Wha s Communiaions? Communiaion involves he ransfer of informaion from one poin o anoher. Three basi elemens Transmier: onvers message ino a form suiable for ransmission Channel: he physial medium, inrodues disorion, noise, inerferene Reeiver: reonsru a reognizable form of he message Speeh Musi Piures Daa 9
Noise in Communiaions Unavoidable presene of noise in he hannel Noise refers o unwaned waves ha disurb ommuniaions Signal is onaminaed by noise along he pah. Exernal noise: inerferene from nearby hannels, humanmade noise, naural noise... Inernal noise: hermal noise, random emission... in eleroni devies Noise is one of he basi faors ha se limis on ommuniaions. A widely used meri is he signal-o-noise power raio SNR SNR= signal power noise power 11
Transmier and Reeiver The ransmier modifies he message signal ino a form suiable for ransmission over he hannel This modifiaion ofen involves modulaion Moving he signal o a high-frequeny arrier up-onversion and varying some parameer of he arrier wave Analog: AM, FM, PM Digial: ASK, FSK, PSK SK: shif keying The reeiver rereaes he original message by demodulaion Reovery is no exa due o noise/disorion The resuling degradaion is influened by he ype of modulaion Design of analog ommuniaion is onepually simple Digial ommuniaion is more effiien and reliable; design is more sophisiaed 1
Objeives of Sysem Design Two primary resoures in ommuniaions Transmied power should be green Channel bandwidh very expensive in he ommerial marke In erain senarios, one resoure may be more imporan han he oher Power limied e.g. deep-spae ommuniaion Bandwidh limied e.g. elephone irui Objeives of a ommuniaion sysem design The message is delivered boh effiienly and reliably, subje o erain design onsrains: power, bandwidh, and os. Effiieny is usually measured by he amoun of messages sen in uni power, uni ime and uni bandwidh. Reliabiliy is expressed in erms of SNR or probabiliy of error. 13
Why Probabiliy/Random Proess? Probabiliy is he ore mahemaial ool for ommuniaion heory. The sohasi model is widely used in he sudy of ommuniaion sysems. Consider a radio ommuniaion sysem where he reeived signal is a random proess in naure: Message is random. No randomness, no informaion. Inerferene is random. Noise is a random proess. And many more delay, phase, fading,... Oher real-world appliaions of probabiliy and random proesses inlude Sok marke modelling, gambling g Brown moion as shown in he previous slide, random walk 7
Probabilisi Coneps Wha is a random variable RV? I is a variable ha akes is values from he oupus of a random experimen. Wha is a random experimen? I is an experimen he ouome of whih anno be predied preisely. All possible idenifiable ouomes of a random experimen onsiue is sample spae Ω. An even is a olleion of possible ouomes of he random experimen. Example For ossing a oin, Ω ={H H, T} For rolling a die, Ω = { 1,,, 6 } 8
Probabiliy Properies P X x i : he probabiliy of he random variable X aking on he value x i The probabiliy of an even o happen is a non-negaive number, wih he following properies: The probabiliy of he even ha inludes all possible ouomes of he experimen is 1. The probabiliy of wo evens ha do no have any ommon ouome is he sum of he probabiliies of he wo evens separaely. Example Roll a die: P X x = k = 1/6 for k = 1,,, 6 9
CDF and PDF The umulaive disribuion funion df of a random variable X is defined as he probabiliy of X aking a value less han he argumen x: Properies F x = P X x X FX = 0, FX = 1 F x F x if x x X 1 X 1 The probabiliy densiy funion pdf is defined as he derivaive of he disribuion funion: f X x = dfx x dx x FX x = fx y dy b P a < X b = FX b FX a = fx y dy f x = sine F x is non - dereasing dfx x X dx 0 X a 30
If x is suffiienly small, Mean and Variane x + x P x < X x + x = f y dy f x x f X y x X Area X f X x x x Mean or expeed value DC level: E[ X] = µ X = x fx x dx Variane power for zero-mean signals: y E[ ]: expeaion operaor X = E [ X X ] = x X f X x dx= E [ X ] X σ µ µ µ 31
Normal Gaussian Disribuion f X x σ 0 m x f F X X x m x = e σ x 1 f π σ 1 for y m σ < x < E[ X ] = m x σ = σ = X 1 e dy σ : rm s value π σ 3
Uniform Disribuion f X x 1 a x b fx x = b a 0 elsewhere 0 x < a x a FX x = a x b b a 1 x > b a + b E[ X ] = b a σ X = 1 33
Join Disribuion Join disribuion funion for wo random variables X and Y F x, y = P X x, Y y Join probabiliy densiy funion Properies XY F XY y fxy x, y = x, y xy 1 F, = f uvdudv, d = 1 XY XY f x = f x, y dy X y= XY 3 f x = f x, y dx Y x= XY 4 X, Y are independen f x, y = f x f y XY X Y 5 X, Y are unorrelaed E[ XY] = E[ X] E[ Y] 34
