Comparison of Wiener Filter solution by SVD with decompositions QR and QLP

Transcription

1 Proceedngs of the 6th WSEAS Int Conf on Artfcal Intellgence, Knowledge Engneerng and Data Bases, Corfu Island, Greece, February 6-9, Comparson of Wener Flter soluton by SVD wth decompostons QR and QLP EDWIRDE LUIZ SILVA AND PAULO LISBOA Departamento de Matemátca e Estatístca Unversdade Estadual da Paraíba - UEPB Rua Juvênco Arruda, S/N Campus Unverstáro (Bodocongó) Cep: Campna Grande - Paraíba BRASIL School of Computng and Mathematcal Scences Lverpool John Moores Unversty Byrom Street, Lverpool, L3 3AF ENGLAND Abstract: - hs paper presents a sequental decomposton of the data auto-correlaton matrx usng SVD, QR-method and QLP as pre-processors from the desgn of Wener Flters It s shown that ths approach s effectve for nose reducton by mprovng the SNR whle reducng CPU tme for a range of nput sgnal lengths, flter length and nose varance he theoretcal and practcal aspects of the proposed approach are ntroduced and compared to those obtaned from smulaton results Keywords: - Wener Flter; SNR; SVD; QR; QLP Introducton In ths paper we propose usng the pvoted QLP decomposton of the autocorrelaton matrx A from estmaton of Wener flter coeffcents for nose reducton he effect of the decomposton on SNR (Sgnal-to-Nose Rato) and CPU-tme varatons wth respect to nput sgnal length and nose varance s analyzed and llustrated wth smulated data he rest the paper s organzed as follows Secton presents Wener Flter, Secton 3 presents the detecton of the Numercal rank of the QLP, Secton 4 presents system models theory and reports on the quanttatve performance comparson SVD, QRmethod and QLP, Secton 5 presents expermental results and Secton 6 offers conclusons from ths study Wener Flter he Wener Flter s a well-know adaptve flter consdered to be optmal for nose reducton applcatons he sgnal reconstructon wth a equaton ~ y( = ( b h )( () sgnal f ( Nose η ( b( + h ~ ε( = y( d( Fg Wener Flter y( d( y (n) s the flter output and d(n) s desred, clean sgnal, and b( n) = f ( n) + η( n) s the orgnal sgnal, contamnated by nose he box h ~ represents the flter operaton In ths equaton and n fg, f (n) represents data, y (n) output and d(n) represent the desred or reference We assume that the sgnal and the nose correlaton matrx are uncorrelated, e φ f ( n), η ( n) = 0 he purpose of the flter s to reconstruct sgnal f ( as accurately as possble from b ( wth d( beng

2 Proceedngs of the 6th WSEAS Int Conf on Artfcal Intellgence, Knowledge Engneerng and Data Bases, Corfu Island, Greece, February 6-9, dentcal sgnals Consder the standard lnear equaton to model a sgnal wth nose b( n) = f ( n) + η( n) () We also need to defne a oepltz matrx, whch s formed wth postve values of the auto correlaton matrx A It has constant values along negatveslopng dagonals [7] We can formulate the mean square error as ε ( = E{[ d( y( ] } (3) Insertng () and (3) n (4) and mnmzng ths error, we can wrte ε ( = 0 ~ h = E{ f ( f ( } E{ d( f ( } (4) where E { f ( f ( } = matrx of the nput ' A s the autocorrelaton Defnng Z = E{ d( f ( }, whch we wll call Z- vector, the prevous equaton can be re-wrtten as ~ h = A Z (5) Where the flter h ~ needs to be estmated from the data hs can be acheved, for nstance, va the QLP of A We can expand (3) wth known algebrac rules hen we can well-take the Fourer transform of the expresson to fnd the power spectrum n ~ ε ( = ( f ( x ) b( x ) h ) (6) = Insertng () n (6), we can wrte ε n ~ ( = (( f ( x ) ( f ( x ) + η ( x )) h = Where =,n number of s nputs sgnal f ( he sgnal to nose rato s the quotent between the powers of the sgnal f ( x ) and η ( x ) s the power of the nose n the estmatve the desgn sgnal d( ( f ( x ) ) H ( x ) = (7) ( η( x ) ) Our am n ths study s to combne the use of the Wener flter wth dfferent matrx decompostons appled to the auto correlaton matrx A ) 3 Detecton of the CPU-tme of the QLP, QR and SVD decompostons It s clear that Sngular value decomposton may be easly avoded by computaton the rank-r prncpal ' subspace of A = Ryx R Ryx But the product these three matrces, nvolves addtonal computatonal cost and ths s made worse by the calculaton of the nverse matrx R [] he QR decomposton produces an upper trangular matrx R of the same dmenson as A and a untary matrx Q so that A = QR, R where R =, where R s an upper trangular 0 matrx For the full matrx A, produces an effectve decomposton n whch E s a permutaton vector, so that A (:, E) = QR he dagonal elements of R are called the R -values of A ; the column permutaton E s chosen so that abs( dag( R)) s monotoncally decreasng o motvate the decomposton QLP consder the parttoned R -factor [] κ = κ R 0 R of the pvoted QR decomposton [] Has observed that κ s an underestmate of A A better estmate s the norm l = κ + κκ of the frst row of R We can calculate that norm by post multplyng R by a Householder transformaton H We can wrte [3]

