UNCERTAINTY ON SIGNAL PARAMETER ESTIMATION IN FREQUENCY DOMAIN

Transcription

1 NCERTAINTY ON SIGNAL PARAMETER ESTIMATION IN FREQENCY DOMAIN Consolatna Lguor Departent of Autoaton, Electroagnets, Inforaton Engneerng and Industral Matheatcs, DAEIMI, nversty of Cassno, Italy Abstract In the paper the analytcal evaluaton of the uncertanty on sgnal paraeter estaton n frequency doan s dealt wth. Two dfferent and wdely dffused algorths able to copensate the spectral leakage effects due to asynchronous saplng are consdered. The cobned uncertantes on the fnal results (tones frequency, apltude, and phase due to the propagaton of the uncertanty on the nput saples are analytcally evaluated. Sulaton tests are then carred out n order to valdate the obtaned forulas. Keywords - ncertanty analyss, FFT, Interpolated FFT, algorth uncertanty, spectral leakage. 1. INTRODCTION A easureent result to be usable (coparable wth other easureent results or wth reference value needs a quanttatve ndcaton of ts relablty and qualty. In fact, snce no easureent s perfect, ts result s only an approaton to the value of the easurand and consequently the qualty of the approaton has to be ndcated [1]. In partcular, the Gude to the epresson of ncertanty n Measureent (GM [1] standardses the easureent qualty epresson, defnng the uncertanty as a paraeter, assocated wth the result of a easureent, that characterses the dsperson of the values that could reasonably be attrbuted to the easurand. The uncertanty copletely descrbes the easureent relablty f the result s corrected for all known systeatc effects (bas that sgnfcantly nfluence the estaton. Bas and uncertanty of a easureent value are not always easy to be estated, especally when they concern easureent obtaned by dgtal elaboraton of sapled sgnals. In fact, the etrologcal characterstcs of dgtalsgnal-processng-based nstruents depend not only on the hardware confguraton but also and anly on the specfc software []. In the feld of the spectral wavefor analyss, dgtal technques are very dffuse, especally for real-te applcaton and/or for sgnal paraeter estaton. The contnuos Fourer transfor s approate by eans of the dscrete Fourer transfor (DFT, pleented by the fast Fourer transfor (FFT. To correctly use the resultng spectru, as prevously sad, the user has to assess ts accuracy. Consequently, there s a great nterest n the FFT results charactersaton n ters of bas and uncertanty evaluaton, as proved by the nuerous and valuable paper found n lterature. A world-wde dscusson s open about deternstc errors that cause bas n the results [3]. Other fundaental studes concern the bas effects of the chosen wndow, of asynchronous saplng, of the fnte duraton of the saplng pulses on frequency, apltude, and phase estaton [4]-[6]. Together wth these error analyses, technques or sutable elaboraton are often suggested n order to elnate or reduce the fnal bas of the results (e.g. nterpolaton technques, advanced wndows [7]-[1]. The authors n [11] already tackled the proble of the analytcal evaluaton of DFT and FFT output data uncertanty, accordng to the GM. They used a ethod based on a "whte bo" theoretcal approach []. Wth reference to an sources of uncertanty (quantzaton, te jtter, croprocessor fnte wordlength, they dentfed equatons useful to evaluate the uncertanty n both odule and phase output values, for any hardware confguraton and for any algorth operatng condton (e.g. wndow used, nuber of ponts. In the case of synchronous saplng, the apltude and phase tones uncertanty s equal to the uncertanty of the correspondng estated DFT saple. Vce versa, n the case of asynchronous saplng, the spectral leakage, causes deternstc error on the tones frequency apltude and phase evaluaton, consequently the above equaltes are not vald. To etend the results obtan n [11] also for the sgnal paraeter estaton n case of asynchronous saplng, t s ndspensable to take nto account the effect of spectral leakage and haronc nterference. Soe ethods are reported n lterature that allow deternstc errors on frequency, apltude and phase due to the spectral leakage to be corrected [7], [8], [1]. They allow the deternstc error to be practcally zeroed, whereas uncertanty stll affects the easureent results due to both the uncertanty on the FFT saples and the one on the correcton algorth. In ths paper, two dfferent algorths for sgnal paraeter estaton n frequency doan [7], [8], [1] wll be charactersed wth reference to the obtanable uncertanty. For both algorths, the uncertanty on the fnal results (tones frequency, apltude and phase wll be evaluated cobnng the uncertanty of each FFT saple as n [11] and the uncertanty ntroduced by the specfc correcton algorth. The obtaned analytcal forulas wll be then nuercally verfed by sulaton. Furtherore coparson between the two approaches wll be presented wth reference to dfferent operatng condtons.