Independen vs. Unorrelaed Independen implies Unorrelaed see problem shee Unorrelaed does no imply Independene For normal RVs joinly Gaussian, Unorrelaed implies Independen his is he only exepional ase! An example of unorrelaed bu dependen RV s Le θ be uniformly disribued in [ 0, π ] f θ x = for0 1 x π π Y Lous of X and Y Define RV s X and Y as X = os θ Y = sinθ Clearly, X and Y are no independen. Bu X and Y are unorrelaed: π 1 = θ θdθ = EXY [ ] π os sin 0! 0 X 35
Join df Join pdf Independen Join Disribuion of n RVs F x, x,... x P X x, X x,... X x XX 1... Xn 1 n 1 1 n n f F f FX X x, x,... xn 1... Xn 1 X X... X x x x n 1,,... n 1 x1 x... xn x1, x,... xn = FX x1 FX x x1, x,... xn = f X x1 f X x n... F... f x X1X... X n 1 X n n i.i.d. independen, idenially disribued The random variables are independen and have he same disribuion. Example: ouomes from repeaedly flipping a oin. x X1X... X 1 X n n n = 36
For i.i.d. random variables, Cenral Limi Theorem z = x 1 + x + + x n ends o Gaussian as n goes o infiniy. Exremely useful in ommuniaions. Tha s why noise is usually Gaussian. We ofen say Gaussian noise or Gaussian hannel in ommuniaions. x 1 x 1 + x x 1 + x x 1 + x + + x 3 x 3 + x 4 Illusraion of onvergene o Gaussian disribuion 37
Wha is a Random Proess? A random proess is a ime-varying funion ha assigns he ouome of a random experimen o eah ime insan: X; ω. For a fixed sample pah ω: a random proess is a ime varying funion, e.g., a signal. For fixed : a random proess is a random variable. If ω sans all possible ouomes of he underlying random experimen, we shall ge an ensemble of signals. Noise an ofen be modelled as a Gaussian random proess. 38
An Ensemble of Signals ω 1 ω ω n 39
Saisis of a Random Proess For fixed : he random proess beomes a random variable, wih mean µ E[ X; ω] x f xdx ; X In general, he mean is a funion of. Auoorrelaion funion = = R, E[ X ; ω X ; ω] xy f x, y;, dxdy X = = 1 1 X 1 In general, he auoorrelaion funion is a wo-variable funion. X 40
Saionary Random Proesses A random proess is wide-sense saionary if Is mean does no depend on µ = µ X Is auoorrelaion funion only depends on ime differene R, + τ = R τ X X X In ommuniaions, noise and message signals an ofen be modelled as saionary random proesses. 41
Power Speral Densiy Power speral densiy PSD is a funion ha measures he disribuion of power of a random proess wih frequeny. PSD is only defined for saionary proesses. Wiener-Khinhine relaion: The PSD is equal o he Fourier ransform of is auoorrelaion funion: S X f = R τ e j π f τ X dτ A similar relaion exiss for deerminisi signals Then he average power an be found as P = RX 0 = S X f df The frequeny onen of a proess depends on how rapidly he ampliude hanges as a funion of ime. This an be measured by he auoorrelaion funion. 43
Noise Noise is he unwaned and beyond our onrol waves ha disurb he ransmission of signals. Where does noise ome from? Exernal soures: e.g., amospheri, galai noise, inerferene; Inernal soures: generaed by ommuniaion devies hemselves. This ype of noise represens a basi limiaion on he performane of eleroni ommuniaion sysems. Sho noise: he elerons are disree and are no moving in a oninuous seady flow, so he urren is randomly fluuaing. Thermal noise: aused by he rapid and random moion of elerons wihin a onduor due o hermal agiaion. Boh are ofen saionary and have a zero-mean Gaussian disribuion following from he enral limi heorem. 47
The addiive noise hannel n models all ypes of noise zero mean Whie noise Whie Noise Is power sperum densiy PSD is onsan over all frequenies, i.e., N0 S f =, < < f N Faor 1/ is inluded o indiae ha half he power is assoiaed wih posiive frequenies and half wih negaive. The erm whie is analogous o whie ligh whih onains equal amouns of all frequenies wihin he visible band of EM wave. I s only defined for saionary noise. An infinie bandwidh is a purely heorei assumpion. 48
Whie vs. Gaussian Noise Whie noise PSD S N f R n τ 0 Auoorrelaion funion of n : Rn τ = δ τ Samples a differen ime insans are unorrelaed. Gaussian noise: he disribuion a any ime insan is Gaussian Gaussian noise an be olored Whie noise Gaussian noise Whie noise an be non-gaussian N Gaussian PDF Noneheless, in ommuniaions, i is ypially addiive whie Gaussian noise AWGN. 49
Ideal Low-Pass Whie Noise Suppose whie noise is applied o an ideal low-pass filer of bandwidh B suh ha S N N 0, f B f = 0, oherwise By Wiener-Khinhine relaion, auoorrelaion funion R n τ = E[nn+τ] = N 0 B sinbτ 3.1 where sinx=sinπx/πx sinπx/πx. Samples a Nyquis frequeny B are unorrelaed R n τ = 0, τ = k/b, k = 1 1,, 50
Bandpass Noise Any ommuniaion sysem ha uses arrier modulaion will ypially have a bandpass filer of bandwidh B a he fron-end of he reeiver. n Any noise ha eners he reeiver will herefore be bandpass in naure: is speral magniude is non-zero only for some band onenraed around he arrier frequeny f someimes alled narrowband noise. 51
