Multitaper Power Spectrum Estimation

Transcription

1 Muliapr Powr Spcrum Esimaion im van Drongln Muliapr Powr Spcrum Esimaion im van Drongln, 4. Inroducion Th mos commonly applid chniqu o obain an simad powr spcrum S ˆ f of a sampld im sris x of fini lngh, is o compu h Fourir ransform X j f, followd by muliplicaion by is complx conjuga and scaling by h lngh of h fini obsrvaion, i.. S ˆ f X jf X * jf / s Chapr 7 in van Drongln, 7. Th rliabiliy of his simad powr spcrum is significanly rducd boh by h varianc of h sima S ˆ f a ach frquncy f i.., h spcra usually loo vry noisy and by laag of nrgy across frquncis, craing a bias. Th laag is du o h fac ha w a a fini scion of signal, which quas o muliplying h signal by a rcangular window somims calld a boxcar s hr uppr racs in Fig.. Th Fourir ransform of h rcangular window shows mulipl sid lobs Fig. A and his causs h laag problm. Rcall ha muliplicaion in h im domain is quivaln o a convoluion in h frquncy domain; hnc hr will b a convoluion of h signal ransform and h boxcar s Fourir ransform rsuling in laag of nrgy. A soluion o rduc laag in h frquncy domain is o firs muliply h signal in h im domain wih a non-rcangular window characrizd by a Fourir ransform wih lss nrgy in is sid lobs. Th windowing approach dscribd in his paragraph is xplaind in mor dail in van Drongln 7, Scion 7... To summariz if w hav a squnc x of lngh, a window a, h simad spcrum is givn by: Sˆ f jf x a. Hr and in h following, for convninc of comparison wih Prio al. 7, I adap mos of hir noaion. If h window a is no xplicily prsn in Equaion, or if i is dfind as a rcangular window/boxcar, h spcral sima S ˆ f is calld h priodogram and i should b nod ha in his cas, on may xpc ha bias du o laag will b srong. Of cours, h us of a window also calld a apr affcs h sima by rducing laag bu i dosn chang h varianc of h sima a ach frquncy. A common approach o rduc h varianc is aing an avrag across svral frquncis and/or compuing an avrag Pag

2 Muliapr Powr Spcrum Esimaion im van Drongln spcrum from svral im pochs. Avraging across frquncis rducs spcral rsoluion and using mulipl pochs is undsirabl if h signal may b non-saionary. Figur : Powr spcrum of a fini obsrvaion Exampl of a Gaussian whi nois signal zro man and uni varianc; uppr rac and a rcangular/boxcar window nd rac. hn signal and window ar muliplid, h rsul dpicd in h hird rac mimics a fini obsrvaion of h signal wih a lngh ha is qual o h lngh of h rcangular window. Th boom rac shows h priodogram of h fini obsrvaion. From h boom graph i is obvious ha h priodogram is vry noisy du o h varianc in his sima. Th xpcd powr is qual o h varianc of h Gaussian disribuion from which h sampls in h im domain signal ar drawn on in his xampl. Laag of nrgy across spcral bands, which is also prsn in his sima du o h applid boxcar window, canno b dircly obsrvd in his plo bu can b dducd from h spcrum of h boxcar islf.g. s Fig. 7.4 in van Drongln, 7 whn convolvd wih h signal in h frquncy domain; s x and Scion 7.. in van Drongln 7 for furhr xplanaion. In Scion, w will firs inroduc h us of mulipl aprs and h raional bhind hir drivaion. In Scion 3, w dpic an xampl of h applicaion of his procdur by using a Gaussian auorgrssiv procss wih a nown powr spcrum. From his xampl w conclud ha hr is a nd for opimizaion of h procdur, which is prsnd wihou proof in Scion 4. Finally, in Scion 5, w show a concr xampl of h opimizd muliapr spcral sima in Malab.. Th Us of Taprs Th muliapr approach, firs dscribd in a sminal papr by Thomson 98, improvs h spcral sima by addrssing boh laag and varianc in h sima. Th basic approach is simpl: wha is br han using a apr? You gussd i h answr is: you us mulipl aprs. In his approach, vry apr v ou of a s of K aprs is a bi diffrn and rducs laag of nrgy across frquncis. In addiion, in Thomson s approach, h aprs ar Pag

