Introduction: Analog Communication: Goal: Transmit a message from one location to another.

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1 ECE-5 Phil Schnier January 6, 8 Inroducion: Goal: Transmi a rom one locaion o anoher When is coninuous waveorm analog comm (eg, FM radio), sequence o numbers digial comm (eg, mp ile), hough he sequence o numbers migh represen a coninuous waveorm (as in he case o mp audio) Typical communicaion media: wised pair wire (eg, elephone A ) coaxial cable (eg, TV A,D, daa D ) iber opic cable (eg, eherne D ) EM waves (eg, cellular phones A,D, WiFi D, TV A,D ) waer waves (eg, underwaer nework A,D ) power lines A,D compac disc D hard drive D magneic ape A,D where A = analog and D = digial Noe ha, wheher he is discree-ime or coninuous-ime, he ransmied is coninuous-ime! ECE-5 Phil Schnier January 6, 8 Analog Communicaion: analog modulaor ransmied channel recovered Perec recovery is impossible in he presence o noise! Digial Communicaion: binary coder/ mapper modulaor sampler symbol sequence,-,,- passband sequence pulse shaper channel equalizer passband equalized sequence,-9,,-9 demodulaor demodulaor de-mapper /decoder recovered A digial is convered o an analog coding and pulse-shaping, and hen ransmied using analog modulaion To recover he, he is demodulaed, sampled, and digially processed Perec recovery is possible even in he presence o noise!

2 ECE-5 Phil Schnier January 6, 8 ECE-5 Phil Schnier January 6, 8 Preview o Comm Sysem Componens: Modulaor: Translaes analog o passband : T T c where c is he carrier requency There are wo principal moivaions or doing his: Oen we wan o communicae several s simulaneously (eg, TV, radio, voice) I s diicul or impossible o do his i hey overlap in requency! Wireless EM ransmission/recepion is much easier a higher requencies, since need anenna lengh > λ (λ = c c is wavelengh and c=e8 m/s speed o ligh) sysem ransmission band λ/ VHF (TV) MHz m UHF (TV) GHz cm cellular MHz cm WiFi 4 GHz cm Noice ha pracical anenna lengh deermines where dieren ypes can be ransmied / /T c Coder/Mapper: Coder ransorms sequence o bis ino an error-resilian sequence o coded bis Mapper ransorms coded bis ino discree symbols Ex: I he symbol alphabe is {,,, } and he symbol mapping is be ransmied via bis symbol -, hen ASCII ex would - leer ASCII code symbol sequence a b - - c - - d

3 ECE-5 Phil Schnier January 6, 8 Pulse Shaper: Convers symbol sequence ino a coninuous waveorm In linear modulaion schemes, he ime-n symbol s[n] scales a nt -delayed version o pulse p(): y() = s[n]p( nt ) n T = symbol period Ex: Say symbol sequence is [,,,, ] Then p() : p( T ) : p( T ) : p( T ) : p( 4T ) : T T T 4T 5T 6T s[n]p( nt ) or n =,, 4 y() = n s[n]p( nt ) T T T 4T 5T 6T ECE-5 Phil Schnier January 6, 8 Preliminaries (Ch): Fourier Transorm (FT): Deiniion: Properies: W () = w() = w()e jπ d = F{w()} W ()e jπ d = F {W ()} Lineariy: F{c w () + c w ()} = c W () + c W () { conjugae symmeric W () Real-valued w() W () symmeric around = Bandwidh : bandpass : lowpass : W () W () db -power BW 99%-power BW absolue BW db single-sided BW double-sided BW 5 6

