3 Frequency Domain Representation of Continuous Signals and Systems

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Αρκάδιος Σπανού
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1 3 Frequency Domain Represenaion of Coninuous Signals and Sysems 3. Fourier Series Represenaion of Periodic Signals Exponenial Fourier Series Discree Fourier Specrum / Line Specrum Parseval s Theorem for Periodic Signals Fourier Transform Definiion and Examples Properies of he Fourier Transform Symmery of he Fourier Transform Convoluion Parseval s Theorem for Energy Signals Fourier Transform of Periodic Signals Frequency Domain Descripion of LTI Sysems Frequency Response Bandwidh of Frequency Selecive Sysems Disorionless Transmission Dr. Tanja Karp

2 3. Fourier Series Represenaion of Periodic Signals 3.. Exponenial Fourier Series A large class of periodic signals f T () wih period T and fundamenal frequency = 2π/T can be represened as a sum of harmonic complex exponenial funcions: f T () = X k= F k exp(jk ) wih complex Fourier coefficiens: F k = T Z +T f T () exp( jk ) d Example: Recangular Pulse Train f T () T τ/2 τ/2 T Dr. Tanja Karp 3. Fourier Series Represenaion of Periodic Signals 2

3 Calculaion of Fourier coefficiens F k : F k = T Z +τ/2 τ/2 exp( jk ) d = T " # τ/2 exp( jk ) jk τ/2 = k T exp( jk τ/2) exp(jk τ/2) j = 2 sin(k τ/2) k T = τ T sin(k τ/2) k τ/2 = τ T Sa(k τ/2) = τ T Sa(kπτ/T ) Sa(x) = sin(x)/x: sine-over-argumen funcion 3..2 Discree Fourier Specrum / Line Specrum Definiion: Graph of he (complex) Fourier coefficiens F k as a funcion of he angular frequency. f T () = X k= F k exp(jk ) For a periodic signal, he Fourier specrum exiss only a discree values of : =, ±, ±2, ±3,... Dr. Tanja Karp 3. Fourier Series Represenaion of Periodic Signals 3

4 Example: Recangular Pulse Train T =, τ varies: f T () = < τ/2 τ/2 < < /2, F k = τ sin(kπτ) kπτ = τ Sa(kπτ) F k.2. ampliude specra τ=.2 F k τ=.. F k τ= Dr. Tanja Karp 3. Fourier Series Represenaion of Periodic Signals 4

5 τ =.2, T varies: f T () = <.. < < T/2, F k =.2 T Sa(.2kπ/T ) F k.2. ampliude specra T= F k T=2. F k T= Dr. Tanja Karp 3. Fourier Series Represenaion of Periodic Signals 5

6 Magniude and Phase Specra: For complex Fourier coefficiens F k he magniude and phase is generally ploed separaely resuling in he magniude and phase specrum. Example: Sinusoids f T () = sin( ) = 2j {z} F exp(j ) + exp( j ) 2j {z} F F = 2j = 2 ( j) = 2 exp( jπ/2) = 2 π/2 F = 2j = 2 j = 2 exp(jπ/2) = 2 π/2 /2 F k π/2 F k π/2 Dr. Tanja Karp 3. Fourier Series Represenaion of Periodic Signals 6

7 3..3 Parseval s Theorem for Periodic Signals A periodic signal f T () is a power signal wih average power: P = T Z +T f T () 2 d = T Z +T Exponenial Fourier Series of complex conjugae funcion: Average Power: P = T = f T () X Z +T X k= F k k= F k exp(jk ) A = f T ()f T () d = T h T Z +T Z +T X k= f T () f T () exp( jk ) d {z } F k f T ()f T () d F k exp( jk ) X k= i = F k X k= exp( jk ) d F k 2 Dr. Tanja Karp 3. Fourier Series Represenaion of Periodic Signals 7

8 Example: Sinusoids f T () = sin( ) = 2j {z} F exp(j ) + exp( j ) 2j {z} F Average Power: Z T/2 P = T T/2 sin 2 ( ) d = F 2 + F 2 = 2 Dr. Tanja Karp 3. Fourier Series Represenaion of Periodic Signals 8

9 3.2. Definiion and Examples 3.2 Fourier Transform Frequency represenaion of an aperiodic signal. Fourier Transform: Inverse Fourier Transform: F () = f() exp( j)d f() = 2π F () exp(j) d The Fourier Transform is aken over all imes, i.e. all ime resuluion is los. F () is called Fourier Transform/specral-densiy funcion/(fourier) specrum of f(). I is generally a complex valued funcion. Each poin of F () indicaes he relaive weighing of each frequency. Shor-hand noaion: F () = F{f()}, F () f(), f() F () Sufficien condiion for he exisence of he Fourier Transform: f() d < Dr. Tanja Karp 3.2 Fourier Transform 9

