Fourier Series & The Fourier Transform

Fourier Series & The Fourier Transform Wha is he Fourier Transform? Fourier Cosine Series for even funcions and Sine Series for odd funcions The coninuous limi: he Fourier ransform (and is inverse) The specrum Some examples and heorems f 1 () = ( )exp( ) 2 F ω i ω d ω F( ω) = f( ) exp( iω) d

Wha do we wan from he Fourier Transform? We desire a measure of he frequencies presen in a wave. This will lead o a definiion of he erm, he specrum. Plane waves have only one frequency, ω. Ligh elecric field Time This ligh wave has many frequencies. And he frequency increases in ime (from red o blue). I will be nice if our measure also ells us when each frequency occurs.

Lord Kelvin on Fourier s heorem Fourier s heorem is no only one of he mos beauiful resuls of modern analysis, bu i may be said o furnish an indispensable insrumen in he reamen of nearly every recondie quesion in modern physics. Lord Kelvin

Joseph Fourier, our hero Fourier was obsessed wih he physics of hea and developed he Fourier series and ransform o model hea-flow problems.

Anharmonic waves are sums of sinusoids. Consider he sum of wo sine waves (i.e., harmonic waves) of differen frequencies: The resuling wave is periodic, bu no harmonic. Essenially all waves are anharmonic.

Fourier decomposing funcions Here, we wrie a square wave as a sum of sine waves.

Any funcion can be wrien as he sum of an even and an odd funcion E(-x) = E(x) E(x) Le f(x) be any funcion. Ex ( ) [ f( x) + f( x)]/2 O(x) O(-x) = -O(x) Ox ( ) [ f( x) f( x)]/2 f(x) f( x) = E( x) + O( x)

Fourier Cosine Series Because cos(m) is an even funcion (for all m), we can wrie an even funcion, f(), as: f() = 1 m = F m cos(m) where he se {F m ; m =, 1, } 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 1 if m = n δ mn, if m n

Finding he coefficiens, F m, in a Fourier Cosine Series Fourier Cosine Series: To find F m, muliply each side by cos(m ), where m is anoher ineger, and inegrae: Bu: So: Dropping he from he m: 1 f () = Fm cos( m) m= f () cos(m' ) d = 1 F m cos(m) cos(m' ) d cos( m) cos( m ' ) d m= m= 1 f()cos( m') d = Fm δm m if m = m ' = if m m ', ' Fm = f ()cos( md ) δ mm, ' only he m = m erm conribues yields he coefficiens for any f()!

Fourier Sine Series Because sin(m) is an odd funcion (for all m), we can wrie any odd funcion, f(), as: f () = 1 m= F m sin(m) where he se {F m ; m =, 1, } is a se of coefficiens ha define he series. where we ll only worry abou he funcion f() over he inerval (,).

Finding he coefficiens, F m, in a Fourier Sine Series Fourier Sine Series: f () = 1 m= F m sin(m) To find F m, muliply each side by sin(m ), where m is anoher ineger, and inegrae: Bu: So: 1 f ( ) sin( m' ) d = F m sin( m) sin( m' ) d sin( m) sin( m' ) d m= m= 1 f()sin( m') d = F mδm m if m = m' = if m m', ' δ mm, ' only he m = m erm conribues Dropping he from he m: F m = f()sin( m) d yields he coefficiens for any f()!

Fourier Series So if f() is a general funcion, neiher even nor odd, i can be wrien: 1 1 f () = F cos( m) + F msin( m) m m= m= even componen odd componen where F m = f () cos(m) d and F m = f () sin(m) d

We can plo he coefficiens of a Fourier Series 1 F m vs. m.5 5 1 25 15 2 m 3 We really need wo such plos, one for he cosine series and anoher for he sine series.

Discree Fourier Series vs. Coninuous Fourier Transform Le he ineger m become a real number and le he coefficiens, F m, become a funcion F(m). F(m) F m vs. m m Again, we really need wo such plos, one for he cosine series and anoher for he sine series.

The Fourier Transform Consider he Fourier coefficiens. Le s define a funcion F(m) ha incorporaes boh cosine and sine series coefficiens, wih he sine series disinguished by making i he imaginary componen: F(m) F m if m = f ()cos( m) d i f()sin( m) d Le s now allow f() o range from o, so we ll have o inegrae from o, and le s redefine m o be he frequency, which we ll now call ω: F( ω) = f( )exp( iω) d The Fourier Transform F(ω) is called he Fourier Transform of f(). I conains equivalen informaion o ha in f(). We say ha f() lives in he ime domain, and F(ω) lives in he frequency domain. F(ω) is jus anoher way of looking a a funcion or wave.

