2 int T. is the Fourier transform of f(t) which is the inverse Fourier transform of f. i t e

Transcription

1 PHYS67 Class 3 ourier Transforms In he limi T, he ourier series becomes an inegral ( nt f in T ce f n f f e d, has been replaced by ) where i f e d is he ourier ransform of f() which is he inverse ourier ransform of We can use o indicae he T of a funcion ie f ( f ) ourier ransform of a Dirac dela funcion Consider f δ ( a) This has T a i e δ ( a) e d, which illusraes he general principle ha a narrow signal has a broad specrum Also, from he inverse T a e ( a) δ ( a) e d e d, which gives a useful represenaion of a dela funcion Properies of he ourier ransform ) ourier ransforms of derivaives: f e d f e d Hence n

2 ( f ), ( f ) ( n) n ( f ) ) To derive he ourier ransform properies of inegrals, consider g f s ds Noe ha he indefinie inegral involves a consan of inegraion, which ges los on differeniaion We have f g g g To include he consan of inegraion we need o add a erm cδ ( ), ie f ( s) ds + cδ To see why consider s s s e f ( s) e d g e dds d a a a e e a d e d e d Hence he erm will give g() only up o an addiive consan 3) Translaion ia f + a e 4) Exponenial muliplicaion α e f ( + iα) Parseval s Theorem Examples ) ind he T of f ( ) exp f d f d,

3 i i i i i e e d e e d + e e d e e d + e e d ( ) ( ) ( ) ( ) e e e d + e d + ( ) ( ) ) By applying ourier s inversion heorem show ha cos exp ( ) d + Using he resul from he previous example, cos + isin f e d e d d + + cos d (because he odd par of he inegrand gives zero conribuion o inegral) + cos d + cos Hence exp ( ) d + 3) Use his funcion in he firs example o demonsrae he validiy of Parseval s heorem We have Also ( ) exp f d exp d exp d 4 4 d d θdθ cos θdθ sec ( ) ( an θ) ( cos θ) dθ θ sin θ + + Since he inegrals are he same, Parseval s heorem is valid 3

4 The Heaviside funcion This is defined by Hence, his is a sep funcion: if > a H( a) if < a a Noe his is relaed o he Dirac dela funcion by δ a H a Convoluion and deconvoluion The convoluion of wo funcions is defined by ( ) h z f x g z x dx f g g f Convoluions occur in he heory of how urbulen and hermal moions broaden specral lines (g is Gaussian), and in he use of poin spread funcions which describe he response of an imaging sysem o a poin source Convoluion heorem This relaes he T of he convoluion of wo funcions o heir individual Ts: h ( k) f ( k) g ( k) This is used for deconvoluion or example, if h is he observed disribuion and g is he resoluion funcion, f he rue disribuion is obained from h ( k) ( k) g k Example A linear amplifier produces an oupu ha is he convoluion of he inpu and he response funcion The ourier ransform of he response funcion is i K ( α + ) Deermine he ime variaion of he oupu when is inpu is he Heavyside sep funcion The oupu is 4

5 rom he convoluion heorem Consider H We have ( τ) ( τ) τ h H K d h H K i i e H H e d e d + somehing undeermined To remove his difficuly, consider he Heaviside funcion as a limi τ lim e if > H( τ ) if < We hen have + i e τ H τ lim e e d lim τ τ + τ Going back o he convoluion heorem, we ge i h α + α + Hence Now Hence h e e α α ( e ) α ( + i ) d α + e d α + Differeniaion wih respec o α gives α e e α inally, we ge Noe ( + i ) h e α d 5

6 α d e i i d e e d d K e d d d ( α ) ( α ) + + α d e α K e ( α) d Correlaion funcions Correlaion funcions are useful for analysis of order/disorder, eg for analysis of differences beween crysalline solids, amorphous solids, liquids and gases, and in sudies of urbulence They allow deerminaion of characerisic lengh scales, ime scales, and preferred direcions Cross-correlaion of wo funcions The cross-correlaion of wo funcions is defined by Noe ha ( + ) C z f x g z x dx f g [ g f ]( z) [ f g] ( z) The ourier ransform of he cross-correlaion is relaed o he ourier ransforms of he wo funcions by Auocorrelaion When f g, we have he auocorrelaion C k f k g k ( + ) a z f x f z x dx f f Applying he ourier ransform relaion o he auocorrelaion, we ge he Wiener- Khinchin heorem ikz a ( z) ( k ) ( k ) e dk Hence, he auocorrelaion is he inverse ourier ransform of he energy specrum of f, Φ k k k We can use his o prove Parseval s heorem Se z in he auocorrelaion so ha a f x f x dx f k f k dk 6

7 Example Prove ha he cross-correlaion of he Gaussian and Lorenzian disribuions a f exp, g, τ τ + a has as is ourier ransform he funcion exp τ exp ( a ) The ourier ransform of f() is exp d τ τ exp ( τ ) τ τ τ + i d s exp τ exp ds exp τ The ourier ransform of g() is a g exp ( ) d + a Earlier, we found ha iσ e e dσ + σ By inerchanging and σ, his becomes e σ iσ e + d Making he subsiuion sa, and leing σ a, we find Hence, we see ha s ds a a e e a + s a g e Applying he Wiener-Khinchin heorem, we find ha he ourier ransform of he crosscorrelaion of f() and g() is C f g exp τ a 7