = 10.2*25MHz = 260 MHz 229

Course N

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ulatio

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ommunicatio

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s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…Demodulation

Optimum Structure for single bit demodulation in AWGN

Statistic: Processing of the original data to lower dimensionality

Sufficient Statistics: results in no loss of information for a decision, comparing to original

Course N

otes, Sim

ulatio

n of C

ommunicatio

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s, Sharif,

Sufficient Statistics: results in no loss of information for a decision, comparing to original data � Optimal decision is based on VI (Tp )as a sufficient statistic

Communication System Design Problem: Good Trade-off (Single bit Transmission)

1) Given s0(t), s1(t) and H(f) � Find optimal threshold

2) Given s0(t), s1(t), H(f) and optimal threshold � Compute fidelity of detection: BER

3) Given s0(t), s1(t)� Design H(f) to minimize BER

4) Given the optimum demodulator� Design s0(t), s1(t) to optimize fidelity

5) Design s0(t), s1(t) to optimize fidelity, having desired spectral characteristics

6) Design the system to minimize the cost230

Course N

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ommunicatio

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s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…AWGN

Gaussian Process:

X(w, t) is Gaussian any is Gaussian

Or process time samples are jointly Gaussian

Specified by its mean and auto-covariance functions

∑n

nn twXa ),(

)(),()()],(),([),(

),()(

2

21

2

2121 tmttRtmtwXtwXEttC

twEXtm

XXXXX

X

−=−=

=

Course N

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ulatio

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ommunicatio

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s, Sharif,

Wide Sense Stationary ~ Strictly Stationary:

Zero mean and White:

Infinite power, cannot be Gaussian!231

)(),()()],(),([),( 212121 tmttRtmtwXtwXEttC XXXXX −=−=

2

1221 )(),(

)(

XXXX

XX

mttRttC

mtm

−−=

=

2/)()()(2

)()( 0

20 NfSN

RC XXXX ==== τδστδττ

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…AWGN…

Example:

5GHz WLAN, BW=20MHz, Receiver noise figure F = 6dB

dBmPP

wattP

kTN

kTNBNP

n

F

n

95log10

102.3

10 FdB figure noiseith Receiver w

)290(1038.1Receiver ideal

13

10/

00

23

000

−==

×=

=

×===

−

−

Course N

otes, Sim

ulatio

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ommunicatio

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s, Sharif,

Example:

Wireless LAN 802.11a

OFDM, kb=53 , Tp = 3.2 µsec

Subcarrier frequency spacing fd = 1/2 Tp =156.25 KHz

BPSK potential transmission rate = 16.25 Mbps

Using 64-QAM (6 bits per subcarrier), FEC ¾ : 54 Mbps

232

dBmPP mWndbmn 95log10 ,10, −==

Course N

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ulatio

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ommunicatio

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s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…Q, Error (erf) and Complementary Error Functions (erfc)

≥≤≤−

−===

−=−==

=−==

−−

∞−

−

∞−

∫

∫

∫

xexQe

xerfxQdxexerfc

xerfcxQdxexerf

xerfcxerfdxexQ

xx

x

x

xx

x

x

0)()1(

)(1)2(2)(

)(1)2(21)(

)2/()2/()(

2/12/11

2

0

2

21

21

212/

2

1

22

2

2

2

π

π

π

Course N

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ulatio

n of C

ommunicatio

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s, Sharif,

-3 -2 -1 0 1 2 3-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x

Q(x

), e

rf(x

)

Plot of Q and erf

233

∞→≈

≤>

≥≤≤−

−

−

−−

xexQx

exQx

xexQe

x

x

x

x

x

x

2/)( Large

)(0 Small

0)()1(

2/

2/

21

2/

2

12/

2

11

2

2

22

2 ππ

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…M-ary Hypothesis Testing

A Framework to decide which of M possible hypothesis H1, H2, …,HM

“best” explains an observation Y.