3 Muliapr Powr Spcrum Esimaion im van Drongln orhogonal and hy ar usd o provid K orhogonal sampls of h daa x. Ths sampls ar usd o cra a s of K spcral simas Sˆ f ha can b usd o compu an avrag S f wih rducd varianc: K S f S ˆ f. K Th gnral ida for h applicaion of a apr is as follows. Each apr a is associad wih an simad spcrum ha is givn by quaion. In ordr o mainain corrc valus for oal powr, w assum h aprs ar normalizd such ha a. Furhrmor, h powr spcrum of apr a is A f, and is spcral propris ar imporan bcaus hy drmin, via convoluion, h sima of h powr spcrum of our windowd aprd im sris: / / S f A f ' S f f ' '. 3 o ha w normalizd h yquis frquncy o ½. Rcall ha muliplicaion in h im domain is associad wih convoluion in h frquncy domain; s Scion in van Drongln, 7. A dsirabl apr will hav low ampliud spcral valus for all largr valus of f f '. In ohr words, hs aprs will hav low nrgy in h sid lobs. This lads o simas for S f ha will mainly and corrcly consis of valus clos o f. In his cas, w will hav minimal laag and hus bias in h spcral sima. Th rasoning coninus as follows. Suppos w wan o sima our spcrum wih a rsoluion bandwidh, which is ncssarily s a a valu in bwn h rsoluion of h spcrum / and h maximum of h spcrum, h yquis frquncy. To simplify noaion, w assum ha h sampl inrval is uniy and h yquis frquncy is normalizd a ½. Th fracion of h apr s nrgy wihin h slcd frquncy band is givn by: / A f, / 4 A f Th basic ida is ha w wan o find h aprs wih minimal laag by maximizing, i.. h fracion of h nrgy wihin bandwidh is maximizd. To ma a long sory shor s Appndix for a longr sory, on maximizs by sing h drivaiv of h xprssion in Equaion 4 wih rspc o vcor a qual o zro. This quas o solving h following, wll nown marix ignvalu problm s Appndix for furhr dails: Pag 3

4 Muliapr Powr Spcrum Esimaion im van Drongln D a a, 5 sin ' in which h marix D has componns D, '. o ha his funcion has ' vn symmry. Thrfor D is a symmric marix. Th soluion has ignvalus,,...,,..., and orhonormal ignvcors v, v,..., v,..., v rcall ha w normaliz hm o uniy: v. Th ignvcors v ar so-calld discr prola sphroidal squncs, also calld Slpian squncs Slpian, 978. A prola sphr is an longad sphrical objc wih h polar axis grar han h quaorial diamr. Fig. Th spcrum of a rcangular window boxcar dpicd in panl A has mos nrgy concnrad in h dsird bandwidh indicad bwn h sippld vrical lins, bu also shows a lo of nrgy in h sid lobs. Th funcion in B is h spcrum of an idal window whr all nrgy is locad wihin h inrval bwn w and w. hn a spcrum is convolvd wih h spcra of hs windows, h on in A gnras significan laag and hrfor bias, whras h idal on in B givs a bias-fr sima. As shown in Figurs 3 and 4, vn h bs opimizd window isn idal as in panl B, bu dfinily br han h on from h boxcar window dpicd in panl A. o ha marix D can b considrd as h unscald covarianc marix of h invrs ransform of a rcangular window bwn and in h frquncy domain Fig. B, Appndix. Thrfor, h compuaion of ignvalus and ignvcors of D is h sam as prforming principal componn analysis PCA on h invrs ransform of h idal spcral window s van Drongln for an inroducion o PCA. Sinc h dimnsion of D is, hr ar ignvalus and ignvcors v. Th firs s of componns capur mos of h propris of his idal window, whras h subsqun componns capur lss, so h qusion is which componns o includ whn using hm as aprs. Th firs ignvalu is vry clos o on, so i is associad wih a vry good ignvcor, i.. a apr ha minimizs laag. Sinc w us h discr Fourir ransform for obaining h spcral sima, h numbr of poins P in Pag 4