Example If whie noise wih PSD of N 0 / is passed hrough an ideal bandpass filer, hen he PSD of he noise ha eners he reeiver is given by S N N 0 0, f fc B f = 0, oherwise Auoorrelaion funion R n τ = N 0 BsinBτosπf τ whih follows from 3.1 by applying he frequeny-shif propery of he Fourier ransform g G ω g os ω [ G ω ω + G ω + ω ] 0 0 0 Samples aken a frequeny B are sill unorrelaed. R n τ = 0, τ = k/b, k = 1,, 5
Deomposiion of Bandpass Noise Consider bandpass noise wihin f fc B wih any PSD i.e., no neessarily whie as in he previous example Consider a frequeny slie f a frequenies f k and f k. For f small: n k = ak os π f k + θ k θ k: a random phase assumed independen and uniformly disribued in he range [0, π a k : a random ampliude. f -f k f k 53
Represenaion of Bandpass Noise The omplee bandpass noise waveform n an be onsrued by summing up suh sinusoids over he enire band, i.e., fk f + k f n = n = a os π f+ θ k k k k k k = 3. Now, le f k = f k f + f, and using osa + B = osaosb sinasinb we obain he anonial form of bandpass noise where n = n os πf n sin πf n n s = ak osπ fk f + θk k = ak sinπ fk f + θk k n and n s are baseband signals, ermed he in-phase and quadraure omponen, respeively. s 3.3 54
Exraion and Generaion n and n s are fully represenaive of bandpass noise. a Given bandpass noise, one may exra is in-phase and quadraure omponens using LPF of bandwih B. This is exremely useful in analysis of noise in ommuniaion reeivers. b Given he wo omponens, one may generae bandpass noise. This is useful in ompuer simulaion. n n n s n s 55
Properies of Lowpass Noise If he noise n has zero mean, hen n and n s have zero mean. If he noise n is Gaussian, hen n and n s are Gaussian. If he noise n is saionary, hen n and n s are saionary. If he noise n is Gaussian and is power speral densiy S f is symmeri wih respe o he enral frequeny f, hen n and n s are saisial i i independen. d The omponens n and n s have he same variane = power as n. 56
Power Speral Densiy Furher, eah baseband noise waveform will have he same PSD: This is analogous o SN f f + SN f + f, f B S f = Ss f 3.4 = 0, oherwise g Gω ω g os i ω [ G ω ω + G ω+ ω ] 0 0 0 A rigorous proof an be found in A. Papoulis, Probabiliy, Random Variables, and Sohasi Proesses, MGraw-Hill. The PSD an also be seen from he expressions 3. and 3.3 where eah of n and n s onsiss of a sum of losely spaed base-band sinusoids. 57
Noise Power For ideally filered narrowband noise, he PSD of n and n s s is herefore given by S f=s s f N, 0 f B S f = Ss f = 3.5 0, oherwise Corollary: The average power in eah of he baseband waveforms n and n s is idenial o he average power in he bandpass noise waveform n. For ideally filered narrowband noise, he variane of n and n s is N 0 B eah. 58
Phasor Represenaion We may wrie bandpass noise in he alernaive form: n = nos π f nsin π f s = r os[ π f+ φ] r = n + n : he envelop of he noise s 1 n s φ = an : he phase of he noise n θ π f + φ 59
Disribuion of Envelop and Phase I an be shown ha if n and n s are Gaussiandisribued, hen he magniude r has a Rayleigh disribuion, and he phase φ is uniformly disribued. Wha if a sinusoid Aosπ f is mixed wih noise? Then he magniude will have a Rie disribuion. The proof is deferred o Leure 11, where suh disribuions arise in demodulaion of digial signals. 60
Summary Whie noise: PSD is onsan over an infinie bandwidh. Gaussian noise: PDF is Gaussian. Bandpass noise In-phase and quadraure ompomens n and n s arelow-pass random proesses. n and n s have he same PSD. n and n s have he same variane as he band-pass noise n. Suh properies will be pivoal o he performane analysis of bandpass ommuniaion sysems. The in-phase/quadraure represenaion and phasor represenaion are no only basi o he haraerizaion of bandpass noise iself, bu also o he analysis of bandpass ommuniaion sysems. 61
Noise in Analog Communiaion Sysems How do various analog modulaion shemes perform in he presene of noise? Whih sheme performs bes? How an we measure is performane? Model of an analog ommuniaion sysem Noise PSD: B T is he bandwidh, N 0 / is he double-sided noise PSD 64
SNR We mus find a way o quanify = o measure he performane of a modulaion sheme. We use he signal-o-noise raio SNR a he oupu of he reeiver: SNR o average power of message signal a he reeiver oupu = average power of noise a he reeiver oupu P S P N Normally expressed in deibels db SNR db = 10 log 10 SNR This is o manage he wide range of power levels in ommuniaion sysems In honour of Alexander Bell Example: raio of 3 db; 4 6 db; 10 10dB db If x is power, X db = 10 log 10 x If x is ampliude, X db = 0 log 10 x 65
