Signals and communications fundamentals

Transcription

1 Signals and communicaions undamenals Luca Reggiani hp://home.dei.polimi.i/reggiani/ Diparimeno di Eleronica, Inormazione e Bioingegneria Poliecnico di Milano Signals and communicaions undamenals

2 Oulines. Base Band and RF signals Deiniions Signal Bandwidh, Specrum and Power Noise loor and Signal To Noise Raio. Modulaion Modulaion Basics Modulaion Techniques Signals and communicaions undamenals

3 Maerial Reerences courses Fondameni di segnali e rasmissione Wireless communicaions J. Proakis, Digial Communicaions, McGraw Hill. S. Haykin, "Digial Communicaions", John Wiley & Sons. Signals and communicaions undamenals

4 Base Band and RF signals /3 The communicaion sysem Signals and communicaions undamenals

5 Base Band and RF signals A signal x represens he behavior o physical eniies emperaure, curren, volage as a uncion o one or more variables ime, requency, space. Mono-dimensional signals can be coninuous / discree periodic / aperiodic y=y+kt inverse o period T is he undamenal requency deerminisic / random de. i he signal is ixed by he value o he independen variable rand. i he value can be described only in saisical erms. Signals and communicaions undamenals

6 Signals and communicaions undamenals Base Band and RF signals Mean value d x T x T T T / / lim Energy d x E Isananeous power x p Average power in an inerval T d x T P T T T / /

7 Signals and communicaions undamenals Base Band and RF signals Average power d x T P T T T / / lim Average power o a periodic signal d x T P o o T T o / /

8 Base Band and RF signals Recangular signal x rec Sine and cosine x Acos. x Asin. A = ampliude, o = requency, = phase period T =/ Ininie energy and average power A / Signals and communicaions undamenals

9 Base Band and RF signals Complex exponenial signal Eulero expj cos j sin x Re Im x cos ; x sin. cos sin expj exp j j. expj exp j ; Signals and communicaions undamenals

10 Base Band and RF signals Fourier Transorm and inverse ransorm x X x X j x e d x j X e d X Some properies j Time shi y x Y X e j Frequency shi y x e Y X Signals and communicaions undamenals

11 Base Band and RF signals Muliplicaion in he requency domain z x y d Z X Y Parseval s heorem x * * * y d X Y X Y d E x d X d Energy specral densiy Signals and communicaions undamenals

12 Base Band and RF signals Examples sin T x A rec rect X AT AT sinc T T T Signals and communicaions undamenals

13 Signals and communicaions undamenals Base Band and RF signals cos X x F sin j j X x F

14 Base Band and RF signals Modulaion heorem: requency up-conversion y s cos j j s e e Y S S carrier Base Band Band-pass RF signal Signals and communicaions undamenals

15 Base Band and RF signals Modulaion on orhogonal carriers y s sin j j j s e e Y S S j carrier Base Band Pass Band RF signal Sine and cosine carriers creae wo orhogonal, independen channels on he same specrum porion: direc I and quadraure Q channels Signals and communicaions undamenals

16 Signals and communicaions undamenals Base Band and RF signals Modulaion heorem requency down-conversion sen s y s y Q I cos sen s s y y y Q I cos Complex represenaion y=s cos - s sin = Re[s + j s ] expj = a cos + Equivalen low-pass signal represenaion j e a js s s * S S Y

17 Signals and communicaions undamenals Base Band and RF signals base band down-conversion 4 4 cos cos cos cos cos cos s sen s s sen s s sen s s y z 4 cos 4 cos cos s s sen s sen s sen s sen sen s s sen y z Modulaion heorem

18 Base Band and RF signals RF modulaion: recovery o he base-band signal rom he cos and sin carriers LPF LPF Signals and communicaions undamenals

19 Base Band and RF signals Linear ime invarian LTI sysems Linear i Time invarian i F[ax + bx ] = af[x ]+bf[x ] y=f[x] y-=f[x-] Impulse response Signals and communicaions undamenals

20 Signals and communicaions undamenals Base Band and RF signals Convoluional inegral ] [ h x d x h d h x d F x d x F x F y Response o a complex exponenial / cosine exp exp exp exp H j d j h j d j h y H j j H H j H j H j H j h x y j j x cos exp exp exp exp exp exp exp exp cos

21 Base Band and RF signals Transer uncion requency domain H h e j d - Inpu X = x - Oupu Y = y - Impulse response H = h Y = X H Signals and communicaions undamenals

22 Base Band and RF signals Ampliude / phase disorion Non disorive LTI sysem: y C x Condiions on he ampliude / phase specra jus in he bandwidh o he inpu signal H C e j A C Signals and communicaions undamenals

