Determination of the Confidence Level of PSD Estimation with Given D.O.F. Based on WELCH Algorithm

Transcription

1 Internatonal Conference on Inforaton Technology and Manageent Innovaton (ICITMI 05) Deternaton of the Confdence Level of PSD Estaton wth Gven D.O.F. Based on WELCH Algorth Xue-wang Zhu, *, S-jan Zhang and Qng-ln Lu Insttute of Systes Engneerng, CAEP, Manyang Schuan, Chna Keywords: Confdence level, PSD estaton, WELCH algorth, Degree of freedo, Rando error Abstract. Approach s proposed for deternng the confdence level of the power spectral densty (PSD) estaton wth gve Degree-of-freedo (D.O.F.) based on WELCH algorth, a odfed perodogra estator for estatng PSD. A relatonshp between the rando error, defned as a rato of varance and ean value, and the D.O.F. s acheved frstly due to the analyss results of the statstc characterstc quanttes. Then a quanttatve relaton between the confdence level, descrbed as confdence probablty and confdence nterval, and the D.O.F. s establshed. Soe eaples are provded fnally to deonstrate the proposed ethod. It s showed fro the results that the PSD WELCH estaton wth 8 D.O.F. have probablty 68.4% only to ensure the true PSD wthn 0.88~. tes the estaton value. Introducton In engneerng feld t s custoary to estate power spectral densty (PSD) of rando vbraton sgnal by WELCH algorth[,], one of odfed perodogra PSD estaton ethods whch reduces effectvely varance segent soothng and data overlappng etc. It s proved that PSD estaton by WELCH algorth s consstent result and ts ean tends toward the eact soluton and ts varance to null f segent nuber and data nuber n every subsecton are large enough. In practce the varance of estaton s never equal to zero because of fnte lted nubers of segent and data length per segent. A valuable concept, rando error[3], whch s defned as a rato of the varance and the ean value, has been proposed to descrbe the relatve agntude of the varance of estaton. Hgher the rando error, hgher s the probablty of the estaton far fro the ean value. In other word lower the rando error, hgher s the probablty of the estaton near the ean value. In order to obtan the rando error for a PSD WELCH estaton, a specal rando varable n atheatcs, χ varable[3], s revewed. Rando error can be epressed by a sple functon of the D.O.F. lsted n equaton for A χ varable wth N D.O.F. whch s defned as a square su of N noral Gaussan varables wth zero ean values and unty varance. ε = σ = N () Where σ, are the varance and the ean respectvely. For a gven rando errorε requreent the nuber of D.O.F. s educed easly as equ. for χ varable. N = ε () It s pty that PSD WELCH estator, as a rando varable, cannot be cognzed as a χ varable blndly because of non-unty varance for every ter used to be sued. A PSD WELCH estaton varable wth M segents data can be epressed as a square su of M rando varables whch are real part and agnary part of Fourer spectru wth non-unty varances for each segent data weghted and overlapped. After standardzng those M rando varables the PSD estaton varable can be rewrtten as a lnear transforaton of a χ varable wth M D.O.F. and the rando error can be ganed by coputng varance and ean value. Concdentally, the epressons of rando errors have the sae forula for χ varable and PSD WELCH estaton varable thus equaton () and () can 05. The authors - Publshed by Atlants Press 67

2 be also used to deterne the nuber of D.O.F. of PSD WELCH estaton wth a gven rando error requreent. Confdence level[4] s another descrpton on requrng precson to PSD WELCH estaton whch s defned by confdence probablty and confdence nterval for the certan D.O.F.. The relatonshp between confdence level and D.O.F. need to be establshed. Two knds of descrpton for the relatonshps are dscussed n the paper. (a) To fnd the confdence nterval to a gven confdence probablty for a PSD WELCH estaton wth the certan D.O.F. ; and (b) to fnd the confdence probablty to a gven confdence nterval for a PSD WELCH estaton wth the certan D.O.F.. The rest parts of the paper s organzed as follows. In Sect., WELCH algorth for PSD estaton s revewed frstly for provdng a sued forulaton to evaluate PSD. In Sect. 3 the rando error epresson of the PSD WELCH estaton wth gven D.O.F. s proved. Confdence level analyss s dscussed n Sect.4 for the PSD WELCH estaton wth gven D.O.F. and the relatonshp s developed between D.O.F. and confdence level. The procedures n detal for confdence level analyss and eaples are provded n Sect.5. The concluson s derved n Sect.6. Revew of WELCH algorth for estatng PSD WELCH algorth[][] for estatng the PSD of a rando vbraton s a perodgra spectral estator n whch the data wth gven length s dvded nto segents wth length ncludng overlapped data and the PSD s averaged to the perodgras of every segent wndow weghted. For eaple, 0,,, N-.Then K { } dscrete rando vbraton sgnals wth length N are noted as ( ) ( ) ( ) segents wth length L can be fored where L-D data s overlapped, N=L+D( K- ).For segent the L data are ( n ) =( n+d ), n=0,,,l-,=0,,,k -.After weghtng wth wndow functon w( n ) the perodgra spectral can be estated as: L- L- () -jπ fn SM ( f ) = wn ( ) ( n+d) e (3) LU Where U= wn ( ) s power paraeter factor of wndow functon ( ) L n=0 n=0 w n n order to ensure the estaton no bas ntately. PSD WELCH estaton s the average of K perod gra spectral above, that s: K- ( ) () Sw f = SM ( f ) (4) K =0 It s showed fro equaton () that perodgra spectral for every segent wndowed s proportonal to the square of the Fourer transfor of the sgnals. Hence equaton (4) can be rewrtten as: K { a b } Sw( f ) = γ - X ( f ) + X ( f) (5) =0 Where γ= s constant and X ( f ) X ( f ) α are real part and agnary part respectvely of KLU Fourer transfor to the _th segent sgnal wndowed. Then the PSD WELCH estaton s descrbed as a square su of K varables wth respect to Fourer transfor. Rando error of a PSD WELCH estaton It s known that the rando vbraton sgnal n engneerng can be splfed as Gaussan varable. And the two nduced cople varables by Fourer transforaton are also Gaussan probablty because 68