Transmied Power P T : The ransmied power Limied by: equipmen apabiliy, baery life, os, governmen resriions, inerferene wih oher hannels, green ommuniaions e The higher i is, he more he reeived power P S, he higher he SNR For a fair omparison beween differen modulaion shemes: P T should be he same for all We use he baseband signal o noise raio SNR baseband o alibrae he SNR values we obain 66
A Baseband Communiaion Sysem I does no use modulaion I is suiable for ransmission over wires The power i ransmis is idenial o he message power: PT = P No aenuaion: PS = PT = P The resuls an be exended o band band-pass pass sysems 67
Oupu SNR Average signal = message power P = he area under he riangular urve Assume: Addiive, whie noise wih power speral densiy PSD = N 0 / Average noise power a he reeiver P N = area under he sraigh line = W N 0 / = WN 0 SNR a he reeiver oupu: SNR baseband Noe: Assume no propagaion loss Improve he SNR by: = P T NW inreasing he ransmied power P T, resriing he message bandwidh W, making he hannel/reeiver less noisy N 0. 0 68
Revision: AM General form of an AM signal: s = [ A+ m ]osπf AM A: he ampliude of he arrier f : he arrier frequeny m: he message signal Modulaion index: µ = m p A m p : he peak ampliude of m, i.e., m p = max m 69
Signal Reovery n Reeiver model 1 µ 1 A mp : use an envelope deeor. This is he ase in almos all ommerial AM radio reeivers. Simple irui o make radio reeivers heap. Oherwise: use synhronous deeion = produ deeion = oheren deeion The erms deeion i and demodulaion d are used inerhangeably. 70
Synhronous Deeion Muliply he waveform a he reeiver wih a loal arrier of he same frequeny and phase as he arrier used a he ransmier: os π fs AM = [ A+ m ]os π f = [ A + m ][1 + os4 π f ] = A + m + Use a LPF o reover A + m and finally m Remark: A he reeiver you need a signal perfely synhronized wih he ransmied arrier 71
DSB DSB-SC SC Double-sideband suppressed arrier DSB-SC os s Am f π = Signal reovery: wih synhronous deeion only The reeived noisy signal is os DSB SC s Am f π = The reeived noisy signal is sin os f n f n s n s x π π + = + = i ] [ sin os os sin os f f A f n f n f Am f n f n s s s π π π π π + + = + = sin ]os [ f n f n Am s π π + = y 7
Muliply wih osf : y = os π f x Synhronous Deeion = Amos π f + nos π f nssin4 π f = Am [1 + os4 π f] + n [1 + os4 π f] n sin4 π f Use a LPF o keep ~ y = Am + n Signal power a he reeiver oupu: s P S = E { A m } = A E { m } = A P Power of he noise n reall 3.5: P W = N0df = N W N 0 W 73
SNR a he reeiver oupu: Comparison SNR A P NW To whih ransmied power does his orrespond? AP PT = E{ A m os π f} = So PT SNRo = = SNRDSB SC NW Comparison wih o 0 = 0 P = T baseband DSB = NW SC baseband 0 SNR SNR SNR Conlusion: DSB-SC SC sysem has he same SNR performane as a baseband sysem. 74
SSB Modulaion Consider single lower sideband AM: A A s SSB = m os π f + mˆ sin π f where mˆ is he Hilber ransform of m. mˆ is obained by passing m hrough a linear filer wih ransfer funion jsgnf. mˆ and m have he same power P. The average power is A P/4. 77
Noise in SSB Reeiver signal x = s + n. Apply a band-pass filer on he lower sideband. Sill denoe by n he lower-sideband noise differen from he double-sideband noise in DSB. Using oheren deeion: y = x os π f A A = m + n + m + n os4 π f A + m ˆ n s sin4 π f Afer low-pass filering, A y = m + n 78
Noise Power Noise power for n = ha for band-pass noise = N 0 W halved ompared o DSB reall 3.4 S N f N 0 / 0 -f -f +W 0 f -W f f Lower-sideband noise S N f N 0 / -W 0 W f Baseband noise 79
Oupu SNR Signal power A P/4 SNR a oupu A P SNR SSB = 4N0W For a baseband sysem wih he same ransmied power A P/4 A P = 4N W SNR baseband 0 Conlusion: SSB ahieves he same SNR performane as DSB-SC and he baseband model bu only requires half he band-widh. 80
Sandard AM: Synhronous Deeion Pre-deeion signal: x = [ A + m ]os π f + n = [ A+ m ]osπ f + n osπ f n = [ A+ m + n ]osπ f n sinπ f Muliply wih osπf : y = [ A+ m + n ][1 + os4 π f] n sin4 π f s s s sinπ f LPF ~ y = A + m + n 81
Oupu SNR Signal power a he reeiver oupu: Noise power: P S = E{ m } = P P N 0 = N W SNR a he reeiver oupu: Transmied power SNR o P = = SNR NW 0 AM A P A + P P T = + = 8
Comparison SNR of a baseband signal wih he same ransmied power: Thus: A + P N W SNR baseband = 0 SNR AM = SNR baseband A P + P Noe: P < 1 A + P Conlusion: he performane of sandard AM wih synhronous reovery is worse han ha of a baseband sysem. 83
Model of AM Radio Reeiver AM radio reeiver of he superheerodyne ype Model of AM envelope deeor 84
Envelope Deeion for Sandard AM Phasor diagram of he signals presen a an AM reeiver Envelope y = envelope of x x = [ A + m + n ] + ns Equaion is oo ompliaed Mus use limiing ases o pu i in a form where noise and message are added 85
Small Noise Case 1s Approximaion: a Small Noise Case Then Then Thus n n << [ A + m ] << [ A + m n ] s + y [ A + m + n ] SNR o P = SNR NW And in erms of baseband SNR: P SNR A + P Valid for small noise only! 0 env env SNR baseband Idenial o he posdeeion signal in he ase of synhronous deeion! 86
