SPECTRUM ESTIMATION AND SYSTEM IDENTIFICATION RELYING ON A FOURIER TRANSFORM AND SUPPLEMENT BY DAVID R. BRILLINGER1 AND JOHN W. TUKEY2 TECHNICAL REPORT NO. 5 MARCH 1982 RESEARCH SUPPORTED IN PART BY 1NATIONAL SCIENCE FOUNDATION GRANT PFR-7901642 AND 2AROD DAAG29-79-C-0205 DEPARTMENT OF STATISTICS UNIVERSITY OF CALIFORNIA, BERKELEY SPEC:IRUM ANALYSIS IN fiE PRESENCE OF NO SE SSUES AND UZMPLES David R. BrilUnger Statics tx W -prt Universty ciCliom Berkidcy Califorma John W. Tuke statiCsiss Q Bd l Pie tci Unvty Murray Hil, Now Jersey ant Pto Now Jesey ABSTRACT. 'fl Foimrs tm d str d data ar fminta quantities in thi siaton d ra tu idntication d systansTi ps d ais d tes use cal Fourier tam s, a critical fashia Plilacshy, tediques, flas, erixee, mightming iamples ae aul e Ibc fos is n piq al design, sdaatfic se in er ation rat t i m . presnation miation attempt to u mify emh a types M= p , para s I var rss and Systi estimatin. It is an oeview ra tban a i0alId pre ion empbasizing pWCKph saa x t tfss stories f timse aalysi Keywwds: Systm fica time serics, s, sSl sectra, Fourier andfrm, ifere, siuty r pint pro, t n, t a _ values, ONENTS INTRODUItC*I . a . . . . . . . . . . . . . . a . * SPECTRAL PRECaSION FOR GAUSSIAN DATA ..... . * WHAT CAN BE ESTBLATED? * . . . . . . . . . . . * THE RaLE AND PlALE CF MODELLING *a6 . 0 Is A. PENINGS I . . T . . . . . . . . . . . . . . .. 1. VNTAL ISSEUES PR OE . . . . . . . . . . . . .. la. NCN-ENTAN G CC Y NaNIrRIS lb. SIGNDS PROME SSE . . . . . . . . . . . . .. lc. NISE ISO MPORTANT . . . . . . . . . . . . . . 2. THE VITAL DIS;TINCIION . .0 . . 2a. Nl;S '.T PROCESSES . .0 . . 2b. SICGNA-I-CE PROCESSES a .0 . . 2c. NUME CF SOURCESi a a . . . 2d4 STABILITY IN TiHE NISECIK CASE . . . . . . . . 2e. EXAMPLES CF NON-GAUSSIANTTY . . . . . . . . . 3. SUPERPOSIICO AND LINEAR TlME INVARIANCE; NON- ENTAN 0 .0 . 0 0 6 Is 0 0 0 . 0 0 4. TWO AITFIS TO ESCAPE REALITY 4a THE FEW-PARAMEIER FJNKHOLE 0000 4b. THE GAUSSLAN FUNKHE . . . . . . . 4c GAUSSIAN WHTE NCSE. .E ... 6 4d. ONC-FF WVlHE NOISE 0 . 0 0 0 . . 0 0 4ec MULTICHIRP WHrE NOISE. 4f. COLORED NSES ARE EQUALLY DIVERSE B OIHER BASICS 0 0 . . . . * . . . 0 0 5. ENCLIr4INIG EXA.E . . . . . . Sa. AN EXAM[PLF SERC5COPY . . . . . Sb. AN EXAMPL TIHE NEGAIIVE IMPORTANCE CF STATIOARTY 0000 0 0 . 0 0 0 0. 0 0 I Sc. AN EXAN4PH THE WT4.WAVEGUI3E 0 0 0 5d. AN EXAML1 C fE MEMBRANE NOISE . & TWO= WW mC' 'T"P AM WvnDIA * * * * * u. I xrM rvv rLr I MAO DJvv 0 * * . . a . . . . 6a. CCN'IINUOUS IThE VERSUS DlSCRETE TIME: ALIASING. 6b. COTIINUOUS VERSUS DISCREIE FR CY 0 . 0 6c. MIXCURES (BETWEEN DATA AND NOISE) 0 0 . . 6d. TAPERING * 0 0 0 0 0 6c. PREILTERING 0 * 0 00 0 0 * 0 . 0 0 0 6f. GUADRATIC WINDOWING 0 0 0 . . . . 0 0 . 0 6g. VARIANCE CONSIDERATINS 0 a . . . . . . . . 7. dPlEX-DEMODLATION AND SINGE SIDE BAND TRANSMISSION * * . . * 0 0 0 0 0 0 0 0 0 0 0 7a. MLTIPUCATION AND SMOOTHING 0 0 0 0 0 00 7b. REMODIJLArIIN 0 . . . . . . . . . . . . . . 7c. SSB 0 0 0 0 . . . . . 7d CCMPLEX- CDULAIN AS AN FT P UOR 000 7ec AN EXAMLEFREEO13LAIIONS . . . . . . . . 7f. AN EXAIPE; W WAVES . . . . . 0 . 0 . . a C VECR SPECRA , . . . . . . 0 . . . 0 . . . 0 8. VECIIRCASES 0 . 0 . . 0 0 . . . . . . . . . 8a. cosPE...RA . .. ... * 8b. GUADSPECTRA. 0 . . . 0 0 . . . . . . . . 2 2 3 4 6 6 6 6 6 7 7 8 8 9 10 10 11 11 12 12 12 13 13 13 13 13 0 0 0 14 0 0 0 0 14 o . 0 0 * 14 . 0 . 0 14 . 0 0 0 . 15 0 . 0 . 16 0 . . 0 . 16 . 0 0 0 17 . 0 . 0 . 18 o . . 0 18 0 . 0 0 0 19 * 0 0 0* 0 0 * 0 0 * * * * * a 19 19 20 20 20 20 21 21 21 22 22 0 .0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 e a 0 4 0 0 0 0 0 0 0 0 0 a a a 0 0 0 0 0 0 0 a 0 0 0 0 a 0 0 0 0 0 0 4 0 0 0 0 0 0 0 a 0 a 8c. CROSSSPECTRA . 0 0 0 0 0 0 . 0 0 . 8d. REGRESSI CN XFI . . . . . . . 8c. CHE :E INI S AND ADJUSTMENT 8f. SPIRALLING . . . . . . . . . . . . . . 8g. MULIPLE REGRESSICN . . . . . . . . . 8h. AN EXAMPLE PCLAR UCIT . . . AN EXAMPL EMORCLOGY . . . D. OTIHER C SICNS CF SCOPE . . . . . . . . 9. umRA. H AIMS . a . 9a. THE CALCULAIIONS . . . . . . . . . . 9b. OTHER QUESTICNS, alHER STATISTICS, OamER APPROACH .0 I. . . . . . . . a . . . a . 9c. APPLPICATIONS . . . . . . . . . . . . . 10. HIGHERORDER SPEA . . . . . . . . 10a. IHE CENERAL PRIiEM . . . lOb. BISPECTRA (NaCLJDING CALULTIN lOc. AN EXAMPLE EQUITPENT VBRATICN NOISE 10d. MGER ANALOGS . . . . . a . a . . . 11. PCiNT PROCESSES,ETG . . . . . . . . . lla. THE SPECTRUM . . . . . . . 0 . llb. CRCSSSPCRA AND COHERENCE . . . . llc. AN EXAMPLEO ARE EARTHQUAKES PERIODIC? lld. AN EXACPLES PARTIAL CEERECES CF NEURCN PRO ESSES, . . . . . . . . . . . . . . . . . . . . Ile. ICHEREtORD)ERtSPECrRiA . . . . . . a . . . . . 12. MCHiER THAN UNQ)MEDENSICIAL .rb a 12a. FORMALIS . . . . . . . . . . . . 12b. SPAnIAL PCBLR S . . . . . . . . . . . . . . 12c. MIE (SPATIATEMPORAL) PRCLMS . . . . . . 124 AN EXAMPLE: SWOP (STEREO WAVE CBSERVAnION PROJECT>) . . a . . . . . . . . . a . . . a . * 0 12.. AN EXAMPLE: MOVING EARTHQUAKE SURCES . 121. AN EXAMPLE EVIENCE FOR SCATTERING CF SEISMIC ENERGY.................. 13. IEPLM AND US . . . . . . . . . . . . . . 13a. CECFE NON E.......... 13b. CNE CF EANCESEMPORTANCE CF CE CE c. MCE AN CtE .............. 14. CCE CF IPTS . .. ........... 14a. QIJASI-GAUJSSIAN INPUIS . . . . . . . . . . . 14b AN EXAMLE:SP SPECRUM RADAR ...... 14c. SINUSCTMAL NY'lJIS . . . . a . . . 14d. PULSE PRCBING . . . . . . . . . . . . . . 14c. CCMfPARISCN AND CCMBINATION. . . . . . . . . 14. hFICULT WITH NATURAL INPUTS . . . . . . . lSb. ERRORS ICDUDEi INIPlIS . . . . . . . .I.. lc. AllASING . . .. . . . . . . . . . . . . . 154 TIHE EFSECI CF BLAS . . . . . . . . . . . 16. EXA P ES . . . . . . . . . . . . . . . 16. AN EXAMPLC F3EEBAK IN A NEUROSENSORY SYSTEM as .s 1 . C C C 22 23 23 24 24 24 25 25 25 26 27 27 28 28 29 30 30 31 31 32 33 33 33 34 34 35 36 37 37 37 37 37 38 38 39 39 40 40 40 41 41 41 41 41 42 42 43 43 43 0 0 0 0 0 0 0 a 60 0 0 0 0 4 0 0 a 0 0 a 0 4 0 0 : : 6 0 01 0 0 0. : : 0 0 0 0 0 16b. AN EXAUMZ ISTRUMENTAL SERIES IN MATELURI~~Cs . . . . . . . . Is . . . . . . . . . . . .............................. 44 16c. AN EXM} ON-IJNE S ALCCl1TROL . . . . . . .............................. 44 F. LEADING CASES; IN DATA ANALYSIS; . .............................. 44 17. LEADIXNCGCASE PHLOS0PHY . ...............................44 17a. LEADING S1TUATIONS, Y:UHSE:,N ............................... 45 17b. WHAT THEN? . . . . . . ............................... 45 17c. S RUM NIIIvRON AND INION; CIN KGlEq IN UNEIANDING AND IN USE . ...............................46 17d. AND EULSEHR ................................................... 47 17c. TIHE SU AS A VAGUE CCNCZ........................................... 47 17f. STATICNARnly. .....a............................... 47 17S. CCNFIMbATORY CR ECP1CAORY? . . . . a . .0 . . .............................. 48 G& SOME STAISACAL TFXEINlQUES . . la . . . . . . I* ............................... 48 18. FaLW-IUP WMrH NCaNLlNEAR LEAST SQUARES .a ............................... 48 lga. CO6INUS IN6 NC 1E . a . . 49 I8b. DAMP'ED COS NUSC IN NCl; 49 18&. JI.HER FEW-PARAMEIER A5RA...........................................S 18d. AN E XAMZ VARIn4SCE COPONENTS AND TREND ANALYSISi . . . . . a . . . . . . . . . . . . ............................. 1 19. RCBST/ RESISTANT APPROACH . . . . . . ............................. 1 19a. THE NED 5 ............................. 2 19b. CE APPROACH. . a . . 5 ............................. 3 20. MISSING VALU1ES . . . 5 ............................. 3 2aINII I O . K . . 5 ............................ 3 20b. APPROACHE USING RCBUST TEl HNIOLTS 5 3 H- DB#1ERSITY DIN SEUM ANALYSIS . . . a . 54 21. TBREE MAIN BRtANCHiES 5 4 35. TREASPEr. .... ...S . . . . 5 4 23. IVRS17 CF AIMS . . . 5 4 24. LaVERSI7Y CF BKEHAVUCR . . .......S 25. PARSIMCNY VS. FULLS AND E--I. DINalRY a . ...S I. Al CXNVJGNII} . .* a . a . . . . . . . 