2-D ARMA MODEL PARAMETER ESTIMATION BASED ON THE EQUIVALENT 2-D AR MODEL APPROACH

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1 -D AMA MODEL PAAMETE ESTIMATION BASED ON THE EQUIVALENT -D A MODEL APPOAH Ayın Kıılky Ahmt Hmi Kyrn -mil: yin@hituutr -mil: kyrn@hituutr Istnul Tchnicl Univrsity, Fculty of Elctricl n Elctronics Enginring, Dprtmnt of Elctronics & Tlcommuniction Enginring, 34469, Mslk, Istnul, Turky Ky wors: -D qurtr-pln cusl AMA mol, -D Equivlnt A mol, prmtr stimtion ABSTAT In this stuy, th rltion twn th prmtrs of qurtr-pln cusl -D AMA procss n thir quivlnt -D A procss is consir Bs on this rltionship, nw lgorithm is propos for trmining th -D AMA mol prmtrs from th cofficints of th -D quivlnt A mol otin y pplying -D MYW qution to th procss unr consirtion I INTODUTION Prmtric rprsnttions of two-imnsionl (-D rnom fils r usful in mny pplictions such s img synthsis, clssifiction, n img moling [], [] From this viwpoint, so mny prmtr n spctrl stimtion lgorithms s on two-imnsionl utorgrssiv (-D A mols hv n wily introuc for moling of -D rnom fils Howvr, thr r fw prmtr n spctrl stimtion lgorithms ssocit with two-imnsionl utorgrssiv moving-vrg (-D AMA mols in th tchnicl litrtur [3-5] As in th -D cs, th prmtr stimtion prolm for -D AMA mols is much mor ifficult thn th -D A mols cus of th intrinsic nonlinrity of stimting th two-imnsionl moving-vrg (-D MA prmtrs In spit of this ifficulty -D AMA mol is prfrr to its A or MA countrprt [3] u to th fct tht -D AMA mol usully provis th most ffctiv linr mol of sttionry rnom fils [3-5] In th spctrl omin whil th AMA mols chrctri oth th pks n th vllys, th A mols trmin only th pks n th MA mols inict only th vllys of th homognous rnom fil [6] From this viwpoint, ow n Ogino [3] hv vlop procur for gnrting -D AMA mol In this procur, th A cofficints r stimt s on th wight lst-squrs critrion n th MA prmtrs r otin y using smooth prioogrm Slction of wighting cofficints for th stimtion of A prmtrs n th usg of smooth prioogrm tchniqu for trmining th numrtor polynomil of AMA mol s powr spctrum r som rwcks of this lgorithm Anothr lgorithm is introuc y Zhng n hng [4] This lgorithm is s on th -D AMA spctrl stimtion pproch Hr, th A prmtrs r stimt y two-imnsionl moifi Yul-Wlkr (-D MYW qution n th MA spctrum prmtrs r otin y mploying th rltionship twn MA spctrum n mol prmtrs of -D AMA mol On th othr hn, s in th mtho [3], th MA prmtrs of th consir -D AMA mol is not cquir xplicitly Altrntivly, Zhng [5] hs propos n itrtiv lgorithm for th stimtion of MA prmtrs of n AMA mol This lgorithm is s on th Nwton-pson mtho n MA prmtrs r stimt xplicitly from th -D AMA procss Howvr, computtionl complxity n th timconsuming itrtion procss for stimting MA prmtrs r som isvntgs of th mtho [5] In this stuy, w shll introuc simpl n computtionlly ttrctiv lgorithm for th stimtion of qurtr-pln cusl -D AMA mol s prmtrs y utiliing two-imnsionl quivlnt utorgrssiv (-D EA pproch It is wll known tht -D AMA procss is quivlnt to -D A procss of infinit lngth n it cn xprss y sufficintly high orr -D A procss s in -D cs of [7] For sttionry n rvrsil -D AMA procss, thr is rltionship; AMA(p, p ; q, q A( ; Using this rltion, w propos n lgorithm tht is s on th connction twn th prmtrs of th quivlnt A n th originl AMA procss Th -D EA prmtrs r otin y using th -D MYW qution vill in [4] with som vritions Thn th otin -D EA prmtrs r us in th propos lgorithm so s to gt th -D A n MA prmtrs of -D AMA mol Thus, th -D AMA mol is fully chrctri with our lgorithm Th propos lgorithm cn rgr s n xtnsion of Mrtinlli s -D AMA stimtion tchniqu [7] to th -D cs Our lgorithm givs goo prformnc in stimting th -D AMA prmtrs n whit nois vrinc tht is us to xcit th linr tim-invrint (LTI AMA mol