23 Base Band and RF signals Filers H - c c Cu requency c Low-pass ideal iler Signals and communicaions undamenals

24 Base Band and RF signals Filers H - c c High-pass ideal iler Signals and communicaions undamenals

25 Base Band and RF signals Filers H - o - c - o - o + c o - c o o + c Band-pass ideal iler Cener requency o and RF band c Signals and communicaions undamenals

26 Base Band and RF signals Random signals Communicaion signal received = modulaed signal + noise + inererence r = s + n + i Transmied signal is aleaory because i conains he inormaion unknown a-priori a he receiver Noise = several physical sources, hermal and elecronic noise Inererence = oher channels or oher communicaion sysems Main saisical properies o he process: average and auocorrelaion uncion E[ V ] V V V; dv mv E[ V V ] E[ V V ] V V Vi, V ;, dvi dv RV, Signals and communicaions undamenals

27 Base Band and RF signals Random signals In case o wide-sense saionary process: E[ V ] m V m V E[ V V ] R, R R E[ V V V V V ] Auo-correlaion properies R V E V R V R R V V RV Signals and communicaions undamenals

28 Base Band and RF signals Random signal as inpu o an LTI m y X d m h d m E[ Y ] h E H x x R YY h h R dd R h h XX XX S YY j RYY e d S XX H Signals and communicaions undamenals

29 Base Band and RF signals Random signals Power specral densiy S V R V e j d Wiener-Khinchine heorem S V d P R V E[ V ] Whie processes Signals and communicaions undamenals

30 Base Band and RF signals Thermal noise I is associaed o he random lucuaions o elecrons inside a conducor. Model: v Volage v or curren due o hermal noise is a random process wih average value zero, Gaussian ampliude probabiliy densiy and la specral power densiy ill o some THz equal o N V ktr v a N I ktg kt / R m V = a Signals and communicaions undamenals

31 Base Band and RF signals T is he absolue emperaure Kelvin degrees k = Bolzman consan =.38 x -3 Joule. Power specral densiy N v is measured in [V /Hz] The maximum power absorbed by an exernal load is obained or an adaped load = R P O V R O O V O R V 4R NV In a generic requency inerval we have PO 4 R And hence he available power specral densiy is [W/Hz] N kt / Signals and communicaions undamenals

32 Base Band and RF signals Noise emperaure A he oupu o an elecronic block wih power gain G we obain P NO N G d N G B The noise emperaure o any whie process can be deined as N T E k Signals and communicaions undamenals

33 Base Band and RF signals Base-band or RF ampliier G G G G B - - B P NO N G d N G B Signals and communicaions undamenals

34 Base Band and RF signals Noise equivalen emperaure is compued a he inpu o a sysem. T E T E T E G T E3 G G Signals and communicaions undamenals

35 Base Band and RF signals Noise acor F / noise igure NF I is deined w.r.. a convenional emperaure T = 9 K. P P G P kt BG kt BG F kt NO NI NA E So we have BG T E F T Signals and communicaions undamenals

36 Base Band and RF signals Bandwidh Exac bandwidh = X requency inerval in which he signal requency conen is dieren rom zero X B - - B Base Band signal RF signals Signals and communicaions undamenals

37 Base Band and RF signals Bandwidh Frequency inerval in which he signal requency conen is signiicanly dieren rom 3dB bandwidh: i he ampliude specrum o a signal iler assumes maximum value in, i is he requency a which he ampliude is reduced by a acor X X 3 99% 9% bandwidh: bandwidh conaining 99% 9% o he signal energy X X d.99 d Signals and communicaions undamenals

38 Base Band and RF signals Bandwidh zero bandwidh: requency a which we ind he irs zero o he ampliude specrum /T or he recangular pulse Equivalen noise bandwidh The bandwidh o an LTI sysem can be measured by applying a whie process wih power specral densiy N. P NO B EQ N H d N H d N H d N B H H H d EQ N = N Signals and communicaions undamenals

39 Base Band and RF signals Bandwidh example RC iler H jrc 3dB 9% RC 6.3 RC 99% 64 RC Signals and communicaions undamenals

40 Base Band and RF signals Signal-o-noise raio SNR I is he raio beween he signal carrying inormaion power and he noise +inererence power. I usually measures he qualiy o he recepion o a signal in a communicaion sysem SNR P P R N N P R B P kt R E B N convenionally one-sided is usually reerred as AWGN Addiive Whie Gaussian Noise Signals and communicaions undamenals

41 Modulaion Digial signals Sequence o bis [/] Modulaion Analog signal o be ransmied a a carrier Signals and communicaions undamenals