Large Noise Case Large Noise Case nd Approximaion: b Large Noise Case ] [ m A n + >> Isolae he small quaniy: ] [ m A n + >> + + + = ] [ n n m A y + + + + + = + + + = ] [ n n m A n m A n n m A y s s + + + + + + + = 1 ] [ n n n m A n n m A n n s s s + + + + ] [ 1 ] [ n n n m A n n y s s n n E s n + + + = ] [ 1 E n m A E n s 87 E n
Large Noise Case: Threshold Effe Large Noise Case: Threshold Effe From he phasor diagram: n = E n osθ n Then: x ]os [ 1 E m A E y n n n θ + + Use 1 for 1 1 << + + x x x ]os [ 1 m A E y n n θ + + ]os [ m A E E y n n n n θ + + = Noise is mulipliaive here! No erm proporional o he message! R l h h ld ff b l i l l Resul: a hreshold effe, as below some arrier power level very low A, he performane of he deeor deerioraes very rapidly. 88
Summary A: arrier ampliude, P: power of message signal, N 0 : single-sided PSD of noise, W: message bandwidh. 89
Frequeny Modulaion Fundamenal differene beween AM and FM: AM: message informaion onained in he signal ampliude Addiive noise: orrups direly he modulaed signal. FM: message informaion onained in he signal frequeny he effe of noise on an FM signal is deermined by he exen o whih i hanges he frequeny of he modulaed signal. Consequenly, FM signals is less affeed by noise han AM signals 9
Revision: FM A arrier waveform s = A os[θ i ] where θ i : he insananeous phase angle. When s = A osπf θ i = πf we may say ha dθ 1 dθ = πf f = d π d Generalisaion: insananeous frequeny: f i d θ 1 i π d 93
FM In FM: he insananeous frequeny of he arrier varies linearly wih he message: f i = f + k f m where k f is he frequeny sensiiviy of he modulaor. Hene assuming θ i 0=0: Modulaed signal: Noe: θ = π f τ d τ = π f + π k m τ d τ i 0 i f 0 s = Aos πf + πk f m τ dτ 0 a The envelope is onsan b Signal s is a non-linear funion of he message signal m. 94
Bandwidh of FM m p =max m : peak message ampliude. f k f m p < insananeous frequeny < f + k f m p Define: frequeny deviaion = he deviaion of he insananeous frequeny from he arrier frequeny: f = k f m p Define: deviaion raio: β = ff / W W: he message bandwidh. Small β: FM bandwidh x message bandwidh narrow-band FM Large β: FM bandwidh >> x message bandwidh wide-band FM Carson s rule of humb: B T = Wβ+1 = f + W β <<1 B T W as in AM β >>1 B T ff 95
FM Reeiver n Bandpass filer: removes any signals ouside he bandwidh of f ± B T / he predeeion noise a he reeiver is bandpass wih a bandwidh of B T. FM signal has a onsan envelope use a limier o remove any ampliude variaions Disriminaor: a devie wih oupu proporional o he deviaion in he insananeous frequeny i reovers he message signal Final baseband low-pass filer: has a bandwidh of W i passes he message signal and removes ou-of-band noise. 96
Oupu Signal Power Wihou Noise Insananeous frequeny of he inpu signal: Oupu of disriminaor: So, oupu signal power: f i = f + k P k f f m = k S = f m P : he average power of he message signal P 98
Oupu Signal wih Noise In he presene of addiive noise, he real predeeion signal is x = Aos π f + πkf m τ dτ 0 + n os π f n sin π f s I an be shown by linear argumen again: For high SNR, noise oupu is approximaely independen of he message signal In order o alulae he power of oupu noise, we may assume here is no message i.e., we only have he arrier plus noise presen: ~ x = Aosπf + n osπf n sinπf s 99
Phase Noise Phasor diagram of he FM arrier and noise signals Insananeous phase noise: θ i = an 1 ns A+ n For large arrier power large A: A n n s 1 ns ns θi = an A A Disriminaor oupu = insananeous frequeny: 1 dθi 1 dns f i = = π d π A d 100
Disriminaor Oupu The disriminaor oupu in he presene of boh signal and noise: 1 dns k f m + πa d Wha is he PSD of Fourier heory: n d dns = d if x X f dx hen j π fx f d Differeniaion wih respe o ime = passing he signal hrough a sysem wih ransfer funion of Hf = jπ f 101
I follows from.1 ha Noise PSD So f = H f S i f : PSD of inpu signal S o f : PSD of oupu signal Hf : ransfer funion of he sysem Then: { PSD of nd } = j π f { PSD of ns } S i BT ns = N wihin band ± n = π f N f B / { PSD of } 0 { PSD of } d Afer he LPF, he PSD of noise oupu n o is resried in he band ±W 0 f f π 0 0 1 dns 1 PSD of fi = = f N = N πa d πa A f S N o f = N 0 f W 6.1 A T 10
Power Speral Densiies S Ns f a Power speral densiy of quadraure omponen n s of narrowband noise n. b Power speral densiy of noise n d a he disriminaor oupu. Power speral densiy of noise n o a he reeiver oupu. 103
Noise Power Average noise power a he reeiver oupu: Thus, from 6.1 P N W P = N SN f df W o W f = = W A NW 3 A 0 N 0df 7.1 Average noise power a he oupu of a FM reeiver 1 arrier power A A Noise, alled he quieing effe 3 106
