Polyphase Filters. Section 12.4 Porat

Transcription

1 Polyphase Flters Secto.4 Porat

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.

3 x h3 Effcet FIR Flterg for Decato Flterg : x ˆ x h Decato : xˆ xˆ x h ˆx3 3 ˆ x 3 h4 ˆx4 h5 h6 h7 h8 ˆx5 ˆx6 ˆx7 ˆx8 Do t Copute ˆ x 3 Do t Copute

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

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

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

7 Effcet FIR Flterg for Iterpolato Orgal Flter 3 gets splt to 3 subflters: Polyphase For of FIR Iterpolato The put goes to each subflter Advatage Flter the Iterpolate The output coes fro alteratg betwee the subflter outputs

8 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.

9 Exaple of Polyphase Flters for Decato pt. atlab Code x:; h ; ph:4:ed ph:4:ed ph3:4:ed p3h4:4:ed yflterh,,x; y_decy:4:ed x x5:4:ed x x4:4:ed x x3:4:ed x3 x:4:ed y_poly_decflterp,,xflterp,,xflterp,,xflterp3,,x3

10 .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

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

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

13 ZD Proof of Decato Idetty For the secod syste: G H X V Y H X V!! where Now W W j e 3 / π But / / / } { W H W X W V V Y Use!!

14 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 }

15 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.

16 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!

17 .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

18 .4. Polyphase Rep of Dec cot. Block-Based Idexg: 3 teger Forward Idexg Each row s dexed forward

19 .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

20 .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>

21 .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

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

23 .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

24 .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>

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

26 .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.

27 .4. Polyphase Rep of Dec cot. Frst. soe te-doa trckery: How do we get 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,,, }

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

29 .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.

30 .4.3 Polyphase Rep of Expaso Recall Expaso: x H y Output gve by.9 as y x h Re-Idex usg: l 443 "backwards" teger l Block Idex l I-Block Idex dexes backward through block

31 .4.3 Polyphase Rep of Exp cot. l teger "backwards" l 443 l Expaso Re-Idex Table

32 .4.3 Polyphase Rep of Exp cot. Usg ths re-dexg gves l h x l h x l y h x y For each l such that l we defe: l y v l h q l l } { q x v l l for each l, ths dexg just reads dow a colu of the Expaso Re-Idex Table

33 .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

34 .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!!

35 .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!!

36 .4.3 Polyphase Rep of Exp cot. A equvalet alterate for of ths processg s Skp.4.4 Shows how to do polyphase ethod for ratoal rate chage of /