Introduction to Option Pricing with Fourier Transform: Option Pricing with Exponential Lévy Models

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1 Inroducion o Opion Pricing wih Fourier ransform: Opion Pricing wih Exponenial Lévy Models Kazuhisa Masuda Deparmen of Economics he Graduae Cener, he Ciy Universiy of New York, 365 Fifh Avenue, New York, NY hp://wwwmaxmasudacom/ December 4 i

2 Absrac his sequel is designed as an inroducion o Fourier ransform opion pricing for readers who have zero previous knowledge of Fourier ransform Firs par of his sequel is devoed for he basic undersanding of Fourier ransform and discree Fourier ransform using numerous examples and providing imporan properies Second par of his sequel applies F and DF opion pricing approach for hree exponenial Lévy models: Classic Black-Scholes model which is he only coninuous exponenial Lévy model, Meron jump-diffusion model (976) which is an exponenial Lévy model wih finie arrival rae of jumps, and variance gamma model by Madan, Carr, and Chang (998) which is an exponenial Lévy model wih infinie arrival rae of jumps Some readers may quesion ha wha he need for F opion pricing is since all hree models above can price opions wih closed form formulae he answer is ha hese hree models are special cases of more general exponenial Lévy models Opions canno be priced wih general exponenial Lévy models using he radiional approach of he use of he risk-neural densiy of he erminal sock price because i is no available herefore, Carr and Madan (999) rewrie he opion price in erms of a characerisic funcion of he log erminal sock price by he use of F he advanage of F opion pricing is is generaliy in he sense ha he only hing necessary for F opion pricing is a characerisic funcion of he log erminal sock price his generaliy of F opion pricing speeds up he calibraion and Mone Carlo simulaion wih various exponenial Lévy models I is no doub o us ha F opion pricing will be a sandard opion pricing mehod from now on 4 Kazuhisa Masuda All righs reserved JEL classificaion: G; G; G3 Keywords: Fourier ransforms; Discree Fourier ransforms; Opion pricing; Lévy processes; Random ime change; Jump diffusion model ii

3 Conens Inroducion o he Opion Pricing wih Fourier ransform Prerequisie for Fourier ransform 7 Radian 7 Wavelengh 7 3 Frequency, Angular Frequency, and Period of a Wave 7 4 Sine and Cosine 9 5 Derivaive and Inegral of Sine and Cosine Funcion 6 Series Definiion of Sine and Cosine Funcion 7 Euler s Formula 8 Sine Wave: Sinusoid 3 Fourier ransform 5 3 Definiion of Fourier ransform 5 3 Examples of Fourier ransform 7 3 Double-Sided Exponenial Funcion 7 3 Recangular Pulse 9 33 Dirac s Dela Funcion 34 Gaussian Funcion 35 Cosine Wave 4 36 Sine Wave 6 33 Properies of Fourier ransform 9 33 Dirac s Dela Funcion 9 33 Useful Ideniy: Dirac s Dela Funcion Lineariy of Fourier ransform F of Even and Odd Funcions Symmery of Fourier ransform Differeniaion of Fourier ransform ime Scaling of Fourier ransform ime Shifing of Fourier ransform Convoluion: ime Convoluion heorem 4 33 Frequency-Convoluion heorem 4 33 Frequency Shifing: Modulaion Parseval s Relaion Summary of Imporan Properies of Fourier ransform Exisence of he Fourier Inegral 48 iii

4 4 Characerisic Funcion 49 4 Definiion of a Characerisic Funcion 49 4 Properies of a Characerisic Funcion 5 43 Characerisic Exponen: Cumulan-Generaing Funcion 5 44 Laplace ransform 5 45 Relaionship wih Momen Generaing Funcion 5 46 Summary: How o Calculae Sandardized Momens from Characerisic Funcion and Momen Generaing funcion Examples of Characerisic Funcions 56 5 Discree Fourier ransform 57 5 Inuiive Derivaion of DF: Approximaion of F 57 5 Definiion of DF 59 5 Physiciss Definiion of DF 59 5 Signal Processing Definiion of DF Requiremen of DF 6 54 Sampling heorem and Nyquis Rule: How o Deermine he ime Domain Sampling Inerval 6 53 Examples of DF Sine Wave Double-Sided Exponenial Recangular Pulse Gaussian Funcion Cosine Wave Properies of DF Lineariy of DF DF of Even and Odd Funcions Symmery of DF ime Shifing of DF Frequency Shifing: Modulaion Discree Convoluion: ime Convoluion heorem Discree Frequency-Convoluion heorem Parseval s Relaion Summary of DF Properies 83 6 Lévy Processes 84 6 Definiion of Lévy Process 84 6 Sandard Brownian Moion Process: he Only Coninuous Lévy Process Generaed by A Normal Disribuion Poisson Process and Compound Poisson Process: Finie Aciviy Lévy Processes Generaed by Poisson Disribuion Lévy-Iô Decomposiion and Infinie Aciviy Lévy Process 9 65 Lévy-Khinchin Represenaion 9 iv

5 66 Sable Processes 9 7 Black-Scholes Model as an Exponenial Lévy Model 95 7 Sandard Brownian Moion Process: A Lévy Process Generaed by a Normal Disribuion 95 7 Black-Scholes Disribuional Assumpions on a Sock Price radiional Black-Scholes Opion Pricing: PDE Approach by Hedging radiional Black-Scholes Opion Pricing: Maringale Pricing Approach 75 Alernaive Inerpreaion of Black-Scholes Formula: A Single Inegraion Problem 5 76 Black-Scholes Model as an Exponenial Lévy Model 7 8 Opion Pricing wih Fourier ransform: Black-Scholes Example 9 8 Moivaion 9 8 Derivaion of Call Price wih Fourier ransform: Carr and Madan (999) 83 How o Choose Decay Rae Parameer α: Carr and Madan (999) 3 84 Black-Scholes Model wih Fourier ransform Pricing Mehod 5 85 Derivaion of Near-Mauriy OM Vanilla Opion Pricing Funcion wih Fourier ransform: Carr and Madan (999) 9 86 Derivaion of Call Pricing Funcion wih Discree Fourier ransform (DF): Carr and Madan (999) 6 87 Implemenaion and Performance of DF Pricing Mehod wih Black-Scholes Model 3 88 Summary of Formulae of Opion Price wih Fourier ransform 37 9 Meron (976) Jump-Diffusion Model 39 9 Model ype 39 9 Model Derivaion 4 93 Log Sock Price Process for Meron Jump-Diffusion Model Lévy Measure for Meron Jump-Diffusion Model 5 95 Opion (Coningen Claim) Pricing: PDE Approach by Hedging 5 96 Opion (Coningen Claim) Pricing: Maringale Approach Opion Pricing Example of Meron Jump-Diffusion Model 58 Meron (976) Jump-Diffusion Model wih Fourier ransform Pricing 6 Meron JD Model wih Fourier ransform Pricing Mehod 6 Discree Fourier ransform (DF) Call Pricing Formula wih Meron Jump-Diffusion Model 67 3 Implemenaion and Performance of DF Pricing Mehod v