2 . THE MEASREMENT ALGORITHMS The DFT of an N-pont sequence {(n} weghted wth a wndow functon {w(n} s defned as: 1 jkβ X(k w(n (n e n k... N 1, (1 S n π n where S w(n and β n. Posng: n N 1 1 R(k w(n(ncos( kβ n ; I(k w(n(nsn( kβ n ( S n S n the followng states: X(k R(k ji(k and, consequently, the odule M(k and the phase ϕ(k of each frequency doan saple X(k are gven respectvely by: M(k R (k I (k ; (3 I(k φ(k - arctg. (4 R(k In case of synchronous saplng, naely f for a ultfrequency sgnal (t A sn(πf t γ all the sgnal coponents, f, are ultple of the frequency resoluton, f: f k f (wth k N, the apltude and phase of the th tones can be drectly derved fro the k th saple of the sgnal DFT, M(k, ϕ(k. Vce versa, n the case of asynchronous saplng ( f k f, the above stated relatons are not vald because of the spectral leakage [9], that causes deternstc error on the tones frequency apltude and phase evaluaton. The algorths [7],[8],[1] for correctng the spectral leakage effects evaluate the frequency tones as: f ( k δ f (5 where 1/ δ <1/, and k correspond to the relatve aa n the apltude spectru. Consequently, the pleentaton of these ethods foresees a prelnary analyss of the apltude spectru to search these aa. Then, for each au k, δ s evaluated and fnally the tone frequency, apltude and phase are calculated. Dfferences are present on the algorth used to deterne δ and to evaluate the tone apltude and phase. In the followng, two wdely dffused algorths are recalled. The frst one [7], [8] s based on an nterpolaton of the FFT output and consequently s often called Interpolated FFT (IFFT; t gves very good results once sutable wndows are used. The second one [1] s based on the evaluaton of certan energy paraeters related to the spectral coponent, and s charactersed by hgh accuracy even wth a sall nuber of saples. Both ethods are always applcable but a nu frequency dstance, λ, between two adjacent spectral coponents ust be guaranteed. In fact tones very close to each other ay cause spectral nterference that can hde very low apltude tones snce the errors ntroduced n the tones paraeter evaluaton are heaver for the tone wth lower apltude..1 Interpolated FFT For the spectral coponent f, the δ evaluaton s carred out by consderng the rato, α, between the two largest saples correspondng to the tone peak: M( k ε α (5 M( k 1 f M( k 1 M( k where ε (6. 1 f M( k 1 < M( k Besdes, consderng the wndow frequency spectru (W(k, we have that [7], [8]: W( ε δ α (7. W( δ By atchng relatonshps (5 and (7, we can obtan δ. The apltude, A, and phase, γ, of the th spectral coponent can be estated as follows: M(k A ; (8 W(-δ γ (k - arg(w(δ. (9 Even though the ethod does not pose any restrcton on the wndow eployed, drect relatons can be obtaned usng the weghted cosne wndow [6], naely the wndow H (w(n such that: ( 1 w n a h cos( hβ n. (1 h where H s the nuber of the consdered cosne ters. In fact, for ths class of wndow we have [7], [8]: H 1 a h ( 1 ε δ h ( 1 ε δ h α ε ; (11 δ Sδ H- 1 a h. (1 Sδ h δ h δ are calculated nvertng (11, whlst A and γ can be evaluated fro (8 and (9 as: S π M(k A (13 N δ sen( πδ Sδ π γ φ(k πδ π ; f S δι > else 1. (14 Table I reports the a paraeters and the relatonshp between α and δ for soe wndows [6].. Energy paraeter-based algorth It s based on the evaluaton n the frequency doan of certan energy paraeters related to each spectral coponent of the analysed sgnal [1]. At frst the wndow energy paraeter, E w, s evaluated usng the wndow DFT saples, W(k: E W(k.(15 N w k Tab. I Characterstcs of soe wndows Wndow Rectangular Hannng 4-ters Nuttal H 1 4 Coeffcents a 1 a 1;a a 1/3; a 1-15/3; a 6/3; a 1-1/3 δ α α ε 1 4α 3 ε ε 1 α 1 α 1 α

