Inverse transform. Inverse transform. tri(t) = Λ(t) sinc 2 (f) sinc 2 (t) tri(f) e πt2 e πf 2. e ±iπt2 e ±iπ/4 e iπf 2

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1 ourier Series o ourier ransforms In he limi as he period he ourier series coefficiens become infiniesimely spaced and he periodic ourier series Coninuous ourier ransform he -D emporal ourier ransform: Definiions Valid for finie suppor or periodic signals orward emporal ourier ransform (Hz) G(f) g()e iπf d {g()} Inverse ransform g() G(f)e iπf df {G(f)} Alernae definiion using angular radian frequency ω πf orward emporal ourier ransform (rad/sec) G(ω) g()e iω d {g()} Inverse ransform g() G(ω)e iω dω {G(ω)} {G(ω)} π ω πf dω πdf Noe ha hese funcions are scaled versions of each oher G(ω) G(ω/π) Kelvin Wagner, Universiy of Colorado Linear Sysems 9 34 Kelvin Wagner, Universiy of Colorado Linear Sysems 9 35 Exacly Analogous -D Spaial ourier ransform: Definiions Can similarly define in space using spaial frequency, u f x [lines/mm], analogous o emporal frequency f [Hz], or use wavevecor, k x [rad/mm], analogous o angular frequency ω [rad/sec]. orward -D spaial ourier ransform G(u) g(x)e iπux dx x {g(x)} Inverse -D spaial ourier ransform g(x) G(u)e iπux du x {G(u)} or in erms of wavevecor k x G(k x ) g(x)e ikxx dx {g(x)} x {g(x)} g(x) π G(k x )e ikxx dk x {G(k x )} x {G(k x )} k x {G(k x )} Noe ha hese funcions are scaled versions of each oher G(k x ) G(k x /π) Kelvin Wagner, Universiy of Colorado Linear Sysems 9 36 Well defined coninuous funcions rec() Π() sinc(f) rec( ) Π( ) sinc( f) sinc() rec(f) ri() Λ() sinc (f) sinc () ri(f) e π e πf -dimensional ourier ransforms e ±iπ e ±iπ/4 e iπf e +(πf) e H() +iπf iπf +(πf) sech π sech πf jinc() J (π) (f) Π(f) / f / Singular funcions in or f δ() (f) δ ( a) a δ() a (f) δ( ) e iπ f iπ sgn( f) u() H() δ(f) + iπf e iπf δ(f f ) ( iπ cos(πf ) [δ(f f )+δ(f+f )] sin(πf ) i [δ(f f ) δ(f+f )] comb() comb(f) comb ( ) comb( f) k ( k iπ) δ (k) (f) ) k δ (k) () f k Kelvin Wagner, Universiy of Colorado Linear Sysems 9 37

2 of complex exponenials he mos basic and fundamenal pairs are he complex exponenials and { e iπf } { e iπf } e iπf e iπf d e iπf e iπf d e iπ(f f ) d δ(f f ) e iπ(f+f ) d δ(f + f ) which produce singular specra since when f f he inegrand is oscillaory and inegraes o zero, bu when mached, f f he inegral is unbounded. he inverse ransform is of a singular funcion, bu we can jus use sifing propery {δ(f f )} δ(f f )e iπf d e iπf Using lineariy of he o combine hese resuls we ge and {cos(πf )} { [eiπf + e iπf ] } [δ(f f ) + δ(f + f )] {sin(πf )} { i [eiπf e iπf ] } i [δ(f f ) δ(f + f )] Kelvin Wagner, Universiy of Colorado Linear Sysems 9 38 ourier ransform of complex phase facor imes real cosine e iπ/4 cos(ω o ) and e iπ/8 cos(ω o + π/4) e iπ/4 cos(ω o ) eiπ/4 [δ(ω ω o) + δ(ω + ω o )] e iπ/8 cos(ω o + π/4) ei(π/8+π/4) δ(ω ω o ) + ei(π/8 π/4) δ(ω + ω o ) ourier ransform of Impulse Pair: Cosine and Sine Dualiy gives of symeric emporal impulse pair Kelvin Wagner, Universiy of Colorado Linear Sysems 9 39 of complex exponenial Or we can use equivalen ω-ransform noaion { e iω } and scaled inverse ransform e iω e iω d {δ(ω ω )} π of he dela impulse Dualiy or direc inegraion can be used o evaluae and for a shifed impulse {δ()} {δ( )} e i(ω ω) d πδ(ω ω) δ(ω ω )e iω d π eiω δ()e iω d δ( )e iω d e iω which also comes ou direcly from ime shifing propery Kelvin Wagner, Universiy of Colorado Linear Sysems 9 4 Kelvin Wagner, Universiy of Colorado Linear Sysems 9 4

