Sampling of Continuous-time Signals

Transcription

1 DSP Spig, 7 Samplig of Coiuous-im Sigals Samplig of Coiuous-im Sigals Advaags of digial sigal possig,.g., audio/vido CD. higs o loo a: Coiuous-o-dis C/D Dis-o-oiuous D/C pf osuio Fquy-domai aalysis of samplig poss Samplig a ovsio Piodi Samplig Idal oiuous-o-dis-im C/D ov [ C/D Coiuous-im sigal: Dis-im sigal: [, < <, : samplig piod I hoy, w ba h C/D opaio io wo sps: Idal samplig usig aalog dla fuio impuls Covsio fom impuls ai o dis-im squ Sp a b modld by mahmaial quaio. Sp is a op, o mahmaial modl. I aliy, h loi aalog-o-digial A/D iuis a appoima h idal C/D opaio. his iuiy is o pi; i ao b spli io wo sps. s s Covsio fom impuls ai o dis-im squ [ NCU EE

2 DSP Spig, 7 Samplig of Coiuous-im Sigals Idal samplig s Samplig Idal samplig sigal: impuls ai a aalog sigal s δ, : samplig piod Aalog oiuous-im sigal: Sampld oiuous-im sigal: s s s δ δ δ Fquy-domai Rpsaio of Samplig s S δ Rma: : aalog fquy adias/s s, wh s / ω : dis omalizd fquy adias/sampl ω ; < ω, Sp : Idal Samplig all i aalog domai s < S δ s δ s h sampld sigal spum is h sum of shifd opis of h oigial. Rma: I aalog domai, y f * Y f * Y NCU EE

3 DSP Spig, 7 Samplig of Coiuous-im Sigals Sp : Aalog Impulss o Squ aalog o dis-im No mahmaial modl. h spum of s, s spum of [, Now,. S h Appdi a h d. s, is h sam as h hus, ω ω wo Cass: Rma: I im domai, s ad [ hav h sam spa i fquy domai. >, ad a wo vy diff sigals bu hy o aliasig: s N aliasig: s < N, wh N is h highs ozo fquy ompo of. Af samplig, h plias of ovlap i fquy domai. ha is, h high fquy ompos of ovlap wih h low fquy ompos of s. s F F s N S NCU EE 3

4 DSP Spig, 7 Samplig of Coiuous-im Sigals Nyquis Samplig hom: b a badlimid sigal wih fo N. i.., o ompos a fquis ga ha h [,, ±, ±, K N is uiquly dmid by is sampls, if s Shao N. Nyquis, -- Nyquis fquy N, h badwidh of sigal. -- Nyquis a N, h miimum samplig a wihou disoio. I som boos, Nyquis fquy Nyquis a. < -- Udsamplig: s N -- Ovdamplig: s > N Foui Sis, Foui asfom, Dis-im Foui Sis & Dis-im Foui asfom Foui Sis : piodi oiuous-im sigal wih piod ompl valud d, Pow: P d NCU EE 4

5 DSP Spig, 7 Samplig of Coiuous-im Sigals Foui asfom : oiuous-im sigal ompl valud Egy: d d P d d Rma: Oh Noaios w w w w d dw f f f f df d Rlaioships bw F.S. & F.. F.S. F.. Pow Spum Egy Spum d d lim NCU EE 5

6 DSP Spig, 7 Samplig of Coiuous-im Sigals 3 Piodi Sigal F.S. F.. / Dis-im Foui Sis [: piodi dis-im sigal wih piod N. [ N D.F.S. N ompl valud F.. / N [ N N [ N N Pow: P N N N [ NCU EE 6

7 DSP Spig, 7 Samplig of Coiuous-im Sigals Dis-im Foui asfom [: dis-im sigal [ w DF w F.. / [ w w [ w w dw Egy: E [ w dw Rma: w v.s. w is a fquy-sald vsio of w w NCU EE 7

8 DSP Spig, 7 Samplig of Coiuous-im Sigals NCU EE 8 Rosuio of a Bad-limid Sigal fom Is Sampls -- Pf osuio: ov h oigial oiuous-im sigal wihou disoio,.g., idal lowpass badpass fil Basd o h fquy-domai aalysis, if w a lip o opy of h oigial spum,, wihou disoio, w a ahiv h pf osuio. Fo ampl, w us h idal low-pass fil as h osuio fil. Rma: No ha s is a aalog sigal impulss. ] [.. ov sq samplig s δ. ] [. ] [ o ov impuls s δ { } s h d h d h h ] [ ] [ ] [ λ λ λ δ λ λ λ δ } ] [ { ] [ H H H H H w w Idal low-pass osuio fil: < ohwis H si h Rosuio fil s Cov fom Squ o Impuls ai [

