tsks01&digital&communication course&director lecture&1 ·...

TSKS01&Digital&CommunicationLecture&1

Introduction,&Repetition,&and&Noise&Modeling

Emil&Björnson

Department&of&Electrical&Engineering&(ISY)Division&of&Communication&Systems

Emil&Björnson

! Course&Director! TSKS01&Digital&Communication! TSRT04&Introduction in&Matlab! TSKS12&Multiple Antenna Communication

! Associate Professor&(Universitetslektor)! Docent&in&Communication&Systems,&PhD&in&Telecommunications! Coordinator of Master&programme in&Communication&Systems&and& Master&profile in&Communication&Systems

! Researcher&on&5G&communications (e.g.,&multiple antennacommunications).&Theoretical research&and&inventor.

2017T09T01 TSKS01&Digital&Communication&T Lecture&1 2


TSKS01&Digital&Communication&T Formalities

Information: www.commsys.isy.liu.se/TSKS01Lecturer&&&examiner: Emil&Björnson,&[email protected]: Kamil Senel,&[email protected]&activities: 12&lectures,&12&tutorials,&lab&exercises

Examination: Laboratory&exercises&(1hp):More&details&in&HT2SignTup&on&the&web&(later)

Written&exam&(5hp):1&simple&task&(basics)& Min:&50%2&questions&(5&points&each) Min:&3p4&problems&(5&points&each) Min:&6pPass:&14&points&from&questions&&&problems


Course&AimsAfter&passing&the&course,&the&student&should&! be&able&to&reliably&perform&standard&calculations&regarding&digital&modulation&

and&binary&(linear)&codes&for&error&control&coding.(basics)

! should&be&able,&to&some&extent,&to&perform&calculations&for&solutions&to&practical&engineering&problems&that&arise&in&communication

(primarily-questions)! be&able&to,&with&some&precision,&analyze&and&compare&various&choices&of&

digital&modulation&methods&and&coding&methods&in&terms&of&error&probabilities,&minimum&distances,&throughput,&and&related&concepts.

(problems)! be&able,&to&some&extent,&to&implement&and&evaluate&communication&systems&of&

the&kinds&treated&in&the&course.(laboratory-exercises)

Course&Book

! Introduction&to&Digital&Communication! New&edition&for&2017,&A1671,&SEK&214! Available&for&purchase&in&Building&A,&LiU Service&Center,&entrance&19C

! 207&pages&dedicated&to&this&course

! There&is&also&a&book&with&problemsto&be&solved&at&tutorials (A1672,&SEK&81)

! Older&editions! Version&from&2016&can&be&used,&along&with&the&errata&and&the&new&tutorial&problems&found&on&last&year’s&course&webpage

2017T09T01 TSKS01&Digital&Communication&T Lecture&1 5 2017T09T01 TSKS01&Digital&Communication&T Lecture&1 6

Classic technology" Newspapers, books" Telegraph" Radio and TV broadcast

Information*transfer

What&is&Communication?

Modern digital technology" CD, DVD, Blu-ray" Optical fibres, network cables" Wireless (4G, WiFi)

Why&Digital Communication?

1. Efficient.Source.Description! Information&has&a&builtTin&redundancy&(e.g.,&text,&images,&sound)! Compression:&Represent&information&with&minimal&number&of&bits! Sequence&of&bits:&0&and&1! 100&bits&represent&2"## ≈ 1.2677 ⋅ 10+# different&messages

2. Efficient.Transmission! NyquistTShannon&Sampling&Theorem:&, Hz&signal&↔ 2,.sample/s! Sequence&of&bits&is&used&to&select&these&samples

3. Guaranteed.Quality! Protect&signals&against&errors:&All&or&nothing&is&received


OneTway&Digital&Communication&System

Analog.channel

Source

Destination

Channel.encoder Modulator

Channel.decoder

DeAmodulator

Channel&coding

Error&control

Error&correction

Digital to&analog

Analog&to&digital

Medium

Digital&modulation

Digital&Communication:&An&Exponential&Story

Digital&Communication&Devices&are&Everywhere! PCs,&tablets,&phones,&machineTtoTmachine,&TVs,&etc.! Total&internet&traffic:& 23%&growth/year! Number&of&devices:& 12%&growth/year


Source:&Cisco&VNI&Global&IP&Traffic&Forecast,&2014–2019&

How.can.we.sustain.this.exponential.growth?