3 Phase shifs of complex exponenials Posiive and Negaive Linear Phase acors e iω o δ(ω ωo ) e iω o δ(ω + ωo ) Phae Shifing Complex Exponenial by muliplying by e iφ ie iω o iδ(ω ωo ) ie iω o iδ(ω ωo ) ( X(iω) Π Example ourier ransform of rec rec ( e iω e iω) ω i ω where sinc(f) sin πf πf ( ) ( ) { > Π < ) e iω d e iω d sin ω πf iω e iω ( e iω e +iω) iω sin πf sin π f π f sinc f or using alernaive snc ω sin ω ω we ge X(ω) snc (ω ) ( ) Π sinc( f) ( ) Π sinc( f). Kelvin Wagner, Universiy of Colorado Linear Sysems f -. Kelvin Wagner, Universiy of Colorado Linear Sysems 9 43 Sinc and is widh Picorial ourier ransform of a symmeric rec Kelvin Wagner, Universiy of Colorado Linear Sysems 9 44 Kelvin Wagner, Universiy of Colorado Linear Sysems 9 45

4 Picorial Inverse ourier ransform producing a emporal rec in he limi of infinie bandwidh Dualiy Picorial Inverse ourier ransform of a rec in frequency Kelvin Wagner, Universiy of Colorado Linear Sysems 9 46 Kelvin Wagner, Universiy of Colorado Linear Sysems 9 ime lapse movie of phasor evoluion of each ourier componen of rec in frequency 47 of sinc : odd sin π sin π {sinc()} sinc()e iπf d cos(πf ) d i sin(πf ) d π π sin(π + πf ) sin(π πf ) + d π π sin[π( + f )] ( + f ) sin[π( f )] ( f ) + d ( + f ) ( f ) + f sin[π( + f )] f sin[π( f )] d + d ( + f ) ( f ) +f -f + f f +f -f f + Π() rec () f + f f Where we used sinc(a) d a f +f sgn(f + ) f f sgn( (f Or direcly by dualiy since Π() (+ sinc(f ) Kelvin Wagner, Universiy of Colorado Linear Sysems 9 48 Kelvin Wagner, Universiy of Colorado )) sinc() (+ Π( f ) Π(f ) Linear Sysems 9 49

5 he sign funcion is defined as of sign funcion sgn() u() u() u( ) { < > and {sgn()} no absoluely convergen since sgn() d Mus define in erms of series of funcions ha in he limi become sgn() { e sgn() lim s a () where s a () /a < a e /a > or finie a we have a well defined {s a ()} e /a e iπf d e /a e iπf d e ( /a iπf) d e (/a iπf) d e( /a iπf) /a iπf e(/a iπf) /a iπf /a iπf /a iπf i4πf (/a) + (πf) Giving he of sgn in he limi a { } {sgn()} lim s i4πf a() a () + (πf) iπf Kelvin Wagner, Universiy of Colorado Linear Sysems 9 5 of sign funcion Noe ha an easier roue is o use he derivaive and inegraion properies hus d (sgn() + K) δ() d { } d (sgn() + K) iω{sgn() + K} d iω{sgn()} + iωπkδ(ω) {sgn()} iω πkδ(ω) he consan K mus be zero since sgn() + sgn( ) iω πkδ(ω) + iω πkδ(ω) K {sgn()} iω And from his we can ge he of u() + sgn() {u()} πδ(ω) + iω Kelvin Wagner, Universiy of Colorado Linear Sysems 9 5 of Heavyside sep funcion of /iπ u() d no convergen so Laplace ransform /s no defined along iω axis. Mus insead define in erms of series of funcions ha in he limi become u() { < u() lim u a () where u a () a e /a > Or we can use ha u() + sgn() wih known and lineariy of {u()} { + sgn()} δ(f) + iπf Now using he scaling of a dela o wrie δ(f) δ ( ω π) πδ(ω) Dualiy can be invoked o evaluae {u()} πδ(ω) + iω sgn( f) iπ δ() + u( f) iπ Kelvin Wagner, Universiy of Colorado Linear Sysems 9 5 { } iπ if i iπ e iπf d sin(πf ) d f πf even iπ cos(πf)d i odd sinc(f)d f f sgn( f) iπ sin(πf)d where he infinie inegral (eg area) of a scaled sinc of widh /f is jus given by he widh /f, bu when f is negaive his area of he even sinc is /f, hus sinc(f)d f or dualiy can be invoked from he known {sgn()} /iπ o evaluae sgn( f) iπ Bu in his case he of /iπ is acually he convergen one, so dualiy should be invoked insead o find of sgn(). Kelvin Wagner, Universiy of Colorado Linear Sysems 9 53

