Polyphase Flters Secto.4 Porat
.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.
x h3 Effcet FIR Flterg for Decato Flterg : x ˆ x h Decato : xˆ 3 4 5 6 7 8 9 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
x h3 h6 h9 Effcet FIR Flterg for Decato 3 3 4 5 6 7 8 9 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
Effcet FIR Flterg for Iterpolato Iterpolat o : x ˆ x h 3 4 5 6 7 8 9 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
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
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
Exaple of Polyphase Flters for Decato Cosder egth- Flter w/ 4 : 3 4 5 6 7 8 9 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.
Exaple of Polyphase Flters for Decato pt. atlab Code x:; h 3 4 5 6 7 8 9 ; 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
.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
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!
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!!!
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!!
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 }
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.
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!
.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
.4. Polyphase Rep of Dec cot. Block-Based Idexg: 3 teger Forward Idexg Each row s dexed forward
.4. Polyphase Rep of Dec cot. Use Block Idexg!!!: y h x h h x 44 43 4 x Su up sde each block Su up all Block Results!!!! Su all eleets the th posto of each block
.4. Polyphase Rep of Dec cot. Now, let s terpret ths: Defe for each, - p h th Polyphase Copoet of h Exaple : 3 4 5 6 h:. 4.5 7.7 3 p p p {., 7, } {4,, } {.5,.7, } Each oe s a decated verso of h & the versos are staggered < See Fg..5>
.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
.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
.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
.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>
.4. Polyphase Rep of Dec cot. Secod Way to Vew It show for 3: <Ths s Fg..7 fro Porat s Book>
.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.
.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,,, }
.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
.4. Polyphase Rep of Dec cot. ad after we get: Y { V } { P X } 4444 4444 3 P P P U { } { X } 44 444 3 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.
.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
.4.3 Polyphase Rep of Exp cot. l teger "backwards" l 443 l 3 3 3 3 3 3 Expaso Re-Idex Table
.4.3 Polyphase Rep of Exp cot. Usg ths re-dexg gves l h x l h x l y h x y 4 4 3 4 4 4 4 3 4 4 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
.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
.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!!
.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!!
.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 /