2 II QUATE-PLANE -D AMA MODEL Th propos mtho is s on th qurtr-pln cusl LTI -D AMA mol From this viwpoint, th gnrl sttionry -D AMA procss of orr (p, p ; q, q is moll s th output of -D igitl filtr xcit y whit nois procss Thn th trnsfr function of this filtr is givn y q q A (, h i, p p D (, h i h, i H ( ( + m n ( m, n (, m, n m This is lso th trnsfr function of th qurtr-pln cusl -D AMA mol In (, th cofficints m,n, ( m p, n p, (m, (, n h,j, ( h q, i q, (h,i (, chrctri th A n MA prts of th -D AMA procss, rspctivly W ssum tht th orrs p, p n q, q r known n, Th (n, n th smpl of th procss is givn on th sis of ( y th following iffrnc qution: p p m n ( m, (, m, n q q h i x ( n, n x( n m, n + w( n h, n i ( whr n N, n N n w(n,n is th smpl of ro mn whit gussin nois procss with vrinc σ w N n N corrspon to th numr of smpls gnrt from th procss of x(n, n fin y ( Th procss x(n,n is th output of th most ffctiv qurtrpln cusl LTI systm with trnsfr function ( Th powr spctrl nsity of x(n,n is fin y, h, i n D (, i j i, j (4 A(, i j whr i,j r th prmtrs of EA mol It cn pproch to th xprssion trmin in (4 y using sufficintly high orr -D EA mol From this viwpoint, for ny L n L vlus, th stimtion procss of th -D EA(L, L mol prmtrs is rli y using th -D MYW qution [4] with som vritions Th -D MYW qution givn for -D AMA mol in [4] is rrrng in orr to otin -D EA mol prmtrs in (4 For th -D EA mol of orr (L, L, th -D MYW qution cn trmin y L L r ( l i, m j σ δ ( l, m (5 i j i, j Th qution (5 cn writtn in th mtrix form: w ε (6 Whr is lock-toplit mtrix with imnsion of (L + (L +: L L L L + in which ch of th sumtrics, k, is Toplit mtrix with imnsion of (L + (L +: (7 P( jw, jw q q h i p p m n m n jw h jwi h, i jw jw A(, σ w σ (3 jw w jw D(, + jwm jwn m, n III THE POPOSED METHOD Th lgorithm introuc hr is rli unr th ssumptions of,, q p, n q p This lgorithm is thr-stp pproch: first th -D EA prmtrs r stimt y using -D MYW qution vill in [4] with som vritions; scon th MA prmtrs r otin y sustituting th EA cofficints in th stlish formul; thn th A prmtrs r stimt y using th EA n MA prmtrs cquir in first n scon stp OMPUTATION OF THE -D EA PAAMETES Th A procss quivlnt to th AMA procss cn otin s th symptotic xpnsion of th invrs of ( Tht is, r(, k r(, r( L, r (, r (, r ( L, r( L, r( L +, r(, {r }vlus givn in (8 r th utocorrltion vlus of th osrv t trmin y ( n ths vlus r comput y th following formuls [4]: r( k, ( N k ( N r ( k, k r r ( k, k r ( k, k r( k, ( N k ( N ( k, k k k N k N k n n k, N k N k n n k, x( n, n x( n + k, n x( n, n k k + k + k x( n + k, n For th solution of (6, th -D EA prmtr squnc n th right hn si of (6 is trmin s follows: (8 (9

