READING ASSIGNMENTS. Signal Processing First. Fourier Transform LECTURE OBJECTIVES. This Lecture: Lecture 23 Fourier Transform Properties

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Signl Procssing Firs Lcur 3 Fourir rnsform Propris READING ASSIGNMENS his Lcur: Chpr, Scs. -5 o -9 ls in Scion -9 Ohr Rding: Rciion: Chpr, Scs. - o -9 N Lcurs: Chpr Applicions 3/7/4 3, JH McCllln & RW Schfr 3/7/4 3, JH McCllln & RW Schfr 3 LECURE OBJECIVES Fourir rnsform h Fourir rnsform d Mor mpls of Fourir rnsform pirs Bsic propris of Fourir rnsforms Convoluion propry Muliplicion propry 3/7/4 3, JH McCllln & RW Schfr 4 d π d Fourir Synhsis Invrs rnsform Fourir Anlysis Forwrd rnsform im - Domin Frquncy - Domin 3/7/4 3, JH McCllln & RW Schfr 5

WHY us h Fourir rnsform? Mnipul h Frquncy Spcrum Anlog Communicion Sysms AM: Ampliud Modulion; FM Wh r h Building Blocs? Asrc Lyr, no implmnion Frquncy Rspons Fourir rnsform of h is h Frquncy Rspons h u Idl Filrs: mosly BPFs Frquncy Shifrs Modulors, Mirs or Muliplirs: p h u H + 3/7/4 3, JH McCllln & RW Schfr 6 3/7/4 3, JH McCllln & RW Schfr 7 < / > / sin / / sin π < > 3/7/4 3, JH McCllln & RW Schfr 8 3/7/4 3, JH McCllln & RW Schfr 9

3/7/4 3, JH McCllln & RW Schfr δ 3/7/4 3, JH McCllln & RW Schfr l of Fourir rnsforms c c πδ δ > < π sin / / sin / / > < u + 3/7/4 3, JH McCllln & RW Schfr cos c c c πδ πδ + + 3/7/4 3, JH McCllln & RW Schfr 3 Fourir rnsform of Gnrl Priodic Signl If is priodic wih priod, d sinc hrfor, πδ δ π

Squr Wv Signl + d + d / / / / 3/7/4 3, JH McCllln & RW Schfr 4 π π Squr Wv Fourir rnsform + π δ 3/7/4 3, JH McCllln & RW Schfr 5 l of Esy F Propris Linriy Propry + + Dly Propry d d Frquncy Shifing Scling 3/7/4 3, JH McCllln & RW Schfr 6 Scling Propry d λ λ / shrins; pnds 3/7/4 3, JH McCllln & RW Schfr 7 dλ

Scling Propry Uncriny Principl ry o m shorr hn will g widr Nrrow pulss hv wid ndwidh ry o m nrrowr hn will hv longr durion Cnno simulnously rduc im durion nd ndwidh 3/7/4 3, JH McCllln & RW Schfr 8 3/7/4 3, JH McCllln & RW Schfr 9 Significn F Propris h H p P π Diffrniion Propry d d 3/7/4 3, JH McCllln & RW Schfr Convoluion Propry Convoluion in h im-domin y h hτ τ dτ corrsponds o MULIPLICAION in h frquncy-domin y h Y H Y H 3/7/4 3, JH McCllln & RW Schfr

Convoluion Empl Bndlimid Inpu Signl sinc funcion Idl LPF Lowpss Filr h is sinc Idlly Bndlimid Signl sinπ π π < π > π Oupu is Bndlimid Convolv sincs 3/7/4 3, JH McCllln & RW Schfr 3/7/4 3, JH McCllln & RW Schfr 3 Convoluion Empl h H sinπ π sinπ π sinπ π Cosin Inpu o LI Sysm Y H H [πδ +πδ + ] H πδ + H πδ + y H + H H + H * H cos + H 3/7/4 3, JH McCllln & RW Schfr 4 3/7/4 3, JH McCllln & RW Schfr 5

Idl Lowpss Filr H lp Idl Lowpss Filr H < co > co f co "cuoff frq." co co y if < co y 4 sin 5π π + 4 sin 5π 3π y if > co 3/7/4 3, JH McCllln & RW Schfr 7 3/7/4 3, JH McCllln & RW Schfr 6 Signl Muliplir Modulor Frquncy Shifing Propry p y p Muliplicion in h im-domin corrsponds o convoluion in h frquncy-domin. Y π Y P π θ P θdθ 3/7/4 3, JH McCllln & RW Schfr 8 y sin 7 Y 7 < < +7 π lswhr 3/7/4 3, JH McCllln & RW Schfr 9 d d

y cos Y + + Diffrniion Propry d d d d π π d d d d u u + δ δ u Muliply y + 3/7/4 3, JH McCllln & RW Schfr 3 3/7/4 3, JH McCllln & RW Schfr 3

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