5 6 *E9 _ 2 m. Trme Series INTRODUCTION Tbe auts f this er lrd separately, early an in wcrk, that a rigid "as mIptacs, then mathatic, then uso" aprch to thues for the andysis qf daa was ir safe nor effwte. Ncr ws it the way that the i i ct scie seCn to have b . What -s thsef a rcd bing flible at t beg arni then ing into, g wand when apprqi ate, s h dctiv t ure (p inlvng a few-parameter dseticma). Tbis se p y true Cf many - bl mat - pplicai C sctrum n l b P., imtial inestigtica to be feb i, ig fa al patt fcr the trum, after wich we may - r may mt - be able to go aer to some fat dsoip c f car spt= Oe Cf the eaIry S, mden sectlrum analwysi s 35 years ago, wu the r analys cf aItcmrrdatious cf a tacing radar, publishel, with an danentay-theory-based fit, in Jame et al. (1947). When oanalyzed iaibly in spectrum t is data sl l (a) the low- frjuay hump cxrrup ling to the e thery, whol was wdl-ag fied by the few o,mtants, aud (b) a unaller but quite gte peac near 2 Hertz, wind the few-cawant aalysis had tirely o ld lime have ben may pralll i s m the it ng de As a result of this pewiae we have mt written tis eia per, p arly in its nug sectios, in an umpns-mathema (sidbly ily sIketde) a-coMlicms style. As analysts cf data we are cfte ned with p -ena, with qualitative aspect Cf what the data shows, as wel as with its qtitative asects This has hldped us to be ccmced with the phezxxnaia Cf oar teciies, with thir qalitaie a iq itat aactseistics, ch usually alize mch furthe narrowly nitative bes. sepm are Cten bth vague and impotant, so we wM cften be deibeately vague. Many manm , ad a few a enta l cns, are rdvely n-stadstial. When a singe subc t says Same vowel, af ad ag , t f cn be remarkably imilr. Tbe smm can be tre Cf d zfmive palss frm a singe radar, c for the trace se at the same distant punt doe to uake c the same fault, yeas apart. l are the ice ca fr the analyst. IF he KNOWS this going on, he or she c an yd spectrm anlysi which dten do blwhat is fitted don to a few rm . Tis nts m n, h r, that these techoiques are either ally safe r stae fr use in a new fidd. All uch teiiques have to iilde strcmg bi assumpio,be se they focus cm the coe particular realizatin befcre us. If that realizatic has been sample, as today it will so ftas have been, so that it sts d T va , (1),..., (T) ad if we dcome m > T/2 cEstinct frapiaes, ohs.u a a a, m two d which are lias(sectic 6a) cf each other, thn we can always represent the gen data as x (t ) c = mco C Xt ) as a a bnaticCf jst fr ay cu ts. Ti can be dcme, inprti r, for al m ws ner zero, o all near u , cr all near 98&r. If we at individwu rlzatim, cme by cm, aud do nt add (Cten tacitly) ictive piom, the whole notio Cf a sctum is i trc So we shold not be sur-ised ehat csial mehods ive tacit asumption - assumio that ten do ne hsd. Ib other treme, were met Cf ycor autl' clients have found their data, is cae in which we have cme - or p s a few - reaizadutio, sbt were Itr St is in the eansble or pr a from which the realizatic.s) in syg the WT4 wae guid (section Sc), car intereSt is not in the p 12 idla s laid for a tst, but in uerstandng what irgWaities are likly to ccur, if mad wh s gude is mifr regar service. In sudying ft backgroainl inise im a cific rment (rp wi a view to the deigp Cf a seal tlepbo3 set), or intrest is n i th few muMtes Cf d that we saed, git , and mlyLed, but in the hxrs to ys f bknse yet to conme loIn ing at the anise racte 2 SpeCtrum Analysis ... Pree cf Nose alternative frequezc y bonds, when desiging a radio nmicat systa, mr cn is RU with the past we have analyzed, bt with the future we mmat face. 11e best paradigm we have for lkIng a im sud"nclie" situtio is a realization - essnhble oan, a alization to fimefumtims c the samaion paradigm. As always, is paradigm may esti cur u nty, , i , "the future may be difft". Bu, in "mise-lke" ms, it rudy oestimates Mr u an ty. It is imp t to ratc such like" situatiaily m ed with Gaus (or for pont-proases, MarkDv) nblo - fran the gnai e nes (eg. FDR's vowels, the pulse frcm radar #115, th tc eu frm station 7 when a one.pcund darge is tonted at station 0). Any aprchds to trum estimation wks for signal-like" bl, but oly care and trepdation can give us useu asaer for "misewi oms Tis p is a ixture cf plilqy, te flas, i , and eulightng eapes (with a blas to (e dsoy Cf FhmIIa). The apprch and tio s to unify areas: i) rdinay ime senes, pjxnt eand spatil series ah ) s o and higher order spectra, for ampe. fe read no fed he muat go straight through the paer, but shoid am Xthe Table d Cotsts f ti Cf sial iter to himself. It may nOt be CaWetdy a nt fm the s fics C the papr Imever it is ar dest i that the practitioaer of spectrum anlysis camt kw toomu cf the yi bac x Cf the situation cf conn. Blmi use cf formuas d t p ages lead many astray. HU". The specrum *" Casceptually, the spectrum Cf a imf timo - wheher discrete time r a nuas time wld be d by taling a very long strh C th tim function (much longer tia we atually have), filtering it with a sharp, robard filter, lookn at a piec Cf the result (as lcag, pehps, as acr actual rerd), ad g hw ch gy ( , variasice) is t. At least two tre caases must be dearlyumder Case 1. Ine very long setch d the time fuziction (a) is the caly time fdcion we wan to ansder amd (b) Cossf a (a summation) Cf Rt too many snusdal terms cf su-,taGtially differast frq Here the nar d filter will sect only ocm freqpswy at a trial, and thes we are to ame wM be a perfet s d lTs we have a go aixe Cf king with high psio. Case 2. '113 time fuction is to be regardd as a realizaion cf a Gussian (which for oawemezice we cuud, but n cd t, asine to be ergiic). It is one Cf many pcsible tme functiau; aor analyses are towdard the eties f the amblef these fumctiom, mt toward tlxue Cf a sific realization. Ibe nw d filter will produce a narba Gauss muse, imxt Hlily - speially if we are fortunate - with a mor dtailed trum matchisg reasonbly doly the result C white Guian mmse through the givea filter. In patcular, the amplitue Cf th rut will cge lttle oe time i short pred to the remprocal Cf the filter wdth, and wl be ai worre over time i als log oarl to the cxral cd the filte width. If we look at a stetch C suh a filterd ecrd that prdsconly a few, say 5 to 50, potiom wxe legth is the al dfth flter width, wewll lhave ly a pcxidea cf wvaiable - an the averag - a m nger rowa i od Ild be - lho mh p r such a n aud d wuld ntain Tat is to say, how larg the spectrnum C the GaussiaM prxm ws at (an noa) the frequeicy air filter. And wat we can do whe allowed to bad filter a very long reard (aud sup act a pec as lng as data) is at least o whastbettr than we can posibly do when we have only the actl data. * SPECTRAL PRECISION FOR GAUSSIAN DATA * Mis lower ba Cd ar ity is easly evaluatd We krwn fricm the S pn erem (Nyquist 1928), als as the Cardnal em C l y ly (Witt r 1935), that if we k1w the wbxe Cf a n ba al at eqiad pints, atd by intervals C 1/2W, where Af is the aw id could rc the wAxe narowbd sn by inte-xiatlin. 3 Trim Series4 Ezcqt for Ilekag Cf informaticm"acos the ends df our lcnger narrow bed reccrd (in cue cirectin cr the other), which is cray igble h, the inforiation in aor narcx b signal is aed in t e discre pints, cf which thre are, (within ? 1), k = TI(1/2Af) =2TAf in a erdcfd l ag T. Tlese k ponts will, if we with a reaizatin cf a Gauian p ss, have a jnt Gaus distribution. Tly will te us as much as posible ait tbor vi asoiated with the nrrow~band Gusian prsswhe they are imnq e . If we k at the mum Cf t bi squares, esstially thb best we can do in this case to estimate the poer m ti narrow its average will be ko and its wM be 2kd (as a mltiple of dinsqare an k df fr n). 1e ratio af standard deviaticn to man wM be (2k)s, which falls to 1/12 fo k at 300. 1hus, the poer in our rW bed is to be wn to me sigmfwam figure ( eily aki as stanard deviation average = 1/12) we rqire k 2 300 and hn T 2 3002W =150/WA For rh beh, w arely analyze ayting l this m h data. So we rarely kww even t13 awrage powr, oer nrowish bench, Cf any Gwssian to as ch a wo sgnificat fgure. Wbln we study p -mmui that are at all Gauss p ses, d to 1) wcrk with dimates Cf average spectra am bmdwidtls that are as great as will be usdul, and to 2) pla fcr sua al iit n tfie avages, usually affecting the ft sgifi figu sedarly. bxse wio mt be xxernad with ra cf Gasisn Ine s - cr with spctra cf can crude ttios d Gausian ss - have a wery much mcre di}lt tsk thn tb xe with murpc3itions Cf well., or ce m eately-wdl sSaate ds , i rly wn only a quite sht pao f the d d Citest is available. 