Oupu SNR Sine P = k P, he oupu SNR S f SNR O = P 3 Ak f P S SNR 3 P = N W = Transmied power of an FM waveform: From N P T = P km T f p SNRbaseband = and β = : NW 0 W 3k f P P SNR = SNR = 3β SNR W m 0 A FM baseband baseband p β SNR baseband Valid when he arrier power is large ompared wih he noise power FM ould be muh higher han AM 107
Comparison of Analogue Sysems Assumpions: single-one modulaion, i.e.: m = A m osπ f m ; he message bandwidh W = f m ; for he AM sysem, µ = 1; for he FM sysem, β = 5 whih is wha is used in ommerial FM ransmission, wih f = 75 khz, and W = 15 khz. Wih hese assumpions, we find ha he SNR expressions for he various modulaion shemes beome: SNRDSB SC = SNRbaseband = SNRSSB 1 SNRAM = SNRbaseband 3 3 75 SNR = β SNR = SNR FM baseband baseband wihou pre/deemphasis 114
Conlusions Full AM: The SNR performane is 4.8 db worse han a baseband sysem, and he ransmission bandwidh is B T = W. DSB: The SNR performane is idenial o a baseband b sysem, and he ransmission bandwidh is B T = W. SSB: The SNR performane is again idenial, bu he ransmission bandwidh is only B T = W. T FM: The SNR performane is 15.7 db beer han a baseband sysem, and he ransmission bandwidh is B T = β + 1W = 1W wih pre- and de-emphasis he SNR performane is inreased by abou 13 db wih he same ransmission bandwidh. 116
Blok Diagram of Digial Communiaion 119
Advanages: Why Digial? Digial signals are more immune o hannel noise by using hannel oding perfe deoding is possible! Repeaers along he ransmission pah an dee a digial signal and reransmi a new noise-free signal Digial signals derived from all ypes of analog soures an be represened using a uniform forma Digial signals are easier o proess by using miroproessors and VLSI e.g., digial signal proessors, FPGA Digial sysems are flexible and allow for implemenaion of sophisiaed funions and onrol More and more hings are digial For digial ommuniaion: analog signals are onvered o digial. 10
Sampling How densely should we sample an analog signal so ha we an reprodue is form auraely? A signal he sperum of whih is band-limied o W Hz, an be reonsrued exaly from is samples, if hey are aken uniformly a a rae of R W Hz. Nyquis frequeny: f s = W Hz 11
Quanizaion Quanizaion is he proess of ransforming he sample ampliude ino a disree ampliude aken from a finie se of possible ampliudes. The more levels, he beer approximaion. Don need oo many levels human sense an only dee finie differenes. Quanizers an be of a uniform or nonuniform ype. 1
Quanizaion Noise Quanizaion noise: he error beween he inpu signal and he oupu signal 13
Variane of Quanizaion Noise : gap beween quanizing levels of a uniform quanizer q: Quanizaion error = a random variable in he range q Assume ha i is uniformly disribued over his range: Noise variane N 1, q f Q q = 0, oherwise { } Q P = E e = q f q dq 3 / 3 3 1 / 1 q 1 q dq i / 3 4 4 / = = = = 1 14
SNR Assume: he enoded symbol has n bis he maximum number of quanizing levels is L = n maximum peak-o-peak dynami range of he quanizer = n P: power of he message signal m p = max m : maximum absolue value of he message signal Assume: he message signal fully loads he quanizer: 1 n n 1 m p = = 8.1 SNR a he quanizer oupu: SNR o = P P S P N = P 1P = / 1 15
From 8.1 In db, SNR o SNR mp mp 4m p = = = n 1 n n 1P 3P 4 n SNRo = = 8. m p m n 3P db = 10 log 10 n + 10 log10 m p 3P = 0nlog10 + 10log10 m p 3P = 6n + 10log10 m p p db Hene, eah exra bi in he enoder adds 6 db o he oupu SNR of he quanizer. Reognize he radeoff beween SNR and n i.e., rae, or bandwidh. 16
Pulse-Coded Modulaion PCM Sample he message signal above he Nyquis frequeny Quanize he ampliude of eah sample Enode he disree ampliudes ino a binary odeword Cauion: PCM isn modulaion in he usual sense; i s a ype of Analog-o-Digial Conversion. 18
The PCM Proess 19
Problem Wih Uniform Quanizaion Problem: he oupu SNR is adversely affeed by peak o average power raio. Companding is he orresponding o pre-emphasis and de-emphasis sheme used for FM. Predisor a message signal in order o ahieve beer performane in he presene of noise, and hen remove he disorion a he reeiver. Typially small signal ampliudes our more ofen han large signal ampliudes. The signal does no use he enire range of quanizaion levels available wih equal probabiliies. Small ampliudes are no represened as well as large ampliudes, as hey are more susepible o quanizaion noise. 130
Companding Soluion: Nonuniform quanizaion ha uses quanizaion levels of variable spaing, denser a small signal ampliudes, broader a large ampliudes. A praial soluion o nonuniform quanizaion: Compress he signal firs Quanize i Transmi i Expand i Companding = Compressing + Expanding The exa SNR gain obained wih ompanding depends on he exa form of he ompression used. Wih proper ompanding, he oupu SNR an be made insensiive o peak o average power raio. 131
