ECE 6560 Chapter 2: The Resampling Process

Transcription

1 Capter 2: e Resaplig Process Dr. Bradley J. Bazui Wester iciga Uiversity College of Egieerig ad Applied Scieces Departet of Electrical ad Coputer Egieerig 1903 W. iciga Ave. Kalaazoo I,

2 Capter 2: e Resaplig Process s 4 s D 4 s I 2 2 s 4x dow-saplig Deciatio by 4 2x up-saplig Iterpolatio by 2 Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

3 Uique ways to dow-saple s 4 s D 40 s 4 1 s D 41 We deciatig by, tere are uique places to start e tie-delay differece i startig locatios ust result i a pase rotatio i te frequecy doai. Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

4 A discrete saplig sequece Defiitio s t t e tie saplig sequece ca also be described as a discrete tie saplig sequece usig a iverse discrete Fourier rasfor described as s s exp 2 j s 1, for k 0, oterwise wit k a iteger Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

5 Aliged Saplig Note Fourier rasfor Pair: cob(t) cob(f) Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

6 Fourier Review See Fourier rasfor Review Slides Covered prior to Capter 2 Fourier rasfor Pairs See rasfor Pairs o te Password Web Site Cotiuous tie cotiuous frequecy Fourier trasfor i f Cotiuous tie cotiuous frequecy Fourier trasfor i w Z trasfor Discrete tie cotiuous frequecy Fourier trasfor aplace trasfor Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

7 extbook Fourier rasfor e expressio for te spectru of te sapled data ipulse respose is sow i were te sapled data frequecy variable ω s is deoted by θ wit uits of radias/saple. I tis coordiate syste, te spectru is periodic i 2 π. H exp j Note: is is te DCF versio of te Fourier rasfor (DCF eas discrete-tie cotiuous-frequecy) Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

8 Fourier Cosideratios of Saplig Fro te text S s t t f t exp j f t dt Note: For a ifiite suatio: te results equals 0 for f x = f/f s ot a iteger. It also provides replicas i frequecy based o te saple rate! 2 f t exp j f t S 2 dt f exp j 2 f exp j f f S 2 s Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

9 Fourier Cosideratios of resaplig te sapled sequece Fro te text S s t t f exp j 2 f S f ' s Were f = f ˑ or f = f / Regeeratig te resapled tie eleets te requires S f ' exp j is is te iverse FF your cofortable wit Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

10 A delayed discrete saplig sequece s t r t r e tie saplig sequece ca also be described as a discrete tie saplig sequece usig a iverse discrete Fourier rasfor described as s r exp j r 2 s 1, for r k 0, oterwise wit k a iteger Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

11 No-Aliged Saplig cob(t+) exp(-j2f) x cob(f) Covolutio i tie ultiplicatio i freq. e pase processio is based o te pase offsets fro te cotiuous Fourier trasfor! Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

12 Sigal Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

13 Aliged Sigal Deciatio Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

14 No-Aliged Sigal Deciatio Iagie a twist i te frequecy axis. exp(-j2f) Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

15 atlab Siulatio Cap2_VExaple. Filter Deciated but at te sae saple rate as te origial Filter ad agitude Plots: 2, 5, 8, 11, 14 Deciated Filter at deciated saple rate Filter ad agitude Plots: 3, 6, 9, 12, 15 Deciated Filter at deciated saple rate Pase ad Passbad oly Pase Plots: 4, 7, 10, 13, 16 Figure 30 sows tat te pases are differet based o wic saple te deciatio started o. Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

16 atlab Siulatio Cap2_VExaplev2. Iproved to sow exact frequecy relatiosip (Fs ad Fc) sic fuctio coets i file Icorporates FIR filter costat pase delay offset Pase (degrees) Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

17 atlab Siulatio Cap2_VExaple3. Iterpolatio by 5 exaple 0.15 H1 Deciated Filter 0 Offset Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

18 Wat is a ultirate Filter? A eas to icrease or decrease te saple rate wile processig iput sapled sigals. Up saplig or iterpolatio 1 to P saples Dow saplig or deciatio Q to 1 saples Ratioal rate Cages 1 to P/Q saples Aliasig (dow saplig) or spectral replicatio (up saplig) cosideratios are always preset! Filters ca be used to liit or iiize probles! Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

19 Up ad Dow Saplig Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

20 Filter Deciatio Bad-liit usig a low pass filter Eliiate te possibility of probles fro aliasig Reduce te saple rate usig a deciator You ust pay attetio to te Nyquist rate & aliasig Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