6 wih Meron Jump-Diffusion Model 67 4 Summary of Formulae of Opion Price wih Fourier ransform in Meron Jump-Diffusion Model 7 Variance Gamma (VG) Model by Madan, Carr, and Chang (998) 7 Model ype 7 Model Derivaion 73 Subordinaion heorem of Lévy Processes 73 empered α-sable Subordinaor: General Case 74 3 Gamma Subordinaor (Process): Special Case of empered α-sable Subordinaor When α = 76 4 Re-Parameerizing empered α-sable Subordinaor Using Is Scale-Invariance (Self-Similariy) Propery 77 5 Gamma Subordinaor (Process): Special Case of empered α-sable Subordinaor wih Uni Mean Rae When α = 79 6 Subordinaed Brownian Moion Process wih empered α-sable Subordinaor wih Uni Mean Rae: Normal empered α-sable Process 8 7 Variance Gamma (VG) Process: Subordinaed Brownian Moion Process wih empered -Sable Subordinaor wih Uni Mean Rae: Normal empered -Sable Process 8 3 Lévy Measure for Variance Gamma (VG) Process 87 4 Modeling Sock Price Dynamics wih VG Process and VG Log-Reurn Densiy 89 5 Opion Pricing wih VG Model 93 6 Opion Pricing Example of VG Model 95 7 Lévy Measure of VG Log-Reurn z VG (Variance Gamma) Model wih Fourier ransform Pricing VG Model wih Fourier ransform Pricing Mehod Discree Fourier ransform (DF) Call Pricing Formula wih VG Model 3 Implemenaion and Performance of DF Pricing Mehod wih VG Model 4 Summary of Formulae of Opion Price wih Fourier ransform in VG Model 5 3 Conclusion 6 Appendix 7 A Se heory: Noaion and Basics 7 A Measure 8 A Null Ses 8 vi

7 A Ouer Measure 9 A3 Lebesgue Measurable Ses and Lebesgue Measure A4 σ -field A5 Borel σ-field A6 Probabiliy A3 Sochasic Process 4 A3 Filraion (Informaion Flow) 4 A3 Nonanicipaing (Adaped) Process 4 A4 Maringales 4 A4 General Concep 4 A4 Maringale Asse Pricing 6 A43 Coninuous Maringales, Righ Coninuous Maringales, Square-Inegrable Maringales 6 A5 Poisson Process 7 A5 Exponenial Disribuion 7 A5 Poisson Disribuion 8 A53 Compensaed Poisson Process 9 A6 Oher Disribuions Used 3 A6 Gamma Funcion 3 A6 Incomplee Gamma Funcion 3 A63 Gamma Disribuion 3 A7 Modified Bessel Funcions: Modified Bessel Funcion of he Firs Kind Iv ( z ) and Second Kind K ( v z ) 3 A8 Iô Formula 34 A8 Iô Formula for Brownian Moion 34 A8 Wimo s (998) Rule of humb of Iô Formula for Brownian Moion 36 A83 Iô Formula for Brownian Moion wih Drif 36 A84 Iô Formula for Brownian Moion wih Drif in Higher Dimensions 37 A85 Iô Formula for Jump-Diffusion (Finie Aciviy Lévy) Processes 37 A86 Iô Formula for General (Finie and Infinie Aciviy) Scalar Lévy Processes 38 Bibliography 4 vii

8 [] Inroducion Many of he opion pricing models assume ha a sock price process { S; } follows an exponenial (geomeric) Lévy process: S = S, e L where { L ; } is a Lévy process he reason of he populariy of exponenial (geomeric) Lévy models is is mahemaical racabiliy which comes from he independen and saionary incremens of Lévy processes Classic Black-Scholes (BS) model chooses a Brownian moion wih drif process which is he only coninuous Lévy process as heir choice of a (risk-neural) Lévy process: = + L r σ σ B where { B ; } is a sandard Brownian moion process his specificaion leads o a normally disribued condiional risk-neural log reurn densiy: ln ( S / S ) r σ Q( ln ( S / S) F ) = exp πσ σ BS call price can be simply calculaed as he discouned value of he expeced erminal payoff under risk-neural measure : Q ( ) ( ), r CS (, ) = e S KQ S F ds, () ln S ln S + ( r σ ) where Q( S F ) = exp which is a S πσ σ lognormal densiy K Bu even before BS model was developed, researchers knew ha he empirical log reurn densiy is no normal, i shows excess kurosis and skewness hus, all he opion pricing models afer BS (so called beyond BS) ry o capure excess kurosis and negaive skewness of he risk-neural log reurn densiy by he use of differen echniques

9 his sequel deals wih Meron jump-diffusion model (we call i Meron JD model) and variance gamma model by Madan, Carr, and Chang (998) (we call i VG model) hese are boh exponenial Lévy models of differen ypes Meron s choice of Lévy process is a Brownian moion wih drif process plus a compound Poisson jump process which has a coninuous pah wih occasional jumps: N σ L = ( r λk) + σb + Yi i= Meron JD model can be caegorized as a finie aciviy exponenial Lévy model because he expeced number of jumps per uni of ime (ie inensiy λ ) is finie and small In oher words, he Lévy measure ( dx) of Meron JD model is finie: ( dx ) < he only bu imporan difference beween he BS and he Meron JD model is he addiion of a compound Poisson jump process N i= i Y Meron inroduces hree exra parameers λ (inensiy of he Poisson process), µ (mean log sock price jump size), and δ (sandard deviaion of log sock price jump size) o he original BS framework and conrols he (negaive) skewness and excess kurosis of he log reurn densiy Choice of Lévy process by Madan, Carr, and Chang (998) is a VG process plus a drif: L r+ + VG x κ ) σκ ln θκ ;,, ( θ σ κ) A VG process VG ( x ; θ, σκ, is defined as a sochasic process { X ; } creaed by random ime changing (ie subordinaing) a Brownian moion wih drif process θ+ σ B by a empered -sable subordinaor (ie a gamma subordinaor) { S; } wih uni mean rae: X θ ( S ) + σ B S A VG process VG ( x ; θ, σκ, ) is characerized as a pure jump Lévy process wih infinie arrival rae of jumps In oher words, he Lévy measure of a VG process has an infinie inegral: ( xdx ) = his means ha a VG process has infiniely many small jumps bu a finie number of large jumps VG model inroduces wo exra parameers: One is variance rae parameer

10 κ which conrols he degree of he randomness of he subordinaion and he larger κ implies he faer ails of he log reurn densiy he oher is he drif parameer of he subordinaed Brownian moion process θ which capures he skewness of he log reurn densiy Coninuous Exponenial Lévy models: No Jumps Example: BS Model Exponenial Lévy Models Finie Aciviy Exponenial Lévy Models: Coninuous wih Occasional Disconinuous Pahs Example: Meron JD Model Infinie Aciviy Exponenial Lévy Models: Pure Jump Process Example: VG Model radiionally, boh Meron JD call price and VG call price have been expressed as BS ype closed form funcion using he condiional normaliy of boh models Meron JD call price can be expressed as he weighed average of he BS call price condiional on ha he underlying sock price jumps i imes o he expiry wih weighs being he probabiliy ha he underlying jumps i imes o he expiry Because of he subordinaion srucure X θ ( S ) + σ B of he VG process{ X ; }, he probabiliy densiy of VG S process can be expressed as he condiionally normal by condiioning on he fac ha he realized value of he random ime change by a gamma subordinaor wih uni mean rae is S g: = ( x θ g) VG ( x = S = g ) = exp πσ g σ g Using his condiional normaliy of VG process, he condiional call price can be obained as a BS ype formula afer lenghy and edious process of numerous changes of variables he fac ha a call price can be expressed as a BS ype formula implies ha he model Q ln( S / S ) F Meron JD has a closed form expression for he log reurn densiy ( ) model has a log reurn densiy of he form: λ i e ( λ) σ QMeron ( ln( S / S) F ) = N ln( S / S);( r λ k) iµσ, iδ i= + + i!, S 3