Overall.IP.traffic

InformationTBearing&Signals

! Example:&Mobile&communication! Analog&electromagnetic&signaling! Bandwidth&, Hz! Sample&principle&in&copper&cables,&optical&fibers,&etc.

Nyquist–Shannon.Sampling.Theorem! Signal&is&determined&by&2, real&samples/second&(or&, complex&samples)

! InformationTbearing&signals! Select&the&samples&in&a&particular&way&# Represent&digital&information! Spectral&efficiency:&Number&of&bits&transferred&per&sample&[bit/s/Hz]


Designing&Efficient&Communication&Systems

! Performance&metric:&Bit&rate&[bit/s]! Formula:&..Bit.rate.[bit/s] = Bandwidth. Hz < Spectral.efficiency.[bit/s/Hz]

! How&do&we&increase&the&bit&rate?1. Use&more&bandwidth2. Increase&spectral&efficiency


Wired:& Use&cable&that&handles&wider&bandwidths

Wireless:&Buy&more&spectrum

Each&sample&represents&more&bits• More&sensitive&to&disturbances• Improve&signal&quality• Protect&against&bit&errors

This.course.primary.deals.with.spectral.efficiency!


Preliminary&Lecture&Plan

HT1:! Introduction,&repetition&of&key&results (Lecture-1)! Basic&digital&modulation (Lectures-283)! Detection&in&AWGN&channels (Lectures-385)! Detection&in&dispersive&channels (Lectures-687)

HT2:! Error&control&coding (Lectures-8810)! Practical&aspects&(e.g.,&synchronization) (Lecture-11)! Link&adaptation (Lecture-12)

Outline

! Introduction! Repetition:&Signals&and&systems! Repetition:&Stochastic&variables! Repetition:&Stochastic&processes! Noise&modeling&and&detection&example

! At&the&end:&Examples&related&to&repetition&← Read&yourself!


Repetition:&Signals&and&Systems

SystemD(F) H(F)

Signals: Voltages,.currents,.or.other.measurements

Systems: Manipulate/filter.signals

Complex.exponential: IJKLMN = cos 2QRF + T sin(2QRF)

Unit.step: U(F) .= . V0, F < 0.1, F > 0

Unit.impulse: Z(F):..∫ D F\]\ Z F .^F = D(0)

Property: U F = ∫ Z _N]\ ^_


Special&Outputs

EnergyAfree.systemZ(F) ℎ(F)

Impulse&response:&

General&case:

Energy0free*system:*No*transients,*constant*input*# constant*output

EnergyAfree.systemD(F) H(F)

Linear&Systems

Definition: Let&D"(F) and&DK F be&input&signals&to&a&oneTinput&energyTfree&system,&and&let&H" F and&HK F be&the&corresponding&output&signals.&If

H F = a"H" F + aKHK(F)

is&the&output&corresponding&to&the&inputD F = a"D" F + aKDK F ,

for&any&D"(F), DK F ,&a",&and&aK,&then&the&system&is&referred&to&as&linear.&A&system&that&is&not&linear&is&referred&to&as&non0linear.

Examples: H(F) = D(F − 1) is linearH F = D F

K is non0linear


TimeTInvariant&and&LTI&Systems

TimeTinvariant&systemDefinition:.Let&D(F) be&the&input&to&an&energyTfree&system&and&let&H(F).be&the&corresponding&output.&If&H(F + _) is&the&output&for&the&input&D(F + _) for&any&D(F),&F,&and&_,&then&the&system&is&referred&to&as&time0invariant.&Otherwise&the&system&is&referred&to&as&time0varying.

Examples: H(F) = D(F − 1) is time0invariantH F = cos 2QRF D(F) is time0varying

LTI&systemDefinition:.A&system&that&is&both&linear&and&timeTinvariant&is&referred&to&as&a&linear*time0invariant*(LTI)*system.


