Polyphase Filters. Section 12.4 Porat 1/39

Transcription

1 Polyphase Flters Secto.4 Porat /39

2 .4 Polyphase Flters Polyphase s a way of dog saplg-rate coverso that leads to very effcet pleetatos. But ore tha that, t leads to very geeral vewpots that are useful buldg flter baks. Before we delve to the ath we ca see a lot just by lookg at the structure of the flterg. Of course, we WI eed to do the ath, too, though. /39

3 x Effcet FIR Flterg for Decato Flterg : x x h Decato : x x x h 3 x3 h3 x 3 h4 x4 h5 x5 h6 h7 x6 x7 h8 x8 Do t Copute x 3 Do t Copute 3/39

4 x h6 h9 h Effcet FIR Flterg for Decato x x x x3 x4 x5 x6 x7 x8 x9 x x x Orgal Flter h h h h3 h4 h5 x 3 3 x 3 gets splt to 3 subflters: Polyphase For of FIR Decato x 3 4 x x4 x7 x h h6 x x5 x8 x h h4 Σ x 3 3 x 3 x3 x6 x9 x h h3 Advatage: Decate the Flter 4/39

5 Ieffcet Drect For of FIR Decato x x x h h h h3 h4 h5 3 x 3 x 3 Effcet Polyphase For of FIR Decato Outputs are the Sae x x4 x7 h h6 x x x x x5 x8 h h4 Σ x 3 x 3 x x3 x6 x9 h h3 5/39

6 Exaple of Polyphase Flters for Decato Cosder egth- Flter w/ 4 : h: h h h h3 h4 h5 h6 h7 h8 h9. egth of Polyphase Flters: cel{legth/} cel{/4} 3 : p : h h4 h8 p : h h5 h9 p : h h6 p 3 : h3 h7 x : x x4 x8 x x6. x : x- x3 x7 x x5. x : x- x x6 x x4. x 3 : x-3 x x5 x9 x3. 6/39

7 Exaple of Polyphase Flters for Decato pt. atlab Code Create put sgal ad flter x:; h ; Drect For Ieffcet yflterh,,x; Copute flter output y_decy:4:ed Throw away ueeded output saples Pad eros to ake legth equal to teger ultple of Polyphase For Effcet Select polyphase flters ph:4:ed ph:4:ed ph3:4:ed p3h4:4:ed Select polyphase sgals Put a ero frot to provde the xx:4:ed x-3, x-, ad x- ters x x4:4:ed x x3:4:ed x3 x:4:ed flter each polyphase copoet ad add together y_poly_decflterp,,xflterp,,xflterp,,xflterp3,,x3 7/39

8 Effcet FIR Flterg for Iterpolato Iterpolat o : x x h x 3 h6 h7 h8 h9 h x x x x3 3 6 x 3 7 x 3 8 x 3 9 x 3 x 3 h x 3 8/39

9 Effcet FIR Flterg for Iterpolato Iterpolat o : x x h x 3 x x x x3 3 6 x 3 7 x 3 8 x 3 9 x 3 x 3 x 3 9/39

10 Effcet FIR Flterg for Iterpolato Orgal Flter h h h h3 h4 h5 3 gets splt to 3 subflters: x x x x3 The put goes to each subflter Polyphase For of FIR Iterpolato 6 9 Advatage Flter the Iterpolate h h3 h h4 h h5 x 3 7 x 3 8 x 3 x 3 x 3 x 3 The output coes fro alteratg betwee the subflter outputs /39

11 .4. ultrate Idettes These provde aalyss trcks useful whe dealg wth atheatcal aalyss of ultrate systes. The questo geeral s: How ca we terchage the order of flterg w/ decato/expaso? Decato Idetty Ths detty asserts equalty betwee the followg systes: x H y x H y Ca prove ths ether the Te-Doa or Z-Doa /39

12 TD Proof of Decato Idetty For the frst syste: x y w * h For the secod syste: x G H k w x y H h k x v k h k w k y k g h h /, f /, otherwse teger By Eq..5! /39

13 3/39 TD Proof of Decato Idetty cot. The k k x k h v y Sae as for Syste # " Proved!!! Thus k l k x k h l x l g g x v * Use!

14 4/39 ZD Proof of Decato Idetty For the secod syste: G H X V Y H X V!! where Now W W j e #"! / π But / / / } { W H W X W V V Y Use!! By ZT Result for Decato

15 5/39 ZD Proof of Decato Idetty cot. { } / / X H W X H H W X Y Whch s clearly the sae thg that the frst syste gves: H X {X } Y H {X }

16 Expaso Idetty Ths detty asserts equalty betwee the followg systes: x H w y x v H y Wll gve oly Z-Doa proof here. 6/39

17 7/39 ZD Proof of Expaso Idetty H x y w Frst syste gves: H X W The H X W W Y v Secod syste gves: H x y X X V The H X H V Y Sae!

