Post on 25-Jul-2020
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
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 3
TSKS01&Digital&Communication&T Formalities
Information: www.commsys.isy.liu.se/TSKS01Lecturer&&&examiner: Emil&Björnson,&emil.bjornson@liu.seTutorials: Kamil Senel,&kamil.senel@liu.seTeaching&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
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 4
Course&AimsAfter&passing&the&course,&the&student&should&! be&able&to&reliably&perform&standard&calculations®arding&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¬hing&is&received
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 7 2017T09T01 TSKS01&Digital&Communication&T Lecture&1 8
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
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 9
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]
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 10
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
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 11
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!
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 12
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!
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 13 2017T09T01 TSKS01&Digital&Communication&T Lecture&1 14
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]\ ^_
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 15
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¬&linear&is&referred&to&as&non0linear.
Examples: H(F) = D(F − 1) is linearH F = D F
K is non0linear
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 16
TimeTInvariant&and<I&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.
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 17
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 .
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 18
Output&of&an<I&Systems
Theorem:.Let&D(F) be&the&input&to&an&energyTfree<I&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
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 19
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)
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 20
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<I&systems&are&related&as
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 21
ℎ Ff(R)
D(F)n(R)
H F = (D ∗ ℎ)(F)q R = n R f(R)
Example:&Channel&is&an<I&Filter
Impulse response:&ℎ F = Z F − _" + Z F − _K + Z F − _+ + Z(F − _r)
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 22
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 , ={|.{},~}
{|.{~}
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 23
Sample&space:.Ω
�
Measureable&sample&space:Ωu = n � : .for.some.� ∈ Ω
n �
Stochastic&variable
Event&pEvent&,
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 24
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\
]\
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 25
Quadratic&mean&(power):
á nK = e DKRu D ^D\
]\
VarianceVar n = á n − á n K
..................= á nK − á n K
Common¬ation:éu = á n ,&éå = á q
èuK = Var n
èu is&called&the&standard&deviation
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 26
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
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 27
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¬ation:&&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∑ ] ц]ë†
Ö
K±†,†�† = üRu†
Dâ
õ
â°"
..#&&Independence
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 29
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}
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 30
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&
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 31
+ Detectorn q
µ
n∂
∑0 +2−2
1/2R∏(∑)
How.should.we.“best”.detect.n from.q?
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 32
Stochastic Process
Sample&space:.Ω�"
�K
�+
n"(�", F)
nK(�K, F)
n+(�+, F)
F
F
F
Stochastic.process.=.Stochastic.timeAcontinuous.signal.(π = ∞)
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 33
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 .
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 34
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¬ation
! 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
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 35
Sampling.of.stochastic.process.# Stochastic.variable
Ensemble&Averages“Observe&many&realizations&and&make&an&average”&– functions&of&time
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 36
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
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 37
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 _
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 38
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 39
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
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 40
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
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 41
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¬&given&by&the&timeTshifted input:&TimeTvarying
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 42
Example:Stochastic&variablesΩ = HH,HT, TH, TT
Ωu = −200,−100,+400 n � = ¬+400, � = ff,.........−100, � = √√,..........−200, � ∈ {f√, √f}
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 43
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)
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 44
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)
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 45
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!
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 46
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
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 47
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 ^a = sin F é}
á{nK(F)} = e DKRu(N) D ^D\
]\= e a sin F K
\
]\R} a ^a = 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 ^a = sin F" sin FK á pK
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 48
2017T09T01 TSKS01&Digital&Communication&T Lecture&1 49
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
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