21 Filter Deciatio (2) ie ad Frequecy Doai Plots Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

22 atlab Siulatio Repeat te deciatio exaple but add i a si wave were te optio is available. Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

23 Iterpolatio Filter Iterpolate by zero paddig betwee saples ow Pass Filter to eliiate Spectral replicas A advaced alterative is to use a badpass filter ad keep oly te desired replica! is work for deciatio too! Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

24 Iterpolatio Filterig ie ad Frequecy Doai Plots Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

25 Iterpolatio Exaple See Cap2. Iterpolatig te sae sigal sapled at two differet rates e plots are all based o iterpretatio ad ateatics Iterpolated Cos Waves Frequecy -fs/2 to fs/ Frequecy -4*fs/2 to 4*fs/2 Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

26 Fro te Readig Assiget Crociere, R.E.; Rabier,.R.;, "Iterpolatio ad deciatio of digital sigals A tutorial review," Proceedigs of te IEEE, vol.69, o.3, pp , arc ttp://ieeexplore.ieee.org/stap/stap.jsp?aruber= Sectios I ad II itroduce deciatio ad iterpolatio. Deciatio: te zeroig of saples betwee te desired periodic saple poits (periodic replicatio) ad te te reoval of te zeroed saples (actual deciatio ad frequecy boud cage) Iterpolatio: te iclusio of additioal zeros betwee te existig saple poits (periodic replicatio ad frequecy boud cage) Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

27 Uifor Saplig viewed as a odulatio/ultiplicatio process Sigal x c (t) - cotiuous Sapler s(t) cob of ipulse resposes of period I te Fourier doai, odulatio/ultiplicatio requires covolutio te covolutio of a cob i frequecy wit te origial sigal spectru! Easier to visualize spectral replicatio due to saplig! Just aoter attept to elp visualize wat is goig o. Goig beyod te at is iportat. Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

28 Properties of Resaplers: pusig blocks aroud Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

29 Properties of Resaplers (2): pusig blocks aroud Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

30 Properties of Resaplers (3) ese equivaleces are referred to as oble idetity ey allow operatio reorderig tat ca drastically reduce te required operatio ad/or coputatio rates! If your goig to copute te filter results ad te deciate by trowig te result away, wy coputed it i te first place? Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

31 Properties of Resaplers (4) For relatively prie P ad Q, te followig is peritted. is is applicable for ratioal rate resaplig e typical iitial cofiguratio cosists of iterpolate (filter filter) deciate Note: if appropriate, spectral aliasig would occur! x() ust be badwidt liited! Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

32 Exaples of Resaplig Filters Stadard Dow- Saplig Filter Arcitectures Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

33 Exaples of Resaplig Filters (2) Stadard Up- Saplig Filter Arcitectures Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

34 Useful Perspectives for ultirate Filters A ultirate filter is ot I (iear ie Ivariat) A ultirate filter is a iear ie-varyig (V) process e ipulse respose depeds o wic subfilter is coected to te output port we te iput ipulse is preseted to te filter. Sice te output periodically revisits eac coutator port, it is said tat te ultirate filter is a Periodically ie Varyig (PV) process. Note: tis ca ake it ard to copare iputs ad outputs! You ust kow wic output you are coparig to or you could be geeratig a output betwee iputs wic will ever perfectly atc!? Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

35 Saple Rate Perspective A equivalet filter sapled at two differet saplig rates, 4x Badwidt ad 5x Badwidt Notice sizes Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

36 Sie Wave Exaple Execute atlab ext Saples of ultiple frequecies (Cap2_exaple25.) Saples at ultiple rates (Cap2_exaple26.) Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

37 Fixed ie Iterval Exaple Differet fc ceter frequecies Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

38 Fixed Iterval, Differet Saple Rates Fixed tie iterval, fixed cotiuous tie iput sigal, ad differet saple rates. Notice aliasig i te last pae. Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

39 Nyquist ad te Saplig Process f 2 s f AX s 1 2 f AX For realistic filter a trasitio bad is required! f s wo sided BW Filter rasitio BW f s 2 f AX Filter rasitio BW Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

40 Spectru Saple Rate Cosideratios for real sigals Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

41 Sigle-sided Spectru Badpass Saplig Cosideratios IF Passbad 0 db fs = ADC clock rate Filter rasitio Bad -80 db "Folded" Alias Copoet 0 fs/4 fs/2 3fs/4 fs is figure sow te previous cocept sifted by fs/4 Note tat i te spectral doai, te widts are critical but tat te absolute frequecy locatio are ot. Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