11 ;, exp πb log reurn densiy of he form: where N( x a b) Q where ( x a) b which is a normal densiy VG model has a =, Γ( / ) θ + κ θ κ 4 σ exp x x + θ σ x κ VG ( ln( S / S) F ) K / κ κ σ π κ σ σ κ σκ x ln( S / S) r+ ln θκ he exisence of he closed form κ expression for he log reurn densiy ( ln( S / S) ) () o calculae a call price Q F enables he use of he equaion Bu Meron JD model and VG model are special cases of exponenial Lévy models in he sense ha more general exponenial Lévy models do no have closed form log reurn Q ln( S / S ) F or hey canno be expressed using special funcions of densiies ( ) mahemaics herefore, we canno price plain vanilla opions using he equaion () How do we price opions using general exponenial Lévy models? he answer is o use a very ineresing fac ha characerisic funcions (CF) of general exponenial Lévy processes are always known in closed-forms or can be expressed in erms of special funcions of mahemaics alhough heir probabiliy densiies are no here is one-o-one relaionship beween a probabiliy densiy and a CF (ie CF is jus a Fourier ransform of a probabiliy densiy wih F parameers (,) ) and boh of which uniquely deermine a probabiliy disribuion If we can somehow rewrie () in erms of a characerisic funcion of he condiional erminal sock price S F (ie log of S F o be more precise) insead of is probabiliy densiyq( S F ), we will be able o price opions in general exponenial Lévy models he purpose of his sequel is o inroduce he basics of Fourier ransform opion pricing approach developed by Carr and Madan (999) o he readers who have no previous knowledge of Fourier ransforms Carr and Madan (999) s conribuion was rewriing he equaion () in erms of a CF of he condiional log erminal sock price φ ln S F : ( ) ( ( + ) ) i ( ) αk r e iωk e φ ω α i C (, k) = e dω π () α + α ω + α + ω 4

12 And approximae he equaion () using DF (ie simply by aking a sample of size N ): ( ) ( ) exp( αkn )exp iπn exp iπn / C( kn) k N wj{ exp ( iπ j) ψ( ωj) } exp( i π jn/ N ), (3) N j= which improves significan amoun of compuaional ime his simple operaion is criically imporan in he pricing wih general exponenial Lévy models Why? Because wihou Fourier ransform pricing approach, we canno price opions in general exponenial Lévy models or we have o spend remendous amoun of energy jus o come up wih closed form soluion like he VG model he excellence of F opion pricing is is simpliciy and generaliy and F pricing works as long as a CF of he condiional log erminal sock price S F is obained in closed form he srucure of his sequel is as follows Chaper provides readers wih minimal necessary knowledge before learning Fourier ransform Chaper 3 defines F and hen, numerous examples are presened in order o inuiively undersand he meaning of F We also presen he imporan properies of F hese are no necessary for he beginners, bu we believe hese will help readers as hey proceed o more advanced inegral ransform pricing mehods In Chaper 4, a characerisic funcion is defined and is properies are discussed We show, using several examples, how o obain momens by using a CF (or characerisic exponen) Momen generaing funcion is also deal Chaper 5 gives an inroducion o he discree Fourier ransform which is jus an approximaion of F his approximaion is done by sampling a finie number of poins N of a coninuous ime domain funcion g () wih ime domain sampling inerval (seconds) and sampling a finie number of poins N of a coninuous F G ( ω) wih angular frequency sampling inerval ω Hz In oher words, boh he original coninuous ime domain funcion g () and he original coninuous F G ( ω) are approximaed by a sample of N poins he use of DF improves he compuaion ime by a remendous amoun Following Chaper 3, he examples of DF and is properies are examined In Chaper 6, a Lévy process is defined and is properies are discussed We frequenly use Lévy-Khinchin represenaion o obain a CF of a Lévy process Chaper 7 revisis he Black-Scholes model as an exponenial Lévy model and is basic properies are reviewed Chaper 8 gives Carr and Madan (999) s general F call price and our version of general DF call price hese general F and DF call prices are applied in he BS framework which shows ha he original BS, BS-F, and BS-DF call prices are idenical as expeced Chaper 9 illusraes he derivaion of Meron JD model, facors which deermine he skewness and he excess kurosis of he log reurn densiy, Lévy measure of jump-diffusion process, and he radiional (ie PDE and maringale approach) opion pricing wih Meron JD model Chaper applies he general F and DF call price o he Meron JD model his is simply done by subsiuing he CF of Meron JD log erminal sock price Chaper presens he derivaion of VG model, facors which 5

13 deermine he skewness and he excess kurosis of he log reurn densiy, Lévy measure of VG process, and is closed form and numerical call price Chaper is an applicaion of he general F and DF call price o he VG model which is simply done by subsiuing he CF of VG log erminal sock price Chaper 3 gives concluding remarks 6

14 [] Prerequisie for Fourier ransform In his secion we presen prerequisie knowledge before moving o Fourier ransform [] Radian Radian is he uni of angle A complee circle has π = 6839 radians which is equal o 36 his in urn means ha one radian is equal o: radian 36 = = π [] Wavelengh Wavelengh λ of a waveform is defined as a disance ( d ) beween peaks or roughs In oher words, wavelengh is he disance a which a waveform complees one cycle: disance λ () cycle Le v be he speed (disance/second), and f soon) of a waveform hese are relaed by: be he frequency (cycles/second, explained λ v(disance/second) (disance / cycle) f (cycles/second) () Displacemen Disance Figure : Illusraion of Wavelengh λ [3] Frequency, Angular Frequency, and Period of a Waveform Period of oscillaion of a wave is he seconds (ime) aken for a waveform o complee one wavelengh: 7

15 seconds (3) wavelengh (cycle) Period is by definiion a reciprocal of a frequency Le hen: f be he frequency of a wave (seconds/cycle) f (cycles/second) (4) Frequency f of a wave measures he number of imes for a wave o complee one wavelengh (cycle) per second: number of wavelenghs (cycles) f (5) second By definiion, f is calculaed as a reciprocal of he period of a wave: f (cycles/second) (seconds/cycle) (6) Frequency f is measured in Herz (Hz) Hz wave is a wave which complees one wavelengh (cycle) per second he frequency of he AC (alernaing curren) in US is 6 Hz Human beings can hear frequencies from abou o, Hz (called range of hearing) Alernaively, frequency can be calculaed using he speed v and wavelengh λ of a wave as: v(disance/second) f (cycles/second) (7) λ(disance/cycle) Angular frequency (also called angular speed or radian frequency) ω is a measure of roaion rae (ie he speed a which an objec roaes) he uni of measuremen for ω is radians per second Since one cycle equals π radians, angular frequency ω is calculaed as: π (radians) ω(radians/second) = = π f (cycles/second) (8) (seconds/cycle) 8