3 Proceedngs of the 6th WSEAS Int Conf on Artfcal Intellgence, Knowledge Engneerng and Data Bases, Corfu Island, Greece, February 6-9, RH l = l 0 Rˆ We can obtan an even better value f we nterchange ( Π ) the largest row of R wth the frst l 0 ΠRH = (8) l ˆ R Now f we transpose (8), we see that t s the frst step of pvoted Householder trangularzaton appled to R If we contnue ths reducton and transpose the result, we obtan a trangular decomposton of the form [][3] Π L Q A Π R L P = 0 We wll call ths the pvoted QLP decomposton of A and wll cal the dagonal elements of L the L - values of A he dagonal elements of R are called the R-values of A, those of L are called the L-values of A he way we motvated the pvoted QLP decomposton suggests that t mght provde better approxmatons to the sngular values of the orgnal matrx A than does the pvoted QR decomposton An advantage of the QLP decomposton s that merely apples the program twce, once to A and once to R [] he mplementaton of the Matlab package permts the QLP: [ P, Q, L, pr, pl] = qlp( A ) to determne the numercal rank of matrx A hus, we have a smple QLP algorthm as follows defne matrx A, whch conssts of A = P L Q calculate the orthogonal matrces P and Q whch reduce the matrx A to lower dagonal form 3 dentfy the dagonal of lower-trangular matrx L 4 sort the dagonal elements by sze Calculators the CPU-tme for the decomposton of matrx of oepltz A and also the soluton of lnear systems h ~ = A Z, able shows us that the QLP and QR are on average 3 tmes faster than decomposton SVD able Comparson between the tmes requred to calculate the matrx of oepltz A usng SVD, QR and QLP decompostons In summary, the applcaton of QLP and QR decompostons reduces computatonal complexty whle gvng smlar results to SVD n the calculaton of Wener-Hopf coeffcents 5 System Model and heory here are many ways to compute can wrte A A Wth QLP, we = P Σ Q w = PΣ Q Z (9) hs approach s llustrated wth smulated data usng coeffcents β = [ ] drvng a recursve flter to generate the followng nput sgnal [4] f ( = f ( j) β (5 j) + ε (, = 5 j= where ε ( ~ N(0, σ ) he desred sgnal s gven by the nose-error component d( = = 5 j= f ( j) β (5 j) A total of 500 samples were smulated to mplement the expectaton operator he nput sgnal length was n taken as N = where n =,,, 0 n was lmted to 0 to control the matrx sze and computatonal tme requred for the matrx sze and computatonal tme requred for the recursve algorthms to work effcently for such a large number of realzatons [4] he functon log for a postve number x, the power y to whch some number must be rased to gve x he number s the base of the logarthm he nput data vector f (, desred result d( and the nose η ( vector were all of sze x500 he