3 Then, for each th detected tone on the sgnal spectru, X(k, soe quanttes are evaluated, related to the th spectral coponent energy, and n partcular the energes of the tone, E, of the tone frst dervatve, E, and of the conjugate d syetrc of the tone E : c E M(k ; (16 E M(k ; (17 d E c R(k (18 Snce the wndow transfor concentrates alost all ts energy near ts center frequency, these energy paraeters can be evaluated on few spectral saples, taken n a very narrow frequency band, B, located around the peak. In partcular, B [-K, K], where K s chosen as coprose between a good energy evaluaton and a low λ. The tone characterstcs are obtaned as: E δ d ; (19 E E A S ; ( E w E c cos (β. (1 E 3. THEORETICAL ESTIMATION OF NCERTAINTY To evaluated the cobned standard uncertanty on the tone characterstcs (f, A, γ, the uncertanty propagaton law [1] s appled to the prevously obtaned equatons, consderng known the uncertanty on R(k, M(k, I(k (see append A. As far as the frequency uncertanty, f, s concern, f uncertantes on the saplng frequency and on the au postons k, are absent or neglgble, fro (5 we have: f f δ. ( In the followng for sake of brevty the phase uncertanty evaluaton s not reported. 3.1 Interpolated FFT Frequency uncertanty. At frst the uncertanty on each α s calculated fro (6: α M(k ε M(k M(k M(k ε M(k ε ( (, M(k ε (3 M k ε 1,. (4 M( k M ( k M( k ε M( k (for ( M( k,m ( k ε see B1. The so obtaned uncertanty s used to evaluate the uncertanty on δ : δ c α α. (5 α The senstvty coeffcent, c, strctly depends on the wndow used. Table II reports the values of c for the sae wndows suarsed n Tab. I. Apltude uncertanty. Applyng the uncertanty propagaton law to (13, and reeberng that δ f( α g ( M(k, M(k ε we have: A A M(k M(k M(k ε M(k ε (6 A ( M(k, M(k ε ε where fro (13: π 1 (7 N δ sen( π δ Sδ (8 ε ε sen( πδ πcos( π δ 1 δ sen ( π δ Sδ (9 π H-1 δ ah ( N δ h sen π δ S (δ δ h whereas for and see (4. M( k M( k ε 3. Energy paraeter-based algorth Frequency uncertanty. Fro (19 we have: δ K K K M k 1 M( k E ( ( M k K r 1 M( k ( M( k, M( k r M( k r (3 [ M( k E E M( k ] Apltude uncertanty. Consderng the equaton (: A K K K r 1 M k whereas for ( M k M k ( M( k r M k M k ( ( r ( d M k 1 M k E E W see (31. ( (, M( k r ( Tab. II c for the sae wndows of table I Wndow Rectangular Hannng 4-ters Nuttal (1 α (1 α (1 α. (31 (3 (33

4 δ A Hannng > Hannng > δ Hannng > A Hannng > (a -4 (b N N N N Fg. 1 δ uncertanty (a, and apltude uncertanty (b versus the nuber of pont, N, for dfferent wndows, obtaned wth the IFFT algorth (easured sybols; estated-lne. It s nterestng to note fro these relatonshps that whereas for the nterpolated FFT the uncertanty forulas are lnked to the wndow used va the senstvty coeffcent /, for the other algorth the relatonshps are ndependent for the wndow used. 4. NMERICAL VALIDATION Ths phase allows the results obtaned n the theoretcal analyss to be verfed. It s carred out by runnng the proposed ethods on data sets obtaned by sutable odel of the acqured syste. In partcular, the uncertanty was coputed as the standard devaton of