Y relates to Hi thru a statistical model

Conditional Density of Y Given Hi �

Prior probabilities of Hi�

A decision rule ‘D’ is a mapping from the observation space to the set of hypothesis

)|( iyf

iiHP π=)(

Course N

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ulatio

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ommunicatio

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s, Sharif,

hypothesis

Errors:

� Maximum Likelihood Decision Rule (ML)

As Quality of observations improves, ML is asymptotically optimal

234

c

M

i

ieie PPP −==∑=

11

|π

)|(logmaxarg)|(maxarg)(11

iyfiyfyMiMi

ML≤≤≤≤

==δ

Course N

otes, Sim

ulatio

n of C

ommunicatio

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s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…M-ary Hypothesis Testing…

� Minimum Probability of Error (MPE) Decision Rule, minimizes the average Pe

)(

)|()|(:'

)|(11

|

yf

iyfyHPRuleBayes

dyiyfPP

ii

M

i D

i

M

i

icic

i

π

ππ

=

== ∑ ∫∑==

)|(loglogmaxarg)|(maxarg)(11

iyfiyfy iMi

iMi

MPE +==≤≤≤≤

ππδ

Course N

otes, Sim

ulatio

n of C

ommunicatio

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s, Sharif,

� Maximum Posterior Probability (MAP) Decision Rule

MAP~MPE MAP~ML equal priors

Binary Decision and Likelihood Ratio (LR)

235

)( yf

)|(maxarg)(1

yHPy iMi

MAP≤≤

=δ

1

0

0

1

)0|(

)1|()(

π

π

H

H

yf

yfyL

<

>=

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…Irrelevant / Sufficient Statistics

Observation Y has two parts Y1 and Y2 .

Y2 is irrelevant if means:

Y1 =g(Y) is sufficient if Y2 = Y is irrelevant for hypothesis testing using

)|(~)|(),|()|,()|( 111221 iyfiyfiyyfiyyfiyf ==

iyyfiyyf ∀= ),|(),|( 1212

)),((~

21 YYYgYY ===

Course N

otes, Sim

ulatio

n of C

ommunicatio

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s, Sharif,

WGN again!

n(t) is WGN, Z=<n|u> will be a Gaussian Random Variable, u deterministic, finite energy

Z1 =<n|u1> and

Z2 =<n|u2>

will be jointly Gaussian

236

)),(( 21 YYYgYY ===

><=>><<=

==

==

>==< ∫+∞

∞−

21

2

2121

22

0

2

*

|)||(),cov(

)(,0)(

2/ n(t) of PSD if

)()(|

uuununEZZ

uZVarZE

N

dttutnunZ

σ

σ

σ

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…Signal Space

In signal space, all the signal energy is concentrated in n≤M dimensions

The components of noise orthogonal to the space are irrelevant to decisions

Gramm-Schmidt Orthogonalization

1

111111

|

signals first thespans},..,{basis

0/

><−=

≤=

≠=→=

∑m

mk

ss

kmkB

s

ψψϕ

ψψ

ϕϕϕψϕ

Course N

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ulatio

n of C

ommunicatio

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s, Sharif,

Projection to the signal space� No Signal energy outside the signal space

Inner product is preserved

237

11111

1

111

make and to/ add 0 if

|

+++++

=

+++

=≠

><−= ∑

kkkkmk

ii

i

kkk

BB

ss

ϕϕψϕ

ψψϕ

)()()()(|)(

|)|,...|(

1

1

tytytytyty

yyyyY

s

n

i

iiS

iin

−=><=

>=<><><=

⊥

=

∑ ψψ

ψψψ

><=>< ba |)(|)( tbta

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…Main Theorems

1) Restriction to the Signal Space is Optimal

Ignoring the orthogonal component of y implies no loss of detection performance

Proof:

MitntstyHMiH iiii ,...,1)()()(:~,...,1: =+==+= NsY

ceindependen means Gaussiansfor 0])([

)(|)(),()()()(1

=

>=<−==

⊥

=

⊥⊥ ∑

j

jj

n

j

jj

NtnE

ttnNtNtntnty ψψ

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif,

2) For finite dimensional M-ary hypothesis testing and AWGN

-When Y=y is observed, optimal ML detection is a ‘minimum distance’ rule

-When Y=y is observed and prior Hi probabilities are known, optimal MAP/MPE

238

),0(~,...,1: 2INMiH ii σNNsY =+=

2/|maxargminarg)(2

1

2

1ii

Mii

MiML ssysyy −><=−=

≤≤≤≤δ

iiiMi

iiMi

MPE πσπσδ log2/|maxarglog2minarg)( 22

1

22

1+−><=−−=

≤≤≤≤ssysyy

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…Main Theorems…

Proof: under hypothesis Hi ,Y is a Gaussian random vector

3) Coherent continuous time model for M-ary hypothesis testing and AWGN optimal detector are:

- Optimal ML decision rule (no min distance interpretation)

)2/exp()2(

1)|( 22

2/2| σπσ

iMiiY Hf syy −−=

2−><=δ

Course N

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ulatio

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ommunicatio

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s, Sharif,

- Optimal MAP/MPE

Proof: direct consequence of previous theorems

239

2/|maxarg)(2

1ii

MiML ssyy −><=

≤≤δ

iiiMi

MPE ssyy πσδ log2/|maxarg)( 22

1+−><=

≤≤

>>=<=<

><+>>=<+>=<<

=

⊥⊥

iiS

iiSiSi

ii

sy

sysysyysy

s

sy

s

||

||||

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…Correlators and Matched Filters

Main Theorems …

4) Coherent demodulation for complex envelopes, M-ary in AWGN

- Optimal ML decision rule

)()()0)(*()()(| ,, tsthhydttstysy iMFiMFi

S

ii −==>=< ∫

2/|Remaxarg)(2

,,1

izizzMi

zML ssyy −><=≤≤

δ

Course N

otes, Sim

ulatio

n of C

ommunicatio

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s, Sharif,

- Optimal MAP/MPE

Proof:

240

iizizzMi

zMPE ssyy πσδ log2/|Remaxarg)( 22

,,1

+−><=≤≤

2/)|Re(2/|

,...1),()()(:

real all ,...1),()()(:

2

,,

2

,,

,

,

izizzipipp

zizzi

pippi

ssyssy

MitntstyH

MitntstyH

−><=−><

=+=

=+=

1 Mi≤≤

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…Real pass-band, Complex Envelope and Correlation

Coherent detection can be stated in terms of real-valued vectors

)|Re(||)()(|

)sin()cos()()(

)()(|)sin()cos()()( *

><>=<+>=<>=<

−=

>=<−=

∫

∫

∞+

∞−

+∞

∞−

zzqqdd

cqcd

zzzzcqcd

vuvuvudttvtuvu

tvttvtv

dttvtuvututtutu

ωω

ωω

,, |||Re iqiiddi sysysy ><+>>=<<

Course N

otes, Sim

ulatio

n of C

ommunicatio

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s, Sharif,

No cross coupling between I and Q � Receiver keeps the components separate

Not true for non-coherent receivers… Use magnitude instead of real value…

241

2

,

2

,

2

,, |||Re

iqidi

iqiiddi

sss

sysysy

+=

><+>>=<<

22

2

2

)||()||(|||

noise ignoring ||||||

noise ignoringcos|Re|

><−><+><+><=><

≈><=><

>≈<>=<

+=

dqqdqqddpp

j

pp

j

pp

zz

j

z

vuvuvuvusy

sAssAesy

sAssAesy

nsAey

θ

θ

θ

θ

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…ML Geometrical interpretation

Minimum distance

Perpendicular Bisectors

Course N

otes, Sim

ulatio

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s, Sharif,

242

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…Soft Decision

ML and MAP are of hard Decision type � 1 out of M candidates is the winner

Demodulator can provide more info,

not only the decision but also the “quality of decision” � Soft Decision

Posterior Probabilities are the maximum information demodulator can provide

−−===

==

iiii

ii

HPifHPifi

sentisPHPi

22)2/exp(][)|(][)|(

)|(

nObservatio:]|[]|[)|(

σππ

π

syyyy

yysyy

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif,

243

∑∑ ==−−

−−===

M

j jj

ii

M

j j

ii

HPjf

HPif

f

HPifi

1

22

1)2/exp(

)2/exp(

][)|(

][)|(

)(

][)|()|(

σπ

σππ

sy

sy

y

y

y

yy

4

1

2

2

=

=

σ

σ

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…Single Bit Transmission Fidelity

OOK (On Off Keying or On Off Signaling) and ML rule

Correlation decision statistics v is sufficient

2/| :Thrm2

)()()(:)()(:

