Digital Signal Processing, Fall 2010

Transcription

1 Digital Sigal Proeig, Fall 2 Leture 3: Samplig ad reotrutio, traform aalyi of LTI ytem tem Zheg-ua Ta Departmet of Eletroi Sytem Aalborg Uiverity, Demar t@e.aau.d Coure at a glae MM Direte-time igal ad ytem Sytem MM2 Fourier traform ad Z-traform Samplig ad reotrutio Sytem aalyi DFT/FFT MM5 MM3 Filter deig MM4 2

2 Part I-A: Periodi amplig Part I: amplig ad reotrutio Periodi amplig Frequey domai repreetatio of the amplig Reotrutio Part II: ytem aalyi 3 Periodi amplig From otiuou-time x t to direte-time x[] x[ ] x T, Samplig period Samplig frequey T f / T 2 / T 4 2

3 Two tage I pratie? Mathematially t Impule trai modulator Coverio of the impule trai to a equee x t x t t x t t T t T x T t T x[ ] x T, x t x t d 5 Periodi amplig Tow-tage repreetatio Stritly a mathematial repreetatio that i oveiet for gaiig iight ito amplig i both the time ad frequey domai. Phyial implemetatio i differet. x t a otiuou-time igal, a impule trai, ero exept at T x[] a direte-time equee, time ormaliatio, o expliit iformatio about amplig rate May-to-may i geeral ot ivertible 6 3

4 4 Part I-B: Freq. domai repreet. Part I: amplig ad reotrutio Periodi amplig F d i i f h li Frequey domai repreetatio of the amplig Reotrutio Part II: ytem aalyi 7 Frequey-domai repreetatio From x t to x t T S T t t 2 The Fourier traform of a periodi impule trai i a periodi impule trai. T t T x T T t t x S t t x t x T * 2? 8 The Fourier traform of x t oit of periodi repetitio of the Fourier traform of x t.

5 5 Frequey-domai T T S or Reovery r r Ideal lowpa filter with gai T ad utoff frequey r

6 Aliaig ditortio Due to the overlap amog the opie of, due to 2 ot reoverable by lowpa filterig Aliaig a example x t o t a a. T b. a.2 x r t o t b.2 2 xr t o t 6

7 yquit amplig theorem Give badlimited igal The If 2, for t x with i uiquely determied by it ample x t x[ ] x T, 2 2 T i alled yquit frequey i alled yquit rate 3 Fourier traform of x[] From t to x[ ] x x x t T t T x[ ] x T, From to e By taig otiuou-time Fourier traform of x t x T e T t x t e dt By taig direte-time Fourier traform of x[] e x[ ] e T e e T 4 7

8 Fourier traform of x[] T From Slide 4, e T e From Slide 8, T So, e T T T 2 2 T T T T T i.e. e i imply pya frequey-aled verio of with T x t retai a paig betwee ample equal to the amplig period T while x[] alway ha uity pae. 5 Samplig ad reotrutio of Si Sigal x t o4t T / / T 2 o aliaig x [ ] x T o4 T o2 / 3 o x t 4 4 T e / T with ormalied frequey T / T ow about x t o6t 6 8

9 Part I-C: Reotrutio Part I: amplig ad reotrutio Periodi amplig Frequey domai repreetatio of the amplig Reotrutio Part II: ytem aalyi 7 Requiremet for reotrutio O the bai of the amplig theorem, ample repreet the igal exatly whe: Badlimited it d igal Eough amplig frequey + owledge of the amplig period reover the igal 8 9

10 Reotrutio tep 2 Give x[] ad T, the impule trai i t x T t T x [ ] t T x i.e. the th ample i aoiated with the impule at t=t. The impule trai i filtered by a ideal lowpa CT filter with impule repoe h r t x t x h t T r r r r r e T 9 Ideal lowpa filter Commoly hooe utoff frequy a / / 2 T i t / T h r t t / T 2

11 Ideal lowpa filter iterpolatio CT igal Modulated impule trai x t r i t T / T x[ ] t T / T 2 Ideal direte-to-otiuou-time overter 22

12 direte-to-otiuou-time overter Pratial DAC do ot output a equee of dira impule that, if ideally low-pa filtered, reult i the origial igal before amplig but itead output a equee of pieewie otat value or retagular pule Ideally ampled igal. From Pieewie otat igal typial of a pratial DAC output. 23 Appliatio 24 2

13 Part II Sytem aalyi Part I: amplig ad reotrutio Part II: ytem aalyi Frequey repoe Sytem futio All-pa ytem Miimum-phae ytem Liear ytem with geeralied liear phae 25 Sytem aalyi Three domai Time domai: impule repoe, ovolutio um y [ ] x[ ]* h[ ] x[ ] h[ ] Frequey domai: frequey repoe Y e e e -traform: ytem futio Y LTI ytem i ompleted harateried by 26 3

14 Part II-A: Frequey repoe Part I: amplig ad reotrutio Part II: ytem aalyi Frequey repoe Sytem futio All-pa ytem Miimum-phae ytem Liear ytem with geeralied liear phae 27 Frequey repoe Relatiohip btw Fourier traform of iput ad output Y e e e I polar form Magitude magitude repoe, gai, ditortio Y e e e Phae phae repoe, phae hift, ditortio Y e e e 28 4

