IMGS-6 Solutons to Homework #9. For f [] SINC [] sn[π], use the modulaton theorem to evaluate and sketch π the Fourer transform of f [] f [] f [] (f []) Soluton: We know that F{RECT []} SINC [] so we use the transform-of-a-transform theorem to show that: F{SINC []} RECT [ ] RECT [+] RECT [] because rectangle s even functon. The modulaton theorem says that: F{f [] m []} F [] M [] F SINC [] ª F{SINC [] SINC []} RECT [] RECT [] We have already evaluated the convoluton of the rectangle wth tself: RECT [] RECT [] Z β+ β Z β+ β RECT [ β] RECT [β] dβ dβ for f >+ R β+ dβ f + β R β+ dβ f β f < f >+ + f + + f f < f >+ f + + f f < ( ) RECT
RECT[]*RECT[]...8.6.4. -.5 -. -.5 -. -.5.5..5..5 -. convoluton of RECT [] RECT [] TRI[] Note that the convoluton of the rectangle wth tself s a functon wth area equal to the product of the areas of the two component functons ( ) and support equal to the sum of the supports: ( +). The two components of the pecewse result are lnear functons of.
. Use the result of # to sketch the form of the Fourer transform of f 3 [] f [] f [] f [] (f []) 3 ; you may evaluate the functonal form frst, but you only have to sketch the result. Soluton: F SINC 3 [] ª F SINC [] SINC [] ª RECT [] TRI[] Ths s the convoluton of a trangle and a rectangle; you only have to sketch t. The resultsafunctonwthsupportof3unts(++ 3)andareaof( ). The result has a smaller ampltude at the orgn. It s easy to show that the ampltude of the convoluton at the orgn s the area of the trangle truncated to unt wdth, whch s + 3. You dd not have to evaluate the convoluton, but I dd t below for 4 4 nformaton only: RECT [] TRI[] Z β+ β Z β+ β R RECT [ β] TRI[β] dβ ( + β) dβ R + f > 3 ( β) dβ f << 3 ( + β) dβ + R + ( β) dβ f << R + ( + β) dβ f 3 << f < 3 f > 3 ( 8 3) f << 3 3 4 f << 8 ( +3) f 3 << f < 3 RECT * RECT * RECT...8.6.4. -.5 -. -.5 -. -.5.5..5..5 -. 3
Note that each component part of the pecewse functon s a quadratc functon of. 3. Use the result of # to sketch the form of the Fourer transform of f 3 [] f [] f [] f [] f [] (f []) 4. Thsstheconvolutonofthetranglewthtselforoftherectanglewththeresultof #. Agan, the area of the convoluton s the product of the areas of the component functons ( ) and the support s the sum of the supports ( +++ 4). Agan, you only had to sketch t, but I evaluated t for eercse: RECT [] (RECT [] RECT [] RECT []) f β> 3 Z β+ (β 8 3) f <β< 3 3 RECT [ β] 4 β f <β< β (β 8 +3) f 3 <β< f β< 3 R R R 3 3 4 β dβ + R + f > (β 8 3) dβ f << (β 8 3) dβ f << (β 8 +3) dβ + R + 3 β dβ f << 4 R + 3 8 +3) dβ f << f < f > 6 ( )3 f << 3 + 3 f << 3 + 3 f << 6 ( +)3 f << f < dβ TRI * TRI...8.6.4. -.5 -. -.5 -. -.5.5..5..5 -. 4
Note that each component part of the pecewse functon s a cubc functon of and that the mamum occurs at wth ampltude 3 Just for fun, put all three on the same plot for comparson:...8.6.4. -.5 -. -.5 -. -.5.5..5..5 -. RECT [] RECT [] n green, RECT [] RECT [] RECT [] n blue, and RECT [] RECT [] RECT [] RECT [] n red. 5