113 Jls desaibed below ae intd to be eecie when we information abou sectra Cf mxsoplus-gnl cr purene Fro-amse T13y will wCrk - gving estimata Cf averag aos bands - fc t any srt f data. Wen we are so f nate - ly dthlw because we dl with iua s d the metioa Cf h y bodies (eg. the fides) cr d human activity (eg. rad ar ied to mm ade targets) - as to have mot cf the gY at discrete, wel-separated fra i, other methods may well be me df:ti icr fr tine use. However, as we w icuss, the ne t dscried below are stil likely to be * WHAT CAN BE ESTIMATED? * If we have the sum Cf a few perfect db (Cf a-mant ampitude) becdkdoed in a very low- levld ms, we do ml ned a lot Cf data to reowc the fr i , phases, d amplitues fd the sanuscich. Thit if the me ght be y i n, te the limitatiam just dised apply at leat as stry to l g about te roi, ame if it is at a high lvel. Without very lcIg we will only learn a little about the mse. If, by nntast, the oss is iam, we can oy le l, sinc the mitaticm just discsed al un to all f m s. Un we are i fact rr t, w we st alxU "M g" the re stuaticn ith m dcripti in tems Cf a ma finite number C s, we will be unabe to do a g app y bter than estimate arages cf the ser J er bnhds d well-chosen width - ther narrow mgah to make us really happy a out r frequenc resdution or broad oug to mak us realy hWy abt cm sa ling f tuatio 'l3e geual statements arwe no to the apch to sctrum analysis we discus in tis paper. No -odurecan do i y bettr, 1) in lg down sa g fl or 4 Sperum Analysis ... Pree df Nose 2) in avding aragg over apprcriate fr j Lc ags prvid that tie data is a rizati Cf a stataary ian p s, nce, as is well kmwn, the realiztia values d its mean d tie quadratic fmctics d its c*sevaticm are a set d esie statistics * THE ROLE AND PLACE OF MODELLING * As statiias, we are i, nod with the real wcrld fram which aor diea's data cn ewith al its i anfctaec aspwt We can - and do - mke use df mads but in the spirit df gidm ratler thn trus Thb etical study d fied s, in ser ca los, can be d great lip in sgg wat scrts f alysis to try. It can be a saurce d equay great danger, if we frget bow teuous ar ital assptias were, and start to treat t fac. In almst every e, any ie ly e aid has both (i) to be ncally teted to sec m what ways it fails to be acrect and by how much a f (i) to have tIe thertical es d these failures at lest crudely ames. After all, no ei ty sim modl isesactly arrect. And weVtake ar lfe in ar hands if we tus it withaut g real SeasCa. How are we to test rically o s d uar mids that give a fewasmtant des anipia f sae spectrum? aily by anayzing real data by tciques tha will wcrk, in usdwl and ~m~stood ways, wEn t de is inadequate. Hower fute we may be in having few cxutan mdIlds that give wcrldgality aprximati to real situatics we can cnly 1w that they do this wien we can test tim And thI simplest adequate test is ten te analysis d se spectra by metahls df geral applica a WEther cr not they ae able to dsiga and qerate as if sne fewamtant m d were tIe uth, anycne dply ornn d with spectra who ues few acmtant in um analysis, we beiee, na ls to be able to dichkx ae C ar imto ese mads give him cr bI. Part C tbis dEcking s likely to call f more ally appicable e s Cf um analysis. Often - we woud st alot always - tie will be te sorts Cf preres we are ab to disus You may, if you l, rgard ai appcacd as cuplortxry. Ln cn so, hoer, ya wil Eed to t C loratici as not onmly wat is in the begiing but also what is dcne evey so Cftm, particlarly to whatever aspc f qality cf arimatia by the currut m d have bees lEwly recx~nized as a critial im . 5 Tme Series UST OF NYrATnC f = f m i radian peruit F - T - 1 and wlxe Fouriertans(orm, HI(w) = hr(t)e-m, u is mmtrated ar a=0 !i f raplduy as ( s will, in aniculare be the case fo mcam hT that are a at the ends, rise to say 1 in the ct, th fall to 0, am! a very enowih.) The Faoier ta m Cf the series Y (t) = hr (,)X (t) may be wntX f(o) h' (t)X (t)ed T } 1 T WJH (w - (6.4) For a srs X(t) = csPt, 4d(w) - H'(w-0) + 2Hr(W+p) am! leabge is redued to the tnt tlHat Hr hlas bter resolutmia than Ara above. In pracei, as the dramatiC amples f T (197) and Van SC ld am! Fjlig (1981) shw, ta ng is esetial m the prlmnry timaticm cf spectra Cf umn fm. Haris (1978) s quite a Im Cf tapers with graphs f hT and the s ng Fourier trafcmn Hr. See als Nuttall (1981). Whil iea (6A) sbows that me culd "taper" afer f g the Fouie transfim (at leat if all ca atiam wec to lig enag psia), this is c atinaIly se0sible i many dracmstamcs. It is ber to multiply by hJ pri to Fourier transming, czept in spcal In te drect mthhad cf spectrum estimatiao amc soh the perimkam 18 19 Spects Analysis ... Prue:e cf Nase Itr(,,) 12= T12 Hr 2(2 dr Tt dJ r ( T2 jtr2> w (6S5) st1U2 T T(T) dT( ) Ibe result is invariably NCT a -hM vrn d the (umtaperd) idam (27a )-1 kti(@) 2. Q slmd mt miss the tity to tpr prir to fo g a penui ram in any arcuinstaices where the sec tum is mt KNOWN to be very, wry tardy flat. Priestly (1981) apte 7 is a further rder5:e 6e. PREFILTERING In se that the series X is statiry with me and m spct f(C), t value cf the peirid-.pam (65) is f IHT (-) pf (a)d a. (6.6) INs sew mak it appa t that the moe ay ant the fumctiaa f (-) is, the less biased the estimate. Nar, if the serifs X is a time.invaiant filter with tramfer ffuid A the spetrm- 1 the resultig seis is 4()2f(), and the value 1 its peridcam is If Vi (w-u) j'14 (a) If (c)drt. In many u t it is pcsible to detemiu a filter, bly cigl, sud that the funtati 4 (.)) f(.) is m znrly man (has b tially prewhi f() itsel. Thes remarks suggest the esmae () 1-2f(), whre f(c) is the sptral eimate cf the prewlitatd series. In eswe c filters the seres df intere-s rior to etimating the ctrum a t Izzupmsates for what the filter cid. In practiCe it cam be e ial that sm form 1 piltering be arid aot in trum eumatind. PI eltuing n also help in tam d t m s the vafia lity d a spennu estima-te. 6f. QUADRATIC WINDOWING das d sec estimates the form (2W) Z w (u)Cr(U)e, uf Wr(W-rz)Ir(a)dt (6.7) wher, in the mse the sies has 0 mean cy (u) = T -1 , X (t +u)X (t) , T -I. r is an es c the xcwaimn e fumctix, w (u); u =0,?:1,2,.. is a seque c CnvrgC factors with Fourier tr m Wr(c) w' (u)p(io }, a w Jre l(G) = (2iG)- Id (a)12 is eft iusdamdtheda Ibe t wr (u), a equivalety, d Wr (w) is called ulSing a quadratc wic . The pected value d ft est (6.7) is (2dr)-1f Cf Wr(w I ) I - )A dIf (r)dra with Ar the Faru tramfom d the tr 1 length T intrxwl above. This pec has the sme form as (6.6), im as Cater ad Nuttal (1980) ad Van Schlmd aed Frijing (1981) renarwk oe cn drive a quadric wiw Sud that the sc timat hs tesm pet value as any erad te. The c the two estmates, we, a ten but m always quite cifferest, as are the mpiities to f (ding the uual cdlalaiau). 19 TiMe Series Awthe ciffer.fce is the ikrc it simate, via the uadratic wimdw, cn take cn epgative values. 6g. VARIANCE CONSIDERATIONS Ccnpsradtive dscus se estimates not be a ned cut in term d p d values Wao Ibe iques d tapern a pirt n have b a dvanced as meas d i an the has If spun estimates Cf cume, if tleir we the imates les stable cne can ax! up with a estimate. It is crucial to also dscu thevaiance ci a trum estimate and to providce an est ct that vanc Al specu estimes we have cdsdered are quadratic fActam 1 the data and so can be repesetedas T {(t #)X(t)X(u) g., wbose apected value is f Qor ()f (a)d a wEe~~~~~~~~~~~~~~~~~. or (a5 = 1 qr t^)( In the case that the series X is Gausian its vanc is 2 f f I QT(a) 12f (a)f ($)dcx do. (6.8) See f msample Ams (1971), p. 445. (In many cases this wm p id a dce apprcxiatn as wl.) Epri (6.8) may be ae dFi to derive idt preas fr the, vwariances o the variws spCtrwm estiMates O e al characteristic that p o (6.8) maks dear is that tbe variance o an estimte d c the wihe cse the pq ao f(4) I case where a um falls df racdly, substanIal leakage the low-power regicx may be ected to cune frcin cther fraade Tapn ax pefilt g wi be red to rede the fcs this leakage. 7. COMPLEX-DEMODULATION AND SINGL SIDE BAND TRANSMISSION mmai ideas hre are: w filteig to allow crtain f ncy t features to stai! ut mae dearly, aix! f ncy tanlao to slow down (ar, pcsibly up) cdilating 7a. MULTIPLICATION AND SMOOTHING Cosider a X (t) = cos( ). a hs the tgtric identies 2X (t)cs(ca) = acs([p--wt +) + cs([] p44+) 2X(t)sin(cat) = - sin([ Ap]t ") + sin([+w]t+y) In h case, fcr co a the fli term an the ight-hand side is slowly vayig, while thescu cidflates more rapidly. When these fdmcticxr1 srec thed in time, the irital primat is that thi firt terms wil be weltera and the sewnd dima , in ach case for oear ,& he resuhs d fomig X(t)a ,d a X (t)sn ad than smoocthing Sparaty are the real and imaginary prt a the e late d the seres X at fruy pe dure may be se as amcdshifting the d freiacq min the sies X w e0 is nar , nto fr y - L pas filteing w allows c to loodeti Iy at a d ge d frcy a d rr, if the phim under study is wodving i time, stdy I cixxp' 20 Sectrum Analysis ... Pree d Nase delmcxilate may allow the detectimD d that elutiaL 11 spectru lAoks merdy at tle average behavir ao ts tim perial ct wy. 0a many asions it is ast to display the g ampitude and rather the retanlar cdinates, l the demnul. - p-aying the imtantanos fr pitxy alo Ct pr1e useu Ccznpi odulation leads cirwtly to S es s 11 time averagef tie sque dc the rng amite is rpnal to thE po rum at fq ycI 7b. REMODULATION Ld U(t) V(t) rmilting ic the X(t)oosca, X(t)sinoa epdely, wiEre o is the ar freqmy ied in the p ildation. The seies U(t)a as -V(t)smn a U(t)snat +V(t)csat tend to fluctuate with frua c (ped) tairmatiom fr X to e series provided the Mxxlhin is by actly aeuivalent linw filtrs, tmse liar filtrs The s f la ca pxrnas to mrrow blpass fltei the series r ay , tE secnd to a co_nlinaatin C( r bdp filterng and Hilbert tram 7c. SSB If one stted with a series X(t) - cos(o ) R api demodulated at frquexy c, nsr m, and termciulated at frueiency o, as the aboe isaisn slhws, cm wslld be led back to tIE original seie Sua e W wvrm rAedmodulated at frapiy w, then is led to cos([-Ajt ) whea A -w - (. If ( l nipred to W, t what has hapnd is that tO f3rhasy phushbeenpusbed c nby& lbs is ft down.m ulatio ed SSB tIsacn whaih is r so pouar in CB-radio, (with SSB up e in the llantter and SSB don-cnvasion in the, ivr). ealy mm is Weaver (1956). 