( ) ( ) ( ) Fourier series. ; m is an integer. r(t) is periodic (T>0), r(t+t) = r(t), t Fundamental period T 0 = smallest T. Fundamental frequency ω
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1 Fourier series e jm when m d when m ; m is an ineger. jm jm jm jm e d e e e jm jm jm jm r( is periodi (>, r(+ r(, Fundamenal period smalles Fundamenal frequeny r ( + r ( is periodi hen M M e j M, e j, is a raional number e j has fundamenal period Periodi exension for ime-limied x( { for < and > x n Le r( ( n x ~ x + x( ~ ( n x n even periodi exension of x( x( - x ~ x x( ~ ( n x n odd periodi exension of x(
2 x( - r( e j ; r r e d average or DC value of r( e j j j e d he h Fourier oeffiien of r( + he h harmoni omponen of r( fundamenal omponen of r( r( is brassier or hasher when i has "heavy" high harmonis j j r ( + ( e + e r( is real-valued ( r r j j j r e d re d r e d sin r( + ( a os( + ( b ( a b Re m { } r os { } r sin Consider he h harmoni omponen of r(: d d
3 Example Re{ } Re{ ( Re( j m( ( os jsin } ( Re( os( ( m( sin ( e + e e + e e + e e r p ( + j j j j j j j π a ( j j sin j j j j r e d e d e e j b e e sin π j π a Re{ } sin π π b { } m π r ( + sin π os + sin π os π π π + + π p sin π os r δ ( + π j j r e d e d δ a Re{ } { } b m
4 π r ( + os + os + π os + δ r( is even { r(- r( } - ( a a+ + a j jτ jτ r e d r τ e dτ r τ e dτ Speifially, a a a+ j( j τ r e d r e d ; d, d j τ j τ r( τ e dτ r e d τ τ r( is odd { r(- -r( } - - ( a τ τ τ τ a+ + a j jτ jτ r e d r τ e dτ r τ e dτ Speifially, a a a+ j( j τ r e d r e d ; d, d j τ j τ r( τ e dτ r e d τ τ τ τ τ τ his proof is similar o he prove ha when r( is even, -. he differene is ha r(- -r(, no jus r(, yielding he negaive sign in fron of he final resul. r( is real valued and even 's are all real and - Real valued r( Even r( - hus,. Beause, is real valued. r( is real-valued and odd 's are pure imaginary and - - Real valued r( Odd r( - -
5 hus, -. Beause, is pure imaginary. f r( is oninuous exep possibly for some jumps, and has only finiely many jumps in any bounded -inerval, hen, when is a par of oninuiy (non-jump of r(, Fourier series onverges o r( when is a jump-poin for r(, Fourier series onverges o he mean value of r( aross he jump Gibbs Phenomena N S N ( N e jumps, S N (-graph overshoos r( graph Widh of he "overshoo blip" narrow as N. However, heigh of overshoo doesn' redue Parseval's deniy: r d r j e j( m r r r e m jm m e m m r j( m j( m r d e m d m e d m m j( m when -m ( ( ; e d when -m ( Fourier series and L SSO sysems wih periodi inpus r( e j d H y( is also -periodi SL y( j j H( e d e Coninuous-ime Fourier ransform (`, CF Nonperiodi signal x(
6 j X e d x x e d j X ( Firs, onsider ime-limied x( whih for > n Le r( ( n x. Fourier series: r( e r xe d e hus, j j j ; r e d x e j j j j x x e d e for j j x e d e for Le d,, j j j x x e d e d X e d x( and X ( are `-ransform pair (one or boh equaions (, - hold(s. n order for x( o have a -ransform, x( an' blow up as n Mahad, Fourier ransform an be found easily by Ener he expression o be ransformed. hen, li on he ransform variable. Finally, hoose ransform / Fourier from he Symbolis menu. Mahad reurns a funion in he variable when perform a Fourier ransform. f he expression you are ransforming already onains an, Mahad avoids ambiguiy by reurning a funion in he variable insead. f x( as (exremely well-behaved, d d x exiss for all { x( is infiniely differeniable }, and d