4 Proceedngs of the 6th WSEAS Int Conf on Artfcal Intellgence, Knowledge Engneerng and Data Bases, Corfu Island, Greece, February 6-9, matrx of auto correlaton was created from the nput data, and the oepltz matrx R was created wth 8 postve values of matrx of auto correlaton wo flters at the same frequency were appled to nput sgnal the data X corrupted by the addton of noseη he flters descrbed by the numerator ~ coeffcent vector b = h (:M, ), where M s sgnal flter length and denomnator coeffcent vector formed wth ones 0 samples were taken n order to realze A the expectaton operator he mean acheved SNR was compared for each nput length and also for each flter order Smlarly the CPU-tme was also compared for each sgnal length and flter length All the smulatons of these recursve algorthms are carred out n MALAB 7 envronment runnng n a Pentum(R) 4, 40-GHZ CPU 6 Results usng the decomposton QLP Fg SNR varaton wrt flter order 6 Smulaton results he Fgures, 6 and 0 contan the results of three decompostons for sgnal to nose rato versus varaton flter order, shows us that mantanng the same resultng Sgnal-to-Nose rato for all decompostons Here, the L-values and R-values perform essentally as well as the sngular value S- values he results n the Fgure 3, shows us that CPU tme versus Sgnal length n the QLP and QR decompostons (Fgure 8 and ) reduced the CPU tme from a value of 0048 to 004 hese confrm the results n the table, where the QLP and QR are faster that SVD he computaton of SVD decomposton s expensve For ths reason, ths paper has proposed others alternatves Of these the pvoted QLP and QR decompostons s wdely recommended because of ts smplcty he results n the Fgures 4, shows us that CPU tme wrt flter length n the QLP reduced the CPU tme from a value of 0084 (SVD) and 0047 (QR) to 0043 In other words, there s monotonc and exponentally ncreasng f we keep flter order fxed and vary nput sgnal length [4] A hgher SNR means less nose and better detecton, the QLP (fgure ) has better results n comparson wth SVD (Fgure 0) he Fgure 5 shows us very smlar results for all three decompostons It shows us that f we ncrease the nose varance, CPU tme remans constant because the flter order and nput sgnal length are practcally constant ndependent of decompostons SVD, QR and QLP Fg SNR varaton wrt /p Sgnal length Fg 3 CPU tme wrt /p Sgnal length

5 Proceedngs of the 6th WSEAS Int Conf on Artfcal Intellgence, Knowledge Engneerng and Data Bases, Corfu Island, Greece, February 6-9, 007 Fg 4 CPU tme wrt flter length Fg 7 SNR varaton wrt /p Sgnal length Fg 8 CPU tme wrt /p Sgnal length Fg 5 SNR and CPU tme varaton wrt nose varance 6 Results usng the decomposton SVD Fg 9 CPU tme wrt flter length 63 Results usng the decomposton QR Fg 6 SNR varaton wrt flter order Fg 0 SNR varaton wrt flter order

6 Proceedngs of the 6th WSEAS Int Conf on Artfcal Intellgence, Knowledge Engneerng and Data Bases, Corfu Island, Greece, February 6-9, 007 Sgnal-to-Nose rato he Fgures above shows the behavour of the QLP and QR decompostons reduced the CPU tme sgnfcantly better that SVD he QLP and QR decompostons are mportant when pretends to reduce a matrx sze and a computatonal tme requred n ths our for the recursve algorthm he QLP and QR reduces computatonal complexty drastcally to gve same results that SVD as gven by Wener equaton but t takes very less computatonal tme hs was demonstrated n a nose reducton task usng smulated data Fg SNR varaton wrt /p Sgnal length References: [] Yngbo Hua and Mazar Nkpour, Computng the Reduced Rank Wener Flter by IQMD Senor Member, IEEE Sgnal Processng Letters, Vol6, No 9, 999, pp40-4 [] Stewart, GW, Matrx Algorthms: Basc Decompostons SIAM Publcatons Phladelpha, USA, 998 [3] Sterwart, GW, On an Inexpensve rangular Approxmaton to the Sngular Value Decomposton, A Conference n Honor of G W (Pete) Stewart, 997 pp-6 [4] Heeralal C, Wener flter soluton by Sngular Value Decomposton (SVD) Electrcal & Computer Engneerng Department 006 Fg CPU tme wrt /p Sgnal length Fg 3 CPU tme wrt flter length 6 Concluson he applcaton of QLP decomposton to estmate the autocorrelaton coeffcent n Wener-flter desgn s shown to be effectve by reducng computatonal cost by an order of magntude compared wth the use of SVD, whle mantanng or ncreasng the resultng