DSP First, 2/e. This Lecture: Lecture 16 DTFT Properties. Definition of the DTFT: Forward DTFT. Inverse DTFT. Always periodic with a period of 2 (

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1 DSP First / Ltur 6 Proprtis READING ASSIGNMENTS Tis Ltur: Cptr 7 Stios 7- & 7-3 Otr Rig: Rittio: Cptr 7 EAMPLES Ju J MClll & RW Sfr 3 Ltur Otivs Proprtis of t Covolutio is mpp to Multiplitio Il Filtrs: LPF PF & BPF Rll: is t mt i t grl pt of frqu omi rprsttios T sptrum is ow tiuous futio of ormliz frqu ot ust li sptrum Ju J MClll & RW Sfr 4 Disrt-Tim Fourir Trsform Dfiitio of t : Forwr Ivrs Alws prioi wit prio of Ju J MClll & RW Sfr 5

2 Summr of Pirs Summr of Pirs Dl Impuls Rigt-si Dl Impuls u Rigt si Epotil si si futio is Blimit / si L L L Ju J MClll & RW Sfr 6 si otrwis Cougt Smmtr Cougt Smmtr vlu rl vlu - rl Ju J MClll & RW Sfr 7 M it A l Pl t u Mgitu Agl Plots io EVEN Fut Futio ODD s / s si rt Ju J MClll & RW Sfr 8 Frqu Ri Frqu Ri Proprtis of Proprtis of Lirit Lirit Tim-Dl ps sift S f Frqu-Sift multipl siusoi Ju J MClll & RW Sfr 9

3 Lirit Proof Lirit Proof Ju J MClll & RW Sfr Tim Dl Proprt Proof Tim-Dl Proprt Proof m m m Ju J MClll & RW Sfr Us Proprtis to Fi Us Proprtis to Fi lswr sw Strtg: Eploit Kow Trsform Pir lswr si / g p Ju J MClll & RW Sfr lswr si Us Proprtis to Fi Us Proprtis to Fi lswr si / Strtg: lswr si si Strtg: Eploit Kow Trsform Pir si / Ju J MClll & RW Sfr 3 si

4 Frqu Sift Frqu Sift Ju J MClll & RW Sfr 4 SINC Futio Rtgl pir pir si Ju J MClll & RW Sfr 5 Si tims siusoi: fi fi i? si? s si Frqu siftig up ow is o si multiplitio i t tim omi Ju J MClll & RW Sfr 6 i t tim omi mps Covolutio to Multiplitio Multiplitio LTI Sstm Sstm Ju J MClll & RW Sfr 7

5 IDEAL LowPss Filtr LPF O i t pss Zro i t stop is utoff tff frqu Filtrig wit t IDEAL LPF si / pir SINC Futio Rtgl si Fi t output w t iput is siusoi: s si s?? Multipl t sptrum of t iput s tims t of t filtr to gt Ju J MClll & RW Sfr 8 Ju J MClll & RW Sfr 9 IDEAL igpss Filtr PF IDEAL BPss Filtr BPF O i t pss Zro i t stops O i t pss Zro i t stop PF is mius LPF Ivrs of is lt si P P BPF s two stops B Rt Filtr s o stop two psss. It is o mius BPF BP Ju J MClll & RW Sfr Ju J MClll & RW Sfr

6 M IDEAL BPF from LPF LPF Empl : = si Fi output w si si 3 BPF is frqu sift vrsio of LPF Frqu siftig up ow is o si multiplitio i t tim omi s i si iff BP mi BP i t tim omi si for 3 Iput wit Lss t Filtr s pss Ju J MClll & RW Sfr LPF Empl : = si Fi output w si si 3 si 3 for Filtr ops off Sigl wit Outsi of pss Ju J MClll & RW Sfr 4 Ju J MClll & RW Sfr 3 Wt to ll t t sptrum T provis frqu-omi rprsttio T sptrum C. 3 sists of lis t vrious frquis wit mpl mplitus T sptrum rprsts sigl tt is t sum of mpl potils wr I wt ss is t goig to giv sum of mpl potils? Ju J MClll & RW Sfr 5

7 T Ivrs is sum of mpl ps T ivrs is itgrl A itgrl is sum i.. t limit of Rim sums: N / N lim N T fiit it sum sists of ps t tfrquis N wit mpl mplitus T limit of ts fiit sptr is t ivrs Ju J MClll & RW Sfr 6 Empl: SAMPLES s Sptrum Smpls of r Proportiol to igt of Sptrl Lis 3 N N / N lim N s Ju J MClll & RW Sfr 7