7d. COMPLEX-DEMODULATION AS AN FT PRECURSOR 13 a nce e fast F(nrer ogra (egDigftal Signal P ing Ccmnittee (1979)) allowed a drect aprd to tIe axtuction cltmates, d frqueny dain ame wa lenhy dta t s e avilabe. Fo eamplbe per sctrum acd be estimated by i) forming e FT, dT(c), ii) f nth am Id" (co)I, iii) sn xlingL3pde (113 orossIcm (see Sectin 8c) cf two ries X and Y cald be estimated by i) fe Frs d4(w), 4(w), i) the crmspeiodgm df(w)d(-w), iii) fsmtching the crms- pmodacpam.) 1i st dir approces (eaier ta thc FT) to beconme ccxnpztaally Mi used cxmzpiex dadulationad. 113 power spectrum at f y w was sima by ing U(t)2 + V(t)2 where U(t), V(t) we tft demcdulaes at freailaecy wa (11 arasp was estimated by hing (Ux*VS iV(i))(Ur (t)iV, ( )).) Ccxzzpe dulation s a powerfu tcique for spectral esia n in ral ime and fo the estimato d isger-rder spcra d lthy seies. T i s may be conpted by an FF when . (See engtam a l (1967).) Tley may be sampled at a crude (time) spadng if thr are age cc ng limiao s. 7c. AN EXAMPLE; FREE OSCaLLATIONS 113 res e cl many d cal systm to an impulse is a cr aIn dl deaying 21 Trim Series ? a it}cc(kt 46). (7.1) k-1 l assc d m ove di the system are its freeada For mmy physical systan, e.g. th Earth's p to majcr ert it is i crucial intrest to timate t1 ge feeies -M aud the asaiated quaity f Ots A Q = k /(2%). In the aftermath ci t}e gret Qilan ea ke c May 1960, many s m estmates were x ated fr seirga, a aes esimated bthe faicm d peaksm thme spetra (se Tukey (1966).) B ause c tbc time vi aacter d tbe signal (7.1), e:r, cm ex ulati is a sulbaamly mo eful temique for tis staton Bldt and Blillingr (1979) p t the ult Cf igat a Cil reord for a umbe ci ees. Tbc lcarithm f thennng amplitude is seen to fallcff in a hear fahio tb apriatm preson (71). It is further sem tl deay rate X dept c on the iny y i a iract fashion. 7f. AN EXAMPLE; EDGE WAVES Edgc waves are (water) waves movig sideways to the sxme rater rolling onto the shcre. They are usuly caused by the suapcn i ie dnt and refted wave s. T lrir formatio is a nlinear phAm na ineed if tht fr i ty ci the i t wave is , and ations are favcrable to the formation ci g wvs, than tbc edge waves will have frequcy 1y/2, (a subbroi). (1981) makes tive use ci -ex danodulation to study the growth and decay c edge waves gezrated in a wve tan HE fits tl dcal m to te growth and dcy rates varioas perimmtal cdticmas. M1 edge w s are allowed to reah a staionary state. beir fr y in this state is estimated by cclslatiao, then that fMcy is ed fcr ncxmdlation ci the in tha gwth and decay perd ( a ng to t tring ao and n Cf cf twe gewat, reely). I 1 mn ampitude c the resulting cFds demodlate provides the growth and dy rates intt. Un fimh that tha ref teio at ci the beach is a crucial peramet in the situation. He iscvers that it is tha (amplitude d the) r wave that acites the edge wand drives its growth. By p tting d lates d the edge Wand wave side by side he is able to set thae mlinear transfer cf enrgy fi the latter to tha fMer. CcznpleC da latioan p idedan apie tool here, becase tha penenm under study wes limited to narrow bends (albft essentiall m a) and c g wth time (for tha udies caried act). Tbrcugh cpe u , lim aiat cf a theadical mecmni Was Prided, tha s d u assup In determined. C. V SPE 8. VECTOR CASES Qwe we have two or moe mannm res, r, if yop a vector valued seres, we can do C m e than dmdr tha spr a ci tha i ivid seres, which described wslfergy is cdistrbuted over frq . We can aso mider hxow the erg shared betwen two senes is stributed o er freq. Siu sharing can be in phas, or in tre o as a cmnina cf thase, tharesuing crmLs.spectrum has to be C -valued. Sometim we wat to midaier its vahs in m adtes (mapitude ad phe); s imes in etanglar wordinates ( a ecran ad quachmc). n lust as, i nom-timseries tiss ia- (based n vriancs well as vaanes) is a mom poweu ad sea h tol tanar e co nts, so too crssspa hlp us unlock many o ta r a al do mt - an c . 22 Sp Analysis ... Presec f Noise 8a. COSPECTRA Tables of qarrsquars were aMe widely availabe to allow mzltiplication via the quarter- square idmtty UV (u - ..)2 _ (i )2 This aescnm s"e to dfi3 axi ine the an two ra vaiabes We can defiI3, snilarly, the it n r dther fac a rss f icr an etimate a lizat, by aONpec(xF =V) =-Cc(Xa +x) -4PEc(x -X) Nice that his iplie cwpec(x ,,-x,) C-cATpeC(xj r) so that negative values, alongside pOjisve cnes, ar c . anal cf least- square fitting d y in term d x wwld be to pm X throgh a linear tme invariant filter with fr futica B (a) an try to am e the spectrm cf Y -B[X] If we restric Ster to be tim-symietric ar zero, thus z so S i with B () real valued tha sectrum this differec is, with an davis to- fcr the s a a cmr- fry (a) ) 28 (W)fzr (a) + B2(co)fn (w) whre cur icdoi5, fzr is rea, t is g to the cmpectrum. Tbis is mniiiemd whem -2fxr(a)) +22(w)fxz(w) 0 that is, wen B (w) = fxr (w) fix (a) in etp c analcy to the vxnvan vmiace pressan r an crdinay least-ses regresc In mctecrdogy fic zaq#e, te aI ZU f vzcal a intal wind vecities gives the frauaxy analyis f the Reyndds stresse, whidc mliate the vertl tamfer CI d mtl mientum, ctc*. (03 ry c Pa (1967)). 8b. QUADSPECTRA Wben a bert f rm, H, is muFienly dcaely relizable, the cospcc(H([X]X ) -cspc(X,FI[Y]) = qadzpec(X X) ('TIE Hilbert tramfrm is thin filter with trafe fuiax -i sgn to, see Brillinger (1981), p. 32.) We discus ti cclau f quacpectra bdow as p ort Cf the cacuation cf ccm-spctrum, whre cm- tn cutm + i quacpctrum. 8c. CROSS-SPECTRA We have iy disced estimating a um y sav values Idr(w) p = dy(w) d4(w) cf th pi a cf t taerd Y (t) =h (t)X (t). UI Dw Yj (t) = h (t)Xj (t) 23 Tume Series Y (t)) = h(t)X& (t) are the ed fams cf two simlta seies, with Fourier transfms df(a d!T (a) respectively,wecan form thec rr_mi g a spo am drTj(w) drrk@ which is ex-vplez.alued, ad thn average (locay in w) to estimate a crmssspectMmm Sd. REGRESSION COEFFICIENTS If we return to Y -B[X] ad gve u c training , wefll that the spctrum d this resid series miz when B (X) u SP le -uf (w) f TX (W) spcrun-x (N) fzx (c) This the mnimin ts mctio, or Winer filter, which we might have apcted to play the rde d an array df x -valued regressicua ocA , mt cily ues ust tht, bt is i in strict analcU to the usual (n.timeosezies) deiatc d ressi ts. I'el thinp are imp t to ber: * each atistical Ipt de-ble by firs a semd ns in a n-me-series sttion has an amlca deinable by me, sptra, and crau-spetr * if the lar ept is mnegative, (vaiance, mltiple orra ), the meseries t is a (nm a.nptive) real-vahued fmcti C f y (ilividual spctra, etc.) * if the saler a can have elther gn, (alct evertig dse) the timseries p is a cop e.valued ftio Cf e . fle re armamt n cf lwa.cmet statist has its analcgs, all fti s f f , IRly al avlz alhed. 