7 d d x as, hen boh ( and ( - hold in srong sense X ( as f x( is absoluely inegrable, hen ( holds in srong sense X ( is a bounded funion of ( - hold a leas in he wea sense j x lim X ( e d f x( is square inegrable, hen X ( is also square inegrable boh ( and ( - hold in srong sense Narrow/sharp in wide/mushy in Wide/mushy in narrow/sharp in Gaussian in Gaussian in Le x X (, x X ( and x X ( x X j X e j d x e δ j j Proof e δ d e δ ( Proof j δ e d Proof Use dualiy: f g g π f From, δ d
8 δ ( δ a aδ ( j e ( Proof δ j j δ ( e d e j Proof Use frequeny-shif rule: e x X ( δ (, j e From j e δ. δ( r e R j π δ ( + X + X x x x( e j j e X ime-shif rule Proof j x X e d x : disree sperum j j j x X e d X e e d Frequeny-shif (or modulaion rule j X x e d Proof ( X ( j( j j ( X x e d x e e d δ j Pa ; a> e sin( a a a sin π a j j j j a j a Proof Pa e d e d ( e e a sin ( a
9 sin ( π d x d P ( j j j j j Proof P e d e d ( e e From a P ( sin π g Proof Use dualiy: f g f ( sin ( a sin ( P ( P, jx ime-derivaive rule d d Proof j j x X e d jx e d d d d X d Frequeny-derivaive rule d Proof d j j X x e d jx e d d d jx x( a ime-saling rule Proof For a >, ja xa X e d X a a u ju X e du ; u aw, du adw a a For a <,
10 ja xa X e d u ju X e du ; u aw, du adw a a + u ju X e du a a For very small < a << x(a is mushier, more spread ou han x( u X is a aller, narrower version of X ( a a x( X ( x X ( j X x e d Proof ( j X x e d ( j ( j X x e d x e d x * x( X X Convoluion-in-ime Rule Proof Le ( y x τ x τ dτ j Y y e d ( j x( τ x ( τ dτ e d j x( τ x ( τ e d dτ jτ x( τ e X dτ ;ime-shif rule jτ ( τ τ ( ( X x e d X X ( * ( x x X X
11 Convoluion-in-frequeny Rule??? j Proof Le ( Y X X d y Y e d j X X( d e d j X( e X ( d d X e x d ;frequeny-shif rule j ( x X e d x x j ( f g g Proof Use dualiy: π f From * X X x x π( X * X x x x X ( e u α α e u ( α + j α j ( α+ j e u e d e d α + j α j α j ( α j e u( e d e d α j x X, Or use α now ha e u,??? α, hen e u α + j. α j j e α α + α α α j e e u e u α + j α j α +
12 j + u πδ os sin > + α < α α j j j sgn lim e u e u( lim + + α α α + j u ( sgn + δ πδ( + + j α α Define sgn lim e u e u( d u j d + j Noe δ πδ ( πδ ( + πδ ( + j j e e + j Reall ha e δ Proof Use os ( ( ( ( ( + π δ δ j os X ( + X ( + x j j x os xe + x e hen, use Frequeny-shif/modulaion rule: j e x X. Proof π e α 4α e hus, Gaussian Gaussian. Dualiy: f g ( g f ( α
13 j g f e d ( jpz g( p f z e dz j g f e d ; z, p j f e d Parseval's deniy: X ( x X x x d X d,if x( is square inegrable * y x x x X X Y By definiion j Y y e d? j ( ( hus, x e d X * X X µ * X µ * x d X µ X µ d µ X µ d µ π π x( is real-valued x x X ( X j j j ( X x e d xe d x e d X x( is even x x( X ( is also even X ( X j jτ X x e d x τ e dτ ; τ jτ ( τ τ x e d X x( is odd x x( X ( is also odd X ( X
14 j jτ X x e d x τ e dτ ; τ jτ ( τ τ x e d X x( is real and even so is X ( X X X X X is real X X x( is real and odd X ( is pure imaginary and odd X( X X X ( X is pure imaginary X( X
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