6 of causal decreasing exponenial Gaussian and Complex Gaussian { e a u() } u()e a e iπf d e ( a iπf) d e( a iπf) a iπf a iπf a + iπf As long as e a which requires a > and a convergen causal exponenial his could be performed wih he ω ransfom definiion o obain he equivalen resul { e a u() } e a e iω d e ( a iω) d e( a iω) a iω a iω a + iω Which should be obvious from jus he subsiuion ω πf Noe ha a causal sinusoid does no o a dela, and here is specral spreading. Since he inegral is no convergen, use he modulaion heorem and known of u() { e iπf u() } e iπ(f+f) d e iπ(f+f ) iπ(f + f )? [ δ(f + f ) δ(f) + ] iπf δ(f + f ) + iπ(f + f ) Kelvin Wagner, Universiy of Colorado Linear Sysems 9 54 g() e π e π d n() e (/τ) G(f) e π e iπf d e π( +if) d noe ( + if) + if f e πf e π(+if) d e πf e π d e πf } {{ } + if d d area of normalized uni Gaussian Scaled Gaussian g() e π(/τ) τe π(τf) Complex Gaussian (Chirp wih Gaussian ampliude modulaion) /τ a + ib when a his is a chirp e π(a+ib) e iπb a + ib e πf /(a+ib) ib e +iπf /b Kelvin Wagner, Universiy of Colorado Linear Sysems 9 55 Lineariy Conjugaion Scale Shif Modulaion Derivaive and Inegraion he emporal ourier ransform: Properies ag() + bh() ag(f) + bh(f) g (±) G ( f) g(α) ( ) f α G α g( ) e iπf G(f) e iπf g() G(f f ) d n dx ng() (iπf)n G(f) g(τ)dτ G() G(f) + iπf δ(f) Kelvin Wagner, Universiy of Colorado Linear Sysems 9 56 Parseval s heorem Convoluion Correlaion ourier Inegral he -D ourier ransform: Properies g() d G(f) df g( )h( )d G(f)H(f) g()h() G( )H( )d g( )h ( )d G(f)H (f) g( )g ( )d G(f) { {g()}} {g()} g() { {g()}} g( ) Kelvin Wagner, Universiy of Colorado Linear Sysems 9 57

7 Lineariy Shif heorem he ourier inegral is clearly linear, so ha and {ax() + by()} a{x()} + b{y()} {ax(f) + by (f)} a {X(f)} + b {Y (f)} Lineariy is an implici assumpion in he decomposiion of funcions ino even and odd pars. Similarly implici in decomposiion of signals or ransforms ino real and imaginary pars Example } {e { e u() + e u( ) } + (πf) + jπf + jπf Noe his follows from he Laplace ransform evaluaed a s πf jπf + ( + jπf) + (πf) e iπf X(f)e iπf df x( ) e iπf X(f) X(f)e iπf( Π() sinc(f) Π( ) sinc(f)e iπf Π( /) sinc(f)e iπf {}}{ ) df x( ) x( ) Kelvin Wagner, Universiy of Colorado Linear Sysems 9 58 Kelvin Wagner, Universiy of Colorado Linear Sysems 9 59 ourier ransform of Shifed uncions Conjugaion Kelvin Wagner, Universiy of Colorado Linear Sysems 9 6 Proof le ω ω So when x() is real x() x () x () X ( ω) ( X (ω) x()e d) iω X ( ω ) X ( ω) x ()e iω d x()e iω d X(ω) So we can wrie X(ω) in erms of real and imaginary pars x ()e +iω d X(ω) R{X(ω)} + ii{x(ω)} X r (ω) + ix i (ω) X r (ω) + ix i (ω) X r ( ω) ix i ( ω) So we can conclude for real x() we have Hermiion X(ω) X r (ω) X r ( ω) ix i (ω) ix i ( ω) even odd Kelvin Wagner, Universiy of Colorado Linear Sysems 9 6