Speeh: Appliaions of PCM & Varians PCM: The voie signal is sampled a 8 khz, quanized ino 56 levels 8 bis. Thus, a elephone PCM signal requires 64 kbps. need o redue bandwidh requiremens DPCM differenial PCM: quanize he differene beween onseuive samples; an save 8 o 16 kbps. ADPCM Adapive DPCM an go furher down o 3 kbps. Dela modulaion: 1-bi DPCM wih oversampling; has even lower symbol rae e.g., 4 kbps. Audio CD: 16-bi PCM a 44.1 khz sampling rae. MPEG audio oding: 16-bi PCM a 48 khz sampling rae ompressed o a rae as low as 16 kbps. 133
Summary Digiizaion of signals requires Sampling: a signal of bandwidh W is sampled a he Nyquis frequeny W. Quanizaion: he link beween analog waveforms and digial represenaion. SNR 3P SNRo db = 6n + 10log10 m p Companding an improve SNR. db PCM is a ommon mehod of represening audio signals. In a sri sense, pulse oded modulaion is in fa a rude soure oding ehnique i.e, mehod of digially represening analog informaion. There are more advaned soure oding ompression ehniques in informaion heory. 135
Line Coding The bis of PCM, DPCM e need o be onvered ino some elerial signals. Line oding enodes he bi sream for ransmission hrough a line, or a able. Line oding was used former o he wide spread appliaion of hannel oding and modulaion ehniques. Nowadays, i is used for ommuniaions beween he CPU and peripherals, and for shor-disane baseband ommuniaions, suh as he Eherne. 138
Line Codes Unipolar nonreurn-o-zero NRZ signaling on-off signaling Polar NRZ signaling Unipolar Reurn-o-zero RZ signaling Bipolar RZ signaling Manheser ode 139
Analog and Digial Communiaions Differen goals beween analog and digial ommuniaion sysems: Analog ommuniaion sysems: o reprodue he ransmied waveform auraely. Use signal o noise raio o assess he qualiy of he sysem Digial ommuniaion sysems: he ransmied symbol o be idenified orrely by he reeiver Use he probabiliy of error of he reeiver o assess he qualiy of he sysem 140
Model of Binary Baseband Communiaion Sysem Lowpass filer T T n n T We only onsider binary PCM wih on-off signaling: 0 0 and 1 A wih bi duraion T b. Assume: AWGN hannel: The hannel noise is addiive whie Gaussian, wih a double-sided PSD of N 0 /. The LPF is an ideal filer wih uni gain on [ W, W ]. The signal passes hrough he LPF wihou disorion approximaely. 141
Disribuion of Noise Effe of addiive noise on digial ransmission: a he reeiver, symbol 1 may be misaken for 0, and vie versa. bi errors Wha is he probabiliy of suh an error? Afer he LPF, he predeeion signal is y = s + n s: he binary-valued funion eiher 0 or A vols n: addiive whie Gaussian noise wih zero mean and variane σ W = N / df = N W W 0 Reminder: A sample value N of n is a Gaussian random variable drawn from a probabiliy densiy funion he normal disribuion: 1 n p n N σ σ π σ N = exp = 0, 0 14
Y: a sample value of y Deision If a symbol 0 were ransmied: y = n Y will have a PDF of N 0, σ If a symbol 1 were ransmied: y = A + n Y will have a PDF of N A, σ Use as deision hreshold T : if Y < T, hoose symbol 0 if Y > T, hoose symbol 1 143
Errors Two ases of deision error: a symbol 0 was ransmied, bu a symbol 1 was hosen a symbol 1 was ransmied, bu a symbol 0 was hosen Probabiliy densiy funions for binary daa ransmission in noise: a symbol 0 ransmied, and b symbol 1 ransmied. Here T = A/. 144
Case i Probabiliy of i ourring = Probabiliy of an error, given symbol 0 was ransmied Probabiliy of a 0 o be ransmied in he firs plae: pi = P e0 p 0 where: p 0 : he a priori probabiliy of ransmiing a symbol 0 P e0 : he ondiional probabiliy of error, given ha symbol 0 was ransmied: 1 n σ P = e0 exp T σ π dn 145
Case ii Probabiliy of ii ourring = Probabiliy of an error, given symbol 1 was ransmied Probabiliy of a 1 o be ransmied in he firs plae: pii = P e1 p 1 where: p 1 : he a priori probabiliy of ransmiing a symbol 1 P e1 : he ondiional probabiliy of error, given ha symbol 0 was ransmied: 1 exp σ π n A σ P = T e1 dn 146
Toal error probabiliy: Pe T = p i + p ii = pp + 1 p P 1 e1 1 e0 Toal Error Probabiliy T 1 n A 1 n = p1 exp dn 1 p + 1 exp dn σ T σ π σ π σ Choose T so ha P e T is minimum: dp e T T dt = 0 147
Opimum Threshold p A T A p ln = σ ln = A T A 1 1 1 1 p1 σ p1 1 σ 1 σ p p A ln = T A T = ln + A 1 p A 1 p 1 1 Equi-probable symbols p 1 = p 0 = 1 p 1 T = A/. For equi-probable bl symbols, i an be shown ha P e0 = P e1. Probabiliy of oal error: P = p + p = p P + p P = P = P e i ii 0 e0 1 e1 e0 e1 sine p 0 = p 1 = 1/, and P e0 = P e1. 149
Digial Modulaion Three Basi Forms of Signaling Binary Informaion a Ampliude-shif keying ASK. b Phase-shif keying PSK. Frequeny-shif keying FSK. 157
