24.2. Properties of the Fourier Transform. Introduction. Prerequisites. Learning Outcomes
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1 Properies of he Fourier Transform 4. Inroducion In his Secion we shall learn abou some useful properies of he Fourier ransform which enable us o calculae easily furher ransforms of funcions and also in applicaions such as elecronic communicaion heory. Prerequisies Before saring his Secion you should... Learning Oucomes On compleion you should be able o... be aware of he basic definiions of he Fourier ransform and inverse Fourier ransform sae and use he lineariy propery and he ime and frequency shif properies of Fourier ransforms sae various oher properies of he Fourier ransform 4 HELM (008): Workbook 4: Fourier Transforms
2 . Lineariy properies of he Fourier ransform (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 ransform of he resuling funcion is simply he sum of he individual Fourier ransforms. (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 ransform by he same consan. These properies follow from he definiion of he Fourier ransform and from he properies of inegrals. Examples. F{e u() + 3e u()} = F{e u()} + F{3e u()} = F{e u()} + 3F{e u()}. = + i i If f() = hen f() = 4p 3 () using he sandard resul for F{p a ()}. { oherwise so F () = 4P 3 () = 8 sin 3 Task If f() = { 6 0 oherwise wrie down F (). Your soluion We have f() = 6p () so F () = sin. HELM (008): Secion 4.: Properies of he Fourier Transform 5
3 . Shif properies of he Fourier ransform There are wo basic shif properies of he Fourier ransform: (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 ransform. Example Use he ime-shifing propery o find he Fourier ransform of he funcion { 3 5 g() = 0 oherwise g() 3 5 Figure 4 Soluion g() is a pulse 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 = 4 in he ime shif heorem G() = F{g()} = e 4i sin. 6 HELM (008): Workbook 4: Fourier Transforms
4 Task Verify he resul of Example by direc inegraion. Your soluion G() = 5 3 e i d = [ e i i ] 5 as obained using he ime-shif propery. 3 = e 5i e 3i i ( ) e = e 4i i e i = e 4i sin i, Task Use he frequency shif propery o obain he Fourier ransform of he modulaed wave g() = f() cos 0 where f() is an arbirary signal whose Fourier ransform is F (). Firs rewrie g() in erms of complex exponenials: Your soluion ( e i 0 ) + e i 0 g() = f() = f()ei 0 + f()e i 0 HELM (008): Secion 4.: Properies of he Fourier Transform 7
5 Now use he lineariy propery and he frequency shif propery on each erm o obain G(): 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 ransform Formal inversion of he Fourier ransform, i.e. finding f() for a given F (), is someimes possible using he inversion inegral (4). However, in elemenary cases, we can use a Table of sandard Fourier ransforms ogeher, if necessary, wih he appropriae properies of he Fourier ransform. The following Examples and Tasks involve such inversion. 8 HELM (008): Workbook 4: Fourier Transforms
6 Example 3 Find he inverse Fourier ransform of F () = 0 sin 5 5. Soluion The appearance of he sine funcion implies ha f() is a symmeric recangular pulse. sin a We know he sandard form F{p a ()} = a or F sin a {a a a } = p a(). Puing a = 5 F sin 5 {0 5 } = p 5(). Thus, by he lineariy propery f() = F sin 5 {0 5 } = p 5() f() 5 5 Figure 4 Example 4 Find he inverse Fourier ransform of G() = 0 sin 5 5 exp ( 3i). Soluion The occurrence of he complex exponenial facor in he Fourier ransform suggess he ime-shif propery wih he ime shif 0 = +3 (i.e. a righ shif). From Example 3 F sin 5 {0 5 } = p 5() so g() = F sin 5 {0 5 e 3i } = p 5 ( 3) g() 8 Figure 5 HELM (008): Secion 4.: Properies of he Fourier Transform 9
7 Task Find he inverse Fourier ransform of sin H() = 6 e 4i. Firsly ignore he exponenial facor and find he inverse Fourier ransform of he remaining erms: Your soluion We use he resul: F {a Puing a = gives sin a a } = p a() F sin { } = p () F sin {6 } = 3p () Now ake accoun of he exponenial facor: Your soluion Using he ime-shif heorem for 0 = 4 h() = F sin {6 e 4i } = 3p ( 4) h() HELM (008): Workbook 4: Fourier Transforms
8 Example 5 Find he inverse Fourier ransform of K() = + ( )i 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() is a complex ime-domain signal. Task Find he inverse Fourier ransforms of (a) sin {3( π)} L() = ( π) (b) M() = ei + i Your soluion HELM (008): Secion 4.: Properies of he Fourier Transform
9 (a) Using he frequency shif propery wih 0 = π (b) l() = F {L()} = p 3 ()e iπ Using he ime shif propery wih 0 = m() = e (+) u( + ) m() 4. Furher properies of he Fourier ransform We sae hese properies wihou proof. As usual F () denoes he Fourier ransform 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 F{f()} = F () hen 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() is even his laer is irrelevan. For example, we know ha if f() = p () = Then, by he dualiy propery, since p () is even, { < < 0 oherwise, hen F () = sin. F{ sin } = πp ( ) = πp (). HELM (008): Workbook 4: Fourier Transforms
10 Graphically: p () P () F P () π πp () F Figure 6 Task Recalling he Fourier ransform pair { e > 0 f() = e F () = 4 < 0 4 +, obain he Fourier ransforms of (a) g() = (b) h() = cos (a) Use he lineariy and dualiy properies: Your soluion HELM (008): Secion 4.: Properies of he Fourier Transform 3
11 We have F{f()} F{e } = F{ 4 e } = (by lineariy) 4 + F{ 4 + } = π 4 e = π e = G() (by dualiy). f() F F () g() 4 F π G() (b) Use 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 { e + e + } = H() 4 H() 4 HELM (008): Workbook 4: Fourier Transforms
12 Exercises. Using he superposiion and ime delay heorems and he known resul for he ransform of he recangular pulse p(), obain he Fourier ransforms of each of he signals shown. (a) x a () (b) x b () 0 0 x c () x d () (c) (d) 0 3. Obain he Fourier ransform of he signal f() = e u() + e u() where u() denoes he uni sep funcion. 3. Use he ime-shif propery o obain he Fourier ransform of 3 f() = 0 oherwise Verify your resul using he definiion of he Fourier ransform. 4. Find he inverse Fourier ransforms of (a) F () = 0 sin(5) 5 e 3i (b) F () = 8 sin 3 ei (c) F () = ei i 5. If f() is a signal wih ransform F () obain he Fourier ransform of f() cos( 0 ) cos( 0 ). HELM (008): Secion 4.: Properies of he Fourier Transform 5
13 . X a () = 4 sin( ) cos(3 ) X b () = 4i sin( ) sin(3 ) X c () = [sin() + sin()] X d () = ( sin( 3 ) ) + sin( e 3i/. F () = 3 + i + 3i 3. F () = sin e i { < < 8 4. (a) f() = 0 oherwise { 4 4 < < (b) f() = 0 oherwise { e + < (c) f() = 0 oherwise (using he superposiion propery) 5. F () + 4 [F ( + 0) + F ( 0 )] 6 HELM (008): Workbook 4: Fourier Transforms
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