5 Muliapr Powr Spcrum Esimaion im van Drongln bandwidh is a mulipl of h spcral rsoluion / of h sampl im sris: hus = P/ and P =. As shown in h abl blow, for choics of = 8 and = 4, 3, or, h firs - ignvalus ar all vry clos o uniy Tabl from Par al., 987. Thrfor, as a firs approach, h firs - normalizd ignvcors could b considrd good o us for rducing laag and also by avraging hir rsuls for rducing h varianc of h spcral sima. As you can imagin, hr ar a fw alrnaivs for h avrag procdur of h aprd powr spcra. On migh simply avrag as in Equaion abov which is wha w will discuss in Scion 3, or on migh improv h sima by composing a wighd avrag which givs br rsuls as you will s in Scions 4 and An Exampl L s considr a concr xampl of h us of h muliapr chniqu and analyz a signal ha is gnrad using an xampl of an AR procss x dscribd by Prcival and aldn 993 hir Equaion 46a: x.767x 3.86x.6535x 3.938x 4 6 Hr is a Gaussian hi ois G procss wih zro man and uni varianc. Sinc w us his sris o valua spcral analysis, w wan o compu is powr spcrum using h approach dscribd in Scion 3.4 in van Drongln 7. If w z-ransform h AR xprssion: h ransform of x is X z, and sinc h ransform of h G is E z, w g: X z a z E z b z c z d z a z b z c z d z Pag 5

6 Muliapr Powr Spcrum Esimaion im van Drongln j Th Fourir ransform X j can now b obaind by subsiuion z and h powr spcrum by compuing X X *. This horical powr spcrum w will us as our gold sandard for comparison wih our muliapr simaion. Th following is par of a Malab rouin ha compus and plos h spcrum in db vrsus a frquncy scal bwn and.5 as in Prcival and aldn 993 and in Figs. 3 and 4. % consans of AR4 in Eq 46a in Prcival and aldn, 993 a=.767; b=-3.86; c=.6535; d=-.938; d=; % im sp s o w=:.:pi/d; % scal for frquncy f=w./*pi/d; % frquncy scald bwn and.5 for plo z=xp-j*w*d; % z^- X=./-a*z-b*z.^-c*z.^3-d*z.^4; % Fourir ransform PX=X.*conjX; % Powr LPX=*logPX; % Powr in db figur;plof,lpx % Compar horical spcrum Fig. 3 and 4 in Handou Th procdur w follow for his muliapr analysis is dpicd in Figurs 3 and 4 plos from Prcival and aldn 993, hir Figs In his xampl w dmonsra ha h numbr of aprs usd provids a compromis bwn rducion of laag and varianc. In his xampl = 4 hus w would considr up o svn - ignvcors. Figur 3 dpics h propris of h individual aprs h 8 h on is also shown, including hir spcra and h ffcs of hir applicaion in h im and frquncy domains. In Figur 4, h ffcs of using hs aprs in spcral avrags ar shown. As can b sn in, h s and bs apr rducs laag almos o zro almos no nrgy in h sid lobs bu producs a singl spcral sima associad wih rahr larg varianc. Adding h rsul from a nd apr adds a bi of sid lobs and hus laag bu avraging i wih h s on rducs varianc bcaus now h spcrum is avragd from wo rsuls. By including h rsul from ach addiional apr in h avragd sima, hr is boh a bi of incras of laag and rducion of variabiliy du o h avraging procdur. In his xampl hr sms o b a rasonabl opimum a fiv aprs. A igh aprs h window conains so much nrgy in h sid lobs ha i causs a larg amoun of laag and hus a srong bias in h spcral sima. Th a-hom mssag of his xampl is ha including h rsuls obaind wih h firs - aprs in h avragd spcral sima may sill caus an unaccpabl lvl of laag! This problm will b furhr discussd and addrssd in Scion 4 blow. Pag 6