3 [,,, ;,,, ;;, ] T,, L,,, L,, L, L,, L L, ε [,,,, ] T σ ( w whr ch of th n ε vctors hv (L + (L + componnts Bnfiting from th (6, th solution of th -D EA mol prmtrs is givn y - ε Tking ccount of simplicity of ε, th solution of cn trmin y, σ w f ( in which th f vctor forms from th vlus t th first column of - n it is fin in th form of f [ f f,, f ; f, f,, f ;; f, f f ] T, (,, L,,, L,, L, L,, L, L Th first componnt, of th vctor must chosn so tht, for th convninc of th ssumptions ssum in formr sction Furthrmor, w cn fin th vrinc of th whit gussin nois xcit to th -D AMA mol fin y ( Thus, th vrinc of whit gussin nois is stimt y nfiting from th (-( s follows: σ w, f, (3 f ESTIMATION OF THE -D A AND MA PAAMETES Bnfiting from th ( n (4, unr th ssumption of,, th rltion mong th m,n, k,l, n i,j for ny L n L vlus cn givn s B, L,,, q q q B k, j k, j + k j h,, L,, L, L L, L B, h, h, p, D B,,, + D, p p, p Ech of th mtrics fin in (4 hv th imnsion of (L + (L + Formul (4 is th sir rltion This formul cn us for trmining th qurtr-pln cusl -D AMA prmtrs from thos of th -D EA cofficints otin y ( Th orrs L n L cnnot slct low th vlus of L (p +q, L (p +q Sinc L n L r usully chosn lrgr thn th numr of unknowns, p +q n p +q, rspctivly, q q q th mtching twn th mtrics B k, jk, j + Bh,h, k j h n D-B, rquir in (4, will otin y minimiing of th squr of thir iffrnc with rspct to th -D MA prmtrs From th ssumption of A n MA prmtrs r rl, this minimition oprtion is fin y, ξ ξ k, l q q q,, + B k j k j k j h B h, h, + B, D, for k,,,q ; l,,,q, kl (5 At th n of this procss, it is otin (q + (q +- linr qution sts s much s unknown MA prmtrs Th rsulting systm of linr qutions cn trmin in th mtrix form Thus, th MA prmtrs r otin y solving th following linr systm: whr, q, q q, q,,,,,,,,,,,,,,,,,,,,, q,,, q, q,, q,,, q, q,, q,,, q q, q,, q,,,,,,, q,,,,,, q,,,,, q, q,, g,,, q,,, q, q,, q,,, q, q,, q,,, q q, q,, q,, q,,, q,, q, q,,, q,, q, q,,, q, q, q, q,,,,,, q,,, q,, q, q q, q q, q, q, q q, q, q, q q, q, q, q B k,j ( L + j, L k, k ( L +, L j L k, L j (4 [,, ;,, ;; ] T,, q,, q q,,, q q, g [ g,, g ; g,, g ;; g g ] T,, q,, q q,,, q q, (6 whr th vctor is th sir MA prmtrs Bnfiting from th xprssions fin in (4, th componnts of th vctor g n th mtrix is givn y

4 g p L L L p, r, s, t B p, r ( k, i Bs, t ( k, i + B p, r ( j, l Bs, t ( j, l k i p + l j p + p L L L + B, ( k, i B p, r ( k, i B k i p + l j p + ( j, l (7 p, r, ( j, l B p, r Not tht p, s,,,q ; r, t,,,q At th sm tim (p, r (, n (s, t (, Thn th A prmtrs r riv simply y insrting th stimt MA prmtrs in (4 Thus, w hv q q q m, n m, n + Bk, j ( m, k, j + Bh, ( m, k j h (8 whr m,,,p n n,,,p, (m, (,,, IV SIMULATION ESULTS In orr to tst th fficincy of th propos lgorithm w hv consir two iffrnt xmpls with istinct L n L vlus In ch of th xmpls N n N hv n tkn s (N, N (6, 6 For ch of th xmpls, n stimt of ch of th cofficints of th AMA procss hs n chrctri y th mn n th stnr vitions Th mn vlu xprsss th stimt cofficints of th consir -D AMA mol Th stimt vlus hv n otin y inpnnt runs of th propos lgorithm Furthrmor, th prformnc of th propos lgorithm hs n vlut with rspct to th iffrnt prformnc critri Ths prformnc critri r mgnitu n contour plots of spctrums n th norm of iffrnc mtrix twn th tru n stimt cofficint mtrics corrsponing to th tru n stimt mol prmtrs Th powr spctrums hv n otin y insrting th tru n stimt mol prmtrs in (3 Exmpl : This xmpl ls with th ro-n procss Hr, w hv ppli our lgorithm to th ro-n procss corrspons to th AMA(,;, mol Th orrs of th -D EA mol hv n chosn s (L, L (, n (L, L (5, 5 for th comprison of th stimt rsults with th tru mol prmtrs Th stimt vlus chrctri y th mn n th stnr vitions n th tru vlus hv n shown in Tl I Th mgnitu n contour plots of powr spctrums corrspon to this xmpl hv n illustrt in Figur n Figur Tl II shows th similrity twn tru n stimt vlus Exmpl : This xmpl is rlt to th nrrow-n procss In this xmpl, w hv ppli our lgorithm to th nrrow-n procss corrspons to th AMA(,;, mol Th orrs of -D EA mol h, hv n tkn s (L, L (, n (L, L (7, 7 to compr th stimt rsults with th tru mol prmtrs Th stimt vlus chrctri y th mn n th stnr vitions n th tru vlus hv n shown in Tl III Th mgnitu n contour plots of powr spctrums corrspon to this xmpl hv n illustrt in Figur 3 n Figur 4 Tl IV shows th similrity twn tru n stimt vlus Tl I Sttistics for Bro-n AMA(,;, Procss (L, L (, (L, L (5, 5 Tru Vlus Mn StDv Mn StDv, , , , , , Tru vrinc Estimt vrinc Estimt vrinc Tl II Th norms of iffrnc mtrics corrspon to th tru n stimt mn vlus in Tl I Prformnc ritri (L, L (, (L, L (5, 5 A MA A MA L -norm L -norm L -norm Fronius norm Tl III Sttistics for Nrrow-n AMA(,;, Procss (L, L (, (L, L (7, 7 Tru Vlus Mn StDv Mn StDv, , , , , , Tru vrinc Estimt vrinc Estimt vrinc Tl IV Th norms of iffrnc mtrics corrspon to th tru n stimt mn vlus in Tl III Prformnc ritri (L, L (, (L, L (7, 7 A MA A MA L -norm L -norm L -norm Fronius norm

5 Figur Mgnitu n ontour Plots of th originl powr spctrum Mgnitu n ontour Plots of th stimt powr spctrum y (L, L (, for xmpl Figur Mgnitu n ontour Plots of th originl powr spctrum Mgnitu n ontour Plots of th stimt powr spctrum y (L, L (5, 5 for xmpl Figur 3 Mgnitu n ontour Plots of th originl powr spctrum Mgnitu n ontour Plots of th stimt powr spctrum y (L, L (, for xmpl Figur 4 Mgnitu n ontour Plots of th originl powr spctrum Mgnitu n ontour Plots of th stimt powr spctrum y (L, L (7, 7 for xmpl V ONLUSION A nw lgorithm hs n propos for th prmtr stimtion of qurtr-pln cusl -D AMA mol A rcursiv qution rlting th mol prmtrs of -D AMA procss n thos of corrsponing -D EA cofficints hs n riv -D A n MA prt of n AMA procss r otin from th stimt -D EA cofficints y using this qution Th simultion rsults show tht our lgorithm works wll for th stimtion of th prmtrs of th ro-n n nrrow-n procss with iffrnt L n L vlus For th ro-n procss, th stimt prmtrs hv convrg to th originl ons for th minimum vlus of L n L, i (L, L (, On th contrry, for th nrrow-n procss, th stimt prmtrs hv convrg to th originl cofficints for th vlus tht r iggr thn th minimum vlus of L n L, i (L, L (7, 7 Ths rsults cn osrv from th prformnc critri illustrt in Tl II n Tl IV EFEENES A osnfl, Img Molling, Acmic, 98 J S Lim, Two-Dimnsionl Signl n Img Procssing, Englwoo liff, Prntic-Hll, 99 3 J A ow, K Ogino, Two-imnsionl Spctrl Estimtion, IEEE Trnsction on Acoustics, Spch n Signl Procssing, Vol 9, No 3, pp 396-4, 98 4 X D Zhng, J hng, High solution Two-imnsionl AMA Spctrl Estimtion, IEEE Trnsction on Signl Procssing, Vol 39, No 3, pp , 99 5 X D Zhng, On th stimtion of Two-imnsionl Moving-Avrg Prmtrs, IEEE Trnsction on Automtic ontrol, Vol 36, No pp 96-99, 99 6 D G Mnolkis, V K Ingl, S M Kogon, Sttisticl n Aptiv Signl Procssing, McGrw-Hill, 7 G Mrtinlli, G Orlni, P Burrscno, AMA Estimtion y th lssicl Prictor, IEEE Trnsction on ircuits n Systms, Vol 3, No 5, pp 56-57, 985