10 Example: Gae Funcion Fourier Transform: F () = = Euler s Ideniy: f() = rec(/τ) τ/2 Z τ/2 rec(/τ) rec(/τ) exp( j)d = exp( j)d τ/2 τ/2 j exp( j) = (exp( jτ/2) exp(jτ/2)) j τ/2 cos( τ/2) + j sin( τ/2) cos(τ/2) j sin(τ/2) F () = j τ/2 = 2 sin(τ/2) = τ sin(τ/2) τ/2 = τ Sa(τ/2) rec(/τ) τ Sa(τ/2) Dr. Tanja Karp 3.2 Fourier Transform

11 rec() Sa(.5) Example: Uni Impulse Funcion f () Fourier Transform: F{δ()} = f() = δ() δ() exp( j)d () F() = exp( j) = Dr. Tanja Karp 3.2 Fourier Transform

12 Example: Complex Exponenial Funcion f() = exp(j ) The only frequency componen presen in he signal is. Inverse Fourier Transform: F {δ( )} = 2π δ( ) exp(j)d = 2π exp( ) exp(j ) 2πδ( ) Properies of he Fourier Transform Lineariy / Superposiion: where k f () + k 2 f 2 () k F () + k 2 F 2 () f () F (), f 2 () F 2 (), k, k 2 : arbiray consans Dr. Tanja Karp 3.2 Fourier Transform 2

13 Example: Sinusoidal Signals F {cos( )} cos( ) = 2 exp(j ) + 2 exp( j ) (π) (π) exp(j ) 2πδ( ) exp( j ) 2πδ( + ) F{cos( )} = πδ( ) + πδ( + ) F {sin( )} sin( ) = 2j exp(j ) 2j exp( j ) F{sin( )} = jπδ( ) + jπδ( + ) ( jπ) ( jπ) Complex Conjugae: Proof: F{f ()} = f() F (), f () exp( j)d = f () F ( ) f() exp(j)d = F ( j) Dr. Tanja Karp 3.2 Fourier Transform 3

14 Example: Complex Exponenial Funcion exp(j ) 2πδ( ) [exp(j )] = exp( j ) 2πδ( ) = 2πδ( + ) F {exp( j )} F {exp( j )} (2π) (2π) Coordinae Scaling (Reciprocal Spreading) f(a) a F a, a : real valued consan For a = : f( ) F ( ) Dr. Tanja Karp 3.2 Fourier Transform 4

15 Example: Gae Funcion rec() Sa(.5) rec(2) Sa(.25) Dr. Tanja Karp 3.2 Fourier Transform 5

16 Time Shifing (Delay): f( ) F () exp( j ), : real valued consan Example: Shifed Recangular Pulses f() = rec( + 2) + rec( 2), rec() Sa(/2) f() Sa(/2) exp(j2) exp( j2) = Sa(/2) 2 cos(2) rec(+2)+rec( 2) 2 2 Sa(/2)cos(2) Dr. Tanja Karp 3.2 Fourier Transform 6

17 Frequency Shifing (Modulaion) f() e j F ( ) Example: Ampliude Modulaion of a Triangular Pulse f() = Λ() cos( ) = 2 Λ()(exp(j ) + exp( j )) f() 2 Λ() Sa 2 (/2) Sa 2 (( )/2) + Sa 2 (( + )/2) Λ() cos(25 ).5.5(Sa 2 (( 25)/2+Sa 2 ((+25)/2))) Dr. Tanja Karp 3.2 Fourier Transform 7

18 Differeniaion Example: Gae Funcion d d f() j F () Wha is he Fourier Transform of d rec() = δ( +.5) δ(.5) d Wih: d d Compare o: rec() Sa(/2) rec() jsa(/2) = jsin(/2) /2 = 2j sin(/2) rec() () d d rec().5 ( ) δ( +.5) δ(.5) exp(j/2) exp( j/2) = 2j sin(/2) Dr. Tanja Karp 3.2 Fourier Transform 8

19 Inegraion Z f(τ) dτ j F () + π F () δ() Example: Uni Sep Funcion u(): u() = Z δ(τ) dτ, δ() u() = Z δ(τ) dτ j + π δ() Dr. Tanja Karp 3.2 Fourier Transform 9

20 3.2.3 Symmery of he Fourier Transform f() = f r () + jf i () = f r,e () + f r,o () + j(f i,e () + f i,o ()) f r (): real par of f() f i (): imaginary par of f() f r,e () / f i,e (): even symmery par of he real / imaginary par of f() f r,o () / f i,o (): odd symmery par of he real / imaginary par of f() F () = f() exp( j)d = f() (cos() j sin()) d = = 2 (f r,e () + f r,o () + jf i,e () + jf i,o ()) (cos() j sin()) d f r,e () cos()d 2j {z } even symmery, real + 2j {z } even symmery, imaginary f i,e () cos()d + 2 f r,o () sin()d {z } odd symmery, imaginary f i,o () sin()d {z } odd symmery, real Dr. Tanja Karp 3.2 Fourier Transform 2