22

2

10

0

1

ssyv

tntstyHtntyH

H

H

I

<=>=

<

>>=<

+==

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif,

ML rule is equivalent to MPE if prior probabilities are the same

244

)2

()|()|(

)|,|cov()|var(

)|,|cov()|var(

)|()|(0)|()|(

)|2/()|()|2/()|(

10,

22

1

22

0

2

10

1

2

10

2

0

σ

σ

σ

sQHePHePP

ssnssnsHv

ssnsnHv

ssnsEHvEsyEHvE

HsvPHePHsvPHeP

MLe

I

I

II

II

===

=>+<>+<=

=><><=

=>+<==><=

<=>=

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…Single Bit Transmission Fidelity…

Binary Signaling and ML rule

OOK tosimilarity useor y Replace2/)(|

2/|2/| :Thrm2

)()()(:)()()(:

2

0

2

101

2

00

2

11

1100

1

0

1

ssssyv

ssyssy

tntstyHtntstyH

H

I

H

H

−<

>>−=<

−><<

>−><

+=+=

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif,

Energy per bit, Eb: The smaller the better for a fixed fidelity

245

)2

()2

()|()|(

OOK tosimilarity useor y Replace2/)(|

01

10,

0101

0

σσ

dQ

ssQHePHePP

ssssyv

MLe

H

I

=−

===

−<

>−=<

2/)2

( :EfficiencyPower

symbolslikely Equally 2/)(

0

0

P,

2

P

2

1

2

0

NN

EQP

E

d

ssE

bMLe

b

b

==⇒=

+=

ση

η

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…Single Bit Transmission Fidelity…

1) Performance depends on SNR

2) For fixed SNR, higher ηP is a better signaling (So is the name power efficiency)

)(,2,1E

)(,4,1E

)(2,2/1E

0

0

2

Pb

Pb

E

N

E

e

N

E

e

b

b

b

QP

QP

QP

===

===

===

η

η

η

bE

d2

P =η

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif,

Orthogonal Signaling like FSK, WH

Antipodal like BPSK

246

)(,2,1E 0Pb N

E

ebQP === η

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…M-ary Signaling

In general it is a difficult problem to measure the fidelity: approximations, bounds, …

Performance is determined by ‘signal inner product’ and ‘noise power’

}:{}:{})(:{

,minarg)(

2/|,maxarg)(

1

2

1

ijDDyijZZyiyy

syDDy

ssyZZy

jijiMLi

iiiMi

ML

iiiiMi

ML

≠∀≤=≠∀≥===Γ

−==

−>=<=

≤≤

≤≤

δ

δ

δ

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, Performance is determined by ‘signal inner product’ and ‘noise power’

N is Gaussian � Zj’s are jointly Gaussian and can be expressed in terms of

conditional error can be calculated

by integration of Gaussian over ML region

Rotation and Performance invariant transforms� QT=Q-1

247

2/||

]|,[]|[

2

|

jjjij

jiiie

ssnssZ

ijDDPiyPP

−><+>=<

∃<=Γ∉=

><=

−>=<

kjkj

jjij

ssZZ

sssEZ

|),Cov(

2/|

2

2

σ

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…M-ary Signaling…

Energy per Symbol and Energy per Bit

If “scale” all the signals by A, all inner products are scaled by A2

Define “scale invariant” inner products, that depends only on the shape of constellation

Performance depends only on E /N and the constellation shape

M

EEs

ME S

b

M

i

iS

21

2

log

1== ∑

=

0

2

2||

}/|{: ProductsInner Invariant -Scale

N

E

E

ssss

Ess

b

b

jiji

bji

><=

><

><

σ

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, Performance depends only on Eb/N0 and the constellation shape

For Binary signaling, Fidelity depends only on power efficiency and Eb/N0

Power efficiency depends on the constellation shape too…

248

bbb

P

bji

E

ssssssss

E

ss

E

d

Ess

><−><−><+><=

−==

><

01100011

2

012 ||||

}/|{: ProductsInner Invariant -Scale

η

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…M-ary Signaling…

Example: QPSK Let’s calculate Pe|1

0

22

2

1

1

2

242

)}2

({)2

(2)02/02/(

)2/2/(),(

N

EddE

dE

dQ

dQdNdNPP

dNdNNNsy

bbs

sce|

scsc

=→=→=

−=<+∨<+=

+++=+=

σ

σσ

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif,

Union Bound: Convert the M-ary case to some binary cases

Example: QPSK249

2

00

1 )}2

({)2

(2N

EQ

N

EQPP bb

e|e −==

0

2

22)

