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This presentation describes the Fourier Transform used in different mathematical and physical applications. The presentation is at an Undergraduate in Science (math, physics, engineering) level. Please send comments and suggestions to improvements to solo.hermelin@gmail.com. More presentations can be found at my website http://www.solohermelin.com.

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  • 1. Fourier TransformSOLO HERMELINUpdated: 22.07.07 Run This

2. Fourier Transform+( ) { ( )} ( ) ( )F w = F f t := f t exp - jw t dt-SOLOJean Baptiste JosephFourier1830 - 1768F () is known as Fourier Integral or Fourier Transformand is in general complexF (w ) = Re F (w ) + j ImF (w ) = A (w ) exp [ j f (w )]Using the identities( j t) d d (t)w w = p+exp- 2- = 1 - + + we can find the Inverse Fourier Transform f (t) = F -1 {F (w )}F ( ) ( j t ) d f ( t ) ( j w t ) d t ( j w t )dw1= -w w w +t w t w( ) ( ( )) ( ) ( ) [ ( 0) ( 0)]22exp2exp exp2exp= - = - = - + + -+-+-+-+-+-f j t d d t f t d t t d tf t f tpppThis is true if (sufficient not necessary)1 f (t) and f (t) are piecewise continue in every finite interval2 and ( ) converge, i.e. f (t) is absolute integrable in (-,)f t F -1 F : F exp j t d+w w w w( ) { ( )} ( ) ( )-= =p2( ) ( ) [ ( 0) ( 0)]2+-f t d t t dt f t f t+-f t dtIf f (t) is continuous at t, i.e. f (t-0) = f (t+0) 3. SOLO Fourier TransformProperties of Fourier Transformf (t)F F (w ) F -11 Linearity+{a ( ) a ( )} [a ( ) a ( )] ( w ) a (w ) a (w ) 1 1 2 2 1 1 2 2 1 1 2 2 f t + f t := f t + f t exp - j t dt = F + F-F2 SymmetryF 2p f (-w )F (t) F -1+f (t) F ( w ) ( j w t) d wtf ( ) F (t) ( j t) dt f ( ) F (t) ( j t) dt {F (t)}= = - = - = F-+-+ -p w wpw wpw2 exp2exp2exp:Proof3 Conjugate Functionsf * (t)F ( ) F * -wF -1:Prooff (t) F ( w ) ( j w t) d wF d w* * * ( w ) ( j w t) { f * (t)}= exp - = - = F -12exp2+-- +pp- w w 4. Fourier TransformF F (w ) Properties of Fourier Transform4 ScalingF 1 w +a t t w w t w t:Prooft 1 F : exp exp= - = - d = ( ) { } ( ) ( ) ( ) -+ =Faa aa- f a t f a t j t dt f jF F (-w)dn f (t)+Corollary: for a = -1 ( ) f - tF d(w)nF ( jw) F (w) n( ) { ( )} ( ) ( ) (w ) ( ) ( w ) ( w ) {( w ) (w )}F f t f t j t dt d n nw F : exp w F f t j exp j t dt Fj Fwdnn= = - = - - = --+-SOLOf (t)F -15 Derivativesf (a t) F -1aFa:ProofF -1( j t) f (t) n -F -1wFdnd tnF -1+( ) { ( )} ( ) ( ) ( ) ( ) ( ) ( ) {( w ) (w )}f t F F j t d d n nw w w w f t F w j w j w t d wj Fppd tnn= F -1 = = = F -1-+exp22- exp 5. SOLO Fourier TransformProperties of Fourier Transformf (t)F