Demodulaion Coheren synhronous demodulaion/deeion Use a BPF o reje ou-of-band noise Muliply he inoming waveform wih a osine of he arrier frequeny Use a LPF Requires arrier regeneraion boh frequeny and phase synhronizaion by using a phase-lok loop Nonoheren demodulaion envelope deeion e. Makes no explii effors o esimae he phase 158
ASK Ampliude shif keying ASK = on-off keying OOK s 0 0 = 0 s 1 = A osπ f or s = A osπf, A {0, A} Coheren deeion Assume an ideal band-pass filer wih uni gain on [f W, f +W]. For a praial band-pass filer, W should be inerpreed as he equivalen bandwidh. 159
Pre-deeion signal: Coheren Demodulaion x = s + n = Aos π f + n os π f ns sin π f = [ A + n ]os π f n sin π f s Afer mulipliaion wih osπ f : y A n f n f f = [ A + n ]1 + os4 π f n sin4 π f = [ + ]os π s sin π os π Afer low-pass filering: s y = A + n 160
Bi Error Rae Reminder: The in-phase noise omponen n has he same variane as he original band-pass noise n The reeived signal is idenial o ha for baseband digial ransmission ~ The sample values of y will have PDFs ha are idenial o hose of he baseband ase For ASK he saisis of he reeiver signal are idenial o hose of a baseband sysem The probabiliy of error for ASK is he same as for he baseband ase Assume equiprobable ransmission of 0s and 1s. Then he deision hreshold mus be A/ and he probabiliy of error is given by: P e, ASK A = Q σ 161
PSK Phase shif keying PSK s = A osπ f, A { A, A} Use oheren deeion i again, o evenually ge he deeion signal: y = A + n Probabiliy densiy funions for PSK for equiprobable 0s and 1s in noise use hreshold 0 for deeion: a: symbol 0 ransmied b: symbol 1 ransmied 16
Analysis Condiional error probabiliies: P 1 n+ A = exp dn 0 σ π σ 0 1 n A = exp dn σ π σ e0 P e1 In he firs se n n + A dn = dn and when n = 0, n = A In he seond se n~ 1 Pe exp dn~ 0 = A σ π σ n n A = n+ A dn= dn when n= 0, n = A, and when n=, n = + : 1 A n 1 n Pe 1 = exp 1 dn exp dn = A σ π σ σ π σ 163
So: Bi Error Rae n 1 P P P e0 = e1 = e, PSK = exp dn A σ π σ Change variable of inegraion o z n/σ dn = σdz and when n = A, z = A/σ. Then: 1 = z / A Pe, PSK A e dz = Q π σ σ Remember ha Q 1 x x = exp / d π 164
FSK Frequeny Shif Keying FSK s 0 0 = A osπ f 0, if symbol 0 is ransmied s 1 = A osπ f 1, if symbol 1 is ransmied Symbol reovery: Use wo ses of oheren deeors, one operaing a a frequeny f 0 and he oher a f 1. Coheren FSK demodulaion. The wo BPF s are non-overlapping in frequeny sperum 165
Oupu Eah branh = an ASK deeor LPF oupu on eah branh = AA +noise if symbol presen noise if symbol no presen n 0 : he noise oupu of he op branh n 1 : he noise oupu of he boom branh Eah of hese noise erms has idenial saisis o n. Oupu if a symbol 1 were ransmied y 1 = A + [n 1 n 0 0 ] Oupu if a symbol 0 were ransmied y 0 = A + [n 1 n 0 ] 0 1 0 166
Bi Error Rae for FSK Se deeion hreshold o 0 Differene from PSK: he noise erm is now n 1 n 0. The noises in he wo hannels are independen beause heir spera are non-overlapping. he proof is done in he problem shee. he varianes add. he noise variane has doubled! Replae σ in 17 by σ or σ by σ P e, FSK = Q A σ 167
Comparison of Three Shemes 169
Commen To ahieve he same error probabiliy fixed P e : PSK an be redued by 6 db ompared wih a baseband or ASK sysem a faor of reduion in ampliude FSK an be redued by 3 db ompared wih a baseband or ASK a faor of reduion in ampliude Cauion: The omparison is based on peak SNR. In erms of average SNR, PSK only has a 3 db improvemen over ASK, and FSK has he same performane as ASK 170
Inroduion, Probabiliy and Random Proesses Primary resoures in ommuniaions: power, bandwidh, os Objeives of sysem design: reliabiliy and effiieny Performane measures: SNR or bi error probabiliy Probabiliy disribuion: Uniform disribuion, Gaussian disribuion, Rayleigh disribuion, Riean disribuion Random proess: saionary random proess, auoorrelaion and power speral densiy, Wiener-Khinhine relaion S j τ R τ e π f X f = X dτ 67
Noise Why is noise imporan in ommuniaions? How does noise affe he performane? Wha ypes of noise exis? Whie noise: PSD is onsan over an infinie bandwidh. Gaussian noise: PDF is Gaussian. Addiive whie Gaussian noise Bandlimied noise, bandpass represenaion, baseband noise n and n s, power speral densiy n = n os π f n sin i π f s SN f f + SN f + f, f B S f = Ss f = 0, oherwise 68
Signal-o-noise raio Noise performane of AM SNR = P P S N SNR Baseband ommuniaion model baseband = P T NW 0 AM, DSB-SC, SSB, synhronous deeion, envelope deeion Oupu SNR 69
Noise performane of FM FM modulaor and demodulaor Mehod o deal wih noise in FM: linear argumen a high SNR Derivaion of he oupu SNR, hreshold effe Pre-emphasis and de-emphasis, how hey inrease he oupu SNR 70
Digial ommuniaions PCM: sample, quanize, and enode 3P = + m p Quanizaion noise and SNR SNRo db 6n 10log10 db Companding A/µ-law and line oding Baseband daa ransmission, effes of noise, and probabiliy of error 71