Tes 1 Review Tes 1 Review Proessor Deepa Kundur Universiy o Torono Reerence: Secions: 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7 3.1, 3.2, 3.3, 3.4, 3.5, 3.6 o S. Haykin and M. Moher, Inroducion o Analog & Digial Communicaions, 2nd ed., John iley & Sons, Inc., 2007. ISBN-13 978-0-471-43222-7. Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 1 / 58 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 2 / 58 Communicaion Sysems: Foundaional Theories Chaper 2: Fourier Represenaion o Signals and Sysems Modulaion Theory: piggy-back inormaion-bearing signal on a carrier signal Deecion Theory: esimaing or deecing he inormaion-bearing signal in a reliable manner Probabiliy and Random Processes: model channel noise and uncerainy a receiver Fourier Analysis: view signal and sysem in anoher domain o gain new insighs inormaion source ransmier receiver inormaion consumpion channel Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 3 / 58 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 4 / 58
The Fourier Transorm (FT) Energy Signals Noaion: G( ) = g() = g()e j2π G( )e +j2π g() G( ) G( ) = F [g()] g() = F 1 [G( )] The energy o a signal g() is given by: g() 2 d I g() represens a volage or a curren, hen we say ha his is he energy o he signal across a 1 ohm resisor. hy? Because a curren i() or volage v() exhibis he ollowing energy over a R ohm resisor. E = i 2 ()Rd = v 2 () R d Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 5 / 58 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 6 / 58 Energy Signals and he Fourier Transorm FT Synhesis and Analysis Equaions Pracical physically realizable signals (e.g., energy signals) ha obey: have Fourier ransorms. g() 2 d < G( ) = g() = g()e j2π g() G( ) G( )e +j2π Since he FT is inverible boh g() and G( ) conain he same inormaion, bu describe i in a dieren way. Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 7 / 58 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 8 / 58
FT Synhesis Equaion e j2π = cos(2π) + j sin(2π) cos(2π) g() = G( )e j2π d 0 g() is he sum o scaled complex sinusoids sin(2π) e j2π = cos(2π) + jsin(2π) complex sinusoid 0 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 9 / 58 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 10 / 58 FT Analysis Equaion G( ) and G( ) G( ) = g()e j2π d g() = = G( )e j2π d G( ) e j(2π + G( )) d The analysis equaion represens he inner produc beween g() and e j2π. The analysis equaion saes ha G( ) is a measure o similariy beween g() and e j2π, he complex sinusoid a requency Hz. G( ) dicaes he relaive presence o he sinusoid o requency in g(). G( ) dicaes he relaive alignmen o he sinusoid o requency in g(). Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 11 / 58 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 12 / 58
Low, Mid and High Frequency Signals Imporance o FT Theorems and Properies Q: hich o he ollowing signals appears higher in requency? 1. cos(4 10 6 π + π/3) 2. sin(2π + 10π) + 17 cos 2 (10π) The Fourier ransorm convers a signal or sysem represenaion o he requency-domain, which provides anoher way o visualize a signal or sysem convenien or analysis and design. A: cos(4 10 6 π + π/3). The properies o he Fourier ransorm provide valuable insigh ino how signal operaions in he ime-domain are described in he requency-domain. Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 13 / 58 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 14 / 58 FT Theorems and Properies Time-Bandwidh Produc Propery/Theorem Time Domain Frequency Domain Noaion: g() G( ) g 1 () G 1 ( ) g 2 () G 2 ( ) Lineariy: c 1 g 1 () + c 2 g 2 () c 1 G 1 ( ) + ( c 2 ) G 2 ( ) 1 Dilaion: g(a) a G a Conjugaion: g () G ( ) Dualiy: G() g( ) Time Shiing: g( 0 ) G( )e j2π 0 Frequency Shiing: e j2πc g() G( c) Area Under G( ): g(0) = G( )d Area Under g(): g()d = G(0) d Time Diereniaion: g() j2πg( ) d Time Inegraion : g(τ)dτ 1 j2π G( ) Modulaion Theorem: g 1 ()g 2 () G 1(λ)G 2 ( λ)dλ Convoluion Theorem: g 1(τ)g 2 ( τ) G 1 ( )G 2 ( ) Correlaion Theorem: g 1()g2 ( τ)d G 1( )G 2 ( ) Rayleigh s Energy Theorem: g() 2 d = G( ) 2 d ime-duraion o a signal requency bandwidh = consan T larger -T/2 duraion Arec(/T) A T /2-4/T -3/T -2/T -1/T AT sinc(t) AT 0 1/T null-o-null bandwidh 2/T 3/T 4/T Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 15 / 58 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 16 / 58
Time-Bandwidh Produc ime-duraion o a signal requency bandwidh = consan LTI Sysems and Filering LTI Sysem impulse response he consan depends on he deiniions o duraion and bandwidh and can change wih he shape o signals being considered I can be shown ha: ime-duraion o a signal requency bandwidh 1 4π wih equaliy achieved or a Gaussian pulse. Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 17 / 58 LTI Sysem requency response For sysems ha are linear ime-invarian (LTI), he Fourier ransorm provides a decoupled descripion o he sysem operaion on he inpu signal much like when we diagonalize a marix. This provides a ilering perspecive o how a linear ime-invarian sysem operaes on an inpu signal. The LTI sysem scales he sinusoidal componen corresponding o requency by H( ) providing requency seleciviy. Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 18 / 58 Dirac Dela Funcion Dirac Dela Funcion Deiniion: 1. δ() = 0, 0 2. The area under δ() is uniy: δ()d = 1 Noe: δ(0) = undeined can be inerpreed as he limiing case o a amily o uncions o uni area bu ha become narrower and higher 1 all uncions have 1 uni area T 1 T 2 T 3 T 1 T 2 T 3 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 19 / 58 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 20 / 58
Dirac Dela Funcion Siing Propery: Convoluion wih δ(): g()δ( 0 )d = g( 0 ) The Fourier Transorm and he Dirac Dela δ() 1 cos(2π 1 ) = ej2π1 2 sin(2π 1 ) = ej2π1 2j cosine 1 1 1 δ( ) e j2π0 δ( 0 ) + e j2π1 2 e j2π1 2j 1 2 δ( 1) + 1 2 δ( + 1) 1 2j δ( 1) 1 2j δ( + 1) 1/2 1/2 g() δ( 0 ) = g( 0 ) - 1 0 sine 1 1 0.5j - 1 1 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 21 / 58 0 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 22 / 58-0.5j Fourier Transorms o Periodic Signals g() A g() = c n e j2πn 0 n= G( ) = c n δ( n 0 ) n= sinc c k -5-4 -3 3 4 5-2 -1 0 1 2 k sinc -5-4 -3 3 4 5 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 23 / 58-2 -1 0 2 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 24 / 58
Transmission o Signals Through Linear Sysems LTI Sysem LTI Sysem Transmission o Signals Through Linear Sysems impulse response impulse response LTI Sysem LTI Sysem requency response requency response Time domain: y() = x() h() = Causaliy: h() = 0 or < 0 Sabiliy: h() d < x(τ)h( τ)dτ Freqeuncy domain: x() X ( ) y() Y ( ) h() H( ) Y ( ) = H( ) X ( ) Y ( ) = H( ) }{{} X ( ) req selecive sysem Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 25 / 58 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 26 / 58 Ideal Low-Pass Filers Ideal Low-Pass Filers H LP ( ) = { e j2π 0 B 0 > B STOPBAND PASSBAND STOPBAND -B B h LP () = 2Bsinc(2B( 0 )) H LP ( ) = { e j2π 0 B 0 > B -B B Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 27 / 58 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 28 / 58
