ourier Series & The ourier Transfor Wha is he ourier Transfor? Wha do we wan fro he ourier Transfor? We desire a easure of he frequencies presen in a wave. This will lead o a definiion of he er, he specru. ourier Cosine Series for even funcions and Sine Series for odd funcions The coninuous lii: he ourier ransfor (and is inverse) Plane waves have only one frequency,. Ligh elecric field The specru Soe exaples and heores f ( ) = ( ) exp( ) i d % ( ) = f ( ) exp( i ) This ligh wave has any frequencies. And he frequency increases in ie (fro red o blue). I will be nice if our easure also ells us when each frequency occurs. Tie Lord Kelvin on ourier s heore Joseph ourier, our hero ourier s heore is no only one of he os beauiful resuls of odern analysis, bu i ay be said o furnish an indispensable insruen in he reaen of nearly every recondie quesion in odern physics. ourier was obsessed wih he physics of hea and developed he ourier series and ransfor o odel hea-flow probles. Lord Kelvin
Anharonic waves are sus of sinusoids. Consider he su of wo sine waves (i.e., haronic waves) of differen frequencies: ourier decoposing funcions Here, we wrie a square wave as a su of sine waves. The resuling wave is periodic, bu no haronic. Essenially all waves are anharonic. Any funcion can be wrien as he su of an even and an odd funcion E(-x) = E(x) E(x) O(x) Le f(x) be any funcion. ()[()()]/ ()()() Exfxfx Oxfxfx fxexox + =+ ourier Cosine Series Because cos() is an even funcion (for all ), we can wrie an even funcion, f(), as: f() = = cos() O(-x) = -O(x) f(x) where he se { ; =,, } is a se of coefficiens ha define he series. And where we ll only worry abou he funcion f() over he inerval (,).
The Kronecker dela funcion, n =%& if inding he coefficiens,, in a ourier Cosine Series ourier Cosine Series: f ( ) = cos( ) = To find, uliply each side by cos( ), where is anoher ineger, and inegrae: f () cos( ) % = cos() cos( ) = Bu: if = cos( ) cos( ) = & %, ) ( if So: f ( ) cos( ) =, only he = er conribues & % = Dropping he fro he : = f ( ) cos( ) yields he coefficiens for any f() ourier Sine Series Because sin() is an odd funcion (for all ), we can wrie any odd funcion, f(), as: f () = = sin() where he se { ; =,, } is a se of coefficiens ha define he series. where we ll only worry abou he funcion f() over he inerval (,). inding he coefficiens,, in a ourier Sine Series ourier Sine Series: To find, uliply each side by sin( ), where is anoher ineger, and inegrae: f ( ) sin( ) = sin( ) sin( ) & % & Bu: = if = sin( ) sin( ) = %, ) & ( if So: f () = = = f ( ) sin( ) = %, & sin() only he = er conribues Dropping he fro he : yields he coefficiens = f ( ) sin( ) for any f()
ourier Series We can plo he coefficiens of a ourier Series So if f() is a general funcion, neiher even nor odd, i can be wrien: vs. f ( ) = cos( ) + sin( ) = =.5 even coponen odd coponen where = f () cos() and = f () sin() 5 5 5 We really need wo such plos, one for he cosine series and anoher for he sine series. 3 Discree ourier Series vs. Coninuous ourier Transfor Le he ineger becoe a real nuber and le he coefficiens,, becoe a funcion (). () vs. Again, we really need wo such plos, one for he cosine series and anoher for he sine series. The ourier Transfor Consider he ourier coefficiens. Le s define a funcion () ha incorporaes boh cosine and sine series coefficiens, wih he sine series disinguished by aking i he iaginary coponen: () f ( i = ) cos( ) ( ) sin( ) i f Le s now allow f() o range fro o, so we ll have o inegrae fro o, and le s redefine o be he frequency, which we ll now call : ( ) = f ( ) exp( i ) The ourier Transfor () is called he ourier Transfor of f(). I conains equivalen inforaion o ha in f(). We say ha f() lives in he ie doain, and () lives in he frequency doain. () is jus anoher way of looking a a funcion or wave.
