Uniform DFT Filter Banks 1/27

Transcription

1 .. Ufor FT Flter Baks /27

2 Ufor FT Flter Baks We ll look at 5 versos of FT-based flter baks all but the last two have serous ltatos ad are t practcal. But they gve a ce trasto to the last two versos whch ARE useful ad practcal ethods. Verso # Not P& Verso #2 Not P& Udecated Rect. Wdow Sldg FT Flter Se # Chaels ecated Rect. Wdow Sldg FT Flter Se # Chaels Verso #3 ecated No-Rect Wdow Sldg FT Not P& Flter Se # Chaels Verso #4 ecated Arbtrary Wdow Sldg FT Not P& Flter Se Arbtrary Verso #5 ecated Polyphase Flter FT.. Flter Se Arbtrary Oly Versos 4 & 5 are Practcal ethods 2/27

3 Settg for Versos, 2, & 3 We wll llustrate wth a four chael case: x 4 Chael Flter Bak u u u 2 u 3 H θ H θ H 2 θ H 3 θ H 3 θ π/2 π 3π/2 2π θ H θ H θ H 2 θ Equvalet To -π -π/2 π/2 π θ 3/27

4 Verso #: Sldg FT Flter Bak x pt. CFT Sgal Blocks CFT Cojugate FT u 3 [] u 2 [2] CFT s Te u u u 2 u 3 4 Exaple Frequecy 4/27

5 fferet Vew of Verso # x x u 4 Exaple x[ ] x[ 2] x[ 3] CFT u u 2 u 3 Sequece of FT s u u u 2 u 3 u [ ] ath Vew x[ ] W,,..., Note: Cojugate FT For Fx, The Copute -pt. CFT 5/27

6 6/27 ath Shows ths OES gve Flter Bak ] [ where ] }[ * { ] [ ] [ W g g x W x u Output of ths structure s: Thus, the th output sgal s the lear covoluto of the put sgal wth the pulse respose g. Q: What s the th flter s Trasfer Fucto? W W W G a a a a N N N N 2 2 Use Geo. Su Result

7 7/27 ath Shows co.t Q: What s the th flter s Frequecy Respose? / 2.5 / 2 / 2 f ] / 2 s[.5 ] / 2 s[.5 j j j e e e e G G j π θ π θ π θ π θ π θ θ θ Looks sort of lke sc fucto: rchlet Kerel Cetered at θ 2π/ rad/saple Note: The wdow deteres the shape of the frequecy respose. The rectagular wdow used here akes a poor flter!!!

8 Frequecy Respose of Verso # Flterbak 2 Exaple Poor Passbad Poor Stopbad G θ 5 5 G θ θ/π θ/π Rego of Iterest π θ π Oly shows 3 of the 2 chaels ths exaple 8/27

9 Sythess Bak for Verso # Flterbak x x u v x[ ] u v x[ 2] CFT u 2 v 2 CIFT x[ 3] u 3 v 3 y Aalyss Sythess CIFT: Icludes / ter ot book! 9/27

10 Sythess Bak for Verso # cot. Sequece of FT s u u u 2 u 3 CIFT After elays, Su Up to Get Output x[] x[] x[2] x[3] x[4] x[5] x[6] x[] x[2] x[2] x[3] x[3] x[4] x[4] x[5] x[5] x[6] x[6] x[7] x[7] x[8] x[3] x[4] x[5] x[6] x[7] x[8] x[9] x[3]x[4]x[5] x[6] /27

11 Probles wth Verso # Flter Bak Total saple rate out of aalyss bak s tes put Ths s redudat ad s detretal applcatos lke data copresso Fxed by decatg Verso #2 #5 Frequecy Respose s Very Poor TFT of Rectagular Wdow Thus, stopbad atteuato s very bad ad passbad falls off Fxed by usg o-rectagular wdow Versos #3 #5 Flters UST have sae legth as uber of chaels Fxed Versos #4 & #5 Use SP trck Verso #4 Use Polyphase Structure Verso #5 /27

12 Verso #2: ecate Output Q: Ca we decate each chael s output ad stll be able to get back the orgal sgal after sythess? A: Yes overlappg of the FT wdows s excessve!!! x CFT Sgal Blocks Ca ecate ecato Factor Equals # Chaels Te u u u 2 u 3 Frequecy 2/27

13 x fferet Vew of Verso #2 elayed ad ecated Versos of Iput Sgal Oly dces show Aalyss CFT u u u 2 u 3 v v v 2 v 3 CIFT Sythess y Blocks Ito CFT 3/27

14 fferet Vew of Verso #2 cot. x Aalyss CFT u u u 2 u 3 v v v 2 v 3 CIFT Sythess y Up-shfted ad elayed Versos of CIFT Output Oly dces show /27

15 Verso #3: No-Rectagular Wdow x ust Have Wdow Legth # of Chaels ecato Factor CFT CFT Wdow w ust be o-ero over block Otherwse, ICFT wll gve back a ero sgal value ad ca t recostruct After ICFT, udo wdow by dvdg by wdow values Te u u u 2 u 3 Frequecy 5/27