Convolution

Definition: The&convolution&of&the&signals&a(F) and&c(F) is&denoted&by&(a ∗ c)(F) and&is&defined&as

a ∗ c F = e a _ c F − _ ^_

\

]\

.

The&convolution&is&a&commutative&operation:& a ∗ c F = c ∗ a F .


Output&of&an&LTI&Systems

Theorem:.Let&D(F) be&the&input&to&an&energyTfree&LTI&system&with&impulse&response&ℎ(F),&then&the&output&of&the&system&is

H F = D ∗ ℎ F .

Proof:.Let&f a(F) denote&the&output&for&an&arbitrary&input&a(t),&then

H F = f D(F) = f e D _\

]\Z F − _ .^_

.....Linear=.

e D _\

]\f Z F − _ .^_

..Time−inv

=.

e D _\

]\ℎ F − _ ^_ = D ∗ ℎ F


Frequency&Domain

Fourier&transform: ℱ D(F) = ∫ D F I]JKLMN^F\]\

Exists if ∫ |D F |^F\]\ < ∞

Inverse&transform: ℱ]" n(R) = ∫ n R IJKLMN^R\]\

Common*notation

Spectrum&of&D(F): n(R)

Amplitude&spectrum: |n R |

Phase&spectrum: arg n(R)


Image&from:&https://en.wikipedia.org/wiki/Fourier_transform

D(F)

FR

n(R)

LTI&Systems:&Time&and&Frequency&Domain

Property:.Let&p R = ℱ a(F) and&, R = ℱ c(F) be&the&Fourier&transforms&of&the&signals&a(F) and&c(F),&then

ℱ (a ∗ c)(F) = p R , R .

Property:.The&input&and&output&of&LTI&systems&are&related&as


ℎ Ff(R)

D(F)n(R)

H F = (D ∗ ℎ)(F)q R = n R f(R)

Example:&Channel&is&an&LTI&Filter

Impulse response:&ℎ F = Z F − _" + Z F − _K + Z F − _+ + Z(F − _r)


Time delay:&_"

Time&delay:&_+

Time&delay:&_r

Time&delay:&_K

Only.timeAinvariant.for.a.limited.time.period.(coherence.time)

Probability,&Stochastic&Variable,&and&Events

Total&probability: Pr Ωu = 1

Probability&of&event&p: Pr.{p} ∈ . [0,1]

Joint&probability: Pr.{p, ,}

Conditional&probability: Pr p , ={|.{},~}

{|.{~}


Sample&space:.Ω

�

Measureable&sample&space:Ωu = n � : .for.some.� ∈ Ω

n �

Stochastic&variable

Event&pEvent&,


Probabilities&and&Distributions

Probability&distribution&function:Ån(D) = Pr.{n ≤ D} ∈ [0,1]

Probability&density&function&(PDF):

Rn(D) .=^^D

Ån(D)

Properties: Ån(D).is&nonTdecreasing,

Ån(D) ≥ 0.and&Rn(D) ≥ 0. for&all D,

∫ Ru(D)\]\ ^D. = .1,

Pr D1 < n ≤ D2 = ∫ Ru D ^DÑÖÑÜ

.

Expectation&and&VarianceExpectation&(mean):

á n = e DRu D ^D\

]\

For&discrete&variables:

á n = à DâPr{n = Dâ}�

â

For&functions&of&a&variable:q = ã n

á q = e HRå H ^H\

]\

= á ã(n) = e ã D Ru D ^D\

]\


Quadratic&mean&(power):

á nK = e DKRu D ^D\

]\

VarianceVar n = á n − á n K

..................= á nK − á n K

Common&notation:éu = á n ,&éå = á q

èuK = Var n

èu is&called&the&standard&deviation


Gaussian/Normal&Distribution,&ê(é, èK)Probability&density

Ru D =1

è 2Q� I]Ñ]ë Ö

KíÖ

Probability&distribution&Åu D = ∫ Ru D ^DÑ]\

is&complicated

Mean:&é

Variance:&èK

ìTfunction

ì D = 1 − Åu D = ∫"

KL� I]îÖ

Ö ^F\Ñ for&ê(0,1)