8 Cosine and Sine pars of ourier ransform Scaling x(a) (+ X a inverse ourier ransform of uni heigh rec pulse Noe he even characer of cos ω x(a)e iπf d Sum along ridge does no cancel x(τ )e iπf τ /adτ /a τ a dτ ad x(τ )e iπf τ /adτ a + f iπ fa τ x(τ )e dτ X a a a x(τ )e iπf τ /adτ /a a> a< special case for mirroring in ime sine carrier surface x( ) (+ X( f ) Odd symery of sin ω Normalized ime widh form of he scaling hm x (+ X(f ) Symmeric sums along ω cancel for all giving real par of f () for even { (jω)} Kelvin Wagner, Universiy of Colorado f a Linear Sysems 9 6 Kelvin Wagner, Universiy of Colorado ourier ransform of Scaled uncions Linear Sysems 9 63 Shifed and scaled funcions ind of shifed and scaled recangle by breaking i down ino seps 4 x() 5Π apply scale heorem Π sinc(f ) apply shif heorem o ha resul 4 Π sinc(f ) e iπ4f apply lineariy 4 5Π Kelvin Wagner, Universiy of Colorado Linear Sysems 9 64 Kelvin Wagner, Universiy of Colorado 5sinc(f ) e iπ4f Linear Sysems 9 65

9 Shifed and scaled funcions ind of composie funcion conaining shifed, scaled, phase shifed, and weighed funcions. [ ] 7 sinc e iπ + 4Π[3 + 5]e iπ3 Looks complicaed. Don panic, jus break i down ino individual pars. irs noe he ineresing dualiy his works for even funcions sinc() + Π() Π(f) + sinc(f) or general funcions here is a more complee dualiy x() + X() + x( ) + X( ) X(f) + x( f) + X( f) + x(f) Kelvin Wagner, Universiy of Colorado Linear Sysems 9 66 lineariy combined wih shifed and scaled funcions ind of composie funcion [ ] 7 sinc e iπ + 4Π[3 + 5]e iπ3 Build up he complicaed funcions from is pars ( ) sinc Π(f) Π(3) ( ) 7 sinc ( ) 7 sinc e iπ Π[3( + 5)] Π(f)e iπ7f ( ) f 3 sinc 3 ( ) f 3 sinc e iπ5f 3 Π[(f )]e iπ7(f ) ( Π[3( + 5)]e iπ3 f sinc 3 ) e iπ5(f+3) sinc [ ] 7 e iπ +4Π[3+5]e iπ3 [ ] ( ) Π (f ) e iπ7(f ) sinc f+3 3 e iπ5(f+3) Kelvin Wagner, Universiy of Colorado Linear Sysems 9 67 Dualiy and he ransform of a ransform Dualiy Suppose h() H(f) bu H() is jus a funcion (eg sinc) and ha we wan o know he ourier ransform of ha funcion of ime le g() H() {H()} {g()} G(f) his means ha H( )e iπ( f) d h( f) g()e iπf d {{h()}} h( ) H()e iπf d Dualiy suggess ha propery of ime domain has dual when applied o frequency domain dx(f) d x()e iπf d x()( iπ)e iπf d df df iπx() dx(f) Differeniaion in frequency df e +iπf x() X(f f ) Modulaion heorem Kelvin Wagner, Universiy of Colorado Linear Sysems 9 68 Kelvin Wagner, Universiy of Colorado Linear Sysems 9 69

10 Dualiy: Shifing in ime and Shifing in requency Convoluion heorem y() x(τ)h( τ)dτ h(τ )x( τ )dτ {y()}y (f)x(f)h(f)h(f)x(f) ransform domain shif by τ [ ] [ {}}{ ] Y (iω) x(τ)h( τ)dτ e iω iω d x(τ) e h( τ) d dτ x(τ) [ H(ω)e iωτ] dτ H(ω) x(τ)e iωτ dτ H(ω)X(ω) y() h() x() Y(ω) H(ω)X(ω) Y (f) H(f)X(f) Modulaion heorem : Dualiy - convoluion in frequency domain r() s()p() S(µ)P(ω µ)dµ S(ω) P (ω) π π S(f )P (f f )df S(f) P (f) Kelvin Wagner, Universiy of Colorado Linear Sysems 9 7 Kelvin Wagner, Universiy of Colorado Linear Sysems 9 7 Convoluion heorem Examples Convoluion heorem Examples Simple proofs of convoluion properies Commuaive Associaive ri() Λ() Π() Π() rec () rec () {Λ()} {Π()} {Π()} sinc (f) x() y() y() x() since X(f)Y (f) Y (f)x(f) x() [ y() z() ] [ x() y() ] z() since X(f) [ Y (f)z(f) ] [ X(f)Y (f) ] Z(f) Disribuive x() [ y() + z() ] x() y() + x() z() since X [ Y + Z ] XY + XZ ind he of he convoluion of x() sin πf wih y() δ( + 4) Do convoluion firs hen sin(πf ) δ( + 4) sin[πf ( + 4)] i (δ(f f ) δ(f + f ))e iπ4f Do firs o avoid having o do convoluion where we used sin(πf ) δ( + 4) i (δ(f f ) δ(f + f )) e iπ4f sin(πf ) i (δ(f f ) δ(f + f )) δ( + 4) e iπ4f Kelvin Wagner, Universiy of Colorado Linear Sysems 9 7 Kelvin Wagner, Universiy of Colorado Linear Sysems 9 73

