TSKS01&Digital&Communication Lecture&1
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1 TSKS01&Digital&Communication&K Formalities TSKS01&Digital&Communication Lecture&1 Introduction,&Signal&&&Systems,&and&Probability&Theory Emil&Björnson Department&of&Electrical&Engineering&(ISY) Division&of&Communication&Systems Information: Lecturer&&&examiner: Tutorials: Examination: Antonios Laboratory&exercises&(1hp): Two&4Khour&sessions&in&HT2 SignKup&on&the&web Written&exam&(5hp): 1&simple&task&(basics)& Min:&50% 2&questions&(5&points&each) Min:&3p 4&problems&(5&points&each) Min:&6p Pass:&14&points&from&questions&&&problems 2015K09K04 TSKS01&Digital&Communication&K Lecture&1 2 Course&Aims Overview&of&a&Communication&Situation After&passing&the&course,&the&student&should&! be&able&to&reliably&perform&standard&calculations®arding&digital&modulation,& binary&(linear)&codes&for&error&control&and&source&coding. (basics) Source Sender Channel eceiver Destination! be&able&to&account&for&practical&problems&that&arise&in&communication,&and&be& able&to&account&for&possible&solutions&to&those&problems,&and&to&some&extent& be&able&to&perform&calculations&for&those&solutions. (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&and&related&concepts. (problems) Source:statistics Stochastic:process Markov:chains Auto7correlation Sampling econstruction D/A7conversion Modulation Channel:coding Source:coding Filtering LTI: or:non:lti Fading Noise Intentional: jamming Sampling econstruktion A/D7conversion Demodulation Channel:decoding Source:decoding Filtering Demands:on:quality Cost:functions Application: Purpose! with&experimental&purpose&be&able&to&evaluate&and&to&some&extent&implement& such&communication&systems&that&are&treated&in&the&course. (laboratory-exercises) 2015K09K04 TSKS01&Digital&Communication&K Lecture& K09K04 TSKS01&Digital&Communication&K Lecture&1 4
2 A&OneKway&Telecommunication&System Digital&Communication:&An&Exponential&Story Packing Error&control Digital to&analog Medium Source Source: encoder Channel: encoder Modulator Overall:IP:traffic Source& coding Channel& coding Digital& modulation Analog: channel Digital&Communication&Devices&are&Everywhere Source:&Cisco&VNI&Global&IP& Traffic&Forecast,& &! PCs,&tablets,&phones,&machineKtoKmachine,&TVs,&etc. Destination Source: decoder Unpacking Channel: decoder Error&correction De7 modulator Analog&to&digital! Total&internet&traffic:& 23%&growth/year! Number&of&devices:& 12%&growth/year How:can:we:sustain:this:exponential:growth? 2015K09K04 TSKS01&Digital&Communication&K Lecture& K09K04 TSKS01&Digital&Communication&K Lecture&1 6 InformationKBearing&Signals Designing&Efficient&Communication&Systems! 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)! InformationKbearing&signals! Select&the&samples&in&a&particular&way&" epresent&digital&information! Spectral&efficiency&(SE):&Number&of&bits&transferred&per&sample&[bit/s/Hz]! Performance&metric:&Bit&rate&[bit/s]! Formula:&##Bit#rate#[bit/s] = Bandwidth# Hz 1 Spectral#efficiency#[bit/s/Hz]! How&do&we&increase&the&bit&rate? 1. Use&more&bandwidth 2. Increase&spectral&efficiency Wired:& Use&cable&that&handles& higher&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! 2015K09K04 TSKS01&Digital&Communication&K Lecture& K09K04 TSKS01&Digital&Communication&K Lecture&1 8