16 Figure : Definiion of Angular Frequency Consider a sine wave g () = sin( π (5) ) which is illusraed in Figure 3 for he ime beween and seconds his sine wave has frequency f = 5 Hz (5 cycles per second) and angular frequency ω = π Hz (π radians per second) Is period is f = 5 = (seconds/cycle) ghl= s i n Hπ 5 L ime Hseconds L ( π ) Figure 3: Plo of 5 Hz Sine Wave g () = sin (5) [4] Sine and Cosine Le θ be an angle which is measured counerclockwise from he x -axis along an arc of a uni circle Sine funcion ( sinθ ) is defines as a verical coordinae of he arc endpoin Cosine funcion ( cosθ ) is defined as a horizonal coordinae of he arc endpoin Sine and cosine funcions sinθ and cosθ are periodic funcions wih period π as illusraed in Figure 5: ( ) sinθ = sin θ + πh, 9

17 ( ) cosθ = cos θ + πh, where h is any ineger Figure 4: he Definiion of Sine and Cosine Funcion wih Uni Circle π π π π Angle θ Figure 5: Sine and Cosine Funcion sin(θ) and cos(θ) sin HθL cos HθL Following Pyhagorean heorem, we have he ideniy: sinθ cosθ [5] Derivaive and Inegral of Sine and Cosine Funcion + = (9) Le sin( x ) and cos( x ) be sine and cosine funcions for x R he derivaive of sin( x ) can be expressed as: he derivaive of cos( x ) can be expressed as: dsin( x) = cos( x) () dx

18 he inegral of sin( x ) can be expressed as: dcos( x) = sin( x) () dx sin( x) dx = cos( x) () Refer o any graduae school level rigonomery exbook for proofs [6] Series Definiion of Sine Funcion and Cosine Funcion For any x R : x x x ( ) sin( x) = x + + = x 3! 5! 7! ( )! x x x ( ) cos( x)! 4! 6! ( n)! n n+, (3) n= n n n = + + = x (4) n= Refer o any graduae school level rigonomery exbook for proofs [7] Euler s Formula Euler s formula gives a very imporan relaionship beween he complex (imaginary) exponenial funcion and he rigonomeric funcions For any x R : From (5), varians of Euler s formula are derived: ix e = cos( x) + isin( x) (5) ix e = cos( x) isin( x), (6) ix ix e + e = cos( x), (7) ix ix e e = isin( x) (8) Consider sine and cosine funcions wih complex argumens z hen, Euler s formula ells: n iz iz ( ) n+ e e sin z = Im( e ) = z = (n+ )! i n= n iz iz ( ) n e + e cos z = Re( e ) = z = ( n)! n= iz iz, (9) ()

19 Refer o any graduae school level rigonomery exbook for proofs [8] Sine Wave: Sinusoid Sine wave is generally defined as a funcion of ime (seconds): a ( π ) g () = asin f+ b, () where is he ampliude, f is he fundamenal frequency (cycles/second, Hz), and b changes he phase (angular posiion) of a sinusoid In erms of a fundamenal angular frequency ω (radians/second), a sine wave is defined as (ie ω π f): ( ω ) g () = asin + b () Figure 6 illusraes he role of a fundamenal frequency f When a fundamenal frequency f doubles from ( cycle /second) o (cycles/second), is period becomes half from o / seconds as illusraed in Panel A ghl ime ( A) Hz sine wave sin () versus Hz sine wave sin π () π ) ( ) sin HπHLL sin HπHLL 5 ghl -5 - ( ) B) 3 Hz sine wave sin π (3) ime

20 Figure 6: Plo of a Sine Wave g () sin( π f) Frequency f = wih Differen Fundamenal he role of ampliude a is o increase or decrease he magniude of an oscillaion Figure 7 illusraes how magniude of an oscillaion changes for hree differen ampliudes a = /,, and In audio sudy ampliude a deermines how loud a sound is ghl ime Figure 7: Plo of a Hz Sine Wave g () asin( π ) /,, and sin HπL sin HπL sinhπl = wih Differen Ampliudes a = π π Consider hree Hz sine waves sin( π ), sin( π + ), and sin( π ) A sine wave π π sin( π ) has a phase, sin( π+ ) = cos( π) has a phase π /, and sin( π ) has a phase π / he role of a parameer b is o change he posiion of a waveform by an amoun b as illusraed in Figure 8 sin HπL ghl ime Figure 8: Plo of a Hz Sine Wave g () sin( π b) π/, and - π/ sin Hπ+ π L sin Hπ π L = + wih Differen Phase b =, 3

21 his means ha g () = asin( π f ) is a sinusoid a phase zero and g () = acos( π f ) is a sinusoid a phaseπ / For he purpose of defining a sinusoid, i really does no maer wheher sin ( ) or cos( ) is used 4

22 [3] Fourier ransform (F) [3] Definiion of Fourier ransform We consider Fourier ransform of a funcion g () from a ime domain ino an angular frequency domain ω (radians/second) his follows he convenion in physics In he field of signal processing which is a major applicaion of F, frequency f (cycles /second) is used insead of ω Bu his difference is no imporan because ω and f are measuring he same hing (roaion speed) in differen unis and relaed by (8): ω = π f (3) able 3 gives a clear-cu relaionship beween frequency f and angular frequencyω able 3: Relaionship beween Frequency f and Angular Frequency ω Frequency f (cycles/second) Angular Frequency ω (radians/second) Hz π = 36 Hz Hz π = 36 Hz Hz π = 36 Hz We sar from he mos general definiion of F F from a funcion g () o a funcion G ( ω) (hus, swiching domains from o ω ) is defined using wo arbirary consans a and b called F parameers as: b ibω G( ω) F [ g ()( ] ω) e gd () a ( π ) (3) Inverse Fourier ransform from a funcion G ( ω) o a funcion g ()(hus, swiching domains from ω o ) is defined as (ie he reverse procedure of (3)): b ibω g () F [ G( ω)() ] e ( ω) dω + a ( π ) G (33) For our purpose which is o calculae characerisic funcions, F parameers are se as ( ab, ) = (,) hus, (3) and (33) become: Euler s formula (5) is for R : [ ] iω G( ω) F g ( ) ( ω) e gd ( ), (34) [ ] iω g () F G( ω)() e ( ω) dω π G (35) 5

23 hus, he F of (34) can be rewrien as: [ ] i e = cos+ isin G ( ω) F g ()( ω) cos( ωgd ) () + i sin( ωgd ) () Inuiively speaking, F is a decomposiion of a waveform g ()(ie in ime domain ) ino a sum of sinusoids (ie sine and cosine funcions) of differen frequencies (Hz) which sum o he original waveform In oher words, F enables any funcion in ime domain o be represened by an infinie number of sinusoidal funcions herefore, F is an angular frequency represenaion (ie differen look) of a funcion g () and G ( ω) conains he exac same informaion as he original funcion g () We can check if he inverse Fourier ransform (35) is rue: ωτ ( ) iωτ ( ) g( )( e d ) iω i iω e ( ω) dω e g( τ) dτ e d π G = π ω = τ ω dτ π = g( τ ) δτ ( ) dτ = g (), where we used he ideniy of Dirac s dela funcion (3) which is proven soon Alhough F parameers ( ab, ) = (,) are used for calculaing characerisic funcions, differen pairs of ( ab, ) are used for oher purposes For example, ( ab, ) = (, ) in pure mahemaics: Modern physics uses ( ab, ) = (,): [ ] iω G( ω) F g ( ) ( ω) e gd ( ), [ ] iω g () Fω G( ω)() e ( ω) dω π G iω G( ω) F [ g ( )]( ω) e gd ( ), π [ ] iω g () Fω G( ω)() e ( ω) dω π G In he field of signal processing and mos of he sandard exbooks of F, F parameers ( ab, ) = (, π ) are used (ie hey use frequency f insead of angular frequencyω ): 6