the output values obtaned by consderng a set of 1 nput sgnals corrupted by the nput uncertanty. Snce t was proved that the an source of uncertanty n DFT algorth s the quantzaton [11], the reported results concern nput corrupted only by the quantzaton (see append A. A eanngful evaluaton of the obtaned theoretcal results requres the defnton of soe paraeters concernng: hardware confguraton (nuber of effectve bt, B, full scale and te jtter of the A/D converter, operatve condton (saplng frequency, nuber of elaborated pont, N, used wndow, band, B as well as characterstcs of the nput sgnal. Fgs. 1,, and 3 report the analytcal results and the nuercal ones concernng a 1 V apltude, A, snusodal sgnal (f 55 Hz acqured at 8 Hz wth an 1V full-scale A/D converter and usng for the energy based algorth K5. It has to be noted that the evaluated A represents the nu absolute uncertanty n these operatng condtons havng chosen the A/D full-scale equal to the sgnal apltude. As to the frequency, t was chosen to present δ nstead of f snce δ s always constraned n [-½; ½]. ncertanty evolutons of δ, δ, n functon of N, havng fed B 1 are reported n Fgs. 1(a and (a. Analysng these two fgures the followng consderatons can be carred out: uncertanty decreases wth N; the analytcal evolutons (lnes are n good agreeent wth the easured ones (sybols; the uncertanty obtanable wth the two algorths are very slar even f the δ obtaned by usng the energybased approach s a lttle greater than the one obtaned by usng the nterpolated FFT; Fg. δ uncertanty (a, and apltude uncertanty (b versus the nuber of pont, N, for dfferent wndows, obtaned wth the energy-based approach (easured sybols; estated-lne. as to the wndow used s concerned, better results are obtaned wth the Hannng and rectangular wndows for the energy-based algorth and for the nterpolated FFT respectvely. Ths last result s due to the very sall equvalent nose bandwdth of the rectangular wndow [9]. Obvously n the choce of the wndow to be used also other paraeters (e.g. the scallop loss rate, the sdelobe attenuaton have to be consdered. Analogously Fgs. 1(b and (b show the apltude uncertanty, A, evolutons n functon of the nuber of processed ponts, N. Alost all the prevously stated consderaton are confred wth only lttle dfferences. Wth the rectangular wndow A s always less than the other wndows. The uncertantes obtaned by eans the two algorths are stll very slar. Further results are reported n Fgs. 3, where the δ (a and apltude (b uncertanty versus B, havng fed N 18 are shown for the nterpolated FFT algorth. As you can see, the uncertantes decrease wth B, wth a saturaton for B grater than CONCLSIONS The proposed theoretcal and sulated approach to the evaluaton of uncertanty of sgnal paraeter estated n frequency doan by two dfferent algorths allows soe nterestng consderatons to be carred out: - the uncertanty obtanable wth both the consdered algorths are slar once the wndow has been chosen; - the uncertanty decreases by ncreasng the nuber of elaborated ponts; - the uncertanty due to the quantzaton decreases sgnfcantly by ncreasng the nuber of effectve bt of the A/D up to 14 bt. δ Nuttal o Hannng > (a B A Nuttal o Hannng >.5 (b B Fg. 3 δ uncertanty (a, and apltude uncertanty (b versus the A/D nuber of effectve bt, B, for dfferent wndows, obtaned wth the nterpolated FFT (Measured sybols; estated-lne.