8e. COHERENCE: DEFINITIONS AND ADJUSTMENT If we ask what frati cf serum (Y) r wh sectn (Y -B [X]) is *mnized, or what fraction cf sptrum (X) i when rum (X-C[Y]) is m, the answer is a further fimction d fr uay, 1. the A vrhe, w hre the core-xe is give by cdrem (ci) = r.spctrm ys (ci)aus.spec tumnn (CO) _ f1& (ci) 12 sctrmz (co)p n () fxz (c)fyy (c) (This is the mlc C 1-R 2, whre R is the mtiple cirelaticn d sqared as used in multiple rersuiu studes) If we try to ft y with a x itrdeant to it, we t mimimized wry -bx) to be mal tha RW y. Similaly, if we estimate the crm.sctra and the two sctra by avera the crms- pericikram and tet periocranm , al in the sam way, c res ing to imc averages cf m terms eadi, we will f se estimated a eame whae X is w dly exie Cf to Y. 0 simple adjustmmt adjusted ab-ur- - sscraxce _ 2 where the lmdwidth cf the spectrumn aass um estimates m cfreedm, dos firly wel in aig tuble fran tis s . No plot Cf raw eceraice should ce be without indi the l Mizonal line at "cchrere =V2m be reation_ a-oih-ay clacl a -yh elb -q ajles when we&fi a c.Pl.vahued quntt 24 5Specrum Analysis ... Preseme d Nose * cru sVspectumyx (o) (Spectm (mx) sP r '())* that is essially a ialized cm-ssp r Early ml, aumcn bween xeme ccltremy wa great; st raf raltm MfS BcOIL We fdlo WeVi (1930) in our use the term. 8f. SPIRAL LNG If Y c5ffem f X by delay cr advm by a timeointeval r, the a. the Fourier trnsforms ciffer by e*", b. the spctra are the same, and c. the cros -ruSs rmnn is imes CtrU a C time the cther. As a (theoretical) lascy is e but if sr is large eigh, t facto rotates so fast that averagin over , for detly-vaiable atmfates es ctra a acs-spwtra, may rmeun 0 o a 0 for the rcspectrm. itr essetial Ilty is m the crcs-spctn, where anything res bling (cotq) dr'() + (dI(u))dT(w + + r has a factor CAtn + CFtl + * -+ cls-n whose aslute value is letsh r equal to m ? m,with Efenf t za,wen T is large. It is easy for the demoits df the craspedigram to sral, and if they dcb ampea noes p Ce usle rcassspecta This proa rq ires tnt viglance, esp ly ce it can be i nt o se fr ras in cthersm Tehciques, based cung phas fran mwing avages f the rms.igram to bring indiviwdal values df the cr-periodigram dcse to zero phase, aeaging ults, and reng the pase he ben le by Ceveand and Pm (1975, se Appentir B and Soctia 4). T1is scn e d wrk for all but the dmemat mte alling pre where fitting d rugh phase d ance may be a itc G reuo Cf the cf this t ique. For other apr hdses sea that per and its rfer Sane aut cadne to cntral spraUling should be routuiy wus 8g. MULTIPLE REGRESSION if Y,X1,X2 a...Xa series, we can try to ethe spcu Cf r -BI[Xj - BAX -* B* * [Xk ] Whn we have timat csspectra and tra fci eadc Cf sermfal f y s, we get essaitially the same equatia, a set for eadc frequy tumi, as we Iwd have for miple (mzwtime seies) regresmsa Cf y ca xl, z2, ,** x. Ibe poer ra a cs-s a replace the vaiances and xwaiawes df that case lit acly imprtant da sare a : a. the acssspctra are x lvalud b. the Bc (c) we are to sdve fcr are a exvalued. 8h. AN EXAMPLE: POLARIZED LIGHT lt dassil analyss d aizled l - into rcaly xariz ly d ai -mp zarizd- aCpcmtm-Wmspxs in XY -morcinates at 4? to the 1 and ca, to 25 rme Series6 ciding spectrum ( L)sEtrm() a3pcIml2 (a) ) =UW (OD) quadspectum- w for the ly pxiaized ccnpcmeints and spectrm Coeu (w) - qua um2x(w) for the unpaz ccxa 1 lts (A c m scale factor is ctta natural.) With X, Y miztal a vrtial, te same as c resprxl to right and left rclarly poarized light ftr thfirst two a un upolized light for the tUd. WVie (1930) ses y and iyzarn i light in se deail. 8i. AN EXAMPLE: TOWER METEOROLOGY Pazdsky aod MMcCcrzick (1954) discvered wind -e s roig alomg the grm by cryig out crcss al analyses d riza an vertical qms cl wind velocity at a pnt mn an dt~servan tower. lby fmd that at the prer height oa the tower, the quapctru had ac g in the meg ani t cther in the aftron. lbepiaaon Cf what was g wa t es imase i as the day prrm. If the r pint were in the upper alf cf a roing eddy in the mng and the lower half in the the served switch in wwld occr. D. OTBER ETSICNS OF SCOPE 9. CEPSTRA: THE AIMS R i i ime i occurs for a variety cf e bt mny in cne cf two ways: (i) single rtiats, as in an cho, ai (i) aseiscd reitions, as in human sh which is driven byd he repeated gthvoW chaxds. Echo ctien and itification has a varieqt f a caticm in g Ceqys. Pitch4etectian, the ideitfiatio f the Spc betw repetitio nf 1 cal-dahrd bCUavior, Is cf adable thcao i . (Fr t latter application see Ranio and Sf (1978), ad Schafe (1979).) To detect rlal ition tes edy, we to focus the infomationa thum that we have in car data. This is not fo in a boafcy hand - a great variety d fraquaxies may pirticipate n an echo. (We my have, wver, in mc t cases, to be prq red for ffer m in thu seh, h hs ift, and a t time delay with which an ecI apars at differt frade ) It is mx fo i tim unls the o al a s8gna itsef na y f i time, whl the pr*la may be simpe - mafly, lver, the original sa td as e cxmdabie its ivery sltai nt__iay. (Tame e cf b ior is m uch lcss mn - radar a cr doplerfeas aside - than f ey dep ce.) Echo idtification is difeet ffrn the joblems we have so far Ilt with, requires a new ap chd. Becaue f de e may matter it is natural to bewi by going to thu frsy ide, S we shal sly do. Like thu other Mt ci mp specrum analysis (vetcr s a, uth applied to ordniy or point ocms), a - when uW is ap re - are much more dfective tha e strum alysis. 26 27Spectum Analysis ... Prmee d Nise Ibe crginal efer is Bgert, Healy aixi Tukey (1963); a recnt reiew can be fu in Oiilc~s et al. (1977). See also B t (1966). 9a. THE CALCULATIONS If X (t) has a reas bly ly bdved spectrum f (c) - a caticiai met by chute impulses, by moderatdly oikred Gaussian mue, amn by many intermciastc proaesc - the remsult d aeching it with a ril is to c lv the ignal wth a fumctc having two es, an to mutiply its srum by the mod-square cf the F ier twamdm d the two-spike fwntiaa Ts last is df the form a +b cs wr whre 1 is th time delay, and bla tha strength cf tha eck. be dta's sectrum is df tha form (reascuably -naoch functicn cf w) - (a +b xNw) so t arl t step in isolating T are a. taking los b. lng fcr t nripple in lc(a4b mew) 1I bla is large, fi te ripple (pasibly a lt aled by "rahnics", see below) s d be easy. If bla is i l log(a +b ar) m log a + r - 2 1 (b) a 2a 3 a log a _ 1 2 + { w 3 r + _ ( b )2a w 2(ka)3ccz 3er +**- so that w ow is the nant term d the ripple, (cxs 2r,Cc3 3wr, etc. are ralmics). We kw how to f ripples in a series, we just have to look at its scu In t pr t ase, th senes is a frequency-series (not a timeseries) an its values are lgs d estimated spectra (for atdy now i vals), but we look at its umL It is nvent to maintain the fre y*time stintio, '-cr by saying "c im" ited Cf et hoer . Sot u pten im c a). calcuating a fine-grained um (but usually nt an imscd perzocram), b). taking los c). dte, gtig rid Cf at h slowv d engs in the result by iftering" (the anclg cf 'Utering') d). caating tha e trium f rn the realt, just as we wwld calculate a sectum f a ime To get ifoat a t tI s ze andphae Cf we may do well havin found K specifc 'r, to fit a + b coswr + c dnwr etlh alcm or in un-linati- with thae term, to our filteedi log ctrm If eking is frqu y kat, we may to civide oz final calcuatic n step, (d) above, into seprate calaalatim for cifflert fre regia 27 rme Series 9b. OTHER QUESTIONS, OTHER STATISTICS, OTHER APPROACHES 'flu apprash in the last subsection is that 1 the ai (Bogert et al. 1963) i s focused n g t time delay (cf a single ech, o wn iticise itin). That paper also a the usduls - better wrcm there - df the e autocrelati , (which cffes fr the rum undng the loArithm bdre the final Forier tramform.) Whb phaseF are important, it is n ral, a s a aprd, to fi d th tme delay or delays (quefr.wy or quefeies) fran ther run o auocrdation ad to fit a al cmnusxd d the i d q a a(awo) to the (posibly liftered) log trum, whm the fitted + esimates the phase Cf the ech. Jus as rs um aaysis has prd more dfectiv tha eneseries-at-a-time sum analysis in making peas dearly detectable (eg in unpublied wk an m tary series by Milton Friedman and cne cf the aflxts), is to be tetat crc-tum analysis, in which we omine Frs d lop cf spectra d two senes (multiplying oe by the znla cugate cf the cther, rather taig the squared absdute values f either) may wel be in mang ime ls (quefendes) c-n to both series dearly detectabe. Ti sy ical id e My n, s r, be yet available. Fortunately the whIle area cf ired aws, techniques, and apcrhdes have relviewed by Childers et al. (1977), wIv asze ty in all thre. This r has 86 rfdereia C we to the interested reseaw. It furthr ses the rdeati bet um analysis and the wlxle area Cf deccnviution. 9c. APPLICATIONS dle et al. (1977) stress tr= are d application f t dass ef tecniq sp h, wh theirst ful autcntic ptc C human spech ame fr sch eclmiqes seismc measur, and hydroacoustic masranents, wherm ,in (1975) r m sung dqL dista (250 to 700 nauical mil) pk3iws in the ocean to stanrd deviaticu of < 2%Cf the ctual epth. Othe i ampes can be fon in the faces given by cilders et al. (1977). We taut hrs es le with very brief mentii d a few apicatica in cthe areas, Cf which raie qustians or re geali . Miles (1975) and Syed at al. (1980), a athesm, rqxrt te use Cf tral techniques to ect fr gr d i in rrowbad s ra m eaurnt ts. Pearsn at al. (1978) dsauss the we u f d tral analyis in cctin -with ESR (Eec Spn Resme) analysis f Itited triaylaium caatin radclsi Ibcy beliee that wile "the epsumwn coes D rot ide mae inormafi oa than the ESR etrum fran wich it is dive, it Is a used a in the anly f td r ESR spectra. R er and (1980), appying oatum teciq to able damage studies, fier evidrice (which is hard to evahlate in the albse Cf infmaion a the adeuacy cf tapening in tIr F -T methxad) mntropy sperum anlysis is hepfl at the last stage Cf rum analySis Cearly a c ination cf prical f ing and th etical insight is need to darify this estion. Rathe xtslantial effort i likely to be necessary, but may wel be rewarding. Ran (1975) dscsses the 2dimsial c arum bridly, and ggests its utility with to imageoblurzing prdibs ( g su by amhr in this cim ). Image im e t is difilt augh f this suggestion to be th gng and nting 28 Specru Analysis ... Prec d Nc'se 10. HiGHER-ORDER SPECTRA Power spetra an csspea, beng basal an seaxl.aer statistics (ani thus naturaly called se rcnd- , spctral) an cly be ead to revel so mh a th situatica It is hwn that Gawsian sses are characteized apetdy by tircfirrdst ar sedd mcmnens, but that is a very prticuiar situati. 