24 [ ] πif G( f ) F g ()( f) e gd (), (36) π ( ) F [ ( )]( ) if f G G( ) g f e f df (37) We use (36) and (37) frequenly because his definiion of F is mahemaically simpler o handle for he purpose of proofs In general, Fourier ransform G ( ω) is a complex quaniy: ω ω ω ω i ( ) G( ) = Re( ) + iim( ) = G ( ) e θ ω, (38) where Re( ω ) is he real par of he F G ( ω), Im( ω ) is he imaginary par of he F, G ( ω) is he ampliude of a ime domain funcion g (), and θ ( ω ) is he phase angle of he F G ( ω) G ( ω) and θ ( ω ) can be expressed in erms of Re( ω ) and Im( ω ) as: [3] Examples of Fourier ransform G( ω) = Re ( ω) + Im ( ω ), (39) Im( ω) θω ( ) = an Re( ω) (3) Before discussing imporan properies of F, we presen represenaive examples of F in his secion o ge he feeling of wha F does [3] Double-Sided Exponenial Funcion Consider a double-sided exponenial funcion wih A, α R : From (34): g () = Ae α α ( ) i i i i ( ) e ω g( ) d e ω α G ω = Ae d = A e ω e α d + e ω e d Aα G ( ω) = A + = α + iω α iω α + ω When A = andα = 3, he ime domain funcion 6 G ( ω) = is ploed in Figure ω g () 3 = e and is Fourier ransform 7

25 ghl Hseconds L A) Plo of a double-sided exponenial funcion g () 3 = e 6 5 HωL π π π 4 π Angular Frequency ω Hz Hradians êsecond L 3 B) Plo of F of g () = e in Angular Frequency Domain, G ( ω) HfL Frequency f Hz Hcycles êsecond L 3 C) Plo of F of g () = e in Frequency Domain, G( f ) Figure 3: Plo of Double-Sided Exponenial Funcion ransforms G ( ω) and G( f ) g () and Is Fourier Using he signal processing definiion of F ( ab, ) = (, π ) of he definiion (36), F of g () = Ae α is compued as (which is simply obained by subsiuing ω = π f ino G ( ω) ): 8

26 Aα G ( f ) = α + 4π f 6 When A = andα = 3, G ( f ) = which is ploed in Panel C of Figure π f [3] Recangular Pulse Consider a recangular pulse wih A, R : A - g () =, > which is an even funcion of (symmeric wih respec o ) ix ix From (34) and use Euler s formula (8) e e = isin( x) : iω iω ( e e ) G iω iω ( ω) e g( ) d = A e d A = iω, isin( ω) Asin( ω) = A = iω ω Using he signal processing definiion of F ( ab, ) = (, π ) of he definiion (36), F of a recangular pulse is compued as: G if if Asin( f ) ( f ) e π π π g( ) d A e = d = π f When A = and =, he ime domain funcion g (), Fourier ransform sin( ω) sin(4 π f ) G ( ω) = in angular frequency, and Fourier ransform G ( f ) = in ω π f frequency domain are ploed in Figure 3 9

27 ghl A) Plo of a recangular pulse g () HωL Hseconds L 4 π π π 4 π Angular Frequency ω Hz Hradians êsecond L B) Plo of he F of g () in Angular Frequency Domain, G ( ω) HfL Frequency f Hz Hcycles êsecond L C) Plo of he F of g () in Frequency Domain, G( f ) Figure 3: Plo of Recangular Pulse G( f ) g () and Is Fourier ransforms G ( ω) and [33] Dirac s Dela Funcion (Impulse Funcion) + Consider Dirac s dela funcion scaled by a R (Dirac s dela funcion is discussed in deail in secion 33):

28 g () = aδ () From (34): iω iω i e g d e aδ d ae a G( ω) ( ) = ( ) = ω = Using he signal processing definiion of F ( ab, ) = (, π ) of he definiion (36), F of a scaled Dirac s dela is compued as: if if if f π e g d = π e aδ d = π ae a G( ) ( ) ( ) When a = (ie pure Dirac s dela), he ime domain funcion g () = δ () and Fourier ransforms G ( ω ) = and G( f ) = are ploed in Figure 33 = A) Plo of g () = δ () B) Plo of F of g () in Angular Frequency Domain, G ( ω ) =

29 C) Plo of F of g () in Frequency Domain, G ( f ) = Figure 33: Plo of Dirac s Dela Funcion g () = δ () and Is Fourier ransforms G ( ω) and G( f ) [34] Gaussian Funcion + Consider a Gaussian funcion wih A R : g () = e A From (34): G iω iω A ( ω) e g( ) d = e e d Use Euler s formula (5): { ω ω } A G( ω) = e cos( ) + isin( ) d A A = e cos( ω) d+ i e sin( ω) d π A π A ω /4 A ω /4A = e + i = e Using he signal processing definiion of F ( ab, ) = (, π ) of he definiion (36), F of a Gaussian funcion is compued as: πif πif A π π f / A G( f ) e g( ) d = e e d = e A

30 When A =, he ime domain funcion g () = e and is Fourier ransforms π /8 ( ) e π G ω = ω and G f / ( f) = e π are ploed in Figure 34 Noe ha F of a Gaussian Funcion is anoher Gaussian funcion ghl A) Plo of Gaussian funcion g () = e Hseconds L HωL π π π 4 π Angular Frequency ω Hz Hradians êsecond L B) Plo of F of g () in Angular Frequency Domain, G ( ω) HfL Frequency f Hz Hcycles êsecond L C) Plo of F of g () in Frequency Domain, G( f ) Figure 34: Plo of Gaussian Funcion G( f ) g () = e and Fourier ransforms G ( ω) and 3

31 [35] Cosine Wave g ( ) = Acos( π f ) = Acos( ω) Consider a sinusoid g ( ) = Acos( π f ) = Acos( ω ) Using he signal processing definiion of F ( ab, ) = (, π ) of he definiion (36), F of a cosine wave is compued as: From Euler s formula (7): πif πif ( ) ( ) = cos( π ) G f e gd e A fd π if i ( π f i π f ) G( f ) = A e e + e d πif i π f πif i π f = A e e d e e d + { } ( ) πi f f πi ( f + f ) = A e d e + d { } Use he ideniy (3) of Dirac s dela funcion: i ( x a) ( x a) e ω δ dω π herefore, we obain: G ( f ) = A ( f f) + ( f + f) A A G ( f ) = δ( f f) + δ( f + f), { δ δ } which is wo impulse funcions a f = ± f hus, F of a cosine wave (which is an even funcion) is a real valued even funcion which means G( f ) is symmeric abou f = Nex, in erms of angular frequency ω from (34): G iω iω ( ω) e g( ) d = e Acos( ω ) d From Euler s formula (7): ix ix e + e = cos( x) herefore: 4