3 Fourer transforaton s lnear operator. Hence, PSD WELCH estaton descrbed by equaton (5) s proportonal to square su of K Gaussan varables.the nduced Gaussan varables have zero ean values and non-unty varances f the raw vbraton sgnal s Gaussan wth zero ean value. So the rando error of the estaton can t be obtaned by equaton () drectly. Further t s proven that the two nduced Gaussan varables, X ( f ) X ( f ) α, have the sae varance, notated as λ. Then a standardzng process can be appled to equaton (5) and obtan equaton (6): K ( ) ( ) ( ) X f X f α f θq( f) = θ - Sw = + (6) = 0 λ λ Where θ= γλ s constant, and: Q f s Varable ( ) Q ( f ) ( ) X f X ( f ) = + K α - (7) = 0 λ λ χ varable wth K D.O.F. f all the varables to be sued, X ( f ), α X ( f ) are ndependent. For ths reason the PSD WELCH estaton usng K segents data s defned as the PSD estaton wth K D.O.F.. On the other word, the nuber of D.O.F. for a PSD WELCH estaton s twce of the nuber of segents used to estate the PSD. Followng the an statstc paraeters ean value and varance wll be calculated for a PSD WELCH estaton wth K D.O.F.. ( f ) S E S w ( f ) θe θe Q ( f ) Kθ w = = = θ (8) σ = E ( S ( )) ( ( ( ) w f - E Sw f ) ) S (9) θ σ ( E ( Q = + ( ) f ) ) 4K θ = 4Kθ Q Then the rando error can obtaned easly and the result s concdent wth equaton (). And equaton () can be used to deterne the nuber of D.O.F. of a PSD WELCH estaton wth gven rando error requreent n reason. Confdence level of a PSD WELCH estaton Confdence level s another representaton for descrbng the precson of a PSD WELCH estaton whch s defned as confdence probablty under the certan confdence nterval and confdence nterval under the certan confdence probablty. In Sect. 3 we have stated that PSD WELCH estaton s a lnear transforaton of χ varable whose confdence level can be obtaned easly fro tetbooks or EXCEL functon. Followng the descrpton of confdence level for a PSD WELCH estaton wll be dscussed based on the confdence level analyss ethod of χ varable. Accordng to equaton (6), followng relaton ests: { a ( ) } b P a Sw f b = P Q( f) (0) θ θ P s the confdence probablty under the confdence nterval{ }. Where { } a θ Notes that Q ( f ) when ( ) b θ a S f b, ths eans that: w 69