3 Preliminary&Lecture&Plan HT1:! Introduction&and&repetition& (Lecture-1)! Introduction&to&stochastic&processes (Lecture-2)! Digital&modulation (Lectures-396)! ErrorKcontrol&coding (Lectures-798) HT2:! Practical&aspects (Lectures-9911)! Software&defined&radio&(lab&preparation) (Lecture-12) epetition:&signals&and&systems Signals: Systems: Complex:exponential: Voltages,:currents,:or:other:measurements Manipulate/filter:signals 8(:) System <(:) = >?@AB = cos 2EF: + H sin(2ef:) Unit:step: I(:) #= # J 0, : < 0 #1, : > 0 Unit:impulse: P(:)::: 8 : P : #T: = 8(0) B Property: I : = P U TU 2015K09K04 TSKS01&Digital&Communication&K Lecture& K09K04 TSKS01&Digital&Communication&K Lecture&1 10 Special&Outputs Linear&Systems General&case: Impulse&response:& Step&response:& Energy7free: 8(:) <(:) system Energy'free system:-no-transients,-constant input-" constant output P(:) I(:) Energy7free: system Energy7free: system h(:) W(:) Definition: Let&8 X (:) and&8? : be&input&signals&to&a&onekinput& energykfree&system,&and&let&< X : and&<? : be&the&corresponding& output&signals.&if < : = Y X < X : + Y? <? (:) is&the&output&corresponding&to&the&input 8 : = Y X 8 X : + Y? 8? :, for&any&8 X (:), 8? :,&Y X,&and&Y?,&then&the&system&is&referred&to&as& linear.&a&system&that&is¬&linear&is&referred&to&as&non'linear. Examples: <(:) = 8(: 1) is linear? < : = 8 : is non'linear 2015K09K04 TSKS01&Digital&Communication&K Lecture& K09K04 TSKS01&Digital&Communication&K Lecture&1 12
4 TimeKInvariant&and<I&Systems TimeKinvariant&system Definition::Let&8(:) be&the&input&to&an&energykfree&system&and&let& <(:)#be&the&corresponding&output.&if&<(: + U) is&the&output&for&the& input&8(: + U) for&any&8(:),&:,&and&u,&then&the&system&is&referred&to&as& time'invariant.&otherwise&the&system&is&referred&to&as&time'varying. Convolution Definition: The&convolution&of&the&signals&Y(:) and&[(:) is&denoted& by&(y [)(:) and&is&defined&as Y [ : = ] Y U [ : U TU. Examples: <(:) = 8(: 1) is time'invariant < : = cos 2EF: 8(:) is time'varying The&convolution&is&a&commutative&operation:& Y [ : = [ Y :. LTI&system Definition::A&system&that&is&both&linear&and&timeKinvariant&is&referred& to&as&a&linear-time'invariant-(lti)-system. 2015K09K04 TSKS01&Digital&Communication&K Lecture& K09K04 TSKS01&Digital&Communication&K Lecture&1 14 Output&of<I&Systems Frequency&Domain 8(:) g(f) Theorem::Let&8(:) be&the&input&to&an&energykfree<i&system&with& impulse&response&h(:),&then&the&output&of&the&system&is < : = 8 h :. Fourier&transform: F 8(:) = 8 : = S>?@AB T: Exists if 8 : T: < : F Proof::Let&_ Y(:) denote&the&output&for&an&arbitrary&input&a(t),&then < : = _ 8(:) = _ ] 8 U P : U #TU ##### Linear = ] 8 U _ P : U #TU # Time inv ## = ] 8 U h : U TU = 8 h : # 2015K09K04 TSKS01&Digital&Communication&K Lecture&1 15 Inverse&transform: F SX g(f) = g F = >?@AB TF Common-notation Spectrum&of&8(:): g(f) Amplitude&spectrum: g F Phase&spectrum: arg g(f) 2015K09K04 TSKS01&Digital&Communication&K Lecture&1 16 Image&from:&
