9. Transient Performance of Adaptive Filters
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1 9. rnsnt Prformnc of Adt ltrs Adt fltrs r tm-rnt nd non-lnr stochstc systms th nhrnt lrnn nd trcn blts. f thy mt ndrlyn rrmnts for stdy-stt rformnc nd trcn, thy shold ntlly roch th dfnd rformnc lls n noh tm. Bt ht s noh tm for n nsmbl of dttons? hs s ht ths scton on trnsnt rformnc ll b bot. Most mortnt sctons (from rmbl) 9.4 nd 9.5. S Chtrs to 5 of th on-ln txt boo. 9. Dt Modl sn th dt modl from scton 6 W ssm th dt stsfs th follon condtons () hr xsts ctor o o sch tht d (b) h nos snc { () } s..d. th rnc (c) h snc () s ndndnt of ll j for ll nd j (d) h ntl condton - s ndndnt of ll { d(j), j, (j) } () h rrssor cornc mtrx s R (f) h rndom rbls { d(),, () } h zro mn. h
2 9. Dt-Normlzd Adt ltrs hs scton s concrnd th dt normlzd fltr th ht dts of th form: th d nd hr [ ] s som ost-ld fncton of. or LMS or -NLMS Not: ths s snfcnt chn from th ros strctr nd th dfnton of [()] hch sd to b 9.3 Whtd nry-consrton Rlton W shll b dln th htd ctor norms, s rosly sn th th RLS lorthm drtons. Whr x x x for som rmtn ost-dfnt htn mtrx. h choc of = rsltsn th stndrd cldn norm of x. x x x x x x o stdy th trnsnt rformnc r ntrstd n th thortcl dt rformnc of nd hr th frst trm rlts nd th scond o o h lton of th scond trm ll rr th htd norm R
3 3 ormn htd-norm rson of th nry rlton, sn th roch of scton 6.3. sn th follon dfntons: o o o Notc tht or th drton, strt th n normlzd ht dt o o mltly both sds by Sbstttn lnt xrssons bsd on th rror stmts or, dr th nry rlton by soln nd sbstttn for th trm n [*] nd n trms of th ht norms An sn htd norm, form th htd nry trms of both sds
4 4 xndn th trms of both sds Rrtn to rconzd htd norms Bsd on th Orthoonlty of trms (rrors nd hts), ths rslt n or lntly h th htd nry-consrton rlton Not tht for, th fnl ton s lso corrct. A ht rcrson rlton cn b dfnd s d Sbstttn nto th ht dt ton hs s nrlzd ht rror rcrson tht xsts for ll dt systm mtn th dt modl.
5 5 9.4 Whtd Vrnc Rlton h htd rnc rltonsh no lonr strts th th ssmton of l ht norms, bt stll sbsttts for th -ostror htd stmton rror xndn th ror-htd rror mntd srd trm 4 lmntn common trms nd ddn throh by th htd norm of sn Sbstttn to lmnt () nd xndn sms hr r som trms to dl th htd nd nhtd -ror rrors
6 6 Anlyzn scfc trms or nd sn Sbstttn nd rconzn th orthoonlty of n xctd l of () nd th -ror rrors or s dfnd n th txt th htd norm ortons h sclr ntts cn b ncldd n th htd norm fnctons nd th common hts cn b combnd. rst, dfn hn hs s th n htd rnc rlton. An, t s n xct rltonsh thot ssmtons.
7 7 ndndnc Assmton hs rlton s dffclt to rot d to t s dndnc on th rrsson ctor. h ndndnc ssmton s sd s n ddton to th modl n ordr to lt th trtons. hrfor, ssm tht: h snc { } s ndndnt nd dntclly dstrbtd. h rslts h bn shon to b rsonbl sn ths ssmton, bt n nrl, t s not ld. hs s n ssmton, nd tht s hy trnsnt nlyss s srt from th ros or n Sctons 6, 7, nd 8. Wth ths condton, cn sy tht h stmton rror,, t - s ndndnt of both nd hs follos s th stmton rror,, t - s dndnt on th ros rrsson ctors j nd (j) for j<, hl s fncton of. hn cn sy tht No, th htn mtrcs bcom dtrmnstc nd no lonr mst b trtd s rndom rbls. h ht rnc rlton cn thn b comtd s sn th nrlzd ht rcrson, ros dlod nd th ndndnc ssmton, bcoms
8 8 Wth th lst trm lmnts bn ndndnt (orthoonl) bcoms W no h rcrson n th ht rror nd n. o smmrz: h trms of ntrst for soln th trtons form th follon thorms: horm 9.4.: Wht rnc rlton th ndndnc horm 9.4.3: Mn ht rror rcrson h mltrt xctd ls tht mst b comtd: nd nd hs cn b nsty. So, hr ossbl, chn n coordnts to smlfy th comttons cn b rformd.
9 Connnt Chn of Coordnts W cn s mtrx mnlton to donlzbl th rrsson cornc. R h stmt-brs r thn nrtd by mltlyn by th ntry mtrx nd nd Not tht nd hs rslts n trnsformd rlton tht rlcs rbl-br for th ros rbls. S horm 9.4.4: trnsformd ht-rnc rlton or ny dt fltr of th form (9..), ny rmtn ost-dfnt mtrx, nd for dt stsfyn th modl (9..) nd th ndndnc ssmton (9.4.8), t holds tht: hr th trnsformd rbls r rltd to th ornl rbls s dscrbd. h trnsformd rson of th mn-ht rror rcrson bcoms. h sctons tht follo sho ho th bo rnc nd mn rltons cn b sd to chrctrz trnsnt rformnc of dt-normlzd dt fltrs.
