13.8 Signal Processing Examples

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Randell Dixon
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1 13.8 Sigal Pocessig Eamples E Time-Vaig Chael Estimatio T Mlti Path v(t) (t) Diect Path R Chael chages with time if: Relative motio betwee R, T Reflectos move/chage with time ( t) = T ht ( τ ) v( t τ ) dτ T is the maimm dela Model sig a time-vaig D-T FIR sstem = p k= h k v k Coefficiets chage at each to model time-vaig chael 1

2 I commicatio sstems, mltipath chaels degade pefomace (Ite-smbol itefeece (ISI), flat fadig, feqec-selective fadig, etc.) Need To: Fist estimate the chael coefficiets Secod Bild a Ivese Filte o Eqalize Boad Sceaios: 1. Sigal v(t) beig set is kow ( Taiig Data ). Sigal v(t) beig set is ot kow ( Blid Chael Est. ) Oe method fo sceaio #1 is to se a Kalma Filte: State to be estimated is h = h h p T (Note: h hee is o loge sed to otate the obsevatio model hee)

3 Need State Eqatio: Assme FIR tap coefficiets chage slowl Assmed Kow That is a weakess!! h = Ah 1 + Assme FIR taps ae coelated with each othe < coelated scatteig > A, Q, C h, ae Diagoal cov{h-1} = M-1-1 cov{} 3

4 Need Obsevatio Eqatio: Have measemet model fom covoltio view: p = h k v k + k= T = v h + w Kow taiig sigal w zeo-mea, WGN, σ Obsevatio Mati is made p of the samples of the kow tasmitted sigal State Vecto is the filte coefficiets 4

5 Simple Specific Eample: p = (1 Diect Path, 1 Mltipath) h = Ah A =.999 Tpical Realizatio of Chael Coefficiets.1 Q =.1 Q =cov{} Note: h decas faste ad that the adom petbatio is small Book does t state how the iitial coefficiets wee chose fo this ealizatio 5

6 Kow Tasmitted Sigal Noise-Fee Received Sigal <It is a bit odd that the eceived sigal is lage tha the tasmitted sigal> Nois Received Sigal The vaiace of the oise i the measemet model is σ =.1 6

7 Estimatio Reslts Usig Stadad Kalma Filte ˆ = T Iitializatio: h 1 1 = M 1 1 = 1I σ. 1 h h 1 Tasiet de to wog IC Evetall Tacks Well!! Chose to eflect that little pio kowledge is kow I theo we said that we iitialize to the a pioi mea bt i pactice it is commo to jst pick some abita iitial vale ad set the iitial covaiace qite high this foces the filte to stat ot tstig the data a lot! 7

8 Deca dow elies moe o model Gai is zeo whe sigal is oise ol Kalma Filte Gais Kalma Filte MMSE Filte Pefomace impoves with time 8

9 9 Eample: Rada Taget Tackig State Model: Costat-Velocit A/C Model!#!" $!! $!#!"! "!! $! #!#!" $ v v v v s A s + = = = } cov{ σ σ Q + + = ta 1 w w R β Obsevatio Model: Nois Rage/Beaig Rada Measemets Fo this simple eample. assme: = = } cov{ σ β σ R w C Velocit petbatios de to wid, slight speed coectios, etc. Velocit petbatios de to wid, slight speed coectios, etc. i adias

10 Eteded Kalma Filte Isses Need the followig: 1. Lieaizatio of the obsevatio model (see book fo details) Calclate b had, pogam ito the EKF to be evalated each iteatio. Covaiace of State Divig Noise Assme wid gsts, etc. ae as likel to occ i a diectio w/ same magitde! model as idep. w/ commo vaiace Q = cov{ } = σ σ σ = what??? Note: / = acceleatio fom -1 to So choose σ i m/s so that σ / gives a easoable age of acceleatios fo the tpe of taget epected to tack 1

11 3. Covaiace of Measemet Noise The DSP egiees wokig o the ada sall specif this o bild oties ito the ada to povide time pdated assessmets of age/beaig accac Usall assme to be white ad zeo-mea Ca se CRLBs fo Rage & Beaig " Note: The CRLBs deped o SNR so the Rage & Beaig measemet accac shold get wose whe the taget is fathe awa Ofte assme Rage Eo to be Ucoelated with Beaig Eo " So se C = diag{σ R, σ β } Bt best to deive joit CRLB to see if the ae coelated 11

12 4. Iitializatio Isses Tpicall Covet fist age/beaig ito iitial & vales If ada povides o velocit ifo (i.e. does ot mease Dopple) ca assme zeo velocities Pick a lage iitial MSE to foce KF to be biased " If we follow the above two ideas, the we might pick the MSE fo & based o statistical aalsis of covesio of age/beaig accac ito & accacies Sometimes oe ada gets a had-off fom some othe ada o seso " The othe ada/seso wold likel had-off its last tack vales so se those as ICs fo the iitializig the ew ada " The othe ada/seso wold likel had-off a MSE mease of the qalit its last tack so se that as M-1-1 1

13 State Model Eample Tajectoies: Costat-Velocit A/C Model 5 Red Lie is No-Radom Costat Velocit Tajecto = 1 sec Y positio (m) σ =.316 m/s v 1 = 1 1 =. ( σ =.1 1 = 5 m v m 1 =. m/s /s ) -5 Rada X p os itio (m) 13

14 Obsevatio Model Eample Measemets σ σ R β =.316 m ( σ R =.1 =.1 ad = 5.7 deg ( σ m R ) =.1 ad ) I ealit, these wold get wose whe the taget is fa awa de to a weake eted sigal Red Lies ae Noise-Fee Measemets Rage R (metes) Sample Ide Beaig β (degees) Sample Ide 14

15 Measemets Diectl Give a Poo Tack If we tied to diectl covet the ois age ad beaig measemets ito a tack this is what we d get. Not a ve accate tack!!!!! Need a Kalma Filte!!! Bt Noliea Obsevatio Model so se Eteded KF! Note how the tack gets wose whe fa fom the ada (agle accac covets ito positio accac i a wa that depeds o age) Rada 15

16 Eteded Kalma Filte Gives Bette Tack Note: The EKF was with the coect vales fo Q ad C (i.e., the Q ad C sed to simlate the tajecto ad measemets was sed to implemet the Kalma Filte) Iitializatio: s-1-1 = 5 5 T M-1-1 = 1I Picked Abitail Set lage to asset that little is kow a pioi Rada Iitializatio Afte abot samples the EKF attais tack eve with poo ICs ad the lieaizatio. Tack gets wose ea ed whee measemets ae wose MSE show obtai tack ad show that thigs get wose at the ed 16

17 MSE Plots Show Pefomace Fist a tasiet whee thigs get wose Net the EKF seems to obtai tack Fiall the accac degades de to age magificatio of beaig eos 17

12.6 Sequential LMMSE Estimation

12.6 Sequential LMMSE Estimation 12.6 Sequetial LMMSE Estimatio Same kid if settig as fo Sequetial LS Fied umbe of paametes (but hee they ae modeled as adom) Iceasig umbe of data samples Data Model: [ H[ θ + w[ (+1) 1 p 1 [ [[0] [] ukow