Digital Signal Processing. Digital Signal Processing READING ASSIGNMENTS. License Info for SPFirst Slides. Fourier Transform LECTURE OBJECTIVES

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1 Digil Signl Procssing Digil Signl Procssing Prof. Nizmin AYDIN hp: Lcur Fourir rnsform Propris Licns Info for SPFirs Slids READING ASSIGNMENS his work rlsd undr Criv Commons Licns wih h following rms: Ariuion h licnsor prmis ohrs o copy, disriu, disply, nd prform h work. In rurn, licnss mus giv h originl uhors crdi. Non-Commrcil h licnsor prmis ohrs o copy, disriu, disply, nd prform h work. In rurn, licnss my no us h work for commrcil purposs unlss hy g h licnsor's prmission. Shr Alik h licnsor prmis ohrs o disriu driviv works only undr licns idnicl o h on h govrns h licnsor's work. Full of h Licns his hiddn pg should kp wih h prsnion his Lcur: Chpr, Scs. -5 o -9 ls in Scion -9 Ohr Rding: Rciion: Chpr, Scs. - o -9 N Lcurs: Chpr Applicions LECURE OBJECIVES Fourir rnsform h Fourir rnsform d Mor mpls of Fourir rnsform pirs Bsic propris of Fourir rnsforms Convoluion propry Muliplicion propry d π d im - Domin Fourir Synhsis Invrs rnsform Fourir Anlysis Forwrd rnsform Frquncy - Domin

2 WHY us h Fourir rnsform? Mnipul h Frquncy Spcrum Anlog Communicion Sysms AM: Ampliud Modulion; FM Wh r h Building Blocks? Asrc Lyr, no implmnion Idl Filrs: mosly BPFs Frquncy Shifrs k Modulors, Mirs or Muliplirs: p Frquncy Rspons Fourir rnsform of h is h Frquncy Rspons H u h + u h sin π sin δ l of Fourir rnsforms c c πδ δ π sin sin u +

3 cos πδ + πδ + Fourir rnsform of Gnrl c c c Priodic Signl If is priodic wih priod, k k k k k k hrfor, sinc πδ k d π k δ k k Squr Wv Signl + k k d + k d k k k k k πk πk Squr Wv Fourir rnsform + π k δ k k l of Esy F Propris Linriy Propry + + Dly Propry d d Frquncy Shifing Scling Scling Propry d λ shrinks; λ dλ pnds 3

4 Scling Propry Uncriny Principl ry o mk shorr hn will g widr Nrrow pulss hv wid ndwidh ry o mk nrrowr hn will hv longr durion Cnno simulnously rduc im durion nd ndwidh Significn F Propris Convoluion Propry h H y h p P π Diffrniion Propry d d Y H Convoluion in h im-domin y h hτ τ dτ corrsponds o MULIPLICAION in h frquncydomin Y H Convoluion Empl Idlly Bndlimid Signl Bndlimid Inpu Signl sinc funcion Idl LPF Lowpss Filr h is sinc Oupu is Bndlimid Convolv sincs sinπ π π π π 4

5 Convoluion Empl Cosin Inpu o LI Sysm h H sinπ sinπ π π sinπ π Y H H [πδ +πδ + ] H πδ + H πδ + y H + H H + H * H cos + H Idl Lowpss Filr Idl Lowpss Filr H lp H co co f co "cuoff frq." co y y co if co if co y 4 sin 5π π + 4 sin 5π 3π Signl Muliplir Modulor p y p Y P π Muliplicion in h im-domin corrsponds o convoluion in h frquncy-domin. Y θ P θ dθ π y Frquncy Shifing Propry sin 7 π d Y 7 +7 lswhr d 5

6 y cos Y + + d d Diffrniion Propry d d d π π d d u u + δ d δ u Muliply y + y p Y P π p cos P πδ y cos +πδ + sin Y π [πδ + πδ + ] Y + + Dly Propry d d d d τ τ + d dτ d For mpl, 5 u Srgy for using h F Dvlop s of known Fourir rnsform pirs. Dvlop s of horms or propris of h Fourir rnsform. Dvlop skill in formuling h prolm in ihr h im-domin or h frquncydomin, which vr lds o h simpls soluion. F of Impuls rin h priodic impuls rin is p δ n k k n n π k δ d for ll k P π δ k k 6

7 Convoluion Empl Y H sin Y y h 7