Update. Continuous-Time Frequency Analysis. Frequency Analysis. Frequency Synthesis. prism. white light. spectrum. Coverage:

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1 Updae Coiuous-Time requecy alysis Professor Deepa Kudur Uiversiy of Toroo Coverage: Before Readig Wee: 41 Wih Ema: 42, 43, 44, 51 Today ad Wed: Brief Rev of 41, 42, 43, 44, 51 ad Secios 52, 54, 55 Joh G Proais ad Dimiris G Maolais, Digial Sigal Processig: Priciples, lgorihms, ad pplicaios, 4h ediio, 27 Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 1 / 44 Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 2 / 44 requecy alysis requecy Syhesis whie ligh prism specrum Scieific Desigaio requecy (Hz) for = 8176 C C C C4 (middle C) C C C C C1 C2 C3 C4 C5 C6 C7 C8 Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 3 / 44 Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 4 / 44

2 Complex Siusoids: Coiuous-Time e jω = cos(ω) + j si(ω) cs-ime complex siusoid cos(2π) Complex Siusoids: Discree-Time e jω = cos(ω) + j si(ω) ds-ime complex siusoid cos(2πf) si(2π) si(2πf) Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 5 / 44 Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 6 / 44 Classificaio of ourier Pairs Classificaio of ourier Pairs CTS-TIME DST-TIME Coiuous-Time Discree-Time PERIODIC ourier Series ourier Series (CTS) (DTS) Coiuous-Time Discree-Time PERIODIC ourier Trasform ourier Trasform (CTT) (DTT) CTS-TIME DST-TIME PER x() = = c e j2π x() = 1 = c e j2π/ c = 1 T p T p x()e j2π d c = 1 1 PER x() = 1 X (Ω)ejΩ dω x() = 1 2π X (Ω) = x()e jω d = x()e j2π/ 2π 2π X (ω)ejω dω X (ω) = = x()e jω Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 7 / 44 Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 8 / 44

3 Dualiy Dualiy ime domai recagle covoluio muliplicaio periodic discree aperiodic coiuous frequecy domai recagle muliplicaio covoluio discree periodic coiuous aperiodic CTS-TIME DST-TIME PER x() = = c e j2π x() = 1 = c e j2π/ c = 1 T p T p x()e j2π d c = 1 1 PER x() = 1 X (Ω)ejΩ dω x() = 1 2π X (Ω) = x()e jω d periodic discree aperiodic coiuous = x()e j2π/ 2π 2π X (ω)ejω dω X (ω) = = x()e jω discree periodic coiuous aperiodic Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 9 / 44 Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 1 / 44 Covergece CTS-TIME DST-TIME PER x() = = c e j2π x() = 1 = c e j2π/ c = 1 T p T p x()e j2π d c = 1 1 PER x() = 1 X (Ω)ejΩ dω x() = 1 2π X (Ω) = x()e jω d = x()e j2π/ 2π 2π X (ω)ejω dω X (ω) = = x()e jω The Coiuous-Time ourier Series (CTS) Covergece issues are prevale whe you have ifiie sums ad iegraio Dirichle codiios provide sufficie codiios for covergece of he ourier pair a coiuous pois of he sigal Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 11 / 44 Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 12 / 44

4 Coiuous-Time ourier Series (CTS) or coiuous-ime periodic sigals wih period T p = 1 : CTS: Dirichle Codiios Syhesis equaio: x() = c e j2π = alysis equaio: c = 1 x()e T p T j2π d p 1 x() has a fiie umber of discoiuiies i ay period 2 x() coais a fiie umber of maxima ad miima durig ay period 3 x() is absoluely iegrable i ay period: T p x() d < Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 13 / 44 Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 14 / 44 CTS: Example CTS: Example id he CTS of he followig periodic square wave: x() c c = 1 x()e j2π d = 1 Tp/2 x()e j2π d T p T p T p T p/2 = 1 τ/2 e j2π d = e j2π τ/2 T p τ/2 T p j2π τ/2 [ e j2π τ/2 e +j2πτ/2 ] = πt p 2j [ e j2π τ/2 e j2πτ/2 ] = π 1 2j = si(π τ) π c Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 15 / Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 16 / 44

5 CTS: Example x() CTS: Example or τ = Tp 3 = 1 3 : c = si(π/3) π x() = = c e j2π = x() = si(π/3) e j2π π c c / oe: square wave discoiuiies (eg, = τ/2), c Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 17 / 44 x(τ/2) = si(π/3) c e j2π(τ/2) = π 2 = Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time -2-1 requecy 1 2 alysis 18 / 44 Coiuous-Time ourier Trasform (CTT) The Coiuous-Time ourier Trasform (CTT) or coiuous-ime aperiodic sigals: Syhesis equaio: x() = 1 X (Ω)e jω dω 2π alysis equaio: X (Ω) = x()e jω d Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 19 / 44 Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 2 / 44