The Inverse Fourier Transform The Fourier Transform akes us from f() o F(ω). How abou going back? Recall our formula for he Fourier Series of f() : 1 1 ' () = mcos( ) + msin( ) m= m= f F m F m Now ransform he sums o inegrals from o, and again replace F m wih F(ω). Remembering he fac ha we inroduced a facor of i (and including a facor of 2 ha jus crops up), we have: f 1 ( ) = ( ) exp( ) 2 F ω i ω d ω Inverse Fourier Transform

Fourier Transform Noaion There are several ways o denoe he Fourier ransform of a funcion. If he funcion is labeled by a lower-case leer, such as f, we can wrie: f() F(ω) If he funcion is labeled by an upper-case leer, such as E, we can wrie: E() Y { E()} or: E () % Eω ( ) Someimes, his symbol is used insead of he arrow:

The Specrum We define he specrum, S(ω), of a wave E() o be: S ( ω) Y { E( )} 2 This is he measure of he frequencies presen in a ligh wave.

Example: he Fourier Transform of a recangle funcion: rec() 1/2 1 F( ω) = exp( iω) d = [exp( iω)] iω 1/2 1/2 1/2 1 = [exp( iω / 2) exp( iω/2)] iω = 1 exp( iω / 2) exp( iω/2) ( ω /2) 2i sin( ω/2) = sinc( ω/2) ( ω/2) F(ω) F {rec( ) } = sinc( ω/2) Imaginary Componen = ω

Example: he Fourier Transform of a decaying exponenial: exp(-a) ( > ) F( ω)= exp( a)exp( iω) d = exp( a iω) d = exp( [ a + iω] ) d 1 + 1 = exp( [ a+ iω] ) = [exp( ) exp()] a+ iω a+ iω 1 = [ 1] a+ iω 1 = a + iω 1 F( ω )= i ω ia A complex Lorenzian!

Example: he Fourier Transform of a Gaussian, exp(-a 2 ), is iself! F 2 2 {exp( )} = exp( )exp( ω ) a a i d 2 exp( ω / 4 a) The deails are a HW problem! 2 exp( a ) 2 exp( ω / 4 a) ω

Fourier Transform Symmery Properies Expanding he Fourier ransform of a funcion, f(): F( ω) = [Re{ f ()} + iim{ f ()}] [cos( ω) isin( ω)] d Re{F(ω)} F( ω) = Re{ f( )}cos( ω) d + Im{ f( )}sin( ω) d Expanding more, noing ha: O () d= = if Re{f()} is odd = if Im{f()} is even = if Im{f()} is odd = if Re{f()} is even + i Im{ f ()} cos( ω) d i Re{ f ()}sin( ω) d Even funcions of ω if O() is an odd funcion Odd funcions of ω Im{F(ω)}

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,. δ () if = if δ()

The Dirac dela funcion δ () if = if I s bes o hink of he dela funcion as he limi of a series of peaked coninuous funcions. f m () = m exp[-(m) 2 ]/ δ() f 3 () f 2 () f 1 ()

Dirac δ funcion Properies δ() δ () d = 1 δ( a) f ( ) d = δ( a) f ( a) d = f ( a) exp( ± iω) d = 2 δ( ω) exp[ ± i( ω ω ) ] d = 2 δ( ω ω )

The Fourier Transform of δ() is 1. δ( ) exp( iω) d = exp( iω[]) = 1 δ() 1 And he Fourier Transform of 1 is 2δ(ω): 1 ω 1exp( iω) d = 2 δ( ω) 2δ(ω) ω

The Fourier ransform of exp(iω ) F exp( iω) = exp( iω) exp( iω) d { } = exp( i[ ω ω] ) d = 2 δω ( ω ) Im Re exp(iω ) Y {exp(iω )} ω ω The funcion exp(iω ) is he essenial componen of Fourier analysis. I is a pure frequency.

