Blind Adaptive Filters & Equalization

Transcription

1 Blid Adaptive Filters & Equalizatio

2 Table of otets: Itrodutio about Adaptive Filters Itrodutio about Blid Adaptive Filters Covergee aalysis of LMS algoritm Ho trasforms improve te overgee rate of LMS Wy Wavelet trasform? Our blid equalizatio approaes Simulatio results

3 Equalizatio { a } Pulse Sape Modulator Cael equalizer MF aˆ { } { a } Pulse Sape Modulator Cael MF Equalizer [] aˆ { } Composite Cael []

4 ± ± N N k k k,..., ] [ ] [ ]* [ δ N N i i i ] [ ] [ δ + : Eample

5 Adaptive Equalizatio Delay Iput Sigal Cael Adaptive Equalizer y e Radom oise Geerator ) e a E[ a + y, error k k ) E[ e ] a k k ]

6

7 ] [ ] [ ] [ k k k k e E y e E e e E error N N k k k y ] [ e E error y a e k k k error a + µ ) )

8 ] [ k k e E error N N k k k y y a e k k k error a + µ ) ) T N N T N N W X ],...,,,,..., [ ],...,,,,..., [ + + T W X Y y a e e µ + +

9 Equalizatio, Deovolutio, System ompestatio

10 System Idetifiatio

11 Noise Caellatio

12 Preditio

13 Sidelobe aellatio y N yt)σ i t)

14 a y a a N N yt)σa i i t)

15 Soud Clip Amplitude of Sigal NORMALIZED INPUT VOIC E SIGNAL *.WAV) Normalized *.av file mirosoft format) 9,946 bytes lik ere Number of Samples

16 Graps /o ad / equalizatio WITH NO EQUALIZATION WITH EQUALIZATION.8 Voie Sigal, Wit No Equalizatio Origial Voie Sigal.8 Voie Sigal, Wit Equalizatio Origial Voie Sigal Amplitude of Sigal Amplitude of Sigal Number of Samples Number of Samples Simulatio uder te advisory of Prof. Fotaie doloaded)

17 Blid Equalizatio Delay Iput Sigal Cael Adaptive trasversal equalizer Additive ite Gaussia Noise

18 Blid equalizatio approaes Stoasti gradiet deset approa tat miimizes a ose ost futio over all possible oies of te equalizer oeffiiets i a iterative fasio Higer Order Statistis HOS) metod tat is usig te iger order umulats spread of te uderlyig proess, ad ee to te flatess Approaes tat eploit statistial ylostatioarity iformatio oeffiiets toard teir optimum value, at a give frequey Algoritms tat are based o te maimum likeliood riterio. depeds o te value of te poer spetral desity of te

19 Blid Equalizatio: HOS Blid Equalizatio: HOS First Order Statistis : ) ) )} { ) d f t t E C X d t t f t t t t E C X )) ), ) ) )} ) { ) Seod Order Statistis : ) τ τ τ τ d t t t f t t t t t t E C Tird X )) ), ), ) ) ) )} ) ) { ), Order Statistis : 3) τ τ τ τ τ τ τ τ

20 Blid Equalizatio: HOS Wite Noise Cael [] y[] AWGN ) C y S y f ) H f ) Siput f ) + S f ) C ) y S y f ) H f ) K

21 Blid Equalizatio: HOS Wite Noise Cael [] y[] For Gaussia sigals: AWGN ) C for > C C + 4) y 4) C 4)

22 Fratioally Spaed Equalizer [] tt/p Ym) y m) W l) m lp) + m) l [] [] i [] N [] tt/p tt/p tit/p tnt/p y [m] y [m] y i [m] y N [m] y i m) W l) i m l) + i m) Y l N N + L + ) HW ) N ) N

23 Fratioally Spaed Equalizer Fratioally Spaed Samplig Cylostatioary Output Y N N + L + ) HW ) N ) N

24 Formula: Cael model Cost futio defiitio Updatig equalizer oeffiiets Our proposed Wavelet domai gradiet J ˆ T ) Y ) Y ) g N N σ W H :,)

25 Covergee rate of LMS algoritm It is ell ko tat te overgee beavior of ovetioal LMS algoritm depeds o te eigevalue spread of iput proess Faster overgee rate of LMS algoritm Smaller EIG-spread of iput orrelatio matri

26 EIG-spread of iput orrelatio matri R) vs. flatess of its PSD Covergee rate of filter oeffiiets toard teir optimum value, at a give frequey depeds o te value of te poer spetral desity of te uderlyig proess at tat frequey relative to all oter frequeies. Smaller EIG-spread of iput orrelatio matri PSD flatess of iput sigal

27 EIG-spread of R vs. sape of error surfae: Eample, EIG.

28 Eample, EIG 3

29 Eample 3, EIG

30 Summary: Better overgee rate of LMS algoritm Sape irularity) of error surfae Smaller EIG-spread of iput orrelatio matri PSD flatess of iput sigal

31 Ho trasforms improve te overgee rate of LMS? Bad-partitioig property of Wavelet trasform Trasformed elemets are at least) approimately uorrelated it oe aoter Correlatio matri is loser to a diagoal matri A appropriate ormalizatio a overt te result to a ormalized matri ose EIG spread ill be mu smaller

32 Advatages of usig Wavelet trasform Effiiet trasform algoritms eist e.g. te Mallat algoritm) Trasforms a be implemeted as filter baks it FIR filters Strog matematial foudatios allo te possibility of ustom desigig te avelets e.g. te liftig seme

33 Wavelet trasform algoritm:

34 Matri form implemetatio of Wavelet trasform As metioed i te previous slide, Wavelet trasform osists of to lo-pass ad ig-pass filters Data Widt Lo Pass Hig Pass

35 Wy Wavelet trasform? avelet aalysis filters are ostat-q filters; i.e., te ratio of te badidt to te eter frequey of te bad is ostat

36 bad-partitioig property of Daubeies filters After trasformatio, ea oeffiiet sos te amout of eergy passed from oe of above filters

37 Te badidt of te filters i lo frequeies is arro ompared to te badidt of te filters i iger frequeies Most ommuiatio sigals ave a lo-pass ature It s more probable tat te output of filters otai te same amout of eergy more likely to obtai a flat spetrum.

38 PSD of a typial ommuiatio sigal after differet trasforms

39 Effet of Wavelet trasform o error surfae

40 MSE of a TD-Godard algoritm vs. WD-Godard algoritm

41 Formula: Cael model Cost futio defiitio Updatig equalizer oeffiiets Our proposed Wavelet domai gradiet

42 MSE of a TD-FSE algoritm ompared it WDFSE algoritm

43 Referees Adaptive Filter Teory Simo Hayki Pretie Hall

44 Adaptive Filters Teory ad Appliatios B. Farag- Boroujey iley