4. Determne the condtons that must be satsfed for the followng epresson to be vald: cos [π ] SINC A cos [π ] Soluton: Use the flter theorem: Known transforms: and the scalng theorem: ½ ¾ F f b F{f [] h []} F [] H [] F{cos [π ]} δ [ + ]+ δ [ ] b G [] F{SINC []} RECT [] b F [b ] b F b ½ ¾ F SINC b RECT b µ δ [ + ]+ δ [ ] b RECT b b (δ [ + ]+δ [ ]) RECT b Ths s the product of a par of Drac delta functons an the rectangle. The output functon s a cosne wth ampltude A, so the output spectrum s: G [] F{g []} F{A cos [π ]} A F{cos [π ]} A δ [ + ]+ A δ [ ] The queston s whether these two epressons are equal: b (δ [ +? ]+δ [ ]) RECT b A δ [ + ]+ A δ [ ] Ths obvously s correct IF the frequency-doman rectangle s suffcentlywdetoen- close both Drac delta functons,.e., b > b ( ) < b also note that A b. 6 b
5. Determne the condtons that must be satsfed for the followng epresson to be vald: Soluton: Agan, evaluate the spectra: cos [π ] SINC b B cos [π ] F{cos [π ]} δ [ + ]+ δ [ ] ¾ F ½SINC b TRI b b F{B cos [π ]} B δ [ + ]+ B δ [ ] ¾ µ F ½cos [π ] SINC b δ [ + ]+ µ δ [ ] b TRI b µ b δ [ + ]+ b δ [ ] TRI b h The convoluton wth SINC b wll multply the spectra by the trangle, so the output wll be a cosne wth a reduced ampltude. The defnton of the Trangle s: TRI[] TRI f > f + f f < b TRI[b ] f > b b f b +b f b f < b so use the property of the Drac delta functon n products: µ b δ [ + ]+ b δ [ ] ½ B f b > b f b +b f b f b < f [] δ [ ] f [ ] δ [ ] TRI b b µ (δ [ + ]+δ [ ]) µ µ b b f >b b ( b ) f <b b (δ [ + ]+δ [ ]) 7
6. Consder the convoluton of scaled SINC functons wth wdth parameters that start small and grow for succeedng terms: g [] SINC h h SINC h 3 SINC h 3 n SINC h n SINC (a) Sketch the frst three SINC functons on the same graph, ncludng both the ampltudes and wdth parameters....8.6.4. -8-6 -4-4 6 8 -. -.4 (red) SINC [], (blue) SINC,(green) SINC 3 3 (b) Evaluate the functonal form of and sketch g []. Soluton: use the flter theorem: ½ h ¾ ½ h ¾ ½ h ¾ G [] F SINC F SINC F SINC " # " # " # RECT [] RECT RECT RECT 3 The product of these rectangles wth dfferent wdths and the same unt heght s the sngle narrowest rectangle: " # G [] RECT Now evaluate the nverse Fourer transform to see that the result s the wdest SINC functon g [] h SINC 8
g[]...8.6.4. -3 - - 3 -. g [] SINC SINC SINC n n SINC 9
7. Consder the convoluton of a SINC functon wth wdth b and a SINC functon wth wdth d : SINC SINC g [] A SINC d d b (a) Fnd the condtons on the wdth parameters b, d,anda that must be satsfed for ths epresson to be vald. Soluton: Evaluate n the frequency doman usng the flter theorem F ½ SINC ¾ SINC d b ½ ¾ ¾ F SINC F ½SINC b d b RECT b d TRI d b d RECT b TRI d The output spectrum s: ¾ G [] F ½A SINC d ¾ A F ½SINC d A TRI d The product of the frequency-doman rectangle and trangle must be the trangle, whch means that the wdth parameters must be related: b d d b d b Under these condtons, the output spectrum s: G [] b d RECT b TRI b d TRI d b SINC d A b d (b) Eplan whether t t possble to select wdth parameters and B such that: SINC SINC g [] B SINC d b In ths case, the frequency-doman product of the rectangle and the trangle must be a rectangle; the product of the rectangle and the trangle s a wndowed trangle, andnotarectangle,sothscannotbedone. b