Ideal Low-Pass Filers h LP () = 2Bsinc(2B( 0 )) 2B 0 1/B LTI Sysems, Sinusoids and Ideal Lowpass Filering Q: Suppose he ollowing signals are passed hrough an ideal lowpass iler wih cuo requency such ha 1 < < 2 c. ha are he corresponding oupus: 1. m() = sin(2π 1 + π 4 ) 2. m() = sin(2π 1 + π 4 ) + cos(2π 2 π 5 ) 3. m() = sin(2π 1 + π 4 ) + cos(2π c) 4. s() = sin(2π 1 + π 4 ) cos(2π c) 5. s() = sin(2π 1 + π 4 ) sin(2π c) 6. s() = sin(2π 1 + π 4 ) + cos2 (2π c ) Noe: cos 2 A = 1 2 + 1 2 cos(2a) 7. s() = sin(2π 1 + π 4 ) cos2 (2π c ) 8. s() = sin(2π 1 + π 4 ) cos(2π c) sin(2π c ) Noe: sin A cos B = 1 2 sin(a + B) 1 2 sin(b A) Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 29 / 58 A: 1. sin(2π 1 + π 4 ), 2. sin(2π 1 + π 4 ), 3. sin(2π 1 + π ), 4. 0, 5. 0, 4 6. sin(2π 1 + π 4 ) + 1 2, 7. 1 2 sin(2π 1 + π ), 8. 0. 4 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 30 / 58 Modulaion Chaper 3: Ampliude Modulaion Modulaion: o adjus or adap o a cerain proporion Used o superimpose one signal ono anoher. In modulaion need wo hings: 1. a modulaed signal ha is changed: carrier signal: c() 2. a modulaing signal ha dicaes how o change: message signal: m() Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 31 / 58 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 32 / 58
Ampliude Modulaion Ampliude Modulaion In modulaion need wo hings: 1. a modulaed signal: carrier signal: c() 2. a modulaing signal: message signal: m() carrier: c() = Ac cos(2π c ); phase φ c = 0 is assumed. message: m() (inormaion-bearing signal) assume bandwidh/max req o m() is Three ypes sudied: 1. Ampliude Modulaion (AM) (yes, i has he same name as he class o modulaion echniques) 2. Double Sideband-Suppressed Carrier (DSB-SC) 3. Single Sideband (SSB) Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 33 / 58 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 34 / 58 Ampliude Modulaion (he speciic echnique) Ampliude Modulaion Suppose s AM () = A c [1 + k a m()] cos(2π c ) % Modulaion = 100 max(k a m()) k a m() < 1 (% Modulaion < 100%) [1 + ka m()] > 0, so he envelope o s AM () is always posiive; no phase reversal c he movemen o he message is much slower han he sinusoid Then, m() can be recovered wih an envelope deecor. AM wave + - + Oupu - Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 35 / 58 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 36 / 58
Ampliude Modulaion Double Sideband-Suppressed Carrier s AM () = A c [1 + k a m()] cos(2π c ) s AM () = A c [1 + k a m()] cos(2π c ) = A c cos(2π c ) +k }{{} a A c m() cos(2π c ) }{{} excess energy message-bearing signal s DSB () = A c m() cos(2π c ) Transmiing only he message-bearing componen o he AM signal, requires more a complex (coheren) receiver sysem. Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 37 / 58 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 38 / 58 Double Sideband-Suppressed Carrier Double Sideband-Suppressed Carrier S () AM s DSB () = A c m() cos(2π c ) S () DSB S () USSB upper SSB Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 39 / 58 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 40 / 58
carrier carrier message message ampliude modulaion ampliude modulaion DSB-SC DSB-SC Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 41 / 58 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 42 / 58 Double Sideband-Suppressed Carrier carrier An envelope deecor will no be able o recover m(); i will insead recover m(). message Coheren demodulaion is required. ampliude modulaion s() Produc Modulaor iler v () 0 Demodulaed Signal DSB-SC Local Oscillaor Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 43 / 58 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 44 / 58