32 G d A iω iω A i i A i( ) A i( ) e e d ω ω e e d ω+ ω e d ω = e d + = + ω iω iω iω iω ( ω) = A e cos( ω) d = A e ( e + e ) Use he ideniy of Dirac s dela funcion (3): i ( x a) ( x a) e ω δ dω π hus: A A G ( ω) = πδ ( ω + ω) + πδ ( ω ω) G ( ω) = Aπδ ( ω + ω ) + Aπδ ( ω ω ), which is wo impulse funcions a ω = ± ω Figure 35 illusraes a cosine wave g ( ) = Acos( π f ) = Acos( ω ) and Fs G ( ω) = Aπδ ( ω + ω) + Aπδ ( ω ω) and A A G ( f ) = δ( f f) + δ( f + f) ghl Hseconds L A) Plo of a Cosine Wave g ( ) = Acos( π f ) = Acos( ω ) Ampliude of he wave is given by A 5

33 B) Plo of F of g () in Angular Frequency Domain, G ( ω) C) Plo of F of g () in Frequency Domain, G( f ) Figure 35: Plo of a Cosine Wave g ( ) = Acos( π f ) = Acos( ω ) and Fourier ransforms G ( ω) and G( f ) [36] Sine Wave g ( ) = Asin( π f ) = Asin( ω) Consider a sinusoid g ( ) = Asin( π f ) = Asin( ω ) Using he signal processing definiion of F ( ab, ) = (, π ) of he equaion (36), F of a sine wave is compued as: From Euler s formula (8): πif πif ( ) ( ) = sin( π ) G f e g d e A f d πif i ( π f i π f ) G( f ) = A e e e d i πif i π f πif i π f = A e e d e e d i { } 6

34 { } ( ) πi f f πi ( f + f ) = A e d e i Muliply = i/ i o he righ hand side: d { } ( ) πi f f πi ( f + f ) { } ( ) πi f + f πi ( f f ) d i G ( f ) = A e d e d i i = A e d e Using he ideniy (3) of Dirac s dela funcion, we obain: i G ( f ) = A{ δ( f + f) δ( f f) } A A = i δ( f + f) i δ( f f), which is wo complex impulse funcions a f = ± f which are no symmeric abou f = Nex, in erms of angular frequency ω from (34): Using Euler s formula (8): G iω iω ( ω) e g( ) d = A e sin( ω ) d iω i i A ω ω i i A ω ω iω iω G( ω) = A e ( e e ) d = e e d e e i i i d A A Ai Ai + + = e d e d e d e i i = i i Ai i( ) Ai ω ω i( ω+ ω) = e d e d i( ω ω ) i( ω ω ) i( ω ω ) i( ω ω ) d Use he ideniy (3) of Dirac s dela funcion: Ai Ai G ( ω) = πδ ( ω ω) πδ ( ω + ω) = Aiπδ( ω ω) Ai πδω ( + ω) Figure 36 plos a sine wave g ( ) = Asin( π f ) = Asin( ω ) and is Fourier ransforms A A G ( ω) = Aiπδ ( ω ω) Aiπδ ( ω + ω) andg ( f ) = i δ( f + f) i δ( f f) 7

35 ghl Hseconds L A) Plo of a Sine Wave g ( ) = Asin( π f ) = Asin( ω) B) Plo of F of g () in Angular Frequency Domain, G ( ω) C) Plo of F of g () in Frequency Domain, G( f ) Figure 36: Plo of a Sine Wave g ( ) = Asin( π f ) = Asin( ω ) and Fourier ransforms G ( ω) and G( f ) 8

36 [33] Properies of Fourier ransform We will discuss imporan properies of Fourier ransform in his secion saring from Dirac s dela funcion which is essenial o he undersanding of properies of Fourier ransform [33] Dirac s Dela Funcion (Impulse Funcion) + Consider a funcion of he form wih n R : n hx ( ) = exp( nx) (3) π his funcion is ploed in Figure 37 for hree differen values for n he funcion hx ( ) becomes more and more concenraed around zero as he value of n increases he funcion hx ( ) has a uni inegral: n hxdx ( ) = exp( nx) dx= π hhxl x Figure 37: Plo of A Funcion h(x) for n =, n = /, and n = n= n= è!!!!!! n= Dirac s dela funcion denoed by δ ( x) can be considered as a limi of hx ( ) when n In oher words, δ ( x) is a pulse of unbounded heigh and zero widh wih a uni inegral: δ ( x) dx = Dirac s dela funcion δ ( x) evaluaes o a all x R oher han x = : 9

37 δ () if x = δ ( x) = oherwise where δ () is undefined δ ( x) is called a generalized funcion no a funcion because of undefined δ () herefore, δ ( x) is a disribuion wih compac suppor {} meaning ha δ ( x) does no occur alone bu occurs combined wih any coninuous funcions f( x) and is well defined only when i is inegraed Dirac s dela funcion can be defined more generally by is sampling propery Suppose ha a funcion f( x) is defined a x = Applying δ ( x) o f( x ) yields f () : f ( x) δ ( x) dx= f() his is why Dirac s dela funcion δ ( x) is called a funcional because he use of δ ( x) assigns a number f () o a funcion f( x) More generally for a R : and: or for ε > : δ ( x) has ideniies such as: δ () if x = a δ ( x a) = oherwise f ( x) δ ( x a) dx= f( a), a+ ε f ( x) δ ( x a) dx= f( a) a ε δ ( ax) = δ ( x), a [ )] δ( x a ) = δ( x+ a) + δ( x a a Dirac s dela funcion δ ( x) can be defined as he limi n of a class of dela sequences: δ ( x) = lim δ ( x), n n 3

38 such ha: lim δn( x) f ( x) dx = f (), n where δ ( x ) is a class of dela sequences Examples of δ ( x ) oher han (38) are: n n if -/ n< x< / n δn( x) =, oherwise n δn ( x) = exp( iux) du π, n inx inx e e δn( x) =, π x i sin ( n+ / ) x δn( x) = π sin x / ( ) [33] Useful Ideniy: Dirac s Dela Funcion Dirac s dela funcion δ ( x) has he following very useful ideniy which we have used many imes before: i ( x a) ( x a) e ω δ dω π (3) PROOF n Firs sep is o prove a proposiion for all d =, 3, 4, and j =,,,, ha j depends on d ): d (noe if j = d π i exp jk = j= = (33) d k = d oherwise Firs, we deal an informal proof of a proposiion (33) When d = and j = : πi exp jk = exp k = exp + exp d d π i π i π i d k= k= his is based on Opion Pricing Using Inegral ransforms by Carr, P, Geman, H, Madan, D, and Yor, M 3

39 = { exp ( ) + exp ( )} = When d = and j = : d exp π i i i jk exp π k exp π πi = = + exp d k= d k= = { exp( ) + exp( iπ )} = { } = When d = 3 and j = : d 3 πi πi exp jk = exp k d k= d 3 k= 3 πi πi πi = exp + exp + exp = { exp ( ) + exp ( ) + exp ( )} = 3 When d = 3 and j = : d 3 πi πi exp jk = exp k d k= d 3 k= 3 πi πi πi = exp + exp + exp πi 4πi = exp( ) + exp + exp = { + ( 8665 i 5 ) + ( 8665 i 5 )} 3 = When d = 3 and j = : d 3 πi πi exp jk = exp k d k= d 3 k= 3 πi πi πi = exp + exp + exp πi 8πi = exp( ) + exp + exp