21 f() real, even real, odd imaginary, even imaginary, odd real imaginary even real par, odd imaginary par odd real par, even imaginary par even real par, even imaginary par odd real par, odd imaginary par F () real, even imaginary, odd imaginary, even real, odd even real par, odd imaginary par odd real par, even imaginary par real imaginary even real par, even imaginary par odd real par, odd imaginary par Dr. Tanja Karp 3.2 Fourier Transform 2

22 3.2.4 Convoluion Time Convoluion: f () f 2 () = f (τ)f 2 ( τ)dτ F () F 2 () Example: Triangular Funcion rec() rec() * = Λ() Λ() = rec() rec() Sa(/2) Sa(/2) = Sa 2 (/2) Example: Uni Impulse Funcion f() δ() F () f() f() δ( ) F (j) e j f( ) Dr. Tanja Karp 3.2 Fourier Transform 22

23 Frequency Convoluion: F () F 2 () = F (ν) F 2 ( ν) dν 2π f () f 2 () Example: Ampliude Modulaion of a Triangular Pulse f() = Λ() cos( ) = 2 Λ()(exp(j ) + exp( j )) Λ() Sa 2 (/2), exp(±j ) 2πδ( ) f() 2π Sa2 (/2) (πδ( ) + πδ( + )) Λ() cos(25 ).5.5(Sa 2 (( 25)/2+Sa 2 ((+25)/2))) Dr. Tanja Karp 3.2 Fourier Transform 23

24 3.2.5 Parseval s Theorem for Energy Signals E = f() 2 d = 2π F () 2 d Example: f() = Sa(/2) = sin(/2) /2 2πrec() = F () E = f() 2 d = sin 2 (/2) (/2) 2 d = 2π = 2π Z.5.5 F () 2 d = 2π d = 2π 4π 2 rec 2 () d Dr. Tanja Karp 3.2 Fourier Transform 24

25 3.2.6 Fourier Transform of Periodic Signals Exponenial Fourier Series represenaion of a periodic signal: f T () = X k= F k exp(jk ) Fourier Transform of a periodic signal f T (): F{f T ()} = F{ = X k= X k= F k exp(jk )} = F k 2πδ( k ) X k= F k F{exp(jk )} Example: Sum of Two Cosine Signals f T () = cos( ).5 cos(3 ) =.5 exp(j ) + exp( j ).25 exp(j3 ) + exp( j3 ) F = F =.5 F 3 = F 3 =.25 Dr. Tanja Karp 3.2 Fourier Transform 25

26 3 /2 F k /4 3 (π) F() (π) 3 ( π/2) 3 ( π/2) Dr. Tanja Karp 3.2 Fourier Transform 26

27 3.3 Frequency Domain Descripion of LTI Sysems 3.3. Frequency Response Time Domain: Frequency Domain: f () LTI sysem h() g() F() LTI sysem H() G() h(): Impulse Response H(): Frequency Response g() = f() h() G() = F () H() An LTI sysem does no generae new frequency componens. Example: RC Lowpass Filer v i () v i () R C v o () v o () T T /RC h() Dr. Tanja Karp 3.3 Frequency Domain Descripion of LTI Sysems 27

28 For T =, RC=: Magniude Specrum V i () H() V o () Dr. Tanja Karp 3.3 Frequency Domain Descripion of LTI Sysems 28

29 Phase Specrum 2 V i () H() V o () Dr. Tanja Karp 3.3 Frequency Domain Descripion of LTI Sysems 29

30 3.3.2 Bandwidh of Frequency Selecive Sysems Lowpass Filer: A H() A/ 2 Bandpass Filer: A H() A/ 2 c c 2 2 Bandwidh: B = c Bandwidh: B = 2 Highpass Filer: A H() A/ 2 Bandsop Filer: A H() A/ 2 c c 2 2 Only posiive frequencies are couned for he bandwidh. Dr. Tanja Karp 3.3 Frequency Domain Descripion of LTI Sysems 3

31 3.3.3 Disorionless Transmission A ransmission sysem is called disorionless, if he oupu signal g() is a scaled and delayed copy of he inpu signal f(): g() = K f( ) G() = K F () exp( j ) f () LTI sysem h() g() F() LTI sysem H() G() Sysem Frequency Response: H() = K exp( j ) K H() H() The ransmission sysem mus have a consan magniude response and is phase shif mus be linear wih frequency (resuling in he same delay of all frequency componens of he inpu). Dr. Tanja Karp 3.3 Frequency Domain Descripion of LTI Sysems 3

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