9 DSP Spig, 7 Samplig of Coiuous-im Sigals h [ s NCU EE 9

10 DSP Spig, 7 Samplig of Coiuous-im Sigals Dis-im Possig of Coiuous-im Sigals Dis-im C/D D/C sysm [ y[ y H H ff ω If his is a I sysm, ω ω ω [ y[ : Y H [ : ω ω 3 y[ y : Y H Y ω No ha is iludd i h pssio of Y, ω. his mas physial fquy o omalizd. 4 y : Y H H H H If H is a idal low-pass osuio fil, h H < Y, /, ohwis I oh wods, if is bad-limid ad is idally sampld a a a abov h Nyquis a, ad h osuio fil is h idal low-pass fil, h h quival aalog fil has h sam spum shap of h dis-im fil. H ff H, < /, ohwis NCU EE

11 DSP Spig, 7 Samplig of Coiuous-im Sigals ω Rma: I od o hav h abov quival laio bw H ad H ff, w d i h sysm is I; ii h ipu is badlimid; iii h ipu is sampld wihou aliasig ad h idal impuls ai is usd i samplig; iv h idal osuio fil is usd o podu h aalog oupu. I pai, h abov odiios a oly appoimaly valid a bs. Howv, h a mhods i dsigig h samplig ad h osuio posss o ma h appoimaio b. H ff H w w H H w < is hoss.. H, fo h[ h h impuls spos of h dis-im sysm is a sald, sampld vsio of h. NCU EE

12 DSP Spig, 7 Samplig of Coiuous-im Sigals Coiuous-im Possig of Dis-im Sigals [ Coiu.-im y y[ D/C sysm C/D Y H Y w Y, w,, H H ω < < w < Y w H w w w w H H w < w w H w < o, quivally, H H < Eampl: Noig Dlay H w wδ w < [ y -Δ y[ NCU EE

13 DSP Spig, 7 Samplig of Coiuous-im Sigals Chag h Samplig Ra Usig Dis-im Possig ' [ '[ ' Oigial samplig piod: Nw samplig piod: ' ' Samplig a duio by a ig fao Samplig a ompsso: ' M, wh M is a ig [ [ M] M d [ M [ [ M ] M d Compsso F Dowsamplig F M Aliasig: If h oigial sigal BW is o small ough o m h Nyquis a quim, NCU EE 3

14 DSP Spig, 7 Samplig of Coiuous-im Sigals NCU EE 4 pfilig is dd. h Oigial ω ω h Dowsampld ' ' ' d ω ω + M M M i M M M Old ad w id: M i +,,,,,,,,,,,,,, M i M i d M i M M M M M M M ' ' ' ω ω ω ω M i M i M M i d M M i M ω ω ω h dow-sampld spum sum of shifd plia of h oigial

15 DSP Spig, 7 Sigals Samplig of Coiuous-im Dowsamplig wih aliasig o avoid aliasig w N M < [ ~ [ ] owpass fil ~ [ d [ M ] M Cuoff M M Gal Sysm fo Samplig Ra Rduio by M NCU EE 5

16 DSP Spig, 7 Samplig of Coiuous-im Sigals Iasig samplig a by a ig fao F Dowsamplig F / Samplig a pad ' / i [, wh is a ig [ [ owpass fil i [ Cuoff / shap is ompssd; plias a movd NCU EE 6

17 DSP Spig, 7 Sigals Samplig of Coiuous-im Ias sampls <im-domai> [ ],, ±, ±, [, ohwis [ ] δ[ ] <Fquy-domai> ω No ha [ ] [ ] δ[ ] δ[ ] δ[ ] ω ω ω ω ω Rma: Essially, h hoizoal fquy ais is ompssd. h shap of h spum is o hagd. old _ ω, w_ω ', old _ ω w _ ω Rma: A his poi, w oly is zos io h oigial sigal. I im domai, his sigal dos loo li h oigial. Idal lowpass filig <Fquy-domai> < H i ohwis <im-domai> h i si, a ipolao! [ [ i si [ ] [ / ] / NCU EE 7

18 DSP Spig, 7 Sigals ia ipolaio /, h li [, ohwis Samplig of Coiuous-im H li ω si ω si ω [ [ ] h [ ] li li NCU EE 8

19 DSP Spig, 7 Sigals Samplig of Coiuous-im Chagig samplig a by a aioal fao Ida: Samplig piod ipola io dimaio M Rma: I gal, if h fao is o aioal, go ba o h oiuous sigals. NCU EE 9

20 DSP Spig, 7 Sigals I summay: Samplig of Coiuous-im -- Samplig im-domai Fquy -domai > Pfilig imi badwidh s N Aalog samplig impuls ai Duplia ad shif Aalog o dis δ[ δ ω -- Rosuio im-domai Dis o aalog [ δ δ ω Fquy -domai Ipolaio Rmov a opis -- Dow-samplig im-domai Fquy -domai Pfilig Dop sampls aag id imi badwidh Epad by a fao of M ad duplia is M- opis -- Up-samplig im-domai Fquy -domai Is zos Shi by a fao of Ipolaio Rmov a opis i a piod NCU EE