5 APPENDIX A The results obtaned n [11] are suarsed n the followng. Sad the absolute uncertanty of each saple, we have: R(k 1 R(k w ( cos ( kβ ( S (A1 I(k 1 I(k ( w ( sn kβ ( S and consderng (3 and (4, general epressons for uncertanty on odule and phase can be obtaned: 1 M(k R (k w (cos ( kβ S M (k φ (k I (k w - R(kI(k w 1 4 R S M (k w (cos (sn (k w (sn ( sn ( kβ ( kβ cos( kβ ( kβ I (k ( kβ R(kI(k sn( kβ cos( kβ (A (A3 These equatons can be specfed consderng the uncertantes. The uncertanty on each saple ( due to the quantzaton process perfored by a real A/D converter can be posed equal to q: B q VRange 1 (A4. B s the effectve bt nuber and V Range the A/D range[]. The te jtter, can be odelled by addng to each deal saplng nstant t a rando varable unforly dstrbuted n the nterval [ J τ, J τ ] ; consequently [11] the uncertanty on each saple due to the te jtter s: J a J τ 3 (A5; where a s the frst dervatve of (t n the consdered pont. The cobned uncertanty on results: q (A7. J Consderng only q (A1-(A becoe [11]: CR(k C I(k R(k q; I(k q; q S q S (A8 ( CR (kr (k C I(kI (k M(k q q SM (k Where: C (k R C (k to C R (k we have: I w w (cos (sn ( kβ ( kβ / S C / S,. If C I (k s equal R M(k q (A9 q S (k In Tab. A1 the values of S, C R (k and C I (k for the wndows of Tab. I are reported. APPENDIX B The covarance ((M(p,M(q between two odule saples M(p and M(q can be obtaned consderng the dependence of both fro the sgnal saples as follows: M ( ( ( ( p M( q M p, M q n Tab. A1 - S, C R(k and C I(k values for the sae wndows of Tab. I. ( ( [ R(p cos(p β ] I(pcos(p β M q [ R(qcos(q β I(qcos(q β ] M p k; kn/ k1; kn/±1 k k; kn/± kn- k3; kn/±3 kn-3 all other k Rectangular C R(k SN C I(k Hannng C R(k SN/ C I(k Nuttal C R(k SN*1/3 C I(k (B1 ACKNOWLEDGMENTS The author wshes to thank proff. Govann Betta and Antono Petrosanto for the useful suggestons and Ing. Mara Pacell for the help gven durng the theoretcal analyss. REFERENCES [1] BIPM, IEC, IFCC, ISO, IPAC, IPAP, OIML, Gude to the epresson of uncertanty n easureent, [] G. Betta, C. Lguor, and A. Petrosanto, A structured approach to estate the easureent uncertanty n dgtal sgnal elaboraton algorths, IEE Proceedngs-Part A vol. 146, n.1, 1999, pp [3] R.I. Becker and N. Morrson, The errors n FFT estaton of the Fourer Transfor, IEEE Transactons on Sgnal Processng, vol. 44, n. 8, 1996, pp [4] J. Harrs, On the use of wndows for haronc analyss wth the dscrete Fourer transfor, IEE Proceedngs, vol. 66, n. 1, [5] A. H. Nuttal, Soe wndows wth very good sdelobe behavor, IEEE Trans. on ASSP, vol. 9, n. 1, 1991, pp [6] L. Salvatore and A. Trotta, Flat-top wndows for PWM wavefor processng va DFT, IEE Proceedngs, vol. 135, n. 1, 1988, pp [7] G. Andra, M. Savno, and A. Trotta: Wndows and nterpolaton algorths to prove electrcal easureent accuracy, IEEE Trans. on Instru. and Meas., vol. 88, n. 4, 1989, pp [8] C. Offell and D. Petr, Interpolaton technques for real-te ultfrequency wavefor analyss, IEEE Trans. on I&M., vol. 39, n. 1, 199, pp [9] C. Offell and D. Petr, The nfluence of wndowng on the accuracy of ultfrequency sgnal paraeter estaton, IEEE Trans. on Instru. and Meas., vol. 41, n., 199, pp [1] C. Offell and D. Petr, A frequency doan procedure for accurate real-te sgnal paraeter easureent, IEEE Trans. on I&M., vol. 39, n. 1, 199, pp [11] G. Betta, C. Lguor, and A. Petrosanto: Propagaton of ncertanty n a Dscrete Fourer Transfor Algorth, Measureent, vol. 7, May, pp