42 Modulaion Aracive eaures o modulaion schemes or wireless sysems. High specral eiciency: η = R/W bi/s/hz. This value reurns how eicienly we are using he scarce specrum resource. I depends mainly on he number o levels M in he modulaion; or example, in case o linear modulaions wih use o he raised cosine pulse, i is easy o obain η = logm/ + ρ bi/s/hz.. Compac signal specrum. This is imporan or assuring low inererence on adjacen bandwidhs and eicien use o he assigned specrum. 3. High energy eiciency. This is imporan especially or mobile devices, which are baery powered. The las ampliier sage in he ransmier chain is usually he mos consuming elemen in he device since i mus provide he power o be emied rom he anenna. Thereore i is imporan, i possible, o use ampliiers in working regions characerized by high energy eiciency. Signals and communicaions undamenals

43 Modulaion Aracive eaures o modulaion schemes or wireless sysems 4. Good perormance, in erms o error probabiliy vs. SNR e.g. E B /N bu also in erms o robusness o random ading lucuaions. 5. Limied complexiy, in erms o circuial implemenaion and algorihms. This eaure is imporan or mobile devices, which should guaranee robusness, low coss and low energy consumpion. Some o he lised eaures can be obviously conrasing. For example a high specral eiciency is in conras wih energy eiciency and low complexiy. Signals and communicaions undamenals

44 Modulaion Scheme o he base-band binary communicaion scheme Binay sequence {b k } a insans kt b.t b is called bi ime bi rae R b =/T b. Pulse Ampliude Modulaor PAM ransorms he bi sequence in a pulses sequence wih levels a k, whose values are given by he anipodal represenaion a k i i b b k k b k Encoder a k Tx iler h T x clock Signals and communicaions undamenals

45 Modulaion Scheme o he base-band binary communicaion scheme x a h kt k k T b h T T b a a 3 T b a a 3 a a T b a a k a k kt b s a h kt k k T b Signals and communicaions undamenals

46 Modulaion Scheme o he base-band binary communicaion scheme: eec o he channel Transmied signal s can be modiied by he channel, which is modelled by a iler wih impulse response h. r x hc x r Channel r h + C AWGN w The noise AWGN wih wo-sided PSD N / is summed a he inpu o he receiver. Signals and communicaions undamenals

47 Modulaion Scheme o he base-band binary communicaion scheme: he receiver The receiver is consiued by wo undamenal seps: a iler or limiing he impac o noise, a decisor, which esimaes he ransmied level and hence he ransmied bi. + r Rx iler h R y Sampler a k =kt b y k Decisor bˆk AWGN w Decision hreshold Signals and communicaions undamenals

48 Modulaion Scheme o he base-band binary communicaion scheme: he receiver y k Decisor bˆk i y k > i y k < Threshold The more he sequence bˆk more he sysem is reliable. reproduces correcly he ransmied sequence b k,he BER = Bi Error Rae = percenage o incorrec bis bi error probabiliy Signals and communicaions undamenals

49 Modulaion The opimal receiver The receiver iler H R is designed or maximizing, aer he sampler, he raio beween he signal componen power and he noise componen power. y k = u k + n k u max k N g is he received single pulse u G H N e n H R R j d d Signals and communicaions undamenals

50 Signals and communicaions undamenals Modulaion The opimal receiver The maximum SNR j R e G H max N E u G n is acheived when i.e. he so-called mached iler g h R N d G d H N d H d e G u R R j n

51 Modulaion The example o received recangular pulses A g k g- ka k g-+t = h R kat g = g*h R g T T -T T T Signals and communicaions undamenals

52 Modulaion The SNR E G N is he raio beween he received pulse energy no modulaed by he level E G and he noise power specral densiy N. Noice ha his resul does no depend on g. This raio is dimensionless [J ]=[W]/[Hz] and i is relaed o wo undamenal SNR measures o communicaions sysems: E B /N,E S /N where E B = energy per bi E S = energy per symbol or muli-level modulaions Signals and communicaions undamenals

53 Modulaion Relaion beween E G N and SNR = S/N, deined as raio beween powers, is easy: R/W is he specral eiciency [bi/s/hz], already inroduced. Signals and communicaions undamenals

54 Modulaion The error probabiliy depends on E B / N : P E P E a P E a m A, A k k P a k m P a P E a k m k P a k In he binary case P E P E ak P E ak Considering ha u k a k G j H R e d ak G d a k E G n N H R d N G d N E G Signals and communicaions undamenals

55 Signals and communicaions undamenals Modulaion N E Q N E Q N E Q N E Q E P B G G G B N E Q dn e E n P E a P G E n n G k k G n we obain

56 Modulaion P B E Q E N B - -4 BER or P B E B /N [db] Signals and communicaions undamenals

57 Modulaion Inersymbol inererence ISI A he sampler y a s it i i y a s a s-t a s-t T Signals and communicaions undamenals