21 DSP Spig, 7 Sigals Samplig of Coiuous-im Digial Possig of Aalog Sigals Idal C/D ov appoimaio aalog-o-digial A/D ov Idal D/C ov appoimaio digial-o-aalog D/A ov Pfilig o Avoid Aliasig Idal aialiasig fil: Idal low-pass fil diffiul o implm shap-uoff aalog fils. A soluio: simpl pfil ad ovsamplig followd by shap aialiasig fils i dis-im domai. Rma: Shap uoff aalog fils a psiv ad diffiul o implm. A/D ovsio h ipu oiuous-im sigal is sampld a a vy high samplig a. NCU EE

22 DSP Spig, 7 Sigals Samplig of Coiuous-im A/D Covsio Digial: dis i im ad dis i ampliud Idal sampl-ad-hold: Sampl h ipu aalog sigal ad hold is valu fo sods. [ h h, < <, ohwis a h { a δ } h NCU EE

23 DSP Spig, 7 Sigals Samplig of Coiuous-im Quaizaio: asfom h ipu sampl [ io o of a fii s of psibd valus. ˆ[ Q [, ˆ[ is h quaizd sampl No: Quaizaio is a o-lia opaio. i Uifom quaiz uifomly spad quaizaio lvls; vy popula also alld lia quaiz ii Nouifom quaiz may b mo ffii fo ai appliaios Paams i a quaiz Disio lvls paiio h dyami ag of ipu sigal Quaizaio psaio lvls h oupu valus of a quaiz; a quaizaio lvl pss all sampls bw wo aby disio lvls 3 Full-sal lvl h quaiz ipu dyami ag No: ypially, wh h disio lvls a fis hos, h h bs quaizaio lvls a didd fo a giv ipu pobabiliy disibuio. O h oh had, wh h quaizaio lvls a hos, h bs disio lvls a didd. NCU EE 3

24 DSP Spig, 7 Sigals Samplig of Coiuous-im Quaizaio o aalysis Fo a uifom quaiz, h a wo y paams: i sp siz Δ, ad ii full-sal lvl ± m Assum B+ bis a usd o ps h quaizd valus. m Δ B+ m B Quaizaio o: [ ˆ[ [ quaiizd valu u valu I is la ha Δ Δ < [ <. Saisial haaisis of [: [ is saioay pobabiliy disibuio uhagd [ is uolad wih [ 3 [, [ +], a uolad whi 4 [ has a uifom disibuio h pdig assumpios a appoimaly valid if h sigal is suffiily ompl ad h quaizaio sps a suffiily small, NCU EE 4

25 DSP Spig, 7 Sigals Ma squa o MSE of [ vaia if zo ma σ { } d Δ / Δ E Δ Δ / B ad m -- Epssd i ms of B m σ -- SNR sigal-o-ois aio du o quaizaio B σ σ SNR log log B log σ m Samplig of Coiuous-im σ m Rmas: O bi buys a 6dB SNR impovm. If h ipu is Gaussia, a small pag of h ipu sampls would hav a ampliud ga ha If w hoos m 4 σ. 4σ, SNR 6B. 5dB Fo ampl, a 96dB SNR quis a 6-bi quaiz. NCU EE 5

26 DSP Spig, 7 Sigals Samplig of Coiuous-im D/A Covsio h idal lowpass fil is plad by a paial fil. Eampls of paial fils: zo-od hold ad fis-od hold. Mahmaial modl: DA ˆ[ h DA quaizd ipu * impuls spos of paial ipolaio fil [ h + [ h + ~ Pupos: Fid a ompsaio fil h o ompsa fo h disoio ausd by h o-idal h so ha is oupu ˆ is los o h aalog oigial a. ˆ[ DA Compsad ˆ D/A osuio Cov ~ fil I fquy domai: F [ [ h H [ H H H NCU EE 6

27 DSP Spig, 7 Sigals Samplig of Coiuous-im Baus, a H a [h ipolaio fil H is usd o mov h plias.] If H is o a idal lowpass fil, w dsig a ompsad osuio fil, ~ H, wh H H is h idal lowpass fil. H Zo-od hold h, < <, ohwis o si / / H hus, h ompsad osuio fil is / ~ H si /, /, < / > / Rma: A paial fil ao ahiv his appoimaio. NCU EE 7

28 DSP Spig, 7 Sigals Ovall sysm: Samplig of Coiuous-im a H aa H ~ Y H a H Ai-aliasig Possig Zo-od-hold Compsad osuio. H ff ~ H H H H aa ~ wh P a H H H σ Δ σ NCU EE 8