Can&be&computed&from&table


Example of the&ì Function

ì(1.96) ≈ .2.4998 · 10]K

MultiTDimensional&Stochastic&Variables

Distribution: ÅuÜ,…,uöD", … , Dõ = Pr.{n" ≤ D", … , nõ ≤ Dõ}

Density: RuÜ,…,uöD", … , Dõ =

úö

úÑÜ<<<.úÑöÅuÜ,…,uö

D", … , Dõ

Vector&notation:&&nù = (n", … , nõ),&&&D̅ = (D", … , Dõ),&&&&Åuù D̅ ,&&&&Ruù D̅

Mutual&independence:

Åuù D̅ = üÅu†(Dâ)

õ

â°"

........Ruù D̅ = üRu†(Dâ)

õ

â°"

Covariance&(pairwise):&Cov nâ, nJ = .á nânJ − á nâ á nJ

Pairwise uncorrelated if nâ and&nJ if&Cov nâ, nJ = 0 for&all&£, T2017T09T01 TSKS01&Digital&Communication&T Lecture&1 28

Marginal&distribution&and&density

Example:&Jointly&Gaussian&Variables

Definition: nù = (n", … , nõ)&is&jointly*Gaussian,&denoted&as&ê(é§, Λuù),&if

Ruù D̅ =1

2Q õdet.(Λuù)�

I]"K Ñ̅]ë§ ¶ß§

®Ü Ñ̅]ë§ ©

where&&&&&é§ = E nù ,&&&&&Λuù =´"," ⋯ ´",õ⋮ ⋱ ⋮

´õ," ⋯ ´õ,õ

,&&&&&&´âJ = Cov nâ, nJ .

If&nù = (n", … , nõ)&are&pairwise&uncorrelated&# ´âJ = 0 for&£ ≠ T

#...Ruù D̅ =1

2Q õ´"," ⋯´õ,õ�

I∑ ] Ñ†]ë†

Ö

K±†,†�† = üRu†

Dâ

õ

â°"

..#&&Independence


Bayes’&Theorem

Joint&and&conditional&probability:Pr{n = D, q = H} = Pr n = D q = H Pr{q = H}..................................= Pr.{q = H|n = D} Pr{n = D}

Ru,å D, H = Ru|å D|H Rå H = Rå|u H|D Ru D

Bayes’&theorem (discrete): Pr.{n = D|q = H} ={|.{å°≤|u°Ñ}

{|.{å°≤}Pr.{n = D}

(continuous): Ru|å D|H =M≥|ß ≤|Ñ

M≥ ≤Ru D

(n discrete,&q cont.): Pr.{n = D|q = H} =M≥|ß ≤|Ñ

M≥ ≤Pr.{n = D}


Thomas.Bayes

Image&from:&https://en.wikipedia.org/wiki/Thomas_Bayes

Example:&Additive&Noise&Channel

! Input:&n ∈ −1,+1 with&Pr n = +1 = Pr n = −1 = 1/2

! Output:&q = n +µ

! µ is&a&continuous&stochastic&variable&


+ Detectorn q

µ

n∂

∑0 +2−2

1/2R∏(∑)

How.should.we.“best”.detect.n from.q?


Stochastic Process

Sample&space:.Ω�"

�K

�+

n"(�", F)

nK(�K, F)

n+(�+, F)

F

F

F

Stochastic.process.=.Stochastic.timeAcontinuous.signal.(π = ∞)


Examples&of&Stochastic&Processes

Example.1: Finite&number&of&realizations:

...n(F) = sin.(F + ∫),. ∫ ∈ {0, Q/2, Q, 3Q/2}.

Example.2:..Infinite&number&of&realizations:

...n(F) .= .p · sin.(F), p~ê 0,1 .