15 Ideal lowpa filter a example Frequey repoe,, e, Frequey eletive filter Impule repoe i h [ ], lp h[ ], oaual, aot be implemeted! ow to mae a oaual ytem aual? I geeral, ay oaual FIR ytem a be made aual by aadig it with a uffiietly log delay! But ideal lowpa filter i a IIR ytem!? 29 Phae ditortio ad delay Ideal delay ytem h id [ ] [ d ] Delay ditortio id e e d id e e, id d Ideal lowpa filter with liear phae 3 lp e e, d i d hlp [ ], d,, Liear phae ditortio Ideal lowpa filter i alway oaual! 5

16 Group delay A meaure of the liearity of the phae Coerig the phae ditortio o a arrowbad igal x[ ] [ ]o w For thi iput with petrum oly aroud w, phae effet a be approximated aroud w a the liear approximatio though i reality maybe oliear e d ad the output i approximately y ] e [ ]o Group delay 3 [ d d grd[ e d ] {arg[ e d ]} A example of group delay Figure 5., 5.2,

17 A example of group delay 33 Part II-B: Sytem futio Part I: amplig ad reotrutio Part II: ytem aalyi Frequey repoe Sytem futio All-pa ytem Miimum-phae ytem Liear ytem with geeralied liear phae 34 7

18 8 Sytem futio of LCCDE ytem Liear otat-oeffiiet differee equatio M m m x b y a ] [ ] [ -traform format M m m b m Y M m m m b Y a m 35 M m m d a b a m m d d pole at a ero at a i the deomiator pole at a ero at a i the umerator Stability ad auality Stable h[] abolutely ummable h ROC i l di h i i l ha a ROC iludig the uit irle Caual h[] right ide equee ha a ROC beig outide the outermot pole 36

19 9 Ivere ytem May ytem have ivere, peially ytem with ratioal ytem futio M ] [ ] [ ]* [ ] [ h h g G i i i M i m m d b a d a b 37 Pole beome ero ad vie vera. ROC: mut have overlap btw the two for the ae of G. m m Example 9 9.9,.9.5 So, i ] [.9.5 ] [.5 ] [.5 u u h i 38

20 Part II-C: All-pa ytem Part I: amplig ad reotrutio Part II: ytem aalyi Frequey repoe Sytem futio All-pa ytem Miimum-phae ytem Liear ytem with geeralied liear phae 39 All-pa ytem Coider the followig table ytem futio * a ap a ap e * e a ae * a e e ae ap e all-pa ytem: for whih h the frequey repoe magitude i a otat. 4 2

21 Example: Firt-order all-pa ytem P275 Example Part II-D: Miimum-phae ytem Part I: amplig ad reotrutio Part II: ytem aalyi Frequey repoe Sytem futio All-pa ytem Miimum-phae ytem Liear ytem with geeralied liear phae 42 2

22 Miimum-phae ytem Magitude doe ot uiquely haraterie the ytem Stable ad aual pole iide id uit irle, o retritio o ero Zero are alo iide uit irle ivere ytem i alo table ad aual i may ituatio, we eed ivere ytem! uh ytem are alled miimum-phae ytem explaatio to follow: are table ad aual ad have table ad aual ivere 43 Miimum-phae ad all-pa deompoitio Ay ratioal ytem futio a be expreed a: mi ap Suppoe ha oe ero outide the uit irle at / *, * * miimum-phae all-pa 44 22

23 Frequey repoe ompeatio Whe the ditortio ytem i ot miimum-phae ytem: d d mi ap mi d G d ap Frequey repoe magitude i ompeated Phae repoe i the phae of the all-pa 45 Propertie of miimum-phae ytem From miimum-phae ad all-pa deompoitio mi arg[ e ap ] arg[ mi ] arg[ From the previou figure, the otiuou-phae urve of a all-pa ytem i egative for So hage from miimum-phae to o-miimum- phae +all-pa phae alway dereae the otiuou phae or ireae the egative of the phae alled the phae-lag futio. Miimumphae i more preiely alled miimum phae-lag ytem 46 e ap e ] 23

24 Part II-E: GLP ytem Part I: amplig ad reotrutio Part II: ytem aalyi Frequey repoe Sytem futio All-pa ytem Miimum-phae ytem Liear ytem with geeralied liear phae 47 Deig a ytem with o-ero phae Sytem deig ometime deire Cotat frequey repoe magitude Zero phae, whe ot poible aept phae ditortio, i partiular liear phae ie it oly itrodue time hift oliear phae will hage the hape of the iput igal though havig otat magitude repoe 48 24

25 Ideal delay id e id e e, id e grd[ id e, ] i h id [ ] whe d h id [ ] [ d ] Ideal lowpa with liear phae i d hlp[ ] d 49 Geeralied liear phae Liear phae filter e e e Geeralied liear phae filter e A e A e e i a real futio of, ad are real otat 5 25

26 Summary Part I: amplig ad reotrutio Periodi amplig Frequey domai repreetatio Reotrutio Part II: ytem aalyi Frequey repoe Sytem futio All-pa ytem Miimum-phae ytem Liear ytem with geeralied liear phae 5 Coure at a glae MM Direte-time igal ad ytem Sytem MM2 Fourier traform ad Z-traform Samplig ad reotrutio Sytem aalyi MM3 Filter deig DFT/FFT MM5 MM