13 icxcti highrcd spctra fllows naturally fn ther a dsre to m igercrder m s (Cften as cxml) cf a pre r a d to handle Xt d(ct now -mi3r -ticzns wn the r~s 10a. THE GENERAL PROBLEM Th3 power specftm f() pides t i cv{Y(t4u),Y(t)} f eC f (w)dc u = 0, ? 1,.. Cf thu c fuztiom c thu statiam e series X. 113 hbiectrm, a third. crd sectum,f ( , pid thu re at cwn{X(t4U),Y(t+v),Y(t)} -f f | (w")f (c,v)doIv (10.1) cf thu thirdcckrde umulant (cr rat mwn) cf thu p ss 1 u culant tru c cder k, f (wit oak -J9 -* o) ides thu esatizan cum {X (t +" -1 ,..., (t +U -i) (t)} (10.2) -f ... p(xi(U1oq + u l Q ...J .is.. )( 0. ... .a.1)da... d4 -1 * (10.2) lif pt t cumA cu{X. .. ,Xi}, Cf a grp Cf ra vables proides a measure cd that part o thir pnt statistical ace which is not chibited by the t bhavir cf any k -11 td m It is a pdy in thu izxlividual dementary m s (avag f mon-i1s) that is aim Cf de k ca th Iying X. It is the mplst fumcticn cf the dentry n that vani if any ibgrwp ef w m variables is statisticay i Wdst f the reminngm bs Cf ffwd the gr 113 almulan spectrum Cdr k, provid a (k -1) fr cy link f hw ti (tj, ... ,X(t*)} df k vlues f the Ios aovary in a way not libited by thu d aa cf any k -1 f it has thu d furthe aracterisc f mesurmg an impa aspet if how ar to a far fne, nady Gaussian the grp b s, sinc, fcr Gausia variaes, al aa cf crder grate tha 2 vanishL Applying a liisr atim to a series with cily nu-ader u i leads to eaeisf rder highr th2. Sue Cut hs the ss X (t) m atco(pt -*) + o%caB(pt with ap,, *, . R tan. If i al w are i dt and trbued uniformly a (-,r) then X will be stat y, have zero man al have tral mass at - an wly at - the fr S and . Suppcu that the naw s YQ) X (t) + XQ) (t)2 (103) is fan. Y(t) Wi tin the sdd d X. It Wi furthe incue tems iM Is(2St+2 ), c((-t + j-S, c(( ]t + 4i+QI, Wa(2_t +24). Natice that ctain irrdseralso a lants) d si w s sih as aw{vcfc* t 4)a*st 44)CO([^2] .i.t + 4+IJ} will vmish, beae Cf th way in which O and k aIIIr. This a ('e) Fwrier analysis f cum{Y(t+.u), Y(t4v), Y(t)} 29 Tama Series will de an timtkii that a in Ity, as (10.3), may have aicrred Spectra kf der k, i vgcumlants f crder kg, are prtioulaly pertinat to xmzli3art-"I that are pnymial df crd k -1. ut, jast sectra (of ader 2) are vey helful in d with moi.Cuasian procea:ea ta e ot gserated -Ily, spectra d cder k cn be helpful n deaing with the aeuthr kiids of nacliies than pdyMaial a If the series imc mderaticn is vwtcvald j i ant a ovide ic for n w at fiu. Fr ample the crms4ifrzm rz (W$, Satis Wum { (t 9u )X (t 44x (t)} =|f f c'(""fxyz(w,v)ddxfv lOb. BISPECTRA (INCLUDING CALCULATION) The amplat highr-cder (cat) sctrum is the luspecrin. It is cccernd with the depailawe amg triples frqxs 4 m to zO Su e the statzary seies X has the rai n X(t f e,od (to) fcr Z a scastic measuwe. (See for ampl illinge U (1981).) Then the renaticn (10.1) ind es that crm {dZ (co)Z (v)d (-y)l - f(oog v)dwdidy with X.) the Dirac deta fAmctia. (Fr c,Y all * 0, the msoat hre may be replaced by ave.) Ibs result sggests how co migst form a imate C the tisectrm and a further iratacn fcr it. TM series qp{it}dZ () may be viewe as the resut df v nass filtng the series X at frp cy Ibis fact s t a means f timating the is um (peps the ist estimaticza tedmqu to be ved wetially, s s an et a. (1963)). Qk nar pa iltes the sines at vaicus fis , azt g a edl cf series X(t w). Then averages in time the triple pracs X(t)X(t,V)X(t,7) and X(t,u)X(t,v)X(t,t) with o + v + -0, wheme X' d hesthel tranfcm Cfthe senes X. Hee a negatiw vahl ef co refers to te at Iwli, m re filtrs trat + i - alI e sIgn C Hilbert traifm must be ed fc co < 0. (I the tim age is mt to imate zero, the three nw series ust be behaving in a 1cl a t fasw.) TIe raults are sies tf the real ad im prts Cf the bis um r e . It is dear that the tium may be estimated aeuivalently by ttriple prodcs C a du ates A s taially differet means d g t isum is to prcad by a g a third crder pei am. The tlird arder p xam Cf th stchX(t); - 0,4;T 1 i ded as IT (1,v) - (2or)-T TdT (c*)dT (v)dTr (+v) (In th case that the seies X dost have zerm it is better that thsame mean- ca more aled tale - be s acted to om dT.) The te auf this third crdcr p rkicamis giv by am Ijr(V) = (2i)-T1ff (c (a)Ar (ci) S (,v)dc a d 30 31Spectrm Analysis ... Prec d Nose i wi be Da S (w,v) fr T large sar S nih nr (w,v). Furthe, poravalues eluated at ciffrat a re asympdehitally i et f mi at. Ths suggest fwaming tie eftimate Sr (,) r _2o,v._2 / 2 s e~~~~~~P wharethawghtsWr sm to 1 aml are suhtaSraetmt bTas t13pfiltyax Altermately one migbt rce by noting that pccram values, say Ir (( e ,,. based osdjcn strec3 f sames are Wsmpoial inm%xiLemen. Ibis reakleads to the estimate Sr (av)- 1' (o,vr)IL t-1 It wi have the aty symmetry and pmdcty pkie cry. lOc. AN EXAMPLE: EQUIPMENT VIBRATION NOISE T se is usu in m o daical dmvriaes. W th tth Cf two cogwh are in hi g ceanly, a publ m f rt df the nisc sgnal ed is X (t) m e cm(Pt ") + 8 cm(21 + 0cm(31t)+ with crrting tth sp h w wich th riving wIed is ng ail ametes csam. If a rmg estim d th sis cooit the esimat may be ed tof fluctate b the levi Q2& 1 T2-12/(2v) 2 with T fth lmgth Cf tha time strech oar ich an i-iAv,dual satimatc is amzpitedL As ft pwe irregularly, th pe a y may be epd- to becane ra and the lispectnzm 4to becune 0. That was shown in Sato et al. (19). A poww sprum (Cf rdar 2) is unable to be used as a ciatetic tml hre s te ao Cd po jr at re mc r ma nably camant as tme pass n. The powcr sectum ns e infman n the p ales f th feqts Cf a tme series lOd. HIGfIER ANALOGS The aiantrum crder k is pvm by espreso (102). It may be simated in seval fashimcu Fcr mpe its r part may be estimated by avergig the p t X (t 4.X (t &) ime, whe X(tx) is X and flted nsr fr K X, and X4 + ... + . =0, (but m sibt Cf the, vs ms to 0). The i pat may be estimated by inlucing a HUbert Altenaively it may be estmated by r th k th order pi am in dther t r frajuaE. Ths p-riram is diAnd as Ir (k-@^ -= (2o)-1 'T -,dr (k).dr whe -)* = A + ... + A& -1. Detailsare givm in Bilinger snd Rmbatt (1967). 31 Time series In the case cf a vwtor*valued (multiccoolat) sries, the Fourier values dT may be based an cifferent Is, deA -ig on the qestiod intet, thus poiding estim d ccss pciyseta. 11. POINT PROXESSES, ETC. A stodIastic xt p s is a rand etity wIxe h ical reza dtios are dbly inite sequei~s {s}'1}. Cf pdints along the lne with 'r 5 -r,+i. (Actual relizations involve fiite sequezs_) e il i) the f a s within a gve regin, ii) the at which a m fires, mi) thes at whiCh births cr in sme p atod f int , iv) th times at which cns arne at a sevice facility d v) the ti f in a tmma e storm It is tmstto s N(t) thb nmbef dT in thieNval [/),thnN(t) is a step function imeugby 1 wuuan DC sin- has thu symbdC r tresatation- dt) with 8) the Dirac delta fmction, swng that a pdnt p s may be dieed a gelized tim serie. A marked pnt prs is a acn entity whew hypLtheUical realizaticns areczy infnite seijs {(Nj,lM,)}7_.,d cf pas (rj)W) with, {tr _ a pcint pros aSS with Mj a mark or value attached to the j-th point. It may te on discree c s o ymbdic values. Examples includc i) ai magitudes f , ii) arrival times and wiing t s ricdby individuals in a que, i) birth t an d bon (twins, triplets, etc.) and iv) l d trema f a an ution X(t) a a ed tme value It is im s convemit to poy th c iaon for a marke pont Pint psres and marke pint p es may be used to provide descriptions for the imo dasw d t sies xmtaig bursts d every so Cfta with the pricipal st~iastic variation cesoadng to the la a th tianty. Oe can o sh prams as XMja(t-rj) + @ (l..)(itl..*) with"= (w,..,.o*). Te itude Ct fdis sem to be lap for na (S9.4.") wheX (tir..tz) cixituin conpcmItC d the faom cm(tI + * + tg4 '). The transfrm itsef is sen to be eqivalat to a tl transfcm applied a fr eadc Cf dth arpmts d X. Mm fdling limit, when it ists, may be rug to be the power qtrum at wave vwctr U, *M Lave', r Idr (a) 12 Q ~may wrte dr (0 p = ( IC 'rI X (t+) >(t) gE-0 with t m (ti...,tx),Uc(aa....a), (ss) ' U1 +..