11 Convoluion heorem Example: ime shifing as convoluion wih shifed impulse Correlaion heorem he correlaion funcion measures he similariy beween funcions x() y() x(τ)y (τ )dτ x(τ + )y (τ )dτ inroducing he impulse response h() y ( ) we can wrie his as a convoluion x() h() x(τ)h( τ)dτ x(τ)y ( [ τ])dτ x(τ)y (τ )dτ x() y() Now we can use he convoluion heorem and he relaion y ( ) Y (ω) o wrie he correlaion heorem x() y() x() h() X(ω)H(ω) X(ω)Y (ω) X(f)Y (f) when x() y( ) and for a fla bandwidh B code ( Y (f) Π [ f B] ) we ge y( ) y() { Y (f)e iπf Y (f) } { Y (f) e iπf} sinc[b( )] Auocorrelaion heorem x() x() x (τ)x(τ + )dτ X(f) Kelvin Wagner, Universiy of Colorado Linear Sysems 9 74 Kelvin Wagner, Universiy of Colorado Linear Sysems 9 75 Modulaion heorem Differeniaion and Inegraion { e iω f() } (ω ω ) Now using sandard ideniies cos ω [eiω + e iω ] we ge {cos(ω )f()} [ ] (ω ω ) + (ω + ω ) And using sin ω i [eiω e iω ] we ge {sin(ω )f()} [ ] (ω ω ) (ω + ω ) i f f f f Differeniaion dx() d d d Inegraion y() X(f)e iπf d [X(f)(iπf)]e iπf d dx() d iπfx(f) x(τ)dτ x() u() u() x() u(τ)x( τ)dτ d { } X(f)e iπf d X(f) d { } e iπf d d d x( τ)dτ Now using he convoluion heorem [ ] Y (f) X(f)U(f) X(f) iπf + δ(f) dx() d iωx(ω) x(τ)u( τ)dτ X(f) iπf + X() δ(f) x(τ)dτ X(f) iπf + X() δ(f) X(ω) iω + X()πδ(ω) x(τ)dτ Kelvin Wagner, Universiy of Colorado Linear Sysems 9 76 Kelvin Wagner, Universiy of Colorado Linear Sysems 9 77

12 of riangle uncion and is Derivaive Picorial of Inegraion Kelvin Wagner, Universiy of Colorado Linear Sysems 9 78 Kelvin Wagner, Universiy of Colorado Linear Sysems 9 79 Differeniaion in requency Example of using differeniaion Dualiy in is f and ω form gives X() x( f) Differeniaion in ime dx() iπfx(f) d Dual is differeniaion in frequency iπx() dx(f) df dx() d Now we can use his o evaluae of causal ramp, remember hus X() πx( ω) iωx(ω) ix() dx(ω) dω u() πδ(ω) + iω u() i du(ω) dω i d [ πδ(ω) + ] iπδ (ω) ω dω iω g() Π(.5) + Π( +.5) ( ).5 g () Π δ( ) sincfe iπf e iπf G () (f) Inegraion propery G(f) iπf G() (f) ( iπf sin πf πf e iπf e iπf ( ) e iπf (πf) ) g() ( eiπf e iπf iπf iπf g () - e iπf e iπf ) [ e iπf (πf) + i πf e iπf iπf e iπf [ e iπf + iπfe iπf] [ e iω + iωe iω] (πf) ω ] Kelvin Wagner, Universiy of Colorado Linear Sysems 9 8 Kelvin Wagner, Universiy of Colorado Linear Sysems 9 8