58 Modulaion Inersymbol inererence ISI ISI y kt ai s kt it ak s ai s k i T i Useul signal ik ISI ISI a a y yt yt a ISI T Signals and communicaions undamenals

59 Signals and communicaions undamenals Modulaion Nyquis s crierium Addiional consrains are necessary or avoiding he generaion o inersymbol inererence ISI: One o he mos diused amily o pulses ha respecs he Nyquis s crierium is he raised cosine: /W roll-o k k kt s sin 4 W W W W W W S M M M M M M

60 Modulaion where W M =/T and is beween and W M. B T W M WM T W = roll-o acor In he ime domain..8.6 = =.5 =..4. /T /T Signals and communicaions undamenals

61 Modulaion Muli-level modulaions Each pulses carries n bis -> he levels have o be M = n The energy per bi is relaed o he symbol energy The bi rae is relaed o he symbol rae {+/-, +/- 3, +/- 5,.., +/- M-} E R B E S log M B log M R S The received power is relaed o he energy P R E B R B E S R S The bandwidh depends on he symbol rae W T R S Signals and communicaions undamenals

62 Modulaion Muli-level modulaions Increasing he number o levels decreases he bandwidh and increases he specral eiciency R B W log M The BER is relaed o he SER For paricular modulaions P B E log M P S E N P S E Signals and communicaions undamenals

63 Modulaion From base-band o RF pass-band modulaions When we shi a signal around a carrier, he occupied bandwidh doubles bu we can exploi independen channels I/Q W RF T T Signals and communicaions undamenals

64 Modulaion Pass-band modulaions In general, given a carrier waveorm see he generic band-pass signal Agcos + = Ag cos cos - Ag sin sin we can modulae: he ampliude he phase he requency Signals and communicaions undamenals

65 Modulaion BPSK Binary Phase Shi Keying P B E P S E Q E N B Signals and communicaions undamenals

66 Signals and communicaions undamenals Modulaion M-QAM Quadraure Ampliude Modulaion 6-QAM signal space consellaion n n S d Q d Q E P N E Q d Q G n S E G E In general M = n 3 4 N M ne Q E P B S

67 Modulaion M-QAM Quadraure Ampliude Modulaion M-QAM modulaor Signals and communicaions undamenals

68 Modulaion M-QAM Quadraure Ampliude Modulaion 6-QAM demodulaor Signals and communicaions undamenals

69 Modulaion M-PSK Phase Shi Keying gcos + k n k M P E Q E N s sin M Signal space or 8-PSK Signals and communicaions undamenals

70 Modulaion M-PSK Phase Shi Keying 8-PSK demodulaor Signals and communicaions undamenals

71 Signals and communicaions undamenals Modulaion Binary FSK Frequency Shi Keying FSK can be demodulaed in a coheren or non coheren way w/o phase recovery bi : s = Agcosπ + Φ bi : s = Agcosπ + Φ = =/T or assuring orhogonaliy beween pulses N E Q E P E P B S B N E S B B e E P E P

72 Modulaion Binary FSK Frequency Shi Keying non coheren demodulaor Signals and communicaions undamenals

73 Modulaion Comparison beween digial modulaions Signals and communicaions undamenals

74 Modulaion Comparison beween digial modulaions Signals and communicaions undamenals

75 Modulaion Comparison beween digial modulaions Signals and communicaions undamenals

76 Modulaion Comparison beween digial modulaions R/W 8 R C W log N P S W 6 4 Shannon M-QAM M-PSK 3 6 M-ASK E N B / R W R / W M-FSK P B 5 3 E B N Signals and communicaions undamenals

77 Impairmens a he receiver Non-lineariies ampliiers LO phase noise Quanizaion noise RF componens response I/Q imbalance LO and ilers ampliude and phase imbalance Synchronizaion sampling ime, requency, phase or coheren receivers Channel esimaion errors equalizaion Signals and communicaions undamenals

78 Impairmens a he receiver The modulaion error raio MER is a measure o he signal-o-noise raio SNR in a digially modulaed signal. MER over N symbols is deined as: where MER I i is he I componen o he j-h symbol received N Ii I i Qi Qi Q i is he Q componen o he j-h symbol received I i is he ideal I componen o he j-h symbol received Q i is he ideal Q componen o he j-h symbol received. N i N N i I i Q i Signals and communicaions undamenals

79 Impairmens a he receiver Error vecor magniude EVM is a measuremen o demodulaor perormance in he presence o impairmens. The rms o he error beween he received and expeced symbol locaions is compared o a reerence maximum value. sin w = measured symbol locaion by he demodulaor e = error vecor v = ideal symbol locaion cos phase error EVM N v N i MAX e Signals and communicaions undamenals