7 Muliapr Powr Spcrum Esimaion im van Drongln Figur 3: Exampl of h propris of h individual aprs = 4 Th lf column in his s of plos shows igh discr prola sphroidal aprs. Th, igh aprs = - 8 ar dpicd in subsqun rows. Th panls in h nd column ar h producs of h im sris compud wih Equaion 6 muliplid wih hs aprs. Th 3 rd column shows h spcra of h aprs. Th righ column dpics h sima of h spcra of h windowd im sris; h blac lin is h horical spcrum and h gry noisy lins ar h simas of h windowd signals. Obviously h nrgy of h sid lobs and h associad laag spcially in h rang from incrass wih. This Figur is a combinaion of Figs in Prcival and aldn 993. Pag 7

8 Muliapr Powr Spcrum Esimaion im van Drongln Fig. 4: Avragd ffcs of h applicaion of aprs Spcral simas of h im sris of Eq 6 by using h sam igh aprs shown in Figur 3. In his cas h lf and hird columns show h spcra of h combind aprs. o ha h sid lobs significanly incras wih. Th scond and righ columns show h avragd muliapr simas gry, noisy lins suprimposd on h horical spcrum blac lins. ih incrasing, h bias incrass du o h nrgy in h sid lobs and associad laag, spcially in h rang.3.5, bu h varianc diminishs. In his xampl hr sms o b a good compromis bwn rducd varianc and laag a = 5. This Figur is a combinaion of Figs. 34 and 34 in Prcival and aldn Muliapr Spcrum wih Opimizd ighs Th xampl in Figurs 3 and 4 shows ha h us of - aprs, in his cas svn, producs oo much bias and ha lss aprs in his cas fiv aprs giv a br rsul. Using fwr aprs howvr lads o sub-opimal rducion of h varianc sinc w avrag ovr lss spcra. On soluion o his problm is o furhr opimiz h wighs of h aprs o minimiz laag bias, so ha all aprs can b usd o rduc varianc of h simad spcrum. In shor w wan o invsiga h us of a wighd avrag as an alrnaiv o a simpl avrag as in h prvious xampl in Scion 3. As dscribd by mulipl auhors.g. Prcival and aldn, 993; Prio al., 7, wighing of h rsul by ach apr may b prformd by hir ignvalus, or on migh consruc an opimizd daa adapiv procdur. Th following prsns hs procdurs wihou proof bu wih som bacground on how hy ar accomplishd. In h prvious implmnaion of h muliapr approach all ignvcors wr normalizd o uniy. Associad wih ach ignvcor, w also hav is ignvalu. ihou furhr proof, on could imagin ha his ignvalu could b mployd as a wigh whn compuing h avragd Pag 8

9 Muliapr Powr Spcrum Esimaion im van Drongln powr spcrum sima S f. Equaion is hn modifid by wighing h individual conribuions of ach aprd sima Sˆ f by is associad ignvalu : S f K K Sˆ f 7 As i appars, h approach of wighing h simas by hir ignvalus may sill no giv h bs simas. Using a rgularizaion approach, Thompson 98 dvlopd a daa adapiv mhod o wigh h individual spcral simas ha conribu o h avrag. This lads o h following xprssion for h spcral sima: S K d Sˆ f f K, 8a d in which h wighs d ar frquncy dpndn and compud by using: S f d f S f, 8b wih h varianc of h im domain signal x. Howvr, h big unnown in h xprssion for d is h spcrum S f! Ofn h firs wo ignspcra Sˆ f for = and, provid an iniial sima for S f and h adapiv wighs ar hn found iraivly for furhr dails, s Thompson, 98; Prcival and aldn, An Exampl in Malab Th approach o compu a muliapr sima of h spcrum is implmnd in h Malab pmm command. In Malab yp: hlp pmm o g h following dscripion of h procdur. pmm Powr Spcral Dnsiy PSD sima via h Thomson muliapr mhod MTM. Pxx = pmmx rurns h PSD of a discr-im signal vcor X in h vcor Pxx. Pxx is h disribuion of powr pr uni frquncy. Th frquncy is xprssd in unis of radians/sampl. pmm uss a dfaul FFT lngh qual o h grar of 56 and h nx powr of grar han h lngh of X. Th FFT lngh drmins h lngh of Pxx. For ral signals, pmm rurns h on-sidd PSD by dfaul; for complx signals, i rurns h wo-sidd PSD. o ha a on-sidd Pag 9