5 LTI&Systems:&Time&and&Frequency&Domain Probability,&Stochastic&Variable,&and&Events Property::Let&i F = F Y(:) and&! F = F [(:) be&the&fourier& transforms&of&the&signals&y(:) and&[(:),&then F (Y [)(:) = i F! F. Property::The&input&and&output&of<I&systems&are&related&as 8(:) g(f) h : _(F) < : = (8 h)(:) j F = g F _(F) Sample&space:#Ω w g w Stochastic&variable Event&i Total&probability: Pr Ω m = 1 Probability&of&event&i: Pr#{i # [0,1] Joint&probability: Pr#{i,! Conditional&probability: Pr i! = st#{u,v st#{v Event&! Measureable&sample&space: Ω m = g w :#for#some#w Ω 2015K09K04 TSKS01&Digital&Communication&K Lecture& K09K04 TSKS01&Digital&Communication&K Lecture&1 18 Probabilities&and&Distributions Example Game&based&on&tossing&two&fair&coins: Two heads +400 Two tails 100& One tail,&one head 200 Probability&distribution&function: y g (8) = Pr#{g 8 [0,1] Probability&density&function&(PDF): Properties: F g (8) # = T T8 y g(8) y g (8)#is&nonKdecreasing, y g (8) 0#and&F g (8) 0# for&all 8, F m (8) T8# = #1, Pr 8 1 < g 8 2 = F m 8 T8. ~ y m 8 = Ω = HH,HT, TH, TT Ω m = 200, 100, , ######8 < 200################# 0.5, ## < , ### < 400 1, ######8 400#################### = X? I X Ñ I X I(8 400) Ñ F m 8 = Ö Ö y g(8) = X? P X Ñ P X P(8 400) Ñ +400, w =,######### g w = Ä 100, w = ÅÅ,########## 200, w {_Å, Å_ 2015K09K04 TSKS01&Digital&Communication&K Lecture& K09K04 TSKS01&Digital&Communication&K Lecture&1 20
6 Expectation&and&Variance Example&cont d Expectation&(mean): Ü g = ] 8F m 8 T8 For&discrete&variables: Ü g = á 8 à Pr{g = 8 à à For&functions&of&a&variable: j = W g Ü j = ] <F â < T< = Ü W(g) = ] W 8 F m 8 T8 Quadratic&mean&(power): Ü g? = ] 8? F m 8 T8 Variance Var g = Ü g Ü g? ##################= Ü g? Ü g? Common¬ation: ã m = Ü g,&ã â = Ü j å m? = Var g å m is&called&standard&deviation Ü g = ] F m 8 = X P X P X P(8 400)? Ñ Ñ P P P(8 400) T8 4 ###########= = 25 Ü g? = 200? ? ? = Var g = Ü g? Ü g? = K09K04 TSKS01&Digital&Communication&K Lecture& K09K04 TSKS01&Digital&Communication&K Lecture&1 22 Gaussian Distribution,&è(ã, å? ) Example of the&ì Function Probability&density F m 8 = 1 å 2E =S Sê?ë Probability&distribution&F m 8 = is&complicated Mean:&ã Variance:&å? F m 8 T8 ì 8 = 1 y m 8 = ìkfunction X?@ =Sî T: Can&be&computed&from&table for&è(0,1) 2015K09K04 TSKS01&Digital&Communication&K Lecture&1 23 ì(1.96) # S? 2015K09K04 TSKS01&Digital&Communication&K Lecture&1 24
7 Example of the&ì Function cont d Other Common&Distributions Uniform:distribution: 1 F m 8 = Ä[ Y, #Y 8 < [ ####0, ######elsewhere Mean: Variance: öõú? úsö X? ì(8) #= 10 Sò 8# #4.75 For&8 > 0 we&have:&& (1 8 S? ) For&large&8 we&have:&&&&&&ì 8 X X?@ =Sô?@ =Sô Important:&ì(8) decreases&as&8 increases < ì 8 < X?@ =Sô Exponential distribution: F m 8 = 1 ù #=S /ü #I(8) Binary distribution: F m 8 = P 8 Y + 1 P(8 [) Mean: ù Variance: ù? Mean: #Y + 1 [ Variance: 1 [ Y? 2015K09K04 TSKS01&Digital&Communication&K Lecture& K09K04 TSKS01&Digital&Communication&K Lecture&1 26
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