11.11 Two-Channel Filter Banks 1/27

Transcription

1 . Two-Chael Filter Baks /7

2 Two-Chael Filter Baks M We wat to look at methods that are ot based o the DFT I geeral we wat to look at Fig..6 rom Porat ad igure out how to choose i & i to get Perect Reco /7

3 Why Two-Chael Filter Baks? Why Cosider Oly Two Chaels? Simple Place to Start Easier to Derive But ideas exted to more chaels Provide buildig blocks or tree-structure used to get more chaels This last reaso is the most importat!! I our aalysis we will assume: Subbad Processig is Igored i.e., we ocus o perect reco w/o subbad processig Maximal Decimatio is to be used i.e., we set decimatio actor K 3/7

4 Two-Chael Structure to Cosider x LPF PF u u v v Σ y Aalysis Bak Sythesis Bak - -/ / Q: ow to choose,,,? Q: Ca we get Perect Recostructio PR? I so OW?!!! 4/7

5 Two-Chael Properties Write output Y i terms o iput X see what we get & what it tells us Start at iput ad work towards output: x u u v v Σ y U i i X, i, Ca easily show that or ollowed by we get: V i Ui + Ui, i ow puttig ito this gives:, 5/7

6 Two-Chael Properties cot. V i i X + i X, i, ow look at how the output is ormed rom the V i s: x u u v v Σ y Y V + V Usig i this gives: 6/7

7 7/7 Two-Chael Properties cot. X X Y Two Terms: Respose to Iput Sigal X Respose to Alias Compoet X- + X X e X X j e j - X - - X + Alias Compoet -

8 Two-Chael Properties cot. Do t wat X- i output to elimiate: ecessary & suiciet to have: + We ll call this the Aliasig Cacellatio Coditio ACC Oe way to satisy the ACC is to choose the sythesis ilters as: Lowpass ighpass ote: This does ot elimiate aliasig i each chael!!! The ilters are chose so: Aliasig i the chaels cacel each other i the summatio! 8/7

9 9/7 Two-Chael Properties cot. ow that we have the aliasig cacelled. What does the output sigal look like??? Usig our previous results: X X Y Deie X Y F Aalysis-Sythesis Traser Fuctio Output Whe Aliasig as Bee Cacelled:

10 Two-Chael Properties cot. ow do we get Perect Recostructio??? For PR we eed: y c x l F c l F ce jl I we do t have Perect Recostructio: F A e jφ Liear Phase Respose Flat Magitude Respose I A costat, the we have Amplitude Distortio I φ l, the we have Phase Distortio Choosig the ilters to get PR drew much research eort /7

11 Quadrature Mirror Filters QMF QMF was oe o the early attacks o ilter bak desig But it ell short o givig PR. ote that the ACC puts o costraits o the idividual ilters: it says. i the aalysis ilters are this the the sythesis ilters must be that So we are ree to choose the aalysis ilters to try to meet other eeds: ood Stopbad Rejectio Perect Recostructio Liear Phase Etc. QMF Baks were developed i 977 by Esteba & alad as a attempt to desig good -chael FB s.. they are useul but ot perect!!!!! /7

12 Quadrature Mirror Filters cot. Deiitio o QMF: A pair o aalysis ilters are QMF i e j e e j e j j QMF Coditio QMF Coditio is equivalet to: Proo: Shit by.5 both sides o DTFT QMF Coditio ow use symmetry o DTFT X X o Rt ad Side /7

13 Quadrature Mirror Filters cot. This is a mirror image criteria is the mirror image o aroud the poit / Magitudes are eve images / Phases are odd images / 3/7

14 Quadrature Mirror Filters cot. So QMF desig requires: Desigig as a lowpass ilter Desig may strive to meet various spec s e.g., stopbad levels, trasitio widths, etc. The you get rom usig or QMF This esures that is a PF, as desired. ow recall that with ACC satisied the aalysis-sythesis traser uctio is: F Usig the QMF Coditio i this gives: F ote: This T-F is completely speciied i terms o QMF Aalysis-Sythesis Traser Fuctio This makes QMF desig airly easy compared to other FB desig schemes 4/7

