[ ] Review. For a discrete-time periodic signal xn with period N, the Fourier series representation is

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1 Discrt-tim ourir Trsform Rviw or discrt-tim priodic sigl x with priod, th ourir sris rprsttio is x + < > < > x, Rviw or discrt-tim LTI systm with priodic iput sigl, y H ( ) < > < > x H r rfrrd to s th frqucy rspos of th systm h[ ] W will s tht H is th ourir trsform of th impuls rspos H( ) is priodic i with priod 2 π H H h Outli ourir Trsform Wht bout priodic sigls? W will cosidr priodic sigl s priodic sigl with ifiit priod Th rsult is th ourir trsform of discrt-tim sigls or discrt-tim sigl x, th ourir trsform pir is x d x is priodic with priod 2 π Ivrs ourir Trsform ourir Trsform

2 2 rom ourir Sris to ourir Trsform Cosidr sigl x[ ] of fiit durtio for x > 2 Costruct priodic sigl x [ ] tht is sm s x[ ] durig o priod Th ourir sris rprsttio of x < > < > x x, is W will s how th ifiitly log priod tur th ourir rsprsttio ito th ourir trsform

3 3 < > x x x X ( ) ot th rltio: choosig th priod i which x x sic x outsid th priod < >, dfiig x, ourir Trsdorm is th ourir cofficit of th priodic sigl x with priod, d x[ ] is th ourir trsform of th durtio-limitd sigl, x < > < > 2 π otig, d tig th priod, ( ) ()

4 4 As, d th summtio i q bcoms itgrl Howvr th itrvl of itgrtio is fixd to 2 π x d ( 2) As, x x for y fiit vlu of Eq 2 bcoms x d Rcll tht th rqucy rspos H Ivrs ourir Trsform of y discrt-tim LTI systm is priodic i with priod 2 π Th proprty pplis to y discrt-tim sigl x is priodic i with priod of 2 π Also is priodic i with priod of 2 π Thrfor is priodic i with priod of 2 π W c t y itrvl of lgth 2 π i th itgrl of th ivrs ourir trsform or discrrt-tim sigl x, th ourir trsform pir is x d x d x[ ] is rfrrd to s th spctrum of x Ivrs ourir Trsform ourir Trsform is th mplitud of th sigl t frqucy

5 5 T of Boxcr id th ourir trsform of th followig boxcr sigl x ( ) ( 2 + ) ( + ) si si multiplyig to both umr d dom is show i th figur for 2 is prodic i with priod 2 π is rl d v Thr r zro crossigs btw d π

6 6 Approximtio of th Sigl or discrrt-tim sigl x, th ivrs ourir trsform is x d tig th pricipl priod, π X( ) d π W c pproximt xby prtil spctrum W ˆ x d, W π W As W π, x x ˆ Approximtio IT, Imouls δ [ ] Lt x Th W c pproximt x W by si si ˆ W W W x d π π W W

7 7 ourir trsform of Complx Expotil Sigls Rviw I cs of th cotiuous-tim complx xpotil, t ( ) πδ 2 or discrt-tim sigls, th ourir trsform must b priodic δ proof W will prov by showig 2 ( 2 ) πδ π δ ( ) δ ( ) d Choos y priod of itrvl 2 π Th itrvl cotis oly o impuls δ ( ) δ ( ) d ( + )

8 8 T of Cos Cosidr x cos with 5 id th T of x 5 5 x cos π π πδ + πδ or π < π, is priodic with priod 2 π πδ + πδ + 5 5

9 9 T of Priodic Sigls Rviw or cotiuous-tim priodic sigls, t 2 π δ( ) T or discrt-tim priodic sigl x with priod, < > 2 π δ ( ) proof Cosidr discrt-tim priodic sigl x rprsttio b with priod Lt its ourir sris 2 π x p < > or simplicity, w will choos < >< > Tig th T of qp, δ ( ) ( ) δ ( ) δ itrchgig th ordr of summtios, 2 π substitutig, δ

10 X ( ) π ( ) 2 2 π ( ) 2 δ ( ) δ

11 T of Impuls Tri Cosidr th followig tri of impulss i th tim domi δ [ ] x id th T of x Solutio x x Sic x is priodic, w c rprst s ourir sris with ourir cofficits < > 2 π x[ ] δ π ( ) 2 2 π δ is tri of impulss sprtd with gp of Proprty 2 π δ[ ] δ( ) or th spcil cs of, x is tri of impulss sprtd with gp of 2 π Proprty δ ( ) x