Discrete-Time Complex Exponential Sequence.

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1 Discrt-Tim Cmplx Exptial Squc. x ] Cα, whr C adα ar i gral cmplx umbrs. Altrativly w ca xprss th squc i th fllwig frm : - x ] C β, whrα β. Althugh this frm is similar t th ctiuus - tim xptial sigal w hav dscribd prviusly, th frmr frm is prfrrd wh dalig with th discrt - tim squc. 1

2 Ral Exptial Sigal x ] C * α whrα > 1. g. x ] 2*1. 1 2

3 Ral Exptial Sigal x ] C * α whr 0 < α < 1 x ] 2*0. 9 3

4 Ral Exptial Sigal x ] C * α whr -1< α < 0 x ] 2*( 0.9) 4

5 Ral Exptial Sigal x ] C * α whrα < 1 x ] 2*( 1.1) 5

6 Ral Exptial Sigal x ] C * α whrα 1 x ] 2*( 1) 6

7 Ral Exptial Sigal Ral-valud discrt xptials ar usd t dscrib:- 1) Ppulati grwth as fucti f grati. 2) Ttal rtur ivstmt as a fucti f day, mth r quartr. 7

8 8 Siusidal Sigals j j j j j j A A A j A x x C x β ϖ ϖ β ) cs( si cs - rlati : Frm Eulr's radias. f hav uits ad th bth Takig as dimsilss, ). cs( ] : sigal is clsly rlatd t siusidal sigal This. ] b purly a imagiary umbr. j 1& lt C, ]

9 Discrt-tim Siusidal Sigals x ] cs(2π /12) x ] cs(8π / 31) x ] cs( / 6) 9

10 Discrt-tim Siusidal Sigals Ths discrt-tim sigals pssssd:- 1) Ifiit ttal rgy 2) Fiit avrag pwr. 10

11 Gral Cmplx Exptial Sigals Th gral discrt - tim cmplx xptial ca b itrprtd i trms f ral xptials ad siusidal sigals. Writig C ad α i plar frm : - C C jθ, x ] Cα α α C α j cs( + θ ) + j C α α 1, ral & imagiary parts ar siusidal. a < 1, siusidal dcayig xptially, a > 1, siusidal grwig xptially.. si( + θ ) 11

12 Gral Cmplx Exptial Sigals α >1 α <1 12

13 2) Pridicity Prprtis f Discrttim Cmplx Exptials Tw prprtis f 1)Th Largr is j t is ctiuus -, th highr is th rat f pridic fr ay valu f. tim cutrpart j t scillati. Thr ar diffrcs i ach f th abv prprtis fr th discrt-tim j cas f. 13

14 Pridicity Prprtis f Discrt-tim Cmplx Exptials Csidr th discrt - tim cmplx xptial with frqucy j( + 2π ) j2π j j Frm this w cclud that th xptial at frqucy + 2π : This is vry diffrt frm th ctiuus - tim cas whrby th sigals ar all distict fr all distict valus f.. + 2π is th sam as that at frqucy. Similarly at frqucis ± 2π, ± 4π,ad s. Bcaus f this pridicity f 2π, w d ly t csidr frqucy itrval f 2π i th cas fr discrt - tim sigals. 14

15 Pridicity Prprtis f Discrt-tim Cmplx Exptials Bcausf this implid pridicity f discrt- tim sigal, th sigal as j ds t hav a ctiually icrasigrat f scillati is icrasdi magitud. Icrasig frm 0 (d.c.,cstat squc, scillati) th scillati icrassutil π, thraftr th scillati willdcrast 0 i..a cstat squcr d.c.sigalat 2π. 15

16 Pridicity Prprtis f Discrt-tim Cmplx Exptials Thrfr,lw frqucis ccursat 0, ± 2π, ± v multiplf π. High frqucis ar at t fr π, dd multiplfπ, ± π, ± 3π, ± dd multiplf π. ( th sigal scillatsrapidly,chagigsig at ach pit i tim. jπ jπ ) ( 1), 16

17 Pridicity Prprtis f Discrt-tim Cmplx Exptials x ] cs( 0 ) 1 x ] cs( π / 8 ) x ] cs( π / 4 ) x ] cs( π ) x ] cs( π / 2 ) x ] cs( 3π / 2 ) x ] cs( 7π / 4 ) x ] cs( 15 π / 8 ) x ] cs( 2π ) 17

18 Pridicity Prprtis f Discrt-tim Cmplx Exptials Scd prprty ccrs th pridicity f th discrt-tim cmplx xptial. I rdr fr j ( + ) i.. j j must b a, r quivaltly multipl f j This mas that th sigal t b pridic with prid 2π. j 2π m, r quvaltly 2π 1. is pridic if / 2π is a ratial umbr ad is t pridic thrwis. This is als tru fr th discrt - tim siusids. m, > 0, 18

19 19 Discrt-tim Siusidal Sigals /12, 2 pridic bcaus /12) cs(2 ] π π π x , / 8 pridic bcaus 31) / cs(8 ] π π π x umbr ratial 2 6, 1/ t pridic bcaus 6) / cs( ] π x

20 Fudamtal Prid & Frqucy f discrt-tim cmplx xptial If x] 2π Its fudamtal frqucy is, m Th fudamtal prid is writt as : - 2π m( ) j is pridic with fudamtal prid, 20

21 Cmparis f th sigal j t j ad j t Distict sigals fr distict valus f. j Idtical sigals fr valus f sparatd by multipls f 2π Pridic fr ay chic f. Fudamtal frqucy Fudamtal 0 : udfid 2π 0 : prid Pridic ly if fr sm itgrs > 2 π m 0 ad 0 Fudamtal frqucy m Fudamtal prid 0 : udfid 2π 0 : m( ) 21, m

22 Harmically rlatd pridic xptial squc Csidrig pridic xptials with cmm prid sampls: k ] jk (2π / ), fr k 0, ± 1,... This st f sigals pssss frqucis which ar multipls f 2π / 22

23 Harmically rlatd pridic xptial squc I ctiuus-tim cas jk ( 2π / T ) t ar all distict sigals fr k 0, ± 1, ± 2,... k + ] j( k + )(2π / ) jk(2π/) j2π k ] 23

24 24 Harmically rlatd pridic xptial squc Thrfr, thr ar ly distict pridic xptials i th discrt harmic squcs. ] ) ] th abv.(.g. f t ]is idtical Ay thr ]..., ], ] 1, ] 0 k / 1) ( 2 1 / 4 2 / 2 1 j j j π π π

2.2.C Analogy between electronic excitations in an atom and the mechanical motion of a forced harmonic oscillator"

2.2.C Analogy between electronic excitations in an atom and the mechanical motion of a forced harmonic oscillator ..C Analgy btwn lctrnic xcitatins in an atm and th mchanical mtin f a frcd harmnic scillatr" Hw t chs th valu f th crrspnding spring cnstant k? Rsnant Absrptin Mchanical rsnanc W idntify th mchanical rsnanc