4 a S w( f b ) = θ θ = E Q ( ) ( ) ( ) () f E Sw f E Q f Equaton () gves the confdence nterval wth gven probablty for PSD WELCH estaton wth gven D.O.F. by usng the nterval of χ varable wth the sae probablty and the sae D.O.F.. Rewrte Equaton () to obtan: E S ( ) ( ) w f Sw f E Sw( f) (a) ( ) ( ) Sw f E Sw f Sw( f) (b) Equaton (a) s the confdence nterval of the estaton and (b) the confdence nterval of the ean value of the estaton. Procedure of confdence level analyss and eercses The procedure for deternng confdence nterval (, ) to certan probablty α of the PSD WELCH estaton wth gven D.O.F. (N) s lke ths: Accordng to the probablty α, to fnd the correspondng one-sde probabltes, for χ varable wth the sae D.O.F., that s: a P Q ( f ) = b a P Q ( f ) =,whle b P Q ( f ) = α () θ θ θ θ Let syetry est, ( ) b ( ) a P Q f P Q f P Q( f) a = = θ θ θ,then: + α α =, = (3) Deterne the confdence nterval [ a θ, b θ ] of the χ varable wth the sae probablty α and the sae D.O.F. N. Ether the tetbooks on probablty or EXCEL tool be can used for ths a. Sply ecel functon gves the results: a θ =CHIINV(,N) and b θ =CHIINV(,N) (4) Applyng equaton () n Sect.4 the confdence nterval can be obtaned: =CHIINV(,N)/N, =CHIINV(,N)/N (5) Applyng equaton (a) or (b) the nterval of estaton value or ean value of the PSD WELCH estaton wth N D.O.F. s obtaned. The procedure for deternng confdence probablty α to certan nterval [, ] of the PSD WELCH estaton wth gven D.O.F. (N) s lsted followng: a) Construct the confdence ntervals of the χ varable wth the sae probablty and sae D.O.F.. a N θ =, b N θ = (6) b) Deterne the confdence probablty and of the χ varable wth D.O.F. N to confdence ntervals a, θ and b, θ.ecel functons are: =CHIDIST( a θ,n), =CHIDIST( b θ,n) (7) c) Deterne the confdence probablty α of the χ varable wth N D.O.F to the nterval a, b θ θ whch s the sae as the confdence probablty of the PSD WELCH estaton. 60

5 α = (8) Eaple : Deterne the confdence ntervals to varous confdence probabltes for the PSD WELCH estaton wth 8 D.O.F.. Such a PSD Welch estaton s satsfy wth the precson requreent of the standard MIL-STD-80F[5] MIL-STD-80G[6]. Table lsts the results for several typcal probabltes. Notes that the rando error of 8 D.O.F. WELCH PSD s.5% accordng to equaton (), the confdence probablty s about 68% wth respect to the estaton whch fluctuates wthn tes rando error around the ean value. Slarly the probabltes are about 95% and 99.9% to tes and 3 tes rando error fluctuaton for the estaton. Further Table and Table 3 lsted the confdence levels of the PSD WELCH estatons of 56 D.O.F. and 5 D.O.F.. The probabltes reach about 85% and 95% to the estatons wthn tes rando error fluctuaton. Table the confdence level of the PSD WELCH estaton wth 8 D.O.F. α 60.0% % % % % % Table the confdence level of the PSD WELCH estaton wth 56 D.O.F. α 60.0% % % % % % % Table 3 the confdence level of the PSD WELCH estaton wth 5 D.O.F. α 60.0% % % % % % Concluson and dscusson A approach and procedure s proposed for deternaton of the confdence level to the PSE WELCH estaton based on a specal rando varable, χ varable. The relatonshp s establshed between the PSD estaton and χ varable. The procedures are suggested for two descrptons of confdence level and the eaples are provded. It s showed that under 95% confdence probablty the confdence nterval of 8 D.O.F PSD WELCH estaton s [0.77,.6].On other words such an estaton fluctuates around the ean value between [0.77,.6] tes. On sae condtons the range s [0.83,.8] te for the PSD estaton wth 56 D.O.F. The ltatons need to be llunate for the ndependence and syetry. In ths paper PSD WELCH estaton s descrbed as a cobnaton of χ varable on the condton of ndependence for all date segents sued. In fact dependence ests certanly because of overlappng. In the 6

6 calculaton of one-sde probabltes and of the χ varable syetry has been appled whch can also effect the quantty of the results. Reference [] P.D. Welch, The use of fast Fourer transfor for the estaton of power spectra: a ethod based on te averagng over short, odfed perodogras, IEEE Trans.Audo Electro acoustcs 5 () (967) [] Zhqang Sun, Hongchu Chen, Yanpng Chen, Applcaton of perodogra and Welch based spectral estaton to vorte frequency etracton, 0 Second Internatonal Conference on Intellgent Syste Desgn and Engneerng Applcaton [3] Newland DE, An ntroducton to rando vbratons and spectral analyss. Wley, New York. 984 [4] Polts, Roano, J.P., La, T.L.,.Bootstrap confdence bands for spectra and cross-spectra. IEEE Transactons on Sgnal Processng 40(99):06 5 [5] MIL-STD-80F Departent of Defense Test Method Standard For Envronental Engneerng Consderatons And Laboratory Tests, Method 54.5 vbraton [6] MIL-STD-80G Departent of Defense Test Method Standard For Envronental Engneerng Consderatons And Laboratory Tests, Method 54.6 vbraton Xue-wang Zhu, tel: 39865, E-al: zhuw@caep.cn 6