The Fourier ransform of cos(ω ) F cos( ω) = cos( ω) exp( iω) d { } 1 2 = [ ] + exp( iω ) exp( iω ) exp( iω) d 1 1 exp( i[ ω ω] ) d exp( i[ ω ω] ) d 2 2 = + + = δω ( ω) + δω ( + ω) cos(ω ) F {cos( ω)} ω +ω ω

The Modulaion Theorem: The Fourier Transform of E() cos(ω ) F E( )cos( ω) = E( )cos( ω) exp( iω) d { } 1 = E( ) exp( iω) + exp( iω) exp( iω) d 2 1 1 E()exp( i[ ω ω]) d E()exp( i[ ω ω]) d 2 2 = + + F 1 1 E ()cos( ω) = E% ( ω ω) + E% ( ω+ ω) 2 2 { } Example: E() = exp(- 2 ) E ()cos( ω ) F { E ()cos( ω ) } -ω ω ω

Scale Theorem The Fourier ransform F { f ( a)} = F( ω/ a) / a of a scaled funcion, f(a): Proof: F F { f ( a)} = f ( a) exp( iω ) d Assuming a >, change variables: u = a { f ( a)} = f ( u)exp( iω [ u/ a]) du / a = = f ( u)exp( i [ ω/ a] u) du / a F( ω/ a)/ a If a <, he limis flip when we change variables, inroducing a minus sign, hence he absolue value.

f() F(ω) The Scale Theorem in acion Shor pulse ω The shorer he pulse, he broader he specrum! Mediumlengh pulse ω This is he essence of he Uncerainy Principle! Long pulse ω

The Fourier Transform of a sum of wo funcions F { af() + bg ()} = af { f( )} + bf { g( )} f() g() F(ω) G(ω) ω ω Also, consans facor ou. f()+g() F(ω) + G(ω) ω

Shif Theorem The Fourier ransform of a shifed funcion, f ( a): F f( a) = exp( ω i a) F( ω) { } Proof : F ( ) { } f a = f( a)exp( iω) d Change variables : u = a f( u)exp( iω[ u+ a]) du = exp( iωa) f( u)exp( iωu) du = exp( ω i a) F( ω)

Fourier Transform wih respec o space If f(x) is a funcion of posiion, F( k) = f( x) exp( ikx) dx x Y {f(x)} = F(k) We refer o k as he spaial frequency. k Everyhing we ve said abou Fourier ransforms beween he and ω domains also applies o he x and k domains.

The 2D Fourier Transform Y (2) {f(x,y)} = F(k x,k y ) f(x,y) = f(x,y) exp[-i(k x x+k y y)] dx dy x y If f(x,y) = f x (x) f y (y), Y (2) {f(x,y)} hen he 2D FT splis ino wo 1D FT's. Bu his doesn always happen.

The Pulse Widh There are many definiions of he "widh" or lengh of a wave or pulse. Δ The effecive widh is he widh of a recangle whose heigh and area are he same as hose of he pulse. Effecive widh Area / heigh: f() 1 Δeff f () d f () (Abs value is unnecessary for inensiy.) Δ eff Advanage: I s easy o undersand. Disadvanages: The Abs value is inconvenien. We mus inegrae o ±.

The rms pulse widh The roo-mean-squared widh or rms widh: Δ 2 f() d Δrms f() d 1/2 The rms widh is he second-order momen. Advanages: Inegrals are ofen easy o do analyically. Disadvanages: I weighs wings even more heavily, so i s difficul o use for experimens, which can' scan o ± )

The Full-Widh- Half-Maximum Full-widh-half-maximum is he disance beween he half-maximum poins. 1.5 Δ FWHM Advanages: Experimenally easy. Disadvanages: I ignores saellie pulses wih heighs < 49.99% of he peak! Δ FWHM Also: we can define hese widhs in erms of f() or of is inensiy, f() 2. Define specral widhs (Δω) similarly in he frequency domain ( ω).

The Uncerainy Principle The Uncerainy Principle says ha he produc of a funcion's widhs in he ime domain (Δ) and he frequency domain (Δω) has a minimum. Define he widhs assuming f() and F(ω) peak a : 1 1 Δ ( ) ( ) f() f d Δω F() F ω d ω 1 1 F() Δ f() d f()exp( i[]) d f () = f() = f() 1 1 2 f () Δω F( ) d F( )exp( i d F() ω ω = ω ω ω F() []) = F() Combining resuls: (Differen definiions of he widhs and he Fourier Transform yield differen consans.) f() F() Δω Δ 2 or: Δω Δ 2 Δν Δ 1 F() f()

The Uncerainy Principle For he rms widh, Δω Δ ½ There s an uncerainy relaion for x and k: Δk Δx ½