+ c sig tht is ctu wi be ap-rqriate when one has the statanasity adti: c (a) =cov{X(t4U),X(t)} = cw{Y(X), X(O)} for all t. (Qier w is Priestly (1981), Capter 9.) In scne mtazc the a c an UF +u? +...+ 4 alone, when the p* s is sad to be istropMc. The axints t must be an smilar scales, bdore io a puiably be a phyicaly reascable smpo. F statimy bir p e data (X (t)y (t)) one can C the ccr function cjy (a) c WV{X(t+),Y(t)} , (12.1) the amspectrum 35 Irim Series fxy (co) = (2iT) - (mg)}Cry (u), (12.2) athe MA c (a) (2 = I y (2L f( /fxz (*)fn (c). he providemf-ess dthedee d lizar time invariant reiatibip betw the two dacta pois In the case d (12.1), (122) aen has the a . 9 , (u) = f J f (i(u)}fz (r)da-4* 12.3) 1Ie gezic farm df a ia timeinaiat filten m 1tcim cl time has the farm Y (t) a a(t-)X (a) with a(-) the i e sc and A (a) - > a(u)ep-i()} the tra fer fumcticn. If t ses X hs per spctrum f(), tetat cf Y will be ^ (a) Ffzz (a), redidng t ual ratiozmip in the me = 1. Suppce that e d the "me coainates are f in a shuy cf the bivaate series (X(t)Y (t)). It is d intert to mderstandthe ri hip betwe thepectra d the full srie and tlxo fr h a tim siC . ra h is appre:nt fa (123). Ilfxed cvcimt arrespcziD to = 0 in (123). In ae the craspeum t red ser is swe to have farm f J .ffx(u)dw,4o@g if o , ...o1 apcr d to th me cad es fed 12b. SPATIAL PROBLEMS 'l case d a atil array X(zv), with zy ng , es an impartt yet drect, the imesi cse Stan&ng (r fn) waves , +y ) i d lding ame d arrays tiaf the Foner a rc A tis lding eample, ider X made up d a finte munb d infeg sa (cs en) wa X(x jY) c' cM( yx v,yA,) + cse We are naturally ite in estimating the wammbrs *, -yj d the imivdual waves the xrrupxxing pvers a?/2. flf Fouie tramfarm is a bedc t far Here the direct tafarm is by say r-1 r-1 I X(xy). p(I,+W)} - 0 + 1 (A p)AT( 6k+) 2 zat^b(+*v( X wth AT (Q) -M Z np-i)tI} a e rlier. Tbe itu d tafcm wm peak ar the wve vetms (0j,j) as desi It is dew w b se d A pp dr the will be leakcage ac ) ad IL As in the -m oM cs M will nee to taper, f far 36 Spectum Analysis ... Preseice cf Nose crzmple dTQh) ( ,v Zhr (x y)Xp(xH (Xx +w)} ax'V with hT (x y) a suoc*h fwti vanishing vwyw e Actide the ckain cf bervation c the array. Qe will wt its Fourier tramfrm HT(' hT (x jy) W (h +i y )} to be =ated (0,0) a)i to die ci raOf d1y as IXI, IALI imrem. In the cos that the arTay X is staticariy wth zero mn an d ctrumf (,1) acm has, fcr dT above, ave1IdT (Xi) F =f f wcRr (KX Pg I r I r (X3ajk0 Pf (L) d ad 0 suggesting that ame migh base an estimate o IdT 12. It fuh suggests the esil d f tar in the se S () is natarly cmt=a. Suppose xt that the i cs array X iS iso Ic, C(U,V) =C0(X(-+Uj-v)x(x)} depasiing ci2 + alcm 113 tlanCf is that tr X iXs X2 + i i Spe&any, if C(U ,V) 2(+v fNO fJo(r V ) g2+ r g (r)dr 2o 0 wih Jo a B usd fwtic. T1e trmsfcrmastic bee is a Hankd tIasoIatfi . his rewlt leads to an it r the 2-dimmmiu in our alility to em sch an i r spectrum. (See &ilhinger (1970)). 12c. MIXED (SPATIAL-TEMPORAL) PROBLEMS lIe are a vady d mdic stal roes, in which a o i emn may be wepsented as X (x vy ), whre x ares ptial irdtes alt istime. For ezample et u$ -frey +&t-kt) may be viewed as a wave m gin cirecticim -7/ with velcity 18V Vf Ihe first two xrdinatest x xy, h e have quite diff t diaract an the r ni es t. (In some stuatiom it may be able to u the giin x ady, but th t and(x2+y2 arbiy inoledthrr~.) If ae has but a singltime slice, say t 0, d p thfl th sitatiens the fr9m d the previous stio On the other hand, in many ntuatiom me wi have much lnger ds terms i the than for the x "V cdcnate. In terms mp it may pove rcaable to tisage the largest dnerad t-value t to b, hu this may net be s Mible for x asd y. ( with t and a ingle x where this is reluant have b amdee by Betherto and McWiiams (1980).) Ihe Faoier tramfom that me might te fran such a data se w d be dr (kpp) 7- hT (x y J)X(x ;,t)inp{-i (4 +y 4w)} with hr a t g fumcti vanshing oif the ain di derlva The power ctru f (NW) might be estimatda by aveaging IdT 2. R fes arise A ning the spay ft resuclting estimatef ( ) as a hmctic 1 3 variables. 37 rune series spe s um algletimesIic say X(x y ,0) is give by J f (Xw)d i.e. if amly a single slice is availabe, it shwld be remeinbered that ts trum a has estimated is a marginal (nt a axdticnal) versia Cf the a e p sc tm. Im ant detail in the fil sectn may aot be available in a marginal M 12d. AN EXAMPLE: SWOP (STEREO WAVE OBSER VATION PROJECT) At 1700 (NIT en 25, 1954 stea cto t iqs we employed to meamse the state Cf the in the Atlantic at abm 39?N, 63.5W. Baiy, two Fictures c the sea s c we tk imtanltwuy at kwwn altitudes ad Eta rt. Frcm te pictes the water heights at seleted points were . AD tdd 5400 er s wr mae, with x a y s ng Cf 30' over a 2700' by 1800' ectangle. 113 data wa creted for =ds by least squares fitting d a pane 1be autcr was st imated at each e-nbination d 90 la in the X -cireticm and 60 lags in the YdrectiraL Ibs was Forier trasfomed and the tranom n by an tiam d the a ing he ated shoed a m pea zing to the lcally ges aedsea and adg to se swel. By sical r g (that wave mergeated is driven f lerwd) the auhors (Cote et al (1960)) were able to igiuish the drectin eruUpnag to a peak fron its antipcxl (which wwld have ,ated the same rum). 12e. AN EXAMPLE:MOVINGEARTHQUAKE SOURCES Bdt et at (1982) e en d s ptial s t analyse C seveal mr etqkes recorded by an array d 2 as nd three icirdes a( thirc m Cter). Spaal tra are ti ed for eari time e s f th asmams and s al (temporal) freque6 . P s, to the cirectia Cf the asouce the vdaty cf the wves are fou in the a. 13 laticm Cf the p for f the evns is foaud to shift with the ime gment analyzed Ibis shift may pride the first nental measurntC iw a sa s slatin moves along a rutrn fault. 12f. AN EXAMPLE: EVIDENCE FOR SCATTERING OF SEISMIC ENERGY Spatial spectra have bes used to pride stra dae for bacrkcattering Cf ergy duing the passage Cf fsarc waves. Ald aud n (1975) p an simate Cf the tial ctnrm for the intial grp Cf r waves (in the fre2 1020 Hetz) a g after an plio 11= data we the e i _ Corded at the Larc Aperature S iumc Array (LASA) in Mantana. l3 esti mate shows a il p ca C rgy in the cirectica Cf the blast, g at an apprqriate velocty, and little dse Aid and 3t aet t a timate Cf the spatial sctrum f t lter arrmng waves (the axih) is ne shews Uy aving frxn al cirecticus with sod sufce ciis. e aga s um analysis d ra secments has tispa ed res an i at sacstif F x L SYSTEM )UETICA1ION We cscuss ln&er system identificatin this pt. 13. INPUTS AND OTPUITS n many a o time series are subjectd, naturally or arificially, to qperatiom Thes ceratn s may be physial or a tai l. te the t f the trsformato is a t series itself. In this case t nes as th input and the tra dsei as the output. 1be ac m d (input, qei, oz} is r d to as a system 1e p cf system identificatn is that Cf de mg a uf di for the cpeatc gv strces Cf 38 39Spe,rum Analysis ... Praenxe cf Ncse t and awrczxing ottpl. 13a. ONE OF EACH. NO NOISE in the rimvlest cse a linear timinriaht syst has a sng1c i and a ringle que tpt. Eaples iclude: Y (t) = [X(t +1) + X (t) + X (-1)]/3 YQ() -X(t+1) -X(t) Y(t) - j fX(t-) , with 0 < p< 1. .4 As tios in Secia 3, arch t re tyically dcaractaizal by a ingle fuxti cf fr y, I tr fncticx i seres is X (t) - piat}, then tI cxaM iaut seres is Y(t) B(wo) ip{iL}, with B tE tansafer mctia Far tIe rnm s aboveB(w) [1+2 acm Y3 1/[1-p p(i*] repciey. 