40 = ) = { ( i ) ( i } (Formal) PROOF of a proposiion (33) Rewrie as he below: π i exp jk = d d d d d k β, (34) k= k= π i where β = exp j When j =, d π i β = exp = hus, from (34): d d d d k k β = = = d d d k= k= k= When j, consider he erm d k = k β which is a geomeric series: S d d k β β β 3 β d β d β (35) k = = = Muliply β o (3): 3 d d d β S d β β β β β β = (36) Subrac (36) from (35): d ( β ) = β S d S d d β = (37) ( β ) Noe ha for j : From (37): d d π i β = exp j = exp( π i j ) = d 33

41 S d d k = β = = ( β ) k = From (34): d π i exp jk = = d k = d d Now we have compleed he proof of a proposiion (33) and we use his now Muliply d o boh sides of a proposiion (33): π i dj= exp jk (38) d = k = d As he limi d and plug j = x a, (38) becomes: i f ( x a) = df (39) δ ( x a) e π Conver frequency f ino angular frequency ω by he equaion (3) which is π ω = = π f From (39): iω ( x a) x a e d δ ( ) ω π = his complees he proof of an ideniy (3) [333] Lineariy of Fourier ransform Consider ime domain funcions f() and g () which have Fourier ransforms F ( ω) and G ( ω) defined by he equaion (34) hen: Or, we can wrie: { } iω iω iω af() + bg() e d = a f() e d+ b g() e d = af( ω) + bg ( ω) (3) [ af + bg] = a [ f ] + b [ g] F () ()( ω) F ()( ω) F ()( ω) 34

42 Lineariy means wo hings: homogeneiy and addiiviy Homogeneiy of Fourier ransform indicaes ha if he ampliude is changed in one domain by a, he ampliude in he oher domain changes by an exacly he same amoun a (ie scaling propery): [ af ] ω = a [ f ] F ()( ) F ()( ω) Addiiviy of Fourier ransform indicaes ha an addiion in one domain corresponds o an addiion in oher domain: [334] F of Even and Odd Funcions [ f + g ] = [ f ] + [ g ] F () ()( ω) F ()( ω) F ()( ω) A funcion g( x) is said o be even if for x R : g( x) = g( x), (3) which implies ha even funcions are symmeric wih respec o verical axis Examples of even funcions are illusraed in Panel A of Figure 38 A funcion g( x) is said o be odd if for x R : g( x) = g( x), (3) which implies ha odd funcions are symmeric wih respec o he origin Examples of odd funcions are illusraed in Panel B of Figure 38 ghxl 5 5 A) Even Funcions: gx= ( ), x gx ( ) = x, and g( x) cos( π x) = x cos HπxL 35

43 ghxl 4 - B) Odd Funcions: g( x) = x, x gx ( ) Figure 38: Plo of Even and Odd Funcions 3 = x, and g( x) sin( π x) = x x3 sin HπxL here are several imporan properies of even and odd funcions he sum of even funcions is even and he sum of odd funcions is odd he produc of wo even funcions is even and he produc of wo odd funcions is also even he produc of an even and an odd funcion is odd Le even( x ) be an even funcion and odd( x) be an odd funcion Inegral properies of even and odd funcions are: A odd( x) dx =, (33) A A A even( x) dx = even( x) dx (34) Consider F of an even funcion g () From he definiion (34) and use Euler s formula: A iω G( ω) e g() d = cos( ω) g() d + i sin( ω) g() d (35) Since he imaginary par sin( ω ) gd ( ) is zero (because sin( ω g ) ( ) is odd and use he inegral propery (33)), F G ( ω) is real and symmeric wih respec oω = In oher words, F of an even funcion is also even Nex, consider F of an odd funcion g () Since he erm cos( ω) gd ( ) in (35) becomes zero ( cos( g ) ( ) ω is odd and he inegral propery (33)), F is given as: G iω ( ω) e gd ( ) = i sin( ωgd ) ( ) 36

44 his means ha F of an odd funcion G ( ω) is complex and asymmeric wih respec oω = his propery is also illusraed in he secion 35 and 36 [335] Symmery of Fourier ransform By he definiion of an inverse Fourier ransform (35): Change he variable in he inegraion o y : Consider π f( ) : i f () = e ω ( ω) dω π F iy f () = e ( ) π F y dy iy π f ( ) = e F ( y) dy (36) We can say ha he righ hand side of (36) is by definiion he Fourier ransform of a funcion F ( y) Replace by ω and y by and (36) becomes: iω π f ( ω) = e F ( ) d (37) he equaion (37) is called a symmery propery of F I means ha if a funcion has a F F ( ω), F () has a F π f ( ω) ( F (), π f ( ω) ) is anoher F pair In oher words, if ( f(), ( ω) ) f() F is a F pair, his symmery propery of F can be shown wih mahemaically simpler form in he frequency domain Hz (cycles/second) By he definiion of an inverse F (37): f πif g () e ( f) df G Change he variable in he inegraion o y : Consider g( ) : πiy g () e ( ydy ) G 37

45 πiy g( ) e G( y) dy (38) We can say ha he righ hand side of (38) is by definiion he F of a funcion G( y) Replace by f and y by and (38) becomes: πif g( f) e G( ) d (39) We can sae in his case ha if ( g (), G ( f) ) is a F pair, ( (), g( f) ) pair [336] Differeniaion of Fourier ransform By he definiion of an inverse Fourier ransform (35): Differeniae wih respec o : i f () = e ω ( ω) dω π F G is anoher F iω f() e i = dω ( ω) = dω( iω ( ω) ) e ω π F π F i = iω ( ω) e ω d π F ω (33) By he definiion of an inverse F (35), he equaion (33) becomes: f() = iω f() (33) Equaion (33) ells us ha F of f()/ is equal o a F of f() muliplied by iω : [ f ] ω = iω [ f ] F ()/ ( ) F ()( ω ) (33) Nex, consider F in frequency domain f By he definiion of an inverse F (37): πif g () e ( f) df G Differeniae wih respec o : g () πif = πif e ( f ) df ifg G = π ( ) (333) 38

46 Equaion (333) ells us ha F of g ()/ is equal o a Fourier ransform of muliplied by π if : [ ] = π [ ] [337] ime Scaling of Fourier ransform F g ()/ ( f) iff g ()( f) g () Consider a ime domain funcion (34): f() and is Fourier ransform F ( ω) by he equaion iω F ( ω) = e f( ) d hen, F of a funcion f( a) (ie scaled by a real non-zero consan a ) can be expressed in erms of F ( ω) as: a ω iω F ( ) = e f ( a) d a (334) PROOF Se a = s When a > : When a < : iω iωs/ a s ( ω / ) i a s e f ( a) d = e f () s d = e f () s ds a a ω = F ( ) a a iω iωs/ a s ( ω / ) i a s e f ( a) d = e f ( s) d = e f ( s) ds a a ω = F ( ) a a Similarly, F of a funcion ga ( ) can be expressed in erms of G( f ) in frequency domain f as: a f πif G ( ) = e g( a) d a 39

47 [338] ime Shifing of Fourier ransform Consider a funcion f() and is Fourier ransform F ( ω) by he definiion (34): iω F ( ω) = e f( ) d hen, F of a funcion f ( ) (ie ime is shifed by R ) can be expressed in erms of F ( ω) as: i i e ω f( ) d = e ω F ( ω ) (335) PROOF Se = : * iω (* ( ) ) i ω + e f d = e f( *) d( * + ) iω iω* = e e f(*) d * = e iω F ( ω ) Nex, consider F in frequency domain f F of a ime domain funcion g () is defined by he definiion (36) as: G ( ) πif ( ) f e g d hen, F of a funcion g ( ) (ie ime is shifed by R ) can be expressed in erms of G( f ) as: PROOF Se = : * π if π e g( ) d = e if G( f) (336) πif ( * ) ( ) if e g d π = e + g( *) d( * + πif πif* πif e e g(*) d* e G( f) = = ) 4