Examples&of&Stochastic&Processes&cont’d

Example.3:..Infinite&number&of&realizations:

n F = ∑ pΩã(F − æ)�Ω ,.........ã F = V

cos.(QF), |F| < 1/2.0, ............elsewhere

Each&.pæ. is&independent&and&ê(0,1)

One&realization:

Sampling&of&Stochastic&Process:&ê Time&InstantsVector&notation

! Time&instants: F̅ = (F", … , Fõ)

! Stochastic&variable: n(F̅) = n(F"), … , n(Fõ)

! Realizations: D̅ = (D", … , Dõ)

ê time&instants! Distribution: Åu(N̅) D̅ = Pr.{n(F") ≤ D", … , n(Fõ) ≤ Dõ}

! Density: Ru(N̅) D̅ =úö

úÑÜ⋯úÑöÅu(N̅) D̅


Sampling.of.stochastic.process.# Stochastic.variable

Ensemble&Averages“Observe&many&realizations&and&make&an&average”&– functions&of&time


Expectation&(mean):éu F = á n(F)

= e DRu(N) D ^D\

]\

Quadratic&mean&(power):

á nK(F) = e DKRu(N) D ^D\

]\

Variance

èu(N)K = á n(F) − éu F

K

.....= á nK(F) − éuK F

AutoTcorrelation&function&(ACF):øu F", FK = á n F" n(FK) =

e e D"DKRu NÜ ,u NÖ D", DK\

]\^D"^DK

\

]\

Symmetry:&&øu F", FK = øu FK, F"

Power:& øu F, F = á nK(F)

Special&case:&p is&stochastic&var.n F = ã(F, p)

øu F", FK =∫ ã(F", p)ã(FK, p)R} a\]\ ^a


StrictTSense&Stationarity

Stationarity is&statistical&invariance&to&a&shift&of&the&time&origin.

Definition:Consider&time&instants&F̅ = (F", … , Fõ) and&shifted&time&instances&Uù = F̅ + Δ = (F" + Δ,… , Fõ + Δ).&The&process&n(F) is&said&to&be&strict0sense*stationary*(SSS)&if

Åu(N̅) D̅ = Åu(¡§) D̅

holds&for&all&ê and&all&choices&of&F̅ and&Δ.

Equivalence:

Åu(N̅) D̅ = Åu(¡§) D̅ ! Ru(N̅) D̅ = Ru(¡§) D̅

WideTSense&Stationarity

Definition:.A&stochastic&process&n(F) is&said&to&be&wide0sense*stationary*(WSS)&if

! Mean&satisfies&éu F = éu F + Δ for&all&Δ.! ACF&satisfies&øu F", FK = øu F" + Δ, FK + Δ for&all&Δ.

Interpretation:Constant&mean,&ACF&only&depends&on&time&difference&_ = F" − FK

Notation: Mean&éu

ACF&øu _



Gaussian&Processes

Recall: nù = (n", … , nõ)&is&jointly*Gaussian,&denoted&as&ê(é§, Λuù),&if

Ruù D̅ =1

2Q õdet.(Λuù)�

I]"K Ñ̅]ë§ ¶ß§

®Ü Ñ̅]ë§ ©

Definition: A&stochastic&process&is&called&Gaussian if&n(F̅) =n(F"), … , n(Fõ) is&jointly&Gaussian&for&any&F̅ = (F", … , Fõ)

Theorem: A&Gaussian&process&that&is&wideTsense&stationary&is&also&stationary&in&the&strict&sense.

EXAMPLESRepetition


Example:&Linear&system

Is&the&system&H F = D(F − 1) linear?! Consider&D(F) = a"D" F + aKDK(F)! Then:&H F = a"D" F − 1 + aKDK F − 1 = a"H" F + aKHK(F)

! Linear&combination&of&the&outputs&of&D" F and&DK F :&Linear&system

Is&the&system&H F = D FK linear?

! Consider&D(F) = a"D" F + aKDK(F)! Then:&

H F = a"D" F + aKDK(F)& K = a"KD"

K F + aKKDK

K F + a"aKD" F DK F≠ a"D"

K F + aKDKK(F)

! Not&a&linear&combination&of&the&outputs&of&D" F and&DK F :&NonTlinear


Example:&TimeTinvariant&system

Is&the&system&H F = D(F − 1) timeTinvariant?! Note:&H F + _ = D F + _ − 1 = D( F + _ − 1)

! TimeTshifted output&is&given&by&equally timeTshifted input:&TimeTinvariant

Is&the&system&H F = cos 2QRF D(F) timeTinvariant?! Note:&H F + _ = cos 2QR(F + _) D(F + _) ≠ cos 2QRF D(F + _)

! TimeTshifted output&is&not&given&by&the&timeTshifted input:&TimeTvarying


Example:Stochastic&variablesΩ = HH,HT, TH, TT

Ωu = −200,−100,+400 n � = ¬+400, � = ff,.........−100, � = √√,..........−200, � ∈ {f√, √f}


Game&based&on&tossing&two&fair&coins:Two heads +400Two tails –100&One tail,&one head –200

Determine distribution&and&density:

Åu D =

0, ......D < −200.................0.5, .. −200 ≤ D < −1000.75, ... −100 ≤ D < 4001, ......D ≥ 400....................

