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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
otes, Sim
ulatio
n of C
ommunicatio
n System
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
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 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
n System
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
otes, Sim
ulatio
n of C
ommunicatio
n System
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
otes, Sim
ulatio
n of C
ommunicatio
n System
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
otes, Sim
ulatio
n of C
ommunicatio
n System
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
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 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
otes, Sim
ulatio
n of C
ommunicatio
n System
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
otes, Sim
ulatio
n of C
ommunicatio
n System
s, Sharif,
Sign changes
when run length 2 happens
Example 4) RZ code
228
1)(0)( 10 fortsforts ±±
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: 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
n System
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
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…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
n System
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
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
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
otes, Sim
ulatio
n of C
ommunicatio
n System
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
n of C
ommunicatio
n System
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
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…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
otes, Sim
ulatio
n of C
ommunicatio
n System
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
otes, Sim
ulatio
n of C
ommunicatio
n System
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
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…
� 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
n System
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
n System
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
otes, Sim
ulatio
n of C
ommunicatio
n System
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
otes, Sim
ulatio
n of C
ommunicatio
n System
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
n System
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
n System
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
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ulatio
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ommunicatio
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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
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ulatio
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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