The Inverse ourier Transfor The ourier Transfor akes us fro f() o (). How abou going back? Recall our forula for he ourier Series of f() : ( ) = cos( ) + sin( ) f = = Now ransfor he sus o inegrals fro o, and again replace wih (). Reebering he fac ha we inroduced a facor of i (and including a facor of ha jus crops up), we have: f ( ) = ( ) exp( ) i d % Inverse ourier Transfor ourier Transfor Noaion There are several ways o denoe he ourier ransfor of a funcion. If he funcion is labeled by a lower-case leer, such as f, we can wrie: f() () If he funcion is labeled by an upper-case leer, such as E, we can wrie: E( ) { E( )} Soeies, his sybol is used insead of he arrow: or: E( ) E () The Specru We define he specru, S(), of a wave E() o be: S( ) { E( )} Exaple: he ourier Transfor of a recangle funcion: rec() / ( ) = exp( i ) = [exp( i )] i / = [exp( i / ) exp( i /)] i = exp( i / ) exp( i /) ( /) i sin( /) = sinc( /) (/) / / () This is he easure of he frequencies presen in a ligh wave. {rec( ) } = sinc( /) Iaginary Coponen =
Exaple: he ourier Transfor of a decaying exponenial: exp(-a) ( > ) ( ) = exp( a)exp( i ) = exp( a i ) = exp( [ a + i ] ) + = exp( [ a + i ] ) = [exp() exp()] a + i a + i = [ ] a + i = a + i () = i ia A coplex Lorenzian Exaple: he ourier Transfor of a Gaussian, exp(-a ), is iself {exp( )} = exp( )exp( ) % a a i exp( / 4 a) exp( a ) The deails are a HW proble exp( / 4 a) ourier Transfor Syery Properies Expanding he ourier ransfor of a funcion, f(): ( ) = [Re{ f ( )} + i I{ f ( )}] [cos( ) i sin( )] Expanding ore, noing ha: O( ) = if O() is an odd funcion = if Re{f()} is odd = if I{f()} is even %Re{()} ( ) = Re{ f ( )} cos( ) + I{ f ( )} sin( ) = if I{f()} is odd = if Re{f()} is even + i I{ f ( )} cos( ) i Re{ f ( )} sin( ) & & Even funcions of Odd funcions of %I{()} The Dirac dela funcion Unlike he Kronecker dela-funcion, which is a funcion of wo inegers, he Dirac dela funcion is a funcion of a real variable,. % () & if = if ()
The Dirac dela funcion if = ( ) % if & Dirac ()funcion Properies () I s bes o hink of he dela funcion as he lii of a series of peaked coninuous funcions. f () = exp[-() ]/* () f 3 () f () f () ( ) = ( a) f ( ) = ( a) f ( a) = f ( a) % % exp( ± i ) = ( ) exp[ ± i( %&) ] = ( %&) The ourier Transfor of () is. % ( ) exp( i ) = exp( i []) = () The ourier ransfor of exp(i ) { } exp( i ) = exp( i ) exp( i ) = exp( i[ ] ) = ( ) And he ourier Transfor of is *(): & % exp( % i ) = ( ) () I Re exp(i ) {exp(i )} The funcion exp(i ) is he essenial coponen of ourier analysis. I is a pure frequency.
The ourier ransfor of cos( ) { } cos( ) = cos( ) exp( i ) = [ exp( i ) + exp( i ) ] exp( i ) = exp( i[ ] ) + exp( i[ + ] ) = ( ) + ( + ) cos( ) {cos( )} + The Modulaion Theore: The ourier Transfor of E() cos( ) { } E( )cos( ) = E( )cos( ) exp( i ) ( = E( ) exp( i ) + exp( i ) exp( i ) & % = E( )exp( i[ ] ) + E( )exp( i[ + ] ) Exaple: E() = exp(- ) E( )cos( ) = E ( ) + E ( + ) { } E( )cos( ) { E( )cos( ) } - Scale Theore The ourier ransfor of a scaled funcion, f(a): { f ( a)} = ( / a) / a Proof: { f ( a)} = f ( a) exp( i ) Assuing a >, change variables: u = a { f ( a)} = f ( u) exp( i [ u/ a]) du / a = f ( u) exp( i [ / a] u) du / a (/)/ aa = If a <, he liis flip when we change variables, inroducing a inus sign, hence he absolue value. The Scale Theore in acion The shorer he pulse, he broader he specru This is he essence of he Uncerainy Principle Shor pulse Mediulengh pulse Long pulse f() ()
The ourier Transfor of a su of wo funcions { a f ( ) + b g( )} = a{ f ( )} + b { g( )} Also, consans facor ou. f() g() f()+g() () G() () + G() Shif Theore The ourier ransfor of a shifed funcion, f ( a) : Proof : { ( )} f a = f ( a)exp( i ) Change variables : u = a { } f ( u)exp( i [ u + a]) du = exp( i a) f ( u)exp( i u) du = exp( i a) ( ) f ( a) = exp( i a) ( ) ourier Transfor wih respec o space The D ourier Transfor If f(x) is a funcion of posiion, () {f(x,y)} = (k x,k y ) f(x,y) ( k) = f ( x) exp( ikx) dx {f(x)} = (k) x = f(x,y) exp[-i(k x x+k y y)] dx dy If f(x,y) = f x (x) f y (y), y x () {f(x,y)} We refer o k as he spaial frequency. k hen he D T splis ino wo D Ts. Everyhing we ve said abou ourier ransfors beween he and doains also applies o he x and k doains. Bu his doesn always happen.
The Pulse Wih + The rs pulse wih + There are any definiions of he wih or lengh of a wave or pulse. The effecive wih is he wih of a recangle whose heigh and area are he sae as hose of he pulse. Effecive wih Area / heigh: eff f ( ) f () % (Abs value is unnecessary for inensiy.) f() + eff The roo-ean-squared wih or rs wih: % + f ( ) & rs ( % & % & % + f ( ) & ) * / The rs wih is he second-order oen. Advanage: I s easy o undersand. Disadvanages: The Abs value is inconvenien. We us inegrae o ±. Advanages: Inegrals are ofen easy o do analyically. Disadvanages: I weighs wings even ore heavily, so i s difficul o use for experiens, which can scan o ± ) The ull-wih- Half-Maxiu ull-wih-half-axiu is he disance beween he half-axiu poins. Advanages: Experienally easy. Disadvanages: I ignores saellie pulses wih heighs < 49.99% of he peak + WHM Also: we can define hese wihs in ers of f() or of is inensiy, f(). Define specral wihs (+) siilarly in he frequency doain ( )..5 + WHM The Uncerainy Principle The Uncerainy Principle says ha he produc of a funcions wihs in he ie doain (+) and he frequency doain (+) has a iniu. Define he wihs assuing f() and () peak a : % ( ) ( ) f () & f % () & d () f ( ) f ( )exp( i[] ) f () % = f () % = f () f () % & ( ) d ( )exp( i d () = () []) = () (Differen definiions of he wihs and he ourier Transfor yield differen consans.) Cobining resuls: f () () or: () f ()
The Uncerainy Principle or he rs wih, + + There s an uncerainy relaion for x and k: +k +x