2()(

)2

()2

||||()|()|}{(

N

E

E

dddQiP

dQ

ssQiZZPiZZPP

b

b

ijij

i ij

ij

e

ij

ij

ij

ji

ij

jijiij

e|i

=≤

=−

=<≤<∪=

∑ ∑

∑∑∑

≠

≠≠≠≠

σσπ

σσ

xN

EQ

N

EQP bb

e largefor only First term )4

()2

(200

+≤

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…M-ary Signaling…

Union bound is loose specially for big constellations

Intelligent Union Bound

NML(i) the indices of all neighbors of signal si (except i) that characterize Γi

Those that define the hyper planes that construct Γi

Tighter bound2

22)

2(

N

E

E

dddQP bijijij

e|i =≤ ∑σσ

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif,

QPSK Example: only first term

Nearest Neighbors Approximation

common approach for getting a better and quicker estimate

Good for regular signal sets that each one has some nearest neighbors at distance dmin

250

0)( 22)

2(

NEQP

biNj

e|i

ML

=≤ ∑∈ σσ

∑=

=

≈≈

M

i

dd

b

b

dede|i

iNM

N

N

E

E

dQNP

dQiNP

1

0

2

minmin

)(1

)2

()2

()(

minmin

minmin σ

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…M-ary Signaling…

We have simple estimates now!

Power efficiency

Similar to binary case

ηP is more important than Ndmin � a way to compare different schemes

Example: 16-QAM

)2

(0

2

min

min N

EQNP

E

d bPde

b

P

ηη ≈→=

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, Example: 16-QAM

For the shown scaling dmin=2

Es= ave I2+ave Q2 = 5+5=10

Eb=10/ log216 = 2.5 � ηP = 1.6

Comparing to QPSK ηP = 4 about 4dB better

Bandwidth efficiency of16-QAM is higher, ηB = 2 comparing to η

B = 1

251

316/)384244(min

=×+×+×=dN

DMB /logfreedom of esbits/degre ofnumber 2==η

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…M-ary Signaling…

Performance analysis of equal energy M-ary orthogonal signaling

Extreme in power–bandwidth tradeoff

M-dimensional signal space, signal vectors each along one axis, Es=1

log

/loglog2

2,log/1

22

2

min2

ME

MMM

dME

BP

b

==

==

ηη

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif,

It can be shown that

Information theoretic limit that we saw before

No Communication possible for Eb/N0< -1.6 dB

252

)log

()1(0

2

N

MEQMP b

e −≤→

<

>=→⇒∞→

2ln/,2

1

2ln/,0

,00

0

NE

NE

PMb

b

eBη

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…M-ary Signaling…

Bit Level Demodulation

We decided how to detect symbols, log2M bits in each

Symbol error rate depends on SNR and constellation shape

BER Depends also on how to assign bits to symbols, bit mapping

Gray coding: 2n-ary constellation, each point is represented by a binary string

b=(b1,..,bn) Bit representation of b and b’ which are nearest neighbors differ

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, b=(b1,..,bn) Bit representation of b and b’ which are nearest neighbors differ

only in 1 bit. It is not easy/possible to map gray codes to any constellation

Use ‘min distance to point b that differ in the ith bit’

253

)2

)((),(

2

1)(

CodingGray no)(1

CodingGray with )2

(

min

1

0

min σ

η

bb

b

dQiNbinerrorP

binerrorPn

P

N

EQP

dni

n

i

ibe

bPbe

∑

∑

≈

=

≈

=

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…Link Budget analysis

Making choices about transmission power, antenna gains, quality of receiver

circuitry, range, … to design a communication link

a) From bit rate Rb and the signal constellation� minimum symbol rate �minimum Nyquist BW = Bmin= Rs� B=(1+a)Bmin Excess BW to improve ISI

b) Constellation and the desired bit error probability � Eb /N0� required SNR

Receiver noise figure F(dB)� noise power� min required received signal B

R

N

ESNR b

rb

req )(0

=

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif,

c) Receiver noise figure F(dB)� noise power� min required received signal power (Receiver Sensitivity in reverse calculation)

d) Calculation of the transmission power: antenna gains, carrier wavelength and line-of-sight distance (to calculate path loss)

Other path loss expressions might be used for other environments.