6 Coiuous-Time ourier Trasform (CTT) CTT: Dirichle Codiios Cyclic frequecy ca also be used Syhesis equaio: x() = alysis equaio: X ( ) = X ( )e j2π d x()e j2π d llowig T p i CTS Dirichle codiios: 1 x() has a fiie umber of fiie discoiuiies 2 x() has a fiie umber of maxima ad miima 3 x() is absoluely iegrable: x() d < Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 21 / 44 Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 22 / 44 CTT: Example CTT: Example id he CTS of he followig periodic square wave: x() X (Ω) = = e jω jω x()e jω d = τ/2 τ/2 τ/2 τ/2 = 2 si(ωτ/2) Ω e jω d c c Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 23 / Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 24 / 44

7 CTT: Example x() X (Ω) = 2 si(ωτ/2) Ω The Discree-Time ourier Series X() (DTS) Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 25 / 44 Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 26 / 44 Discree-Time ourier Series (DTS) or discree-ime periodic sigals wih period : DTS: Covergece Codiios Syhesis equaio: 1 x() = c e j2π/ = oe due o fiie sums alysis equaio: c = 1 1 x()e j2π/ = Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 27 / 44 Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 28 / 44

8 DTS vs CTS: Why a fiie sum? 1 x() = c e j2π/ vs x() = c e j2π = = Coiuous-ime siusoids are uique for di frequecies; e j 2 3 π 16 j e 3 π Discree-ime siusoids wih cyclic frequecies a ieger umber apar are he same; e j 2 3 π = e j 16 3 π Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 29 / 44 Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 3 / 44 DTS vs CTS: Why a fiie sum? Harmoically Relaed Ds-Time Siusoids There are oly di ds-ime harmoics s () 1 x() = c e j2π/ vs x() = Cosider = = s () = e j2π/, =, ±1, ±2, s () is periodic e e j2π/ = e j2πf where f = raioal c e j2π There are oly di ds-ime harmoics s (): =, 1, 2,, 1 f s () harmoics are uique for =, 1, 2,, 1 Ouside his rage of, he cyclic frequecies are iegers apar hus resulig i he same siusoids as for =, 1, 2,, 1 Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 31 / 44 Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 32 / 44

9 DTS: Example DTS: Example id he DTS of he followig periodic square wave: x() - -L L c c = 1 1 x()e j2π/ = 1 = 1 = L e j2π/ = 1 = L = ej2πl/ = ej2πl/ = 2L m= si(2πl/) si(π/) /2 x()e j2π/ = /2 2L e j2π(m L)/ m= e j2πm/ = ej2πl/ 1 (e j2π/ ) 2L+1 1 (e j2π/ ) = 2L m= (e j2π/ ) m e j2πl/ e j2πl/ e jπ/ e jπ/ 2j 2j c Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 33 / 44 DTS: Example x() Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 34 / 44 or L = 3 ad = 18: c = si(π/3) -L 18 si(π/18) - L The Discree-Time ourier Trasform c (DTT) c oe: c is periodic wih period Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 35 / 44 Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 36 / 44

10 Discree-Time ourier Trasform (DTT) or discree-ime aperiodic sigals: DTT: Covergece Codiios Syhesis equaio: x() = 1 X (ω)e jω dω 2π 2π alysis equaio: = x() < X (ω) = x()e jω = Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 37 / 44 Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 38 / 44 DTT: Example DTT: Example id he DTT of he followig recagle fucio: x() -L L X (ω) = = = 2L m= x()e jω = L e jω = = L 2L (e jω ) (m L) = e jωl (e jω ) m jωl 1 e jω(2l+1) 2j = e 1 e jω 2j = si(ωl) si(ω/2) m= L (e jω ) = L Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 39 / 44 Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 4 / 44

11 DTT: Example x() or L = 3: X (ω) = si(ωl) -L L si(ω/2) x() x() CTS CTT c c X() x() DTS c c - -L L oe: X (ω) is periodic wih period 2π x() DTT -L L Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 41 / 44 Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 42 / 44 DTT Theorems ad Properies DTT Symmery Properies Propery Time Domai requecy Domai oaio: x() X (ω) x 1 () X 1 (ω) x 2 () X 1 (ω) Lieariy: a 1 x 1 () + a 2 x 2 () a 1 X 1 (ω) + a 2 X 2 (ω) Time shifig: x( ) e jω X (ω) Time reversal x( ) X ( ω) Covoluio: x 1 () x 2 () X 1 (ω)x 2 (ω) Correlaio: r x1x 2 (l) = x 1 (l) x 2 ( l) S x1x 2 (ω) = X 1 (ω)x 2 ( ω) = X 1 (ω)x2 (ω) [if x 2() real] Wieer-Khichie: r xx (l) = x(l) x( l) S xx (ω) = X (ω) 2 amog ohers Time Sequece x() x () x ( ) x( ) x R () jx I () x() real x e() = 1 2 [x() + x ( )] x o() = 1 2 [x() x ( )] DTT X (ω) X ( ω) X (ω) X ( ω) X e (ω) = 1 2 [X (ω) + X ( ω)] X o (ω) = 1 2 [X (ω) X ( ω)] X (ω) = X ( ω) X R (ω) = X R ( ω) X I (ω) = X I ( ω) X (ω) = X ( ω) X (ω) = X ( ω) X R (ω) jx I (ω) Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 43 / 44 Professor Deepa Kudur (Uiversiy of Toroo) Coiuous-Time requecy alysis 44 / 44