48 [339] Convoluion: ime Convoluion heorem Convoluion of ime domain funcions f() and g () over a finie inerval [, ] is defined as: f g f() τ g( τ) dτ (337) Convoluion of ime domain funcions f() and g () over an infinie inerval [, ] is defined as: f g f( τ ) g( τ) dτ = g( τ) f( τ) dτ (338) Convoluion can be considered as an inegral which measures he amoun of overlapping of one funcion g () when g () is shifed over anoher funcion f() Websie by mahworld provides an excellen descripion of convoluion wih animaion For example, suppose f() and g () are Gaussian funcions: ( µ ) f() = exp{ }, πσ σ ( µ ) g () = exp{ } πσ σ hen, he convoluion of wo Gaussian funcions is calculaed as from (338): f ( ( µ + µ )) g = exp{ }, πσ ( + σ ) ( σ + σ ) which is anoher Gaussian funcion he convoluion f g of wo Gaussians for he case µ =, µ =, σ =, and σ = is ploed in Figure 38 4

49 Densiy 4 3 f g f g Figure 38: Plo of wo Gaussian funcions f and g and heir convoluion f g Consider ime domain funcions f() and g () wih Fourier ransforms F ( ω) and G ( ω) F of he convoluion of f() and g () in he ime domain is equal o he muliplicaion in he angular frequency domain (called ime-convoluion heorem): ( f g) ( f g d ) F F ( τ ) ( τ) τ = F( ω) G ( ω) (339) PROOF ( ) ( ) = f ( τ) dτ ( g( τ) e ) iω d iω F f g F f( τ ) g( τ) dτ = e f( τ) g( τ) dτd Use he ime shifing propery of Fourier ransform of he equaion (335): ( ) ( iωτ ) F i f g = f() τ dτ e G ( ω) = G ( ω) f() τ e ωτ d τ = F( ω) G ( ω) [33] Frequency-Convoluion heorem Consider ime domain funcions f() and g () wih Fourier ransforms F ( ω) and G ( ω) Frequency-convoluion heorem saes ha convoluion in he angular frequency domain (scaled by ) is equal o he muliplicaion in he ime domain In oher words, F of π he produc f () g() in he ime domain is equal o he convoluion F( ω) G ( ω) (scaled by ) in he angular frequency domain: π 4

50 F[ f ( g ) ( )]( ω) = F( ω) G( ω) ( ϖ) ( ω ϖ) d π π F G ϖ (34) PROOF here are several differen ways o prove he frequency convoluion heorem Bu we prove his by showing ha he inverse F of he convoluion F G in he angular frequency domain is equal o he muliplicaion f () g() (scaled by π ) in he ime domain Following he definiion of inverse F (35): ω [ ] i F F( ω) G( ω) ( ) e ω F( ω) G( ω) dω π = iω e ( ϖ ) ( ω ϖ) dϖdω π F G i dϖ ( ϖ) e ω = F ( ω ) d π G ϖ ω Using he frequency shifing (modulaion) propery of F (discussed in secion 33): [ ] = iϖ Fω F( ω) G( ω) ( ) dϖf( ϖ) e g( ) iϖ = g () dϖf ( ϖ) e = g () π f() Nex, consider F in he frequency domain (37): f Following he definiion of inverse F [ ] πif f ( f ) ( f) ( ) e ( f) ( f) df F F G F G πif e ( f *) ( f f *) df * df F G = πif ( ) = df * F( f *) e G ( f f *) df Using he frequency shifing (modulaion) propery of F (discussed in secion 33): [ ] = F F( f ) G( f)() df * F ( f*) e g() πif * f πif * = g () df* F ( f*) e = f() g () 43

51 [33] Frequency Shifing: Modulaion Consider ime domain funcion f() wih Fourier ransform F ( ω) If F F ( ω) is shifed by ω R in he angular frequency domain, hen he inverse F i by e ω : f() is muliplied e iω f() = F ( ω ω ), (34) iω ( e f ) F () = F ( ω ω ) (34) PROOF Le s = ω ω From he definiion of an inverse Fourier ransform (35): F ( ( ω ω ) ) iω i s ω e ( ) d e ( s) d( s π ω ω ω F = + π F + ω iω is iω = e e () s ds e () π F = f ) Similarly, if F ( f ) is shifed by F if g () is muliplied by : f R in he frequency domain, hen he inverse F [33] Parseval s Relaion e π πif e g() = [ ( f f )] πif ( ) F G, (343) F e g() = G ( f f ) (344) Le f() and g () be L -complex funcions A L -funcion can be informally considered as a square inegrable funcion (ie A funcion f() is said o be square- inegrable if f () d <) Le F ( ω) and G ( ω) be he Fourier ransforms of and g () defined by (34): iω F ( ω) = e f( ) d, iω G ( ω) = e g( ) d f() Le g () be a complex conjugae of f() and G ( ω) be a complex conjugae of F ( ω) : A complex conjugae of a complex number z a+ bi is z a bi 44

52 f () = f() g(), F( ω) = F( ω) G ( ω) hen, Parseval s relaion is: f () d = ( ω) dω π f () gd () = ( ) ( ) d π ω ω ω F, F G (345) Parseval s relaion in he case of F parameers ( ab, ) = (,) indicaes ha here is a very simple relaionship beween he power of a signal funcion f() compued in signal space or ransform spaceω of he form (345) Parseval s relaion becomes simpler when considered in he frequency domain f insead of angular frequency domainω Le F ( f ) and G ( f ) be he Fourier ransforms of f() and g () defined by (36): ( ) πif ( ) F f e f d, ( ) πif ( ) G f e g d Le g () be a complex conjugae of f() and G ( f ) be a complex conjugae of F ( f ): f () = f() g(), F( f ) = F( f) G ( f) hen, Parseval s relaion is: f () d = F ( f) df, f ( gd ) ( ) = F( f) G ( fdf ) (346) his version of Parseval s relaion means ha he power of a signal funcion wheher i is compued in signal space or in ransform space f PROOF f() is same 45

53 Using inverse Fourier ransforms of (35): f () d = f() g() d = e F( ω) dω e ( ω') dω' d π G π i( ω' ω) = ( ω) ( ω') e d dω' d π F G π ω iω iω' Use he ideniy of Dirac s dela funcion (3): iω ( a) δ ( a) e dω π hus: f () d = F( ω) G( ω') δ( ω' ω) ' d ω d ω π f ( ) d = F( ω) G( ω) ( ) d ω = F ω d ω π π [333] Summary of Imporan Properies of Fourier ransform able 3: Summary of Imporan Properies of Fourier ransform in Angular Frequency Domain ω Hz (radians/second) y F [ y] Propery ime Domain Funcion () Fourier ransform Lineariy af () + bg() af( ω) + bg ( ω) ()( ω) Even Funcion Odd Funcion f() is even F ( ω) R f() is odd F ( ω) I Symmery F () π f ( ω) Differeniaion df () d k d f() k d iωf ( ω) k ( iω) F ( ω) ime Scaling f( a) a ω F ( ) a 46

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