10 9.5 rnsnt Prformnc of LMS or th LMS fltr d h n [] fncton sn th non-br rltonshs for th trms of ntrst th th xctd ls tht mst b comtd: nd nd By dfnton, R nd R r r h lst momnt s mor dffclt nd ll b rformd s to ossbl css, Gssn nd Non-Gssn.
11 Gssn Rrssors Assm tht th rrssor rss from crclr Gssn dstrbton coordnt chn ll rslt n donl cornc mtrx. So tht th ros ltons bcom nd r r hn n comtn th mor dffclt trm, th sclr trm cn b mod n th comtton s sn Lmm.B.3 on. 46 hch rrs donl cornc mtrx nd comlx rrssors, r mortnt Not: or rl dt nd rrssor,, Lmm.B. on. 44 shold b sd nd h follon ssms comlx rrssor. r sn th rslts for th thr trms, no h r r Not tht ll lmnts r non bsd on th ssmtons rosly md nd cn b trtly comtd s thortcl rformnc cr. W cn no rojct thortclly th rformnc of n dt systm bsd on -ror nold of th Gssn ntr of th rrssor nd th Gssn sttstcs. Also, f s donl thn s s ll nd frthr smlfcton cn b md to comt ctors nstd of donl mtrx. Smlfctons for donl mtrcs follo
12 Donl Notton Allo th stndrd mtrx to ctor dfntons to xst (th MALAB df of d) d nd d nd d nd d hn d d d d r d As both mtrcs r donl d d d r or or fnlly hr ntrstn rslt, thr s lnr ctor rlton btn sccss sm trtons for d d h rltons cn thn b smmrzd s nd Prorts of th smmrzd tons. Mn Bhor Mn-Sr Bhor Stt-Sc Stblty Stdy-Stt Prformnc
13 3 Mn Bhor By ssmton, th ntl condton for th trton of th ht rror s ot ot nd ot ot h xctd mn l s thn ot hn, bsd on th trt ton th mn bcoms ot As rosly shon for th LMS lorthm, ths lcs bond on th st sz. or M for :, rr mx nd thn ot lm Mn-Sr Bhor By ssmton, lt th ntl l for sm br b th dntty mtrx or col sn th rnc rcrson nd rnc roton rcrson s hn bc rojctn 3 3 contnn ntl th ntl condtons r comtd s ot ot
14 4 No tht h th bc rojcton to th ntl condtons, cn dlo th forrd smmton of trms. ot rom ths rslt, not tht for th - ht rror ot hn trnn ths nto n trt ton ot ot ot ot nd ldn to ot hs rcrs rltonsh cn b sd to nrt th thortcl cr dsrd. Gn nd ot nd ot h comttons ndd r, strtn t =: b ot ot c c b
15 5 Stt-Sc, Mn-Sr Stblty h systm ll b stbl f th nls of -br l btn - nd +. or B A Condtons: -br s ost dfnt. hn only mst orry bot th mxmm nl of -br. rom lnr lbr, t s nls ll b bondd by on f nd only f B A of mx o conr n th mn nd mn-sr sns, B A of of mx mx, mn bt t cn b ron tht of B A of mx mx thrfor t s sffcnt tht B A of mx Drn th mxmm st sz. Lttn B A of mx h l r ntrstd n s th mnmm l for hch dt B A dt dt dt dt dt dt
16 6 sn th forml X Y Y X dt dt dt dt hs dfns th to condtons rosly sttd. n ddton, th scond trm s sclr. dt h l tht sts th dtrmnnt to zro thn mst com from th scond trm Not tht, bn sclr, ths s th lnt of nd sn th dfnton of s nty colmn ctor, th smmton of th donl trms rodcd hn mltlyn by bcoms M As fncton otntlly comlctd to comt, form n t fncton nd thn srch for hr t os to s th mxmm. M f t my b notd tht t th ont hr th fncton ll roch nfnty! Bt ths s smly th ont hr mx As sttd bfor, ths corrsonds to ont hr th t fncton s >>, byond hr t old l for or bond on th st sz. h t fncton s nothr mns to comt hthr th dt systm ll conr! or f, th LMS fltr ll b mn-sr stbl. S horm nd horm : Stblty of comlx nd rl LMS.
17 Rl s. Comlx rrssors Not tht for rl rrssors, th fncton bcoms M M f Gon bc frthr, th trt tons mst lso b djstd s d r r d d r nd chn n th l B hr th n fnctons s A B for mx of A B 7
18 Stdy-Stt Prformnc Bsd on th thortcl nlyss, h connntly slctd ntl condtons nd lmnts tht smlfy th solton n trms of th donl comttons. W cn lso do ths for th stdy stt rsonss for th MS nd MSD or mn-srd dton of th fltr. MS lm MSD lm MS lm h ls r hr nd MSD lm MSD lm MS lm D D MS lm D D D h tons dfnd llo s to rform smlton of th xctd MSD, MS, nd MS nd comr thm to dt drn smlton. hs s ht th Comtr Problm s ll bot! 9.5. Non-Gssn Rrssor nstd of sclr ssmtons, ll bts r off. Kroncr Prodcts sd n th drton nd dscsson s y to jontly xnd to mtrcs from n m x n nd m b x n b r nto (m x m b ) x (n x n b ) mtrx! Mtlb rforms ths orton ( ron(a,b) ) s rt D of th comtr rojct 8
19 Andx 9.D Conrnc m of Adt ltrs sfl dscsson of ho nold of th trnsnt rformnc cn b sd to stmt th conrnc tm. Andx 9. Lrnn Bhor of Adt ltrs hs scton ttmts to brn to lht som hnomn tht chrctrz th lrnn cblts of th dt fltr hn lrr st szs r sd. 9
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