15 5/7 Quadrature Mirror Filters cot. Q: Ca we get Perect Recostructio rom QMF Filter Baks???? Fact: FIR QMF s ca oly give PR whe the ilter has o more tha coeiciets. Proo: Recall that or the QMF case the FB traser uctio is: F But this ca be re-writte as: F + The requiremet or PR is F c l which implies that we eed: l l c c + or some c, c, l, l Solvig this leads to: l l l l c c c c + These are -tap FIR Filters Bit o big jump here see other otes or better proo

16 Quadrature Mirror Filters cot. So, oly way to get PR rom QMF is to use -tap ilters: Serious Limitatio But o-pr QMF s are still used. ote: IIR QMF s do t have this limitatio but have their ow drawbacks e.g., oliear phase Two-Tap FIR ilters have poor passbad, stopbad, & trasitio bad perormace: i db / 6/7

17 7/7 Quadrature Mirror Filters cot. Result: I has liear phase, the the resultig QMF aalysissythesis FB is ree o phase distortio i.e., the FB also has liear phase. ote: Must use FIR to get liear phase Proo: Let be a th order FIR ilter with liear phase, the j e A.5 Amplitude Fuctio real valued, but ca go egative Liear Phase The we have: j j j e A e e A.5 /.5 ad j e A A F

18 Quadrature Mirror Filters cot. Aside: ca t be chose to be eve valued!!! ere s why: From stadard FT properties eve magitude: A A Thus F / A / i is eve!! Thus, eve causes complete atteuatio at /. BAD!!! Back to the proo Sice is odd we get: F A + A e j FB s Magitude Respose FB s Phase Respose The FB has liear phase. Ed o Proo 8/7

19 Quadrature Mirror Filters cot. Result: A liear-phase QMF FB will have amplitude distortio except or the -tap case. The amplitude distortio is give by D amp F + QMF Desig Method: Use umerical optimiatio techiques to id a liear-phase that miimies D amp. <See Books Etirely o Multirate & Filterbaks> 9/7

20 Perect Recostructio Filter Baks With careul desig you ca get QMF s to give early perect reco To get perect recostructio we must replace the QMF coditio by a dieret coditio. Recall QMF: ew Coditio: is FIR Order Filters satisyig this: Cojugate Quadrature Filters CQF The CQF coditio is equivalet to: g g /7

21 /7 PR Filter Baks cot. Proo o this equivalece: Let / g w W See 3.. i P&M Let g g w w W W See 3..5 i P&M Let 3 3 g w w W W See 3..9 i P&M with a - Fially ote that: / 3 g g W W W <Ed o Proo>

22 /7 PR Filter Baks cot. Restrictig to be Odd as we did or QMF gives: F I this, the we get PR! } { * W w m e w m e e w w w W m m j m m j j F ote this result: Ad. W W

23 3/7 PR Filter Baks cot. Sice or PR we wat + Usig these two results gives the equivalet equatio: A ilter satisyig this is called Power Symmetric

24 PR Filter Baks cot. Desig Procedure or CQF-PR Filter Bak. Fid a odd-order, power symmetric LPF with passbad o,/ ad havig desired stopbad atteuatio ives. Use CQF Coditio to get 3. Use Aliasig Cacellatio Criteria to get: Step # is Challegig!!!! See books etirely o Multirate & Filterbaks 4/7

25 Tree-Structured Filter Baks Q: What do we have ater a two-chael split & decimatio? Ideal Filter or Illustratio X / / Ideal Filter or Illustratio / / You et The Desired al-bads, But Spread Over,!!!! All Ready to Re-Apply the SAME -Bad Split ote: o Aliased terms due to Ideal Filters 5/7

26 A Full-Tree Cascade o Two-Bad Filter Baks Uiormly Spaced Filter Bak x u u u u 3 Ch. 3 Ch. Ch. Ch. 6/7

27 There are other ways to prue x P-Prued Tree Cascade o Two-Bad Filter Baks Octave-Bad Filter Bak Used to Implemet Wavelet Trasorm u u u u 3 Ch. 3 Ch. Ch. Ch. Each higher bad is twice as wide 7/7