Tracking Performance of the MMax Conjugate Gradient Algorithm Bei Xie and Tamal Bose

Transcription

1 racki Performace of the MMa Cojuate Gradiet Alorithm Bei Xie ad amal Bose Bradley Det. of Electrical ad Comuter Eieeri Viriia ech

2 Outlie Motivatio Backroud Cojuate Gradiet CG Alorithm Partial Udate Methods Partial Udate CG Alorithm racki Performace Aalysis Simulatios Summary

3 Motivatio Ho to reduce the comutatioal comleity of a adative filter? Solutios: Usi artial udate PU methods. What are the tracki erformace by usi artial udate methods to CG?

4 Cojuate Gradiet Alorithm b Solve system ith form Equivalet to fid a to miimize the cost fuctio 1 J b he radiet of the cost fuctio is: b J he residual vector is defied as: b J

5 A lie search method for miimizi the cost fuctio has the form: 1 is the directio vector CG chooses the directio is cojuately orthooal to the revious directios m 0, m

6 No e fid to miimize he residual vector is also equal to he residual vector is orthooal to the revious directio vectors, 1 J b J 1 1 b 1 1 b b m m, 0

7 CG chooses the directio i the form of Use Polak-ibière P method, 1 β β P method is chose because it is a o-reset method ad erforms better for o-costat matri CG ith P method usually coveres faster tha Fletcher- eeves F method

8 CG Alorithm i adative filter system A basic adative filter system model is d * v d is the desired sial =[,-1,,-N+1] is the iut data vector of a uko system * =[ 1*, *,, N* ] is the imulse resose vector of the uko system * is costat for time-ivariat system * chaes for time-varyi sytem v is a hite oise

9 o estimate the ad b i he eoetially decayi data ido is used b d i i d i i i i i i b b λ is the foretti factor

10 , 1 1 β β d b he CG alorithm i a adative filter system is summarized as: Iitial coditios: 0=0, 0=0, 1=0 η is used to uaratee coverece

11 Partial Udate PU Methods Udate art of the eihts to save the comutatioal comleity Each udate ste, udate M<N coefficiets Basic PU methods iclude eriodic, sequetial, stochastic, ad MMa methods he eriodic method: udate the eihts at every S th iteratio ad coy the eihts at the other iteratios, here S M N

12 he sequetial method: choose the subset of the eihts i a roud-robi fashio. he stochastic method: is a radomized versio of the sequetial method. Usually a uiformly distributed radom rocess ill be alied. he MMa method: the elemets of the eiht are udated accordi to the ositio of the M larest elemets of the iut vector.

13 ˆ ˆ , ˆ 1 ˆ ˆ 1 ˆ ˆ β β d Partial Udate CG Alorithm he artial udate CG alorithm i a adative filter system is summarized as:

14 01,, ˆ 1 1 i M i i i i k N k k N M M I I he umber of multilicatios of CG is 3N +10N+3 er samle he umber of multilicatios of PU CG is N +M +9N+M+3 er samle

15 he MMa method: the elemets of the iut are chose accordi to the ositio of the M larest elemets of the iut vector i k 1 0 if k otherise ma 1l N l, M SOLINE method is used for comariso Mma CG eeds + lo N comarisos

16 racki Performace Aalysis of PU CG Desired sial becomes: d ime-varyi system * uses a first-order Markov model * * is very close to uity is rocess oise * v 1

17 Assumtios: he coefficiet error -* is small ad ideedet of the iut sial at steady state White oise v is ideedet of the iut sial ad is ideedet of rocess oise η Iut sial is ideedet of both v ad η

18 At steady state, the MSE of PU CG ith correlated iut is 1 1 ~ ˆ ~ 1 1 v v tr d E e E ˆ ˆ ˆ ˆ 1 ~ ~ ~ 1 ˆ ~ E E E

19 ~ ˆ ~ ˆ 4 ~ ˆ v E tr tr tr tr N e E v v v At steady state, the MSE of PU CG ith hite iut is Variace of oise

20 1 1 1 tr N e E v v For MMa method ad hite iut, κ <1, κ is close to 1 ~ ˆ, For hite rocess oise η v v N e E

21 Simulatios System idetificatio model he iitial imulse resose of uko system is 16-order N=16 FI filter *=[0.01,0.0,-0.04,-0.08,0.15,-0.3,0.45, 0.6, 0.6,0.45,-0.3,0.15,-0.08,-0.04,0.0,0.01]

22 he variace of the iut oise σ v = Parameter λ=0.9 ad η=0.6 White iut, variace is 1 i Markov model is

23 racki Performace of MMa CG MMa CG, M=8, = MMa CG, M=4, = MMa CG, M=8, =0.001 MMa CG, M=4, = MMa CG, M=8, = MMa CG, M=4, = MSEdB MSEdB samles samles Comariso of MSE of MMa CG for varyi rocess oise η, M=8 Comariso of MSE of MMa CG for varyi rocess oise η, M=4

24 he MSE of MMa CG icreases he the rocess oise icreases. he variace of the MSE icreases he the rocess oise icreases. he artial udate leth does ot have much effect o the MSE results. he artial udate leth oly affects the coverece rate. he coverece rate decreases as the artial udate leth decreases.

25 able 1. he simulated MSE ad theoretical MSE of MMa CG for varyi rocess oise η, M = 8. Process oise σ η Simulated MSE db heoretical MSE db able. he simulated MSE ad theoretical MSE of MMa CG for varyi rocess oise η, M = 4. Process oise σ η Simulated MSE db heoretical MSE db he theoretical results match the simulated results.

26 Performace comariso betee MMa CG ad MMa LS CG MMa CG M=8 LS MMa LS M= CG MMa CG M=4 LS MMa LS M=4 MSEdB MSEdB samles Comariso of MSE of MMa CG ith CG, LS, MMa LS for hite iut, N=16, M= samles Comariso of MSE of MMa CG ith CG, LS, MMa LS for hite iut, N=16, M=4. After 000 samles/iteratios ass, the uko system is chaed by multilyi all coefficiets by -1. he artial udate leth oly affects the coverece rate at the beii i this case.

27 able 3. he comutatioal comleities of CG, MMa CG, LS, ad MMa LS. Alorithms Number of multilicatios er symbol Number of comarisos er symbol CG N= MMa CG N= MMa CG N= LS N= MMa LS N= MMa LS N=

28 Summary he tracki erformace of the MMa CG is aalyzed heoretical mea-square erformace is derived for hite ad correlated iuts he tracki erformace of MMa CG is comared ith CG, LS, MMa LS by usi comuter simulatios he MMa CG alorithm ca achieve similar erformace to the full-udate CG hile reduci comutatioal comleity siificatly he MMa CG alorithm ca achieve similar erformace to the MMa LS hile havi loer comutatioal comleity