24.2. Properties of the Fourier Transform. Introduction. Prerequisites. Learning Outcomes. Before starting this Section you should...

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1 Properies of he Fourier Transform 4. Inroducion Prerequisies Before saring his Secion you should... Learning Oucomes Afer compleing his Secion you should be able o...

2 . Lineariy Properies of he Fourier Transform (i) If f(), g() are funcions wih ransforms F (), G(), respecively, hen F{f()+g()} = F ()+G() i.e. if we add funcions hen he Fourier Transform of he resuling funcion is simply he sum of he individual Fourier Transforms. (ii) If k is any consan, F{kf()} = kf() i.e. if we muliply a funcion by any consan hen we mus muliply he Fourier Transform by he same consan. These properies follow from he definiion of he Fourier Transform and properies of inegrals. Examples. F{e u()+3e u()} = F{e u()} + F{3e u()} = F{e u()} +3F{e u()} = +i + 3 +i { If f() = 0 oherwise hen f() = 4p 3 () and so F () = 4P 3 () = 8 sin 3 using he sandard resul for F{p a ()}. If f() = { 6 0 oherwise wrie down F (). Your soluion We have f() =6p () sof () = sin. HELM (VERSION : March 8, 004): Workbook Level

3 . Shif properies of he Fourier Transform There are wo basic shif properies of he Fourier Transform: (i) Time shif propery: F{f( 0 )} = e i 0 F () (ii) Frequency shif propery F{e i 0 f()} = F ( 0 ). Here 0, 0 are consans. In words, shifing (or ranslaing) a funcion in one domain corresponds o a muliplicaion by a complex exponenial funcion in he oher domain. We omi he proofs of hese properies which follow from he definiion of he Fourier Transform. Example Use he ime-shifing propery o find he Fourier Transform of he funcion { 3 5 g() = 0 oherwise g() 3 5 Soluion g() isapulse of widh and can be obained by shifing he symmerical recangular pulse { p () = 0 oherwise by 4 unis o he righ. Hence by puing 0 =4in he ime shif heorem G() =F{g()} = e 4i sin. Verify he above resul by direc inegraion. 3 HELM (VERSION : March 8, 004): Workbook Level

4 Your soluion We have G() = 5 3 e i d [ i]5 = e 5i e 3i = e 4i i e i e = i 3 i ( e ) i ( e = e 4i i e i i = e 4i sin, ) as obained using he ime-shif propery. Use he frequency shif propery o obain he Fourier Transform of he modulaed wave g() =f() cos 0 where f() is an arbirary signal whose Fourier Transform is F (). Firs rewrie g() in erms of complex exponenials. Your soluion We have g() =f() ( e i0 + e i 0 ) = f()ei 0 + f()e i 0 Now use he lineariy propery and he frequency shif propery on each erm o obain G(). HELM (VERSION : March 8, 004): Workbook Level 4

5 Your soluion We have, by lineariy F{g()} = F{f()ei 0 } + F{f()e i 0 } and by he frequency shif propery G() = F ( 0)+ F ( + 0). F () G() Inversion of he Fourier Transform Formal inversion of he Fourier Transform, i.e. finding f() for a given F () issomeimes possible using he inversion inegral (4). However, in elemenary cases, we can use a Table of sandard Fourier Transforms ogeher, if necessary, wih he appropriae properies of he Fourier Transform. sin 5 Example Find he inverse Fourier Transform of F () = HELM (VERSION : March 8, 004): Workbook Level

6 Soluion The appearance of he sine funcion implies ha f() isasymmeric recangular pulse. We know he sandard form sin a F{p a ()} =a a or F sin a {a a } = p a(). Puing a =5 F sin 5 {0 5 } = p 5(). Thus, by he lineariy propery f() =F {0 f() sin 5 5 } =p 5() 5 5 sin 5 Example Find he inverse Fourier Transform of G() =0 exp ( 3i). 5 Soluion The occurrence of he complex exponenial facor in he FT suggess he ime-shif propery wih he ime shif 0 =+3(i.e. a righ shif). From he previous example so F {0 sin 5 5 } =p 5() g() =F sin 5 {0 5 e 3i } =p 5 ( 3) g() 8 HELM (VERSION : March 8, 004): Workbook Level 6

7 Find he inverse Fourier Transform of sin H() =6 e 4i. Firsly ignore he exponenial facor and de-fourier (o coin a phrase) he remaining erms: Your soluion We have so puing a = F { F sin a {a a } = p a() sin } = p () F {6 sin } =3p () Now ake accoun of he exponenial facor: Your soluion Using he ime-shif heorem for 0 =4 h() =F {6 sin e 4i } =3p ( 4) h() 3 6 Example Find he inverse Fourier Transform of K() = +( )i 7 HELM (VERSION : March 8, 004): Workbook Level

8 Soluion The presence of he erm ( ) insead of suggess he frequency shif propery. Hence, we consider firs ˆK() = +i. The relevan sandard form is F{e α u()} = α +i or F { α +i } = e α u(). Hence, wriing ˆK() = +i ˆk() =e u(). Then, by he frequency shif propery wih 0 = k() =F { +( )i } = e e i u(). Here k() isacomplex ime-domain signal. Find he inverse Fourier Transforms of (i) sin {3( π)} L() = ( π) () M() = ei +i Your soluion HELM (VERSION : March 8, 004): Workbook Level 8

9 (i) Using he frequency shif propery wih 0 =π l() =F {L()} = p 3 ()e iπ (ii) Using he ime shif propery wih 0 = m() =e (+) u( +) m() 4. Furher properies of he Fourier Transform We sae hese properies wihou proof. As usual F () denoes he Fourier Transform of f(). (a) Time differeniaion propery: F{f ()} =if() (Differeniaing a funcion is said o amplify he higher frequency componens because of he addiional muliplying facor ). (b) Frequency differeniaion propery: F{f()} =i df d or F{( i)f()} = df d Noe he symmery beween properies (a) and (b). (c) Dualiy propery: If hen F{f()} = F () F{F ()} =πf( ). Informally, he dualiy propery saes ha we can, apar from he π facor, inerchange he ime and frequency domains provided we pu raher han in he second erm, his corresponding o a reflecion in he verical axis. If f() iseven his laer is irrelevan. Example We know ha if f() =p () = hen F () = sin. { << 0 oherwise, 9 HELM (VERSION : March 8, 004): Workbook Level

10 Then, by he dualiy propery, (since p () iseven). Graphically F{ sin } =πp ( ) =πp () p () P () F P () π πp () F Recalling he Fourier Transform pair f() = { e >0 e <0 (or f() =e for shor) F () = 4 4+, obain he Fourier Transforms of (i) g() = (ii) h() = cos Your soluion for (i). Use he lineariy and dualiy properies. Your soluion HELM (VERSION : March 8, 004): Workbook Level 0

11 We have F{ F{f()} F{e } = F{ 4 e } = (by lineariy) } =π 4 e = π e = G() (by dualiy). f() F () F 4 g() G() F π Your soluion for (ii) using he modulaion propery based on he frequency shif propery. Your soluion We have h() =g() cos. F{g() cos 0 } = (G( 0)+G( + 0 )), so wih 0 = F{h()} = π 4 { e + e + } = H() H() HELM (VERSION : March 8, 004): Workbook Level