16 fferet Vew of Verso #3 x w w w 2 w 3 CFT u u u 2 u 3 v v v 2 v 3 CIFT /w /w /w 2 /w 3 y Aalyss Wdow Values Ca t Be Zero!!!! Sythess 6/27

17 Ver. #4: Arbtrary Se Wd., Sldg FT Verso #3 Has Severe Ltato: Wdow se s set by uber of chaels desred ay force a short wdow flter se But log flters are ofte eeded to get desred frequecy respose To see how to reove ths ltato, back to the ath Vew: Recall ath Vew of Verso # [ ] u x[ ] W,,..., -Pt. FT u [ ] ath Vew of Verso #3 x[ ] w[ ] e j2π /,,..., Wdow Legth # Chaels ec. Factor No-Overlapped Blocks -Pt. FT 7/27

18 u [ ] L x[ Verso #4 cot. ath Vew of Verso #4 ] w[ ] e j2π / ec. Factor # Chaels L Wdow Legth < L,,..., NOT a FT!! L-pt su but.. Freq. Pts x FT FT u u u 2 u 3 8/27

19 If t s NOT a FT, What IS t??!! For each,,, Coplex Susod w/ Frequecy of 2π/ e j2π/ L Pots w[]x[ ] o for Each,,, To Get The Chaels Oe Wdowed Sgal Block For Each Value Σ OK, But How To Copute Ths Effcetly??!! 9/27

20 Here s How To Copute Ths No-FT Effcetly!! A SP TRICK!!!! e j2πk/ L Pots Pots ust have L k w/ k Iteger Sae Isde Each -Pot Block!!! x w Σ Ths IS a -pt. FT of the Su of Blocks!! Aalogy: Arthetc strbuto a d b d c d a b c d 2/27

21 Verso #4: Suary esg of Flter Bak Assue that # of Chaels,, has bee specfed Usually pck as power of two to allow use of FFT Choose Wdow Shape ad Wdow Legth, L, to gve desred passbad ad stopbad characterstcs To eable good flter, pck L > ; also pck L as teger Choose ecato Factor,, as large as possble wthout geeratg excessve ter-bad alasg Algorth Ipleetato Apply L-pt wdow to curret sgal block Break wdowed L-pt block to -pt sub-blocks Add all the -pt sub-blocks together to get a sgle -pt block Copute the -pt FT usg FFT algorth Each FT coeffcet s the curret output of a chael ove the L-pt wdow ahead pots Called Overlap & Add FT-Based Flter Bak See coped pages posted o Blackboard For Sythess: Crochere & Raber, ultrate gtal Sgal Processg, Pretce Hall, /27

22 22/27 Ver. #5: Arb. Se Wd., Polyphase, FT ] [ } * {,...,,, ] [ ] [ ] [ ] [ / 2 g x e g x u g j L Δ π Recall ath Vew of Verso #4 or Chages w g Vew: Each chael of FB cossts of flter g that s a frequecy-shfted verso of a prototype lowpass flter g. All the ufor FBs we ve looked at ca be vewed ths way. I the Frequecy & Z oas ths s: / 2 ] [ ] [ f f / 2 W j W G G G G e g g π θ θ π Frequecy Shft

23 23/27 Ver. #5: evelopet Approach:. Wrte the prototype LPF ts polyphase ters 2. odulate result to get the chael flters 3. Use result to wrte pre-decato chael output 4. Wrte post-decato chael output P G Step # P W W P W W G G Step #2 apply decato detty vew as put X P W U Step #3 Flter the ec { } { } X P W U Step #4 ec the Flter U X G U

24 Ver. #5: Iterpret { } U W P X elay by ecate by Flter w/ th Polyphase Flter -pt CFT over Polyphase Braches x P u P P 2 P 3 CFT u u 2 u 3 24/27

25 Ver. #5: Sythess x P P P 2 P 3 CFT u u u 2 u 3 v v v 2 v 3 CIFT Q Q Q 2 Q 3 y Cacel Each Other Lke P coects to Q Y {{ } } X P Q 25/27

26 26/27 Ver. #5: Sythess oes t Work? { } { } Q P X Y Here s where we were o the last slde: / / Q P W X W Y Use Z-oa result for operato: Use Z-oa result for operato: Ths gves"perfect Recostructo" to get ths... ] [ Wat X c c l l Q P W W X Q P W X W Y δ Requreet for Perfect Recostructo

27 27/27 Ver. #5: Perfect Reco Crtera ] [ IFT of c Q P W l Q P δ Look at what we saw o the last slde:, c Q P l Takg FT of each sde gves a Equvalet PR Crtera: Geeral Flter esgs to eet Ths are HAR!!! We Wo t Cover It Specal Cases: Verso #2 s. P Q, Verso #3 s. P w[] & Q / w[],