7bl value B(w) itI dange in ampitudeai p has e that tI eato ects th e ;pit at frq ac w ais tude ID (wo) is s ca1aI tIE gan eae gB(w)}, is In 1actic, liar ie Cften dbed by the fcrm cf B(w); fcr cm an ide (urealle) band-pan filtw at fr y ) withv ltdwidth A is sifiea by 3(w)-K fcr w-AI '82(M) + (28.2) where ko(m)i m k2(m) 1 U-._ k l(M ) = 7, I Ojos Eqrsa (282) may w be ilat by FFT I we wan to we ws up to A,x md e to apmmate =p(i4 to 1% tl we an use tha fist two tami fer ka s 1,Axh22 514 To mt the same tith tha fist three ta ld reur *hl2 s 39 713 ratio of manbers of intas is 391.14 - 2.7 wAich is greate th ratio Cf amnbs of FFTs, zmnelv 135, so gmg an far a the trd tam may esily be thwlie. Simi calaalatku IVc 1S Tme Series max ratio term = FFI's kaI ratio 2 14 14 3 .39 7 4 .7 5.7 5 1 5 Suggsut it may py to gp to 4 c 5 tnum. If we we the a rcmatio e'* = 9994 + 9567i - .4853 whch is gotolto| cr I S1 fcr-h s 0 s A imtead th leaing terms f the Taylor sies, we can use the first two ter aover 4 the interval. IRmWiing (see eg. Airamon tz ud Steg& 1964, p. 791) the h r-c a cimaw tic er o wla intervas wi allow It will be m shmtly that, in the point trcss case, it may be c Fifatiwe to &o c (x11*atcim ca the time sdc, befre to the f I dain. 3d. THE SECOND-ORDER CASE Cidu the jma 1 fdrMEz stmates d povwer am cas-sa, tramf fctim and aEmMe fc dary seris ant painst 3e. THE POWER SPECTRUM OF AN ORDINARY TIME SERIES Tbc step in an pe rum timatim crwt hse iu: 1. aalys (e. t a , ilg, smplig, Irhiing 2. g. 3. vkizgwith zec, 4. Fonui tranfmatiai, 5. sdar- (to utain the i a), 6. smhiDg (the periclcam), ( time c frajxy). '11 estimate at frequeny w = 2wIU may be witten S ( W) ?4 IjT (w+2) (3.1) where the W. are to 1, wher the ha pmlA by ading U-T zerci, 2 IT(W) = e Z hT(tQ)Xtt) 12?rYhT(t)2 (3.2) is the cram, am! we X' is the sries after P with zercs plays seval rdes. It allows FF7 r iig lixite U to be apled, it avnidk the uni sry rlarizatia 1 the dtta with the aliasng (s ed at the and d Sectcia 6a), amd it l4 the ri 1pa vaus be a finly Wa In samr drnae step 6. my e Pl by: 16 S Itrum Analysis ... Prmae cf Nose 6'. iwe l m 7'. iply by lag Wi 8'. Ua=cm for an indit timatiam eduzre. Ibis maks the s thg dcc a iim - rather than frupxy - & jdy e ie (1 cinxuptatacm may to be din double pjasi.) As am hs the rdatidzuh45 1 T-p4-l etv) =T Xt tf)X t) S.. U 2 Ui r(2/U) i2"w/U} med Ir( w)i,Z (v)uP(iw} pediz in the last fasha is largdy a matter cic e. 113 ait (3.1) is Cs d1 the iakrWam ci the vlxic data strech, as a functian fr=p y. l13re 71 we hda it is e uscil to Pa by ing th ITWa 1T (4f) d the e.th _t cf the data as a fuicd e, to fcrm an itimate L-1 d IT (4e) *-1 If the s cdap, this called a ied intimate. In s r s it may be amwuiet to wib the e nts ally. Ws last imate hs tf form of the p fiteaS d entimate c cd the cicpex- Iatc finate te. If dT(4t) cinxtes the FT of the e-th sctem, then p(iol}dT (t4) fo t in the e.th se ics a ban&d ltered (at foiy w) via aC fte sais. In sme drmtance on will sh pe azm fin both te timaed emy ain Each c the above timats ae quadratic f ?im cf the d arvatiau E oying an FFT can rece cul2ltatic timad rul kU arrc. 3f. THE CROSS -SPECTRUM Su~ae the stretch {X(t)X,(t)}, t0..-1 cf bivaiate iis av-lable. 113 umectn intimate anakglsto (3.1) - (32) isg wO by *. (", [s sohT(t)Xt e,| h 8h()t|/2,W 't) where - ()xoJ- X ' am! Y ' ke theresults ci ynnlnyzq the menes (ruMz trex, jpwiteuing, remaligning, :c.) DIerffait tape may be plyed fc v a Yd'. (Ntice the empl cipuat impied y -i in the fil p.) It is dr tt a FT may ve sd. As in t vi , it may mak e to fam a mate by hing te rs paricxlcpams ci s ci the weie, 17 Tnre Series -1i Ose tEff es mate = i lzr 4 r Or e avalatc, estmates ci tIc trfr ftic, Mx re.ial q trm may be famed. 13 i e e may be esmatd by an essain c the fcrm a() =r-p > i2=p1iP}4(2qpIP)kP(p) vith k (p) a wind functicam! P a pc3tve iteger. Wben the basic series inlved are inucs i tme, im tim a suitable winw fuxtim can be cial. A e again an FF1 ge~afly poms ul. 38. THE POWER SPECTRUM OF A SPATIAL PROXESS dTQ()X =q p(k r)}hT(x y)X(xZ ) ,, e tIe FT ca a piece ci tIe tial array X, with ta i ierted. 113 piram cf this (taped data isgven by IT - dT (S,4 )/2irV hT (x , )2 . To d,tain mtsmate cf tIE por crn cf tE seies X, me simply h this periam n a fimctin d XI The data m have d ei to ipt tIE (reted) FT. Naturally, in an citaa used alterate a h, pid raII might be cunpued fcr (t sgen tf the data athese ir aver tcther. Ilis mgt be , fr oc , if ew a vy lge vdume ci data. In the isotric ase, the pow seetrun is a fucticd c2 + IO ly. lls m that a further y be ried thrb, with a cusquent inmse in stability. Spefically, peieda vahs at fra Xies uwith value di f2 + i my be averged tcgh. Qe e interesting fect that aiws is that estimates with larger 2 + ? are e stable, rlativdy, bomuse mer peidcram values may be averad tcgther. iller (1970) pides a proc d this lsst. 3h. THE POWER SPECTRUM OF A POINT PR XESS '113 power smectn d a inpt may be estimed at frequawy co * Oby inzxt hg the periaWram\ Ir()- (21T)-il F=}2 Alteratively it may be etimated by tcgcther periadcgram based a diffaet stretches ci ,-~~~~~~~~~~~~~~~~~~~~~- In a ecf ait tu out to be faster to ftm tem in an ir fashion, fdlcinvg (11.1). Spafiwlly oe esimates p(u) =Prob{dN(t4U) =1 aI dNQ() = 1) by j5 (u) = Y,r #(Tj-Ja < P121 (3-3) Jo& for tiuwidth I II= me cunptes 18 19 Spec-nn Analysis ... Preee d Nase 1(w) 2-r @ +21r d i WT U 0) U 0) (3 wre , -N(T)IT is an etimate d ti nte cf the md m ie wT is a lig w;. 1I emate (33) may be cc ted rpicly, a it amply incM mtin Eqmios (3A) may be aiud viaan FF- 3i. THE HIGIER-ORDER CASES In ea ldglt-mde a, a the bi- a tri sPecirum, Cmptatiaaati cxd mderati can bexc - meral. 3j. THE BISPECTR UM Set dT(w) - p (-it}hT (t)X (t) wh3en X xt) a~U fti ealyd vahuuL 113 tlrd-crder j~a mIs pv.i by IT (46)= dT (w)dT (vdT' _1$(2ir) hT(t ) 11nlspctumMY be est by abc ts fucti In out this avergig, the p_ricalidty and 'ymzry qtie d be d. 113se ihxle IT, Qj(v) * I'( -) IT(w,v) = IT (42W,v) IT (4,pv+2,W) I'T(wD,) - jT (.j,.) Alwavely, the sum wy be estimated by aeaging twrdcrd ri r u muaed at the s e bi (wv, bt buted s f re s d te data a y a X(t )X (t, AX(t , a f actnao ci t wlre XQ.w) es the reult Cf z ss fteng tXe ere X at r y f (This be ad through by ex dilaticna d anP FTif' ]r 3k. SPECTRA OF ORDER K Suw tht a stretch i m J Mr-valud series {Xj(t) -i ,..j} is iablec fCr ana.yiL Set (wT) -z upit}T(t)X,(t) vt X th j.th sarieL 113 j th kbe ofi crdar, K rrcm n to caoits ae l.. .ac (std fr 1 ) is uvaby w sdaTI(ok *-aT *(, (/(2s-v) 1hT (t)K It is t to ma is b4i by etm a dlnmy argmt (ft pvaa by t makes the sd taticaml awl tand out me dely. 113 ccuqxOiis mawted by teby 19 rme Series 2=iw 213: Wal...8KIT (* + U ' '*9 + U whare the wdgts sum to 1, vadsh if any u" -0, wa e are ic i g. . . . . (i. (t is e m ad that th u have th ferm 2n(inga)/U a n d tads FT's Cf lth U have bea fxmd.) abis dvs tt etimate rtoe tty pA iclity p t timates the aWIag C baud an rate stretcs basd n ow baau series ar o availae. Biifgaer (1965) an Brifinger ad Rmnblatt (1967) we revait s 4. >ENTmCA1cn OF NON1INEAR SYS B Y ENCY METHO16 .Sectin D hot 1 lir time i ant st aild be m by specrum hIs. Tis secticm 0mde sm rIiitw, but timinvariant, stas. 4a. INSTANTANEOUS NONLINEARITY Cider a systm d.aibed by U(t) a (t -)X(u) r(t) s[U(t)J +e(t) with e a nsc seies a() th ipse Cf an ti m iniant filter d wth () a funtiu d a aingle riate. A in t, fit wted by azg (1952) fcr tU case df pdyn i,is that if the inpt X is C an statiy, a,. (c) 5 cA (w)sa (c) with c a Mutant. In other wrdsb if aie les tha system, by cspctrsaral yss, as if it we 1 , tl tha transfer fumcp d*ained is t xivml to tht d the 1;w d thE stan K nbieg (1973) cm to amider the casef rd a fitrs ad veral imtmutazs n^la C Brilier (19) m statistical detai s al t u e df tlet is that if a systa is idntified by caoa-ectral analysis wvith ion iut, th the reltin transfe ftucn can have a smple intopretatia in a much rInader daB Cf i s tn ac might have an1ti. 4b. QUADRATIC SYSTEmS A izaixa cf n 1w tim ariant systan is an bn time-invarint ys, a ystan ith two ibyu tht ctys Ml tiam in ach input sparatdy. Sud a systan is Wiwl, n1 , at can behave i n m-linewr ways, ailarly wI we ac t the same X(t) to both liputs treltin quadratic time-invaiant stn, with a ipt and ac wtpt, c be c1zl aitxty by tha y C[X+Y - C[X-rY is a biw system with inputs X ad . f hi di fcr a X(t) ad r(), C is a quadr timeinmaiant systaa wil have mdc Cf the rqresntatim fa . Let us tu to t ca See that r(t) b(t -u--v)X(u)X(v) +e(t) (4.1) with e e. Lv B () Z -(cow)}b(u,v) aI l U btru futia. Suwe that X is zeo mn, im, statiay with Pame 20 21Spectrwn Analysis ... Pr5em d N sc sectn S,, (4). 7ben tU aas-um cf Y with X saifis Szxr (wiv3 m 2B (--#-)S= (c*)Ssx (v) *(4.2) Ts rit W biz x k i (1961). An dim c 3 my bc fcntd at S., a S,,, ve a cxti . eTt bi-qmpdse rup- fuiicu b, may be t by Foxriff traI f B, usig g e ftcrs n y. If the y (41) is ext to cain a lizterm, a t aQ-u)X(u), the reladiubip (4.2) nins to dIe tans fuzcticu A may be sintiated by ac pwtral yss. Hng et at (1979) ett Mn fit t mc ad drme kms cf t lman pyillary stam by this tque. 4c. COSINUSOIDAL INPUT An irmatiw diqe fcr Ut examiastion cit syit s to tale a pure nLud a us(f +) as iipt. If tit systam is liar timainvim, tlt frequai al will awear th outpt. If is qadratic timanvaiant, thn t freq is 2w awi ar. If it is tiM& invariant ~x~ymm2ial cr der L, then frqe %in ?, 20 ... . p wIll aw r. on cxsios mba i 2, Pf3, my a ar, as with th edp waves in Section 7f. Ibis is an iic tim c a le aimple sxt et nm i Further infmatm thenig t ystanmay be f !by ting a pslir cc(pit +1) + cu(06t +12) ci ds aa input. Subctn inatna frpa s s(W+S0 isn, 2n -2,...; i A= :t: 1,? 2,.. will be iiced by sane (nmi pdyia) neiar systms. lie icati p ci Pt amd to th natural freqa fies th Ut systan very impnat ire. 21