F (w ) F -16 ConvolutionF (w) (w) 1 2 ( ) ( ) ( ) ( ) F * Ff t f t = f t f t -t dt 1 2 1 2 * :F (w) (w) 1 2 F F:Proof( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ))(t ) ( w t ) ( ) ( w ) t (t ) ( w t ) t (w ) (w ) (w ) f f t d j t f f t d dt f j f t j t dt dt t t w t t t t w t t w t tt= - -: exp exp exp1 2 1 2 1 2+ - = f j f u j u du d f j d F F F= - -exp exp exp1 2 1 2 1 2t u= - - - - = - - + - + - - =+- +- +- +- +- Ff (t) f (t) 1 2 F -1+-F -1http://mathworld.wolfram.com/Convolution.htmlThe animations above graphically illustrate the convolution of two rectangle functions (left) and twoGaussians (right). In the plots, the green curve shows the convolution of the blue and red curves as afunction of t, the position indicated by the vertical green line.The gray region indicates the product as a function of g () f (t-) , so its area as a function of t isprecisely the convolution. 6. SOLO Fourier TransformF Properties of Fourier Transformf (t)F (w ) F -1+7 Parsevals Formula 1 ( ) ( ) ( ) ( )f t f t dt F F 2 d-+-= w w wp*2 1*1 2:Proof1 ( ) ( ) ( ) ( ) ( ) ( )f t f t dt F F d F F d 1 2 1 2 1 2 2+( ) ( ) ( )F w = f t exp - jw t dt 1 1-++f t F j t dw w w= -f t f t dt f t F j t d dt F f t j t dt d F F dw w ww w ww w w ( ) ( ) ( ) ( ) exp ( ) ( ) ( ) ( ) ( ) ( )2-+-+-+-+-+-= - = - =ppp2 2exp2*2 1 1*2*2 1*1++F w = f t exp - jw t dt F -w = f t exp jw t dt 1 1 1 1f t f t dt f t F j t d dt F f t j t dt d F F dw w ww w ww w w ( ) ( ) ( ) ( ) exp ( ) ( ) ( ) ( ) ( ) ( )1 2 1 2 2 1 1 2-+-+-+-+-+-= = = -ppp2 2exp2( ) ( ) ( )-p2exp *2*2( ) ( ) ( ) ( ) ( ) ( )-+-+-+-+-= - = -w w wpw w wp12 7. SOLO Signal Duration and BandwidthF Relationships from Parsevals Formulaf (t)F (w ) F -1+1 7 Parsevals Formula ( ) ( ) ( ) ( )f t f t dt F F 2 d-+-= w w wp*2 1*1 2Choose f (t ) = f (t) = (- j t)m s (t) 1 2( ) 1 m( ) 20,1,2,t s t dt d S mw 2 m 2 = w=2+-+-d ndwp( j t ) f (t ) n -F (w)F -1dwFdnn a and use 5Choose ( ) ( ) ( )nnd tdnb f t = f t = d s t f (t )1 2 and use 5 d tnF ( jw) F (w) nF -1( ) +( ) 0,1,2,d s t m1 2 222 = =-+-dt S d nd tnnw w wpc Choose+( ) ( ) ( ) ( ) ( ) ( ) 0,1,2,, 0,1,2,j t s t d s t dt j S d Smw w w* = = 2 = -+-d n mdd tmnnnnm m wwp( ) ( )nnd tf t = d s t 1 f (t) = (- j t)m s (t) 2 8. SOLO Fourier TransformF Properties of Fourier Transformf (t)F (w ) F -18 Shifting: for any a real+- = - - = - + = -F : exp exp expF { f ( t) exp ( j a t)} : = f ( t) exp ( j a t) exp (- j w t) dt = f ( t) exp (- j ( w - a) t) dt = F ( w- a)9 Modulation:ProofF F (w) exp (- j aw ) f ( t) exp ( j a t)Ftt af (t) t 0 cosw F -1F F (w - a)1 F w +w + F w -w[ ( ) ( )] 0 0 2:Proofcosw = 1 w + - wt [ ( j t) ( j t)] 0 0 0 exp exp2f (t - a)F -1F -1{ f ( t a )} f ( t a ) ( j w t ) dt f (t ) ( j w (t a )) d t ( j w a ) F(w )-- =+-+-+-use shifting property with a=0 9. SOLO Fourier TransformF F (w ) ) Properties of Fourier Transform (Summary1 Linearity {a ( ) a ( )} [a ( ) a ( )] ( w ) a (w ) a (w ) 1 1 2 2 1 1 2 2 1 1 2 2 f t + f t := f t + f t exp - j t dt = F + Ff (t - a)F F (w) exp (- j aw)F -1f (t)F -1+-F2 SymmetryF ( ) F * -wF (w)F (w) (w) 1 2 ( ) ( ) ( ) ( ) F * FF F (w) F(w) 1 2 ( ) ( ) 1 ( ) ( )f t f t dt F F 2 d+1 ( ) ( ) ( ) ( ) ( ) ( )f t f t dt F F d F F d 1 2 1 2 1 2 2F (t)F 1 wddn nf (t )-F 2pf (-w)F -13 Conjugate Functions f * (t )F -14 Scaling f (a t )F -1 aFa5 Derivatives ( j t ) f (t ) n -F -1wFdnd tnF ( jw) F (w) nF -16 Convolutionf (t ) f (t ) 1 2 F -1+f t f t = f t f t -t dt 1 2 1 2 * :-F -1+-+-= w w wp*2 1*1 27 Parsevals Formula8 Shifting: for any a real f ( t ) exp ( j a t )F F (w-a)F -1F9 Modulation f (t ) t 0 cosw F -11 F w+w +F w-w[ ( ) ( )] 0 0 2+-+-= - = -w w wpw w wp12 10. SOLO Fourier TransformFourier Transform of Real or Imaginary Functions+f t F -1 F F exp j t dw w w wF ( w ) = F { f ( t )} := f ( t ) exp ( - jw t )dt ( ) { ( )} ( ) ( )-+-= =p2F -1F+( ) ( ) ( ) * - complex conjugateF * w = f * t exp jw t dt-+( ) ( ) ( )F -w = f t exp jw t dt-( )f t real { ( )}f (t) imaginaryf t =( ) ( )Im 0Re { f ( t)} =0= ( ) ( )f t f tf (t) f *(t)*= -F - =Fw w( w ) *(w )*F - = -Ff (t) real F (-w ) = F * (w )f (t) imaginary F (-w ) = -F * (w )Therefore 11. SOLO Fourier TransformFourier Transform of Real or Imaginary Functions (continue 1)f (t) real F (-w ) = F * (w )f (t) imaginary F (-w ) = -F * (w )( ) ( ) ( ) { ( )} ( ) ( )F w = Re F w + j ImF w = F f t := f t exp - jw t dtf (t) real( ) ( )( ) ( ) Re Re F Fw = -wF Fw = - -wIm Imf (t) imaginary ( ) ( )F FIm Imw = - -wF FRe Re( ) ( ) w = -w+-+{ (- )} = (- ) (- w ) = ( ) ( w ) = (-w)F f t f t exp j t dt f t exp j t dt F[ )] -f (t) := 0.5 f (t) + f ( - t= f ( - t) f (t) := 0.5 [ f (t) - f ( - t)] = - f ( -t) even even odd odd +-f (t) real( ) [ ( ) ( )] { ( )} [ ( ) ( )] ( )( ) ( ) ( ) [ ] ( ) { } ( ) ( ) [ ] ( ) f t = f t + f - t f t = F + F - =Fw w w: 0.5 F0.5 Reeven evenf t = f t - f - t f t = F - F - =j Fw w w: 0.5 F0.5 Imodd even( ) [ ( ) ( )] { ( )} [ ( ) ( )] ( )( ) ( ) ( ) [ ] ( ) { } ( ) ( ) [ ] ( ) f t = f t + f - t f t = F + F - =j Fw w w: 0.5 0.5 ImF f (t) imaginaryeven evenf t = f t - f - t f t = F - F - =Fw w w: 0.5 F0.5 Reodd even 12. Fourier TransformwRe F (w )Im F (w )Real & EvenFourier Transform of Real