10 Muliapr Powr Spcrum Esimaion im van Drongln PSD conains h oal powr of h inpu signal. Pxx = pmmx, spcifis as h "im-bandwidh produc" for h discr prola sphroidal squncs or Slpian squncs usd as daa windows. Typical choics for ar, 5/, 3, 7/, or 4. If mpy or omid, dfauls o 4. By dfaul, pmm drops h las apr bcaus is corrsponding ignvalu is significanly smallr han. Thrfor, Th numbr of aprs usd o form Pxx is *-. Pxx = pmmx,,fft spcifis h FFT lngh usd o calcula h PSD simas. For ral X, Pxx has lngh FFT/+ if FFT is vn, and FFT+/ if FFT is odd. For complx X, Pxx always has lngh FFT. If mpy, FFT dfauls o h grar of 56 and h nx powr of grar han h lngh of X. [Pxx,] = pmm... rurns h vcor of normalizd angular frquncis,, a which h PSD is simad. has unis of radians/sampl. For ral signals, spans h inrval [,Pi] whn FFT is vn and [,Pi whn FFT is odd. For complx signals, always spans h inrval [,*Pi. [Pxx,] = pmmx,, whr is a vcor of normalizd frquncis wih or mor lmns compus h PSD a hos frquncis using h Gorzl algorihm. In his cas a wo sidd PSD is rurnd. Th spcifid frquncis in ar roundd o h nars DFT bin commnsura wih h signal's rsoluion. [Pxx,F] = pmm...,fs spcifis a sampling frquncy Fs in Hz and rurns h powr spcral dnsiy in unis of powr pr Hz. F is a vcor of frquncis, in Hz, a which h PSD is simad. For ral signals, F spans h inrval [,Fs/] whn FFT is vn and [,Fs/ whn FFT is odd. For complx signals, F always spans h inrval [,Fs. If Fs is mpy, [], h sampling frquncy dfauls o Hz. [Pxx,F] = pmmx,,f,fs whr F is a vcor of frquncis in Hz wih or mor lmns compus h PSD a hos frquncis using h Gorzl algorihm. In his cas a wo sidd PSD is rurnd. Th spcifid frquncis in F ar roundd o h nars DFT bin commnsura wih h signal's rsoluion. [Pxx,F] = pmm...,fs,mhod uss h algorihm spcifid in mhod for combining h individual spcral simas: 'adap' - Thomson's adapiv non-linar combinaion dfaul. 'uniy' - linar combinaion wih uniy wighs. 'ign' - linar combinaion wih ignvalu wighs. Pag