4 ECE-5 Phil Schnier January 6, 8 Dirac Dela (or coninuous impulse ) δ( ): An ininiely all and hin waveorm wih uni area: δ() limi ha s oen used o kick a sysem and see how i responds Key properies: Siing: w()δ( q)d = w(q) Time-domain impulse δ() has a la specrum: F{δ()} = δ()e jπ d = (or all ) Freq-domain impulse δ() corresponds o a DC waveorm: ECE-5 Phil Schnier January 6, 8 Frequency-Domain Represenaion o Sinusoids: Noice rom he siing propery ha F {δ( o )} = Thus, Euler s equaions δ( o )e jπ d = e jπ o cos(π o ) = ejπ o + e jπ o sin(π o ) = j ejπ o j e jπ o and he Fourier ransorm pair e jπ o δ( o ) imply ha F{cos(π o )} = δ( o) + δ( + o) F{sin(π o )} = j δ( o) j δ( + o) Oen we draw his as F{cos(π o )} = F{sin(π o )} F {δ()} = δ()e jπ d = (or all ) o o 7 8

5 ECE-5 Phil Schnier January 6, 8 Frequency Domain via MATLAB: Fourier ransorm requires evaluaion o an inegral Wha do we do i we can deine/solve he inegral? Generae (rae- T s ) sampled in MATLAB Plo magniude o Discree Fourier Transorm (DFT) using plom (rom course webpage) Square-wave example: = ; _max = ; Ts = /; = :Ts:_max; x = sign(cos(*pi**)); plo(x,ts); Noise-wave example: ampliude magniude ampliude 5!5! ime !5!4!!! 4 5 requency! ECE-5 Phil Schnier January 6, 8 Linear Time-Invarian (LTI) Sysems: An LTI sysem can be described by eiher is impulse response h() or is requency response H() = F{h()} in ime domain : impulse δ() LTI sysem in requency domain : la LTI specrum sysem Inpu/oupu relaionships: h() H() impulse response requency response Time-domain: Convoluion wih impulse response h() x() h() y() y() = h() x() = h( τ)x(τ)dτ Freq-domain: Muliplicaion wih req response H() X() H() Y () Y () = H()X() Linear Filering: Freq-domain illusraion o LPF, BPF, and HPF: _max = ; Ts = /; x = randn(,_max/ts); plo(x,ts); magniude!! ime 8 6 4!5!4!!! 4 5 requency Noice ha plom only plos requencies [ T s, T s ) X() H() = Y () LPF BPF HPF 9

6 ECE-5 Phil Schnier January 6, 8 Lowpass Filers: Ideal non-causal LPF (using sinc(x) := sin(πx) B H() = > B B H() B B B Ideal LPF wih group-delay o : e jπ B H() = > B B B F B h() πx ): h() = B sinc(b) B H() h() B B B F h() = B sinc ( B( ) ) A causal linear-phase LPF wih group-delay o : B H() B B h() B symmery around cener yields linear phase bu MATLAB can give beer causal linear-phase LPFs ECE-5 Phil Schnier January 6, 8 In MATLAB, generae T s -sampled LPF impulse response via where G h = irls(l, [,p,s,], [G,G,,])/Ts; p s /(T s) L+ = impulse response lengh {, p}, {s, } = normalized req pairs {G, G}, {, } = corresp magniude pairs The commands irpm and ir have he same inerace, bu yield slighly dieren resuls (oen worse or our apps) In MATLAB, perorm ilering on T s -sampled x via y = Ts*iler(h,,x); or y = Ts*conv(h,x); _max = ; Ts = /; x = randn(,_max/ts); h = irls(,[,,4,],[,,,])/ts; y = Ts*iler(h,,x); subplo(,,); plo(x,ts, ); ylabel( X() ) subplo(,,); plo(h,ts, ); ylabel( H() ) subplo(,,); plo(y,ts, ); ylabel( Y() ) X() H() Y() 5 5!5!4!!! 4 5 requency 5 x! 5!5!4!!! 4 5 requency 5 5!5!4!!! 4 5 requency Imporan: The rouines irls,irpm,ir generae causal linear-phase ilers wih group delay = L samples Thus, he ilered oupu y will be delayed by L samples relaive o x