or Imaginary Functions (continue 2)tIm f (t)Re f (t)Real & EvenSOLOwRe F (w )Im F (w )Imaginary & OddtIm f (t)Re f (t)Real & OddwRe F (w)Im F (w)Imag. & EventIm f (t)Re f (t)Imag. &EvenwRe F (w )Im F (w )Real & OddtIm f (t)Re f (t)Imag. & Oddf (t) realF (-w ) = F * (w )f (t) f (t) [ f (t) f ( t)] even = := 0.5 + -{ f ( t )} = { f ( t )} = [ F ( ) + F ( -)]evenF F 0.5F F= = -Re Ref (t) real F (-w ) = F * (w )f (t) = f (t) := 0.5 [ f (t) - f ( -t)] odd { f ( t )} { f ( t )} [ F ( ) F ( )]even= = - -j F j FF F 0.5= = - -Im Imf (t) imaginaryw w(w) ( w)w w(w) ( w)F (-w ) = -F * (w )f (t ) f (t) [ f (t ) f ( t )] even = :=0.5 + -{ f ( t )} { f ( t )} [ F ( ) F ( )]even= = + -j F j Fw wF F 0.5(w) ( w)= = -Im Imf (t) imaginary F (-w ) = -F * (w )f (t ) f (t ) [ f (t ) f ( t )] odd = :=0.5 - -{ f ( t )} = { f ( t )} = [ F ( w ) - F ( -w)]evenF F 0.5(w) ( w)F F= = - -Re Re 13. SOLO Fourier Transformf (t) [ f (t) f ( t)] f ( t) even even := 0.5 + - = - f (t) [ f (t) f ( t)] f ( t) odd odd := 0.5 - - = - - f (t) f (t) f (t) even odd = +( ) { ( )} ( ) ( ) [ ( ) ( )] [ ( ) ( )]F w = F f t : = f t exp - j w t dt = f t + f t cos w t -j sinwt dt even odd+[ ( ) ( )] cos ( w ) [ ( ) ( )] sin( w)-+-+-+-= + - +f t f t t dt j f t f t t dteven odd even odd+ + + + ( ) ( w ) ( ) ( w ) ( ) ( w ) ( t )( w t ) t ( ) ( w ) ( ) ( w)( )--+-= + = - + =even even0 0 0 00f t cos t dt f t cos t dt f t cos t dt f cos d f t cos t dt 2 f t cos t dt even evenfteven eveneventt+ +( ) ( ) ( ) ( ) ( ) ( ) ( )cos w cos w cos w t cos ( wt ) t ( ) cos ( w) 0odd odd0 0 0( )0 = + = - + =-+--+-f t t dt f t t dt f t t dt f d f t t dt oddftodd oddoddtt+ + +( ) ( ) ( ) ( ) ( ) ( ) ( )sin w sin w sin w t sin ( wt ) t ( ) sin ( w) 0even even0 0 0( )0 = + = - - + =--+-f t t dt f t t dt f t t dt f d f t t dt evenfteven eveneventt+ + + ( ) ( w ) ( ) ( w ) ( ) ( w ) ( t )( w t ) t ( ) ( w ) ( ) ( w)( )odd odd-+--+-= + = - - + =0 0 0 00f t sin t dt f t sin t dt f t sin t dt f sin d f t sin t dt 2 f t sin t dt odd oddftodd oddoddtt+ + +Therefore ( ) { ( )} ( ) ( ) ( ) ( ) ( ) ( )F w = F f t : = f t exp - j w t dt = 2 f t cos w t dt -2 j f t sin wt dt even odd -0 0+F ( w ) = 0.5 [ F ( w ) + F ( - w )] =2 f ( t ) cos ( w t )dt ( ) [ ( ) ( )] ( ) ( )even even 0+F w = 0.5 F w - F - w =2 j f t sin wt dt odd odd 0Odd and Even Parts 14. Fourier TransformCausal FunctionsA causal functions is a equal zero for negative tSincef t -1 F F j t d F j F t j t d Fw w w w ww w w w( ) = { ( )} = ( ) ( ) = [ ( ) + ( )] [ ( ) +( )] +w w w w ww w w w w[ ( ) ( ) ( ) ( )] [ ( ) ( ) ( ) ( )]-+-+-+-= - + +pppp2