11 Muliapr Powr Spcrum Esimaion im van Drongln [Pxx,Pxxc,F] = pmm...,fs,mhod rurns h 95% confidnc inrval Pxxc for Pxx. [Pxx,Pxxc,F] = pmm...,fs,mhod,p whr P is a scalar bwn and, rurns h P*% confidnc inrval for Pxx. Confidnc inrvals ar compud using a chi-squard approach. Pxxc:, is h lowr bound of h confidnc inrval, Pxxc:, is h uppr bound. If lf mpy or omid, P dfauls o.95. [Pxx,Pxxc,F] = pmmx,e,v,fft,fs,mhod,p is h PSD sima, confidnc inrval, and frquncy vcor from h daa aprs in E and hir concnraions V. Typ HELP DPSS for a dscripion of h marix E and h vcor V. By dfaul, pmm drops h las ignvcor bcaus is corrsponding ignvalu is significanly smallr han. [Pxx,Pxxc,F] = pmmx,dpss_params,fft,fs,mhod,p uss h cll array DPSS_PARAMS conaining h inpu argumns o DPSS lisd in ordr, bu xcluding h firs argumn o compu h daa aprs. For xampl, pmmx,{3.5,'rac'},5, calculas h prola sphroidal squncs for =3.5, FFT=5, and Fs=, and displays h mhod ha DPSS uss for his calculaion. Typ HELP DPSS for ohr opions. [...] = pmm...,'droplastapr',dropflag spcifis whhr pmm should drop h las apr/ignvcor during h calculaion. DROPFLAG can b on of h following valus: [ {ru} fals ]. ru - h las apr/ignvcor is droppd fals - h las apr/ignvcor is prsrvd [...] = pmm...,'wosidd' rurns a wo-sidd PSD of a ral signal X. In his cas, Pxx will hav lngh FFT and will b compud ovr h inrval [,*Pi if Fs is no spcifid and ovr h inrval [,Fs if Fs is spcifid. Alrnaivly, h sring 'wosidd' can b rplacd wih h sring 'onsidd' for a ral signal X. This would rsul in h dfaul bhavior. Th sring inpu argumns may b placd in any posiion in h inpu argumn lis afr h scond inpu argumn, unlss E and V ar spcifid, in which cas h srings may b placd in any posiion afr h hird inpu argumn. pmm... wih no oupu argumns plos h PSD in h currn figur window, wih confidnc inrvals. EXAMPLE: Fs = ; = :/Fs:.3; Pag

12 Muliapr Powr Spcrum Esimaion im van Drongln x = cos*pi**+randnsiz; pmmx,3.5,[],fs; % A cosin of Hz plus nois % Uss h dfaul FFT. Figur 5 dpics how h muliapr compuaion using h dfaul daa adapiv mhod applid o h AR4 im sris wor in Malab, and how is rsul compars o h sandard priodogram. Figur 5: Comparison of h muliapr and boxcar mhods Comparison of h horical spcrum of h AR4 im sris and h priodogram and h muliapr spcrum using h pmm command. o ha h muliapr spcrum, as compard o h priodogram, has lss bias across h whol bandwidh. Th varianc a h lowr frquncis is also lss in h muliapr spcrum bu no so much a h highr frquncis in his xampl. This Figur was mad wih Malab scrip ARspcrum.m. o ha if you us his Malab scrip lisd blow, rsuls can vary du o h random gnraion of h im sris! Th rsuls in Figur 5 wr obaind wih h following Malab scrip ARspcrum.m % ARspcrum.m clar; clos all; % consans of AR4 in Eq 46a in Prcival and aldn, 993 a=.767; b=-3.86; c=.6535; d=-.938; d=; % im sp s o w=:.:pi/d; % scal for frquncy f=w./*pi/d; % frquncy scald bwn and.5 for plo z=xp-j*w*d; % z^- X=./-a*z-b*z.^-c*z.^3-d*z.^4; % Fourir ransform PX=X.*conjX; % Powr LPX=*logPX; % Powr in db figur;plof,lpx % Compar horical spcrum Figs. 3 and 4 in Handou il'spcrum AR4'; xlabl'frquncy'; ylabl'powr db'; Pag

13 Muliapr Powr Spcrum Esimaion im van Drongln %cra a im sris x:4=randn,4; % s iniial valus for i=5:8; % cra im sris in loop xi=a*xi-+b*xi-+c*xi-3+d*xi-4+randn; nd; x=x5:8; % rmov h iniial valus figur;plox; il'insanc of h AR4 im sris'; xlabl'tim'; ylabl'ampliud'; % Us h malab pmm funcion =4; % s o 4 as in h xampl in Figs. 3 & 4 [Pxx,] = pmmx,; F=./*pi/d; % frquncy scald bwn and.5 for plo LPxx=*logPxx*; % compu in db and muliply by o scal as LPX % Th sandard Priodogram Y=ffx; Pyy=Y.*conjY/lnghx; LPyy=*logPyy; % Th following producs Fig. 5 in h handou % o ha rsuls across rials may diffr du o randomnss figur;hold; plof,lpx,'.' plof,lpyy:lnghf plof,lpxx,'r' il'spcra AR4 Thoricalblac, Priodogramblu Muliaprrd'; xlabl'frquncy'; ylabl'powr db'; Pag 3