18 .4. Polyphase Represetato of Decato Now we re-vst ths topc ad do t atheatcally Basc ath Idea: Re-wrte covoluto su s dex & apulate to get parallel flters: x Recall Decato: H y Output gve by.7 as y h x!!! Wrte su s dex block for a coo trck : teger Block Se Couts Blocks Couts Saples Isde a Block 8/39

19 9/39.4. Polyphase Rep of Dec cot. Block-Based Idexg: 3 Each row s dexed forward teger Forward Idexg

20 .4. Polyphase Rep of Dec cot. Use Block Idexg!!!: y h x h h x #&& " &! & x Su up sde each block Su up all Block Results!!!! Su all eleets the th posto of each block /39

21 .4. Polyphase Rep of Dec cot. Now, let s terpret ths: Defe for each, - p h th Polyphase Copoet of h Exaple : h: p p p {., 7, } {4,, } {.5,.7, } Each oe s a decated verso of h & the versos are staggered < See Fg..5> /39

22 Fg..5 fro Porat s Book /39

23 .4. Polyphase Rep of Dec cot. What have we doe? Splt up h to subsequeces where the th subsequece s a decated-by- verso of h Why the ae Polyphase? Recall: Te-Shft TD Phase-Shft FD h e jθ H f θ " Polyphase 3/39

24 4/39.4. Polyphase Rep of Dec cot. Now let s chop up the put slarly: x u Dffers Fro Before: Each row s dexed backward Backward Idexg

25 5/39.4. Polyphase Rep of Dec cot. Now back to the atheatcal developet. Puttg these re-dexed versos to!!!!: { } * u p u p y x u h p x h y To Ipleet Polyphase Decato Chop up flter to sub-flters Chop up sgal to sub-sgals Flter each sub-sgal w/ a sub-flter Add outputs pot-by-pot

26 .4. Polyphase Rep of Dec cot. Two equvalet ways to thk of ths: Frst Way show for 3: Note that Decato occurs Before Flterg Effcet!!! <Ths s Fg..6 fro Porat s Book> 6/39

27 .4. Polyphase Rep of Dec cot. Secod Way to Vew It show for 3: <Ths s Fg..7 fro Porat s Book> 7/39

28 .4. Polyphase Rep of Dec cot. Now we re-aalye ths set-up, but the Z-Doa. Why?.It provdes further aalyss sght. Z-Doa results ofte provde sght to how to: Derve other results Desg Polyphase Flters Etc. 8/39

29 .4. Polyphase Rep of Dec cot. Frst. soe te-doa trckery: How do we get back h fro the p???. Isert - eros betwee each saple. e the up usg delays 3. Add the up Recall Exaple: p p p {., {4, {.5,, 7, }.7, } } {., { 4, {.5,,,,,,, 7,,.7,,,, Expaso!,,,,,,,,, } } } {.,,, 7,,,,, } {, 4,,,,,,, } {,,.5,,,.7,,, } h {., 4,.5, 7,,.7,,, } 9/39

30 3/39.4. Polyphase Rep of Dec cot. Thus. } { p h So. Z-Doa we have: P H Delay Expad Now flter/decate looks lke ths: H X Y V X P H X V

31 .4. Polyphase Rep of Dec cot. ad after we get: Y { V } { P X } #&&&& "&&&&! P P P U { } { X } #&& "&&&! U X X H V By the Decato Idetty By Defto Sgal s Polyphase Copoets Y.whch s the Z-Doa Descrpto of the polyphase decato structure. We have ow developed two dfferet dervatos of the polyphase structure. 3/39

32 .4.3 Polyphase Rep of Expaso Recall Expaso: x H y Output gve by.9 as y x h Re-Idex usg: l #&"&! "backwards" teger l Block Idex l I-Block Idex dexes backward through block 3/39

33 33/39 l Polyphase Rep of Exp cot. l teger "backwards" l #&"&! Expaso Re-Idex Table

34 34/ Polyphase Rep of Exp cot. Usg ths re-dexg gves l h x l h x l y h x y & &! & #& " & &! & #& " for each l, ths dexg just reads dow a colu of the Expaso Re-Idex Table For each l such that l we defe: l y v l h q l l } { q x v l l

35 .4.3 Polyphase Rep of Exp cot. To see ths dexg structure, look at a exaple wth 3: l v v v y y y 3 y y y y5 y4 y3 y8 y7 y6 35/39

36 .4.3 Polyphase Rep of Exp cot. Now how do we get y fro the v l s?? If we terpolate each v l sequece we get 3. y 3 y y3 y6 y y y4 y7 y y y5 y8 Now delay these terpolated sequeces y 3 y y3 y6 y y y4 y7 y y y5 y8 y 3 y y y y y y3 y4 y5 y6 y7 y8 To get y: add up the delayed, terpolated copoets!! 36/39

37 .4.3 Polyphase Rep of Exp cot. Fro ths we see that we ca wrte y l { v l } l Recall: vl { x ql} Ths leads to the followg polyphase pleetato for expaso: Note: Expaso Occurs After Flterg Effcet!! 37/39

38 .4.3 Polyphase Rep of Exp cot. A equvalet alterate for of ths processg s 38/39

39 Skp.4.4 Shows how to do polyphase ethod for ratoal rate chage of / But brefly to chage the rate by factor of / Iterpolate Decate x h h y whch s equvalet to x h y Q: How to pleet ths effcetly usg polyphase deas? If terested: see Ch.3 of Oppehe & o reserve 39/39