Example: 1/R4 decay for cluttered wireless , instead of 1/R2 free space

Link margin to compensate for AWGN or no fading … 254

BN 0

min,min,

10/

0min, log10,10, RXRX

F

nnreqRX PdBmPBkTPPSNRP ===

dBinmdBpathdBiRXdBiTXdbmTXdbmRX

RXTXTXRX

LRLGGPP

RGGPP

,arg,,,,,

222

)(

)16/(

−−++=

= πλ

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…Link Budget analysis Example

5 GHz WLAN, 20 MHz Channel, QPSK + Gray Coding, Excess Bandwidth 33%,

Receiver noise figure 6 dB, Antenna Gains 2 dBi

1) Bit Rate?

2) Receiver Sensitivity for BER=10-6

3) Range for 20dB link margin and 100mW transmit power

Rs=1/T=B/(1+α)=20/(1+0.33)=15 Ms/sec � Rb=Rslog2M=30Mb/secEE2

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif,

s b s 2

QPSK+Gray:

Sensitivity:

Typical Wireless Range255

dBQN

E

N

EQP req

bbe 2.10)10()()

2( 61

21

00

==→= −−

dBmwattkTBNP

dBB

R

N

ESNR

F

n

br

breq

95102.310)290(1038.110

12)20/30(log102.10)(

136.02310/

00

10

0

−=×=×===

≈+==

−−

dBmdBdBmSNRPP dBreqdBmndBmRX 831295(min) ,,, −=+−=+=

mRdBP

LGGPPRL dBdBiRXdBiTXdBmRXdBmTXdBpath

107872022)83(20

(min))( margin,,,,,,

=→=−++−−=

−++−=

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…Link Budget analysis Example 2

Spectrum Analyzer View, BPSK bit rate 4096 Kbit/sec

100KHz Resolution for Spectrum Analyzer

1.2 Equivalent BW � 120 KHz

Eb= P T = P/4096

N = N0/2 × 120KHz

Eb/N0= -10log104096 + 10 log1060 + PdBm - NdBm= -18.34 + 27 = 8.66 dB

27dB

P

N

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif,

Eb/N0= -10log104096 + 10 log1060 + PdBm - NdBm= -18.34 + 27 = 8.66 dB

Power Meter shows P+N’= -88dBm

N0 can be calculated

256

5

2

1063.5)2/83.3erf(5.05.0

)83.3()Q( 0

−×=−=

== QPN

E

eb

dBP

dB

N

E

WN

WE

N

P b

b

bb

3.88)101/10log(1088

67.1101.366.8

2

)2/(

167.1167.1

00

−=++−=

=+=

==′

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…Link Budget analysis Example 3, Self-Synchronizing Scrambler

Scramblers� Make random like data using LFSR

Can sit before/after FEC and just before line code/modulation

Reasons to use scrambler (randomizer)

1) Facilitate the timing recovery, AGC and other adaptive circuits of the receiver by removing long 0 or 1 runs

2) Eliminate the dependence of signal power spectrum to the transmitted data,

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif,

2) Eliminate the dependence of signal power spectrum to the transmitted data, causing it to disperse to meet maximum spectral density requirements (avoiding cross modulation and inter-modulation caused by non-linearities)

Additive Scrambler (synchronous)

Descrambler is the same

Needs a sync-word in each frame

Receiver locates few adjacent frames

To find the LFSR initial state257

15141 −− ++ xx Used in DVB

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…Link Budget analysis Example 3, Self-Synchronizing Scrambler …

Multiplicative Scrambler (Self Synchronizing)

23181 −− ++ xx V. 34 Recommendation

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif,

multiplicative scrambler is recursive and a the descrambler is non-recursive

Application in Simulation and BER Meters

258

TX RXChannel

Given Prototype or Simulation

Scrambler

DescramblerAll one

Binary SequenceLength N

BER= Zeros count / m / Nm=3 for V.34 Scrambler

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011

Digital Communications…Digital Communications…Link Budget analysis Example 3, Self-Synchronizing Scrambler …

Multiplicative Scrambler (Self Synchronizing)…

Application in Simulation and BER Meters…

Download the Simulink example from the course site

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif,

259

Course N

otes, Sim

ulatio

n of C

ommunicatio

n System

s, Sharif, E

E, Im

an G

holam

pour, im

angh@

sharif.ed

u , Fall 2

011