42 Spectral Aalysis Paper Spectral Aalysis Ipleetatios: A Coparative Aalysis of te Widowed Precobier Odd-Frequecy Fourier rasfor, Widowed Dual-Real Radix-2 Fast Fourier rasfor ad No-Widowed FF Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

43 Hoework Exaple Explai Butterwort Filter Order Geerator [N, W] = buttord(2*pi*20e3, 2*pi*24.1e3, 3, 96,'s') Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN

44 ateatical Derivatio Paper: Crociere, R.E.; Rabier,.R.;, "Iterpolatio ad deciatio of digital sigals A tutorial review," Proceedigs of te IEEE, vol.69, o.3, pp , arc extbook: R.E. Crociere ad.r. Rabier, "ultirate Digital Sigal Processig," Pretice-Hall, Ic., Eglewood Cliffs, NJ, 1983, (paperback 1996), ISBN: Cap

45 Saplig Rate Cage Origial Saple Rate at saple rate F odified saple Rate at saple rate F Assue a ratioal rate cage ( deciate, iterpolate) ' ' x x c y g x 45 Notes ad figures are based o or derived fro aterials i Crociere, R.E.; Rabier,.R.;, "Iterpolatio ad deciatio of digital sigals A tutorial review," Proceedigs of te IEEE, vol.69, o.3, pp , arc 1981.

46 e filter g () e filter is liear but tie varyig (for 1). e coefficiets fro (t) are For = 1, it will be a siple digital filter 46 g g g ~ ˆ ˆ g ~ ˆ 1 ˆ Notes ad figures are based o or derived fro aterials i Crociere, R.E.; Rabier,.R.;, "Iterpolatio ad deciatio of digital sigals A tutorial review," Proceedigs of te IEEE, vol.69, o.3, pp , arc 1981.

47 e filter g () e filter is liear but tie varyig (for 1) For = 2 (te coefficiets cage, but repeat every saples) 47 g g ~ ˆ 1 0 ~ 1 2 ~ ˆ ˆ ~ 2 ~ 2 2 ˆ ˆ g g odd eve 2, 1, 0,, r for g g r Notes ad figures are based o or derived fro aterials i Crociere, R.E.; Rabier,.R.;, "Iterpolatio ad deciatio of digital sigals A tutorial review," Proceedigs of te IEEE, vol.69, o.3, pp , arc 1981.

48 Deciatio (, =1) e ew data rate is 1/ te old data rate 48 F F ' ' ' F F g ~ ˆ 1 1 ˆ oterwise F w e H jw, 0 2 ' 2 1, ˆ x y ~ k x w k ~ w y Notes ad figures are based o or derived fro aterials i Crociere, R.E.; Rabier,.R.;, "Iterpolatio ad deciatio of digital sigals A tutorial review," Proceedigs of te IEEE, vol.69, o.3, pp , arc 1981.

49 Deciatio (, =1) (cot.) e suatio of a, spaced filter coefficiet prefilters. If te filters is a fiite legt FIR, Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN x y ~ r r r x r w y r r x r y 1 0 : 1 0 for r x r y r 1 0 y y

50 Iterpolatio (=1, ) e ew data rate is ties te old data rate 50 F F 1 ' ' ' F F oterwise F w e H jw, 0 2 ' 2, ~ k k k x k k w k y ~ ~ oterwise k k x w k 0,, 2, 0,, Notes ad figures are based o or derived fro aterials i Crociere, R.E.; Rabier,.R.;, "Iterpolatio ad deciatio of digital sigals A tutorial review," Proceedigs of te IEEE, vol.69, o.3, pp , arc 1981.

51 Iterpolatio (=1, ) (cot) Expadig te result :, r k k k r x k k r x k r y ~ ~ u g u u u g r ~ ˆ, u v v u v u r x v u v u r x v u r y ~ ~ 1 0 :, v v u k u u r x u g r y Notes ad figures are based o or derived fro aterials i Crociere, R.E.; Rabier,.R.;, "Iterpolatio ad deciatio of digital sigals A tutorial review," Proceedigs of te IEEE, vol.69, o.3, pp , arc u g k u u u g k, ˆ ˆ u u r x u r y ~ uique filters

52 For ore details Crociere, R.E.; Rabier,.R.;, "Iterpolatio ad deciatio of digital sigals A tutorial review," Proceedigs of te IEEE, vol.69, o.3, pp , arc ttp://ieeexplore.ieee.org/stap/stap.jsp?aruber= See te CR_DecAdIterp.pdf slides for te igligts. Notes ad figures are based o or take fro aterials i te course textbook: fredric j. arris, ultirate Sigal Processig for Couicatio Systes, Pretice Hall PR, ISBN