14 Muliapr Powr Spcrum Esimaion im van Drongln Pag 4 Appndix Hr w labora on h sps bwn Equaions 4 and 5. For furhr dails, s also Prcival and aldn 993, p. 4 c. For convninc w rpa quaion 4: / /, f A f A. 4 nd o maximiz his xprssion for. Firs w limi h signal in h im domain ovr h inrval ], [ so ha w can us h following xprssion for h discr Fourir ransform: jf a f A. Using Parsval s horm, w can rwri h dnominaor in Equaion 4 as: a. A- Th numraor in Equaion 4 can also b wrin as no h us of dummy variabls and and complx conjuga signs, *: jf a jf a f A f A ' * * ' ' ow w chang h ordr of summaion and ingraion and combin h ingrals ovr f: ' ' ' ' ' ' * ' ' * a j a jf a jf a jf j A-

15 Muliapr Powr Spcrum Esimaion im van Drongln o ha in h abov xprssion, ach of h wo ingrals lf of h qual sign is h invrs Fourir ransform of a rcangular window in h frquncy domain his window rangs from - o,.g. Fig. B. Th rsuling ingral in Equaion A- gnras h covarianc of his rcangular window: j ' jf ' j ' j ' sin ' j ' ' jsin ' A-3 Combining Equaions A- A-3 o rwri Equaion 4, w g:, ' * sin ' a a ' ' a This can b compacly rwrin in marix/vcor noaion as: a D a. A-4 a a Hr w simplifid noaion for and a, whil marix D is givn by: sin ' D, '. A-5 ' In ordr o rduc laag w nd o maximiz h xprssion for in Equaion A-4. accomplish his by sing h drivaiv of his xprssion wih rspc o a qual o zro. If w simplify Equaion A-4 o h fracion u/v, h xprssion o solv is u ' v uv' / v, which is h sam as solving u ' v uv'. This can b rwrin as u ' / v' u / v, oru' v'. ow w rcall from Equaion A-4 ha u a D a and ha hus u' D a. o ha his is h cas bcaus marix D is symmric and no a funcion of a. Th dnominaor v a a, and is drivaiv v' a. ow w combin h xprssion for u' and v' wih u' v', w divid by and w g Equaion 5, h wll-nown ignvalu xprssion: D a a or D a a 5 Pag 5

16 Muliapr Powr Spcrum Esimaion im van Drongln Rfrncs Par J, Lindbrg CR and Vrnon FL 987 Muliapr spcral analysis of high-frquncy sismograms. J. Gophys. Rs. 9:,675-,684. Prcival DB and aldn AT 993 Spcral Analysis for Physical Applicaions: Muliapr and Convnional Univaria Tchniqus. Cambridg Univrsiy Prss, Cambridg, UK. Prio GA, Parr RL, Thomson DJ, Vrnon FL and Graham RL 7 Rducing h bias of muliapr spcrum simas. Gophys. J. In. 7: Slpian D 978 Prola sphroidal wav funcions, Fourir analysis, and uncrainy V: h discr cas. Bll Sysm Tch. J. 57: Thomson DJ 98 Spcrum simaion and harmonic analysis. Proc. IEEE 7: Van Drongln 7 Signal Procssing for urosciniss: An Inroducion o h Analysis of Physiological Signals. Acadmic Prss, Elsvir, Amsrdam. Van Drongln Signal Procssing for urosciniss: A Companion Volum. Advancd Topics, onlinar Tchniqus and Muli-Channl Analysis. Elsvir, Amsrdam. Homwor. Modify h ARspcrum.m scrip o invsiga h ffc of h adap, uniy, and ign opions in h pmm command. ha do hs opions man and wha can you conclud?. ha is h varianc of h avragd spcral sima S f if w hav K spcral simas f ach wih a varianc of Using Equaion? S Pag 6