="

KU D + 200 +

"

rU D + 100 +

"

rU(D − 400)

Ru D =≈

≈ÑÅn(D)

="

KZ D + 200 +

"

rZ D + 100 +

"

rZ(D − 400)


Example&(cont.):Stochastic&variables

Determine mean and&variance:

á n = e D12Z D + 200 +

14Z D + 100 +

14Z(D − 400) ^D

\

]\

...........= −200 <12− 100 <

14+ 400 <

14= −25

á nK = −200 K <12+ −100 K <

14+ 400 K <

14= 62500

Var n = á nK − á n K = 61875

Ru D ="

KZ D + 200 +

"

rZ D + 100 +

"

rZ(D − 400)


Example:&Determine&mean&and&varianceof&different&distributions

Exponential distribution:

Binary distribution:

Uniform.distribution:

Ru D = ¬1

c − a, .a ≤ D < c

....0, ......elsewhere

Mean: ∆«»

K

Variance: »]∆ Ö

"K

Ru D =1….I]Ñ/ .U(D)

Mean: …

Variance: …K

Ru D = ÀZ D − a + 1 − À Z(D − c)

Mean: À.a + 1 − À c

Variance: À 1 − À c − a K

Example:&Uncorrelated&≠ Independent

The&stochastic&variable&n is&+1,&0,&and&−1 with&equal&probability

Mean&value: éu ="

+⋅ 1 +

"

+⋅ 0 +

"

+⋅ −1 = 0

Consider&the&stochastic&variable&q = nK.&Are&n and&q correlated?ÃÕŒ n, q = á nq −éuéå....................= á n+ − 0

....................=13⋅ 1+ +

13⋅ 0+ +

13⋅ −1 + = 0

Uncorrelated!


Are.n and.q independent?.No,.since.q = nK

Example:&Stochastic&processes

n(F) .= .p · sin.(F), for&some&stochastic&variable&p,&with&Å}(a).

Åu N D = Pr n F ≤ D = Pr p sin F ≤ D

=

Pr p ≤D

sin.(F)= Å}

Dsin.(F)

, ..........F: .sin F > 0

Pr 0 ≤ D = U(D) , ................................F: .sin F = 0

Pr p ≥D

sin.(F)= 1 − Å}

Dsin.(F)

, ..F: .sin F < 0

Ru N D =^^D

Åu N D =

1sin.(F)

R}D

sin.(F).............F: .sin F > 0

Z(D), ............................F: .sin F = 0

−1

sin.(F)R}

Dsin F

, F: .sin F < 0


Example:&Stochastic&processes&(cont.)

n(F) .= .p · sin.(F), for&some&stochastic&variable&p,&with&Å}(a).

Determine&mean&and&ACF:

éu F = e DRu(N) D ^D\

]\= e a

\

]\sin F R} a â = sin F é}

á{nK(F)} = e DKRu(N) D ^D\

]\= e a sin F K

\

]\R} a â = sinK F á pK

èu(N)K = á nK(F) − éu

K F = sinK F á pK − é}K = sinK F è}

K

øu F", FK = e a\

]\sin F" a sin FK R} a â = sin F" sin FK á pK



Example:&Filtering&of&Stochastic&Process

Let n(F) be&a&WSS&process&with øu.F = I] œ

Determine&the&output&power á{qK(F)}:

á qK F = øå 0 = e –å R ^R\

]\= e f R K–u R ^R =

\

]\e –u R ^R

"

]"

= Calculator =2Qarctan.(2Q)

LTI”systemf(R)

n(F) q(F) f R = V1, ...... R ≤ 1.0, elsewhere

www.liu.se

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