Cosas Receiver Cosas Receiver Coheren Demodulaion I-Channel (in-phase coheren deecor) Produc Modulaor iler local oscillaor oupu Demodulaed Signal v () 0 Produc Modulaor v () I iler Demodulaed Signal DSB-SC wave Volage-conrolled Oscillaor Phase Discriminaor DSB-SC wave Volage-conrolled Oscillaor Phase Discriminaor -90 degree Phase Shier -90 degree Phase Shier Produc Modulaor Filer Produc Modulaor v () Q Filer Circui or Phase Locking Q-Channel (quadraure-phase coheren deecor) Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 45 / 58 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 46 / 58 Muliplexing and QAM Quadraure Ampliude Modulaion Muliplexing: o send muliple message simulaneously s() = A c m 1 () cos(2π c ) + A c m 2 () sin(2π c ) Produc Modulaor iler Quadraure Ampliude Muliplexing (QAM): (a.k.a quadraure-carrier muliplexing) ampliude modulaion scheme ha enables wo DSB-SC waves wih independen message signals o occupy he same channel bandwidh (i.e., same requency channel) ye sill be separaed a he receiver. Muliplexed Signal -90 degree Phase Shier Produc Modulaor Filer Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 47 / 58 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 48 / 58
Quadraure Ampliude Modulaion Quadraure Ampliude Modulaion s() = A c m 1 () cos(2π c ) + A c m 2 () sin(2π c ) s() = A c m 1 () cos(2π c ) + A c m 2 () sin(2π c ) S () DSB Suppose m 1 () and m 2 () are wo message signals boh o bandwidh. QAM allows wo messages o be communicaed wihin bandwidh. S () QAM Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 49 / 58 S USSB Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 50 / 58 () upper SSB Is here anoher way o gain his bandwidh eiciency? Single Sideband S LSSB () S () QAM Modulaion: lower SSB S () USSB lower SSB upper SSB s SSB () = A c 2 m() cos(2π c) A c 2 ˆm() sin(2π c) where he negaive (posiive) applies o upper SSB (lower SSB) ˆm() is he Hilber ransorm o m() S LSSB() upper SSB lower SSB Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 51 / 58 M() H() = -j sgn() -j H() j M() m() h() = 1/( ) Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 52 / 58 m()
Single Sideband Single Sideband Modulaion: s SSB () = A c 2 m() cos(2π c) A c 2 ˆm() sin(2π c) Coheren Demodulaion: m() Produc Modulaor Band-pass iler s() s() Produc Modulaor iler v () 0 Demodulaed Signal Local Oscillaor upper SSB H () BP upper SSB Noe: Cosas receiver will work or SSB demodulaion. - c lower SSB lower SSB c Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 53 / 58 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 54 / 58 Comparisons o Ampliude Modulaion Techniques AM: Comparisons o Ampliude Modulaion Techniques DSB-SC: S AM() s AM () = A c [1 + k a m()] cos(2π c ) S AM ( ) = A c 2 [δ( c) + δ( + c )] + k aa c [M( c ) + M( + c )] 2 S AM() s DSB () = A c cos(2π c )m() S DSB ( ) = A c 2 [M( c) + M( + c )] S DSB() S DSB() S USSB () highes power B T = lowes complexiy S USSB () Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 55 / 58 lower power B T = higher complexiy S LSSB () Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 56 / 58
S DSB() Comparisons o Ampliude Modulaion Techniques SSB: Comparisons o Ampliude Modulaion Techniques SSB: S USSB () upper SSB s USSB () = A c 2 m() cos(2π c) A c 2 ˆm() sin(2π c) { Ac S USSB ( ) = 2 [M( c) + M( + c )] c 0 < c s LSSB () = A c 2 m() cos(2π c) + A c 2 ˆm() sin(2π c) { 0 > c S LSSB ( ) = A c 2 [M( c) + M( + c )] c S LSSB () lower SSB lowes power B T = highes complexiy Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 57 / 58 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 58 / 58