Test 1 Review. Test 1 Review. Communication Systems: Foundational Theories. Communication System. Reference: Sections and

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1 Tes 1 Review Tes 1 Review Proessor Deepa Kundur Universiy o Torono Reerence: Secions and o S. Haykin and M. Moher, Inroducion o Analog & Digial Communicaions, 2nd ed., John iley & Sons, Inc., ISBN Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 1 / 44 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 2 / 44 Communicaion Sysem Communicaion Sysems: Foundaional Theories Modulaion Theory: piggy-back inormaion-bearing signal on a carrier signal inormaion source ransmier receiver inormaion consumpion Deecion Theory: esimaing or deecing he inormaion-bearing signal in a reliable manner Probabiliy and Random Processes: model channel noise and uncerainy a receiver channel 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 / 44 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 4 / 44

2 The Fourier Transorm (FT) Dirichle Condiions Noaion: G( ) = g() = g()e j2π G( )e +j2π g() G( ) G( ) = F [g()] g() = F 1 [G( )] For he Fourier ransorm o g() o exis, i is suicien ha: 1. he uncion g() is single-valued, wih inie number o minima or maxima in any inie inerval 2. he uncion g() has a inie number o disconinuiies in any inie inerval 3. he uncion g() is absoluely inegrable g() d < Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 5 / 44 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 6 / 44 Energy Signals Energy Signals and he Fourier Transorm 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. Pracical physically realizable signals (e.g., energy signals) obey: have Fourier ransorms. g() 2 d < E = i 2 ()Rd = v 2 () R d Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 7 / 44 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 8 / 44

3 FT Synhesis and Analysis Equaions FT Synhesis Equaion 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. g() = G( )e j2π d g() is he sum o scaled complex sinusoids e j2π = cos(2π) + jsin(2π) complex sinusoid Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 9 / 44 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 10 / 44 e j2π = cos(2π) + j sin(2π) FT Analysis Equaion cos(2π) G( ) = g()e j2π d 0 sin(2π) 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. 0 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 11 / 44 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 12 / 44

4 G( ) and G( ) Imporance o FT Theorems and Properies g() = = G( )e j2π d G( ) e j(2π + G( )) d G( ) dicaes he relaive presence o he sinusoid o requency in g(). G( ) dicaes he relaive alignmen o he sinusoid o requency in g(). e live in he ime-domain. However, someimes viewing inormaion signals or sysem operaion as uncion o ime does no easily provide insigh. 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. The properies o he Fourier ransorm provided 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 / 44 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 14 / 44 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 / 44 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 16 / 44

5 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 / 44 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. Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 18 / 44 LTI Sysems and Filering Dirac Dela Funcion g 1 (τ)g 2 ( τ) G 1 ( )G 2 ( ) The convoluion heorem 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. Deiniion: 1. δ() = 0, 0 2. The area under δ() is uniy: δ()d = 1 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 19 / 44 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 20 / 44

6 Dirac Dela Funcion Dirac Dela Funcion can be inerpreed as he limiing case o a amily o uncions o uni area bu ha become narrower Siing Propery: 1 all uncions have uni area 1 g()δ( 0 )d = g( 0 ) Convoluion wih δ(): g() δ() = g() T 1 T 2 T 3 T 1 T 2 T 3 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 21 / 44 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 22 / 44 The Fourier Transorm and he Dirac Dela Fourier Transorms o Periodic Signals δ() 1 1 δ( ) e j2π 0 δ( 0 ) cos(2π 1 ) 1 2 δ( 1) δ( + 1) sin(2π 1 ) 1 2j δ( 1) 1 2j δ( + 1) g() = c n e j2πn 0 n= G( ) = c n δ( n 0 ) n= Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 23 / 44 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 24 / 44

7 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 25 / 44 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 26 / 44 Ideal Low-Pass Filers Ampliude Modulaion h LP () = 2Bsinc(2B( 0 )) 2B 0 1/B carrier: c() = A c cos(2π c ) message: m() assume bandwidh/max req o m() is Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 27 / 44 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 28 / 44

8 Ampliude Modulaion AM 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) s AM () = A c [1 + k a m()] cos(2π c ) % Modulaion = 100 max(k a m()) Suppose k a m() < 1 c 3. Single Sideband (SSB) Then, m() can be recovered wih an envelope deecor. Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 29 / 44 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 30 / 44 AM DSB-SC s AM () = A c [1 + k a m()] cos(2π c ) s DSB () = A c cos(2π c )m() Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 31 / 44 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 32 / 44

9 carrier carrier message message ampliude modulaion ampliude modulaion DSB-SC DSB-SC Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 33 / 44 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 34 / 44 DSB-SC carrier An envelope deecor will no be able o 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 35 / 44 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 36 / 44

10 Cosas Receiver Phase synchronizaion is required. Quadraure Ampliude Modulaion s() = A c m 1 () cos(2π c ) + A c m 2 () sin(2π c ) Coheren Demodulaion Produc Modulaor iler local oscillaor oupu Demodulaed Signal v () 0 Produc Modulaor iler DSB-SC wave -90 degree Phase Shier Volage-conrolled Oscillaor Phase Discriminaor Muliplexed Signal -90 degree Phase Shier Produc Modulaor Filer Circui or Phase Locking Produc Modulaor Filer Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 37 / 44 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 38 / 44 SSB SSB Modulaion: s SSB () = A c 2 m() cos(2π c) A c 2 ˆm() sin(2π c) Coheren Demodulaion: s() Produc Modulaor iler v () 0 Demodulaed Signal m() Produc Modulaor Band-pass iler s() Local Oscillaor Noe: Cosas receiver will work or SSB demodulaion. Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 39 / 44 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 40 / 44

11 Comparisons o Ampliude Modulaion Techniques AM: Comparisons o Ampliude Modulaion Techniques DSB-SC: S() 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() s DSB () = A c cos(2π c )m() S DSB ( ) = A c 2 [M( c) + M( + c )] S() S() S() highes power B T = lowes complexiy S() Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 41 / 44 S() Comparisons o Ampliude Modulaion Techniques SSB: lower power B T = S() higher complexiy S() Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 42 / 44 S() Comparisons o Ampliude Modulaion Techniques SSB: S() 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() lower SSB 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 lowes power B T = highes complexiy Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 43 / 44 Proessor Deepa Kundur (Universiy o Torono) Tes 1 Review 44 / 44